Molecular Mechanics and Ab Initio Simulations of Silicon (111) Surface Reconstructions, Semiconductors and Semiconductor Superlattices, H Abstraction for Nanotechnology, Polysilane, and Growth of CVD Diamond Citation Musgrave, Charles Bruce (1995) Molecular Mechanics and Ab Initio Simulations of Silicon (111) Surface Reconstructions, Semiconductors and Semiconductor Superlattices, H Abstraction for Nanotechnology, Polysilane, and Growth of CVD Diamond. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7khv-pb17. https://resolver.caltech.edu/CaltechETD:etd-10192007-104223 Abstract This thesis describes the application of ab initio and molecular mechanics quantum chemical methods to several problems in the materials and surface sciences. Chapter 1 reviews these methods. Chapter 2 details the application of these methods to study the reaction rate of a proposed mechanism for growth of CVD diamond. Chapter 3 uses high level ab initio methods to study the feasibility of a hydrogen abstraction tool for nanotechnology. Chapter 4 uses ab initio methods together with experimental data to develop a force field potential to model polysilane polymers. Chapter 5 is comprised of the development of atomistic potentials to describe semiconductors and their superlattices and interfaces. The approach of Chapter 5 is extended in Chaper 6 by combining the bulk force field with force field parameters developed from the Biased Hessian Method applied to unique clusters to model the reconstructions of the Si (111) surface. Chapter 7 concludes this thesis with a description of the Generalized London Potential which was developed to accurately model chemical reactions at the accuracy of high level configuration interaction methods, but with the practicality of molecular mechanics. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Materials Science Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Materials Science Thesis Availability: Public (worldwide access) Research Advisor(s): Goddard, William A., III Thesis Committee: Goddard, William A., III (chair) Goodwin, David G. Johnson, William L. McGill, Thomas C. Okumura, Mitchio Defense Date: 29 September 1994 Additional Information: Thesis title in 1995 commencement program -- Molecular Mechanics and Ab Initio Simulations of Silicon (III) Surface Reconstructions, Semiconductors and Semiconductor Superlattices, H Abstraction for Nanotechnology, Polysilane, and Growth of CVD Diamond -- differs slightly from thesis file (PDF). Record Number: CaltechETD:etd-10192007-104223 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10192007-104223 DOI: 10.7907/7khv-pb17 ORCID: Author ORCID Musgrave, Charles Bruce 0000-0002-5732-3180 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4182 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 02 Nov 2007 Last Modified: 16 Apr 2021 23:33 Thesis Files PDF - Final Version See Usage Policy. 19MB Repository Staff Only: item control page

Molecular Mechanics and Ab Initio Simulations of Silicon (111) Surface Reconstructions, Semiconductors and Semiconductor Superlattices, H Abstraction for Nanotechnology, Polysilane, and Growth of CVD Diamond Thesis by Charles Bruce Musgrave In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1995 Defended September 29, 1994 11 © Charles B. Musgrave 1995 All Rights Reserved iii Table of Contents Page Chapters Acknowledgements Abstract ......................................................... v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vl Introduction .............................................................. . 1 Chapter 1 Review of Methods ............................................. 8 Chapter 2 SR-SOR Step for Growth of CVD Diamond .................... 30 Chapter 3 Ab initio Study of H Abstraction in Nanotechnology ........... 64 Chapter 4 The Hessian Biased Force Field for Polysilane Polymers ........ 93 Chapter 5 Force Fields for Semiconductors and Their Superlattices ...... 149 Chapter 6 Silicon (111) Surface Reconstructions ......................... 232 Chapter 7 Generalized London Potential for Hydrocarbon Reactions ..... 269 iv To my parents V Acknowledgements I have benefited from the help and support of many people during my stay at Caltech. My advisor, Bill Goddard acted as a wise mentor, and inspiring example of unquenched enthusiasm for science. I thank him for his support and friendship. K. T. Lim, Rick Muller, and Terry Coley were always helpful in answering my computer questions. I also want to thank Jim Gerdy, Erik Bierwagen, Yue Jin Guo, Mario Blanco, Ersan Demiralp, Jamil Tahir-Kheli, Naoki Karasawa, Xiaojie Chen and Jean-Marc Langlois for sharing their expertise with me. The entire Goddard group has contributed greatly to my research here at Caltech and without them this work would have proven much more difficult. I have also benefited from many collaborators both in and out of the Goddard group. These scientists not only provided scientific insight, mentorship, and fulfilling interaction, but also friendship. I am grateful to each of them; Ralph Merkle of Xerox, Steve Harris of General Motors, Eric Drexler of the Foresight Institute, Jenna Zinck of Hughes, and Jason Perry, Siddharth Dasgupta, Jinsong Hu, and Bob Donnelly of the Goddard group. I'd also like to thank my friends who never quite knew what I was working on, but supported me anyway. Jessica and Jean were especially good to me. I especially appreciate Luanne Tseng for her patience, and friendship. My brother Richard and sisters Adriana and Gabriela have each given me their support. Without the support and efforts of my parents none of this would be possible. To them I am ever thankful. VI Abstract This thesis describes the application of ab initio and molecular mechanics quantum chemical methods to several problems in the materials and surface sciences. Chapter 1 reviews these methods. Chapter 2 details the application of these methods to study the reaction rate of a proposed mechanism for growth of CVD diamond. Chapter 3 uses high level ab initio methods to study the feasibility of a hydrogen abstraction tool for nanotechnology. Chapter 4 uses ab initio methods together with experimental data to develop a force field potential to model polysilane polymers. Chapter 5 is comprised of the development of atomistic potentials to describe semiconductors and their superlattices and interfaces. The approach of Chapter 5 is extended in Chaper 6 by combining the bulk force field with force field parameters developed from the Biased Hessian Method applied to unique clusters to model the reconstructions of the Si (111) surface. Chapter 7 concludes this thesis with a description of the Generalized London Potential which was developed to accurately model chemical reactions at the accuracy of high level configuration interaction methods, but with the practicality of molecular mechanics. 1.0 Introduction Advances in quantum chemistry and molecular dynamics have progressed to the point where materials behavior predictions from computational atomistic simulations are being realized. Much progress has also been made in solving the more challenging problems involved in simulating materials processing. Advances in computer hardware together with advances in theory have lead to application of computational materials science to numerous important industrial and scientific problems, however many issues remain. Our goal was to develop and apply molecular mechanics and quantum chemistry methods to simulate and predict the materials properties. We have chosen to simulate several challenging materials problems, which required the development and extension of techniques and which illustrate the success of our approach; (i) Si (111) 1 - 3 surface reconstruction, (ii) chemical vapor deposition (CVD) growth of diamond, (iii) properties of bulk semiconductors and their (iv) interfaces and superlattices, (v) H abstraction for nanotechnology, and (vi) polysilane polymers. In the case of surface reconstruction, the primary goals are to determine the atomic structure of the surface and its stability. For the various Si(lll) surface reconstructions the differences in stability are very small and therefore only methods that are highly accurate should be employed in order to correctly resolve the energetic differences between the various structures. This is the case in many other systems as well. Advanced ab initio methods with high accuracy are the natural choices for studying systems where slight differences in geometry or energy need to be resolved. 4 However, except in the few cases of surfaces with small unit cells, ab initio quantum chemical methods are not practical since they scale as Ns where N is the number of electrons and s is at least 4, and depends on the method employed. Furthermore, few of the most accurate methods that have been developed for molecules and clusters have been extended to crystalline systems. One quantum based method that shows promise in simulating large systems, such as the recon- 2 struction of surfaces is density functional theory (DFT), 5 ,6 especially if the method includes the local density approximation, the gradient correction and algorithms to reduce the scaling factor. Currently there is insufficient experience with these methods and their accuracy for such systems is yet to be determined, however, preliminary results are promising. These methods can be applied to relatively large unit cells for single point energy calculations. However, these calculations are computationally intensive, particularly optimization of geometries, and computation of Hessians for frequency predictions. Quantum chemical methods are not practical for (i) or other large systems because they include the electrons in the simulations. 7 Molecular mechanics methods were developed to overcome this problem. In molecular mechanics, the atoms of the system are treated as classical particles that interact with each other via two, three, four, ... , n-body interactions. The series of interactions is usually truncated at a small n, like 4 or 5. The potentials that model these interactions take on various forms. In the method8 we employ these are represented by spring-like valence terms for atoms connected by a series of bonds. Including bond terms, bond angle terms, dihedral torsion terms, coupling terms and nonbond terms. For atoms not connected by a string of bonds, there are Coulombic and van der Waals terms. Each type of specific interaction has its own spring constant. In bulk silicon Si-Si bonds are modelled by the same potential. Bonds between a Si with three-fold coordination and a four-fold coordinated Si would have a different spring constant. Since molecular mechanics includes only interactions between the atoms as classical particles in a system, the effects of the electrons are averaged out. The number of interactions is therefore relatively small leading to a relatively simple algebraic expression for the energy of the system, in contrast to the much more complicated Schrodinger equation. The molecular mechanics total energy can be calculated by summing the individual energy terms. The properties of the system can be calculated from the 3 expression for the total energy. For example, forces are first derivatives of the energy with respect to the atomic coordinates, and vibrational frequencies are the square roots of the diagonalized second derivative matrix. Consequently, the energy of the system, and thus the system properties, are functions of the spring constants: This set of spring constants, called the force field or potential, can be optimized so as to reproduce the system properties. Often this procedure is empirical and involves varying the force constants to reproduce the experimental properties of the system. This is often very involved and can consume a major portion of the research activity. Once an accurate force field is available, it can be used to study arbitrary systems containing the interactions modelled by the force field. For moderately sized systems, less than a few thousand atoms, simulations of geometry optimization, phonon dispersion curves, vibrational frequencies, elastic constants, and thermodynamic properties are all carried out on inexpensive work stations over periods of less than several days. The development of Cell-Multipole Methods 9 has enabled simulations of systems with up to 1 to 10 million atoms at present. The advantages of molecular mechanics are (a), it can handle large systems relatively quickly and (b), the classical nature of the energy and other properties make it easy to understand, especially if valence-like terms are used to model the interactions. On the other hand, there are several drawbacks. One arises from the same feature that gives it its advantage; since electrons are not included in the simulation, quantum effects are neglected. The most important quantum effect is that of bond dissociation and formation. During a chemical reaction major changes to the electronic configuration of the system occur as what were once bonding interactions become antibonding or nonbonding interactions. So as a system nears a transition state, neglect of the Pauli principle leads to unphysical results, for example excess covalent bonds. _This problem will make simulation of (ii), CVD diamond film growth and (v), H abstraction reactions unreliable. One way to circumvent this disadvantage is to (a), combine molecular mechanics and ab initio methods for a simulation in such a way that ab initio methods are applied to the portion of the 4 system undergoing reaction, and molecular mechanics is applied to large numbers of atoms in the nonreacting portion of the system which usually affect the energy through strain effects. Another way to circumvent this drawback is to (b) develop for force field expressions that take into account the Pauli Principle. One such approach is the Generalized London Potential method developed by Donnelly 10 and applied to hydrocarbon reactions in collaboration with Musgrave. 11 In chapter 2 we report the results of approach (a) to study the CVD growth of diamond and in chapter 7 we report the efforts of approach (b). The reactions involved in the mechanisms to grow diamond are similar to the reactions envisioned to make diamond using nanotechnology. In chapter 3 we describe the simulations of hydrogen abstraction and carbon addition for making diamond using nanotechnology concepts. Another disadvantage with the molecular mechanics approach is that the approximate method entirely depends on (c) the availability of a force field. Assuming the force field has been developed, (d) it may not be transferable to the system under study. That is, although the force field may model experimental properties of one system well, it may break down when transferred to a similar system. Additionally, (e) there may not be sufficient experimental properties to determine an accurate potential through an empirical fit. We use the Biased Hessian Method of Dasgupta and Goddard 12 for developing force fields for molecules to overcome (c) and (d). The Biased Hessian Method increases the number of constraints on a force field by requiring it to fit the vibrational frequencies, and also the normal modes which are the ab initio vibrational eigenvectors. The force field is obtained by minimizing an error function which depends on the fit to the normal modes, the experimental frequencies and the ab initio normal modes. In the simulation of (i), Si surface reconstructions and (vi), polysilane polymers we are faced with systems for which insufficient experimental information exists to apply the Biased Hessian method in the standard way. In the case of Si (111) surface reconstruction, clusters needed to model the surface have not been experimentally detected. In the case of developing force fields for polysilane polymers, 5 the experimental vibrational frequencies are incomplete for some of the oligomers needed to develop the atomic potentials. Therefore in these cases the biased Hessian method cannot be used to develop a force field that reproduces the experimental information, since the experimental information currently does not exist. In chapter 6 we describe a method based on developing scales that describe the ratio of the force constants fit to experimental frequencies to force constants that were fit to the ab initio frequencies. We then describe how we apply these scales to the force field for the experimentally unobserved cluster by applying the scales to the FF fit to ab initio frequencies. We then apply this scaled force field to study the reconstructions of the Si ( 111) surface. Chapter 4 describes how (e), the lack of experimental information was over come to simulate (vi), polysilane polymers. Crystalline polysilane is not experimentally studied, because no crystalline samples have been prepared. We developed force fields for the SinH 2 n+ 2 oligomers which can then be used on the extended, periodic system. However, the set of experimental frequencies is incomplete for Si3Hs and n-Si4H10. Consequently, we develop a scaling method which uses scales between experimental frequencies and ab initio frequencies. The scales are averages of these ratios for a particular class of vibrations within each molecule. If one of the modes within that class is not experimentally known, then we apply the scale from that class and molecule to the ab initio frequency of that mode. These scaling methods are thought to be accurate since the standard deviation of the ratios within each molecule is very small. We test the method by applying the force field to larger oligomers of polysilane and comparing to the scaled ab initio frequencies. The n-Si 4Hrn potential is then applied to the polysilane crystal. Polymers are simple molecular extensions of oligomers and application of a force field from an oligomer to the extended system is expected to include the important physics that determines the material's properties. On the other hand, crystalline systems that are not molecular crystals, for example the diamond and zinc-blende crystals cannot be treated this way. To simulate the semiconductor (iv), materials we need to develop force fields directly for periodic systems. Our 6 molecular mechanics simulations of (i), Si (111) surface reconstructions (ii), growth of CVD diamond and (iv), semiconductor interfaces and superlattices involves combining bulk potentials with atomic interactions that either model the surface configurations, the interface configurations, or a surface chemical reaction. We develop potentials to model periodic systems, specifically (iii), the semiconductor crystals using a method similar to the Biased Hessian approach. In the periodic case, we do not use a normal mode description (although this will soon be possible). We fit to the special points in the ~ direction of the phonon dispersion curve and to the elastic constants. To complete the force field to simulate (iv) we use an extrapolation scheme to extend the potentials from the group IV, III/V or II/VI crystals to include terms necessary to describe the interfaces between various combinations of the materials. We describe our work on semiconductor force fields, interfaces and superlattices (iii and iv), in chapter 5. 7 2.0 References 1. K. E. Khor and S. Das Sarma, Phys. Rev. B 40, 1319 (1989). 2. D. Vanderbilt, Phys. Rev. B 36, 6209 (1987). 3. H.-X. Wang and R. P. Messmer, Phys. Rev. B, 41, 9241 (1990). 4. Bauschlicher, C. W. and Langhoff, S. R., Chem. Phys. Lett., 135, 67, (1987); Taylor, P. R. and Partridge, H., J. Phys. Chem., 91, 6148, (1987). For an excellent review see Bauschlicher, C. W., Langhoff, S. R., and Taylor, P.R., Adv. Chem. Phys., 77, 103, (1990). 5. W. Pickett, H. Krakauer and P. Allen, Phys. Rev. B38, 2721 (1988). 6. J. Ihm and M. Cohen, Phys. Rev. B21, 3754 (1980). 7. C. B. Musgrave, J. K. Perry, R. C. Merkle, and W. A. Goddard III, Nanotechnology 2, 187 (1991). 8. S. L. Mayo, B. D. Olafson and W. A. Goddard III, J. Phys. Chem. 92, 7488 (1990). 9. H. Q. Ding, N. Karasawa, and W. A. Goddard, J. Chem. Phys., 97, 4309 (1992). 10. R. E. Donnelly, C. B. Musgrave, and W. A. Goddard III, to be submitted. 11. C. B. Musgrave, R. E. Donnelly, and W. A. Goddard III, to be submitted. 12. S. Dasgupta and W. A. Goddard III, J. Chem. Phys. 90, 7207 (1989). 8 Chapter 1 Review of Methods 9 1.0 Review of Methods This chapter gives a brief review of the methods used in this thesis, including ab initio methods, molecular mechanics, and the Biased Hessian method. 1.1 Review of Ab Initio Quantum Chemistry Methods Ab initio quantum chemistry theory is used to find approximate solutions to the electronic Schrodinger equation 1iiI! = EiI! (1) for the electronic wave function iI!, where the Hamiltonian of the system is given by (2) Here the first sum includes the kinetic energy of the electrons, the second sum includes the Coulombic interaction between the electrons and nuclei and the third sum includes the Coulombic interactions of the electrons with each other. ZA are the nuclear charges, n the number of electrons and M the number of atoms in the molecule. In (2) we have ignored the nuclear kinetic energy since the nuclei are assumed to be at rest relative to the motion of the electrons. This approximation works well in practice and is called the Born-Oppenheimer approximation. The Coulombic interactions between the nuclei themselves are also not included since they remain constant with the assumption of fixed nuclei. The wave function will then be a function of the electron coordinates and only parametrically dependent on the nuclear coordinates. To generate a potential energy surface one calculates the wave function energy at various input nuclear coordinates. To calculate the forces on the molecule, one takes derivatives with respect to the nuclear coordinates. To calculate a Hessian, one computes the matrix of second derivatives and to extract the vibrational frequencies and eigenvectors one diagonalizes the Hessian. The solution to the many-electron Schrodinger equation will be an antisymmetrized n-fold product of spin orbitals. Since electrons are Fermions they must 10 satisfy the Pauli principle which states that no two electrons can occupy the same spin orbital. This constraint on the solution is imposed by antisymmetrizing the wave function by making '¥ a Slater determinant or sum of Slater determinants. Constructing wave functions from Slater determinants also insures that the electrons are indistinguishable. 1.2 The Hartree-Fock Wave Function The simplest wave function which obeys the Pauli Principle and can describe the ground state of the electronic Hamiltonian is, of course the single determinant \.Li. The single determinant W that best approximates a solution to (1) is called the Hartree-Fock wave function. Now (3) can be varied until it solves(l), the HF solution. This is done by applying the variation principle which states that the best description of the wave function is the one that minimizes the energy. To apply this in practice involves introducing parameters which vary the shape of the spin orbitals. The most common way of doing this is by using a set of atomic orbitals c/>µ on each of the atoms as a basis for describing the spatial portion of the spin orbitals. Each basis function, in this case a fixed atomic orbital, is mixed into the spin orbital by an amount determined by its orbital coefficient, Cµi: K 'l/Ji(r) = L Cµic/>µ(r). (4) µ=1 Increasing the size of the basis set allows us to better describe the electronic orbitals, however at a greatly increased computational cost. It is best then to be judicious in the basis set expansion, including sufficient functions so as to accurately describe the wave function, but not so many terms as to make the calculation too costly. Usually these types of calculations follow the law of diminishing returns as each additional basis function of a wisely chosen basis set improves the description of '1T less than the preceding function. 11 Applying the variational principle by minimizing the expectation value of the energy E = < wHF I H I wHF > (5) with respect to the set of coefficients C µi leads to a pseudo-one particle equation (6) where HHF = h + I)2Jj - kj) (7) j includes a sum over the Coulomb (Jj) and exchange (Kj) interactions with the other particles. The one electron operator h is a function of the coordinates of only one electron. It therefore includes the electronic kinetic energy and the Coulombic interactions between the electrons and the nuclei. Equation (6), known as the Hartree-Fock equation is in eigenvalue form. The operator HHF is known as the Fock operator. The spin orbitals are the eigenfunctions and the energies of the spin orbitals are the eigenvalues. The spin functions 'I/Ji are expansions of the basis functions, so the HF equation can be transformed into a matrix equation by multiplying (6) on the left by </>µ(1) and the appropriate spin function, substituting in (4), and integrating. This leaves K I: Cvi < </>µ(1) I HHF (1) I </>v(l) >= Ei < </>µ(1) I </>v(l) > . (8) V The matrix (9) is defined as the overlap matrix and the matrix (10) is called the Fock matrix. We also define the matrix of expansion coefficients C and the diagonal matrix £ of eigenvalues Ei which are the orbital energies. So 12 intergrating the HF equations and using the above definitions allows us to write the equation FC = SC£. (11) This matrix equation is known as the Roothaan equation. The matrices are of size K x K, where K is the number of basis functions so the time required to solve such an equation grows rapidly with K. Since the Fock operator of the HF and Roothaan equations depends on the eigenfunctions through the Coulomb and exchange operators, the equations are nonlinear. This makes it necessary to use an iterative procedure to find solutions to (6) or (11). 1.3 The SCF Procedure The iterative procedure used for solving these equations is known as the self consistent field (SCF) procedure. First to minimize the energy we require that the first order variation of the energy with respect to orbital variation be zero. We expand the energy to second order in changes to the orbitals and derive (12) where variations in the orbitals are restricted to be orthogonal to the orbital being varied. The SCF procedure involves setting the shape of all the spin orbitals at a starting guess value by estimating C. This defines the Hartree-Fock potential for this starting guess wave function and we can obtain the new expansion coefficients and orbital energies by diagonalizing the Fock matrix. If the new expansion coefficients are the same as the original ones, then the wave function is converged. If they are significantly different then we use the orbitals described by this intermediate C to start the procedure over again and repeat until self-consistent. The n lowest energy spin orbitals are occupied and the 2k-n remaining orbitals are unoccupied and referred to as virtual orbitals. The Hartree-Fock method does not allow the electrons to dynamically correlate their motions because they interact with each other only in an average way. This approximation makes HF very fast 13 and still capable of describing many processes reasonably well. On the other hand, many physical properties will be poorly described, for example the dissociation of bonds. Because the electrons do not correlate their motions dynamically, as a bond breaks upon atom separation, the electrons continue to doubly occupy the bonding orbital, which becomes nonbonding as the bond breaks. What is required is that the doubly occupied orbital that describes the bond of the molecule become two singly occupied nonoverlapping orbitals of the fragments. So near equilibrium HF describes the ground state relatively well, but it poorly describes strains, bond dissociation, excited states, chemical reactions, etc. Furthermore, this deficiency leads to overestimates in the HF vibrational frequencies, typically 10 to 20% too large. Much of quantum chemistry research has been devoted to improving upon HF by including additional electron correlation. The inclusion of additional correlation leads to a series of improved theories, all using HF as their starting point. These include the Generalized Valence Bond 1 (GVB), Complete Active Space SCF 2 (CASSCF), Configuration Interaction (CI), Multi Reference CI (MRCI), Multiconfiguration SCF (MCSCF), and second order M0ller-Plesset perturbation theory3 (MP2) wave functions. 1.4 Configuration Interaction Wave Functions The HF wave function, and all wave functions that use it as a starting point explicitly include all of the electron-electron interactions in 1i, N 2 L ;__ i>j=l (13) iJ but solve for the wave function in terms of a hierarchy of increasingly accurate solutions. The Hartree-Fock wave function was limited to a single determinant. Although HF satisfied the Pauli principle, there is no reason additional Slater determinants (electronic configurations) cannot be allowed to be included in the wave function. The configurations are obtained by replacing one or more of the occupied HF spin orbitals with unoccupied virtual orbitals. The resulting Slater determinant 14 is called an excited determinant. If one orbital is replaced, then the excitation is referred to as a single excitation, two orbitals replaced leads to a double excitation, etc. At this point we can do a series expansion for the ground state wave function in the Slater determinants to make what is known as the CI wave function w = Co'1>'o + L C1 '1>'1 + L C2'112 + ... + single double L Cn Wn-tuple • (14) n-tuple As we include additional configurations, we can improve upon the description of the wave function, however the computational costs of determining the coefficients of the Slater determinants increases with the length of the series. Therefore, this series is usually truncated very early, for example at the double excitations in many cases, and even this leads to very large calculations for moderately sized molecules. The coefficients are solved for by applying the variational principle to the CI wave function and optimizing the configuration coefficients to minimize the energy of the system. The HF*SD CI wave function is an example of CI method that is truncated at the level of double excitations. Here all single and double excitations of the valence electrons are allowed into the virtual orbitals with reference to only the HF configuration. 1.5 MRCI Wave Functions The MRCI wave function is similar to the CI wave function just described, except that rather than just including excitations of the valence electrons into the virtual orbitals from the HF determinant, we include valence excitations to the virtual orbitals with reference to other important configurations (references). If the CI wave function is not truncated, then the MRCI and CI wave functions are degenerate. However, since we truncate the CI series at doubles in most cases, the excitations from the references, which are excited Slater configurations themselves can contribute significantly to lowering the energy of the wave function. Allowing these extra references from which we can excite from greatly improves the CI description of transition states where several electronic states mix into the transition state wave function. For example, in the case of H transfer between methyls the 15 important configurations are the HF configuration, the antibond configuration with one node and the antibond configuration with two nodes. The GVB*SD CI wave function is a multireference CI that includes up to double excitations. This MRCI wave function is a high accuracy approximation to complete CL The improvement over the HF*SD CI wave function for the description of transition states is shown in chapter 3. As the MRCI wave function is calculated on larger and larger systems a drawback is encountered in that the correlation energy included in the calculation decreases as the size of the system increases. For example, the MRCI energy of two infinitely separated molecules will be greater than the sum of the MRCI energies of the component molecules. This is because a greater fraction of the valence space is allowed to be excited in the smaller calculation where the larger calculation is restricted to a larger fraction by the truncation in the series. As the series of excitations is extended this effect becomes less important, but the expense of the calculation becomes prohibitively large. Because MRCI calculations, from only a few references and truncated at the doubles level, can still be prohibitively expensive we have in some cases used smaller CI calculations which do a good job of approximating the results of GVB*SD CL 4 , 5 These wave functions restrict the orbitals from which excitations can be made. For example, in the correlation consistent CI ( CCCI) 6 wave function all single and double excitations of the active electrons (active electrons are those in orbitals that undergo significant change during a reaction, for example the electrons in a breaking bond) and all single excitations of the other valence electrons into the virtual space relative to the three GVB configurations are allowed (thus double excitations from the valence space is not allowed). Since double excitations from the valence space are so numerous and do not contribute to the correlation as significantly as the allowed excitations, the essential physics of a reaction is retained while greatly improving the cost of the calculation. An equivalent way of writing the CCCI wave function would be GVB*(SDactive + Svalence). A similar wave function, called the dissociation consistent CI (DCCI) 7 wave 16 function can be constructed which allows double excitations which are the product of single excitations from the valence space with single excitations from the active space. When higher order excitations are allowed an even larger hierarchy of wave functions can be made. Often, the set of wave functions in this hierarchy are tested on a small system analogous to the one under study by comparing to experiment. We can then determine which of them best describes the phenomena of interest without being computationally expensive. In some cases a method may seem to give accurate results, although the quality of the results are fortuitous. Therefore care must be taken when using such a method. It is therefore often useful to apply the method to small model systems that are well understood or experimentally measured. 1.6 MCSCF Wave Functions In addition to optimizing the configuration coefficients it is possible to reoptimize the orbital coefficients (therefore the orbitals from which you build the configurations are no longer confined to be the HF orbitals). This type of solution is referred to as a MCSCF wave function. Once the orbital's coefficients are reoptimized, then the configuration coefficients can be reoptimized until self consistency is obtained. This procedure allows one to obtain much of the correlation energy with far fewer configurations than a MRCI or CI. The CASSCF or GVBCI-SCF 2 wave function is an example of an MCSCF wave functions. In the CASSCF wave function an active space is picked. This space consists of orbitals that vary significantly during the process being studied. For example, in the case of H transfer between methyls the active space would include the C-HC orbitals along the symmetry axis with zero, one and two nodes. The electrons are the radical of the reactant, the radical of the product and the H electron. All symmetry and spin allowed configurations of these three active electrons in the three orbitals are generated. Thus a CASSCF makes no restrictions on the excitations the active electrons can make within the active space. 17 1.7 The Generalized Valence Bond Wave Function Probably the simplest and most economical way of improving the HF wave function is the generalized valence bond wave function developed by Goddard and coworkers. A brief summary of the GVB approach as described by Bobrowicz 1 is given here to illustrate the method. The main aim of the GVB wave function is to describe bond dissociation properly. To remove the HF deficiency GVB allows bonds to be described by two singly occupied, overlapping orbitals which make up a GVB pair. This removes the condition HF imposes on the bonding orbital; that it remain doubly occupied during the dissociation process. The GVB pair can then be written as (15) where A is the antisymmetrizer and where the overlap, S12 between the two orbitals describing the bond is not zero (16) Each orbital is associated with one of the atoms participating in the bond. For example, ¢ 1 is localized on atom 1 and ¢ 2 on atom 2. As these orbitals are optimized to describe the bond properly their shape changes to describe the polarization of charge. For simplicity we focus here on the perfect pairing form of the wave function GVB-PP which has the form W = A ['1f core Wopen Wpair] , (17) where ncore/2 Wcore = IT (<Pia) ( </Ji/3) (18) i=l contains all doubly occupied orbitals in the core space of the molecule however, not the doubly occupied bonding orbitals. The function nopen/2 Wopen = IT i=l (<Pia) (19) 18 is the product of all the singly occupied orbitals. To maximize the exchange interaction (Hund's Rule) the singly occupied orbitals will have the same spin. The core and open shell electrons are treated exactly as they are for the wavefunction. npair Wpair = II (C ic/>igc/>ig - Cuic/>iuc/>iu) (a/3- f3a), 9 (20) i=l describes all GVB pairs. Here we have rewritten Wpair so that the two orbitals which make up the spatial part of the bond are a linear combination of the symmetric bonding HF-like orbital and antisymmetric antibonding HF-like orbital. The symmetric orbital c/>ig of pair i is just the sum of ¢ 1 + ¢ 2 , while the antisymmetric orbital c/>iu is the difference ¢1 - ¢2 and therefore has a node through the bond. The general energy expression for GVB-PP wave functions is DCC E = DCC L 2Jihii + L (aijJij + bijKij) i,j (21) where h, J, and Kare the standard one-electron, Coulomb, and exchange energies. This expression for the energy is similar to the HF expression except that the coefficients of the Fock, Coulomb and Exchange energies are now functions of the pair coefficients Cgi and Cui• The coefficient of the Fock energy is called the orbital occupation coefficient · Ji and given by Ji = 1 1 Ji = 2 Ji = (Ci ) 2 i is a core orbital i is an open orbital (22) i is a pair orbital with pair coefficient Ci, The coefficients for the two-electron Coulomb and exchange operators can be written as 19 except that 1 bij = - 2 if i and j are both open orbitals aii = Ii if i is a pair orbital bii = 0 if i is a pair orbital aii = 0 if i and j are in the same pair bii = CiCi if i and j are in the same pair. With the energy expression (21), the general condition for convergence is that the first-order change in the energy due to orbital changes be zero, leading to (12) i where Fi is the generalized Fock operator for orbital </Ji, occ Fi = fih + L aijJi + bijKi, (23) j Since Fi depends on the orbitals, this equation is nonlinear and we solve for (12) iteratively until self-consistency is achieved. 1.8 Summary The preceding review of ab initio quantum chemistry methods illustrates the type of calculations that are done on small molecules. Many other methods have been developed to be more accurate or faster. Often however, these methods have deficiencies which make them inappropriate for application to certain classes of problems. The various semi-empirical methods for example are very fast, and work well for unstrained molecules. However, in the case of transition states or strained systems the method gives highly inaccurate results. The pseudospectral methods were also developed to be much faster than standard QC methods. These methods however have so far proven very accurate for the systems studied. An exhaustive review of the methods available today would span many large volumes and because 20 this review is only meant to cover some introductory material we stop our review of ab initio methods here. 2.0 Review of Molecular Mechanics A molecular mechanics calculation begins by defining the coordinate information of a given list of atoms. The form of the atomic potential used in this thesis also requires that a list of the bonds in the system be specified. Given the coordinate and coordination information the program (MSI/Polygraf) 8 makes a list of all bond, angle, dihedral torsions, cross, and non-bond interactions. Each interaction is classified by the atom types involved and an energy term is assigned for that interaction. The molecular mechanics energy can then be calculated by summing this series of energy terms and the properties of the system can be calculated from the energy function. The energy of the system and thus the system properties are functions of the spring constants in each valence term of the energy expression. 2.1 Form of the Force Field In some molecular mechanics methods the potential is only dependent on the atom type, and geometry. Thus the coordination must be determined implicitly from the geometry. In the valence form of the potential the valence interactions are determined based on the connectivity input, thus atoms with equivalent nuclei, but with different coordination numbers have different force constants and thus different functions describing their interactions with other atoms, while in the nonvalence form they have the same force constants. The advantage to the valence method is that it greatly simplifies the functional form of the potential and facilitates physical understanding of the calculation. The general form of the force field 9 we use is (1) where Eval includes all terms involving bonds and angles where EQ describes electrostatic interactions, and Evdw describes the van der Waals nonbond interactions. 21 The valence term is taken as (2) which includes all terms involving bonds between atoms and the coupling of these bonds. When the notation MSXX is used to label the force field it indicates that both one-center and two-center cross terms are included, the MS stands for materials simulations. 2.2 Electrostatics The electrostatic term is written as I EQ - C - ~ qiqj coul L.,; R·. i>j (3) iJ involving partial charges qi (in units of the electron charge, lei) on the various atoms. [Here Ccoul = 332.0637 ensures that energies are in kcal/mol with distances in A.] 2.3 van der Waals The van der Waals (vdW) potential represents the long-range attraction (London dispersion) and short-range repulsion (Pauli orthogonalization of nonbonded electrons). Such terms are often ignored in force fields (their effects being embedded in valence terms). The vdW interactions are described with the Lennard-Jones 12-6 form I Evdw = L....t ~ D1!~W [p:-: 12 _ 2p:-:6] iJ iJ iJ ' (4a) i>j where (4b) In addition we assumed the standard combination rule (5a) 22 (5b) 2.4 Nonbond Exclusions The summation over i and j in (3) and (4) excludes bonded atoms [1-2 cases] and next-nearest neighbors [1-3 cases]. It is assumed that these interactions (which would be partially shielded) are included in the valence interaction terms for bonds and angles, respectively. We found that the next-next-nearest neighbors [1-4 cases] are very important in determining the lattice constant and include them plus all longer interactions. 2.5 Bond Terms We take Eb 0nd as a sum over all bond pairs where each has the form of a Morse function, 9 (6) with (7) and (8) This describes anharmonicity and allows a proper description of bond dissociation. There are three independent parameters Re, kn, and Dn; however Dn is not sensitive to the data used in the fits. 2.6 Angle Terms For each atom J there are six angle terms I - J - K. The functional form for the diagonal angle term is taken as 9 (9) where the force constant is (10) 23 This form (9) leads correctly to dE/d0 = 0 for 0 = 0, 180° and has a barrier of Egarrier = C [1 + COS 0e)2 . 2 (11) 2.7 Angle Cross Terms (!-Center) For each angle term I - J - K, we include in Ecross the couplings between the bonds (I J - J K) and the coupling between each bond and the angle (I J with I J K and J K with I J K). These have the form of 9 (12) where R1 is the I - J distance and R2 is the J - K distance and (13) where there are two terms, one for R = R1 J and the other for R = RJ K. D R0 is related to the force constant as (14) 2.8 Torsion Terms Consider the bond J - K. The dihedrals involving various atoms I bonded to J and L bonded to K are given the energy dependence described by 9 1 Etor = ½or [1 + COS m¢) , (15) where ¢ is the dihedral angles <PIJKL· This has minima (with zero energy) for m¢ = 180°, and maxima (with energy ½or) for 0° (cis), and m¢ = 360°. We sum (16) over all DIJ=(C1 - l)x(CJ - 1) dihedral terms, where Ci is the coordination number of atom i, and divide by D1 J so that ½or is the total barrier. 24 2.9 Additional Cross Terms For two angles sharing a common bond and apex, the one-center angle-angle cross terms has the form 9 (16) The two-center angle-angle cross term involves the coupling between two angles sharing a common bond, but not a common apex, 9 (17a) where the force constant is (17b) Thus for a sequence of four bonds, I - J - K - L, 01 = 0IJK, and 02 = 0JKL• For a given J - K there are D1 J possibilities. 2.10 Vibrational States Periodic Systems (Phonons) Using the above force constants to build an energy expression allows us to manipulate the energy to calculate system properties. This section describes the general concepts necessary to compute phonon dispersion curves. The detailed substitutions of the derivatives of the energy etc. are straightforward, although often tedious. Consider the energy expanded near equilibrium, these calculations expand R·i -R~i +8R·i E(Ri) = E(Rf) + (18) L (EI)e 8Ri + ~ L (Eijt 8Ri8Rj, i (19) iJ Here the force at equilibrium is zero (20) 25 and (21) is the Hessian. Newton's equation of motion becomes (22) where a and /3 are the x, y, z components of the atomic coordinates and I, j 1, ... , N. Assuming periodic motion with frequency w, (23) leads to the eigenvalue equation w2 M1( bR~I) = L 4>ad,,6J ( bRij) eik-(R13J-Ra1). (24) ,6J For the r point this leads to the eigenvalue equation (25) where (26) (27) (with win cm-1, mass in atomic mass units, R in A, and energy in kcal/mol, then Cjreq = 108.5913) and (28) describes the lh vibrational mode. 3.0 The Biased Hessian Method The calculation of properties using molecular mechanics methods depends on the quality and availability of a force field. Often a force field is applied to a 26 system only to find that it is inadequate for the phenomena of interest. At this point it is necessary to acquire an improved force field. One way to do this is to use the Biased Hessian method of Dasgupta and Goddard. 10 The energy expression of a molecule can be expanded as: where the force on the i th component is: (30) and a2E Hij = 8Ri8Rj (31) is the Hessian. From ab initio HF wave functions we calculate 11 a full Hessian (32) where Rai is the a component (x, y, z) of the coordinates of atom i. After mass weighting, - HF 1 H m,·13·=--;::== J ✓MiMj HHF ·13· m, J' (33) the vibrational modes {Uff F} and vibrational frequencies {vff F} are obtained by solving (34) where ,HF /Ii = (CJregViHF)2 (35) and C freq = 108.5913 converts units so that energies are in kcal/mol, distances are in A, frequencies are in cm- 1 , and masses are in atomic mass units ( C 12 has mass = 12.0000 amu). This Hessian provides an enormous number of constraints 27 useful in determining the force field. Thus there are g(g + 1)/2 independent pieces of information [for example, 666 for Si 4 H 10 ], where g = 3N - 6 is the number of degrees of freedom. In contrast, fitting just the frequencies leads to only g conditions [36 for n-Si 4 H 10 ]. However, at the HF level the calculated frequencies, vfIF, are 10-20% too high. This led to the development of the Hessian Biased method 6 for FF parameterization in which the force field is fit to the biased Hessian (36) where U is the transpose and Aexp is the diagonal matrix based on experimental frequencies , exp Ai = (C freq Viexp)2 • (37) This Hessian has the property that, HHBuHF = uHFAexp, that is, the eigenvalues match experiment (or a combination of experimental frequencies and scaled ab initio frequencies) while the eigenfunctions match HF theory. Thus, HHB has the best available information on the vibrational modes. The method does require accurate mode assignments. To optimize the force field we minimize the error function 3N ERR =Wgeom 3N L (8ED + Whess L (8Eij) 2 2 i=l 6 6 i=l i<j=l (38) + Wstrs L (8~i) 2 + Wcij L (8Cij) 2 where 8 denotes the difference between the quantities calculated from the force field and the reference values from experiment (N is the number of atoms). Here EI is the energy gradient (negative force, which should be zero for each atom), and E:'j is the second derivative of the energy (Hessian). For periodic cases we ignore the Hessian and include ~i, which is the lattice stress (6 independent values of which should be zero), and Cij, the elastic constants (3 independent values for cubic crystals). 28 The Hessian term is transformed to the eigenvalue form and replaced by 3N Wfreq L i=l 3N (8vi) 2 + Waffdiag L (8Hij) 2 • (39) i<j=l In general the optimum geometry at the HF level differs slightly from experiment, raising the question as to which structure is used in (10). We use the structures optimized at the Hartree-Fock level of theory. Previously 12 we advocated the use of the experimental structure for determining force constants from the ab initio calculations primarily because the internuclear separations (which strongly affect the Hessian) reflect the experimental system. However, in molecules with low frequency torsions, a slight difference in structure can cause a noticeable rotational contamination of the torsional modes. Since we want to use frequency scaling parameters to compare various molecules, it is better to derive the frequencies for all molecules at the ab initio minima (rather than at the experimental minimum for molecules where experimental geometries are available and the ab initio minimum for those where experimental geometries are not available). Fortunately, the differences between the ab initio and experimental geometries are small. 29 4.0 References 1. Bobrowicz, F. W. and Goddard, W. A. in Methods of Electronic Structure Theory (Ed. Schaefer, H.F.), 79, 1977. 2. Smith, D. and Goddard III, W. A., J. Am. Chem. Soc. 109, 5580 (1987). 3. C. M0ller and M. S. Plesset, Physical Review, 46, 618 (1934). 4. C. B. Musgrave, J. K. Perry, R. C. Merkle and W. A. Goddard, III, Nanotechnology 2, 187 (1991). 5. C. B. Musgrave, S. J. Harris and W. A. Goddard, III, to be submitted. 6. Carter, E. A. and Goddard, W. A., J. Chem. Phys., 88, 1752, (1988). 7. Carter, E. A. and Goddard, W. A., J. Chem. Phys., 88, 3132, (1988). 8. These calculations used POLYGRAF from Molecular Simulations Inc. (Burlington Mass). 9. S. L. Mayo, B. D. Olafson and W. A. Goddard III, J. Phys. Chem. 92, 7488 (1990). 10. S. Dasgupta and W. A. Goddard III, J. Chem. Phys. 90, 7207 (1989). 11. Gaussian86: M. J. Frisch, J. S. Binkley, H.B. Schlegel, K. Raghavachari, C. F. Melius, R. L. Martin, J. J. P. Stewart, F. W. Bobrowicz, C. M. Rohlfing, L. R. Kahn, D. J. Defrees, R. Seeger, R. A. Whiteside, D. J. Fox, E. M. Fleuder, and J. A. Pople, Carnegie-Mellon Quantum Chemistry publishing Unit, Pittsburgh,PA, 1984. 12. N. Karasawa, S. Dasgupta, and W. A. Goddard III, J. Phys. Chem. 95, 3358 (1991). 30 Chapter 2 SR-SOR Step for Growth of CVD Diamond 31 Abstract Recombination of a Surface-Radical with a Surface-Olefin (SR-SOR) to form a six-membered ring is a critical step in the current mechanism for chemical vapor deposition (CVD) growth of the diamond (100) surface. We compare the potential energy surfaces (PES) and activation barriers (Eact) calculated with various ab initio methods for the SR-SOR recombination step. The PES of SR-SOR-C 3 H 7 is calculated at various levels of correlation to compare the resultant geometries of the reactant and transition states and to compare the topology of the PES. The ab initio calculations of the C3 H7 PES include Hartree-Fock (HF), second order M0ller-Plesset (MP2), Generalized Valence Bond-Configuration Interaction (G VBCI or CASSCF) and GVB times singles and doubles CI (GVB*SD CI). We estimate the rate constant for SR-SOR by combining quantum chemistry calculations, molecular mechanics calculations, and transition state theory. The cluster models were corrected for steric interactions of the cluster with the rest of the surface and for strain effects on the lattice. We also compare the barrier and geometries obtained when using two different clusters; (i) C 3 H 7 , and (ii) C 10 H 15 to model the surface. HF and MP2 were used to model the reactants, and saddle point of the larger cluster modeling the surface. Our results show that the barrier calculated for the larger cluster was slightly lower than the barrier of the small cluster. The ab initio Hessian matrix was diagonalized to find the vibrational frequencies, which were used to construct a partition function for calculating the entropy. Transition state theory was used to obtain the rate constant, k = 2 x 10 13 e- 8800 / RT sec- 1 • This implies that under normal growth conditions SR-SOR is fast compared with competing gas-surface reactions. 32 1.0 Introduction Because of strong industrial interest in developing low pressure technologies for synthesizing diamond, there has been considerable effort in elucidating the fundamental mechanisms of diamond film growth. 1 - 3 The understanding of the kinetics and thermochemistry of carbon-hydrogen systems is so developed that it has been expected that chemical vapor deposition (CVD) of diamond could be understood at a level of detail far greater than that of most other CVD processes. The assumption usually made is that the gas phase reactions are a good analogue to reactions on a diamond surface. The assumption depends on whether the electronic rearrangement that occurs during reaction is local and to what extent the surface constrains the reaction. To calculate a rate constant for SR-SOR on this surface we combine several methods. The strategy is to use ab initio methods on small clusters where they are less expensive and where the accuracy in modelling the local electronic structure of the system is critical. Furthermore, molecular mechanics techniques were employed where they were most accurate and cost effective; in estimating the strain energies of the reaction on a lattice of approximately 500 atoms. This hybrid technique gives an alternative, more accurate method to semi-empirical methods when studying the energies of large systems where only a small portion is undergoing significant electronic structure rearrangement. For tetrahedral electronic materials (Si, GaAs, ... ) the (100) surface has generally been the surface of interest for commercial growth of thin films by CVD and MBE technology. Hence the mechanisms for diamond growth on this surface is of interest. Both AFM and STM measurements have established that the stable surface is hydrogen terminated with a 2 x 1 reconstruction as indicated in Figure la. A close-up of the 5-membered rings (denoted as Cs) is shown in Figure lb. This Cs ring must be opened and converted to a 6-membered ring (Cs to C 6 ) during growth. Such a process is potentially rate determining, and hence an understanding of the mechanism is relevant. In this chapter we use ab initio quantum chemistry theory, molecular mechanics, and transition state theory to analyze the critical Surface 33 Radical-Surface Olefin Recombination (SR-SOR) step in the proposed mechanism, and we estimate its rate constant on the diamond surface. A number of experiments have shown that methyl radicals are generally the dominant gas phase precursor species reacting with the surface to grow diamond, 4 -s and several detailed chemical kinetics mechanisms have been proposed to analyze growth on various idealized diamond surfaces. 9 - 15 For the most part, these models have relied on the assumption that a growing diamond surface behaves like an alkane and that the chemistry on diamond surfaces is controlled by the local electronic environment, as in alkanes. In addition to assuming behavior similar to that of alkanes, often rate constants for gas phase reactions have been used to predict the reaction rates of analogous reactions taking place on the growing CVD diamond surface. This assumption holds in most instances, however dynamical constraints imposed by the diamond surface restrict nuclear motions such that transition states can only be reached by straining the surrounding structure. Furthermore, several of these mechanisms rely on the ability of semi-empirical methods to predict transition state barriers. Recently, Garrison et al. 16 proposed dimer opening reactions for the (100)-(2xl):H diamond surface based on molecular dynamics simulations using Brenner's hydrocarbon potential. 17 These reactions form part of a complete chemical kinetics mechanism which can make quantitative predictions for the growth rates in hot filament, microwave plasma, flame and plasma jet CVD systems. The virtue of the Garrison mechanism is that it consists entirely of reactions which have well-known analogs in hydrocarbon chemistry, immediately making the mechanism plausible. 2.0 The Mechanism i. The Garrison mechanism (Figure 2) commences with addition of a CH3 radical to a surface radical site (Figure 2b to Figure 2c), forming the structure denoted as C5M. ii. The next step (Figure 2c to Figure 2d) is abstraction of a hydrogen atom 34 from the CH3 group to make the radical denoted as C5 M*. 'tz't. Through the ,8-scission electronic rearrangement (Figure 2d to Figure 2e), this structure isomerizes to another radical, denoted as C5d. Here the dimer bond breaks and a double bond forms between the carbon atom (C,a) of the original structure and the carbon atom (Ca) that started as an adsorbed methyl (now the a-carbon of the olefin). Such unimolecular reactions are generally fast compared with bimolecular reactions. iv. The final SR-SOR step in the mechanism (Figure 2e to Figure 2f) 1s mtramolecular attack of the radical carbon (Cr) of C 5d with the Ca of the double bond. This leads to a 6-membered ring denoted as C6, containing a radical site (C,a). It is this step that we examine herein. Gas phase reactions involving radical attack at a doubly bonded carbon, such as (1) H2Ca = C,aH2 + ·CrH3 --t H2Co: - C,aH2, are typically very fast and proceed with almost no barrier. However, on the surface two factors considerably decrease the rate: i. the initial equilibrium distance between the radical carbon, Cr, and the acarbon, Ca, is 2.85A, almost twice the normal CC bond distance, and ii. Cr is tightly constrained by the lattice from moving toward Ca, while Ca cannot move toward Cr without straining the Ca-C,a 1r-bond. Although all the steps in the mechanism have analogous gas phase reactions, some of the reactions are modified by the strains introduced when putting these reactions on a surface. Thus, on the growing diamond surface the SR-SOR reaction step (2) could have a significant barrier. The question is whether the constrained SR-SOR reaction (6), is sufficiently slow on the diamond surface as to limit the rate of di- 35 amond formation. We concentrate on examining the critical step in the Garrison mechanism: the attack of the radical carbon of Cd at the CH2 end of the double bond, leading to a 6-membered ring containing a radical. We refer to the radical carbon as C 1 , the a-carbon as C2 and the ,B-carbon of the olefin as Ca, as shown in Figure 3. We refer to the final step as the Surface Radical-Surface Olefin Recombination Step (SR-SOR). In the gas phase the analogue of the SR-SOR ring closing step is the attack of an olefin by a methyl. Reactions involving radical attack at a doubly bonded carbon, such as (1) typically have almost no barrier. This reaction in particular proceeds with a very small barrier in the gas phase as the methyl is free to approach the olefin and the resulting er-bond more than compensates for the breaking of the 1r-bond throughout the reaction. However, on the surface two factors may considerably increase the barrier. (i) The equilibrium distance between the radical carbon, C 1 and the a-carbon, C2 is 2.85A, almost twice the normal CC bond distance; (ii) C 1 is tightly constrained by the lattice from moving toward C 2 , while C 2 , being the tail of the olefin cannot move toward the radical carbon without moving out of the plane defined by the double bond and the two sub-surface carbons below Ca, straining or perhaps breaking the 1r-bond. Thus, on the surface the constraints imposed by the lattice force the reaction to proceed over a significant activation barrier. The surface does not allow the radical to freely engage the olefin and for the reaction to proceed the tail of the olefin must bend towards the radical. This process involves the straining of the 1r-bond without any significant contribution from the er bond as it is too far away to stablize the transition state. Barriers are the result of bonds dissociating while new bonds are only partially formed and thus unable to contribute enough to the stability of the system to make up for the energy to break the original bond. Because it is the bond dissociation process that is poorly described by more 36 approximate methods, the transition state is also poorly described. Because transition states are not well described by simple wave functions or by semiempirical wave functions that are parameterized to describe unstrained systems and thus do not adequately include the effects necessary to describe transition states it is important to determine what is the minimum level of correlation necessary to adequately describe various transition states to make quantitative predictions of reaction rates. In this chapter we compare the results of various ab initio quantum chemistry methods in predicting the barriers and PES of the critical Surface Radical-Surface Olefin Recombination (SR-SOR) step in the proposed mechanism. The questions we ask are; (i) what is the appropriate level of correlation necessary to accurately predict the PES and reaction barriers, (ii) what size cluster is needed to model the relevant parts of the surface and (iii) what are the characteristics of various methods that lead to the variation in reaction barriers. 3.0 Calculational Details 3.1.1 Ab Initio Quantum Chemistry The simplest wave function used is the wave function in which each molecular orbital is doubly occupied (except the radical orbital). HF should give a qualitative picture of the SR-SOR reaction under study. HF will tend to over estimate reaction barriers since it generally over estimates stretching frequencies of bonds due to its poor description of the bond breaking process (a doubly occupied orbital of the molecule must become two singly occupied orbitals for the fragments). We can remedy this problem by removing the HF constraint that a bond be described with a single doubly occupied orbital. Additionally, wave functions needed to describe transition states can be significantly more complex than those needed to describe minima. HF, being a single determinant wave function precludes the contributions of important configurations along the reaction path. At the transition state, the electronic structure includes a resonance between the reactant and product states, which requires a multi-configuration wave function for proper description of the 37 partially broken bonds and partially formed bonds. Thus, to obtain more accurate potential energy surfaces for reactions such as (1) or (2) than HF, it is essential to include all important electron correlation effects that change during the reaction. For this purpose we use the GVB-CI method. GVB 18 ,19 is the simplest wave function which describes bond dissociation properly, by allowing each bond to be described with two singly occupied, overlapping orbitals leading to a proper description of dissociation. In the GVB-CI method all electrons involved in bonds that change during the reaction are correlated, and all other pairs of electrons are calculated self consistently. For Reactions (1) and (2) there are three such electrons. For the reactant these correspond to the 1r-bond electrons and the radical electron of the a-carbon. In the product they correspond to the new C - Ca-bond electrons and the radical electron on the C 3 • There are two ways to spin pair these three electrons (one corresponding to reactant and the other to the product) and the GVB-CI method calculates all orbitals self consistently (these three plus all other electrons) while optimizing the spin coupling. Simply, all symmetry- and spin-allowed configurations of three active electrons in three orbitals are generated. Although this wave function is an improvement on the description of the reaction by HF, it will still tend to over estimate the reaction barriers. After calculating the GVB-CI wave function, we then allow all single and double excitations from the three GVB-CI configurations to all possible virtual orbitals, which is called a GVB*SD CL This calculation is rather computationally intensive and was not attempted for the larger C10H 15 cluster. GVB*SD CI has been shown to accurately approximate the results of very complete CI calculations. 20 To describe the reaction path we considered 40 positions of the critical carbon, C 2 in the cluster shown in Figure 3. These points were selected to describe the reactant and saddle points (the exit channel was not examined, although the geometry and energy of the product, C 6 * was calculated). Calculations were carried out at the points (R12 , R 23 ) shown in Figure 4. R 12 corresponds to the distance parallel to the surface from the initial position of C2, while R23 refers to the dis- 38 tance from the original position of C 2 perpendicular to the surface. We fixed C 1, C3 and the two hydrogens on C 1 and on C3 representing the bulk atoms. Then for each value for R 12 and R 23 , we optimized the position of the other three hydrogens. This optimization was carried out at the MP2 level of M0ller-Plesset perturbation theory. 21 MP2 is based on the Hartree-Fock wave function and although through perturbations it can include additional electron correlation not included in HF, it may not include all the important electron correlation needed to describe transition states well. Hence, the energies and forces on C1, C2 and C3 may not be accurate in the transition region. However the C - H bonds should be well described with MP2 and hence the H positions are expected to be accurate. For each such geometry (R 12 and R 23 ) from MP2 we calculated the GVB-CI and GVB*SD CI energies, denoted GVB-CIMP2 and GVB*SD CIMP2· In addition to the calculations performed on the C3H 7 cluster, we perform ab initio calculations on a larger cluster to estimate the effects of cluster size on the reaction barriers. The cluster consists of 10 carbon atoms and 15 hydrogens (as shown in Figure lb), where the product is bicyclononane with a radical at one of the tertiary carbons. We consider only two geometries. (i) The reactant geometry and; (ii) The transition state geometry. In these calculations, the positions of C 1, C2 and C3 and the hydrogen atoms bonded to these three carbon atoms were all optimized, which gives a better model of the relaxation on the surface than the smaller cluster. The remaining 7 carbon atoms and 12 hydrogen atoms were held fixed to model the lattice. This optimization was carried out at the MP2 level of M0ller-Plesset perturbation theory. 21 Using the MP2 optimized geometries we calculate the GVB-CIMP 2 reactant and saddle point energies to obtain MP2 and GVB-CIM P2 barriers. For all GVB calculations on the 3-carbon cluster we use the Dunning22 and Huzinaga23 double ( basis set plus diffuse s and p functions (( 8 = 0.0474, (P = 0.0365) plus one set of d polarization functions (( = 0.75). The triple zeta contraction of the 6s set was used for hydrogens. The 6-31G** basis set was used for the 39 MP2 optimizations on the 3-carbon cluster while 6-31G* was used for the MP2 optimizations on the 10-carbon cluster. MP2 was performed using GAUSSIAN 92 24 on Hewlett Packard 730s. The HF, GVB-CI, and GVB*SD CI calculations on FPS 522 and CRAY YMP computers used the GVB 25 and MOLECULE SWEDEN Suites of programs. 26 3.1.2 Potential Energy Surface Using the HF, GVB-CIMP2, MP2 and GVB*SD CIMP2 energies versus the reaction coordinates we calculate the potential energy surfaces shown respectively in Figures 5a-d, with the reaction paths shown by the dashed curves. The reaction energy versus distance along the reaction paths are shown in Figures 6a-d. The barriers and transition state geometries are shown in Table 1. The calculated reaction barrier generally falls with increasing levels of correlation. Hartree-Fock (HF) leads to a barrier about a factor of two too high and to a poor location of the transition state geometry. The Eact = 9.3 kcal/mol from the GVB*SD CI is expected to be about 1 kcal/mol too high from residual errors due to incompleteness of the basis set and the CI expansion. Thus for the SR-SOR process, we obtain a corrected GVB*SD CI activation barrier of E~~f ct ~ 8.3kcal / mol. (3) We calculated the zero point energy contribution to the vibrational adiabatic barrier by computing the MP2 vibrational frequencies at the transition and initial states. This adds 0.13 kcal/mol to the activation barrier, leading to E~/Jster ~ 8.4kcal / mol. (4) Table 2 lists the contributions to the energy barrier for the SR-SOR step. 3.2 Discussion and Results of Ab Initio Calculations Table 1 lists the reaction barriers and geometries for the transition state from various levels of calculation on the C3 H7 cluster. The calculated reaction barrier 40 falls with increasing levels of correlation, except for the MP2 case. Thus, Ef!cf = 18.1 kcal/mol, E~r 2 =10.0 kcal/mol, E';!cYB-CI = 10.5 kcal/mol and EfXB*SDCI = 9.3 kcal/mol. As expected, HF leads to a barrier much too high (by almost 10 kcals/mol) and to a poor location of the transition state geometry. This error is exponentiated in calculating the reaction rate and can lead to errors in the rate constant on the order of 100 at CVD diamond growth temperatures. We find that the MP2 level of correlation leads to a good value for the barrier (higher than the GVB*SD CI by only 0.5 kcal/mol) and transition state geometry, but favors a shorter n-bond and a larger C 1 - C 2 distance in the initial state than the GVB-CI and GVB*SD CI levels by about 0.02A. and 0.0511, respectively. Since for the GVB-CIMp 2 and GVB*SD CIMP 2 PES we use the positions of the three movable hydrogens calculated from the constrained MP2 geometry optimizations, the bias of MP2 towards a shorter 7r-bond in the initial state may leave moderate amounts of strain in these three hydrogen's positions. To determine the size of this effect and to further refine the transition state barriers, several single point energies were calculated for each of the three H degrees of freedom and fit to a quadratic at the transition and initial states. For each degree of freedom, we find that the MP2 geometry accurately describes the H angles for GVB-CI and GVB*SD CI at the the transition and initial states, despite its inaccuracy in predicting the position of C 2 for the initial state. The refined GVB*SD CI geometries, for example lead to a barrier only 0.1 kcal/mol less that the GVB*SD CIMP 2 barrier. With the exception of HF, the transition state geometries change relatively little despite the changes in the reaction barriers. The transition state geometries predicted by all four methods keep the C 2 - C 3 7r-bond and the plane of the adsorbed CH2 group coplanar up through the transition state. This positions the tail of the olefin for attack by the radical while maintaining the 7r-bond and therefore leads to a rather gently sloping entrance channel. The accuracy of the MP2 barrier is considered fortuitous, and its ability to accurately approximate the GVB*SD CI barrier in general remains 41 undetermined. The effects of cluster size were determined from ab initio calculations on a C 10 H 15 cluster. We consider only two states. (i) The reactant geometry and; (ii) The transition state geometry. The two carbon atoms, C1 and C3 are allowed to optimize their positions, in contrast to C3H7. Because these two carbon atoms change hybridization during the reaction they are expected to relax towards the surface from their positions in C5. However, the amount of relaxation is reduced because although the change in hybridization causes the relaxation, the constraint that each of the subsurface atoms already have 4 bonds reduces the increase in the order of the bonds of C 1 and C3 with the sub-surface carbon atoms, thus reducing the strain imposed on the surrounding lattice. For the initial state of SR-SOR for example, the radical carbon would generally like to relax into the surface, increasing its bond order with the sub-surface carbons. However, orthogonality constraints between the p-orbital of the radical electron and bonds between the two subsurface carbons and the surrounding lattice reduce relaxation into the surface. The MP2 geometry optimization results in moderate relaxation of C1 and C3 towards the surface, and increases in the distances between C 1 and C3 and C1 and C2 from their positions in Cd-C3H7. The effect is larger for geometries where C1 and C3 have sp 2 character, and is reduced as these atoms change hybridization to sp3 during the reaction. Thus, the relaxation towards the surface is largest for the reactant geometry and smaller for the transition state, although because the transition state maintains a large amount of sp 2 character on the surface carbons the effect is only slightly smaller for the transition state. The remaining 7 carbon atoms and 12 hydrogen atoms were held fixed, thus providing some estimate of the constraints of the lattice on the reaction barrier. The optimization results in an MP2 barrier for the reaction on this cluster of 11.2 kcal/mol. The approximately 1 kcal/mol increase in the barrier results in a reduction of the reaction rate by approximately 40 percent at 1200k, since rates depend exponentially on the barriers. The differences are attributed to the additional relaxation of the strains near the reacting part of 42 the cluster where hybridization changes are significant. In addition, portions of the surface which do not directly participate in the reaction, but which have significant changes in their electronic structure during the reactions, including a portion of the strain effects which were included in the molecular mechanics calculations on the smaller cluster are better estimated. The improvement in the model when C -C -C bond angles of the surface are modeled by C - C - C bond angles in the cluster, rather than H -C-H bond angles, as in C 3 H1 allows for the improved description of the hybridization changes and also better models the strains in these bond angles. Our results show that HF provides a poor description of the transition state. This is similar to the results seen with HF on other transition states. Previous work, for example showed that for hydrogen exchange reactions, HF greatly over estimates the experimental reaction barrier and barriers predicted by other more sophisticated techniques. 27 Their results show that HF over estimates the experimental results by approximately 250 percent (this includes some error due to the incompleteness of the basis set). Not only does HF greatly over estimate the barrier, but it also predicts a poor transition geometry. The over estimate of the barrier is reduced by including correlation with MP2, GVB, GVB-CI, and GVB*SD CI. Although the MP2 perturbation wave function is based on the HF wave function, it predicts a barrier and transition state very similar to that of GVB-CI and GVB*SD CI with the largest error being in the description of the initial state. Although its high-accuracy here is most likely fortuitous, the MP2 description of the reaction is much superior to HF and should generally be considered as the minimum level calculation necessary for quantitative prediction of reaction barriers. Even sophisticated, computationally expensive calculations like GVB*SD CI wave functions over estimate the barriers. Musgrave et al. showed the over estimate of GVB*SD CI in the hydrogen exchange reactions were still IO to 20 percent too large (after taking into account the difference in zero point energy). 27 The effect is partially due to the incompleteness of the basis set and partially due to the incompleteness of the configuration expansion. The over estimate due to the truncation of the CI becomes more prominent as the 43 clusters studied become larger due to the size consistency problems of CI methods. We estimate that at the cluster size studied and the level of basis set used we over estimate the barrier by approximately 1 kcal/mol based on comparisons of GVB*SD CI results with experimental results on other systems. 27 Semi-empirical methods (e.g. MNDO) that are parameterized to describe stable molecules may also provide poor descriptions of transition states and reaction barriers. For example, Valone showed that these methods produced large errors in the reaction barrier for H transfer between two methyls. 28 Although semiempirical methods are capable of modeling large systems, it is not recommended that they be used for modelling transition states where the approximations used for the methods become less valid. The most frequent hypothesis used in modeling CVD diamond growth kinetics has been that the chemistry of diamond is similar to that of analogous hydrocarbon molecules. For example, gas-surface rate constants for diamond growth have been taken more or less directly from gas phase data. 11 •12 This hypothesis has been verified experimentally for the H abstraction reaction from a diamond surface. In this work we examined a reaction between a radical and a 1r-bond. In the gas phase this reaction has hardly any barrier. However, in our case the reaction is subject to severe dynamical constraints introduced by putting it on the surface of the very rigid diamond lattice. Our results show that there is a significant difference in activation barriers between the reaction on the diamond surface and the analogous reaction in the gas phase. Thus, the analogy between diamond and the alkane hydrocarbons may in general be a better approximation for gas-surface reactions than for some surface-surface reactions, like SR-SOR. On the other hand, SR-SOR-like sites are present on the (111) surface, the reconstructed (110) surface and at steps. We can thus extend the results of the (100)-(2xl):H SR-SOR to approximate the nature of the transition states of other SR-SOR reactions. We conclude that the analogy is potentially very useful, but it must be used with considerable care. 44 3.3 Molecular Mechanics Calculations Strain imposed on the lattice by the reacting surface species as calculated with molecular mechanics using the MSXX many-body force field fit to the diamond phonon dispersion curves and elastic constants. 29 To model the reaction we increased the cubic unit cell by a factor of 4 in the z direction and by factors of 5 in the x and y directions (leading to 800 atoms in the unit cell). We then cleaved the (001) surface, 29 leading to a slab which was then hydrogenated. The cluster atoms optimized in the quantum chemistry calculations were held fixed at the initial, final, and transition state geometries, while the remaining atoms of the slab were optimized. 29 Strains and steric interactions included in the ab initio cluster calculations were not included in the molecular mechanics energy to avoid double counting of this portion of the energy. The lattice strain energy (not including van der Waals interactions) and the van der Waals energy were calculated for the transition state and the reactant state. These energies do not include the interactions already included in the ab initio calculations on the cluster. The steric interaction with the surrounding lattice decreases in the transition state relative to the initial state. The strain imposed on the lattice at the transition state relative to the reactant state is calculated to add 0. 73 kcal/mol to the barrier while the van der Waals interactions subtract 0.34 kcal/mol from the barrier. Thus the van der Waals and strain effects are small. The total of 0.39 kcal/mol adds to the net electronic structure barrier to yield a net E!~;Jace = 8.8 kcal/mol. (5) Cell structure and atomic coordinates were updated at each optimization cycle. The atomic coordinates were optimized30 using conjugate gradient techniques until the RMS force per degree of freedom was less than 0.01 (kcal/mol) / A. 3.4 Transition State Theory We determined the entropy change LlS+ between reactants and the transition state from vibrational mode analyses that included the effects of the constraints on 45 the vibrational levels during the reaction. First, the Hessian (second derivative matrix) was calculated using MP2 theory at each geometry of the C 3 H 4 cluster. 24 Next, the 3N - 6 vibrational levels were calculated by diagonalizing the (mass-weighted) ° Hessian. 3 Finally, these energy levels were used to construct a partition function ° from which the entropy was calculated. 3 For the transition state the imaginary vibrational frequency was ignored. The TST pre-exponential factor is given by A= ( ekiT) eb.st /R, (6) where lne = 1, kB is the Boltzmann constant, Tis the temperature, his the Planck constant, R is the gas constant, and ~st is the change in entropy between Cd and the transition state. We estimated the entropies for the initial, transition, and final states of reaction (6), leading to ~st = -2.6 cal/mol-K and A= 5.6 X 10 12 (7) from equation (9). Combining (4) with (8) leads to a total rate constant for SR-SOR of ksR-SOR = 5.6 X 1012 e-SSOO/ RT sec- 1 • (8) The unimolecular reaction (6) on the diamond surface competes primarily with abstraction and addition reactions involving gas phase H atoms. To examine the relative importance of these competing gas phase reactions, we compare reaction (6) with reaction (12) (9) in which the radical site recombines with a gas phase H atom. At 1200K the characteristic time scale for (6) is l/ksR-SOR = 7x 10- 12 sec. In contrast, assuming k 12 is in the range 31 10 13 to 10 14 cm3 /mole-sec with a H atom concentration31 of 106 to 10 10 atoms/cm3 , leads to characteristic time scales of 10- 3 to 10- 8 sec. 46 Thus the SR-SOR reaction is 104 to 109 times faster than other steps in the growth process, indicating that it does not affect the rate of diamond growth in the Garrison mechanism. 4.0 Conclusion Different levels of correlation predict reaction barriers and potential energy surfaces with some variation. The largest errors were those produced by the HF method which was expected, considering that it describes the bond dissociation process poorly. We conclude that HF is a poor method for predicting quantitative aspects of transition states. Although HF greatly over estimates the barriers, it does so consistently. Semi-empirical methods, on the other hand give highly inconsistent results 7 and are not recommended for the prediction of transition states or other highly strained systems. MP2, was not expected to estimate the results of higher level methods so well. We attribute much of this improvement over HF to fortuitous cancellation of errors, however even in the general case, MP2 is a major improvement over HF. Furthermore, MP2 did not reproduce the initial state geometries of GVBCI and GVB*SD CI. This inaccuracy in the geometry leads us to believe that the result for the activation barrier at the MP2 level was fortuitous. Although the accuracy of MP2 is fortuitous, we recommend it as the minimum level of correlation needed to calculate transition barriers with reasonable accuracy. The GVB-CI result shows that as expected, the inclusion of all correlation within the active space greatly improves the description of the transition state over HF and provides a greatly improved activation energy. Although the activation energy is higher than that predicted by MP2, we contend that in general, GVB-CI provides a more robust method for predicting transition states than MP2. GVB-CI, however, requires moderately CPU intensive calculations that in all but the smallest systems would be impractical for computing the PES. GVB*SD CI is an even more computationally intensive method that is well established at predicting the results of complete CI calculations. It describes additional dispersion and gives better approximations to 47 the potential energy surface. For quantitative accuracy GVB*SD CI is the method of choice. This becomes even more true when reaction rates are to be calculated from activation barriers. We make our best estimate of the rate for the SR-SOR step by combining the barrier from GVB*SD CI ab initio calculations with strain energies from molecular mechanics and entropic contributions from transition state theory. The rate equation we predict of ksR-SOR = 5.6 x 10 12 e- 8800 1RT sec- 1 predicts that the SR-SOR recombination step does not limit diamond growth under the range of conditions for which diamond is grown, rather the activation of surface sites by H abstraction will be the limiting step. The methods of combining quantum chemical calculations with classical molecular mechanics calculations is useful in that it concentrates the power of accurate ab initio methods on the structure undergoing reaction while adding in strain effects of the surrounding surface using inexpensive molecular mechanics. Methods that treat a large enough portion of the surface to include strain effects are necessarily approximate to avoid large computational costs, often leading to misleading results, especially for reaction barriers. 48 6.0 References *To whom correspondence should be addressed. 1. J. C. Angus and C. C Hayman, Science 241, 913 (1988). 2. K. E. Spear, J. Am. Ceram. Soc. 72, 171 (1989). 3. W. A. Yarbrough and R. Messier, Science 247, 688 (1990). 4. C. J. Chu, M. P. D'Evelyn, R.H. Hauge, and J. L. Margrave, J. Appl. Phys. 70, 1695 (1991). 5. C. E. Johnson, W. A. Weimer, and F. M. Cerio, J. Mat. Res. 7, 1427 (1992). 6. S. J. Harris and A. M. Weiner, Thin Solid Films 212, 201 (1992). 7. S. J. Harris, A. M. Weiner, T. A. Perry, J. Appl. Phys. 70, 1385 (1991). 8. W. A. Yarbrough, K. Tankala, T. J. Debray, Mat. Res. 7, 379 (1992). 9. S. J. Harris, Appl. Phys. Lett. 56, 2298 (1990). 10. S. J. Harris and D. N. Belton, Thin Solid Films 212, 193 (1992). 11. D. N. Belton and S. J. Harris, J. Chem. Phys. 96, 2371 (1992). 12. M. Frenklach and H. Wang, Phys. Rev. B. 43, 1520 (1991). 13. M. E. Coltrin and D. S. Dandy, J. Appl. Phys. 74, 5803 (1993). 14. W. A. Yarbrough, in Diamond Optics IV, A. Feldman and S. Holly Editors (SPIE, Bellingham, Washington, 1991), p. 1534. 15. S. J. Harris and D. G. Goodwin, J. Phys. Chem. 97, 23 (1993). 16. B. J. Garrison, E. J. Dawnkaski, D. Srivastava, and D. W. Brenner, Science 255, 835 (1992). 17. D. W. Brenner, Phys. Rev. B. 42, 9458, (1990). 18. F. W. Bobrowicz and W. A. Goddard, III, Methods of Electronic Structure Theory ed. HF Schaefer (New York: Plenum) p. 79. 19. W. J. Hunt, P. J. Hay and W. A. Goddard, III, J. Chem. Phys. 57, 738 (1972). 20. C. W. Bauschlicher and S. R. Langhoff, Chem. Phys. Lett. 135, 67 (1987). 21. C. M0ller and M. S. Plesset, Physical Review, 46, 618 (1934). 22. T. H. Dunning, J. Chem. Phys. 53, 2823 (1970). 49 23. S. J. Huzinaga, Chem. Phys. 42, 1293, (1965). 24. M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J. B. Foresman, B. G. Johnson, H. B. Schlegel, M. A. Robb, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. K. Fox, D. J. Defrees, J. Baker, J. J. P. Stewart, and J. A. Pople, Gaussian 92 Revision, Gaussian, Inc. Pittsburgh, PA, (1992). 25. W. A. Goddard, III, unpublished. 26. J. Almlof, C. W. Bauschlicher, M. R. A. Blomberg, D. P. Chong, A. Heiberg, S. R. Langhoff, P-A Malmqvist, A. P. Rendell, B. 0. Roos, P. E. M. Siegbahn and P. R. Taylor MOLECULE SWEEDEN (PROGRAM), unpublished. 27. C. B. Musgrave, J. K. Perry, R. C. Merkle and W. A. Goddard, III, Nan- otechnology 2, 187 (1991). 28. S. M. Valone, in Proc. 1990 NATO Adv. Study Inst. on Diamond and Diamond-Like Films and Coatings, (1990). 29. C. B. Musgrave, J. Hu, W. A. Goddard, III, to be submitted. 30. These calculations used POLYGRAF from Molecular Simulations Inc. (Burlington Mass). 31. D. G. Goodwin, J. Appl. Phys. 74, 6895 (1993). 50 Table 1. Transition state geometries and barriers at various levels of correlation. Level Barrier (kcal/mol) HF 18.1 -0.58 -0.075 MP2 10.0 -0.54 -0.042 CASSCF 10.5 -0.50 -0.023 GVB*SD CI 9.3 -0.51 -0.025 51 Table 2. Contributions to the activation energy for the SR-SOR step. Calculation Energy (kcal/mol) a. Quantum Chemical Calculations GVB*SD CI 9.30 Corrections (Basis set and Correlation) -1.0 Differential Zero Point Energy 0.13 Net Electronic Structure Barrier 8.43 b. Force Field Calculations Strain from Balance of Surface 0.73 van der Waals Interactions -0.34 c. Total Eact 8.82 52 Figure Captions Figure 1. (a) Dimer paired structure of the H stabilized C(lO0) surface; (b) Enlargement of (a). Unshaded = H, shaded = Bulk C, hatched = bulk terminating C. Figure 2. The Brenner-Garrison mechanism for dimer ring opening during CVD growth of C(lO0). Figure 3. The C3H7 cluster used to generate the PES surfaces. Figure 4. The points R12 and R23 used to generate the PES surfaces of the SR-SOR step on the C3H1 cluster. Figure 5a. The energy surface for the SR-SOR reaction from CASSCF calcula- tions. The reactant site is denoted as 0.0. The saddle point for the reaction is denoted as 10.5. The product is far to left and top of the figure. The contour spacing is 1.15 kcal/mol. Figure 5b. The energy surface for the SR-SOR reaction from MP2 calculations. The reactant site is denoted as 0.0. The saddle point for the reaction is denoted as 10.0. The product is far to the left and top of the figure. The contour spacing is 1.15 kcal/mol. Figure 5c. The energy surface for the SR-SOR reaction from GVB-SD CI calcu- lations. The reactant site is denoted as 0.0. The saddle point for the reaction is denoted as 9.3. The product is far to the left and top of the figure. The contour spacing is 1.15 kcal/mol. Figures 6a-c The energy along the reaction paths for the CASSCF, MP2 and GVB*SD CI reaction surfaces, respectively. 53 Figure 1 54 Figure 1 55 Figure 2 (b) Cs* site (a) Cs site ~-scission SR-SOR (e) Cs/ site (d) CsM* site 56 Figure 3 57 Figure 4 Pairs of R1 and R2 used for ab initio calculations. y 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.8 • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • -0.4 0 • -0.6 • -0.2 X / lnltlal state • • • • 0.2 0.4 y 38.0 @ Figure 5 25.5~ X ---------------+0.0 <:.11 00 59 60 +<=? ,o I I I 'I'' I I I ' I I I I I >< 61 Figure 6 Energy Along Reaction Path 10 -.€ -@ 0 8 tU V ~ 6 cu = 4 ~ 2 0 -i"'=r"""'T""'T"""T""T""T""'T"""lr"""'l""""l"""'T"'"'T"""'l'__,,.._......,.._ _ _....1 10 15 20 0 5 25 Distance 62 Figure 6 . Energy along Reaction Path 10 to 6 CIJ ~ 4 2 o~-==::;::;..._______. . . . . . . . . . . . . . . . . ~ 0 2 4 6 8 10 12 14 16 18 20 Distance 63 Figure 6 Energy Along Reaction Path 10 9 -.€ - 8 0 ns 7 V ~ 6 :>-. 5 ~ cu = 4 i;.::i 3 2 1 0 0 5 10 15 Distance 20 25 64 Chapter 3 Ab initio Study of H Abstraction in Nanotechnology 65 Abstract Processes which use mechanical positioning of reactive species to control chemical reactions by either providing activation energy or selecting between alternative reaction pathways will allow us to construct a wide range of complex molecular structures. An example of such a process is the abstraction of hydrogen from diamond surfaces by a radical species attached to a mechanical positioning device for synthesis of atomically precise diamond-like structures. In the design of a nanoscale, site specific hydrogen abstraction tool, we suggest the use of an alkynyl radical tip. Using ab initio quantum chemistry techniques including electron correlation we model the abstraction of hydrogen from dihydrogen, methane, acetylene, benzene and isobutane by the acetylene radical. Of these systems, isobutane serves as a good model of the diamond (111) surface. By conservative estimates, the abstraction barrier is small (less than 7. 7kcal/mol) in all cases except for acetylene and zero in the case of isobutane. Thermal vibrations at room temperature should be sufficient to supply the small activation energy. Several methods of creating the radical in a controlled vacuum setting should be feasible. Thermal, mechanical, optical and chemical energy sources could all be used either to activate a pre-cursor, which could be used once and thrown away, or alternatively to remove the hydrogen from the tip, thus refreshing the abstraction tool for a second use. We show how nanofabrication processes can be accurately, and inexpensively designed in a computational framework. 66 1.0 Introduction Mechanical positioning of reactive species can be used to convert mechanical energy to chemical energy to select between alternative reactions, or to provide activation energy. Mechanosynthesis is the employment of these mechanochemical processes to synthesize molecular structures. 1 Atomically precise mechanosynthesis promises to let us manufacture complex systems of molecular machinery. Examples include: self-replicating assemblers,2 molecular scale surgical systems,2 computers made with molecular logic elements, 3 and macroscopic machines made of diamondlike materials. 1 Construction of such systems will require the ability to precisely manipulate structure on an atomic level. The great specificity of the chemical reactions required to synthesize designs with specific atomic structures should be achievable with mechanochemical tools capable of positioning the reactive moieties with sub-angstrom accuracy. Mechanochemistry allows alternative reaction transition states to be selected by maneuvering the reactive species in to a position where the chosen reaction has the smallest barriers. Such positional control requires that the tool exert forces and torques on the reactive molecule to move it over the potential energy surface of interaction with the workpiece. Feynman in a talk entitled "There is Plenty of Room at the Bottom," 1960 is credited with first pointing out that the laws of physics said nothing about the impossibility of using mechanical means to direct chemical reactions to synthesize molecules and materials. Applying positional control to reactions will require that the tool have certain properties to make synthesis reliable, feasible, and practical. The tool must (a) have the proper chemical properties (b) be relatively small to reduce steric interactions with the workpiece (c) be capable of remaining chemically and mechanically stable under thermal motions and strains induced during positioning (d) be bound to a system which can transfer forces and torques to the 67 reactive portion of the tool (e) be selective between alternative reactions and (f) be easily made. Molecular tips attached to atomic force microscope (AFM) tips, scanning tunneling microscope (STM) tips, or molecular robotic arms have been suggested. 4 Because construction of atomically precise machinery might require about as many unit operations as there are atoms in the system, it is important that reactions be fast. To increase the speed of reactions with moderate barriers, forces can be exerted between the work piece and reactive species to effectively increase the pressure on the system, reducing the barrier height. Moderate reductions in the barrier heights lead to substantial increases in the reaction rate because thermal vibrations have an exponential Boltzman probability of overcoming the reaction barrier. Mechanochemistry not only reduces the barriers by converting mechanical energy to chemical energy, but also maximizes the effective reactive concentration by positioning the reactive moieties to best advantage. These speed enhancing steps together with multiple mechanochemical machines working simultaneously can compensate for the loss of parallelism when compared against solution based reactions. Mechanochemical synthesis also increases the range of synthetic steps that can be used to build novel structures by the use of applied torques; a moiety attached to both the tool and the workpiece can be twisted, for example, to break 1r-bonds. 1 Nanomachines made of complex specific arrangements of diamond-like material offer several advantages. First, diamond is light, and stiff. Macroscopic machines could be made stronger and simultaneously much lighter, making such activities as air and space travel substantially more practical. Moving parts of such machines would be lighter and therefore, faster. Furthermore, hydrocarbons are abundant, making raw materials readily available and inexpensive. Stiffness is not only a desirable property of finished machines it is also useful during construction since the material surrounding the reactive site on the workpiece must be stiff. This 68 allows it to withstand the compressive forces that might be needed to reduce reaction barriers, to withstand the tensile forces during moiety abstraction, and to withstand torques applied to break 1r-bonds. Building machines of diamond will include maneuvering hydrocarbons into reactive sites, torsion of structures, insertions into bonds, and preparation of reactive sites by removing unwanted moieties to create radical sites. Abstraction of hydrogen is likely to be the most repeated step and common to building a wide range of molecular structures, including diamond-like structures. Highly reactive species are commonly thought to play a crucial role in the chemical vapor deposition (CVD) synthesis of diamond. 5 - 7 The abstraction of hydrogen via any of several radicals is one of the central mechanisms involved in the growth of diamond. It is not unreasonable, therefore, to expect that the atomically precise synthesis of diamond-like materials will utilize site-specific hydrogen abstraction via a radical as one of the main steps. Drexler has proposed using a molecular tip made of an ethynyl radical 1 bound to a mechanical base on the tool (Figure 1). While many radicals exist, the desire for a simple, general, positionally accurate and sterically undemanding hydrogen abstraction tool can be used to narrow the search to a structure which (a) has a very high affinity for hydrogen (b) is not encumbered by surrounding groups (c) can be made part of an extended structure which can be used as a "handle" for positioning and can be attached to an STM or AFM tip (d) is mechanically and chemically stable during positioning (e) is selective between alternative reactions such as abstraction of a neighboring hydrogen or bonding to a nearby carbon atom and (f) is easily made or regenerated. Perhaps the most natural structure in this regard is the ethynyl radical. The C-H bond in acetylene is one of the strongest bonds to hydrogen; thus, the ethynyl radical formed by removing this hydrogen is likely to have a higher affinity for hydrogen 69 than almost any other chemical structure. Further, the ethynyl radical can easily be incorporated into structures which provide a high degree of steric exposure. A structure resembling the propynyl radical, but with the carbon furthest from the radical site embedded in an extended diamond-like structure (Figure I) provides both excellent steric exposure to the radical and a "handle" for positioning the radical for the desired abstraction. Attachment of the tool to an STM or AFM tip may develop from technology designed for attachment of proteins to surfaces. 4 Drexler showed that the bending stiffness of an ethynyl- like tip attached to an adamantyl group is ~ 6 N/m and can be increased to ~ 65 N/m by building up a surround- ing collar. 1 If the reaction requires application of mechanical force to supplement thermal energy, then bending stiffness may need to be increased. Stiffness also is desirable to achieve selectivity. STM and AFM positing is stable to more than sub-angstrom accuracy. However, bending modes of the ethynyl tip will be active at moderate temperatures. If during positioning of the tip, the bending of the radical and displacement of the AFM or STM relative to the workpiece positions the reactive portion of the tip near a branched transition state, for example (one pathway leading to abstracting the neighboring hydrogen), then selectivity is reduced. Drexler has shown that at worst at room temperature with a bending stiffness of 20 N/m and transition states separated by 1.2A the unwanted reaction rate is less than 10-12 times the rate of the target reaction. 1 Transition states between neighboring hydrogens on the (111) surface of diamond are separated by 2.5A, and transition states for other possible reactions in diamond-like structures also generally exceed 1.2A, making mechanochemical reactions highly selective. The strong C-H bond of alkynes (127-132 kcal/mol) 8 - 12 should give large exothermicities and small barriers (rapid reactions) for alkynyl radical abstraction of hydrogen from weaker sp 2 and sp 3 hybridized C-H bonds (see Table 1). The large exothermicity would also give 70 a small reverse reaction rate constant. Compressive mechanical forces could be applied to supplement thermal energy in cases where the barriers are large; however, care must be taken so that the alkynyl radical tip does not bend away from the transition state. In the cases we study, the barriers are such that this is not an issue. Several methods of creating the radical should be feasible. The process of creating the radical should take place in an inert environment: vacuum, helium, or some other extremely non-reactive system would be appropriate. The activation energy required to create the abstraction tool could be provided from thermal, mechanical, optical, or chemical sources. There are two obvious approaches. In the first, a pre-cursor compound is activated to create the abstraction tool. The tool is then used once and discarded. A second pre-cursor would then be activated to abstract a second hydrogen. Thus, in a functioning system using this approach, a steady supply of the pre-cursor would be required as well as a method for disposing of the used abstraction tools. In the second approach, the abstraction tool would be refreshed by the removal of the hydrogen after each use. Of course, the ethynyl radical was selected on the basis of its strong C-H bond, so removal of the hydrogen might at first seem paradoxical. However, there are several methods of solving this problem. One would be to first weaken the C-H bond, and then abstract the hydrogen from the abstraction tool using a weaker radical. Drexler 1 proposed that the C-H bond could be weakened by positioning a weak radical near the carbon atom. A second weak radical could then abstract the hydrogen from the tip. An alternative to the attack by two weak radicals strategy would be photo exciting the acetylene to obtain the 7r7r* state which would rearrange to the structure with a weak C-H bond and thereby allow removal of the hydrogen. Our primary concern is to analyze the energy barriers associated with hydrogen abstraction using alkynyl radicals to determine the feasibility of such a tool, rather than to analyze 71 the methods of creating such a tool. We model the chemically active site of the tool by the acetylene radical and determine the transition state geometry and activation energy for transferring the hydrogen from several species: H 2, CH4 , C 2H 2, C 6 H5, and CH(CH3)3. The geometry of the various transition states can be used to position a working hydrogen abstraction tool for fast reaction and without bending the tip. The barrier height itself can be used to calculate an abstraction rate at a given temperature and thus, how long the abstraction tool must remain at the transition state until the probability that abstraction has occurred reaches a given value. Various levels of generalized valence bond (GVB) and configuration interaction ( CI) ab initio calculations are used. To calibrate the accuracy of these calculations, we consider the abstraction barriers and transition states for hydrogen transfer between methyl and methane and for hydrogen transfer between Hand H2 as compared to other theoretical and experimental results. 13 - 15 2.0 Results The barrier to the acetylene radical abstraction of hydrogen from isobutane (sp 3 carbon) is conservatively estimated to be less than 0.45 kcals/mol (Figure II). Reaction barriers calculated at various levels of correlation are shown in Table 2. The barrier to abstraction of hydrogen by the acetylene radical from benzene (sp 2 carbon) is estimated to be less than 7. 7 kcals/mol. The Hartree-Fock times Singles and Doubles Configuration Interaction (HF*SD CI) consistently overestimates the Generalized Valence Bond times Singles and Doubles Configuration Interaction (GVB*SD CI) barriers while the Dissociation Consistent Configuration Interaction (DCCI) consistently underestimates the GVB*SD CI barriers. The GVB*SD CI barriers will be conservatively high due to a lack of a third diffuse p function, zero point corrections, and lack of more correlation of the valence electrons. Transition states were optimized at the Correlation Consistent Configuration Interaction 72 (CCCI) level (Table 3). Although CCCI does not accurately predict activation barriers, it does accurately describe the transition state geometries. For the largest cases, the number of spin eigenfunctions in the Configuration Interaction (CI) calculation grows beyond our computational capabilities (Table 4). This makes calculation of abstraction from benzene and isobutane at the GVB*SD CI level impractical; however, the overestimated, yet small barriers at the HF*SD CI level shows that the acetylene radical hydrogen abstraction is feasible; thermal vibrations at room temperature providing sufficient energy to overcome the barriers. Table 5 shows the exothermicities for the various abstractions. There is little difference in the accuracy of the methods in predicting the exothermicities because all the methods describe bound states rather well. Note that the exothermicities are for reactions where the product radical species are not allowed to relax. This describes abstraction from surfaces where relaxation is constrained. Exothermicities for gas phase reactions will be higher. The transition state is poorly described by many semi-empirical methods and by ab initio methods with insufficient electron correlation and small basis sets and large variation of the predicted barriers can be seen in Table 2. 3.0 Calculational Details Standard ab initio quantum chemistry methods are employed and results are given for several levels of calculation. The simplest wave function used is the wave function in which each molecular orbital is doubly occupied. This single configuration (one determinant) Hartree-Fock (HF) wave function is the lowest energy antisymmetrized n-fold product of molecular orbitals and should give a qualitative picture of the hydrogen abstraction reactions studied. HF will tend to overestimate the abstraction barrier since the radical-hydrogen stretching frequency is too high due to the poor description by HF of the bond breaking process (a doubly occupied orbital of the molecule must become two singly occupied orbitals for the 73 fragments). This problem is remedied by using a GVB (Generalized Valence Bond) wave function, 16 which allows each bond to be described with two singly occupied, overlapping orbitals leading to a proper description of dissociation. When solved for self-consistently, this calculation is termed a Generalized Valence Bond Configuration Interaction Self Consistent Field (GVBCI-SCF) or equivalently a Complete Active Space Self Consistent Field (CASSCF). Simply, all symmetry and spin allowed configurations of three active electrons in three orbitals are generated. These electrons are the radical of the reactant, the hydrogen and the radical of the product. All other orbitals (considered inactive) are doubly occupied as in Hartree-Fock. It is found that three configurations are all that is necessary to adequately describe the transition state. These are the dominant configuration (with the Hartree-Fock occupations of the orbitals) and the single and double excitations of the electrons in the doubly occupied R 1 -H-R 2 bonding orbital to the empty R1-H-R2 antibonding orbital. The R 1 -R 2 antibonding orbital (with a node at the hydrogen center) is singly occupied in all three configurations. While this level of calculation includes the most important correlation, it will still tend to overestimate the abstraction barrier and, thus, will only serve as a zeroth order wave function for large CI ( Configuration Interaction) expansions which will account for additional dispersion. Ideally, we would like to do a CI calculation in which all single and double excitations of the valence electrons are made into the virtual orbitals with reference to the three most important configurations describing the abstraction. This type of multi-reference CI (called a GVB*SD CI) has been well established at approximating proximating results of complete CI calculations. 17 However, this CI has not been carried out for the largest cases, abstraction of hydrogen from isobutane and benzene by the acetylene radical. Thus we have considered some smaller CI calculations which will do a good job in approximating the barriers for the larger CI. 74 The first of these is the CCCI wave function. 18 , 19 It involves making all single and double excitations of the active electrons and all single excitations of the other valence electrons into the virtual space relative to the three GVB references, or simply GVB*(SDactive + Svalence)- The second CI, called a DCCI (Dissociation Consistent Configuration Interaction), will add the double excitations which are the product of a single excitation of an active electron and a single excitation of a valence electron, or GVB*(SDactive *Svalence + Svalence). The third CI does all single and double excitations of the valence electrons (active and inactive) relative to only one reference, a calculation called HF*SD CI (or equivalently 1 reference SDCI). Table 4 shows the sizes of the CI expansions in terms of the number of spin eigenfunctions (SEFUs) for each of the systems studied. The HF*SD CI already approaches the limits of our programs 2 million SEF's) in the cases of isobutane and benzene, for which the GVB*SD CI is not possible. In all other cases, however, the GVB*SD Cl's are of small to medium size and will serve as benchmarks to calibrate the accuracy of the smaller Cl's. The standard basis sets of Dunning/Huzinaga are used. 20 , 21 Their double zeta contraction of the 9s5p set is used on all carbons, with the addition of one set of d polarization functions ((d=0. 75). On the active carbons, diffuse s and p functions ((8 =0.0474 and (p=0.0365) are also added. For active hydrogens or hydrogens bound to active carbons (in the case of methane), the triple zeta contraction of the 6s set is used, supplemented with a p polarization function ((p=0.60). For all other hydrogens, the double zeta contraction of the 4s set is used, scaled by a factor of 1.2. The basic geometries of the various systems studied are illustrated schematically in Figure III. The geometries will be optimized at the CCCI level. The orbital optimization at the GVBCI-SCF level is the most time consuming step, so this CI will be a simple correction to that wave function. It would be impractical to do a full geometry optimization, so certain constraints are assumed. Namely, 75 only the relevant parameters to the description of the hydrogen abstraction (RiH and R2H or combinations thereof) will be optimized. In the case of abstraction on from methane (by the methyl radical or by the acetylene radical), the H-C-Habs bond angle is also optimized since this angle changes from 109.5° for methane to 90° for the methyl radical. For isobutane, we would expect a small relaxation from a tetrahedral C-C-Habs bond angle to something more planar at the transition state and in the radical species. But Page and Brenner, 22 in their work on abstraction of hydrogen from isobutane by atomic hydrogen, found that full relaxation of the t-butyl species reduced the abstraction barrier by only 1. 7 kcal/mol at the GVBCISCF level. However, since the goal of this work is to show the feasibility of using an alkynyl radical tip as a hydrogen abstraction tool, a conservative overestimate of the abstraction barriers is acceptable. So the C-C-Habs bond angle is fixed to 109.5° in these calculations. All other radicals are expected to show little or no relaxation and are fixed to the experimental values of their hydrogen bound counterparts. All calculations are run with the GVB 23 and MOLECULE/SWEDEN 24 suites of programs on the Caltech group's Alliant FX/80 and FPS 500. 4.0 Discussion 4.1 H-H-H: A great deal of theoretical work has been done on this system, 15 and, due to its simplicity, it provides a good test of our hydrogen basis set and, to a lesser extent, our methods. While the calculated equilibrium bond distance in H 2 compares favorably to experiment (0.74A vs. 0.74144A 25 ), the calculated dissociation energy for H2 (De) is 105.4 kcal/mol as compared to the experimental number of 108.6 kcal/mol. 25 This discrepancy of 3 kcal/mol is chiefly due to the lack of a second p polarization function. In contrast, the transition state is well described by the GVB*SD CI (in this case only, the CCCI, DCCI and GVB*SD CI are equiva- 76 lent since there are no valence electrons in addition to the three active electrons). The optimized geometry has a H-H distance of 0.94A ( compared to Liu's value of 0.930A 15 ) and a barrier of 10.3 kcal/mol (compared to Bauschlicher's value of 9.56 kcal/mol using a large ANO basis set 15 ). The geometries for all the systems studied are listed in Table 3 and the abstraction barriers are listed in Table 2. An error of less than 1 kcal in the barrier is adequate for the calculations at hand, particularly since the barrier is overestimated. It should be noted, however, that the HartreeFock barrier is well off the mark at 24.3 kcal/mol and that the apparently good result at the GVBCI-SCF level (9.9 kcal/mol) is primarily due to the weak H-H bond strength (87.5 kcal/mol) at this level. The HF*SD CI number including the Davidson correction is in fortuitous agreement with the reference barrier height of 9.56 kcal/mol. This is rather symptomatic of the Davidson correction, which can often overestimate the contributions from additional correlation. The methyl-methane hydrogen transfer reaction is perhaps more representative as a test case of the systems in which we are most interested. There has been less theoretical work done on this system 14 but a reliable experimental number for the abstraction barrier of 14.2 kcal/mol 13 gives a good benchmark for us to work with. Theoretical investigations into this reaction have not been successful in obtaining quantitative accuracy in the barrier height. The best calculations overestimate the barrier by 5-6 kcal/mol. Part of this is due to some assumptions made in the calculations, namely the neglect of zero-point corrections, the BornOppenheimer approximation and temperature effects. However, the sum of these effects should only lower the theoretical activation energy by 1-2 kcal/mol (see Sana, et.al. 14 ). Our results agree well with previous theoretical work. The optimized transition state geometry is virtually identical to the full gradient optimized structure of 77 Wunsch, et al., 14 which was done at the Hartree-Fock level. The calculated barrier is 20.4 kcal/mol, higher than experiment by 6.2 kcal/mol. Test calculations with a large ANO basis set 26 only lowered the barrier to 19.2 kcal/mol, indicating that the discrepancy between theory and experiment is likely a correlation problem rather than a basis set problem. The importance of ionic terms such as are probably underestimated in the CI calculations due to biases against anionic states and would require additional correlation of the non-active valence electrons. This could be a formidable task even for such a small system and would not be possible with our current code for the larger cases we wish to study. However, again, since these factors all tend to lead to an overestimate of the barrier height the results for abstraction of hydrogen by the acetylene radical can be considered a conservative upper limit to the actual barrier height. These calculations on the methyl-methane system also offer a comparison of the smaller Cl's to the GVB*SD standard. We find the CCCI result (29.8 kcal/mol) to be comparable to the GVBCI-SCF number (27.8 kcal/mol), being slightly higher due to the stronger C-H bond at the CCCI level (113.2 kcal/mol vs. 97.6 kcal/mol at the GVBCI-SCF level and 107.2 kcal/mol at the GVB*SD CI level). This indicates that correlation of the non-active electrons is important in obtaining quantitative accuracy for the hydrogen abstraction barriers. The DCCI, which includes only limited correlation of the inactive electrons, underestimates the GVB*SD CI barrier by 2.9 kcal/mol. Alternatively, the HF*SD CI, which sacrifices some of the active electron correlation, overestimates this barrier by 2.1 kcal/mol. The combination of these two CI calculations should offer an upper and lower limit to the GVB*SD CI for those large cases where that CI is not feasible. 78 4.2 H-H-CCH, CH 3 -H-CCH, C 6 H 5 -H-CCH and HCC-H-CCH Results of calculations on these systems underscore the results of methylmethane. In particular, Hartree-Fock greatly overestimates the activation energies, GVBCI-SCF and CCCI offer some improvement but still overestimate these barriers, and DCCI and HF*SD CI bracket the results of the GVB*SD CI. In the case of abstraction from methane and benzene, we found the geometries were most conveniently optimized by using the coordinate system R 1H + R2H and R 1H - R 2H. In the case of abstraction from H 2 , it was easier to optimize the transition state in terms of the coordinates R 1H and R 2 H. This was likely due to the fact that the barrier was small at the CCCI level and that it occurred quite early, with only an 8% increase in the H-H bond length. To a large degree the barrier height and the position of the barrier is determined by the exothermicity of the reaction. Other properties, such as the polarizability of the bonds, play a role as well. The largest barrier (and latest transition state) was for abstraction from acetylene, with an activation energy of 14.6 kcal/mol. This is as expected, since this particular reaction is thermoneutral. The calculated exothermicities of the other reactions are listed in Table 5 (see also Table 1, for the experimental bond dissociation energies). H 2 and CH4 are the most exothermic and have the smallest barriers. These barriers maybe considered negligible as the calculations on H + H 2 and methyl-methane showed the numbers to be overestimated. Abstraction from benzene is less exothermic and shows a small but non-negligible barrier. In addition, the phenyl-H bond is stretched 14% at the transition state in comparison to an 11 % stretch of the metyl-H bond. All of these results correlate with the exothermicities of the reactions. 1.3.2 (CH 3 )3C-H-CCH: Finding the transition state for abstraction from isobutane proved to be quite difficult. The potential energy surface has many of the same features as that of H-H- 79 CCH, in particular, a small, early barrier which leads to non-quadratic behavior in the region of the saddle point. Due to the computational costs of these calculations, it was necessary to sacrifice some accuracy in order to find the transition state. In the end, the values of R 1 H=L2A and R2H=l.5A are in agreement with results for abstraction from methane and benzene by the acetylene radical. The large CI calculations strongly indicate that there is no barrier for abstraction of hydrogen, the more conservative number giving a barrier of only 0.45 kcal/mol. This again correlates with the large exothermicity of this reaction, which is calculated from snap bond energies. If one considers that the t-butyl group should relax somewhat at the CCCI transition state and, thus, lower the energy of the barrier still, the argument for the absence of a barrier becomes even more persuasive. Since this system is a good model for the hydrogenated diamond ( 111) surface C-H bond, we conclude that no barrier exists to abstraction of hydrogen from this surface by acetylene. Barriers on other surfaces of diamond are likely nonexistent or negligibly small. 4.4 Abstraction from Acetylene: Now that it has been established that an alkynyl tipped hydrogen abstraction tool would be able to abstract hydrogen from diamond surfaces with little or no thermodynamic hindrances, it would be desirable to find a method for removing the hydrogen from the tip. A simple alternative, but less elegant strategy, is to make a new tip for each abstraction and dispose of the tool after use. What makes the acetylene good at abstracting hydrogen is the strength of its C-H bond. However, this bond is quite weak in the 3Bu excited state (see Table 6). We calculate a bond strength of 41. 7 kcal/mol doing a GVBCI-SCF in which all 10 valence electrons are active in 10 orbitals, followed by a Multi-Reference times Singles and Doubles Configuration Interaction (MR*SD CI) in which all configurations in the GVBCI 80 with coefficients > 0.05 are chosen as references. In the case of the triplet excited state, there are 4 references and, in the case of dissociated H + CCH, there are 6 references. The geometry for the excited state is optimized at the MR *SD CI level. The molecule is not linear in this state, having a C-C-H bond angle of 132.0°. The C-C bond length also increases to 1.38A from 1.20A for the ground state, reflecting the double bond character of this bond. The weakening of the C-C bond in the excited state leads directly to the weakening of the C-H bonds, as the triple bond character can be restored upon dissociation of one of the C-H bonds. The weakening of the C-H bond leaves the acetylene prone to abstraction, making it easy to remove the hydrogen and refresh the tip. So photoexcitation of the alkynyl tip from its ground state to the lBu excited state, followed by relaxation to the triplet would facilitate the breaking of the tip-hydrogen bond. Drexler also made an alternative proposal for removing the hydrogen from the tip by destabilizing the H-tip bond with a second tip and disposing of the H into a H sink. 1 5.0 Conclusion We model the abstraction of hydrogen from H2, CH4, C2H2, C6H6, and CH(CH 3 )3 by the acetylene radical using accurate CI ab initio quantum chemistry techniques. From our results, conservative estimates show that the reaction barriers for abstraction from sp 3 hybridized carbons are negligible, or zero for the case of isobutane. The barriers are small for sp 2 hybridized carbons and slightly larger for sp hybridized carbons. Therefore, abstraction tools based on ethynyl radical molecular tips should reliably and rapidly abstract hydrogen from most carbon structures at moderate temperatures. We also find that the hydrogen bond to the 7r7r* excited ethynyl tip is relatively weak, and therefore can be broken to refresh the tip. We also wish to show how nanofabrication processes can be accurately and inexpensively designed in a computational framework. 81 6.0 References 1. Drexler, K. E., Nanosystems: Molecular Machinery, Manufacturing, and Computation, John Wiley & Sons, scheduled for publication in 1992. 2. Drexler, K. E., Engines of Creation, Anchor Press/Doubleday, (New York), 1986. 3. Drexler, K. E., Molecular Electronic Devices, Elsevier Science Publishers B.V. North-Holland (Amsterdam) , 1988. 4. Drexler, K. E., J. Vac. Sci. Technol., B9 (2), 1394, (1991). 5. Celii, F. G., Pehrsson, P. E., Wang, H. and Butler, J. E., Appl. Phys. Lett., 52, 2043, (1988). 6. Harris, S. J., Weiner, A. M. and Perry, T. A., Appl. Phys. Lett., 53, 1605, (1988); Harris, S. J. and Weiner, A. M., J. Appl. Phys., 67, 6520, (1990). 7. Goodwin, D. G. and Gavillet, G. G., J. Appl. Phys., 68, 6393, (1989). 8. Wodtke, A. M. and Lee, Y. T., J. Phys. Chem., 89, 4744, (1985). 9. Ervin, K. M., Gronert, S., Barlow, S. E., Gilles, M. K., Harrison, A. G., Bierbaum, V. M., Depuy, C. H., Lineberger, W. C. and Ellison, G. B., J. Am. Chem. Soc., 112, 5750, (1990). 10. Segal, J., Wen, Y., Lavi, R., Singer, R. and Wittig, C.J., J. Phys. Chem., 95, 8078, (1991). 11. Green, P. G., Kinsey, J. L. and Field, R. W., J. Chem. Phys., 91, 5160, (1989). 12. Segal, J., Lavi, R., Wen, Y. and Wittig, C.J., J. Phys. Chem., 93, 7287, (1989). 13. Tedder, J. M., Angew. Chem. Int. Ed. Engl., 21, 401, (1982). 14. Wunsch, E., Lluch, J. M., Oliva, A. and Bertran, J., J. Chem. Soc. Perkin Trans. II, 211, (1987); Sana, M., Leroy, G. and Villaveces, J. L., Theor. 82 Chim. Acta., 65, 109, (1984); Rayez-Meaume, M. T., Dannenberg, J. J. and Whitten, J. L., J. Am. Chem. Soc., 100, 747, (1978). 15. Liu, B. J., Chem. Phys., 58, 1925, (1973); Bauschlicher, C. W., Langhoff, S. R. and Partridge, H., Chem. Phys. Lett., 170, 345, (1990) and references therein; Siegbahn, P. and Liu, B. J., Chem. Phys., 68, 2457, (1978). 16. Bobrowicz, F. W. and Goddard, W. A. in Methods of Electronic Structure Theory (Ed. Schaefer, H.F.), 79, 1977. 17. Bauschlicher, C. W. and Langhoff, S. R., Chem. Phys. Lett., 135, 67, (1987); Taylor, P. R. and Partridge, H., J. Phys. Chem., 91, 6148, (1987). For an excellent review see Bauschlicher, C. W., Langhoff, S. R., and Taylor, P.R., Adv. Chem. Phys., 77, 103, (1990). 18. Carter, E. A. and Goddard, W. A., J. Chem. Phys., 88, 1752, (1988). 19. Carter, E. A. and Goddard, W. A., J. Chem. Phys., 88, 3132, (1988). 20. Dunning, T. H., J. Chem. Phys., 53, 2823, (1970). 21. Huzinaga, S. J., Chem. Phys. 42, 1293, (1965). 22. Page, M. and Brenner, D. W., J. Am. Chem. Soc., 113, 3270, (1991). 23. Goddard III, W. A., et al., Caltech, unpublished. 24. MOLECULE-SWEDEN is an electronic structure program system written by Almlof, J., Bauschlicher, C. W., Blomberg, M. R. A., Chong, D. P., Heiberg, A., Langhoff, S. R., Malmqvist, P-A; Rendell, A. P., Roos, B. O.,Siegbahn, P. E. M. and Taylor, P. R. 25. Huber, K. P. and Herzberg, G., Molecular Structure IV: Constants of Diatomic Molecules, Van Nostrand Reinhold Company (New York), 1979. 26. Almlof, J. and Taylor, P. R., J. Chem. Phys., 86, 4070, (1987). 27. McMillen, D. F. and Golden, D. M., Ann. Rev. Phys. Chem., 33, 493, (1982). 83 Table 1. Experimental bond dissociation energies (kcal/mol). R-H Do [reference] H-H 102.3 [25] CH3 -H 105.1 [27] (CH3)g - H 93.2 (27] C6H5-H 110.9 [27] HCC-H 126.6 [11], 131.3 [9] 84 Table 2. Barriers for the hydrogen abstraction reaction, R 1 -H + R 2 --+ R 1 +H-R2 , calculated at the CCCI optimized geometry (kcal/mol). Negative energies indicate that the CCCI transition state geometry is lower in energy than the reactants. Numbers in parentheses are the barriers when the Davidson correction is included. R1 -H-R2 HF GVBCI-SCF CCCI H-H-H 24.3 9.9 10.3 CH3-H-CH3 34.9 27.8 29.8 H-H-CCH 11.6 8.0 5.4 CH3 -H-CCH 14.6 8.6 10.2 (CH3)3-H-CCH 11.0 5.9 8.0 C5H5-H-CCH 18.3 12.0 14.3 HCC-H-CCH 30.0 22.9 24.1 R1 -H-R2 DCCI HF*SD CI GVB*SD CI H-H-H 10.3 10.5 (9.6t 10.3 (10.3) CH3-H-CH3 17.5 22.5 (19.5) 20.4 (18.8) H-H-CCH 0.8 4.5 (3.2) 3.3 (2.7) CH3 -H-CCH -2.9 4.2 (2.0) 2.2 (1.3) (CH3)3 - H - CCH -7.0 0.45 (-2.78) C5H5-H-CCH -0.7 7.7 (4.1) HCC-H-CCH 11.6 17.0 (13.7) 14.6 (12.9) 85 Table 3, Transition state geometries optimized at the CCCI level. See text for constraints on these geometries. Ri -H-R2 R1H R2H H-H-H 0.94 0.94 CH3-H-CH3 1.36 1.36 H-H-CCH 0.80 1.61 CH3-H-CCH 1.22 1.48 (CH3)3-H-CCH 1.20 1.50 C5H5-H-CCH 1.24 1.42 HCC-H-CCH 1.28 1.28 H-C-C angle 105.2 105.5 86 Table 4. Number of spin eigenfunctions in each of the CI calculations for the transition state. If the transition state has higher symmetry, then the number of SEF is doubled for the product or reactant state. R1 -H-R2 CCCI DCCI HF*SD CI GVB*SD CI H-H-H 191 191 139 191 CH3-H-CH3 4914 71666 79428 310778 H-H-CCH 2160 18234 15189 53048 CH3-H-CCH 9477 150557 (CH3)3-H-CCH 21676 695842 1675566 C6H5 -H- CCH 17265 587343 587343 1514151 HCC-H-CCH 2697 42201 55211 221805 87 Table 5. Calculated exothermicities (kcal/mol). Numbers in parentheses include the Davidson correction. R1 -H-R2 HF GVBCI-SCF CCCI H-H-CCH 33.9 31.6 31.8 CH3-H-CCH 28.9 28.4 27.0 (CH3)3 - H - CCH 28.5 27.9 24.2 C6Hs -H-CCH 18.7 18.2 15.4 R1 -H-R2 DCCI HF*SD CI GVB*SD CI H-H-CCH 30.1 30.0 (27.5) 29.9 (28.1) CH3-H-CCH 29.2 26.9 (25.5) 26.8 (25.4) (CH3)3 - H - CCH 27.4 26.8 (25.9) C6Hs-H-CCH 18.9 18.3 (18.0) 88 Table 6. Calculated relevant energetics of ground-state acetylene and the excited-state triplet (kcal/mol). R1 -H-R2 R1H R2H H-C-C angle Te( 3 Bu +- 1 ~ ; ) 72.3 92.0 90.2 De( 1 ~ : HCC - H) 116.5 122.3 131.9 De( 3 BuHCC - H) 44.2 30.3 41.7 89 Figure Captions Figure 1. Acetylene radical hydrogen abstraction molecular tip attached to diamond-like material. Figure 2. Schematic diagram showing the transition barrier and exothermicity of the acetylene radical abstraction of hydrogen from isobutane. Figure 3. Geometries for the transition states. 90 Figure 1: Acetylene radical hydrogen abstraction molecular tip attached to diamond-like material. ~\ _____/ ........_c~ __. ~~ CH3 / CH3,,~-H CH3 -26.8 kcals/mol Dc=c-H 0.45 kcals/mol CH3 / CH3,~C) CH3 H-c=c-H Figure 2: Schematic diagram showing the transition barrier and exothermicity of the acetylene radical abstraction of hydrogen from isobuatane. c.o ..... 92 Figure 3: Geometries for the transition states System Symmetry H-H-H H-H-c:=c-H H-c:=c-H-c:=c-H H-c:=c-H 93 Chapter 4 The Hessian Biased Force Field for Polysilane Polymers 94 Abstract We report a force field (FF) suitable for molecular dynamics simulations of polysilane polymers. This FF, denoted MSXX, was developed using the Hessian biased method to describe accurately the vibrational states, the ab initio torsional potential energy surface, and the ab initio electrostatic charges of polysilane oligomers. This MSXX FF was used to calculate various spectroscopic and mechanical properties of the polysilane crystal. Stress-strain curves and surface energies are reported. Gibbs molecular dynamics calculations (Nose, Rahman-Parrinello) were used to predict various materials properties at higher temperatures. Phonon dispersion curves and elastic constants were calculated at various temperatures. Although this polymer is of increasing industrial interest we could find no experimental data with which to compare these predictions. 95 I. Int:roduction Polysilane polymers, -[-SiH2 -]n- denoted herein as P(SiH), have become of significant interest. The substituted derivatives of P(SiH) can be used as precursors to SiC ceramics, as nonlinear optical materials, as semiconducting polymers, and as photoresists. 1 The polysilane polymers are amorphous and their structural, physical, and electronic properties are not well characterized. 2 - 5 In order to predict such properties, we have developed a force field expected to be accurate for predicting structural, mechanical, vibrational, and thermodynamic properties. Section II develops the force field and Section III applies this force field to the prediction of various properties for P(SiH) oligomers. The properties of P(SiH) crystals are predicted in Section IV. II. Development of the MSXX Force Field II.A Introduction The general form of the force field is taken as (1) Here EQ - C - ~ qiqj coul L....t R·. i>j (2) iJ represents the Coulombic interactions between partial charges on the various atoms ( Ccoul = 332.0637 converts units so that the R is in A and EQ is in kcal/mol), (3) represents the long-range attraction (London dispersion) and short-range repulsion (Pauli orthogonalization of nonbonded electrons), and (4) 96 represents all terms involving bonds between atoms and coupling behavior of these bonds. Our general approach to developing force fields is to emphasize the use of accurate quantum chemical calculations on model systems. Thus the atomic charges [for EQ] and the torsional potentials about single bonds [for Etorsion] are taken directly from Hartree-Fock (HF) calculations using good basis sets. The force constant parameters important in describing valence interactions (Ebond, Eangle, Ecross) are taken from the Hessian (second derivative of energy with respect to atomic coordinates) calculated from HF wavefunctions. However the eigenvalues of this Hessian are modified [the Hessian Biased 6 FF, HBFF] since HF vibrational frequencies are too high. Herein we derive the HBFF for P(SiH) using the model systems: SiH4, Si2H5, Si3Hs, n-Si4H10. Only the vdW terms, (3), are not based on HF calculations. The vdW parameters for Si and H were obtained from the Dreiding force field 7 which were based on fits to experimental structural data for simple solids and on extrapolations. 11.B Calculations For each model system we carried out HF calculations using the 6-31G** basis set. The geometry was optimized at the HF level (using Gaussian 92 8 and PS-GVBl 9 ) and this geometry was used in determining the force field. Comparing to experiment the HF geometry leads to errors of about 0.02 A in Si-Si distances, 0.01 A in Si-H distances, and 0.5° in bond angles (see Table 1). We use the Potential Derived Charges (PDQ) as the partial atomic charges. 10 PDQ are derived by (i) calculating the electron density distribution, p(r), from the HF wavefunction, (ii) using p( r) to calculate the electrostatic potential on a set of 97 grid points around the molecule, (5) and (iii) determining the set of atomic point charges on the various atoms to optimally fit this electrostatic potential, (6) at grid points outside the vdW radii. The PDQ charges (from Gaussian 92) as well as Mulliken populations are shown in Table 2. Based on these calculations we recommend in Table 2 the charges for P(SiH) chains. 11.C The Biased Hessian Method From ab initio HF wavefunctions we calculate 8 a full Hessian (7) where Rad is the a component (x, y, z) of the coordinates of atom i. After mass weighting, l HHF H- HF . (3 . = --;:::::::;:::;;::::::::::: . f3 . ai, 1 JMiMj ai, 1' (8) the vibrational modes {Uf F} and vibrational frequencies {vflF} are obtained by solving (9a) where ,HF Ai = (Cfregl.liHF)2 (9b) and C freq = 108.5913 converts units so that energies are in kcal/mol, distances are in A, frequencies are in cm-1, and masses are in atomic mass units (C 12 has mass= 98 12.0000 amu). This Hessian provides g(g + 1)/2 independent pieces of information [666 for Si 4 H 10 ], where g = 3N - 6 is the number of degrees of freedom. These constraints are sufficient to determine the force field. In contrast, fitting just the frequencies leads to only g conditions [36 for n-Si4H10]. However, at the HF level the calculated frequencies, vfl F, are 10-20% too high. This led to the development of the Hessian Biased method6 for FF parameterization in which the force field is fit to the biased Hessian HHB = uHFAexpfjHF (10) where U is the transpose and ,\exp is the diagonal matrix based on experimental frequencies , exp "'i = (Cf reqViexp)2 • (11) This Hessian has the property that, HHBuHF = uHF..\exp, that is, the eigenvalues match experiment while the eigenfunctions match HF theory. Thus, HHB has the best available information on the vibrational modes. In general the optimum geometry at the HF level differs slightly from experiment, raising the question of which structure to use in (10). We use the structures optimized at the Hartree-Fock level of theory. Previously6 •11 we advocated the use of the experimental structure for determining force constants from the ab initio calculations primarily because the internuclear separations (which strongly affect the Hessian) reflect the experimental system. However, in molecules with low frequency torsions, a slight difference in structure can cause a noticeable rotational contamination of the torsional modes. Since we want to use frequency scaling parameters to compare various molecules, it is better to derive the frequencies for all molecules at the ab initio minima (rather than at the experimental minimum for molecules where experimental geometries are available and the ab initio minimum for those where 99 experimental geometries are not available). Fortunately, as indicated in Table 1, the differences between the ab initio and experimental geometries are small. II.D The Potential Energy Surface for Torsions The distribution of conformations in a polymer and the rates of conformational transitions have a strong effect on the properties (moduli, glass temperature); hence, it is critical that the FF lead to the correct relative energies of the minima (e.g., trans versus gauche) and of the barrier heights between them. Thus torsional FF parameters are particularly important for describing amorphous polymers. We use ab initio calculations to provide the torsional potential energy surface. With the 6-31G** basis, the torsional potentials calculated from HF wavefunctions are adequate. 12 The HF calculations lead to a total torsional potential function EH F ( </>) which we want to fit with the FF, (12) Here EBa~ contains all parts of Eval except for the torsional term including </J. We ' have already specified how EQ and EvdW are to be calculated, and the dependence of EBa~ on bonds and angles will be determined in Section D. Thus we define the ' torsional potential as (13) where (14) contains all terms except the torsion, </J. In determining Et 0 r(</>) from (13), the simplest procedure would be to fix all bonds and angles so that only the torsional angle </> changes [thus EBa~ (</>) would be ' 100 constant]. However such rigid rotations about bonds sometimes lead to bad contacts (very short distances between nonbonded atoms). At a bad contact the ab initio wavefunctions readjusts the molecular orbitals to minimize repulsion which invalidates the assumption of constant energy in the bond and angle terms. In addition the EvdW derived from fits to experiment may not accurately describe the inner repulsive wall. Consequently (12) is defined by fixing </J and optimizing all other degrees of freedom for each conformation to obtain the adiabatic torsional potential. Thus for the HF wavefunction we optimize the other geometric parameters at each </J and for EF F (<p) we do the same. With this procedure EF F (<p) depends on the FF for bonds and angles, which is determined from fitting HHB (Section ILE). Thus, to determine the nontorsional parts of the FF using HBFF, we use approximate torsional parameters and put zero weights on fitting the torsional modes. Then after determining Et 0 r(</J) from (13) we redo the HBFF using Et 0 r(</J) and then use the final bond and angle FF to calculate a new EF F (<p) and hence a final Etor (<p). We started with the Dreiding FF and found that one such iteration was generally sufficient. An alternative to the above procedure is to use the same geometry for both the HF and FF calculations. This could be optimized either for HF or for the FF. The problem is that the HF structure is generally not optimum for the HBFF. Hence fitting to (13) to determine Et 0 r(</J) would lead to residual forces (due to bond and angle terms in the unoptimized structure). Thus at the top of the barrier, the total forces on the molecule from either HF or HBFF would be nonzero so that it would not be a true maximum. The HF wavefunction was calculated by fixing the dihedrals of interest (in increments of 15°) and optimizing all other degrees of freedom, leading to the results in Figures 1-3 and Table 3. 101 For each dihedral, the geometry was optimized using the force field with the dihedral constrained (zero barrier used for this torsion). As indicated in (13) the true torsional energy Et 0 r(c/>) [or Et 0 r(cp 1 , ¢ 2 ), in the case of Si3 H 8 ] is defined as the difference between EH F(cp) and EF F(cp) with the torsion excluded. For SiH 8 we considered the two-dimensional surface where the two torsions (¢ 1 and ¢ 2 ) are changed independently. In this case a bicubic spline was fitted to Etor (¢ 1 , ¢ 1 ) (15) to generate a denser grid which was in turn fitted to a Fourier series of torsional terms 1 P Etor(c/>) = 2 L Km[l+cosmcp] = Vo+ L 21 Vmcosmcp, P m=O (16) m=l For a single term this becomes (17) where Km is the barrier. For a given J - K dihedral there are 3 choices for atom I bonded to J and 3 choices for atom L bonded to K, leading to 9 possible I - J - K - L terms. In POLYGRAF 20 the energy for this torsion is written as t LL 3 E:tk = 3 E}°JKL (c/>IJKL). l=l L=l Thus each torsional energy E}jKL (cp) is written as if the whole barrier were due to this term, but it is scaled to 1/9 of this value. Tables 3a-c show EHF(c/>), and EFF(cp) [or EHF(c/> 1 , ¢ 2 ) and EFF(c/> 1 , ¢ 2 ) in the case of ShHs]. The fit to the ab initio energy surface was Boltzmann weighted so that the errors near the minima are smaller than the errors near the maxima (since 102 the higher barrier regions will be sampled less in molecular dynamics simulations). We found that a single K 3 term (barrier of 0.94 kcal/mol) for H - Si - Si - H was sufficient for Si2H6. For Si3H6, we assumed the same H - Si - Si - H term and found that for H - Si - Si - Si a single K 3 term (barrier of 0.806 kcal/mol) was sufficient. For n - Si 4 H 10 we assumed the H - Si - Si - Hand H - Si - Si - Si terms from Si2H6 and Si3H 8 and added Si - Si - Si - Si. Here we found a 3-term potential (with K 1 , K2, and K3) was sufficient. The resultant PES are plotted in Figures 1-3 for HF and HBFF. 11.E Valence Force Field Terms The bond and angle part of the valence FF is written as We take Eb 0nd as a sum over all bond pairs, each of which has the form of a Morse function, 7 (18a) with (18b) and (18c) This includes anharmonicity and allows bond dissociation. Here there are three independent parameters Re, kR, and DR. However DR is not sensitive to the Hessian or geometry; Hence, we choose DR based on the experimental bond energy (it was not optimized). 103 We take Bangle as a sum over all six angle terms I - J - K for each atom J, where each angle term is described with the cosine angle form, 7 . C 2 Ecosine(0) = - [cos0- cos Be] (19a) 2 with C = k0 (19b) sin 2 Be This form leads correctly to dE / d0 = 0 for 0 = 0, 180° and has a barrier of Ebarrier = C [1 + COS 0e]2. (20) 2 We found that bond-bond cross-terms (21) sharing an apex atom (e.g., IJ and JK for the atoms I and K bonded to J) are generally useful when the two are equivalent (e.g., Si-H/Si-H at SiH3 or SiH2 groups). However, Si - Si/ Si - Hat the SiH3 group does not have much effect on the force field. This is because the splitting between equivalent terms is dominated by off-diagonal interactions whereas analogous couplings for inequivalent bonds can be built into the force field by modifying the diagonal terms. Thus we include bond-bond cross terms only for equivalent bonds. Bond-angle cross terms (e.g., bond IJ with angle IJK), E1BA = Dre (r1 - rf) (cos 01 - cos en (22) are necessary for a good description of the vibrations. For a given J - J - K there are two such terms, one for r I J and one for r J K. In addition we find that one-center angle-angle cross terms (involving bonds defining two angles sharing a common bond) are important, E1AA = F00 ( cos 01 - cos en (cos 02 - cos BD . (23) 104 The sign and magnitude of these terms are difficult to predict a priori. Hence we started with various combinations of sign and magnitude and allowed the optimization to determine both the sign and magnitude of these coupling constants. For long chain molecules, we found that 2-center angle-angle cross terms are also important, (24) Thus for the dihedral J - J - K - L, 01 corresponds to the I J K angle and 02 corresponds to J KL. For disilane such terms determine the splitting between rocking modes of different symmetries. In the case of polyethylene, 11 they are essential in reproducing the stiffness in the chain direction. For group IV and III/V solids (e.g., diamond, silicon, GaAs, etc.) they are necessary to describe the mode softening of the transverse acoustic (TA) mode near the zone boundary. 13 In P(SiH), we find that they are important for predicting the modulus in the chain direction. For semiconductors the factor !(¢) is taken as !(¢) = ½[1 - j cos(¢)] so that f(O 0 ) = £(180 °) = 1 and f(60 °) = £(240 °) = 0. This is because only trans coupling is important. For P(SiH) f( </>) is taken as f( ¢) = 1 - cos(¢). Summarizing we take II.F Scaling of Ab Initio Frequencies For both Si 3 H 8 and n-Si 4 H 10 the set of experimental frequencies is incomplete. In order to estimate of the experimentally undetermined modes, we scale the ab initio frequencies for these modes using scale factors (the ratio of the experimental frequency to the ab initio frequency) obtained from observed modes. This works well because within a particular class of vibrations the scale factor is nearly 105 constant, even between different molecules. For example, it is 0.901 ± 0.016 for the SiH bending modes of Si 3 Hs, (A further refinement is to calculate separate average scales for the rocking, wagging, and scissor Sill-bending modes.) Table 4 shows the scale factors derived for each molecule. Because the scales vary little from molecule to molecule, the scaling procedure can be applied even to molecules with no experimental frequencies. Table 5 shows the Si 5 H 1 2 scaled frequencies (from HF) and those predicted by the MSXX (from n - Si 4 H10). Here the MSXX vibrations differ from the HF scaled values by 13 cm- 1 (rms error), about the same as for cases where experiment is available. Torsions do not follow the same scaling trend as bending and stretching modes. However we base all torsional parameters on the HF torsion curves ( Section III.C) rather than on scaled torsional vibrations. 11.G Optimization of Parameters We used the program FFOPT 14 developed by Yamasaki, Dasgupta, and Goddard to optimize the valence HBFF parameters. This uses Singular Value Decomposition (SVD) and emphasizes changes in parameters that most affect the properties of interest (and to eliminates parameter redundancies). III. Polysilane Oligomers III.A SiH4 The vibrational frequencies for silane are shown in Table 6 and the silane force field is shown in Table 7a. HBFF exactly reproduces the geometry and the experimental frequencies. 15 The MSXX (from n - Si 4 H 10 ) leads to slightly low (about 1%) frequencies for silane, and the inaccuracy in applying the MSXX to disilane and trisilane is much smaller. (Table 6 also shows the accuracy of the silane HBFF for predicting the frequencies of deuterated silane. 15 ) This indicates 106 the robustness of the MSXX. The MSXX FF parameters from the P(SiH) oligomers are shown in Table 7a (the calculated oligomer geometries are in Table 1). The MSXX FF leads to excellent geometries and frequencies. 111.B Si2H6 Disilane is the largest oligomer of P(SiH) having a complete set of experimental vibrational assignments. 16 In addition, all the isotopic shifts, except for the torsional mode, are available for fully deuterated disilane. Table 8 shows predicted frequencies, where we find an RMS difference of 4.8 cm- 1 between HBFF and experiment. 16 The HBFF accurately reproduces the geometry as shown in Table 1. The validity of the force field is shown by its accurate prediction of the Si 2 D 6 experimental frequencies 17 as shown in Table 8. Included in the complete set of frequencies for disilane is the torsional frequency (determined indirectly from a two photon process). However we do not use this in our fit. As discussed above the torsional potential is fitted to the entire torsional potential surface. Disilane is the only P(SiH) oligomer for which we can compare the torsional frequency for the single bond rotational potential to experimental data. Figure 1 and Table 3a show the accurate match of the ab initio and HBFF torsional potentials over the entire dihedral angle range. This accuracy comes at the modest cost of an error of 8.2 cm- 1 in the torsional frequency. 111.C Si3Hs Silylpropane is the smallest model compound containing the [-SiH2 - ] repeating unit of P(SiH). Some experimental data (13 of 27 modes) 18 are available on the vibrational frequencies of Si 3 H 8 , but a complete set of experimental frequencies for the isolated molecule in the gas phase is not available. Consequently, 107 for each class we considered only the experimentally known vibrations to obtain the accurate scale factor for that class, which was then applied to all ab initio frequencies to predict the complete set of "experimental" frequencies. Using scales from within the same molecule leads to very small standard deviations of the scales and thus is likely to yield accurate scaling of the experimentally unassigned modes. The average error between the HBFF frequencies and the experimentally available frequencies 18 is 7.7 cm- 1 (6.0 cm- 1 including the scaled HF frequencies). Table 9 also shows the frequencies for two deuterated species to compare with future experimental results. We find that fitting the force field to experiment requires the use of different valence terms for the SiH2 hydrogens and the SiH3 hydrogens (we use the H - Si - Si - H torsional potential transferred from disilane). Using the MSXX FF from n - Si4H10 to calculate the modes of Si 3H 8 leads to good accuracy, especially for the geometry (Table 1). This justifies the transferability of the HBFF between molecules and indicates that the HBFF description of P(SiH) or the other P(SiH) oligomers should be accurate. The torsional modes were not obtained from HBFF. In the Si 3 H 8 force field we include the disilane H-Si-Si-H torsional potential and fit a Fourier series of torsions to the combinations of rotations of the two H-Si-Si-Si dihedrals. Figure 2 and Table 3b show the accuracy of the HBFF torsional potential energy surface. 111.D n-Si4H10 n-Si4H10 is the smallest oligomer which includes all the valence terms necessary to model larger oligomers and P(SiH). Again only a partial assignment from experimental spectra 18 is available for n-silylbutane. We follow the same procedure used for scaling of the silylpropane frequencies. Table 10 shows the HF frequencies, the experimental assigned frequencies and the scales used for modes not observed. Table 10 also shows the narrow range for similar scales. The HBFF procedure re- 108 produces the experimental frequencies and scaled HF frequencies to 6.2 cm- 1 rms error. The geometry is accurately reproduced as shown in Table 1. For n - Si 4 H 10 we also develop the MSXX-R FF which does not distinguish between the SiH3 and SiH2. We find that MSXX-R is only slightly worse than the MSXX, as shown in Table 10. We use the valence terms describing the central silicons of MSXX to describe the larger oligomers and the polysilane crystals. The nsilylbutane HBFF not only includes all the terms to simulate the smaller oligomers, it also includes all the terms necessary to model the larger oligomers and P(SiH) and thus can be used to predict the properties of P(SiH) of arbitrary chain length. The torsional mode was not scaled, but was calculated by fitting a Fourier series to the rotations of the molecule about the Si-Si-Si-Si dihedral. The H-Si-Si-Si and H-Si-Si-H torsional potentials were transferred from silylpropane and disilane, respectively and not varied. Figure 3 and Table 3c show the HBFF and the ab initio torsional potential. 111.E n-Si5H12 The n-Si5H12 molecule has not been observed experimentally. We calculate HF /6-31G** and MP2/6-31G** vibrational frequencies to validate the MSXX force field (from Si4H10), Scales from the smaller oligomers are applied to the n-Si 5H 12 HF vibrational frequencies to obtain scaled HF frequencies which can be compared to frequencies calculated using the MSXX. Table 5 shows the MSXX frequencies and the scaled HF frequencies. The RMS difference is less than 12 cm- 1 with most modes differing less than 2%. Either method provides an acceptable prediction of experimental frequencies; however scaled ab initio calculations require substantially greater computational time. MP2 ab initio calculations take substantially more time than scaled HF calculations but still overestimate experiment by 3-8% (Table 109 5), with the exception of the low energy skeletal modes. Since scaled HF and MSXX predictions are generally within a percent of experimental frequencies, we conclude that MSXX is the most cost-effective approach to predicting frequencies in good agreement with experiment. Table 11 compares the geometries for n-Si 5 H 12 from HF and from the MSXX. The differences are small, with the largest error (1.1 °) for the Si - Si - Si central angle. IV. Polysilane We base the FF for polysilane polymers on the parameters for the central atoms of the HBFF for n - Si 4 H 10 in Table 7a. We refer to this as the MSXX FF (see Table 7b). The charges for polysilane polymer were based on the PDQ atomic charges for the central SiH2 group of n - Si 5 H 12 (See Table 2). Particularly important for calculations on P(SiH) polymers are the Si-Si-Si bends denoted as accordion modes in Tables 5, 9, and 10. This mode is prominent on the Raman spectra for long alkanes and extrapolating the frequency of this mode to infinite alkanes leads to an excellent prediction of the Young's modulus (see reference 11 for discussion). The MSXX FF was used to calculate the properties of the crystal built with the all trans conformation of P(SiH). This is analogous to polyethylene (PE) except that structures with both one and two chains per cell were considered. Table 12 shows the structure and mechanical properties of the crystal at 0, 77 and 300K. The bulk modulus at OK is 13.10 GPa. The Young's moduli calculated are 11.94, 110.57, and 18.64 GPa at OK which compare to 9.0, 337.0, and 9.4 for PE. PE is much stiffer along the chain direction than P(SiH) while being softer perpendicular to the chains at low temperatures and being similar in stiffness to P(SiH) perpendicular to the chains at temperatures above 300K. Because of the anharmonicity of vdW and 110 electrostatic interactions between chains, the Young's moduli perpendicular to the chain direction decrease dramatically with temperature (by ~ 65% for Oto 300K). The decrease in the Young's moduli of polyethylene is much smaller (~ 35% for 0 to 300K). The elastic constants of P(SiH) behave similarly to those of polyethylene. Thus C 22 , the elastic constant along the chain direction, decreases by only 15% from 0 to 300K. In PE the decrease is 5.9%. C 11 and C33 for deformations perpendicular to the chain direction show a decrease of 65% while in PE the decrease is~ 40%. The relative decrease in the deformation properties with temperature is large relative to polyethylene (see Table 12) both along the chain direction and perpendicular to it indicating that both the valence and non-bond interactions are more anharmonic for P(SiH) than for PE. The properties at 77K and 300K were calculated by averaging the crystal structures from the last 20 ps of a 30 ps Gibbs molecular dynamics calculation (Nose plus Rahman-Parrinello 19 ) at several temperatures to calculate the thermal expansion tensor. Using the thermal expansion tensor we calculated the lattice parameters at the desired temperatures and calculated the properties of the crystal at that desired temperature. (After reminimizing the atomic positions for the new lattice constants.) The cohesive energy of the crystal is calculated to be 3.816 kcal/mol-SiH2 (Table 13). This compares with 1.8701 kcal/mol-CH 2 for polyethylene calculated by Karasawa et al. 11 also using the biased Hessian approach. Their result compared favorably to experiment where the cohesive energy is measured to be 1.84 kcal/molCH2. The larger elastic constants of PSi compared to PE perpendicular to the chain direction indicate the relatively stronger non-bond interactions and thus a larger cohesive energy. Tables 14 and 15 show the vibrational frequencies for the crystal and for the 111 infinite single chain, respectively. Calculations by Cui et al. 5 on single unpacked chains (ignoring vdW and Coulomb) are within 5-30 cm- 1 of our results. Our results show that the inclusion of coulombic and vdW forces in the MSXX FF strongly affects interactions between neighboring chains, as shown by the changes in the predicted vibrational frequencies between the packed and unpacked P(SiH). Our single chain frequencies are in better agreement with many of the experimental frequencies (Table 15) than our packed P(SiH) frequencies. This may be because the samples are not very crystalline (including silicon-like clusters and metastable gauche configurations), leading to an inefficiently packed local structure better approximated by a single chain. Figure 4 shows the phonon dispersion curves for the all-trans polysilane crystal (there are no experimental numbers). The low energy bands depend strongly upon the van der Waals and coulombic interactions as well as the torsional force constant. In the case of PE (Karasawa et al. 11 ) the MSXX led to average errors of 7.9 cm- 1 for n-C4 H 10 and 24 cm- 1 for the crystal. The MSXX force field reproduces the vibrational frequencies of n-Si 4 H 10 with an error of 6 cm- 1 but we expect larger errors for the crystal. Figure 5a shows stress-strain curves for stresses along the unit cell axes (perpendicular to the chain) In these calculations we used a super cell consisting of 32 primitive cells. Although the Young's modulii perpendicular to the chain direction are larger for P(SiH) than for PE the yield stress is similar. This is the result of the higher anharmonicity of the P(SiH) non-bond interactions which fall off faster that those of PE. In the case of PE, shear perpendicular to the chain direction eventually leads to a transition to a metastable monoclinic phase. Shearing P(SiH) using our force field did not lead to any stable phases. Minimization from any structure arrived at by shearing always led back to the orthorhombic unit cell. 112 The surface energy is calculated by stretching the crystal perpendicular to the surface until the crystal breaks (Figure 5b). From the curves in Figure 6 we derive surface energies of 63.5 dyn/cm for the (100) surface and 66.9 dyn/cm for the (001) surface (using a conversion factor of 694.8 to convert from kcal/mol-A 2 to erg/cm3 ). This compares with 106.8 dyn/cm (100) and 109.2 dyn/cm (001) for the analogous surfaces of polyethylene. The (100) surface has two SiH 2 groups per unit cell, leading to a surface energy per SiH2 of 1.698 kcal/mol while the (001) surface has 4 SiH2 groups per unit cell leading to a surface energy of 1.609 kcal/mol. This compares with 0.938 kcal/mol and 0. 720 kcal/mol for the analogous surfaces of polyethylene. We know of no experimental information on such properties for P(SiH). Assuming only nearest-neighbor fiber-fiber interactions, we would expect a surface energy of ½the cohesive energy of the crystal. This would predict a surface energy of 1.42 kcal/mol for both the (100) and (001) surfaces, which is low by~ 15 %. Figure 6 shows the predicted heat capacity ( Cv), entropy, enthalpy and free energy versus temperature (again, we know of no experimental data). V. Summary In this paper we develop the MSXX force field for P(SiH) that should be accurate and useful for a variety of structural, thermodynamic, spectroscopic, mechanical, and surface properties. The method follows the procedure developed by Karasawa et al. 11 for PE (modified for incomplete experimental vibrational data). For PE the substantial amount of data on experimental properties validated the accuracy attained with this procedure. For P(SiH) sample quality is poor and this procedure is used to predict the properties of crystalline P(SiH) in advance of experiment. 113 Spectroscopic force fields (SFF) usually omit the electrostatic (Q) and van der Waals (vdW) nonbond terms since the geometry is considered fixed. For molecular dynamics (MD) calculations the FF must describe how the energy changes with geometry, and hence the MSXX FF includes these intramolecular nonbond terms. They are small for P(SiH) oligomers except for torsional modes, where they make significant contributions. On the other hand, the bulk properties of P(SiH) depend greatly on the intermolecular interactions. The HBFF 6 combines experimental vibrational frequencies with the normal mode description of the vibrations from ab initio calculations. For P(SiH) oligomer systems it was necessary to extend the HBFF approach to handle systems with incomplete spectroscopic information through scaling. The validity of the resulting MSXX force field is tested by calculating the properties for n-Si 5 H 12 , where we find the modes to be within 2% of the scaled ab initio (HF /6-31G**) results (no experimental data are available). MSXX predicts a Si-Si-Si bond angle 1.13 ° larger than HF, which although not a large error, will manifest itself in predicting a slightly larger bond angle for the crystal. Despite the slight difference in the geometry the MSXX predicts the vibrational frequencies accurately. The MSXX predictions should be useful for predicting the spectra of larger oligomers where assignments are incomplete (particularly for torsional modes and skeletal modes) and for the P(SiH) condensed phase where there are no assignments. We used the Nose formalism 19 of canonical molecular dynamics to extract thermodynamic properties of polysilane from the MD simulations. Properties at various temperatures were averaged from dynamics calculations and thermal expansion coefficients were derived for temperatures up to 450K. Other properties were calculated using the unit cell at the desired temperature. 114 References 1. R. D. Miller and J. Michl, Chem. Rev. 89, 1359 (1989) and references therein. 2. J. R. Damewood Jr. and R. West, Macromolecules 18, 159 (1985). 3. J. M. Zeigler, Synth. Met. 28, C581 (1989). 4. J. W. Mintmire, Phys. Rev. B 39, 13350 (1989). 5. C. X. Cui and M. Kertesz, Macromolecules 25, 1103, (1992). 6. S. Dasgupta, and W.A. Goddard III, J. Phys. Chem. 90, 7207 (1989). 7. S. L. Mayo, B. D. Olafson, and W. A. Goddard III, J. Phys. Chem. 92, 7488 (1990). 8. M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J.B. Foresman, B. G. Johnson, H.B. Schlegel, M.A. Robb, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. K. Fox, D. J. Defrees, J. Baker, J. J. P. Stewart, and J. A. Pople, Gaussian 92 Revision, Gaussian, Inc. Pittsburgh, PA, (1992). 9. PS-GVB, Murco Ringnalda, Jean-Marc Langlois, Burnham H. Greeley, Thomas V. Russo, Richard P. Muller, Bryan Marten, Youngdo Won, Robert E. Donnelly, Jr., W. Thomas Pollard, Gregory H. Miller, William A. Goddard III, and Richard A. Friesner, PS-GVB vl.03, Schroedinger, Inc., 1994. M. N. Ringnalda, Y. Won, and R. A. Friesner, J. Chem. Phys. 93, 3397 (1990). J-M. Langlois, R.P. Muller, T. R. Coley, W. A. Goddard III, M. N. Ringnalda, Y. Won, and R. A. Friesner, J. Chem. Phys. 92, 7488 (1990). 10. L. E. Chirlian and M. M. Francl, J. Comp. Chem., 8, 894 (1987). 11. N. Karasawa, S. Dasgupta, and W. A. Goddard III, J. Phys. Chem. 95, 2260 (1991). 12. D. Cremer, J. S. Binkley, J. A. Pople, and W. J. Hehre, J. Am. Chem. Soc. 115 96, 6900 (1974). 13. H. L. McMurry, A. W. Solbrig Jr., J. K. Boyter, and C. Noble, J. Phys. Chem. Solids 28, 2359 (1967). 14. T. Yamasaki, S. Dasgupta, and W.A. Goddard, III, J. Chem. Phys., submitted. 15. H. W. Kattenberg and A. Oskam, J. Mol. Spectry. 49, 52 (1974). 16. J. R. Durig and J. S. Church, J. Chem. Phys. 73, 4784 (1980). 17. D. C. McKean, Spectrochimica Acta 48A, 1335 (1992). 18. F. Feher and H. Fisher, Naturwissenschaften 51, 461, (1969). 19. (a) T. Cagin, W. A. Goddard III, and M. L. Ary, J. Comp. Polymer Physics I, 241 (1991). (b) T. Cagin, N. Karasawa, S. Dasgupta, and W. A. Goddard III, Mat. Res. Soc. Symp. Proc. 278, 61 (1992). (c) S. Nose, J. Chem. Phys. 81, 511 (1984). (d) M. Parrinello and A. Rahman, J. Appl. Phys. 52, 7182 (1981). 20. POLYGRAF is from Molecular Simulations Inc., Burlington, Massachusetts. 116 Table 1. Structural parameters. Distances are in A, angles are in degrees. Molecule SiH3 SiH4 Si2H6 Si3Hs n - Si4H10 Featureb,e MSXXc FF HF/6-31G** Si-H H-Si-H Si-H Si-Si H-Si H- Si- Si Si - Sic He - Sic Hip - Si Hop - Si Si - Sic - Si Hip - Si - Sic Hop - Si - Sic He - Sic - Si Hop - Si - Sic - Si Si - Sic Sic - Sic He - Sic Hip - Si Hop - Si Si - Sic - Sic Hip - Si - Sic H 0 p - Si- Si He - Sic - Si Hop - Si - Sic - Sic He - Sic - Sic - Si 1.476 111.01 1.476 2.353 1.479 110.35 2.358 1.482 1.479 1.479 112.73 110.74 110.17 109.07 59.77 2.358 2.362 1.481 1.480 1.479 113.06 110.86 110.01 109.22 59.76 58.32 1.476 111.01 1.476 2.353 1.479 110.35 2.357 1.482 1.478 1.479 112.56 110.69 110.16 109.12 59.77 2.357 2.361 1.482 1.479 1.479 112.73 110.73 110.08 109.30 59.79 58.48 n-Si4H10 FFd Experiment ala 1.476 2.356 1.479 110.50 2.351 1.482 1.477 1.478 108.83 110.84 109.75 109.98 59.65 References 15 and 16. b Sic and He denote atoms of SiH2 groups. These are used in the P(SiH) FF. c Separate FF for each molecule, parameters in Table 6a. d Using the MSXX FF of n-Si 4H 10 to calculate the structure of the other molecules. e ip and op denote in-plane and out-of-plane hydrogens, respectively. a 110.6 1.481 2.331 1.492 110.3 117 Table 2. Atomic charges (electron units) from HF calculations on linear (all trans) chains. Based on these results we recommend the following: (i) Qsi = 0.30, QH = -0.15 for SiH2 of P(SiH). (ii) Qsi = 0.44, QH = -0.14 for terminal SiH3 groups, and (iii) Qsi = 0.23 and QH = -0.125 for SiH2 next to a terminal group. Si2H6 Atoma PDQ Mulliken Si3Hs PDQ Mulliken Si4H10 PDQ SisH12 PDQ Mulliken Si (SiH2)' 0.2971 0.2878 H (SiH2)' -0.1485 -0.1508 Mulliken Si (SiH 2) 0.1550 0.2287 0.2211 0.2565 0.2287 0.2617 H (SiH2) -0.1042 -0.1491 -0.1274 -0.1501 -0.1260 -0.1495 Si (SiH3) 0.3936 0.4752 0.4714 0.5096 0.4829 0.5175 0.4494 0.5138 H (SiH3)-ip -0.1312 -0.1584 -0.1369 -0.1567 -0.1458 -0.1569 -0.1370 -0.1570 -0.1424 -0.159 -0.1518 -0.1584 -0.1446 -0.1584 H (SiH3)-op a Primes signify central SiH2. ip and op denote in-plane and out-of-plane hydrogens, respectively. 118 Table 3a. The torsional potential (kcal/mol) for Si 2 H 6 • All bonds and angles were optimized at each¢. E}7; is described with a single 3-fold term (17) with a barrier K 3 = 0.94 kcal/mol (see Table 7a). HF MSXX 60.000 0.00000 0.00000 45.000 0.13500 0.14490 30.000 0.47000 0.49150 15.000 0.82300 0.83280 0.0000 0.97400 0.97290 ¢ 119 Table 3b. The torsional potential (kcal/mol) of Si 3 H 8 for various values of the two dihedral angles, ¢ 1 and ¢ 2 • Same conventions as Table 3a. Efo{ (<p) is described using the H - Si - Si - H torsion from Table 3a plus a single 3-fold H - Si- Si - Si torsion with barrier of K3 = 0.806 kcal/mol (see Table 7a). MSXX Torsional Energies HF Torsional Energies ¢1 ¢2 180 165 150 135 120 180 165 150 135 120 180 0 0.1205 0.4241 0.7504 0.8995 0 0.1150 0.3389 0.6884 0.8161 165 0.1205 0.2478 0.5568 0.8839 1.0278 0.1150 0.2058 0.4897 0.8111 0.9728 150 0.4241 0.5568 0.8734 1.2069 1.3511 0.3389 0.4897 0.8082 1.1716 1.3585 135 0.7504 0.8839 1.2069 1.5505 1.7011 0.6884 0.8111 1.1716 1.5656 1.7533 120 0.8995 1.0278 1.3511 1.7011 1.8601 0.8161 0.9728 1.3586 1.7533 1.9196 105 0.7504 0.8839 1.2069 1.5505 1.7011 0.6884 0.8111 1.1716 1.5656 1.7533 90 0.4241 0.5568 0.8734 1.2069 1.3511 0.3389 0.4897 0.8082 1.1716 1.3585 75 0.1205 0.2478 0.5568 0.8839 1.0278 0.1150 0.2058 0.4897 0.811 0.9727 60 0.0000 0.1205 0.4241 0.7504 0.8995 0.0000 0.1150 0.3389 0.6884 0.8161 45 0.1205 0.2478 0.5568 0.8839 1.0278 0.1150 0.2058 0.4897 0.8111 0.9727 30 0.4241 0.5568 0.8734 1.2069 1.3511 0.3389 0.4897 0.8082 1.1716 1.3585 15 0.7504 0.8839 1.2069 1.5505 1.7011 0.6884 0.8111 1.1716 1.5656 1.7533 0 0.8995 1.0278 1.3511 1.7011 1.8601 0.8161 0.9728 1.3586 1.7533 1.9196 120 Table 3c. The torsional potential (kcal/mol) for central torsion angle of n-Si4H10. For each ¢ all other structural parameters were optimized (at either the HF or FF level). Same conventions as Table 3a. The Etar ( ¢) was described with the H - Si - Si - H and H - Si - Si - Si terms from Tables 3a and 3b plus a three term Si - Si - Si - Si potential (see Table 7a). The last column is the Si-Si-Si-Si torsional potential (after eliminating the other nonbond and valence torsions). ¢ HF MSXX Etor 0 1.5631 1.5628 0.7189 15 1.3380 1.3302 0.6682 30 0.8461 0.8345 0.5271 45 0.3893 0.3891 0.3455 60 0.1539 0.1542 0.1891 75 0.1639 0.1504 0.0734 90 0.3238 0.3189 -0.0324 105 0.5261 0.5376 -0.1005 120 0.6735 0.6458 -0.0657 135 0.5359 0.5478 -0.0787 150 0.3190 0.3110 -0.0247 165 0.0986 0.0883 0.0018 180 0.0000 0.0000 0.0000 121 Table 4. Scale factors (ratio of experimental value to HF value) for estimating experimental frequency from HF vibrational frequencies. Standard deviations are in parentheses. Mode SiH4 Si2H6 Si3Hs n-Si4H10 Si-H Stretch 0.9255 0.9113 0.9163 0.9140 (0.0035) (0.0215) (0.0015) (0.001) 0.9125 0.8972 0.9009 0.9081 (0.0191) (0.0126) (0.0162) (.0203) 0.9100 0.9370 0.9177 (0.0226) (0.0055) Si-H Bend Si-Si Stretch Si-Si-Si Bend 1.025 122 Table 5. Predicted vibrational frequencies (cm- 1) for n - SisH12Mode Sym Character MSXX Scaled HF MP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Error B2 A2 Torsion (Si-Si) Torsion (Si-Si) Si-Si-Si bend Torsion (SiH3) Torsion (SiH3) Si-Si-Si bend Si-Si-Si bend SiH2 rock SiH2 rock Si-Si Stretch twist-rock Si-Si stretch Si-Si stretch Si-Si stretch twist SiH3 rock SiH3 rock SiH2 rock twist wag twist wag twist wag SiH3 s-def SiH3 s-def SiH2 scissor SiH2 scissor SiH2 scissor SiH3 def SiH3 def SiH3 def SiH3 def SiH2 stretch SiH2 stretch SiH2 stretch SiH2 stretch SiH2 stretch SiH2 stretch SiH3 stretch SiH3 stretch SiH3 stretch SiH3 stretch SiH3 stretch SiH3 stretch 24 26 47 93 95 104 138 303 322 366 379 392 446 457 464 505 515 568 646 665 705 732 740 797 892 892 926 929 933 939 939 941 941 2112 2113 2122 2122 2123 2124 2142 2142 2142 2142 2144 2144 4.0 23 26 45 89 90 105 137 301 321 366 376 397 441 464 461 491 526 567 647 655 703 733 742 799 890 895 927 929 931 937 938 941 944 2117 2119 2120 2121 2125 2128 2141 2142 2142 2143 2144 2146 0 23 26 49 89 90 115 131 331 353 399 414 433 482 506 508 542 581 624 711 721 773 806 816 879 982 988 1019 1022 1024 1034 1034 1038 1041 2315 2318 2319 2320 2325 2328 2343 2343 2343 2344 2345 2348 116.7 52 43 56 111 112 112 124 300 329 383 391 414 463 483 487 510 550 596 674 678 736 768 770 830 931 938 972 973 978 993 993 997 999 2285 2287 2290 2294 2299 2304 2305 2306 2319 2319 2319 2320 92.8 A1 A2 B2 B1 A1 B2 A2 A1 B2 B1 A1 A2 B1 B1 A1 B2 A2 B1 B2 A2 A1 B1 B1 A1 A1 B1 A1 A2 B2 B1 A1 B2 A2 B1 A1 B2 A1 B1 A1 B1 A1 A2 B2 2187.0 2191.0 2187.0 2191.0 Al T2 3 4 Reference 15. 6-31G** basis. a b rms error 0.0 975.0 975.0 E 2 914.0 914.0 T2 1 Char Expera Sym Mode HBFF Table 6. Vibrational frequencies ( cm- 1 ) for SiH4 • 140.6 2360.6 2370.0 1053.4 1017.1 HFb 116.0 2349.9 2339.3 1017.6 973.7 MP2b 25.5 2154.2 2151.6 970.7 883.1 MSXX 0.928 0.923 0.926 1598 1563 685 674 10.7 1585.8 1546.4 689.5 669.9 SiD4 SiD4 Exper/HF 0.899 MCXX Expera Scale Iv I-' c,., Re Dn 6 Re Dnb I<n 396.3 1.476 92.6 SiH4 1 Be 1(3 1 Foo 1(3 Foo -0.178 -0.673 -0.333 -1.259 0.940 -14.749 50.91 (64.90) 117.2 (109.46) -9.17 -5.91) 4.05 4.20) 39.30 43.02) 120.78 110.62) -8.57 -2.69j -9.23 -7.02 42.53 122.55 -6.19 15.21 55.96 113.1 -10.16 2.39 42.25 115.14 16.14 -10.67 0.086 (-1.203) -1.51 -1.413 ~0.311) -3. 00 -0.233 0.806 -2.33 0.940 -15.062 399.4 ~381.8j 1.468 1.477 92.6 256.7 2.344 73.7 Si3Hf 399.8 1.471 92.6 276.7 2.369 73.7 Si2H6 -0.349 -0.55 (-0.57) -1.32 -1.56 -4.775 0.328 4.60 6.32 -1.02 -13.2 0.806 r-806) -7.35 -9.95) 0.940 0.940) -15.468 -15.288) 5.86 3.54 0.46 -9.28 0.806 r-806) -7.93 -9.36) 0.940 0.940) -13.401 -13.401) -0.67 (-0.46, -0.51) -1.621 (-1.583) -1.108 (-1.184, -2.204) 50.4 ?9.2) 118.1 111.6) 2.49 (-12.0) 4.43 (3.51) 35.4 (43.1, 42.2) 123.3 (114.4, 114.0) -6.05 (-5.67, 5.74) -2.04 (-7.47, -13.94) 26.1 ~75.9) 126.0 114.6) -22.9, -11.1, (-18.6) 0.515 (0.518) 377.7 ?85.7~ 1.479 1.474 92.6 267.9 ~253.5~ 2.321 2.363 73.7 n - Si5H12 56.39 113.6 113.4 -11.2 -1.46) 3.34 4.46) 45.0 (33.3, 41.6) 117.2 (118.5, 115.8) -5.34 ~-6.41, -4.16) -10.54 -10.64, -2.68) 35.4 126.6 -15.6 (-18.7) 0.635 r6.05l 395.8 ?83.1 ~ 1.468 1.476 92.6 278.8 ?61.1~ 2.330 2.328 73.7 n - Si4H10a avalues in parenthesis refer to terms involving the central hydrogens. 6 Not optimized. ccentral and terminal groups described using the same parameters. 1 Center angle-angle see equation (24) Si Si-H-H Goo Si Si-Si-H Goo' Si H-Si-H Go0' Si H-Si-Si Goo 5.717 Si H-H-H Goo H-Si-Si-H H-Si-Si-Si Foo 1(3 K2 Dro Krr ~or~i~ns_ see equations (17) and (23) S1-S1-S1-S1 R.1 Si-Si-Si Dno (SiSi) Dn 0 (SiH) Ko Be 73.7 Angles see equations (18), (20), and (21) 68.35 H-Si-H Ko 110.4 Be -1.90 Dno 4.30 !(RR' Si-Si-H Ko Si-Si Bonds see equation (17) Si-H Kn Valence Terms Table 7a. Parameters for the MSXX FF of various polysilane oligomers. t'-' 1--' fl:,,. 125 Table 7b. The MSXX FF for polysilane polymers. The subscript on the Si indicates the number of H atoms attached. All quantities are in (kcal/mol), A, radian units except for 0e which is in degrees. Bonds H-Si4 H-Si3 H-Si2 Si3-Si3 Si3-Si2 Si2-Si2 Angles H-Si4-H H-Si3-H H-Si2-H Sh-Si3-H Si2-Si3-H Si3-Si2-H Si2-Si2-H Si3-Si2-Si3 Si2-Si2-Sh Si2-Si2-Si2 Torsions H-Si-Si-H H-Si-Si-Si Si-Si-Si-Si 1-Center Angle Angle Si4 H-H-H Si3 H-H-H Si3 Si-H-H Si2 Si-H-H Si2 Si-Si-H Re Kr Dr 1.476 1.478 1.476 2.369 2.330 2.328 396.3 395.8 383.1 276.7 278.8 261.1 92.6 92.6 92.6 73.7 73.7 73.7 0e K0 Krr DR10 DR20 110.4 113.6 113.4 115.14 117.2 118.5 115.8 122.55 126.6 68.35 56.39 56.05 42.25 45.0 33.3 41.6 42.53 35.4 4.30 3.34 4.46 15.21 0.635 -1.90 -11.20 -1.46 16.14 -5.34 -6.41 -4.16 -6.19 -15.6 -1.90 -11.20 -1.46 -10.67 -10.54 -10.64 -2.68 -6.19 -18.7 Ko 0.940 0.806 2.785 K1 K2 K3 0.940 0.806 0.46 F00 G00 5.717 -0.349 -1.108 -4.775 -1.6 5.86 3.54 -13.4 -9.36 -9.28 126 Table 7c. Van der Waals parameters (reference 7) used for all MSXX force fields, Evdw = Dv (p- 12 - 2p- 6 ) where p = R/ Rv. The off-diagonal parameters Dv and Rv (Si · · · H) are obtained from the diagonal parameters by using the geometric mean. vdW parameters Si H Rv (A) 4.270 3.195 Dv (kcal/mol) 0.310 0.0152 A1u Eu A1 9 Eg A2u A1 9 Eu Eg A2u A1 9 Eg Eu 1 2 3 4 5 6 7 8 9 10 11 12 rmse HFa 134.3 416.6 464.0 697.1 947.9 1031.5 1043.1 1027.5 2337.1 2352.7 2337.3 2346.7 126.6 Char torsion SiH3 rock Si-Si Str SiH3 rock SiH3 def SiH3 def SiH3 def SiH3 def SiH3 Str SiH3 Str SiH3 Str SiH3 Str 0.931 0.910 0.936 0.897 0.890 0.881 0.891 0.914 0.880 0.915 0.922 0.928 Scale9 143.6 385.7 458.9 666.8 900.5 980.1 984.7 1001.3 2309.7 2319.5 2323.3 2331.7 101.5 MP2 125.0 379.3 434.2 625.2 843.5 909.0 929.3 939.6 2145.3 2152.0 2155.6 2178.6 g Vexper I VHF· . 133.2 379.7 435.2 625.2 844.9 907.0 931.5 937.1 2152.4 2154.9 2166.2 2168.2 4.8 HBFF Experb . Si2H6 a 6-31G** basis b Reference 16. c Reference 5. d Reference 17. e Excluding the torsional frequency, a1u, which was not fit. f Using the HBFF for Si2H6. Sym Mode Table 8. Vibrational frequencies (cm- 1 ) for Si2H6. 400 418 644 856 922 964 952 2145 2160 2154 2164 16.5 Cui et al.c 133.3 377.9 418.9 630.9 870.1 872.9 939.9 942.4 2137.5 2139.7 2137.4 2139.0 14 MSXX 94.2 271.3 379.2 475.4 622.1 730.7 668.6 668.9 1531.0 1534.8 1564.1 1566.0 19.5 Si2D6 HBFFf 277.0 407.5 474.7 624.9 683.1 667.0 682.9 1549.3 1547.5 1569.0 1585.0 Experd Si2D6 ~ ..... t..:> A2 B2 Al B2 Al A2 Bl Bl Al B2 A2 Bl Bl Al Al A2 B2 Bl Al B2 Al A2 Bl B2 Al Bl Al l 2 3 4 5 6 7 torsion torsion Si-Si-Si bendc SiH2 rock Si- Si twist Si -Si wag SiH3 rock SiH3 rock SiH2 twist SiH2 wag SiH3 s-def SiH3 s-def SiH2 scissor SiH3 d-def SiH3 d-def SiH3 d-def SiH3 d-def SiH2 str SiH3 str SiH3 str SiHa str SiH3 str SiHa str SiH3 str SiH3 str Char 77.3 99.2 109.3 353.0 411.3 464.l 485.3 502.l 625.0 657.4 777.8 810.7 983.0 989.9 1025.9 1032.3 1036.2 1034.6 1044.8 2321.3 2320.0 2338.3 2339.7 2345.0 2344.4 2343.5 2350.1 122.5 HFe 0.915 0.916 0.918 0.903 0.921 0.932 0.904 0.888 0.905 0.883 0.891 0.953 1.025 Scale 107.6 108.0 120.0 291.9 395.8 430.9 458.5 477.2 596.4 624.5 730.l 751.9 931.8 939.2 973.5 991.9 993.6 995.6 1001.4 2302.3 2305.9 2315.4 2317.7 2318.9 2319.7 2340.7 2352.7 102.9 MP2e 89.l 94.7 113.4 321.4 394.l 422.8 447.9 476.l 545.6 600.0 703.5 714.7 883.6 887.2 929.1 935.9 936.2 937.7 940.1 2127.8 2130.3 2145.5 2145.8 2146.2 2146.6 2149.2 2149.6 6.0d (7.7) f~~l !93 937 3 935 944 . (2128) 2130b ~2143~ 2144 2148 6 -~2149~ 2148 2150 l 112 [317] 392 6 (420) 447b 468b 565b 584 6 704 6 716b. 876b (895) 926 6 HBFF Expera 107 337 371 476 445 446 582 606 733 740 881 897 940 954 959 959 965 2145 2137 2149 2156 2158 2159 2163 2164 17 Cui FF et al. 5 88.7 95.2 109.9 324.6 375.9 421.2 447.2 470.7 539.0 603.0 693.4 748.9 859.2 862.3 927.2 939.5 939.9 940.9 943.1 2127.3 2128.3 2144.5 2144.3 2145.3 2145.3 2146.6 2147.1 12.65 MSXX 63.4 69.9 103.9 232.3 303.l 344.l 354.2 400.5 430.7 465.1 510.7 567.5 658.5 657.9 674.3 670.5 670.9 671.8 672.7 1537.4 1524.6 1549.6 1527.7 1550.1 1529.l 1551.5 1551.9 HBFF Si3Ds 89.l 94.3 112.6 253.7 387.9 357.5 389.9 460.6 544.9 577.4 591.8 612.9 883.6 672.1 887.2 935.6 936.l 937.6 939.8 1537.2 1525.2 2145.5 2145.8 2146.1 2146.6 2149.l 2149.5 HBFF Si3D2H6 a The values in parenthesis and brackets are not available from experiment and were scaled from HF. The values in parentheses used scaling from corresponding modes of Si3H6 , the values in brackets (torsions) used the scale for Si2H6. 6 Reference 18. · c Accordion mode. d Excludint the two torsional frequencies fit using the. full PES. e 6-31G** asis. 20 21 22 23 24 25 26 27 rms 19 12 13 14 15 16 17 18 11 9 10 8 Sym Mode Table 9. Vibrational frequencies (cm- 1 ) for Si 3Hs. l:v 00 ..... J Bg Ag Bu Ag Ag Bu Bg Au Bu Ag Bg Au Bu Ag Bg Ag Bu Ag Bu Ag 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 rms Torsion ~i-Si) Si-Si-Si end Torsion (SiHa) Torsion lSiH3) Si-Si-Si end SiH2 rock twist-rock Si-Si stretch Si-Si strett tj.st-roc S1- 1 stretc wag SiHa rock SiH2 rock twist wag twist wag SiHa s-def SiHa s-def SiH2 scissor SiH2 scissor SiHa d-def SiHa d-def SiHa d-def SiHa d-def SiH2 str SiH2 str SiH2 str SiH2 str SiHa str SiHa str SiH 3 str SiHa str SiHa str SiHa str Char 9Y~:3 721.0 492.4 527.5 581.6 ui~ 23.6 73.1 97.6 102.7 141.7 307.6 355.5 357.1 MSXX-R 98g:l 743.8 527.8 577.8 1ij8:i 493.1 32.1 73.8 92.5 97.8 ~31.5 307.8 350.2 358.7 MSXX g3~~ 744c 542c 577.Ob 1~~~ 473c 24.8° 73.2: 80.3 88.0b 139c 306.4b 349.4b 361c Exper 24.8 73.2 86.3 94.6 132.8 337.0 384.3 394.6 HFe 0.915 1.047 Scale 47.5 77.2 110.0 115.5 126.2 310.9 362.0 379.6 MP2 753 780 883 891 941 943 956 957 963 961 2142 2148 2136 2139 2159 2158 2152 2158 2161 2160 16d g~i 498 556 587 0.909 0.910 495.3 566.3 608.9 fgl~ 8:~i1 f83:f 798.7 0.932 761.5 525.8 595.9 636.5 n1 t~ti ij:BU t~i:i 130 320 368 357 Cui et al.I 784.lb 782.5 782.1 862.5 815.6 374c 865.6 871.4 981.1 0.891 931.4 869c 868.4 874.1 988.9 0.879 939.4 917c 924.1 924.9 1024.0 0.896 975.1 933c 924.4 927.3 1021.8 0.913 975.8 937.6b 940.5 939.8 1034.1 993.5 938.Ob 941.2 940.5 1034.7 993.7 941.7b 941.7 941.5 1040.6 997.1 943.4b 942.4 942.2 1038.7 998.6 2119.5b 2123.7 2120.5 2316.8 2286.4 2125.7b 2125.2 2121.0 2317.6 2287.5 212O.6b 2125.8 2121.6 2318.0 2295.2 212Oc 2126.4 2123.5 2321.5 0.913 2301.6 2143c 2139.3 2141.7 2341.8 0.915 2303.8 2144c 2139.5 2141.9 2345.0 0.914 2306.8 2143.6b 2139.5 2143.4 2318.3 2343.2 2142.9b 2139.7 2144.8 2342.4 2320.4 2144.lb 2140.9 2143.6 2343.8 2319.6 2148.5b 2148.2 2151.5 2348.5 2319.0 0.0 7.3 (7.1) 6.2 (6.0) 119.4 95.3 a Ab initio frequencies. bAb initio frequencies scaled by n-Si 4H 10 frequencies. cReference 16. dExcluding the torsional frequencies (not fitted). e6-31G** basis. !Reference 5. 9 The accordion mode. ~~ B! Ag Bg rn 1~ 12 13 14 Au Bu Au Bg Ag Au Bg Ag 1 2 3 4 5 6 7 8 Jq Sym Mode Table 10, Vibrational frequencies (cm- 1 for n-butylsilane, n-Si4H10- Exper includes the best estimate of experiment using scaled HF results. The MSXX force eld distinguishes between the SiH2 and SiH3 centers. The MSXX-R FF does not distinguish. c.o ...... N> 130 Table 11. Predicted structure of n-Si 5 H 12 • Distances are in A, angles are in degrees . a MSXX HFa .6. Sic - Bice 2.365 2.361 0.004 Sic - Si 2.355 2.357 0.002 He - Sic 1.480 1.482 0.002 Hee - Bice 1.479 1.482 0.003 Hip - Si 1.478 1.479 0.001 Hop - Si 1.478 1.479 0.001 Si - Sic - Bice 112.1 112.69 0.59 Sic - Bice - Sic 114.16 113.03 1.13 Hip - Si - Sic 110.06 110.63 0.57 Hop - Si - Sic 110.04 110.14 0.10 Hee - Bice - Hee 106.63 107.25 0.62 He - Sic - Bice 110.06 108.93 1.13 Hop - Si - Sic - Bice 59.95 59.81 0.14 6-31G** basis. 131 Table 12. Properties of P(SiH) at OK, 77K, and 300K from Gibbs dynamics for a 2x4x4 supercell (containing 384 atoms). Property OK 77K 300K a 8.422 8.526 8.833 b 3.966 3.9769 3.955 C 4.685 4.733 4.929 13.095 10.412 4.594 Yx 11.94 9.586 4.720 Yy 110.57 107.64 98.384 Yz 18.64 14.750 5.789 Cn 15.60 12.424 5.937 C22 C33 C12 C13 121.30 116.501 102.85 22.97 18.172 7.088 12.73 10.335 5.045 8.17 6.486 2.765 C23 C44 9.24 7.374 3.369 16.23 13.995 8.322 Css 7.34 5.936 2.876 c66 19.30 16.299 8.906 unitcell (A)a Bulk Modulus (GPa) B Young's Moduli (GPa) Elastic Constants (GPa) a Structure was constrained to remain orthorhombic during the dynamics. 132 Table 13. Predicted cohesive energy (kcal/mol SiH2) for polysilane crystal. The cohe- sive energy at OK of 3.82 kcal/mol SiH2 compares with 1.87 kcal/mol CH2 for polyethylene crystal. Total Energy Zero-point Energya at OK At minimum a Lattice Enthalpy Isolated Chain 3.224 8.568 11.792 Crystal (2 chain) -1.055 9.031 7.976 Cohesive Energy (2 chain) 4.279 -0.463 3.816 Crystal (1 chain) -1.231 9.053 7.822 Cohesive Energy (1 chain) 4.455 -0.485 3.970 Using 5 x 5 x 5 points in the Brillouin zone. 133 Table 14. Vibrational frequencies (cm- 1 ) for polysilane crystal P(SiH) with two chains per unit cell. Only the values for k = 0 (r point) are shown. The chain direction is [010]. Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Description interchain interchain interchain skeletal torsion skeletal torsion SiH2 rock SiH2 rock Si-Si-Si bend Si-Si-Si bend Si-Si stretch Si-Si stretch SiH2 rock SiH2 twist SiH2 rock SiH2 twist SiH2 wag SiH2 wag SiH2 twist SiH2 wag SiH2 twist SiH2 scissor SiH2 scissor SiH2 wag SiH2 scissor SiH2 scissor SiH stretch SiH stretch SiH stretch SiH stretch SiH stretch SiH stretch SiH stretch SiH stretch Polarization Direction [010] [100] [001] 63.6 95.7 98.1 152.2 160.9 320.9 357.7 398.7 403.2 509.5 532.3 573.5 573.5 583.7 648.5 702.1 771.7 840.6 871.5 927.5 948.2 950.6 954.5 954.7 964.7 2150.2 2151.8 2152.2 2153.1 2154.5 2155.3 2161.3 2164.3 63.6 95.7 98.1 152.2 160.9 320.9 348.9 398.7 403.2 509.5 532.3 573.5 573.5 583.7 649.0 742.5 771.7 840.6 871.5 927.5 948.2 950.6 940.1 954.7 964.7 2150.2 2151.6 2151.8 2153.1 2154.5 2155.3 2161.3 2159.0 63.6 96.3 98.1 152.2 160.9 343.8 348.9 398.7 403.2 509.5 532.3 573.5 573.5 583.7 648.5 702.1 771.7 840.6 871.5 927.5 953.8 950.6 940.1 954.7 964.7 2150.2 2151.6 2153.9 2153.1 2160.1 2155.3 2161.3 2159.0 134 Table 15. Vibrational frequencies (cm- 1 ) of P(SiH). The first two columns are for a single infinite chain, while last two columns are for the crystal, with 1 chain/unit cell. Isolated Chain Crystal (single chain) a Mode Sym Description Cui et al.a MSXX MSXX 1 Au SiH2 rock 311 307.7 367.9 2 Ag Si-Si-Si bend 413 396.8 407.2 3 Ag Si-Si stretch 469 493.1 526.7 4 Au SiH2 twist 549 556.8 613.0 5 Bg SiH 2 rock 532 562.5 585.1 6 Bu SiH 2 wag 604 685.0 692.9 7 Bg SiH 2 twist 769 795.2 842.4 8 Ag SiH2 wag 808 852.1 929.8 9 Ag Scissor 943 936.4 949.4 909 10 Bu Scissor 953 935.6 974.7 905 11 Bg SiH stretch 2140 2128.3 2146.6 2155 12 Bu SiH stretch 2139 2128.5 2151.0 2100 13 Au SiH stretch 2158 2132.6 2153.6 2100 14 Ag SiH stretch 2150 2130.3 2156.5 2115 Reference 5. Exper 480 135 Table 16. Packing of P(SiH) and Polyethylene. R is the C-H (1.096A) or Si-H (1.48A) bond distance and R is the C-C (1.553A) or Si-Si (2.35A) bond distance for PE and P (SiH), respectively. Lattice constant P(SiH) PE a 8.422 7.121 b (chain) 3.966 2.546 C 4.685 4.851 a/R 5.69 6.50 b/R 1.69 1.64 c/R 3.17 4.43 unitcena 136 Figure Captions Figure 1. Torsional potential (kcal/mol) of Si 2 H 6 from HF calculations and from the MSXX FF. All bonds and angles are optimized for each ¢. The total EH F ( ¢) and EF F ( ¢) is plotted. Figure 2. Torsional potential (kcal/mol) of Si 3 H 8 from (a) HF calculations and (b) the MSXX FF. Each is plotted versus the two dihedral angles ¢1 = H -Si- Si2 - Si 3 and ¢ 2 = Si - Si 2 - Si 3 - H (abscissa and respective ordinate). All other bonds and angles are optimized for each ¢1 and ¢2 Figure 3. Torsional potential (kcal/mol) for the central Si- Si bond of H 3 Si - SiH2 - SiH2 - SiH3 from HF and HBFF. All other bonds and angles are optimized for each¢. Figures 4a-b. The calculated phonon modes (cm- 1) of crystalline P(SiH) for the (010) and [001) directions (the chain direction is (010)). Only the modes below 1000 cm- 1 are shown. Figure 5. (a) The calculated stress-strain curves for crystalline P(SiH) in directions perpendicular to the chain axis. (b) The strain energy (kcal/mol) as a function of strain. Figure 6. The thermochemical properties as a function of temperature (using a 5 by 5 by 5 set of phonons in the Brillouin zone). (a) The heat capacity (Cv). (b) The Helmholtz free energy. (c) The internal energy. (d) The entropy. 137 Figure 1 1 ,/' 0.9 / ! / 0.8 .., / / I / ,! ; I 0.7 --E 0 ftS y ~ >,, ~ 0.6 MSXX 0.5 /I ti 0.4 HF Q> = ~ 0.3 0.2 0.1 0 0 30 60 90 0 (degrees) 120 150 180 138 Figure 2a HF 0 10 20 30 40 so 60 70 80 01 90 100 110 120 130 140 150 160 170 180 0 0 ,-t 0 N 0 C"') 0 -=:I' 0 10 0 \0 0 0 0 R 00 °' 0 ,-t 02 0 ,-t ,-t 0 N ,-t 0 0 ,-t ,-t C"') s,:j< 0 10 ,-t 0 \0 ,-t 0 R 00 ,-t ,-t 139 Figure 2b MSXX 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 0 0 '""' 0 N 0 Cf) 0 s::t< 0 II) 0 \Cl 0 t--. 0 00 0 °' 140 Figure 3 1.6 1.4 -;§ -£o 0 ftS t.J 1.2 1 ~ HF 0.8 QJ c ~ 0.6 0.4 0.2 0 0 20 40 · 60 80 100 0 (degrees) 120 140 160 180 141 Figure 4a 1000 sciss { wag twist 900 wag twist 800 wag wag -e 700 l""I I {,J twist :>.. {,J c:: (IJ = 0"' (IJ 1-1 j;.I.c Si-Si stretch { 500 300 200 R (IJ ............ 100 {,J ns ..,;i 0 0 r 0.1 0.4 0.5 Wavevector in units of [Ore/a O] X 0.2 0.3 142 Figure 4b 130-------------.--------------. 160t============r::R Ag 140 -e - 120 ,-I I V >u-. 100 i::: (lJ ::i O"' (lJ "'" j;.r.i 80 60 R+ 40 20 0-+-----.----.---......---...---~-----,---..---.....---------1 0 -0.5 0.5 Wavevector in units of [O n/2 O] 143 Figure Sa 1----------.-----------------------, 0.5 \ 0 - . ... res rJ-4 c., -0.5 Cl.) Cl.) Q.I 1,.1 en -1 -1.5 -2 -2.5--+-i-.-,-,-,-.-,-.-,-+-,-.-,...,...,.__,....-,,-,-,...,...,.-,-,...,...,...,...,...,...,...,...,.....,.....,.............,......,..............,.....,....,..,1 -10 -5 0 5 10 15 Strain(%) 20 25 30 35 40 144 Figure 5b 1.8 -- --- -- Strain a (100) -..E 0 ~ y ~ >.. ~ 1.6 1.4 1.2 1 GJ = 0.8 = i;,r,J - •..-I (-:1 161 en 0.6 0.4 0.2 0 -0.2 -10 -5 0 5 10 15 % Strain 20 25 30 35 40 145 Figure 6a 12 -] -u ~ I 0 tel V 10 8 6 ?- 4 2 0 0 50 100 150 200 250 300 350 400 T(K) 146 Figure 6b -0.5 -e 0 -1 ::::::i ft! ~ ~ -1.5 ~ -2 -2.5-1-r-.,..,...,...,..,..,...,......,...,...,..,...,...,..,.......,...,...,..,....,..,...,...,...,...,..,...,...,..,......,...,...,..,...,...,..,-11 0 50 100 150 200 250 300 350 400 T (K) 147 Figure 6c - 2 0 ~ 1.5 fl:! l;J ~ ::> 1 0.5 0-4-,..,..,...i:::;;..........................................._ _ _ _ _ _.,....,l 0 50 100 150 200 250 300 350 400 T (K) 148 Figure 6d 12 10 -;:§ 8 0 f'CS 6 V Cf.) 4 2 0 0 50 100 150 200 250 300 350 400 T (K) 149 Chapter 5 Force Fields for Semiconductors and Their Superlattices 150 Abstract We develop the MSXX force field (FF) for molecular dynamics (MD) simulations of the group IV diamond materials, and group II/VI and III/V zinc-blende semiconductors. This was developed to fit the structure, elastic constants, and the phonons of the crystal. The MSXX force field for zinc-blende materials contains only 6 adjustable force constants, 2 geometric parameters and an atomic partial charge, yet it accurately describes the experimental crystal structure, elastic constants, and phonon dispersion curves, and is suitable for predictions of the strained structures at heterojunctions and for MD. The MSXX FF is used to calculate the interface phonons of various superlattices and the Ge/Si ordered alloy where we find excellent agreement with experiment. 151 1.0 Introduction Ab initio quantum mechanics can provide an accurate description of the structures, mechanical properties, and electrical properties of semiconductor materials. Unfortunately, despite recent advances in first principles methods for large systems 1 and for periodic systems, 2 such simulations are impractical for many important applications to semiconductor materials. Consequently, we have developed a consistent set of valence force fields that should be useful for molecular dynamics simulations of semiconductors. This MSXX (FF) is derived using empirical data on lattice constants, elastic constants, and phonon states. Herein we report the MSXX FF for the nine III/V systems with III= Al, Ga, In and V = P, As, Sb, the five II/VI systems with II = Zn, Cd, Hg, and for VI = S, Se, Te, and the group IV systems, diamond, silicon and germanium. A prime motivation in developing the MSXX FF is to describe the strain effects and vibrational modes at heterojunction interfaces and superlattices. We illustrate the approach by predicting the vibrational interface states in several superlattices. These results show that the MSXX FF provides an accurate description of the strains and interface phonons. The form of the MSXX FF is described in Section 2 and the optimization of the parameters is described in Section 3. Comparison to experimental phonon dispersion data is made in Section 4. The calculations of the superlattice phonon dispersion curves are given in Section 5. 2.0 The MSXX Force Field Chapter 1 describes the general form of the force field. Here we describe the MSXX force field as specifically applied to semiconductor materials. The general form of the force field is taken as 3 (1) 152 where E Q - C ~ QiQj coul L__ R .. i>j (2) iJ represents the Coulombic interactions between partial charges on the various atoms ( Ccoul = 332.0637 ensures that Eis in kcal/mol where R in A), Evdw = L Eijdw ( Rij) (3) i>j represents the long-range attraction (London dispersion) and short-range repulsion (Pauli orthogonalization of nonbonded electrons) and (4) represents all terms involving bonds between atoms and coupling behavior of these bonds. This type of FF has been used to describe polymers (polyethylene, 4 polyvinylidene £1.uoride, 5 polysilane, 6 nylon 7 ), Si 3 N 4 ceramics, 8 and many other systems. It is denoted as MSXX to indicate that it is for materials simulations and that it includes both 1 center and 2 center cross terms. 2.1 Bond Terms We take Eb 0nd as a sum over all bond pairs where each has the form of a Morse function, 3 (5) with (6) and a=~- (7) This includes anharmonicity and allows a proper description of bond dissociation. We choose the Morse form (5) over the more common harmonic description (8) 153 in order to better describe anharmonic effects, e.g. thermal expansion. Equation (5) contains three independent parameters Re, kR, and DR, The structure, elastic constant, and phonons are sensitive to Re and kR but not to DR. Consequently we choose DR based on the experimental atomization energies. 9 2.2 Angle Terms We take Bangle as a sum over all angles 1-J-K for each atom J here each angle term is described with the cosine angle form, 3 . C 2 Ecosine(0) = - [cos0 - cos0e] 2 (9) where the force constant is ke = Csin 2 0e = ( ~:!) ee. This form leads correctly to dE / d0 = 0 at 0 = 0 and 180° with a barrier (at 180° )of Ebarrier = C [1 + cos0e]2 . 2 One might restrict 0e to be 0e = 109.471 °, the tetrahedral value, since this is the optimum geometry in these crystals. However we have optimized both 0e and k 0 • 2.3 Cross Terms We find, generally, that bond-bond cross-terms (11) sharing an apex atom (e.g. I J and J K) are required to describe the coupling of equivalent bonds. 4 Since bonds sharing a common apex are identical for the zinc- blende structures, we include such terms. For two bonds I - J and J - K sharing a common atom, there are two bond-angle cross terms of the form 3 (12) 154 We find these to be necessary for III-V systems. If three (or more) bonds share a common atom (say I - J, K - J, L - J), we find that the one center angle-angle cross terms (e.g coupling of the I J K and K J L angles ) 3 (13) k001 = F00 sin 0i sin 02 are not necessary to describe the structure, elastic constants, and phonon spectra of III/V semiconductors~ On the other hand we do find two-center angle-angle coupling to be important. Thus for three sequential bonds I - J, J - K, and K - L, the coupling of the IJK angle (01) with the JKL angle (02 ) 3 •10 is important when the I J and KL bonds are trans (dihedral angle, ¢ = 180°) but not when they are gauche (¢ = 60°,300°). Consequently we define this coupling term as (14) where 1 2 3 3 !(</>) = - - - cos</> and k002 = G00 sin 0i sin 02 which leads to /(180°) = +1 and /(60°) = /(300°) = 0. McMurry 10 first introduced such a coupling term to describe the TA mode softening for k approaching the Brillouin zone boundary in Si. Such terms are generally required to describe the vibrations of long chain molecules (such as polysilane 6 and polyethylene 4 ) that 155 feature a trans zig-zag chain. Similarly they are needed to describe the stiffness along the < 110 > chain directions of the diamond and zinc-blende materials. Summarizing the cross terms are taken as (15) 2 .4 Torsion Terms Generally one would include terms of the form 3 (16) for describing the torsional barrier in a tetrahedral system. Such barriers describe the observed preference for staggered dihedral angles and lead to chair-like six membered rings rather than the boat conformation. Thus such interactions would prefer the sphalerite (cubic) or zinc-blende structure over wurzite (where 1/4 of the bonds have eclipsed ligands). However, we find that the structure, elastic constants, and phonon dispersion are not sensitive to vtor and hence do not include it here. We do find electrostatic effects lead to a slight (0.1 kcal/mol) disfavoring of the Wurzite structure with respect to zinc-blende (see below). 2.5 Electrostatics A valence force field without electrostatic terms leads to degenerate LO and TO modes at the r point, thus the group IV materials have degenerate optical r points. The observed splitting in these levels for the compound semiconductors results from the macroscopic electric field arising from the macroscopic displacement of charge in the LO mode as k ---+ 0. To describe this splitting requires the electrostatic term in (2). Indeed, fitting the LO-TO splitting leads to unique charges in the range expected from the electronegativity differences of the III/V atoms. [We use the Accuracy Bonded Convergence Acceleration (ABCA) 11 procedure to sum the Coulombic interactions.] 156 2.6 Van der Waals The energetics of small displacements in a tetrahedral crystal do not require explicit van der Waals interactions. However we intend to use these potentials for materials with surfaces, dislocations, vacancies, etc. where there may be other atoms or molecules. Consequently we want to include the van der Waals interactions required to describe interactions with nonbonded molecules. To do this we use the parameters from the Universal Force Field, 12 which includes vdw parameters for all elements up through Lr (element 103). 3.0 Force Field Optimization We optimized the MSXX force field by minimizing the error between the calculated and experimental properties (lattice parameter, elastic constants, and phonon special points) of the crystal with respect to variations in the force field parameters. 3.1 The Error Function and Weights The error function is 13 3N-6 Serr= Wjorce L (oE:) i=l 3N-6 2 +WJreq L (ovi) i=l 6 2 +Wstress L(O~i) 2 +Welas i=l 6 L (<5Cij) i<=j=l (16) where o denotes the difference between the calculated quantities from the force field and from experiment. Here N is the number of atoms, EI is the gradient of the energy (the force), vi are the frequencies of the phonons (r and X point only), ~i are the stresses, and Cij are the elastic constants (only Cu, C 12, C44 are unique for cubic crystals). We use the HB-SVD method 13 •14 to minimize the error, Serr• HB-SVD also handles redundancy in force field parameters so that the parameters are changed only when they significantly improve the fit. We choose the weights to ensure that the forces on the atoms and the stresses on the crystal are zero. These weights were selected with three primary goals: 2 157 1. We want to reproduce the crystal geometry and fit the experimental phonon frequencies and elastic constants as closely as possible. There is generally a trade off between these two errors. We choose the weights such that neither is too large, but with more emphasis on the phonon frequencies. The reason is that the low frequency properties associated with the elastic constants have less bearing on the localized strains surrounding defects and surfaces. 2. We want to ensure the intuitive nature of the valence force field by having physically meaningful values for the force constants. This led us to include only the significant cross terms, as discussed above. 3. We want the resulting force constants to show regular behavior as we move across and down the periodic table. This is so that reliable predictions can be made for mixed systems such as heterojunctions. The weights chosen for (1) and (2) lead to this regular behavior. 3.2 Phonon Dispersion After fitting the force field, we used it to predict the complete phonon dispersion curves along the < 100 > and < 111 > directions. Expanding the energy about equilibrium leads to (17) with The force at equilibrium is zero (18) and (19) is the Hessian. 158 The equations of motion become M 82[bRa1(t)] = - ~ H I ~ f)t2 al,f]J [bR j]J (t)] j]J where the subscripts J and J run from 1 to the number of atoms N, and the subscripts a and /3 run though the x, y, z components of the atomic coordinates. The time periodic eigenstates have the form where w 2 M1(bR~ 1) = LHaI,f]J (bRiJ) eik(rr31 -r r). 0 (20) j]J Taking k in < 100 > and < 111 > directions and solving the equation (20) gives the corresponding frequencies. 3.3 Charge Equilibration For tetrahedral crystals, the splitting of the LO and TO modes at the r point is quite sensitive to the charge difference between the cation and anion. This arises from the macroscopic dipole for the LO mode near k = 0. (With no charge difference these modes are degenerate.) We develop a electronegativity scale based on the splitting of the r points. The resulting charges represent the electronegativity difference in the elements. Thus qvI = 0.88, 0.80, 0.76 for ZnS, ZnSe, ZnTe indicates that the electronegativities are in the sequence S =Se> Te. The qn = 0.76, 0.86, 0.74 for ZnTe, CdTe, and HeTe indicate that the electronegativities are in the sequence Hg > Zn > Cd. This is in rough agreement with the Pauling values as optimized by Allred 15 : 2.58, 2.55, 2.3 for S, Se, Te and 1.65, 1.69, 2.00 for Zn, Cd, Hg. For the more interesting cases of interfaces and defects, we expect charge readjustments but there is not sufficient experimental data to determine the modified charges. Thus we have used the charge equilibration ( QEq) procedures of 159 Rappe and Goddard 12 but with the readjusted atomic electronegativities to fit the MSXX FF. In QEq16 the energy of an atom A is assumed to be (21) where 1 XA = 2 (JP + EA) (22) is called the electronegativity and 1 -JAA =IP-EA 2 (23) is called the hardness (or idempotency). Then in a molecule or crystal the total energy is taken to have the form E = LEA (QA)+ A L QAQBJAB (RAB) (24) A=B where JAB(RAB) is the Coulomb potential between spherical charge distributions on A and B with radii RA and RB, respectively. For crystals the second term in (42) must be evaluated using the Ewald procedure. The QEq Ewald program was rewritten by Karasawa and Goddard 17 to calculate the ionic charges for periodic systems. For group II the standard QEq parameters 9 , 16 lead to the results in Table 1. The discrepancy with the MSXX values are probably due to the special nature of the group II atoms where the ground state s 2 is used to define XA and J AA but, the crystal involves sp3 hybridization. Consequently we have readjusted the Xa for group II to agree with MSXX. The results are shown in Table 1. With this readjustment of QEq parameters we can now predict charges at interfaces, impurities, etc. 160 4.0 Results 4.1 Group IV Materials The experimental phonon frequencies and elastic constants of the group IV materials are listed in Table 2 and were selected from reference 18 which provides all the elastic constants and special phonon points needed for fitting the FF for all the materials. We choose to fit to the phonon frequencies to the neutron scattering experiment for each case when available. For elastic constants we choose to fit to values that were in best agreement with other experimental techniques (this means we fit to data that was not extreme relative to other experiments). This was done to minimize the possibility of fitting to a piece of experimental data having a large error. The optimized parameters for the MSXX force fields are listed in Table 3, and the van der Waals parameters (not optimized) are listed in Table 4. The parameters change monotonically going from C, to Si, to Ge. Thus Kr, K0, and K00 2 decrease moving down the periodic table (since the bonds get weaker). The cross terms (except the 2 center angle-angle term) do not demonstrate this behavior. We can also see that the charges decrease as the bonds gets weaker. The predicted phonon dispersion curves are compared with experiment in Figures 1-3. We plot the FF phonon dispersion curves in the <100>, <110> and <111> directions and the neutron scattering data in these directions. Since the data used to fit the curves was the elastic constants and the special points at r and X the <100> direction demonstrates the accuracy of the fitting procedure for fitting to the entire branch while the <110> and <111> directions demonstrate the ability of the model to predict values which are not included within the fit. The discrepancy between the neutron scattering data and the theory is remarkably small. Overall the deviation from experiment is quite small. The ability of MSXX FF to reproduce phonon dispersion with such few parameters adds credibility to the fundamental appropriateness of the description underlying this model. This 161 suggests that this force field may be useful in predicting the strains and mechanical properties of these systems. 4.2 Group IH/V Materials The experimental phonon frequencies and elastic constants of the group III/V materials are listed in Table 5 and were selected from reference 18 which provides all the elastic constants and special phonon points needed for fitting the FF for all the materials except AlP for which the X points are unavailable. The optimized parameters for the MSXX force fields are listed in Table 6, and the van der Waals parameters (not optimized) are listed in Table 7. The parameters change monotonically going from AlP to AlAs to AlSb, from GaP to GaAs to GaSb and from InP to InAs and InSb. Thus Kr and Ke decrease moving down the periodic table (since the bonds get weaker). The cross terms do not demonstrate this behavior. We can also see that the charges decrease as the bonds gets weaker. The predicted phonon dispersion curves are compared with experiment in Figures 4-12. The best data is for GaAs. We plot the FF phonon dispersion curves in the <100>, <110> and <111> directions and the neutron scattering data in these directions. The discrepancy between the neutron scattering data and the theory is small. The largest differences occur along the TA branches in the <110> direction (which is also the branch for which the neutron data has large error bars. To extend the MSXX FF to the prediction of polarization properties, we will add a covalent shell description to the valence force field (as was done for PVDF 5 We could also improve the fits of MSXX by using additional points from the phonon dispersion curve in the set of constraints. However, the current fit is acceptable for all systems. 4.3 Group II/VI Materials The experimental phonon frequencies and elastic constants of the group II/VI materials are listed in Table 8 and were selected from reference 18. The optimized parameters for the MSXX force fields are listed in Table 9, and the van der Waals parameters (not optimized) are listed in Table 10. The predicted phonon dispersion 162 curves are compared with experiment in Figures 13-17. Again, as in the case of the group III/V materials the largest errors occur along the TA branches in the <110> direction. 5.0 Superlattices Phonon dispersion curves for superlattices have long been an active research area. Since Colvard et al. first observed the folded acoustic phonons, 19 much progress has been made both in theory and experiment. Theoretical work has included the elastic model and Fourier transform analysis which have been proposed to explain the folding of the acoustic branch, and the degeneracy at the Brillouin zone center and boundary. 20 The alternating linear chain model has been employed to explain the confined modes. 20 And a theory treating the dielectric modulation has been used to explain the interface modes. 20 All these models deal only with the dispersion curves along the axis direction. Complicated, non-intuitive, numerical models such as the Valence Overlap Shell model, etc., have been necessary for the full 3-D description of the superlattice phonons. Our approach offers an advantage in that it is much simpler and physically intuitive. Another complicated issue is the effect of strain on the superlattice phonons which is discussed by Jusserand and Cardona. 20 The acoustic phonon folding, zone edge and zone center splittings are observed experimentally as are the confined optical modes and interface modes. A general scheme to model the superlattice phonons is to utilize the models for bulk materials and adapt them to the superlattice. As pointed out in, 20 3 problems should be addressed: (1) The choice of proper lattice dynamical models for the bulk constituents. (2) The transferablity of the bulk model parameters to the superlattice case, which is reasonable and straight-forward for local quantities or very short range interactions, but which becomes more difficult for long range forces. (3) Numerical difficulties arising from the size of the secular equations in the case of thick layer superlattices. In this section we extend the molecular mechanics FF to calculate the phonon 163 dispersion curves for the superlattices. In the force fields, the local quantities such as bond lengths and angles, bond stretching and angle bending force constants, are all directly transferred, the long range van der Waals forces are also directly transferred. We used the electronegativity of the atoms to calculate the charge on the atoms to determine the Coulomb forces. However, some of the interface local parameters are unique to the interface and we use an averaging method to interpolate the interface potential. We justify the scheme in the following section on detailed calculations and with comparisons to experiment in section 5.3. The MSXX model is a 3-D model, capable of dealing with all crystallographic directions and interfaces, and with difficult problems as strain, etc. It is also readily to modified to simulate surface roughness. The parameters involved are few and so the calculation is relatively fast. 5.1 Calculation Details: Transfer and Extension of the Force Field We have developed MSXX force fields for the bulk materials elsewhere, 21 as described in sections 3 and 4 and showed that our force field reproduces the whole range of the dispersion curves well. We directly transfer the force fields to the superlattice after adding interface interactions interpolated from the bulk FF. There are two types of superlattices: one type is (AB)n 1 (AC)n 2 , and the other is (AB)n 1 (CD)n 2 • The first type has a common atom on the interface, while the second one doesn't. Here we only deal with the first type. The second type is very similar, and will be investigated in future work. For the first type, the superlattice consists of layers of AB and AC, e.g. (HgTe)n 1 (CdTe)n 2 . From the bulk FF we have the parameters involving only AB and AC. For the atoms on the interface, the force fields are different from both of the bulk materials as new three and four body ( and higher) interactions are involved. We derive the new interactions using a simple averaging technique. For the Angle B-A-C; We take the angle to be the geometric mean, i.e: 164 We also do this for the angle bending force constant: Ke,BAC = JK0,BAB * Ke,cAC· The other terms needed to describe the interface are the cross terms and these become more slightly more complicated. For the B-A-C angle we need two bond-cross-angle terms which we obtain from the related parameters in the original bulk material. However, the original bond-cross-angle terms are symmetric for angle B-A-B and C-A-C, so the obvious way to extend it to the B-A-C is to use them directly: Kre BAG = Kre ' BAB Kre CAB = Kre 'GAG ' ' and for the bond-bond cross-term, we can also take the geometric mean: Krr,BAC = JKrr,BAB * Krr,CAC• In developing the MSXX FF for the II/VI and III/V systems we did not use torsion terms and likewise we don't include them here. However, we did use the two-center angle-cross-angle terms, for the superlattices we obtain from the twocenter angle-cross-angle terms in the chain B-A-C-A and C-A-B-A of the component materials taking the geometric mean of the two-center angle-cross-angle terms BA-B-A and C-A-C-A : Kangang,BACA = JKangang,BABA * Kangang,CACA Kangang,CABA = JKangang,BABA * Kangang,CACA· With these interpolations we get all the force field parameters we need, (Table 11 shows the additional terms needed for the ZnSe/ZnTe superlattice, as an example). 165 5,2 Superlattice Dispersion Curve Calculation The superlattice dispersion is calculated in the same way we calculated the bulk phonon dispersion curves. 21 We build a unit cell and apply periodic boundary conditions. However, there are several concerns about building the cell. (1) The number of atoms in the cell should be as small as possible to reduce the expense of the calculation. In the superlattice (AB)n 1 (AC)n 2 , if n 1 + n2 = even, the space group is P4m2(D 2 d5 ), and if n 1 + n 2 = odd, the space group is I4m2(D 2 d9 ). Both lattices are 4m2(D2d)- In the n 1 + n 2 = even case it is easy to build a unit cell which contains the smallest number of atoms. And in the n 1 + n 2 = odd case, we have to double the cell size so that the cell can repeat itself. This increases the computer time required to compute the vibrations. (2) The II-VI superlattices are usually strained. In the commensurate case the lattice constant in-plane is constrained by the substrate or buffer layer. In the free standing case, the lattice constant in-plane can be different from that of substrate, and the stress is released. When we build the lattice unit cell, we can fix the in-plane lattice constant to be that of substrate or buffer to simulate the commensurate case; or we can optimize all the lattice parameters to simulate the free standing case. (3) The charge on the atoms may be different from the bulk materials. For example, we expect the charge on the interface atom to be intermediate between the two component materials. We assigned the charge by doing Ewald charge equilibration: First, we used the fitted Mulliken eletronegativities of the atoms that reproduce the charges of the bulk material, and then use those eletronegativities to calculate the charges on the atoms of the superlattice unit cell. Indeed, we see that the interface atom has a charge which is intermediate to the charges of the like atoms in the two component materials, and the charge transition is entirely confined to the interface layer and the two monolayers adjacent to it. 4) Usually the unit cell is under stress; Both the short range and long range interactions will contribute stress to the unit cell. So we fix the in-plane lattice constant to the substrate lattice constant and do charge equilibration and energy minimization of the cell iteratively until convergence. We 166 then obtain the cell size and the right charge. We also obtain the in-plane stress. Usually 3 or 4 iterations will suffice. It is also possible to free the in-plane lattice constant and iterate to get the cell without strain for free standing superlattices. 5.3 Results Our model is first compared to the available experimental data. Unfortunately, the experimental phonon frequencies are scant and incomplete. The available data for ZnS/ZnSe are from Cui, 22 for ZnSe/ZnTe from Cui, 23 Shen, 24 Ozaki, 25 and Wu, 26 and for CdTe/ZnTe from Menendez. 27 In the simulation, we calculated the layers of the composite material and constructed the corresponding unit cell. It is required that the number of layers of the composite be an integer. For the commensurate case we require the in-plane lattice constant to be clamped by the buffer layer or substrate layer; while for the free-standing case we allow the lattice completely relax, optimizing the unit cell without constraints. Another assumption we made is that the superlattices have perfect interfaces. We show that the interface phonons are insensitive to surface roughness and then only consider perfect interfaces. The results are summarized in Tables 12-14. Both the commensurate(stressed) and freestanding (relaxed) cases are shown. The stressed cases should be compared to the experiment. For ZnS/ZnSe (Table 12) it is not clear whether the sample is commensurate or free-standing. We find basically the prediction of our model is in agreement with experiment. However, there are also 3 discrepancies. (1) For the ZnS/ZnSe LA mode we have to use a (ZnSh2(ZnSeh2 unit cell to reasonably reproduce the experimental results, instead of (ZnS)i 0 (ZnSe)i 0 unit cell which has the layer thickness reported in the experiment. This could possibly be explained by an error in the thickness measurement in the experiment. (2) The prediction of a few of the interface modes deviates from experiment. In the first sample of CdTe/ZnTe, Menendez 27 assigned the Raman scattering peaks at 155.92 cm- 1 and 199.69 cm- 1 as interface modes, and in the second sample of CdTe/ZnTe, Menendez 27 attributed the Raman scattering peaks at 153.4 cm- 1 and 192.1 cm- 1 to interface modes. However, our simulations clearly show that there is no interface 167 modes around those frequencies, instead we found interface modes at 176 cm- 1 and 179 cm-1, and they are independent of the slab thickness. This may not be explained by the roughness of the interface of the experimental sample, because the surface roughness should not affect the optical phonons drastically. On the other hand it could be explained by an error in the experimental assignment. (3) For the 184 cm- 1 LO frequency in sample (1) of CdTe/ZnTe, Menendez 27 assigned it to be LO modes in the CdTe. We didn't find any LO modes of CdTe around that frequency, instead we found that there is a ZnTe confined LO mode at 185.3 cm- 1 so we assign this mode to the experimental peak at 184 cm- 1 as a confined LO mode in ZnTe. Other than that our model agrees well with experiment. The above comparison of our model and experiment shows that the model 1s accurate and therefore useful for making predictions, which we make in the form of phonon dispersion curves for the superlattices. The dispersion curves (AB)n1(AC)n2 for the ZnS/ZnSe, ZnSe/ZnTe,CdTe/ZnTe and HgTe/CdTe superlattices are shown for nl = 1, n2 = 1 in Figures 18-21, for n 1 = 2, n2 = 2 in Figures 22-25, for n 1 = 4, n 2 = 4 in Figures 26-29, for n 1 = 10, n 2 = 2 in Figures 30-33. Dispersion curves along both parallel and perpendicular to the interface are shown. 5.4 Acoustic Phonon Folding and Splitting Here we compare the superlattice dispersion curves with those of the bulk materials. For illustrative purposes we used the simplest case (ZnSe)i(ZnTe)i and compare it to the bulk ZnSe and ZnTe in figure 34. From the figure we notice that the dispersion curve of the superlattice is constructed by folding the composite dispersion curves and then mixing the corresponding branches, e.g. the TA and LA modes of the superlattice of (ZnSe)i(ZnTe)i is a mixture of the folded TA(LA) modes of ZnSe and ZnTe. If we compare the dispersion of (ZnSe)i(ZnTe)i to that of (ZnSe)2(ZnTe)2, we immediately observe that the dispersion curves of (ZnSe)2(ZnTe)2 are just the folded dispersion curves of (ZnSe)i(ZnTe)i. The bulk materials' TA modes flatten near the zone boundary, and when they are folded, 168 they remain essentially flat at the new BZ boundary. When the acoustic branches are folded, the once degenerate branches split. The splitting increases as the energy of the branch increases. Also shown in Figure 18-33 are the in-plane dispersion curves. The TA modes are split, as expected, and one of the TA modes is degenerate with the LA mode at the zone boundary. This is because for the in-plane wave vector, the motion of the LA modes is in-plane, while for TA modes, the motion of one of the branches is in-plane and the other is perpendicular to the plane. At zone boundary, the in-plane part of the higher energy TA mode takes on LA mode character. 5.5 Optical Confined Modes We also observe the folding of the optical modes of the bulk composite dispersion curves. When the number n 1 and n 2 are small, the optical dispersion curves remain curved. However, as the number of atomic layers in the superlattice increases, the optical modes flatten, and the "folding" becomes harder and harder to see, and it becomes more appropriate to treat them as confined modes. In our simulation we distinguish the TO and LO modes by their multiplicity: TO modes are doubly degenerate and LO modes are non-degenerate. The in-plane optical dispersion curves are also shown in the Figure 18-33. However, in this case the double degeneracy of the TO modes is lifted and the resulting branch becomes a mixture of the TO and LO modes of the bulk component materials. However, a clear identification is difficult. Notice at the reciprocal cell zone center, the phonon frequencies are different for branches, in different directions. That is the r points have split depending on the direction of the wavevector (thus the polarization direction). This polarization dependency is entirely due to the charge of the atoms. If the ionic charge is removed, all the corresponding bra~ches are continuous along different directions. With an ionic charge a finite jump will occur as K passes through r for some of the optical modes. This behavior also exists in other anisotropic materials such as hep crystals, and it is called "Angular dispersion." It doesn't exist in materials with the fee lattice 169 because of the cubic symmetry. 5.6 Interface Phonons Interface modes in semiconductor superlattices have been reported by several studies. In our simulations, we can directly distinguish the interface modes from the non-interface modes by analyzing the displacement vectors of the atoms, making assignments much easier. We list the interface modes in various superlattices in Table 15. For comparison, the typical motion of the folded TA and LA modes, the confined TO and LO modes, and the interface modes of (ZnSe) 4(ZnTe)4 are shown in Figure 35. Analysis of the normal mode character of the interface modes listed in Table 15 indicates that all the interface modes are TO modes of the superlattice, being confined to the interface, and being doubly degenerate. The motion of the atoms is in-plane, and only the atoms on the interface and immediately adjacent to it move. The frequency is characteristic of the interface, almost independent of the layer thickness except in the (AB)i(AC)i case, where the slab is very thin and there is no clear difference between interface atoms and inner layer atoms. To verify this assumption we constructed several other superlattices with different slab thicknesses, and found the assumption that interface modes are independent of layer thickness to be true. The result is shown in Table 15. This means that the interface phonon can be employed to characterize the interface properties for crystal growth. Another important issue is that interface phonons are insensitive to surface roughness. Since surface phonons are essentially TO modes, they come from folding of the bulk material optical branches, and surface roughness just means that the length of the reciprocal lattice perpendicular to the interface is different at different parts of the superlattice, the lattice parameter being just the average of the entire interface. However, the TO phonons are almost independent of the reciprocal length, and so the effect of the interface roughness is minimal. In order to test this, we constructed a supercell of (ZnSe) 4(ZnTe) 4 and made the interface roughness vary from 30% to 50%, and we found the interface modes varied in the small range 170 of 192.6 to 194.6 cm- 1 . 5. 7 Elastic Constants of the Superlattice The elastic constants of the superlattice are shown in Table 16 along with the bulk materials. The superlattice has lower symmetry and the elastic constant matrix has the relation Cn = C22 , C31 = C32, C44 = C55. We have 6 unique (C11, C12, C33, C13, C44, C 66 ) non-vanishing components. We find that the elastic constants are usually between those of the component materials. Currently, no measurement of elastic constants for superlattice has been reported. 6.0 Discussion A number of approaches have been proposed to describing the interatomic interactions of semiconductor crystals. This includes the bond charge models of Martin 28 and Weber 29 and the quasiparticle valence bond force field of Messmer. 30 These models require a greater number of parameters (for example Messmer's FF requires 10 parameters for Si while MSXX requires only 7 parameters) and lead to fairly complicated equations. These models succeed in modeling the phonon spectra of the diamond lattice and zinc-blende semiconductors, although with some cost in accuracy and simplicity. These models may provide a good model of the polarization effects in these materials. The Kane potential31 models such properties through the interaction of his dipoles and quadrupoles with the polarization field. The MSXX FF is essentially as accurate as the experimental data to which it was fitted. Although more accurate and more complete experimental data is becoming available there is still significant uncertainty in the experimental values for some properties of some of these materials. These uncertainties often arise from some of the indirect methods required to measure the phonon spectra. Force fields capable of describing the strain fields, geometric structures, and vibrational properties of distorted systems must be capable of reproducing the interatomic potential energy surface for moderate strains and also the couplings of interatomic interactions. It is possible to reproduce the phonon spectra of a zinc- 171 blende crystal to relatively high accuracy using harmonic potentials. However, harmonic potentials are inappropriate for describing the breaking of a bond. Thus even though the phonon dispersion curve is reproduced, the strain fields may not be sufficient for accurate modeling of highly distorted structures. For this reason we use Morse potentials to approximate the anharmonic effects of strained bonds as bonds are broken. This should lead to a good description of thermal expansion. Furthermore, the use a Morse potentials for bonds should allow phonon-phonon couplings to be described in the harmonic description of the interatomic interactions. The MSXX FF includes the coupling of coplanar angles that share a common side, but not a common apex. This term (also used by Kane 31 ) was first implemented by McMurry 10 to describe the flattening of the TA modes on approaching the X and r points along the < 100 > and < 111 > directions. Formally a fifth neighbor interaction, it allows for through bond coupling of the accordion chains along < 110 >. Furthermore, Kane 31 observed that the introduction of an impurity in the diamond lattice polarizes the charge distribution with high directionality along the < 110 > chains. Again, the form of the coplanar-two-center-angle-angle coupling leads to an effective model and an intuitive picture of this effect through the coupling of adjacent bond angles along the < 110 > direction. This effect emphasizes the necessity for such terms. Because this coupling depends on the large delocalization of bond charge along < 100 > chains associated with through bond coupling, we expect the coupling to fall quickly as the bond angles become noncoplanar. For the perfect crystals this is irrelevant, but for strain fields that break the crystal symmetry and structures sampled with molecular dynamics at higher temperatures, the damping of the coplanar angle-angle coupling is necessary. Although the coupling between the two coplanar angles of the trans dihedral is critical to the description of the TA mode, we find that the dihedral angle torsion force constant itself is unnecessary for our fits. It might play a role in distinguishing between the sphalerite and wurzite forms of the crystal, and if so it would be important for describing of the stacking fault energy. 172 7 .0 Conclusion The simple MSXX valence force field with just a few parameters leads to accurate structural, mechanical, and vibrational properties of the zinc-blende semiconductors. The systematic behavior of the force constants allows one to interpolate force constants for tertiary interactions, as illustrated for the II/VI superlattices. The MSXX FF describes anharmonic interactions in bonds and should be useful for describing the thermal expansion, temperature dependence of the phonons, and phonon-phonon interactions. 173 References 1. R. Friesner, J. Chem. Phys. 85, 1462, (1986); R. Friesner, J. Chem. Phys. 86, 3522, (1987); M. Ringnalda, Y. Won and R. Friesner, J. Chem. Phys. 92, 1163, (1990). 2. C. Pisani, R. Dovesi and C. Roetti, Hartree-Fock ab initio Treatment of Crystalline Systems, Lecture Notes in Chemistry 48, 1 (Springer-Verlag, Berlin, 1988). 3. S. L. Mayo, B. D. Olafson and W. A. Goddard III, J. Phys. Chem. 92, 7488 (1990). 4. N. Karasawa, S. Dasgupta, and W. A. Goddard III, J. Phys. Chem. 95, 3358 (1991). 5. N. Karasawa and W. A. Goddard III, Macromolecules 25, 7268, (1992). 6. C. B. Musgrave, S. Dasgupta, and W. A. Goddard III, The Hessian Biased Force Field for Polysilanes, To be submitted to Macromolecules. 7. S. Dasgupta, W. B. Hammond, and W. A. Goddard III, A New HydrogenBond Potential for Peptides: Crystal Structure and Properties of Nylon. 8. J. A. Wendel and W. A. Goddard III, J. Chem. Phys. 97, 5048 (1992). 9. NBS Technical Note 270-3, selected values of chemical thermodynamic prop- erties U. S. Department of Commerce, National Bureau of Standards. 10. H. L. McMurry, A. W. Solbrig Jr., J. K. Boyter and C. Noble, J. Phys Chem. Solids 28 2359 (1967). 11. N. Karasawa and W. A. Goddard III, J. Phys. Chem. 93, 7320 (1989). 12. A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard III, and W. M. Skiff, J. Am. Chem. Soc. 114, 10024 (1992). 13. S. Dasgupta and W. A. Goddard III, J. Chem. Phys. 90, 7207 (1989). 14. T. Yamasaki, S. Dasgupta, and W. A. Goddard III, Hessian Biased Force Fields: IL The Singular Value Decomposition (SVD) Based Least Squares Method for Optimization and Analysis of Force Field Parameters, submitted 174 to J. Chem. Phys. 15. Allred and Provin, Pauling electronegativities. 16. A. K. Rappe and W. A. Goddard III, J. Phys. Chem. 95, 3358 (1991). 17. N. Karasawa and W. A. Goddard III, unpublished. See N. Karasawa Thesis, Applied Physics 1991. 18. K. H. Hellwege, Landolt-Bornstein Numerical Data and Functional Relation- ships in Science and Technology Springer-Verlag (1982). 19. C. Colvard, R. Merlin, M. V. Klein, A. C. Gossard, Phys. Rev. Lett., 43, 298 (1980). 20. M. Cardona and G. Guntherodt, Light Scattering in Solids, V: Superlattices and Other Microstructures, Springer-Verlag, 49 (1989). 21. J. Hu, C. B. Musgrave, W. A. Goddard III, to be published. 22. J. Cui, H. Wang, F. Gan and A. Li, J. Crys. Grwth., 111 811 (1991). 23. J. Cui, H. Wang, F. Gan, J. Appl. Phys 72, 1521 (1992). 24. A. Shen, J. Cui, H. Wang and Z. Wang Superlattices and Microstructures, 12, (1992). 25. H. Ozaki, D. Suzuki, K. Imai, and K. Kumazaki, Phys. Stat. Sol (a) 133, 523 (1992). 26. Y. H. Wu, H. Yang, A. Ishida, and H. Fujiyasu, Appl. Phys. Lett., 54, 239 (1989). 27. J. Menendez, A. Pinczuk, J.P. Valladares, R. D. Feldman and R. F. Austin, Appl. Phys. Lett., 50, 1101 (1987). 28. R. M. Martin, Phys. Rev. B 1, 4005 (1970). 29. W. Weber, Phys. Rev. Lett. 33, 371 (1974). 30. H. X. Wang and R. P. Messmer, Phys. Rev. B 41 (1990); 31. E. 0. Kane, Phys. Rev. B 31, 7865 (1985). 175 Table 1. Ewald charge equilibration parameters and the comparison of the charges resulted from phonon gap fitting and charge equilibration. Atom Zn Cd (a) Modified QEq Parameters Electroneg (eV) 0.900 0.376 4.285 3.957 Hardness (eV) (b) Predicted CHARRM ZnSe Material ZnS Calculated Cation q 0.90 0.79 Fitted Cation q 0.88 0.80 Hg s Se Te 0.850 4.160 6.928 4.486 6.428 4.131 5.816 3.526 ZnTe 0.77 0.76 CdTe 0.86 0.86 HgTe 0.74 0.74 176 Table 2. Experimental lattice constants, phonon frequencies, elastic constants, used in determining the MSXX FF (from reference 8, see discussion in Section 4.1). Comparison with theoretical predictions. Quantity Lattice Parameters (A) Experiment MSXX Elastic Constants (GPa) Cn exper MSXX C12 exper MSXX C44 exper MSXX rms Cij deviation Phonons (cm- 1) TA(X) exper MSXX LA-LO(X) exper MSXX TO(X) exper MSXX I'15 exper MSXX rms phonon deviation Diamond Silicon Germanium 3.5610 3.5610 5.4310 5.4310 5.6507 5.6507 1076.0 1083.5 125.0 125.1 576.0 573.4 2.61 167.5 170.7 65.0 64.2 80.1 78.6 3.38 129.0 138.2 48.0 47.1 67.0 64.5 0.51 803.6 808.8 1078.1 1069.4 1194.8 1161.8 1333.9 1341.7 0.40 150.4 150.6 415.0 412.7 463.7 463.8 518.1 518.7 2.69 80.1 80.2 240.5 238.7 275.5 275.4 298.9 300.4 0.94 177 Table 3. MSXX force field parameters for the group IV materials. Diamond Silicon Germanium kR [kcal/(molA 2 )] 576.55 193.75 180.04 Re (A) 1.522 2.380 2.465 DRa (kcal/mol) 110.0 73.7 73.7 ke [kcal/(mol rad 2 )] 118.60 33.61 21.76 0 (degree) 109.41 105.05 109.47 kre [kcal/(molA rad)] -59.56 -14.76 -14.86 krr [kcal/(molA 2 )] 8.05 3.39 0.56 Quantity (a) Bond (c) Angle (e) Two Center Angle-Angle Cross Term Gee [kcal/(mol rad 2 )] -27.54 -25.14 -23.35 178 Table 4. van der Waals parameters (from reference 23). See equations (4) and (5). Atom C Si Ge Rvdw (A) 3.883 4.270 4.270 nvdw (kcal/mol) 0.0844 0.3100 0.3100 179 Table 5. Lattice Constant, elastic constants, and phonon frequency from experiment and from the MSXX force field. Numbers in square braces were not used in the fit (they were extrapolations from trends in other materials and are included for comparison). AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb 6.1355 5.4506 5.6419 6.0940 5.8687 6.0584 6.4788 (a) Lattice Constants (A) 5.467 a 5.6611 (b) Elastic Constants (GPa) 132.0 125.0 87.69 141.20 118.1 88.39 102.0 83.29 66.69 128.3 124.0 86.46 142.16 122.9 93.50 106.6 85.11 68.56 63.00 53.40 43.41 62.53 53.20 40.33 57.6 45.26 36.45 67.42 57.01 41.63 61.13 51.89 36.63 57.4 42.71 35.12 46.00 54.20 40.76 70.47 59.40 43.16 46.0 39.59 30.30 MSXX 65.71 57.70 39.39 69.44 57.57 40.52 44.9 37.07 29.04 rms Cij 11.86 2.65 1.47 1.15 3.06 3.95 2.73 2.32 1.51 c11 exp MSXX C12 exp MSXX C44 exp (c) Phonon Frequencies (cm- 1 ) TA(X) exp [51.91] 108.9 70.00 106.7 81.7 56.6 68.33 53.0 37.3 MSXX LA(X) exp 147.60 109.2 70.30 106.7 81.9 56.7 68.9 53.0 37.5 [47.67] 221.8 155.00 249.0 225.0 166.3 193.33 160.0 143.3 MSXX TO(X) exp 356.60 216.6 152.90 249.0 221.9 163.3 182.8 160.6 141.3 [436.10] 334.0 296.00 353.5 256.3 212.0 323.33 216.0 179.3 MSXX 437.20 333.9 296.30 353.9 256.7 213.1 323.5 217.0 179.9 LO(X) exp [371.54] 402.5 341.00 366.2 240.0 211.7 331.67 203.0 158.3 MSXX 395.50 400.8 346.60 365.8 236.8 209.3 326.6 201.7 156.4 TO(r) exp 439.40 360.9 323.40 366.2 271.0 232.3 304.00 217.3 184.7 MSXX LO(r) exp 439.30 363.5 320.80 365.5 272.2 231.7 307.8 215.6 185.0 501.00 404.1 344.40 404.1 293.0 235.0 346.00 238.6 196.7 MSXX 500.80 406.4 341.80 403.5 294.0 [240.5] 349.3 237.1 196.9 0.16 2.64 2.87 0.44 1.94 1.81 5.17 1.17 1.16 rms Vi 180 Table 6. Optimized parameters for the MSXX force field. AlAs AlSb GaP GaAs GaSb InP InAs InSb 0.83 0.78 0.60 0.72 0.64 0.46 0.83 0.72 0.61 kn 143.36 101.67 94.996 107.51 99.10 85.68 93.77 78.26 75.82 Re 2.480 2.630 2.779 2.559 2.631 2.805 2.772 2.865 2.998 Dn a 48.25 44.53 40.80 40.60 38.88 34.73 38.64 36.11 32.04 AIP (a) Charge (e) Cation (e) {b) Bond {c) Angle at III k0 32.15 36.882 28.97 50.09 34.69 36.85 32.29 34.91 21.15 0e 124.96 114.87 114.80 110.46 107.16 109.09 114.13 112.15 112.27 Kr0 -17.57 -15.84 -10.02 -21.58 -15.24 -11.79 -21.09 -18.54 -6.49 Knn 7.10 3.33 2.71 4.19 4.70 4.071 2.55 3.25 1.61 {d) Angle at V Ko 29.61 41.53 25.10 33.87 43.30 35.57 47.19 33.71 30.19 0e 125.05 119.79 118.23 113.14 112.20 99.55 112.26 107.93 108.84 Kr0 -26.28 -14.74 -10.67 -19.22 -17.76 -10.35 -20.03 -13.42 -17.17 Krr 7.25 16.67 8.84 9.47 6.27 1.96 10.48 4.11 2.76 -17.44 -17.45 -18.99 -18.37 -18.77 -14.95 -14.89 -16.37 ( e) Cross Term K002 a -15.08 Dn is based on atomization data at 298.15K from reference 21. 181 Table 7. Van der Waals parameters from the universal force field (reference 5). Al Ga In p As Sb Rv (A) 4.499 4.383 4.465 4.147 4.230 4.420 Dv Pkcal/mol) 0.505 0.415 0.599 0.305 0.309 0.449 182 Table 8. Experimental lattice constants, phonon frequencies, elastic constants, used in determining the MSXX FF (from reference 8, see discussion· in Section 4.1). Comparison with theoritical predictions. Quantity ZnS Lattice Parameters (A) Experiment 5.410 MSXX 5.410 Elastic Constants (GPa) 104.62 Cu exper MSXX 107.37 65.33 C12 exper MSXX 62.18 46.50 C44 exper MSXX 43.85 rms Cij deviation 2.61 Phonons (cm- 1) TA(X) exper 89.67 MSXX 89.68 LA(X) exper 211.33 MSXX 210.60 TO(X) exper 315.67 MSXX 315.60 LO(X) exper 330.00 MSXX 329.50 TO(r) exper 276.67 MSXX 277.20 LO(r) exper 348.00 MSXX 348.60 rms phonon deviation 0.40 ZnSe ZnTe CdTe HgTe 5.6687 5.6687 6.1037 6.1037 6.481 6.481 6.461 6.461 86.00 90.49 51.10 47.91 40.20 38.14 3.38 71.30 72.17 40.70 40.83 31.20 31.08 0.51 53.30 49.33 36.50 31.85 20.44 18.42 3.37 59.71 53.71 41.54 32.89 22.59 18.25 2.70 70.00 70.40 194.00 189.20 219.00 219.60 213.00 208.20 213.00 214.40 253.00 255.50 2.69 54.0 54.10 143.0 141.10 173.7 173.7 183.7 182.40 176.7 177.50 206.7 207.50 0.94 35.0 34.90 97.0 125.00 148.00 147.80 15.86 15.86 85.00 84.80 133.93 133.80 85.00 84.80 118.00 118.60 138.00 138.50 0.72 125.30 140.50 138.80 167.0 168.30 0.80 183 Table 9. MSXX force field parameters for the II-VI zinc blende materials. ZnS ZnSe ZnTe CdTe HgTe 0.88 0.80 0.76 0.86 0.74 kn [kcal/(molA 2 )] 100.51 71.72 68.51 62.32 55.44 Re (A) 2.519 2.686 2.809 2.971 3.005 Dna (kcal/mol) 36.78 31.13 26.59 23.97 21.35 Quantity (a) Charge (e) Cation {b) Bond ( c) Angle, Group II Atom at Apex k0 [kcal/(mol rad 2 )] 34.33 29.85 17.02 17.98 20.13 0 (degree) 119.23 113.60 116.35 128.08 116.03 kr0 [kcal/(molA rad)] -24.01 -16.51 -17.92 -14.22 -2 0.51 krr [kcal/(molA 2 )] 13.01 7.61 9.11 6.11 8.43 kr0 [kcal/(molA rad)] -23.21 -15.40 -9.63 -11.34 -7 .39 krr [kcal/(molA 2 )] 4.63 8.07 6.46 3.70 0.45 -11.40 -9.12 -11.11 (e) Two Center Angle-Angle Cross Term G00 [kcal/(mol rad 2 )] -10.20 -10.47 a Dn is based on atomization energies at 298.15K from reference 30. 184 Table 10. van der Waals parameters (from reference 23). See equations (4) and (5). Atom Zn Cd Hg s Se Te Rvdw (A) 2.763 2.848 2.705 4.035 4.205 4.470 nvdw (kcal/mol) 0.124 0.228 0.385 0.274 0.291 0.398 185 Table 11. Additional force field terms for the ZnSe/ZnTe superlattice. (a) Angle Se-Zn-Te k0 = 22.54 kcal/mol 0e =114.97 degree kr0 = -17.92 kcal/(molA) for R = Se - Zn kr0 = -16.51 kcal/(molA) for R = Zn - Te krr = 8.33 kcal/(molA 2 ) (b) Two-Center Angle-Angle G00,Zn-Se-Zn-Te = G00,zn-Te-Zn-Se = -10.93 kcal/mol 186 Table 12. Comparison of the ZnS/ZnSe, 22 superlattice predictions and the available experimental data. (ZnS)i2(ZnSe)i2 LA modes (q=0.07) 19.14 23.94 40 47.87 64.79 74 stressed 22.10 25.44 45.59 48.96 68.55 71.96 relaxed 21.30 24.51 43.94 47.19 66.07 69.35 (Sample dzns = 26.A., dznse = 29.A., N = 150, buffer 1µ ZnS, GaAs substrate) TO modes in ZnS in ZnSe experiment 270.1 214.6 stressed 263.9 214.2 relaxed 272.8 216.4 (Sample dzns = 12.A., dznse = 12.A., N = 100, buffer lµm ZnSe, substrate GaAs) 187 Table 13. Comparison of the ZnS/ZnTe, 22 ,23 superlattice predictions and the available experimental data. (ZnSe)10(ZnTe)io LO modes in ZnSe 228.5 233.5 239 242.5 244 246 stressed 227.8 232.1 235.8 238.7 240.9 242.2 (relaxed) 228.0 232.3 236.0 238.9 241.1 242.4 Sample dznSe = 27 A, dznTe = 29.A., N = 100, substrate InP (ZnSe)2(ZnTe)26 LA modes(k=0.09) 11.59 14.49 24.15 28.21 stressed 13.8 14.2 27.6 28.2 (relaxed) 12.91 15.49 27.01 29.60 Sample ZnSe 5 layers , ZnTe 52 layers, total dsL = 85.3.A, buffer 0.5.A. ZnTe, N = 80 188 Table 14. Comparison of the CdTe/ZnTe 27 superlattice predictions and the available experimental data. (CdTe)3(ZnTe)5 LO modes in Sublattice CdTe CdTe ZnTe ZnTe experiment 172.88 184 203.87 205.5 stressed 175.7 185.3 205.5 207.1 (relaxed) 173.7 182.5 202.3 204.5 Sample: dcdTe = 21A, dznTe = 27 A, N = 21, buffer 0.9µ Cdo.1Zno_gTe, substrate GaAs < 100 > (CdTeh(ZnTe)n LO modes in CdTe Layer L02 L04 L05 LOs experiment 180 175.84 171 165.86 (stressed) 179.54 176.92 172.74 167.16 relaxed 177.41 174.78 170.60 165.03 Sample dcdTe = 5lA, dznTe = 61.A, N = 15, buffer 0.9µ Cdo.1Zno_gTe, substrate GaAs < 100 > 189 Table 15. Interface modes for several II/VI superlattices. superlattice IF mode cm- 1 IF mode cm- 1 (ZnS)2(ZnSe)2 223.0 (ZnS)4(ZnSe)4 225.9 (ZnS)io(ZnSe)2 226.1 (ZnS)io(ZnSe)io 225.9 (ZnS)i2(ZnSe)i2 225.9 (ZnSe)2(ZnTe)2 191.6 195.1 (ZnSe)4(ZnTe)4 192.6 193.6 (ZnSe)io(ZnTe)2 194.4 197.7 (ZnSe)10(ZnTe)10 193.8 195.3 (ZnSe)is(ZnTe)20 192.2 193.5 (CdTe)2(ZnTe)2 170.2 (CdTe)4(ZnTe)4 169.1 (CdTe)a(ZnTe)s 170.3 (CdTeh(ZnTe)n 170.1 (CdTe)10(ZnTe)2 170.7 (HgTe)2(CdTe)2 137.6 151.6 (HgTe)4(CdTe)4 135.1 152.6 (HgTe)s(CdTe)s 135.3 152.3 (HgTe)io(CdTe)2 138.8 151.8 All interface modes are of TO character. 190 Table 16. Elastic constants of the bulk materials, force field fitted results and the predicted results for superlattice. Name MSXX ZnS Cn 104.50 MSXX 107.37 C12 65.30 ZnSe 85.90 90.49 ZnTe 71.30 CdTe MSXX 62.18 C44 46.00 50.60 47.91 40.60 38.14 72.17 40.70 40.83 31.20 31.08 53.30 49.33 36.50 31.85 20.44 18.42 HgTe 50.80 53.71 35.80 32.89 20.50 18.25 Name C12 34.44 Caa 98.82 C1a 54.11 C44 41.01 c66 ( ZnS h ( ZnSe h Cn 117.13 22.01 (ZnS)2(ZnSe)2 118.39 35.22 98.43 54.84 40.46 22.05 (ZnSe)4(ZnTe)4 118.90 35.58 97.95 55.05 40.01 22.12 (ZnS)10(ZnSe)2 125.97 41.01 104.99 61.20 41.51 22.24 (ZnSe h (ZnTe h (ZnSe)2(ZnTe)2 95.49 28.07 81.86 44.62 33.82 17.75 95.66 28.51 82.15 44.70 33.76 18.04 (ZnSe )4 (ZnTe )4 95.67 28.57 81.88 44.66 33.90 18.23 (ZnSe ho(ZnTe )2 103.00 29.65 87.74 46.47 36.72 20.25 (CdTe h (ZnTe h 70.43 24.54 62.19 36.66 23.59 11.22 (CdTe)2(ZnTe)2 70.78 24.65 62.58 36.88 23.31 11.57 (CdTe)4(ZnTe)4 70.87 24.69 62.81 36.96 23.11 11.71 (CdTe)106(ZnTe)2 62.37 23.12 54.28 33.64 19.94 9.62 (HgTe)1(CdTe)1 60.16 23.28 52.06 32.37 18.16 9.39 (HgTe)2(CdTe)2 60.30 23.50 51.78 32.39 18.21 9.49 (HgTe)4(CdTe)4 60.25 23.62 51.70 32.41 18.22 9.52 (HgTe) 10 (CdTe )2 61.21 24.65 53.16 32.81 18.21 10.11 Experimental lattice constant and elastic constant taken from reference. 18 43.85 191 Table 1 7. Cell geometries of the superlattices. Name ZnS ZnSe ZnTe CdTe HgTe Name (ZnS)i(ZnSe)i (ZnS)2(ZnSe)2 (ZnSe )4 (ZnTe )4 (ZnS)io(ZnSe)2 (ZnSe )i (ZnTe )i (ZnSe)2(ZnTe)2 (ZnSe)4(ZnTe)4 (ZnSe)io(ZnTe)2 ( CdTe )i (ZnTe )i (CdTe)2(ZnTe)2 (CdTe)4(ZnTe)4 (CdTe)106(ZnTe)2 (HgTe)1(CdTe)1 (HgTe)2(CdTe)2 (HgTe )4( CdTe )4 (HgTe)i 0 (CdTe)2 a 5.41 5.6687 6.1037 6.481 6.461 a;; dsL 5.5324 5.5308 5.5321 5.4321 5.8748 5.8687 5.8673 5.7389 6.2413 6.2421 6.2408 6.3880 6.4531 6.4615 6.4641 6.4591 5.5457 11.0631 22.0902 32.5660 5.9100 11.8163 23.6193 34.5737 6.3539 12.6978 25.4010 38.7556 6.4760 12.9465 25.8917 38.7802 Experimental lattice constant and elastic constant taken from reference. 18 192 Figure Captions Figure 1. Phonon dispersion curve for diamond. Figure 2. Phonon dispersion curve for silicon. Figure 3. Phonon dispersion curve for germanium. Figure 4. Phonon dispersion curve for AlP. Figure 5. Phonon dispersion curve for AlAs. Figure 6. Phonon dispersion curve for AlSb. Figure 7. Phonon dispersion curve for GaP. Figure 8. Phonon dispersion curve for GaAs. Figure 9. Phonon dispersion curve for GaSb. Figure 10. Phonon dispersion curve for InP. Figure 11. Phonon dispersion curve for InAs. Figure 12. Phonon dispersion curve for InSb. Figure 13. Phonon dispersion curve for ZnS. Figure 14. Phonon dispersion curve for ZnSe. Figure 15. Phonon dispersion curve for ZnTe. Figure 16. Phonon dispersion curve for CdTe. Figure 1 7. Phonon dispersion curve for HgSe. Figure 18. Phonon dispersion curve for HgTe. Figures 19-22. Superlattice phonons of ZnS/ZnSe, ZnSe/ZnTe, CdTe/ZnTe, and HgTe/CdTe grown in the [111] direction. Figures 23-38. Phonons of ZnS/ZnSe, ZnSe/ZnTe, CdTe/ZnTe, and HgTe/CdTe for superlattices with layer thicknesses of 1, 2, 4 and 10 grown in the [100] direction. Figures 39. Comparison of folded phonons with phonons of component materials for ZnSe/ZnTe. 193 Figure 1. Phonon Dispersion Curve for Diamond A 1400-r-------"T""""------..----------, 1200 -... 1000 .... I E u 800 >, C) Cl) C w 600 400 200 0-+----,--......--.--...-----+----..L--.----,.--....ll&--.....-------' X K Wavevector (2rr/a) r 194 Figure 2 Phonon Dispersion Curve for Si A 600-.-------....---,,------...-------- -c - 11""' I u > 0) I,,, a, C w X K Wavevector (2rrla.) r 195 Figure 3. Phonon Dispersion Curve for Ge A ti 350----------..-------------- - 250 'ii"" I E u ::,,.. 200 C'l !I.,, (I) C w 150 X K Wavevector (2rua) r 196 600-----------------500 200 0 - t - - - r - - - , - - - , - - - - , - - - ~ ~........- - - . - - . . - - - - - l aQ.5 n/2 [111] 0 0.5 1t/2 [100] 197 Figure 5. Phonon Dispersion Curves for AIAs 450------------.--------400 - 350 ,... I E u 300 >,, u C Cl) :::::s er Cl) 250 ... LL 200 150 100 50 0-+--....----.---....----JIC;...__--,-_ _ _ _ _ _~ -0.5 7t/2a [111) 0 0.5 7t/2a [1 0 OJ 198 Figure 6. Phonon Dispersion Curves for AISb 350 300 - 250 '11""' i E u >,,. 0 C 200 m ::I C" m ii,,,, IU. 150 100 501-------- O-t---r----r---,---,--.....,_,,3'1'--__,,..-.....---...---,,--1 -0,5 7t/2 [111] 0 0.5 rc/2 [100] 199 Fugure 7, Phonon Dispersion Curves for GaP 450......-----------------------. 400 350 - 300 'lF g E u ~ u 250 C @) :::s er u. @) 11,,, 200 150 100 50 0-+--......----.----,---,--~:..........--,--.....-----0.5 0 -0.5 n/2 [111] n/2 [100] 200 Figure 8. Phonon Dispersion Curves for GaAs 300 250 ,... I E CJ >CJ 200 C Cl) ::, O" ! LL 150 100 50 0--1---,--r---..,....--,-~~.,...._,---,---.--l--.....----,-----,..---,---I 1 0.5 -0.5 0 7t/2 [100] 7t/2 [111] 201 Figure 9" Phonon Dispersion Curves for GaSb 200 150 100 0-1-----r---,.--,------,---~---.------.--..---.....----I m0.5 n/2a [111] 0 0.5 n/2a [100] 202 Figure 10. Phonon Dispersion Curves for lnP 350 ♦ ♦ ♦ ♦ ♦ + + + 300 -5 - ,... I 250 >uC G) ::, er ! LL 200 150 100 50-;....----- 0-+------,--------,.-----.::li+c--....---------,-----l 0.5 0 -0.5 rr/2 [100) rr/2 [111) 203 Figure 11. Phonon Dispersion Curves for lnAs 250-------------,-------------, 200 - ,.... I E u ~ 150 C a, ::::, O" e LL 100 50 04-----------____;;:i~------------~ 0.5 0 -0.5 ,r/2 [100] ,r/2 [111] 204 Figure 12. Phonon Dispersion Curves for lnSb 180 - 160 -2- 140 ,... I E ~ C Cl) :I er e IL 120 100 80 60 40 20 0-+----,-----,----,.--,---___::iloj,£--...,.....-....----,-----.-----1 -0.5 ,r/2 [111] 0 0.5 ,r/2 [100] 0 -r: 0 50 100 150 200 250 300 350 400 l • ___.___~ ~ K Figure 13. Phonon Dispersion Curve for ZnS fl r • •• • C11 tv 0 206 ♦ Cl) Cl) C ... .e N ♦, Cl) c::::, <I ♦, 0 C 0 ♦ .I! Cl) a. en c C 0 C 0 s= A. -... "flt Cl) ::, 0) u:: 8 8 ,- C") L-W:> 0 207 ~ .. N .e ~::::, 0 C 0 .i!! Cl) a. Cl) Q C 0 C 0 .r::. A. '-° l-W:> 208 • • • • ~ .. 0 .e <1 CD 2: ::, 0 C ...0CD ·;;; . 0. en c C 0 C 0 .&:. • 0,,. -"°.. CD ::, C) i:i: L·W:> 209 ♦ ♦ Cl) Cl) .. C) :c 0 <l Cl) ~ ::, 0 C 0 .i!! Cl) - a. Cl) c C 0 C 0 .c a. -"... Cl) ::, C) ii: L-W:> 210 !0) - :c .e Cl) ~ ::, 0 C -~ 0 • • • • • • • • • <] Cl) c.Cl) c C 0 C 0 .c • D. -co Cl) ::, 0) u:: • • • 0 L·W:> 211 Figure 19 ZnS1/ZnSe1 superlattice (grown on <111> surface) phonon 350 300 250 .... e 200 0 150 100 50 x-0 0-z 212 Figure 20 ZnSel/ZnTel superlattice (grown on <111> surface) phonon 250 -e 1so (J 100 so x-0 0-z 213 Figure 21 CdTe1/ZnTe1 superlattice (grown on <111> surface) phonon .... s E 0 x-0 0-z 214 Figure 22 HgTe1/CdTe1 superlattice (grown on <111> surface) phonon 180 160 140 120 ..... e 100 0 80 60 40 20 x-0 0-z 215 Figure 23 ZnS 1/ZnSe 1 superlattice phonon 350 300 250 .... E' 200 0 150 100 50 216 Figure 24 ZnS2/ZnSe2 superlattice phonon 350 250 .... eu 200 -+--~--150 100 x-0 0-z 217 Figure 25 ZnS4/ZnSe4 superlattice phonon 350 250 -e u 200 150 100 x-0 0-z 218 Figure 26 Zn510/Zn5e2 superlattice phonon 350 300 250 -e 200 C, 150 100 50 x-0 0-z 219 Figure 27 ZnSe 1/ZnTe 1 superlattice phonon I E () x-0 0-z 220 Figure 28 ZnSe2/ZnTe2 superlattice phonon I E CJ x-0 0-z 221 Figure 29 ZnSe4/ZnTe4 superlattice phonon .... n E 0 x-0 0-z 222 Figure 30 ZnSe 10/ZnTe2 superlattice phonon -Eu I x-0 0-z 223 Figure 31 CdTe 1/ZnTe 1 superlattice phonon I E () x-0 0-z 224 Figure 32 CdTe2/ZnTe2 superlattice phonon -E I (J x-0 0-z 225 Figure 33 CdTe4/ZnTe4 superlattice phonon -u I E x-0 0-z 226 Figure 34 CdTe 10/ZnTe2 superlattice phonon ... I E 0 x-0 0-z 227 Figure 35 HgTe 1/C~e 1 superlattice phonon 180 160 140 120 -u I E 100 80 40 20 x-0 0-z 228 Figure 36 HgTe2/CdTe2 superlattice phonon 200 _ _ _ _ _ _ _ _ _...,..,....._ _ _ _ _ _ _ __ 180 .....a E 0 x-0 0-z 229 Figure 37 HgTe4/CdTe4 superlattice phonon 200 - - - - - - - - - - - - , , - , - - - - - - - - - - , 180 -E I CJ x-0 0-z 230 Figure 38 HgTe 10/CdTe2 superlattice phonon 180 160 140 120 -e (.) 100 80 60 40 20 x-0 0-z . 50-I 100~ 150-I I / ZnSe I / 0.5 I I I \ I ·0.5 0I x 0 1c: I ZnSe1/ZnTe1 I so.............-'\ I 100~ / z I 0.5 I ,..-I . 200~ 200-I ~1-C- 250 250 I 300,-------.-------, . 300 .........- - - - - - - - - - - , Figure 39 ZnTe 0.5 J /_j 100 150 200 250 300,------------, 1:-:) ~ .... 232 Chapter 6 Silicon (111) Surface Reconstructions 233 Abstract We use the biased Hessian method in conjunction with ab initio generalized valence bond (GVB) calculations on Si clusters to derive the MSXX force field (FF) for silicon (111) surfaces. This FF is tested by calculating the atomic geometries and relative energies of the (2n+l) x (2n+l) DAS models of the Si (111) reconstruction. We find that the (5 x 5) and (7 x 7) surfaces are most stable. 234 1.0 Introduction Since Schlier and Farnsworth 1 first observed the 7th order spots in the LEED pattern in 1959, numerous models of the Si (111)-7 x 7 surface reconstruction have been proposed based on the results of various experiments and theoretical calculations. 2 - 5 Strong evidence from STM experiments have led to wide acceptance of the Dimer Adatom Surface (DAS) model Takayangi 7 •8 which involves dimers, adsorbed atoms, and a stacking fault. However the large number of atoms in the surface unit cell has impeded detailed ab initio theoretical study. Indeed with approximately 100 atoms in just the first two layers of the 7 x 7 unit cells, it has been difficult to unambiguously determine the structure from LEED and RHEED experiments. Angle resolved X-ray photoemission spectroscopy (ARXPS), X-ray diffraction, ion scattering, transmission electron diffraction, and scanning tunneling microscopy each give clues to the atomic structure. However, no one technique has conclusively determined the atomic arrangement of the Si (111)-7x7 surface. McRae 9 suggested a model containing a stacking fault in one of the two sub-cells which was supported by Bennet's 10 medium energy ion scattering results. Himpsel3 and McRae 9 each described how 12 atom rings around corner holes, two 8 atom rings around shallow holes on the sub-cell borders, and dimers along the sub-cell borders resulted from removing the broken bonds across the stacking fault boundary. These features are all observed by STM. 5 Harrison2 first proposed an adatom model. However, it was STM that first gave strong evidence that each unit cell contained 12 adatoms and led Bennig 11 to propose such a model. Takayangi 7 •8 combined the stacking fault-dimer idea with the adatom model to arrive at the DAS model. The calculated intensities of over one hundred atomic arrangements were compared to the TED pattern to refine the atom positions until the projected surface geometry was well described. Figure 1 shows the projected geometry of the DAS model, where the detailed atom positions are from the MSXX force field described below. The unit cell consists of two triangular sub-cells. The surface double layer of one sub-cell has the wurtzite stacking and the other sub-cell the sphalerite stacking. The broken bonds needed to create the stacking fault are removed by dimers that form at the sub-cell boundaries. The cell corner contains a large hole with a twelve atom ring and one atom in the 235 center with a dangling bond. Holes with eight atom rings separate dimers except at the cell corners. Twelve adatoms occupy the T 4 sites, removing a net of 24 dangling bonds. Three rest atoms in each sub-cell remain each with a dangling bond. Twelve dangling bonds from the adatoms, six from the rest atoms and one in the cell corner account for the 19 dangling bonds per unit cell in the Si ( 111 )- 7 x 7 DAS model. This structure eliminates 30 dangling bonds of the 49 dangling bonds for the unreconstructed Si ( 111) surface, accounting for most of the stability of this reconstruction. The essential features of the DAS model are observed by STM. The small and large holes are easily seen as depressions in the STM micrographs. The twelve adatom dangling bonds produce tunneling current at a different tunneling voltage than the dangling bonds of the six rest atoms which are seen as twelve bright spots at one voltage and six at another voltage, respectively. 5 There remain, however, many questions. The 7 x 7 structure does not result uniquely from the DAS reconstruction scheme. Thus one can form a family of (2n + 1) x (2n + 1) DAS surfaces: 3 x 3, 5 x 5, 7 x 7, 9 x 9, etc. Of these, the 7 x 7 reconstruction is observed for the most part, although small amounts of the 5 x 5 surface reconstruction has been found to co-exist with the 7 x 7 surface in some temperature ranges. To fully understand Si surface reconstruction, we need a more quantitative description of the relative stabilities of such structures. Due to the small surface normal momentum required to make the diffraction experiment surface sensitive, TED like LEED provides little information about the vertical positions of the atoms. However, unlike LEED, dynamical effects are much less important for TED giving it an advantage for determining structure. Clearly the surface normal positions must be determined in order to obtain quantitative understanding of Si reconstruction. Other experimental techniques or theoretical calculations more sensitive to the surface normal positions can be used to refine the vertical atomic positions of the DAS model. Several investigators have proposed refined atomic positions for the DAS model. 12 - 17 A key configuration is the adatom in the T 4 site. This leads to a trigonal bipyramid with a sub-surface atom directly below the adatom. The distance between these two atoms, R13(T4 ), is referred to as the 1-3 or nonbond distance. Proposed values of R 13 (T4 ) have ranged from 2.45 to 3.lA, indicating the uncer- 236 tainty remaining in the structure characterization. From ab initio calculations on clusters, we obtain R 13 (T4 ) = 2.52 to 2.56A (depending on the site) which we believe is accurate to 0.05A. The Si-Si bond length of the dimer has also been subject to considerable uncertainty. We find values of 2.46 to 2.50A depending upon the site. This is stretched by 0.11 to 0.15A from the bulk value. The large unit cell has made impractical the use of ab initio methods to study the surface. However, ab initio calculations can be performed on clusters of moderate size that model specific configurations of the surface. Messmer 18 used the GVB ab initio method to calculate the geometries of various clusters to estimate the atomic geometry of the Si (111)-7 x 7 surface. Using GVB calculations with the interstitial electron model, 19 he also developed a silicon FF which was applied to study the Si (111)-7 x 7 surface. 20 Other methods include the density functional theory 21 •22 (LDA) and semi-empirical methods such as modified intermediate neglect of differential overlap (MINDO ) 23 and tight binding energy minimization. 13 •24 Khor et al. calculated the energies of several of the DAS (2n + 1) x (2n + 1) reconstructions using LDA and found the experimentally observed 7 x 7 surface to be slightly higher in energy than the 5 x 5 surface. 14 The 5 x 5 surface was observed experimentally (in small areas) after low temperature anneals of the cleaved Si (111)2 x 1 surface. 25 The predicted energy difference 0.008 eV /lxl cell. is probably less than the uncertainty in the method, so that these calculation do not show conclusively which surface is lowest. Chadi 13 used a semi-empirical tight binding based energy minimization method to calculate the structure and found the DAS 7 x 7 model to be energetically more stable than 5 x 5 by 0.008 eV /lxl cell and more stable than the Si (111) c-2 x 8 dimer chain model by 0.223 eV /lxl cell. We have used the Biased Hessian force field method 26 •27 to combine the second derivatives (Hessian) from ab initio calculations with the experimental geometry and spectroscopic data to generate an accurate force field (MSXX). This force field is then used to calculate the atomic structures and energies of the 3 x 3, 5 x 5, 7 x 7 and 9 x 9 Si (111) DAS models, the Si (111) c-2 x 8 structure, two Si (111)-2 x 2 structures, two Si (111)-v'J x J3 surfaces and the perfect Si (111) surface. 237 2.0 Method 2.1 The MSXX Force Field The force field of a molecule or surface can be described by the following expression: which includes bond stretch (Eb), angle bend (Ea), dihedral angle torsion (Et), umbrella inversion(E'¢ ), cross (Ex), van der Waals (EvdW ), and electrostatic (EQ) terms. 2.1.1 Bonds Bond stretching is described by the Morse potential, r' E]J = Db [e-ab(R-Rb) - 1 (2) where R is the length of bond IJ, Rb and Db are the position and depth of the well, and kb = 2Dba:i is the force constant. 2.1.2 Angles Angle bending is described by the harmonic cosine expression, (3) where 0 is the angle between bonds IJ and JK, 0a is the equilibrium angle, and k0 = C sin2 0a is the diagonal force constant. 2.1.3 Torsions The dependence of the energy on the dihedral angle between bonds IJ, JK, and KL is described by the three fold potential, (4) where cp is the torsional angle, and Vi is the barrier. 2.1.4 Inversions Given an atom I bonded to exactly three other atoms J, K, and L, we can include an inversion to describe the energy associated with the umbrella motion. We use the harmonic expression (5) 238 where 1/; is the angle between the IL bond and the IJK plane. Here the barrier to planarization is 2.1.5 Cross Terms Bond-angle and bond-bond cross terms are given by the following expression, Eax = D1(cos0 - cos0a)(R1 - Rbl) + D2(cos0- cos0a)(R2 - Rb2) (6) + Krr(R1 - Rb1)(R2 - Rb2) associated with each angle term (6), where R 1 and R 2 are the lengths of the IJ and JK bonds, kre = -D sin 0a is the angle-stretch force constant, and krr is the stretch-stretch force constant. One-center angle-angle cross terms are of the form, E1aa = G(cos0IJK - cos0aJJK)(cos0IJL - cos0aJJL), (7) where k100 = G sin 0aI J K sin 0al J L is the force constant for two angle terms (IJK and IJL) sharing a common central bond (IJ) and a common central atom (J). Two-center angle-angle terms are described by, E2aa = f(c/>)F(cos0IJK - cos0aIJK)(cos0JKL - cos0aJKL) (8) where k200 = f(c/>)Fsin0aIJK sin0aJKL is the force constant for angle terms (IJK and JKL) in which the central atoms (J and K) are bonded to each other. This interaction is important when IJ and KL are trans (dihedral angle, ¢ = 180°), but not when they are gauche ( ¢ = 60°, 300°). Consequently we define this coupling term as 1 3 2 3 f (¢) = - - - cos ¢. (9) These cross terms are considered collectively as (10) 239 2.1.6 Charges The electrostatic nonbond interaction ( EQ) is described using the Coulomb . expression (11) where Q1 is the charge (electron units) on center I, E = 1, and 1/Eo = 332.0637 converts units to give E in kcal/mol when R1 J is the distance in A. The charges used for each molecule were calculated from the Hartree-Fock wave function by using the Mulliken populations. 2.1.7 van der Waals The van der Waals part of the nonbond interaction (3) for atoms I and J is described using the exponential-6 potential (12) 2.2 The Biased Hessian Method The energy expression of a molecule can be expanded as: where the force on the i th component is: 8E Fi= - 8Ri' (14) and 82E Hij = 8Ri8Ri (15) is the Hessian. To calculate the vibrational frequencies we mass weight Hij to form Hij = Hij (MiMj )- 1/ 2 and diagonalize HU= U.t\, (16) 240 where = (CvibVi) 2 Ai gives the vibrational frequencies (with energies in kcal/mol, distances in A, and masses in AMU, Cvib = 108.5918 gives Vi in cm- 1 and the columns of U give the vibrational eigenfunctions. To extract a force field we could require that the force field reproduce this Hessian and the experimental geometry. However, the HartreeFock wave function predicts frequencies that are 10% to 20% too large and thus force constants derived from the theoretical Hessian would be too large. The biased Hessian 26 ,33 alternative combines the normal mode description (U matrix) from Hartree-Fock with the experimental frequencies to form a biased Hessian. That is, . using (17) we construct 'HBH = uHFAexpfjHF (18a) (where- rv indicates transpose), which leads to (18b) We then fit the FF parameters to 'HBH_ The (3N - 6)(3N - 5)/2 independent elements of JiBH provide constraints on the force field parameters. Requiring the force field to also reproduce the experimental geometry leads to 3N - 6 additional constraints on 8EFF\ ( oR· - ) i - O. (19) EQ The MSXX force field is obtained by simultaneously minimizing the error in fitting (20) and (21). 2.3 Electronic Structure Calculations Hartree-Fock calculations were performed using GAUSSIAN 86 28 to determine the theoretical second derivatives. We used the experimental structure where available. For others we used the HF optimized geometry. We used the 6-311G** basis (valence triple zeta on both Si and H with d polarization functions on Si and p polarization functions on H). 241 2.4 Scaling Theoretical Force Fields The MSXX force field for the Si (111 )- 7 x 7 surface is obtained from combining the force fields of Si 4 H 9 (Figure 2), Si 5 H7 (Figure 3), and bulk silicon. First the Biased Hessian method was used to calculate the force fields for SiH 3 , SiH 4 and Si2H5. Because no experimental frequencies or geometries were available for Si 4 H 9 , we used the following approach. We calculated theoretical force fields for SiH3 , SiH4 , and Si 2 H 6 in the same manner as the BH force fields except that we fit to the Hartree-Fock frequencies. Comparing the force constants for the theoretical force fields and the experimentally biased force fields for each of the three molecules led to scaling rules for each type of term in the force field. This works well for bonds, angles, and cross terms but not for inversion and torsion. We then calculated the Hessian for Si4 H 9 at the Hartree-Fock optimized geometry and fit the force field to the Hartree-Fock frequencies and geometry. The various terms in this force field were then scaled using the scaling rules obtained from SiH3 , SiH4 , and Si 2 H 6 • (It is sufficient to use only the scaling rules for disilane.) The energy versus distance curves for Si 4 H 9 from the Hartree-Fock wave function from the MSXX FF are compared in Figure 4. In order to calculate the force field for the Si 5 H 7 , we must add additional force constants to the Si 4 H 9 force field. These terms are calculated from bulk Si calculations. Zur et al. 29 reported a five parameter force field for bulk silicon obtained from fitting the vibrational modes of Si 5 H 12 calculated using the GVB-configuration interaction (GVB-CI) wave function. We use this FF as a starting point and optimize the parameters to fit the phonons and the elastic constants. Using this FF (Table 1) we reproduce the phonon dispersion frequencies and elastic constants found by Zur et al. However, the MSXX FF also includes van der Waals interactions (which tend to stiffen the lattice), and hence we reoptimized the bulk force field (to calculate accurate elastic constants, and phonons). vVe started with the Zur force field optimized with van der Waals interactions and added two center angle angle and torsion terms. The parameters were optimized to fit the elastic constants and phonons (Table 2). 242 In agreement with McMurry 30 we find that the two center angle angle force constant (with couplings only between planar angles)ref is necessary to describe the very flat transverse acoustic mode along the delta direction, Figure 5. The resulting bulk silicon force field is similar to that of Turbino. 31 We use the same terms except the addition of the van der Waals interaction. The torsion term (7) has little effect on the phonons and elastic constants. Instead we calculate V3 from the difference in energy between the wurtzite and sphalerite forms of silicon obtained from first principles local density calculations by Cohen et al. 32 The energy difference between these two forms is due to one of every four torsions being in the eclipsed rather than the staggered position. There are two torsions per atom and one quarter of these are responsible for the energy difference between the two phases. Thus the torsional barrier is one half of the energy difference per atom, or V3 = 0. 738 kcal/mol. The van der Waals interactions also affect the energy difference between the two forms of silicon. Thus we find that V3 = 0.51 kcal/mol reproduces the difference 0.016 ev per atom as obtained by Cohen from DFT calculations. 31 Because the Si (111)-7 x 7 reconstructed surface contains a stacking fault, V3 plays a role in determining the energetics of the surface reconstruction. On adding the bulk force constants to those of Si 4 H 9 to complete the force field for Si 5 H 7 , we then calculate the energy of the Si 5 H 7 cluster as a function of the cap silicon position along the C 3 axis. The MSXX FF accurately reproduces this curve, including the equilibrium 1-3 distance (Figure 6). This requires an equilibrium angle for the Si 3 - Si 2 - Si 3 (see Figure 3) angle terms of 115° (HF calculations on Si 4 H 9 lead to 122°). The HF equilibrium 1-3 distance of 2.906A was fit to within 0.02A. With the MSXX force field for Si 5 H 7 , we optimized the geometries of the (3 x 3), (5 x 5), (7 x 7), and (9 x 9) DAS Si (111) reconstructed surfaces, the centered (2 x 8), and three (2 x 3) surfaces, and the relaxed perfectly terminated Si ( 111) surface. 2.5 Optimization All calculations were carried out using POLYGRAF, 33 an interactive molecular simulations package for molecular mechanics and molecular dynamics of crystals. 243 The atomic coordinates were optimized (using conjugate gradient techniques) until the RMS force became less than 0.01 (kcal/mol) / Ao At the optimized structure, all stress components are less than 0.005 GPa. We modelled the surface using 6 layers of silicon (three double layers). The bottom four layers having undergone no reconstruction. The sixth layer of silicon atoms has its bonding saturated with hydrogen atoms. The surface unit cell is constrained to an integral multiple of the lateral size of the bulk unit cell. 3.0 Results 3.1 SiH4 Table 3 compares the MSXX frequencies with experiment. Table 4 shows the MSXX force field, the theoretical force field, and the scales. The MSXX FF leads to forces less than 0.005 kcal/molA at the experimental geometry. 3.2 Si2H6 rable 5 lists the MSXX force constants and Table 6 compares the frequencies from the MSXX and theoretical force fields for Si 2 H 6 • 3.3 SiH3 The MSXX force field for silyl radical in Table 7 leads to the frequencies and geometry in Table 8. Table 7 also gives the scaling factor for this force field (the ratio of the force constants fitting the experimental frequencies to the force constants fitted the HF frequencies). The stretch stretch cross terms do not scale the same as found for silane and disilane. The inversion force constant scales differently than the other valence terms. 3.4 Si4H9 and SisH1 Table 9 shows the force constants for Si4 H9 from fitting to the HF frequencies. To obtain the MSXX we used the scaled force constants from Si 2 H 6 plus (i) the inversion term was fitted to the HF energy curve for inversion and (ii) we adjusted the equilibrium Si 3 Si 2 Si 3 angle to fit the geometry. The IVISXX force field at the experimental geometry leads to an rms force of less than 0.08 kcal/molA. For Si 5 H 7 we used the MSXX force field for Si 4 H 9 plus additional terms from the bulk force field (Table 10). We find that we can obtain the HF optimized 244 geometry for this cluster and the HF energy curve for the Si 2 position along the C 3 axis by adjusting the equilibrium Si 3 - Si2 - Si3 bond angle. 3.5 DAS Structures 3.5.1 The R 1 3(T 4) Distance The adatom geometry for the Si ( 111 )- 7 x 7 reconstructed surface is of most interest. In the DAS model the 12 adatoms sit directly above second layer atoms in T 4 sites. The repulsive interaction between the adatom and the third layer atom tends to push the adatom outwards. On the other hand, the four coordinated silicon of the second and third layer atoms prefer tetrahedral bond angles and the triply coordinated adatom prefers angles near 113°. This tends to limit the R 13 (T4 ) distance. Ichmiya 17 found a back bond distance of 2.56A using RHEED. Tong et al. 15 used dynamical LEED and reported a back bond distance that ranged from 2.49A to 2.56A (depending on whether the adatom was on the faulted half, unfaulted half, near a corner or near the center). Tromp and van Loenen 12 found that the back bond distance must be approximately 2.6A using medium energy ion scattering. Robinson 16 used XRD experiments to estimate a back bond distance of 2.64A. Qian and Chadi 13 used a tight binding energy minimization calculation to determine the atomic structure. They reported a back bond distance of 3.lA. Messmer 18 , 20 reported a 1-3 distance of 2.52A based on GVB calculations and 2.45A from calculations on the surface using a silicon potential based on the interstitial electron model. We find R 13 (T4 ) = 2.52A to 2.56A, depending on the location of the adatom within the unit cell. 3.5.2 The R34(T4) Distance Tong finds R 34 (T 4), the bond length between the sub-surface eclipsed atom in the T 4 site and the atom directly below it, to be 2.13A or 2.18A (with uncertainties of O.lA), depending on whether the bond is in the faulted or the unfaulted side of the unit cell respectively. We find R 34 (T4 ) = 2.25A. 3.5.3 The Dimer Distance In addition to the adatom structures the dimer distances are also important. Tong 15 finds a dimer distance of 2.45A using dynamical LEED. Khor and Das Sarma 14 using an empirical potential and Robinson 16 using X-ray diffraction find 245 the dimer distance to be 2.49A. We find a dimer distance of 2.46A if the dimer is not adjacent to a cell corner, and 2.50A if it is next to a cell corner. The equilibrium bond distance in silicon is 2.35A, showing that the dimer is in tension. 3.5.4 Relative Energy of (2n + 1) x (2n + 1) DAS Structures Optimizing the structures of each of the (3x3), (5x5), (7x7), and (9x9) DAS models of the Si (111) reconstruction, we find that the (5x5) structure has the lowest energy, with (7x 7) slightly higher in energy. We find that the strain energies decreases as the size of the DAS unit cell increases (Table 10). Although the 5 x 5 unit cell has a higher tensile stress than the 7 x 7 reconstruction, the lower density of 0.0244 dangling bonds/ A 2 compared to 0.0263 dangling bonds/ A 2 for the 7 x 7 reconstruction accounts for part of the stability of the 5 x 5 cell over the 7 x 7 cell. The surface energy 0.8995 kcal/molA 2 due to the dangling bonds for the 5x5 unit cell is 0.0694 kcal/molA 2 less than the surface energy of 0.9689 kcal/molA 2 due to dangling bonds for the 7x 7 surface. Here we assign an energy per dangling bond equal t_o half the energy required to break a bond in bulk silicon (73. 7/2 kcal/mo!). On the other hand, assigning an energy per dangling bond equal to half the average energy required to break all the bonds in the silicon lattice (54.3/2 kcal/mol) gives a surface energy difference due to the dangling bonds of 0.0511 kcal/molA 2 • The 5x5 reconstruction is still slightly lower than the 7x7 by 0.0055 kcal/molA 2 • We assign an energy of 73.7/2 kcal/mol per dangling bond. Although the 3x3 DAS structure has the lowest dangling bond density (0.0226 / A2 ) of the DAS structures, it leads to stresses too large for stability. Several investigators have reported finding the 5 x 5 DAS structure to be slightly more stable than the 7x7 DAS structure. Table 10 shows the relative energies of the (2n + 1) x (2n + 1) structures from our results, and those of Khor et al. and Vanderbilt. It is unclear which structure is the lowest energy structure since both are experimentally observed on the same samples and the calculated energy difference ·is so small that the difference may be within the error of the calculation. 4.0 Discussion Because the MSXX force field differentiates between three coordinated and four coordinated silicon atoms, it can model both the geometric and elastic proper- 246 ties of the bulk as well as the energetics of surface atoms with large angle strains. This makes MSXX appropriate for the modeling of the silicon (111) surface because local adatom strains and the strain of the surface unit cell on the bulk both contribute to the energetic stability. Earlier force fields 34 - 36 are successful in particular areas, for example in the reconstruction of the (100) surface, modeling the bulk elastic properties, and melting. However, they are inappropriate for the reconstruction of the (111) surfacee The empirical potential used by Khor and Das Sarma 14 successfully describes the DAS structures. The MSXX potential has the advantage that it is partitioned into physically significant valence energy terms. Additionally, being derived from theoretical and experimental studies of small clusters, we can get a quantitative test of the concepts. Although the energy difference between the 5 x 5 and 7 x 7 DAS surfaces is too close to resolve with our current level of theory, the stability with respect to other (2n + 1) x (2n + 1) structures and the unreconstructed surface is successfully demonstrated. The geometric description given by the MSXX FF is in excellent agreement with the available experimental data on the 7x 7 surface. This is evident in the agreement between this study and experiment for R 13 (T4 ), the dimer length, and the overall strain in the system. 5.0 Conclusion Using the Biased Hessian method, we developed the MSXX FF for silicon and use it to analyze the energetics and structures for the DAS (2n + 1) x (2n + 1) reconstructions of the Si (111) surface. For n = 1 to 4, we find that the tensile stress increases as the unit cell size decreases. We find the 3x3 and 9x9 structures to be stable relative to the relaxed perfect (111) surface but much less stable than 7x7 surface. We find the 5x5 and 7x7 surfaces to be nearly equal in energy (in agreement with some agreement with some previous findings) with the 5 x 5 surface slightly more stable. For the 7x7 surface we find dimer distances of 2.50A and 2.46A for dimers adjacent to cell corners and dimers between small eight atom ringed holes respectively, in agreement with X-ray diffraction studies. We find R 13 (T4 ) = 2.52A (for the distance between the adatom and the eclipsed atom in the T 4 ) for the faulted subcell center, 2.54A for the site unfaulted subcell center, 2.53A in the faulted sub- 247 cell corner, and 2.56A in the unfaulted subcell corner. These distances compare well with RHEED and LEED data. This MSXX FF is appropriate for calculating structures containing a stacking fault since it describes the energy difference between the sphalerite and wurtzite forms of silicon. The elastic constants, and phonon dispersion curves are accurately calculated by the MSXX FF including the TA mode :flattening. Of most importance is the ability of this potential to describe the energetics, geometry and vibrations of surface silicon atoms in both T4 and H 3 sites. 248 7 .O References *To whom correspondence should be addressed. 1. R. E. Schlier and H. E. Farnsworth, J. Chem. Phys. 30, 917 (1959). 2. W. A. Harrison, Surf. Sci. 55, 1 (1976). 3. F. J. Himpsel, Phys. Rev. B 27, 7782 (1983). 4. E. G. McRae, Surface Sci. 147, 663 (1984). 5. R. J. Hamers, R. M. Tromp, and J. Ee Demuth, Phys. Rev. Lett. 56, 1972 (1986). 6. D. J. Chadi, R. S. Bauer, R.H. Williams, G. V. Hansson, R. Z. Bachrach, J. C. Mikkelson Jr., F. Houzay, G. M. Guichar, R. Pinchaux, and Y. Petroff, Phys. Rev. Lett. 44, 799 (1980). 7. K. Takayanagi, Y. Tanishiro, M. Takahashi, and S. Takahashi, J. Vac. Sci.. Technol. A 3, 1502 (1985). 8. K. Takayanagi, Y. Tanishiro, S. Takahashi, and M. Takahashi, Surf. Sci. 164, 367 (1986). 9. E. G. McRae, Phys. Rev. B 28, 2305 (1983). 10. P. A. Bennett, L. C. Feldman, Y. Kuk, E. G. McRae, and J. E. Rowe, Phys. Rev. B 28, 3656 (1983). 11. G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Phys. Rev. Lett. 50, 120 (1983). 12. R. M. Tromp, and E. J. Van Loenen, Surface Sci. 155, 441 (1985). 13. G. X. Qian and D.J. Chadi, Phys. Rev. B 35, 1288 (1987). 14. K. E. Khor and S. Das Sarma, Phys. Rev. B 40, 1319 (1989). 15. H. Huang, S. Y. Tong, W. E. Packard, and M. B. Webb, Phys. Lett. A 130, 166 (1988). 16. I. K. Robinson, J. Vac. Sci. Technol. A 6, 1966 (1988). 17. A. Ichimiya, Surface Sci. .192, L893 (1987). 18. C. H. Patterson and R. P. Messmer, Chem. Phys. Lett. 192, 277 (1992). 19. M. H. McAdon and W. A. Goddard III, Phys. Rev. Lett. 55, 2563 (1985); M. H. McAdon and W. A. Goddard III, J. Phys. Chem. 91, 2607 (1987)s 20. H.-X. Wang and R. P. Messmer, Phys. Rev. B, 41, 9241 (1990). 21. D. Vanderbilt, Phys. Rev. B 36, 6209 (1987). 249 22. D. Vanderbilt, Phys. Rev. Lett. 59, 1456 (1987). 23. P. Badziag and W. S. Verwoerd, Phys. Rev. B 40, 1023 (1989). 24. G. X. Qian and D.J, Chadi, J. Vac. Sci. TechnoL 16, 1290 (1979). 25. RJ.G. Uhrberg, E. Landemark, and L. S. 0. Johansson, Phys. Rev. B 39, 13525 (1989). 26. S. Dasgupta and W. A. Goddard III, J. Chem. Phys. 90, 7207 (1989). 27. T. Yamasaki, S. Dasgupta, and W. A. Goddard, III, J. Chern. Phys., submitted. 28. Gaussian86: M. J. Frisch, J. S. Binkley, H. B. Schlegel, K. Raghavachari, C. F. Melius, R. L. Martin, J. J. P. Stewart, F. W. Bobrowicz, C. M. Rohlfing, L. R. Kahn, D. J. Defrees, R. Seeger, R. A. Whiteside, D. J. Fox, E. M. Fleuder, and J. A. Pople, Carnegie-Mellon Quantum Chemistry publishing Unit, Pittsburgh, PA, 1984. 29. A. Zur, Ph.D. Thesis, California Institute of Technology (198). 30. H. L. McMurry, A. W. Solbrig Jr., J. K. Boyter and C. Noble, J. Phys. Chern. Solids 28, 2359 (1967). 31. R. Tubino, L. Piseri, and G. Zerbi, J. Chern. Phys. 56, 1022 (1971). 32. M. T. Yin and M. L. Cohen, Phys. Rev B 26, 5668 (1982). 33. Polygraf is from Molecular Simulations Inc., Burlington, Massachusetts. 34. Fe H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985). 35. B. W. Dodson Phys. Rev. B 35, 2795 (1987). 36. J. Tersoff, Phys. Rev. Lett. 56, 632 (1986). 250 Table 1. Force constants for bulk silicon. MSXX van der Waals Bond Stretch [eq (5)] Angle Bend [eq (6)] Angle Cross Terms [eq (8)] Torsion [eq ( 7)] 2-C A-A [eqs (9) and (11)] Dv 0.310 Rv 4.270 Rb 2.381 kb 193.0936 Db (73. 70) ke 31.2682 0a 105.0467 ksiae -14.8184 krr 3.6001 V3 0.5100 Fsia :Si3Si3 :Si3 -25.7242 251 Table 2a. Elastic constants (in GPa) and the bulk modulus (B) of silicon from the MSXX FF compared with experiment. Parameter MSXX Experiment 170.7 167.5 64.2 65.0 78.6 80.1 99.1 99.2 Table 2b. Phonon frequencies (in THz) of silicon from the MSXX FF and experiment. Phonon MSXX Experiment LA-LO(X1) 12.38 12.44 TA(X3) 4.51 4.51 TO(X4) 13.92 13.90 TO-LO(r2s) 15.56 15.53 252 Table 3. Vibrational frequencies of SiH4 • Expt. MSXX HF Scale SiH3 d-deform. 914 914 1017.1 0.899 SiH 3 d-deform. 975 975 1053.4 0.926 SiH s-str. 2187 2187 2370.0 0.923 SiH d-str. 2191 2191 2360.6 0.928 0 140.3 Modes l~vlrms 253 Table 4. SiH4 FF. MSXX HF Scale Rb 1.479 1.478 1.000 kb 400.0273 459.1177 0.871 Db (92.60) (92.60) 1.000 ke 67.5345 79.8951 0.845 0a 110.9769 112.2213 0.989 Bond Stretch [eq (5)] Si-H Angle Bend [eq (6)] H-Si-H Angle Cross Terms [eq (9)] H-Si-H kHe -10.2182 -11.1244 0.919 krr 4.021 6.030 0.667 One-Center Angle-Angle Cross Terms [eqs (9) and (10)] GsiH:HH 5.3177 7.3658 0.722 254 MSXX HF Scale Rb 1.4782 1.4781 1.000 kb 393.9586 455.8698 0.864 Db (92.60) (92.60) 1.000 Rb 2.3366 2.3517 0.994 kb 286.9195 342.3163 0.838 Db (73.70) (73. 70) 1.000 k0 57.3767 69.2450 0.829 0a 111.9874 112.6561 0.994 k0 43.3213 53.2658 0.813 0a 114.6627 116.1750 0.987 V3 0.848 1.3542 0.626 kH0 -3.5257 -3.6203 0.974 krr 3.2252 4.8795 0.661 ksi0 19.8201 26.7122 0.742 kH0 -3.7978 -4.6882 0.810 ksiH 1.3292 1.5511 0.857 Bond Stretch [eq (5)] Si-H Si-Si Angle Bend [eq (6)] H-Si-H Si-Si-H Torsion [eq ( 7)] H-Si-Si-H Angle Cross Terms [eq (9)] H-Si-H Si-Si-H One-Center Angle-Angle Cross Terms [eqs (9) and (10)] GsiH:HH -0.6003 -0.6424 0.934 GsiSi:HH -0.1914 -0.2321 0.825 GsiH:SiH 0.3563 0.4043 0.881 Two-Center Angle-Angle Cross Terms [eqs (9) and (11)] FH:SiSi:H -16.4782 -19.9482 0.826 255 Table 6$ Vibrational frequencies of Si2H5. Modes Expt. MSXX HF Scale Torsion 125 127 134.3 0.931 SiH 3-rock 379 378 416.6 0.910 Si-Si stretch 432 431 464.0 0.936 SiH 3-rock 628 625 697.1 0.897 SiH 3 s-deform. 844 854 947.9 0.890 SiH3 cl-deform. 941 946 1031.5 0.881 SiH 3 s-deform. 920 925 1043.1 0.891 SiH 3 d-deformG 940 931 1027.5 0.914 SiH 3 d-str. 2155.00 2166 2337.1 0.880 SiH 3 s-str. 2154.30 2158 2352. 7 0.915 SiH3 d-str. 2178.60 2166 2337.3 0.922 SiH3 s-str. 2163.00 2161 2346.7 0.928 l~vlavg 6.66 256 Table 7. SiH3 FF. MSXX HF Scale Rb 1.4830 1.4830 1.000 kb 323.6191 443.7879 0.729 Db (92.60) (92.60) 1.000 k0 50.9033 52.9548 0.961 0a 110.5997 110.5997 1.000 Bond Stretch [eq (5)] Si-H Angle Bend [eq (6)] H-Si-H Angle Cross Terms [eq (9)] H-Si-H kH0 -13.1042 -15.2668 0.858 krr -0.1102 4.0105 -0.027 One-Center Angle-Angle Cross Terms [eqs (9) and (10)] GsiH:HH -9.7116 -12.6668 0.767 65.9534 61.5300 1.072 Inversion [eq (8)] Fsi:HHH 257 Table 8. Vibrational frequencies frequencies of SiH 3 • Expt. MSXX HF Scale SiH3 deform. 926 926 871.2 1.063 SiH 3 deform. 996 996 1016.5 0.9800 SiH str. 1955 1955 2342.4 0.836 SiH str. 1999 1999 2361.0 0.847 0 266.6 Modes l~vlrms 258 MSXX HF 2.395 2.395 215.845 257.5712 (73. 70) (73. 70) 1.483 1.483 382.387 442.578 (92.60) (92.60) 1.485 1.484 381.585 441.650 (92.60) (92.60) 58.256 70.273 110.953 110.953 54.843 66.156 111.562 111.562 55.267 67.979 116.838 116.838 49. 711 61.146 112.207 112.207 50.8023 62.488 122.671 124.2864 Bond Stretch [eq (5)] Rb kb Db Rb kb Db Rb kb Db Si2-Si3 Si3-Hi Si3-Ho Angle Bend [eq (6)] Hi-Si3-Ho Ho-Si3-Ho Si2-Si3-Hi Si2-Si3-Ho Si3-Si2-Si3 k0 0a k0 0a k0 0a k0 0a k0 0a Torsion [eq (7)] Si3-Si2-Si3-Hi V3 · 1.020 1.630 Si3-Si2-Si3-Ho V3 2.126 3.396 -K'l/J 1.500 0.9700 1l/J e 81.0000 38.5060 Inversion [eq (8)] Si2 :Si3-Si3-Si3 259 Table 9b. Si 4 H 9 FF continued. MSXX HF Angle Cross Terms [eq (9)] Hi-Si3-Ho kH 0 -2.672 -2.602 -2.787 -2.862 kHi0 3.457 5.230 kHiHo -1.899 -1.950 Ho-Si3-Ho kH 0 5.052 3.339 krr 10. 730 14.461 Si2-Si3-Ho ksi20 -2.363 -1.914 kH 0 1.958 2.285 ksi2Ho 1.106 0.821 Si2-Si3-Hi ksi20 -4.522 -3.662 kHi0 5.974 5.120 ksi3Hi 7.598 5.637 Si3-Si2-Si3 ksi30 0.908 1.060 krr One-Center Angle-Angle Cross Terms [eqs (9) and (10)] 0 0 0 G Si2Si3 :Si3Si3 GsiaSi2:HiHo Gsi3Si2:H0Ho Gsi3Hi:HoHo GsiaHo:HiHo GsiaHo:Si2Hi 0. 7610 6.388 2.645 -0.756 -0.783 -2.476 0.992 7.251 3.002 ~0.810 -0.838 -3.001 -2.031 -2.461 Gsi3Hi:Si2Ho -2.771 -3.001 Two-Center Angle-Angle Cross Terms [eqs (9) and (11)] Fsi 3 :Si 2 Si 3 :Hi -30.891 -37.398 -19.970 -16.495 Gsi3Ho:Si2Ho 260 Table 10. Relative energies (kcal/mol) of the (2n+l)x(2n+l) DAS structures and the relaxed (111) surface from this work, Khor and Das Sarma, 14 Vanderbilt 24 and Chadi. 13 Structure This work Khor Chadi DAS3x3 -0.310 -0.326 DAS5x5 -0.341 -0.344 -0.395 DAS 7x7 -0.333 -0.335 -0.403 DAS 9x9 -0.328 -0.325 261 Table 11. Additional Surface Force Constants. Angle Bend [eq (6)] Si2-Si3-Si3 ke 11.0000 Si2-Si3-Si2 ke 11.0000 0a 109.4712 Angle Cross Terms [eq (8)] Si2-Si3-Si3 ksiae -17.0000 Si2-Si3-Si2 ksia0 -17.0000 krr 0.0000 Si3-Si2-Si3-Si3 V3 0.5100 Si2-Si3-Si3-Si3 V3 0.5100 Torsion [eq (7)] Two-Center Angle-Angle Cross Terms (eqs (9) and (11)] Fsis:Si2Si3:Si3 -24.3179 -24.3179 262 Figure Captions Figure lo Top view of the optimized 7 x 7 DAS model. Blue shaded atoms denote silicon atoms in the corner of the unit cell with a dangling bond. Yellow shaded atoms are surface silicons in H 3 sites and with a dangling bond. Silicon adatoms in T 4 sites have a dangling bond and are shaded red. All other atoms are Si, shaded orange and have four bonds. Figure 2. The Si 4 H 9 cluster. Figure 3. The Si 5 H1 cluster. Figure 4o The ab initio and force field energies plotted versus the central Si atom position for Si4H9. Figure 5. Phonon dispersion curve for SL Figure 6. The ab initio and force field energies plotted versus the cap Si atom position for Si5H7. 263 Blue - Corner Si (dangling bond) Figure 1. DAS 7x7 Si (111) Surface 264 Figure 2 265 Figure 3 266 Figure 4 Si4h9 Energy vs. R 14 ···-·-········+·-·-··--····+-··········-·-· ········-··-····· ·········-······- ······-··-····· ···--········-··t 10 6 i = i .. - - . . I············-··+-l-··· ..... t~·. · -· · · l _ _ rt, ···-··········r·················t .. .............. ···-············· ············-····· ··················· ····-··-····-....i ···-·····-···---r '.',.............. ·-················ ······-·····-·-· ········-········· ... · · · · · · · · · · · · -· · ·-· ·-· · · ·-· -· · · ·. I.· · · -· · · · · !· · · · ·-' ',.:.- -◄ l i ' I 2 -2 -6 -10 -14 --0~5 -0.4-0.3-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 R (A) 267 Figure 5. Phonon Dispersion Curve for Si A X K Wavevector {2n/a) r ~ 2.4 ' JJJ = cu '-4 -50 --45 -40 : : : . 2.5 . : • . : • ... . L -~ ,, 2.6 2.7 2.8 ·····I··········+······ . ( ___ ,__________ ········I· ...•• : : ·r. . . · ~ : Z(A) 2.9 3 "' - L. . . . -',i'r·········· 3.2 i ! : . 3.4 .. •.......... . : 3.3 • ••••....•• : .......•.•. , 3.1 : . ~ ~·· : ...... -r : . i .••········t··· : ···,· ,. ··:i . : . __,,,.,.,. ....... .;. ............, : i -35 bO : : :::::::t.\·\.l. . . . . . J.. ·····+········+··········-r-·········--r--··········rJ~.:. ..l...........t . . . . ·+······t···· ··t· ·· · l . . . . . - - - l' t'\: .. .i :i .L..........:f . p" ~- - - - - - .i ··· ·+i. · · ·· ·..tL... ·.......·\~r.~··..·······1···········-r-·· i. l. ······•·,l. . _ . . . ~ ~ . . . · · .f ..········+ · · · · .... il,' . . i r i i i \··1······,····· J... ···· ·· ~ --o--Ab initio - • - Force Field -·--1---·1-·-:~~[~~~:~~,~: ~:~:1:~:~~~~,~~~:~~1~~~~:~1~~~~1~~~-· -30 -25 -20 ~ ..._., y fU ,-,.I Cl) .......... e 0 ,-,.I -15 -10 Figure 6. Si5H 7 Energy vs Z m 00 tv 269 Chapter 7 The Generalized London Force Field for Hydrocarbon Reactions 270 Abstract In this chapter we use the Generalized London Force Field (GLFF) method to derive the potential energy surface (PES) for the exchange reaction: CH4 + H --+ CH3 + H 2 • The GLFF includes the effect of the Pauli Exclusion Principle on the PES of chemical reactions. We find excellent agreement with the PES from ab initio quantum chemical reactions (GVB*SD CI)o 271 1.0 Introduction In chapters 2 and 3 we described the calculation of transition barriers for reactions of diamond surfaces (for CVD growth and nanotechnology). The calculation of the potential energy surface for these reactions to determine the transition state geometry and the activation barrier required a large quantity of CPU time on a Cray supercomputer. This chapter describes a method for calculating the same PES, but at a very small fraction of the computer cost. Chemical reactions such as (1) generally lead to a reaction barrier [~9.6 kcal/mol for (1)] much less than the bond energy [10% for (1)]. There are clearly subtle interplays of bonding and antibonding factors where the bond is never really broken, but rather as one bond is dissociating the other is forming. Thus the transition state is a resonance between the reactant and product states. The transition state barrier is a direct consequence of the Pauli principle which allows only two electrons per orbital. As the third electron (of the H) approaches the D 2 , it must remain orthogonal to the bond pair on the D 2 ( to satisfy Pauli), leading to antibonding character and hence a barrier. Such reaction barriers are adequately described with the modern methods of quantum chemistry, however they generally require intensive calculations, especially if quantitative accuracy is required. Furthermore, the generation of a PES is even more intensive since to generate the surface the wave function must be calculated at a grid of geometries along the reaction path. In contrast, the FF for a Molecular Dynamics (MD) study would use a bonding interaction (say a Morse function) between each particle. This incorrectly predicts that the H 3 molecule is stable, with a triangular geometry and a bond energy of ~100 kcal/mol rather than a barrier of 10 kcal/mol and a linear transition state. This property prevents the simulation of reaction dynamics since the description of the reaction surface is poorly described. For a proper description of the reaction surface, must include the effects of the Pauli principle, which are rarely incorporated into MM potentials. Of course quantum chemistry calculations automatically include the Pauli Principle. However 272 z. the quantum chemistry (QC) requires a great deal of work for each geometry and we would like a way to fit the QC with a general potential function ii. the size of the QC calculation grows rapidly with the the size of the system, making it impractical for many cases of interest and we would like to have a way of predicting the interaction energies in the absence of QC. We have developed a general procedure, 1 - 3 the Generalized London Force Field (GLFF), for accomplishing both objectives. This methodology allows a general procedure for all reactions and permits development of routines for using this procedure with commercial molecular simulation codes such as POLYGRAF. 4 The GLFF methodology is straightforward for any simple (three-electron) radical reaction, e.g. ·A+ X - B--+ A - X + ·B. (2) including surface reactions such as •A+ H - S1 (3) The same formalism can also be used for simple four electron reactions, such as metathesis A A-B ⇒ and insertion I B I C-D C D A-B A B ⇒ C=D I (4) I C-D (5) (e.g. Ziegler-N atta polymerization). 2.0 The Generalized London Force Field 2.1 Spin Coupling and the London Potential Consider the bonding in the reactant (Figure la), transition state (Figure lb), and product (Figure le) as given in Figure 1. 273 The consequences of the Pauli principle can be directly expressed in terms of the spin couplings shown schematically in the middle row of Figure 1 and more explicitly in the bottom row. For the reactant configuration of (1) the Valence Bond (VB) wave function is A{cpzc/Jcc/Jr[a(a/3- /30:)]} where the electrons on the D2 always have opposite spins, leading to the singlet or bonding state of D2. (Here cpz, c/Jc, c/>r refer to atomic orbitals on the left, center, and right atoms.) However the spin on the H is sometimes the same and sometimes different than the spin on the D atoms. This leads to an H - D interaction that contains both antibonding (or triplet) interactions and bonding (or singlet) interactions. Analysis of the VB wave function shows that the HD interaction is 75% triplet and 25% singlet. The result of these triplet or anti bonding terms is that the energy increases as H approaches D 2 • For the transition state the VB wave function is (6) which describes the resonance of H -D D· with H • D-D. Analysis of the wave function shows that the outer orbitals are always triplet coupled whereas the middle atom has interactions with the outer two atoms that are 75% singlet and 25% triplet. Thus during the reaction the interactions between the H and central D changes from 25% singlet (reactant), to 75% singlet (transition state), to 100% singlet (product) and the other interactions change correspondingly. Thus to include the effect of the Pauli principle, the simplest description is E(R1, R2, R3) = L Eij(Rij) (7) i>j where (8) Here f s and JT are the fractions of singlet and triplet character and Es and ET describe the bonding and antibonding two-body interactions. Defining the classical and exchange energies as (9) 274 and the spin coupling fractions as (10) the energy (7) can be rewritten as E= L [Ei](Rj) + COS/ijEij]. (11) i>j The three spin coupling angles in (11), rij, are related to each other (by ± 2; ) so that there is only one degree of freedom. Requiring that , be optimum, 8E / 8, = 0, leads to (12) Since (12) depends only on the exchange energies, Ex(Rij), which depend only on the distances, the optimum spin coupling is uniquely determined by the geometry. Thus the full effect of the Pauli principle is included by using (12) with (11), leading to EL= Ef +E~z+E~z_ [(Ef) 2 1 + (E2) + (E3) - Ef E 2 - E 2E 2 - E 2E 3 - Ef E 3] • 2 2 2 (13) This expression was originally derived by London. 5 - 7 Table 1 shows the special cases of the energy for various ,1's. 2.2 Generalization of the London Force Field Using the exact two body functions for the triplet and singlet states of H 2 in the London expression (1), leads to a potential surface that is qualitatively correct. Thus for H 3 it leads to a barrier of 12.4 kcal/mol rather than 9.6 kcal/mol 8 and a saddle point geometry of 1. 7922 bohr rather than 1. 7757 bohr. In addition the barrier is too narrow having too low an energy for particles along the reaction path far from the transition state. Donnelly et a1. 1 , 2 showed that the problem at 275 the saddle point involves three body corrections when all three orbitals have high overlap. By substituting the two-body VB energy expressions with S~b # 0 into the three-body VB energy expressions, also with S;,b -=I=- 0, and retaining terms in overlap squared only, the first order overlap correction to (13) was found to be 1 ' 2 The problem with (11) is that it considers overlaps Sij as addition to the exchange energies. This is not acceptable since we will generally not have overlaps. However Sfj is proportional to E 0 so that S'fj in (11) can be replaced with S~== 8° E~iJ 1,J ' (15) where 8° is a scaling constant. Using just this one parameter, the overestimate of the energy barrier is removed. The resulting three-body correction depends only on Ex. For larger distances the energy for (12) is too attractive because interference between the London dispersion (vdw attraction) terms of different pairs of atoms is ignored. The dynamic correlation effects (responsible for this attraction) between atoms 1 and 2 interfere with the 1-3 and 2-3 dispersions, leading to less attraction. This dispersion correction has the form Lldisp = -{i L Ej (sfk + sJk) (16) i>j where Ef is the (negative) dispersion energy of pair i, and 8d is a constant. A good estimate of the dispersion energy is found from the difference between the exact energy (experiment or configuration interaction) and the VB or GVB energy. The Generalized London Force Field (GLFF) has the form EGLFF(R1, R2, R3) = L [E5 (Rij) + COS"fijEij(Rij)] + Ll3 + Lldisp 1 (17a) i>j [Egt (Rij) + COS"fij Eij] - L (sfk + SJz) [ ~ Eij + {jd Ej] k#i,j (17b) 276 E3 1(Rij) + Eij COS"fij - ~ L (SJk + SJ1) k#i,j - 8dEt L (S[k+ SJ1) k#i,j (17c) where 8° is given by (16). 3.0 Ab Initio Calculations To derive ab initio potentials (PES) that include the essential effects important in describing saddle points of chemical reactions, one must employ QC methods that accurately describe bold dissociation. Chemically accurate descriptions of the PES generally require a large computational cost. Our goal is to predict accurate reaction surfaces using a simple method practical for molecular dynamics simulations. We will illustrate the procedure for the reaction (21) Elsewhere 11 we required extensive GVB-CI calculations on (2). These studies involved Generalized Valence Bond (GVB) calculations followed by configuration interaction through single and double excitations from all GVB configurations. This is denoted as GVB*SD CL These studies used an extended basis set [Dunning and Huzinaga 9 , 10 doublezeta contraction of the 9s5p carbon basis with one additional set of d polarization functions (rJd =0.75). We also add diffuses and p functions (rJ 8 = 0.0474 and (rJP = 0.0365). The hydrogens were described using the triple-zeta contraction of the 6s set with one additional set of p polarization functions (rJP = 0.60)]. The final potential surface is shown in Figures 2. This method has been applied to the (22) and (23) reactions using H · ·H, H · · · CH3 , and CH3 · · · CH3 two-body interactions from QC. The potential surface for (20) is shown in Figure 3. The barrier from GLFF 277 is 10.6 kcal/mol, in excellent agreement with the 10.9 kcal/mol from extensive QC calculations on the reaction (17). 4.0 Discussion The approach outlined above can be used for any three electron (or four electron) exchange reaction, A- B + ·C ~·A+ B - C. The first step is to obtain the singlet and triplet states for A-B and for B-CG For best results we should do QC calculations at various distances for A-B and B-C separately. With no additional data on the A-B-C system, the London Equation would predict the full potential surface in Figure 2. This leads to a barrier rv25% too high and transition state bond distances about 3% too long. If there were no additional information, this could be used to simulate the reactions. Better yet we might do a quantum mechanical calculation at a single point near the predicted saddle point and another single point half way to dissociation. This would determine 8° and od, allowing an accurate description of most geometries. Thus the G LFF allows a little bit of QC information to predict a great deal about the potential surface. Alternatively with no QC data on the reacting units, we could use data, say, on the barrier height from experiment to estimate the overlap correction constant (8°). In addition, if there were no QC data on the A-B bond and antibond, one could get a qualitative estimate by using a Morse curve for the singlet or bond state (which requires only Re, ke, and De) and we could use an antimorse curve for the triplet or anti bond state (no additional data). 5.0 Application to Hydrocarbon Reactions We have shown that the Generalized London Potential is effective in modeling H2 + H and CH3 + H by comparing the GLFF results to accurate ab initio calculations. Because quantum chemistry calculations of potentials of systems with more than several electrons are so impractical most systems have little available data. We developed the GLP with the intention that it could accurately predict the 278 entire potential energy surface using only two-body potentials plus a limited amount of three-body information. Our results indicate that the GLP is indeed well suited for such predictions. An accurate PES can be derived in a matter of hours or minutes, on a small computer where quantum chemical calculations have taken tens of hours of super computer CPU for PES derivation on similar systems. 11 , 12 Actually, the GLP may work best to compliment ab initio calculations in reducing the amount of work required to find the saddle point. An expected sequence would be: 1. Calculation of two-body curves. 2. Use of London Potential to predict saddle point. 3. Use of GLP using parameters estimated from similar systems to improve estimate of saddle point. 4. Ab initio calculation of several energies near saddle point. 5. Incorporation of new information into GLP to improve estimates of entire potential energy surface and saddle point. 6. Possible refinement of GLP parameters using a small number of additional ·ab initio points. 7. Use of GLP for simulations. Our overall objective is to perform accurate reaction dynamics with molecular modeling where reaction rates would be accurately modelled and mechanisms could be studied, and extracted from the dynamics trajectories. We are especially interested in applications to surface reconstructions, growth of thin films, polymer and crystalline systems. Important carbon-based examples exist for all of these cases. We first examine the case of hydrogen exchange among hydrocarbons~ We first considered the abstraction of one hydrogen from methane by an atomic hydrogen: Actually, this reaction as written is uphill in energy by several kcal/mol, however this is not true for most hydrogen abstractions from hydrocarbons by a lone hydrogen atom. Thus, we write this equation in this direction, however it is more appropriate to write it in the opposite direction for this system only. With the C H 4 bond angles fixed, we first calculate a first approximation of hydrogen ab- straction from a diamond surface. Of course, an improved model would consider 279 abstraction from isobutane or larger clusters. Allowing the CH4 bond angles to relax gives this reaction an extra degree of freedom not present in H 3 • This better describes smaller systems and polymers where relaxation is not constrained by the surrounding structure as in a surface reaction. For both the fixed and variable angle cases the two-body interaction potentials are required. We present them next before their use in the GLP. 5.1 Application to the CH4 + H-+ CH3 + H2 Reaction The GLFF is expressed in terms of two body potentials. Thus for (18) we need to consider the (19) and H+H (20) potentials. We carried out calculations on (18) and (19) at the same level of basis set 9 , 10 and electron correlation as previously used for ab initio studies of (17). The details of the calculations are in section 3.0. The potential curves are shown in Figures 2 and 3. In (18) we considered three values 0HcH = 90°, 105°, 120°. For consistency we used a harmonic fit to predict the choice of 0HcN for each particular RcH• The constants 8° and [Jd were chosen by fitting to the saddle point and 50 % dissociated energies for the reaction. 5.2 Comparison to ab initio calculations The GVB-CI calculations optimized the structure near the saddle point for reaction (17). As a first step we used the transition state geometries from these calculations to calculate the London and GLFF energies. The results are in Table 2. Here we see the LFF leads to an energy barrier too high by 2.1 kcal/mol or 15%. Using 8d = 0.003 from the H 3 studies leads to EAcT = 12.63kcal/mol. Adjusting to match the ab initio surface leads to EAcT = 12.9kcal/mol. In the remainder of this chapter we use 8d = 0.003 from H 3 • 280 To obtain a full potential surface, we first considered the same angle 0HcH = 104° for all RcH and RHH· This leads to the London and GLFF PES in Figures 2 and 3. We then used the spin coupling at each geometry to determine the optimum 0HcH by minimizing the energy (8) with respect to the angle. We recalculated the London and GLFF potential surfaces shown in Figures 2 and 3. Very little change occurs in the spin coupling so that additional iterations were not needed. We conclude that GLFF gives an excellent description of the QC potential surface. 5.3 H2 and CH4 Potentials When all of our information is based on ab initio calculations, it is best to use a consistent level of calculation for all potential surfaces. The C H 4 + H calculations of Mus grave et al. on the saddle point of CH4 +H 2 are not to high accuracy, however a similar level calculation is feasible on larger systems. Calculations at a similar level of accuracy were done for H 2 and CH4 1 and are reported in Tables 3 and 4 for H2 ·and CH4, respectively. We include the tetrahedral geometries for the fixed-angle, surface-like atom case. For a lone CH4 , the angle between the H whose bond is being stretched and the other hydrogens relaxes from tetrahedral to planar during the course of the reaction with H. For lone CH3 , about 7 kcal/mol is required to go from planar to tetrahedral. Calculations of the snap bond energies at various angles are possible which can then be used to do the general relaxed angle case. To simplify matters here, we report the energies at 104° which is about the optimized saddle point geometry. 3 , 11 5.4 Fixed Angle CH4 + H We consider the first approximation of using CH4 with the bond angle fixed at tetrahedral and calculate the resulting potential energy surface. We start from spline fits to the H 2 and CH4 energies of the previous subsection. The London Potential for this system is shown in Figure 2. The C H 3 + H2 limit is taken as the zero of energy. While the bond energies of CH4 and H2 are only several kcal/mol apart, the additional energy to bend CH3 to tetrahedral makes the energy difference in the channels about 9 kcal/mol. 281 As expected, the saddle point is closer to the higher energy H 2 channel. The corresponding spin coupling is shown in Figure 1. This shows that the "transition state" as defined by the spin coupling (, = 0) is even further into the H 2 valley than the saddle point. The smooth variation of , along the reaction path is again seen as for H 3 in Figure 4 and for CH 4 + Hin Figure 5. The potential energy surface is improved by assuming that the parameters of the H 3 GLP are about on the order of those for CH4 + H. Given the similar bond energies, this should be a reasonable approximation. The corrections from second order overlap and dispersion for H 3 lead to the new surface of Figure 3. Thus, we have estimated the entire potential energy surface for CH4 + H using only two-body curves and our knowledge of the London Equation error for H3. With more experience in the variation of the GLP parameters across various systems, we can hopefully establish a methodology of creating such surfaces to a reasonable accuracy with little more than two-body input. 5.5 Relaxed Angle CH4 + H In order to compare the CH4 + H results of Reference 2 with the GLP results, we need to allow the C H 4 bond angle to relax as the reaction proceeds. The singlet and triplet energies at several angles for each CH3 - H distance could be calculated in order to obtain the angular dependence. If the angle relaxed to its singlet minimum, it would be planar at far distances, near tetrahedral at the bond distance, and slightly beyond tetrahedral at shorter distances. At the saddle point distance of about 2.6 bohr, the energy of the singlet is a minimum for an angle between 106 and 107°. This is in contrast to the optimized angle of 103 . 8° of Reference 11. The discrepancy arose because the energy minimum with respect to angle for the triplet is much nearer planar. From the spin coupling ti, we know the mix of singlet and triplet character for a given pair of atoms i: E,; == jf E! + f~E~ II 'l, 'l, 'l, 'l, • This is the function to be minimized with respect to bond angle at each geometric configuration. Even without knowledge of the CH4 triplet and singlet energies at various angles and distances, we can still estimate the energy at the saddle point. Using 282 the 104° energies of CH4 from Subsection A, the saddle point for the relaxed angle C H 4 + H reaction can be found. This was done for the London Potential and the GLP with second order corrections based on the H 3 parameters. Table 5 shows that predicted GLP energy is within 0.3 kcal/mol of the ab initio value. Separate H 2 and CH4 corrections would improve the saddle point position. The information that went into this prediction includes only two-body H 2 and CH4 potentials, an estimate of two GLP parameters from H 3 , and an estimate of the CH4 bond angle at the saddle point. This last piece of information is also derivable independently by knowing the angular dependence of the separate singlet and triplet curves at the saddle point distance. In addition to this saddle point information, we now have an estimate of the potential energy surface over the entire range of geometries. This can be used to add reactive dynamics to molecular modeling simulations. 6.0 Conclusions We use the Generalized London Force Field ( GLFF) method to derive the potential energy surface (PES) for the exchange reaction: CH4 + H - t CH3 + H 2 • We find that we are able to accurately determine the PES when including dispersion and three body interaction corrections. The GLFF includes the effect of the Pauli Exclusion Principle on the PES of chemical reactions and greatly speeds up the calculation of accurate PES over quantum chemical methods. Our results show excellent agreement with the PES from ab initio quantum chemical reactions (GVB*SD CI) where primarily only two-body interactions were included. To describe other PES for other reactions a set of accurate two-body potentials are requiredc With these potentials it is hoped that a large number of reactions could be described accurately and with vastly reduced computational effort. 283 References 1. R. E. Donnelly, Ph. D. Thesis, California Institute of Technology, (1992). 2. R. E. Donnelly, C. B. Musgrave, and W. A. Goddard III, to be submitted. 3. C. B. Musgrave, R. E. Donnelly, and W. A. Goddard III, to be submitted. 4. These calculations used POLYGRAF from Molecular Simulations Inc. (Burlington Mass). 5. J. C. Slater, "Quantum Theory of Molecules and Solids" Vol. 1 (McGraw-Hill Book Company, Inc., New York, 1963). Page 53 contains two "typos": add e-w after a 2 (1 + w - ½w 2 ) on -a 2 (K + S) line and replace ½w 2 with ½w 3 on aK line. Also note that Rydberg (not Hartree) energy units are used. 6. F. London, Probleme der Modernen Physik-Sommerfeld Festschrift, (S. Herzel, Leipzig, (1928). 7. Ztschr. f. Elektrochem, 35, 552 (1929). 8. Huber, K. P. and Herzberg, G., Molecular Structure IV: Constants. of Diatomic Molecules, Van Nostrand Reinhold Company (New York), 1979. 9. Dunning, T. H., J. Chem. Phys., 53, 2823, (1970). 10. Huzinaga, S. J., Chem. Phys. 42, 1293, (1965). 11. C. B. Musgrave, J. K. Perry, R. C. Merkle, and W. A. Goddard III, Nanotechnology 2, 187 (1991). 284 Table 1. Special Cases of Energy for Various 1's. 0 7r 3 21r 3 7r 7r 3 285 Table 2. Geometries and transition barriers for CH4 + H · ---+ • CH3 + H 2 (energies in kcal/rnol). RcH Reaction 109.47 1.09 CX) TS 104.0 1.459 0.868 Product 90.0 CX) 0.7 GVB-CI London GLFF 12.91 15.00 12.64 286 Table 3. Two-Body Potentials for H 2 • Distances are in A (note that so far we have dealt mainly in bohr), and energies are in Hartrees. R Es ET 0.4 -0.96387083 0.5 -1.09183076 -0.57411048 0.6 -1.14695975 -0.68374487 0.7 -1.16604573 -0. 75632627 0.74 -1.16781588 0.8 -1.16641530 -0.80860353 0.9 -1.15698075 -0.84849745 1.0 -1.14262579 -0.87990173 1.2 -1.10924298 -0.92499045 1.4 -1.07765713 1.5 -1.06391438 1.6 -1.05181371 1.8 -1.03262628 -0.98315379 2.0 -1.01952662 -0.99000541 2.5 -1. 00460386 -0.99738858 3.0 -1. 00090262 -0.99930581 5.0 -0.99988367 -0.99988112 50.0 -0.99988096 -0.99988096 -0.96391211 287 Table 4. Two-Body Potentials for CH4 • Distances are in A, and energies are in Hartrees. Tetrahedral 104° Es ET Es ET 0.80 -40.25584 723 -39.87551304 -40.2504 7509 -39.86470574 0.90 -40.32346617 -39. 94289320 -40.31871254 -39.92518046 1.00 -40.35288348 -39.95152791 -40.34895780 -39. 96925500 1.05 -40.35863390 -39.96678868 -40.35518093 -39. 98352944 1.09 -40.36023885 -39.97693589 -40.35718587 -39. 99279529 1.15 -40.35887945 -39.99059157 -40.35645516 -40.00497786 1.20 -40.35516185 -40.00201942 -40.35328124 -40.01514904 1.25. -40.34974849 -40.01428129 -40.34842353 -40.02631230 1.30 -40.34311129 -40.02727004 -40.34234854 -40.03843393 1.40 -40.32756398 -40. 05334920 -40.32792504 -40.06325829 1.50 -40.31062243 -40.07708134 -40.31207546 -40.08607365 2.00 -40.23606332 -40.14743236 -40.24161279 -40.15436516 2.50 -40.19695719 -40.16986183 -40.20385300 -40.17646641 3.00 -40.18315200 -40.17612653 -40.19006031 -40.18274 728 4.00 -40.17853794 -40.17815402 -40.18525719 -40 .18483343 6.00 -40.17824863 -40.17824772 -40 .18493831 -40.18493718 50.00 -40.17824051 -40.17824051 -40.18493026 -40.18493025 R 288 Table 5. Saddle Points for CH4 + H. Distances are in atomic units, and saddle point energies are from H 2 channel in kcal/mol. R1sp London 2.6095 1. 7435 14.9974 GLP 2.6448 1.6998 12.6379 2.63 1.74 12.91 Reference 39 289 Figure Captions Figure 1. The product, transition and reactant states of a H exchange reaction. Figure 2. CH4 + H London Potential Contour Plot. The contour values shown are the powers of two times ±3 kcal/mol and zero. Axes are distances in atomic units. H2 channel is to the upper left and CH4 channel is to the bottom right. Figure 3. CH4 + H GLP Contour Plot. The contour values shown are the powers of two times ±3 kcal/mol and zero. Axes are distances in atomic units. H 2 channel is to the upper left and C H 4 channel is to the bottom right. Figure 4. Contour Plot of, for H 2 + H. Contours are spaced every 10 degrees with negative contours endashed. Figure 5. Contour Plot of, for CH4 + H. Contours are spaced every 10 degrees with negative contours endashed. 290 4.8 l'J .::i,. C1' CS;» 4.0 CS;» ~ 96 CS) CS) -1:J 1.7570 bohr Esp /Cb N 3.2 - Rsp . - ~ -e 9.8000 kcal/mole 0.0210 hartree/bohr2 0.1067 harlree/bohr2 CS) -0.0580 hartree/bohr 2 . CT 6\ \ 2.4 \ \ \ 24.0 \ 24.0 \ ' ' -..... 1.6 6.00 ' 6.00 24.0 24.0 0.8 0.8 1 .6 2.4 3.2 4.0 4.8 Figure 1. Truhlar and Horowitz (LSTH) Contour Diagram. Reported saddle point properties are actually those of CI calculations of Liu et al. 291 5.6 . cg "l;f' C\I I\.) b,. 4.8 cg 0IS} cg IS} ~ .,c, . 4.0 R1sP - 2.6863 bohr R2sp - 1.6913 bohr Esp - 12.8396 kcal/mole CS) "'=I' C\I • (\.) &a .G 3.2 24. 0 - - - - - - - - 2 4 . 0 - - - - - - 1 6.00 0.0 2.4 1.6 0.8 1.6 2.4 3.2 4.0 4.8 Figure 2 C H 4 + H London Potential Contour Plot. The contour values ·shown are the powers of two times ±3 kcal/mole and zero. Axes are distances in atomic units. H2 channel is to the upper left and C H4 channel is to the bottom right. 292 5.6 tS;a CSl ..0 4.8 = 2. 7585 bohr R25 p = 1.6405 bohr Esp = 10.6080 kcal/mole R 1 sp cg 4.0 'l ~ C\I I N . .(l, &a CS\ G \ ' 3.2 \ \ ' 24. 0 - - - - - - - - 2 4 . 0 - - - ~ \ \ --------===6.0.00 0 - - - - - 1 \ ' 2.4 0~ - -b.00 --~ ----------=~;--;;-:::----=-~6~-0~0~=-=-=-=-:::i ---- :::::'.~-------I 1 •6 0.8 1.6 2.4 3.2 4.0 4.8 Figure 3 C H 4 + H GLP Contour Ploto The contour values shown are the powers of two times ±3 kcal/mole and. zero. Axes are distances in atomic units. H2 channel is to the upper left and C ...1I1 channel is to the bottom right. 293 5.6 y / / / / / / 4.8 . ~ / / / 4.0 / ,re~- ~ / / I / / I 3.2 / / / I / 2.4 / / / / 0.8 I I I I I / / / / / / 1.6 / I 't;,.(b •. / / / I / / I I I ~ , / I I / (l) I ~/J; I / / I / / / / / / I ,~~- I / / / I / (l) / / I / / / / , ~~· / / / (b / / / / / / / / / / / / / / Cb. / / ,,.,; / / / / / 1.6 2.4 3.2 4.0 4.8 Contour Plot of --y for CH4 + H. Contours are spaced every 10 degrees with negative contours endashed. Figure 4 294 4.8 / / / 4.0 / / / / / . / / / / ~- / /'C (b / 2.4 / / / / / / / / 1.6 / ~/ / / ~- (b / / 0.8 0.8 1.6 / / / / / / / / / / / / I / / ,""/; I ~ / / I / / / / / / ~ / / / / ¢) (b. / / (b. / / / / ,'bi / / / / / / / ~ / / / / / / / , / / / (b / ~¢)· / / 3.2 / / / / / 2.4 3.2 4.0 4.8 Figure 5. Contour Plot of, for Linear H 3 • Contours are spaced every 10 degrees with negative contours endashed.