Angle Resolved Photoemission Calculations Citation Sheard, Sarah A. (1979) Angle Resolved Photoemission Calculations. Master's thesis, California Institute of Technology. doi:10.7907/JECM-5E41. https://resolver.caltech.edu/CaltechTHESIS:04292010-100441032 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Weinberg, William Henry Thesis Committee: Unknown, Unknown Defense Date: 1 September 1978 Record Number: CaltechTHESIS:04292010-100441032 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04292010-100441032 DOI: 10.7907/JECM-5E41 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5753 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 07 Jul 2010 15:18 Last Modified: 27 Nov 2024 00:59 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Angle Resolved Photoemission Calculations Thesis by Sarah A. Sheard - In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California June 1979 (Submitted September 1, 1978) to women in science tt Acknowledgments This thesis would not have been possible without the support and encouragement of my advisor, Dr. W. Henry Weinberg. Dr. Stephen L. Cunningham and Dr. John T. Yates were also invaluable for the theoretical and experimental portions of my work, respectively. I thank them all for their help. And I owe my sanity to two friends, Tom Stevens and my coworker Jenna Zinck; for their friendship I offer heartfeld thanks. To others, too numerous to-mention individually, including the Booth Computing Center and the Noyes Glass Shop, my gratitude. I also thank Kathy Lewis for her excellent typing. In addition, I would like to acknowledge the support of the Danforth and National Science Foundations for graduate fellowships and the National Science Foundation for research support. Sally Sheard tit Table of Contents Angle-Resolved Photoemission. . . 2 6 ee eee ee ee ete ee Introduction Formalism . . eo «© © © © © © e@e© © © @© -# @© #© © @ @ @ @ &#® @ #© @# @ e © «© © e@© e#& &® © © © @ @© #@ e@ @ Suggestions for Future Work... . o # ee # © #¢ e© @ #© @ © @ @ References. . . 2. 2 6 + ew ee : | APPENDIX 1. Angle-Resolved Photoemission Programs “APPENDIX 2. Laser-Induced Chemisorption Studies APPENDIX 3. Propositions .... 1... 2 1 ee eee ee ew eens i. ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY INTRODUCTION When high energy electromagnetic radiation interacts with molecules or solids, electrons are emitted. The study of these electrons is called photoelectron spectroscopy, or photoemission when dealing with metals. Angle-integrated photoelectron spectroscopy has been used routinely since the early sixties! to probe electronic states of molecules in the gas phase. Under such conditions, the random orientation of the molecules causes almost all angular dependences due to molecular properties to be averaged out. As a result, this technique produces peaks in electron intensity as a function of energy only, and is fairly straightforward to interpret. But a great deal of information is lost by this angular averaging. Recently, interest has grown in extracting and analyzing this angular information. In order to measure angular dependences, one must first orient the molecules coherently. Hence, it becomes apparent that this technique is well suited to studying chemisorptive systems, where the metal surface orients the molecules by breaking the symmetry of their environment. In many cases, chemisorption perturbs the molecular orbital energies little enough that fruitful comparisons can be made with gas phase studies. Furthermore, surface scientists have long been asking questions that angle-resolved photoemission is likely to help answer. Being more sensitive to the metal surface than to the bulk crystal, photoemission can a. examine the admolecules while they are chemisorbed. Angle~integrated ultraviolet photoemission can determine the energetic positions of their valence orbitals but it can say very little, or nothing, about their bonding character, symmetry, or orientations. One must consider angular dependences to extract this information. 1. Previous Work Interest in the application of angle-resolved photoelectron spectros- copy to chemisorption surged with the publication of a paper which calculated the spectrum expected from a single isolated C0 molecule.* James Davenport used Xo approximations for the initial state molecular orbitals and for the final state electron waves. Different angular distributions were calculated for different initial state orbitals. Such a calculation can be compared to measured photoemission distributions of any oriented CO molecules, and specifically to CO molecules adsorbed on metal surfaces. In the comparison, one could expect to find information on the "tip angles" of CO with respect to the surface, and on the extent of orbital mixing. The subject of orbital mixing has been extensively pursued by J. W. Gadzuk.?**9° Using Green's functions, he has studied the "surface- molecule" limit in which molecular orbitals of the adparticle mix with | surface bands to produce a localized state called the surface-molecule. Photoemission from this state will then reflect this orbital mixing and can presumably be applied to distinguish “atop" sites from bridged and fourfold sites. Two major problems exist with his approach. One is that 7 the final state is treated as a plane wave, which others’ have shown cannot reproduce experimental spectra. The second problem is that the extent of orbital mixing must be used aS a parameter. This means dis- tributions must be calculated with different values of the mixing parameter as well as of orbital energies and orientations. It is very . difficult and time consuming to try to fit this many parameters to the experimental data. A different approach was used by Liebsch,2>9 and by Tong and Van Hove , 2° who account for the multiple scattering effects of the metal substrate. The electrons emitted from the adsorbate toward the metal are scattered and reflected back toward the detector, producing interference effects related to the symmetry of the surface. Although this approach best approximates the true final state of the electron, only atomic overlayers have been treated. Thus, initial state molecular orbitals have not been handled. A synthesis of many of the above methods was provided by Jacobi, et al.’ For an atom with two orbitals of symmetry a, and e, they tried several types of approximations, the best of which included LCAO initial state molecular orbitals and a LEED-type (scattered wave) final state. This approximation was compared to one with a plane wave final state and separately to one with an atomic-orbital initial state (no mixing). In the first case, the curves fit the experimental data quite well. In the other. two, however, there was moderate to severe disagreement, particularly when the final state was a plane wave. Other papers of this group consider polarization changes, symmetry effects, and variations in A at the surface + and calculate distributions for a commonly encountered chemisorption system, CO on ni(110). 22 The good agreement reached here between theory 4. and experiment paves the way for further studies of this type. 2. This Work In our laboratory, we have currently the capability to perform angle-resolved photoemission experiments. We would like to analyze the experimental data by comparing it to theoretical curves calculated for a number of possible surface structures. In this way, we hope to use angle- resolved photoemission spectroscopy to deduce the configurations and properties of chemisorbed overlayers. For this reason, I have written a series of computer programs which calculate photoemission distributions from various atomic orbitals at arbitrary energies and polarizations. These programs and the writeups explaining specifically how each works are included in the appendix. FORMALISM Photoemission is treated here in the context of first order per- turbation theory. The full Hamiltonian of the system is a sum of an unperturbed Hamiltonian and an interactional correction: H = HOO sy, (1) where the perturbation V represents the interaction of the incident radiation with the admolecule plus substrate system. Classically, the 13 full one-electron Hamiltonian can be written where p is the momentum operator for the electron, A is the vector ~ potential of the radiation, is the scalar potential, and v' represents all potentials due to the unperturbed system. Expanding the square and th removing the zero” -order Hamiltonian, we have . 2 V = 5=[p-A+A-p] +25 A amc ss ~ ONE ome’ ot eo. (3) Several assumptions will simplify this expression. If the wavelength of the radiative field is large with respect to the wavefunctions describing the system, then A can be approximated as a constant using aiken ~ 1 (the dipole approximation). ‘Then A commutes with Pp» so that pA = Asp. Since the perturbation, represented by Ais small, the az term can be neglected, We can also set the scalar potential » to zero since, in the absence of sources, one can always choose the Coulomb gauge. Then the perturbation is simply V = = (Arp). It is desirable to transform the derivative operator p to the multi- plicative operator r. Using the fact that [Hor] = Hr-rH = Bp, (4) if the potential term v' does not depend on velocity, we find y= eh caup - ArH] . (5) Perturbation theory requires expanding the "true" perturbed. states wy in terms of the unperturbed eigenstates yl), In this case, the expansion coefficients become — ph vp yl0)> Gos ee (6) Substitution of Eq. (5) into the matrix element leads to an alternate expression for the perturbation: -ien V = —3- (Eg - E;)(A-r) . (7) © amc ui ue We must now find an expression which will incorporate the angular dependence of the photoemitted electrons into the transition probabilities. Assuming V is independent of time, we can use Fermi'’s Golden Rule for the transition rates: 19 Woe Ki <belArr|y,> 1? o ,(e(0) + hy), (8) 7. where we have substituted Avr for V, K represents a collection of con- stants, and energy conservation is accounted for by including the density of final states only at the energy of the initial state plus hv. Now what we seek is the total final state wavefunction, which includes angular information. If the total final state wavefunction is denoted by yw, the quantity Lv_(0,0) 2 (9) tells us the electron current reaching a detector located at (9,4) per unit time (since we are only interested in the angular dependences , and not in the absolute magnitudes of this function, we can neglect normal- ization and constants). This total wavefunction is a sum of the final states which are represented in Eq. (8). The sum must weigh these final States $¢ according to their likelihood of being occupied, which is the square root of the transition rate w. Thus YF ,, lee: <Peg lA éty> (10) m where the sums extend over all possible values of the angular momentum quantum numbers. The values of the matrix elements tell us which final state transitions are allowed; since we are considering only electric dipote-allowed transitions, this means AR = + 1 and we need only consider final states of s(2' = 0) or d(z' = 2) symmetry. Therefore the sum will have at most six terns. | The wavefunctions of an atom can be separated into radial and angular parts: [gp > = Regt) Ygrgi (04), and 14> = Ry lt) Yan ( Od)» (12) and the dot product A-r can be similarly broken down: Aer = A,r sinécos$ + Avr singsing + A,r cose . (12) The radial and angular integrals are separable, A,SinGcos > mem . + nes “Gey: A 0104 9? Re os |r {Ry o> tm! +ASindsing Yom? ? (13) +A cos | so that the electron intensity measured at the detector is 2 | [vel” = i radial integra] (2"))- } (angular integral (2! ,m'))- Q' m' 2 ee (14) | Note that the radial integrals are independent of the angular momentum substate; for a transition from a p state to a d state thus means all 5d states will have the same radial weighing factor. The angular integrals are calculated analytically; expressions of the Form <¥ 91,1 [Sinecose|Y > can be reduced to Clebsch-Gordan coefficients. The coefficients are zero for certain non-allowed transitions, such as Py * doi it is here that the m' values are important. _ THE PROGRAMS Two programs are used to solve for the final detector intensity as in Eq. .(14). A radial program calculates two radial integrals, one for s final states and one for d final states. Then an angle program uses these integrals as input, calculates the final detector intensity for the specified polarization, initial state, final state energy, and detector locations, and plots’ the specified intensity-versus-angle plots. These two types of programs are discussed separately. 1. Radial Programs. These programs calculate the integrals co “y. . 2 [, Regi rea oer dr where R represents the radial part of the corresponding initial or final state. We need expressions for the initial state and the Final state, which ate quite different since the first is a bound state of negative energy and the second is an outgoing electron wave of positive kinetic | energy. | The initial state has been calculated for many different atoms by E. Clementi using self-consistent field methods. A linear combination of Slater orbitals of the form was used, where the coefficients C and the exponents x were optimized to produce the minimum energy. The exact form of C is given in Reference 14 and is different for 2p and 3p orbitals. The orbitals are given as a 10. linear combination of four basis orbitals in the first case and eight in the second; these modifications necessitate slightly different programs for first row and second row atoms. The final state is quite different. . It must be calculated numerically as a solution to the radial Schroedinger equation. This is possible only because the final state energy is positive and a continuum of final states is available. The radial Schroedinger equation is By transforming this equation into two first order differential equations, we can use existing computer integration routines to give us the value of ® for any r.. For program input we need the final state energy E, which is arbitrary; the potential V, which can be interpolated from a table of ‘atomic potentials (we used values from Ref. 15); and. the value and slope of the wavefunction at any one point. The last quantities are calculated as. in Ref. 16. Thus, a table of values of the final state wavefunction versus radial distance r, and an analytical expression for the initial state can be obtained. The integral can then be approximated as n j=l where Ar is the distance between successive values of r, and the sum goes to an arbitrarily chosen cut-off point, after which any further contri- bution to the integral is neglected. 11, Since the values of the final wavefunction depend on the angular momentum 2, that entire calculation is performed twice, once for 2 = 0 and once for 2 = 2. The first number is called RADINT(1) and the second RADINT(2). The radial program will repeat the calculation for any number of energies; the output is a deck of cards each of which contains | the energy, RADINT(1), and: RADINT(2). These cards are then ready to be fed into the angular program where they form a table from which the radial integrals at any energy can be interpolated. Thus, one needs to run the | radial programs only once for any one atom; the angular program can run at any number of intermediate energies on the same input cards. 2. Angular. Programs These programs calculate the detector intensity at specified angles using Eq. (14). The radial integrals used are the ones calculated and punched onto cards by theradial programs. The final states on are Coulomb waves,the angular dependence of which is described by the spherical ‘harmonics and the radial dependence of which is set equal to one since it does not affect the detector intensity. Thus, the program calculates RADINT(2) <¥ out |AySinecoso|Y > + <Youm [Aysinesing |¥ 0. m! g=1,2 + £¥ pig IR yo050 Yop? | Ygis (808) ’ where the values of RADINT(2), 2, m, Aye Aye and A, are specified as input. The three angular integrals reduce to simple numbers multiplied by the components of A. The program calculates all the integrals and multiplies 12. by RADINT(1) or RADINT(2), as appropriate, to obtain a set of coefficients which depend on the final state parameters %' and m'. Then it cycles through a series of angle loops wherein it calculates the values of the six spherical harmonics Yo'm! at the specified angle, multiplies by the proper coefficient, . and sums and squares these numbers. Note that the angular integrals and all quantities derived from them are complex numbers; the last operation is. actually taking the square of the magnitude of W_l0s) SO as to produce a real number. After finishing the angle loops, the program has three arrays: two contain angle values, and the third contains the detector currents at these angles. Plotting sequences normalize all distributions to a maximum value of unity and give a two-dimensional plot of current as a function of one angle at a fixed value of the other angle. 13. DATA The radial programs were run for six atoms: C, N, 0, Si, P, and S. The values of the overlap integrals and their ratios are shown in Tables 1 and 2. The first column in these tables is the kinetic energy in eV of the outgoing electron (ENG); the second column is the radial overlap integral for the s final state (RAD1); the third column is the overlap integral for the d final state (RAD2); and the last column is the ratio of the d integral to the s integral (RAD3), with the last three columns expressed in scientific notation. Only the first four digits are signif- icant. The values of the radial integrals and their ratios as a function of energy are plotted in Figs. 1 and 2. Fig. 1 shows the values for carbon, nitrogen, and oxygen for energies from zero to 100 eV. The first row of plots is the overlap integral for the final state s-wave, the second raw is for the d-wave, and the third is the ratio of d-wave to S-wave integrals. Note that the ordinates have different scales, with the d-wave values being up to eight times the S-wave values. There are no units shown because only the relative values are important. Fig. 2 is essentially the same as Fig. 1, plotting the radial integrals and their ratios as. a function of energy for the second-row atoms. Note that here the horizontal axis is not at zero for the first and third rows, since there are also negative values of the integrals here. In all cases the integrals converge to zero at high energies, which is reasonable since at high energies the final states oscillate very rapidly, causing the positive and negative parts of the integral to cancel. The reason these integrals can be negative where the integrals 14. for the first row atoms are always positive is that the radial part of the 3p orbital is negative near the nucleus and positive later, whereas that of the 2p orbital has no radial node. | Figs. 3 and 4 show the values of the one initial and two final state wave functions for oxygen and sulfur at 10 eV, and the products Wy beers which are the radial integrands. The horizontal axis is radial distance from the nucleus, given in atomic units (0.529 A). The functions were calculated and integrated to 10 Bohr for first row atoms and to 15 Bohr for second row atoms, since the remainder of space contributes little to the integral. The vertical axis shows the relative magnitudes of the three wave functions and the relative values of the two integrands. Note the shapes of the initial state wavefunctions for the two cases. The signs of the initial states are unimportant. since all values are eventually squared, but the - fact that the s and d wavefunctions can overlap the initial state with opposite signs in Fig. 4 is important. Where that is the case, the out- going waves add with opposite phase, the difference being that between ¥, + Vy and Vs - bye This effect will become clearer in the discussion of angular plots. . Another important feature of the radial plots is that due to the sign change in the d-wave overlap integral, there is a point where the ratio of d- to s-wave integrals is zero. At this point, the outgoing electron is exclusively s-wave in character. This is an important point since the. emission should be spherically symmetric at this energy. It should thus be possible to separate emissions from two initial states of different symmetries if one has this property since all angular dependences will then be due -to electrons from the other. 15. The angular plots emphasize these points. For each atom, I have plotted the polar and azimuthal angle dependences for photoemission from a Pp, orbital using x-polarized, z-polarized, and xz-unpolarized light. These plots are shown in Figs. 5-10 for all six atoms at an arbitrarily chosen energy of 10.0 eV. The top plots are the polar angle dependences in the xz-plane. The bottom plots show the azimuthal angle dependence in the xy-plane. These plots are at a constant polar angle of 10 degrees. Note that the unpolarized light produces a distribution similar to the z-polarized light; this is because the emission is much stronger for z-polarized than x-polarized light. The only difference between the atoms in these plots is the values of the radial integrals which were input into the program. Thus, the z-polarized light emission from carbon (Fig. 5), which has a higher d-to-s ratio that oxygen (Fig. 7), looks more like a d 2 orbital than does that from oxygen. What happens is that the outgoing s-wave adds to the appropriate combination of d-waves (which is only d 2 for this combination of initial state and polarization) before squaring, which means that phase factors are important. For positive values of the d-to-s ratio, the larger is the proportion of s-wave, and the smaller is. the small lobe on this plot compared to the large lobe. This trend can be seen in the C-N-O series (Figs. 5,6, and 7). If the d-to-s ratio is negative, however, the s-wave will add con- structively to the small lobe and destructively to the large lobe. Thus, the appearance of photoemission from Si, P, and S (Figs. 8, 9, and 10) for this polarization is very different from the first row atoms. Emission is a minimum at small polar angles, whereas it is a maximum in the other case (Figs. 5-7). This trend would be expected to change at 16. different energies since the radial integral ratio changes from negative ‘to positive. Figs. 11 and 12 show the shapes of the z-polarized light distributions for oxygen and sulfur, respectively, at various energies. Note that the oxygen distributions retain the same shape for all energies but the sulfur plots change dramatically, looking more like the oxygen plots at higher energies where the d-to-s ratio reaches its largest positive value. | It is thus apparent that within the approximations used, the shape of the angular distribution of photoelectrons depends on three factors: the atom (and orbital) from which the electron is emitted, the energy with which it leaves, and the polarization of the exciting radiation. The First two factors determine the ratio of d-wave to s-wave phases and intensities in the final state and the third determines which d-waves will be used. This treatment is a useful starting point for calculation of photoelectron distributions expected from chemisorption systems. 17. SUGGESTIONS FOR FUTURE WORK Two types of extensions to this work would be valuable. One 7s to add the capability to treat molecules, and the other is to consider sub- strate multiple scattering effects in a LEED-type final state. The second is perhaps more straightforward as our research group has possession of programs written for this purpose by Michel Van Hove and S. Y. Tong. The programs are long and complicated and appear to suffer from convergence problems. If they can be fixed to run properly, it should be straightforward to adapt them to our uses. Treating molecules will be more difficult. One must have an analytic or numerical expression for the initial state and must be able to compensate for the reduced symmetry of a molecule over an atom. In other words, the final state for CO will no Tonger be a sum of six spherical harmonics. It is suggested that future workers either examine the scattered-wave-Xa- approach of Davenport” and of Johnson?” 3,4,5 or the Green's-function approach of Gadzuk. In the latter case, some further data wil] be needed to pinpoint the extent of orbital mixing before valuable comparisons with experiment can be made. 18. REFERENCES 1. Eastman, D. E. and Nathan, M. I., Physics Today (April 1975) 44-51. 2. Davenport, d., Phys. Rev. Lett. 36 (1976) 945-949. 3. Gadzuk, J. W., Surf. Sci. 53 (1975) 132-143, 4. Gadzuk, J. W., Solid State Commun. 15 (1974) 1011-1016. 5. Gadzuk, J. W., Phys. Rev. B. 10 (1976) 5030-5044. 6. Gadzuk, J. W., Phys. Rev. B. 12 (1975) 5608-5614. 7. Jacobi, K., Scheffler, M., Kambe, K., and Forstmann, F., Solid State Commun. 22 (1977) 17-20. 8. Liebsch, A., Phys. Rev. Lett. 38 (1977) 248-251, 9. Liebsch, A., Phys. Rev. B. 13 (1976) 544-555, 10. Tong, S. Y., and Van Hove, M. A., Solid State Commun. 19 (1976) 543- 546. ll. Scheffler, M., Kambe, K., and Forstmann, F., Solid State Commun. 25 (1978) 93-99. 12. Horn, K., Bradshaw, A. M., and Jacobi, K., Surf. Sci. 72 (1978) 719- 732. 13. Schiff, L. I. Quantum Mechanics, 3rd ed., Sections 24, 37, and 44. McGraw-Hill (New York, 1968). 14, Clementi, E., IBM J. Res. and Devel. 9 (1965) 2-19 (Supplement). 15. Lu, C. C., Carlson, T. A., Malik, F. B., Tucker, T. €., and Nestor, C. W., dr., Atomic Data 3 (1971) 1-31. 16. Pendry, J. B. Low Eneray Electron Diffraction, Appendix C4, Academic (New York, 1974). 17. Johnson, K. H., J. Chem. Phys. 45 (1966) 3085-3095. CARBON NITROGEN OXYGEN ENG 1.000000 5.000000 10.000000 15.000000 20.000000 25.0C0000 30.000000 35.000000 40000000 452000000 50.000000 55.000000 60.000000 65.000000 _ 70.000000 75000000 80.000000 85.000000 90.000000 95060000 _100.000000 ENG 1.000000 5.000000 10.000000 15.000000 20.000000 25.000000 30.000000 352000000 - 40000000 454000000 50.000000 55.000000 60.000000 65000000 __70.000000 75.000000 80.000000 85.000000 90.000000 95.000000 100.000000__ ENG 1.000000 52000000 10.000000 _, 1452000000 20.000000 252060000 30.000000 35000000 40.000C00 452000000 50.000000 55.000000 60.000000 65.000000 .70.2000000 -75.000000 80.000000 85.000000 90.000000 95.000000 100.000000 19, RAD 0.261L94E 00 0.17526E 00 0.12580E 00 0.95002E-01 0.71842E-01 0.55990E-01 0.46163E-01 0.39644E-01 0.34202E-01 0.28978E-01 0.24158E-O1 0.20233E-OL 0.17449F-O1 0.15655E-O1 0214445E-01 0.13389E-O1 0.12213E-01 0.10857E-O1 0.9429 2E-02 0.81121E-02 0. 70621E-02_ RAD1 0.19612E 00 0.14447E 00 0.10700 00 0.83012E-O1 0.66196E-01 0.54199E-O1 0.45496E-O1 0.38885E-O1 .0233556E-01 0.29102E-01 0-25369E-O1 0.22296E-01 0.19804E-01 O.L7776E-O1 RAD2 0.19848E Q1 0-10168E Ol RAG Oo TS7TLE 0.58019E O.5374BE CO Oe42724E 0.32121E 00 0.20333E 00 Oe 13540E GQ 0.95252E-C1 0-69913E-Cl 0.52420E-01 0.39645E-O1 0.30317E-61 0.23699E-O1 0.19085E-CG1 0.15789E~OL 0-13254E-O1 0.11142E-01 Oe 931 CTE-C2 0. 77452E-02 0.6470TE-02 0« 5493 8E-02 0247813E-02 RAD2 O.1LL402E OL 0-68457E 00 0.40499E 00 0.26047E 00 OlL7688E 00 0.12528E 00 0.91953E-O1 0.69454E-01 0-53618E-O1 0.42095E-OL1 0-33810E 0-28302E Oe 241 83E 0.20634E. 0.17635E 0215326E ‘0-13681E 0212549E -OellTL3E 0.10938E o.1l0case .0.91758E 0.83216E 0. 762 35E O.TL341E 0.68624E Oe67724E (0+677T04E RAD3 0.-58140E 0.47385E 0.3750E. 0-31378E O.26721E O5e23115E O-20211E O-17861E 04159786. 0.14465E 0.33548E-01 0.13224E 0-27133E-01 0.22263E-01 0.18504E-01 Oe LOQTTE-Cl 0015541E-01 0.96665 0.14592E-OL 0.613156E-O1 0.13249E-01 0.11207E~01 G.LZ01L6E~OL 0.96018E-02 0.10902E=01 0.82799E~02 0.99167 E~02 0.71930E-02 0-90696E~ 02 0.62988E-02 0.69449 00 RACL RAD2 | 0.16029E 00 0.79974E 00 0.12152E 00 0.50226E CO 0492756E-O1 0.31332E 00 0. 73669E-01 0«21090E 00 0.59984E-O1 0.14890E CO 0.50006E-01 0.10905E 00 0.42562E-01 0.82337E-O1 0.36742E-01 0463724E-01 0.31969E-01 0.50271E-O1 0.27963E-Ol 0.40264E-C1 0.24610E-O1 0.32689E-Cl 0.21840E-01 0.26888E-O1 0.19567E-O1 0.22394E-OL 0.17680E~01 0.18861E-O1 0. 16067£~01 0.16033E-O1 0.14643E-01 0.1373 1E-O1 0.13356E-01 0.11833E-01 0.12190€-01 0.10257E-O1 Oe11145E-O1 0. &9438E-C2 0.10227E-01 0.78493E-02 0.94339E~02 0.69343E-02 0-121 70E —0.11241E 0.10409E 0-90158E 0284588E 0.7989 7E 0275946E 0.72534E _ . RADB 0.49894 0.41332E 0.33779E _0028626E 00 24824E 0.21807E 0+19345E 0.17343E 0.15725E 0.14399E 0.13283E 0.12311E 0.11445E 0.10668E 0.99788E 0.93776E 0-88595E 0.84138E 0. 80249E 0.76753€ 0.73504E Radial integrals (RAD1 and. RAD2) and their ratio (RADS) for carbon, nitrogen, - ENG and oxygen at different energies Table 1. SILICON PHOSPHORUS SULFUR 20, ENG RADI RAD2 RAD3 16 000000-0. 32160E€ 00 0.311106. 00-0.96735E 00 5.000000-0.19017F 00 0.89514E-01-0.47072E 00 10.006000-0.10977E 00 0.16705E-01-0.15218E: 00 15» 000000- 02 741 00E-01-0014027E~02 0.18930E-01 20.900000-0.53039E-01-0.66184E-02 0.12478F 00 25-000000-0, 39560E-01-0474054E-02 Oe18719E 00 30. 000000-0. 31 962E-01-0.69903E-02 0.21871E 00 35 -000000-0 26434E-01-04.64760E-02 0.24499E 00 40. 000000-0, 21406E-01-0.57899F-02 0.27048E 00 » 45 ¢Q00000-0..17602E-91-0249990E-02. 0.28400EF 00 50.000000-0.15270E-01-0.43428E-02 0.28440E 00 55_ 000000-0, 13565E-01-0e 36651E~02 0228494E 00 60.000000-0.11 757E~01-0.34444E-02 0.29296E 00 65.000000-0.99228£-02-0.30131E-02.0,30365E 00 70.000000- 0. 85122£-02-0.26145E~02 0.30714E 00 ... 132900000-0.76866F-02-0623072E-02 0.30016£ 00 80.000000-0. 71 766E-02-0.20876E-02 0+29090E 00 85.000000-0, 66171£-02-0,19050E-02 0.28788E 00 90.000000-0.58813£-02-0.17190E-02 0.29228E 00 95» 000000- 0, 51182E-02-0.15302E-02 0629897E 00 (100. 000000-0.45501E-02-0.13646E-02 0-29990€ 00 ENG RADL. RAD2 1,000000-0.25557E 00 0.17240E o0-0.67455¢ Qo. 5 -000000-0.17344E 00 0.64822E-01~-0.37373E 00 10.000000-0. 11147E 00 O2:17901E-01-Ge16060E 00 15 6000000~-04 78614E-O1 022394 7E-02-0030461E-01 20 .000000-0. 57606E-01-0.31751E-02 0.55116E-01 25 0 000000-0e 44023E-0 1-00 48284E-02 0210968E 00 ‘30 6000000-0. 35596E-01-0.51320E-02 0.14417E. 00 35 «000000-0. 29333E~01-0.50205E-02 Oe17115E 00 40 .000000-0. 241 80E~01~0.46565E-02 0.19258E 00 45 .000000~-0.20378E-01-0241840E-02 0.20532E 00 50 .000000-0. 17 763E-01-0037594E-02 Ove21164E 00 55 0000000-0. 15662E-01-0.34087E-02 O.21764E 00 60.000000-0.13663E-01-0.30795E-02 0.22539E 00 65 e000000-O« LL E57E-01-0627514E-02 0523204E 00 70 0000000-0-10475E-01-0424531LE-02 0.23418E 00 7§ «000000-0. 951 T0E-02-0.22116E-02 0.23238E 00 80 .000000-0. 87595E-02-0.20194E-02 0.23054E 00 85 .000000-0.79896E-02-0.18490E-02 0,.23143E 00 90. 000000-0. 71 724E-02-0.16822E-02 0023453E 90 95 2000000-0664211E-02-0015222E-02 0.23706E 00 100. 000000 --0-56531 E-02-0 L382 7E-02 0423624E 00 RAD1 RAD2 RAD3 le 900000- 0.22142E 00 0.11008EF 00-0.49713E 00 5.000000-0.15230E 00 0.44849£-01-0.29448E 00 10.000000-0.10206EF 00 0.14457E&-01-0.14165E 00 15.000000-0.73222E-01 0631 767E-02-0,43385E-01 _. 202 000000- 0 54838E-O1-00 1261L4E-02 0.23003E-01 25.000000-0-.42851E-01-0.29120F-02 0,.67955E-O1 3.0.900000-0.348 75E~01~0.34562E-02 0.99103E-01 35.000000-0, 29003E-01-0. 35533E-02 0.12252E 00 - 40.000000-0. 2445 7E-01-0.34265E-02. 0.1401 0F 00 - 452000000-0.21044E-01-0.32037E-02 0.15224E 00 50. 000000- 0, 18469E-01-0.29701E-02 0,16081E 00 55 .000000-0.16350£-01-0.27489E-02 0o16812E 00 60.000000- 0. 14480E-01-0+25314E-02 0617482E 00 65.000000- 0. 12867E-01-0.23169E-02 0.18007E 00 70.000000-0.11561£-01-0.21179F~02 0.18319E 00 752 000000-0+ 10522E-01-0019442E-02 0.18477E 00 80.000000-0.96376E~02-0,17933E-02 0.18607E 00 B5.000000-0. 88161F-02~0.16561E-02 0.18785E 00 90. 000000-0, 80406£-02-0.15266E-02 0.18986E 00 95 .000000~0. 73480E-02-0.14054E-02 O.19127E 00 100.000000-0.67723£-02-0.12968E-02 0.19148E 00 Table 2. 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The four programs which follow are of three types: radial programs (RADOXYG, RADSULF), an angular program (ANGSULF), and a tabulation and plot program (RADTABL). The first two are to be used for first-row and second-row atoms, respectively, as the manner of calculating the initial state differs in the two cases. The third program can be used for all atoms and requires as input the radial integrals. which are calculated and punched onto cards in the radial programs. This version assumes one uses. radial integrals. at an energy for which there are integrals calcu- lated; alternately, one could call the subroutine RADIAL to interpolate from a table. In this case, the table should be read into the variables ENERGY, SDUM, and KDUM, which are already dimensioned. Plots from this . program are shown in Fig. 10; the same program with different input data. plotted Figs. 5 through 12. The last program collects the radial integral values calculated in the radial programs for up to eight atoms. It calculates the ratio of the d-wave final state integral to the s-wave final state integral for each atom and energy, lists the values in a table, and plots the integrals and their ratios as a function of energy. This program plotted Figs. 1 and 2 and printed Tables 1 and 2. The data is not shown since it consists mainly of the output cards from the radial programs. 34. Program 1. J7RADOXYG JOR (92052 WHYS CHE) sSHEARD s TIMF=(} 950} 7* JOBPAPM TOC=)5 4 EXEC FORTG A/SYSPLTON 1D SYSOUT=IN» eGSW4) 7/FORT op # : c 4. INITIAL STATEs GIVEN AS A LINFAR COMBINATION OF SLATFR OPATTALS. c - THIS PROGRAM IS FOR. 2P ORATTALS»s USING 4 TOTAL ORAITALS AS RFAL LPLUSI DIMENSION RADINT (2) OTMENSTON 889 (150) VV (1509 5¥(2) oY (2) DIMENSION YPT(8N0) SUM] (800) »StM2 (800) DIMENSION AN (3) ; DIMENSION XPLT (AN) o YPLT (800) sPHINIT (800) sACOEF (4) eAEXP (4) COMMON Fe LPLUSI> Zs L COMMON/VAL /RR 9 VV gNPOT EXTERNAL. NERIV DATA DD/0e 92 004/ 401 FORMAT (2Xs2F 15,8) 499° FORMAT (3X0 "OXYGEN 2P*) 500 FORMAT (?F15-8) 501: FORMAT (13) 502 FORMAT (F7.4) 503 FORMAT(* ATOMIC NUMBER = oF4.1) 5N& FORMAT (SOXe'L =%eI3s? NUM =¢45T4) S07 FORMAT. (* ENERGYCEV) Po1EXe*RANINTS L=O0'914Xs *L=2?) 508 FORMAT(F IS. 79LSX9E LS 07 92X9E1 5079 3NXeTHTNORM =9F1567) 509 FORMAT (3615.8) 530 FORMAT ( 1Xe*R=*oFlBe7Tot VL) =" oF 1507 et DY C1) 24 9E 1467) 540 FORMAT ( 21X9*Y(2)2% E1507 tDY (2) = E1467) 550 FORMAT ( ‘ERROR OCCURRFDs K=-ls N1D NOT CONVERGE) N=2 . RSTART=0,009 C=1.0 RMAX=15.0 EPS=0.0 WRITE (69499) WRITE (75499) READ (59502) 2 WRITE (69503) 2 READ (59501) NPOT. WRITE (69501) NPOT READ (50500) CORROCT) oVV(LT)) eo T=1eNeOT) WRITE (60500) CCRR(T) sVVCI)) o I=1eNPOT) RFAD (50401) (CACNEF (5) eAEXP (J) ) eo J=1 94) WRITE (60491) ((ACOEF (J) sAEXP (J) ) eJ=194) ON 149 J=le4 149 ACOFF(J)= ACOEF (J) #2.#SQRT(AEXP (J) ) ##5/SQRT (34) WRITE (fe507) R=RSTART. VO=(YNTERP (RReVVeReNPOTS3)—7)/R REAN. (55501) NENG DN 941 JU=1eNENG REAN (52592). FEV — = EEV/27.18 L=9 18 CONTINUF c FINAL WAVFFUNCTTON 20 c 35, R=RSTA PBLUST CEL OAT (La1) FL=FLOAT(L) ALPHA=(7##2=LPLHS1*® (VO4F))/0(2,4F1 +3.) #LPLUS1) BETA=Z# (7442—(3,8FL 44.) #(VOFED) PS (RZeMLPLIIS] © (FL 420) (2 04FL 425) ) VY CL) =C#R #41 PLUST# C1 eo (Z#K/LPLUS) ) +ALPHAPRE#2—-RETARR G3) Y¥ (2) =C#P HS. PLUS) ® (=7/LPLUS142, #0, PHAHR~3,FHETAPR OH?) K=]} I=] CALL MONNEQ (DEATV 9K s29RsYaNVenR, FPS) IF (Kk .LT.0) GO TO 100 PHI=Y(1)/R XPLT(I)=R YPLT(T) =PHI I=I+) IF (R -olFe RMAX) GO TO 20 NUM=]=1] , WRITE (64504) LoNUM INITIAL WAVEFUNCTION C SUM OVER RASTS SET» PHINIT(T)=VALUE. OF INITIAL WAVEFUNCTION AT R(T) 15] 153 154 176 rao 179 39 IF (JueGTe1) GO TO 154 IF (L.EQ.?) GO To 154 TNORM = 0.0 NO 153 1=1—9NUM PHINIT(1)=0,0 ON 151 J=1le4 Se~AEXP (J) °XPLTAT) IF (S.LT.-?0.0) GO TO 151 PHINIT(T) =PHINIT(T) + XPLT (1) *ACOFF (J) SEXP (S) CONTINUF TNORM=AMAX1 (TNORMsPHINIT (19) CONT INUF CONTINUE SUM=0.0 IF (L.E0,0) GO To 189 DO 176 T=1_NUM TNORM=AMAX1 (TNORMs YPLT (1) ) S=YPLT (1) *PHINIT (2) ®XPLT(1) #8 3¢NR SUM2 (1) =S SA=ABS (S) TES=AMAX] (TESSSA) SUM=SUM+S CONTINUE 60 TO 39 DO°179 T=1,NUM TNORM=AMAX) (TNORMyYPLT(T)) S=YPLT (1) #PHINIT (1) *XPLT (1) ##3¢np SUM1 (1) = SA=ABS (5S) TES=AMAX1 (TESSSA) YPT(T)=YPLTCI) SUM=SUM+S RADINT (1) = SUM L=? 69 TO 18 RADINT (2) = SUM WRITE (69508). EEVeRADINT (1) sRADINT (2) » TNORM WPITE (79599) EEVe RADINT(1) 5 RANTNT (2) 36. 00 177 T=1sNUM SUM1 C1) =SUM1 C1) ATES SUM? (1) =SUM?2 (1) /TES - PHINT TCT) =PHINIT (CT) ZTNORM YPLT (I) =YPOLT(T) 7 TNORM YPT(L)=YPT (CT) 7TNORM 177 CONTINUE CALL VLAREL (9. 94 e90e9l5Se92Se915etP (BOHR) § 98 e098 (F390) 8,2) CALL LAREL (Me Oe emlesleslOesmle"Cts001) CALL LABFL (009B.5e0eelSeetSes—l5e PRODUCT! 9740) CALL PRTNUM(10.91.590.21 9EEVe® (RHENFRGY =s9F7.393H EV) 990.) CALL XYPLOT(NUMyXPLT sPHINITsOe915,9— 8910290090) CALL XYPLOT (NUMsXPLT oSUM1 900915 09—8e591659DD 90) CALL XYPLOT (NUM ee XPLT ¢SUM2 0009150 9—AeSele590D 90) CALL XYPLOT (NUM, XPLT¢YPT 90e015.9~ OeBele2eDDe9) CALL XYPLOT(NUMsXPLToYPLT o0eelScom OoAhele2oMDol) 941 CONTINUE GO TO 3A 100 CONTINUE WRITE (60530) ReV(1) 9DYC1) WRITE (69540) Y(2) sDY(2) WRITE (659550) 30 CONTINUE stop END SUBROUTINE DERIV (NeRo¥sDY) REAL LPLUSI INTEGER Z COMMON Es LPLUS1» Zs Lt COMMON/VAL /RR o VV oNPOT DIMENSION Y(2) sNY¥ (2) eRR(150) sVV (150) OY(1)=Y(2)4Y 01) #LPLUSI/R c c INTERPOLATE V FROM TABLE c V=-YNTERP (RP 9VVeReNPOT*3)/R DY (2)=2,0%(V~E) #Y¥(1)-¥ (2) #LPLUSI/2 RETURN END 7/DATA DD * Be OXYGEN 32 OXYGEN POTENTIAL 0.90031514 7.99606272 0.90066715 7.99160426 0.901246639 7,98476778 0.00263A62 796648492 0.00318278 7.95948065 0.90408677 7.94778195 0.90524752 7293265102 0.99673795 7.91305331 0.00865170 7.RB763256 0.01340001 7823468717 0.92075434 7.72185905 0293214495 7.561432A0 0.04677N62 7..25465541 0.06805085 7296054509 0.09301449 4.733A2093 9,1353352A 6223175934 0, 18498140 0,?52R3960 0434559075 0.47236455 0.40653N66 0,.82902912 1213314845 1.548A3030 1.75595464# 1.86824596 1.98873747 3.0 4.0 5.0 31.10908815 2P 0616372 2P 0257600 2P 0433392, 2P 0601495 10.0 4/ 5.71564367 §.10512453 4.41601050" 3.7179002%4 3216473780 “2645977490 1.°0001923 1.763203466 1.%8792133 1.91999747 Se iLedicaali natant and so s273900 1615360 1.79600 3.43790 7.90700 37. 38. Program 2. ZJ/RADSULF JOB (9ANS2AeWHW eCHE) sSHF ARN se TIMF=(1 550) 7* JQNBPARPM INCH15 4/ EXEC FOPTG 7/SYSPLTON DD SYSOQUT=N 7/F ORT 401 499 ‘500 501 502 503 504 S507 508 509 530 540 550 149 150 148 no * RFAL LPLUS) DIMENSION RADINT(2) DIMENSION PP(159) oVVE1SO) 6 ¥ (2) NY (2) NTMENSION On (3) DIMENSION XPLT (80M) sYPLT (800) ePHINIT CANN) DIMENSION ACOEF.(8) sAEXP (8) DIMENSION YPT(800) sSUM] (800) Stim? (800) COMMON Fefl_PLUSI 5Zok COMMON/ VAIL /RR oVV eNPOT EXTERNAL NERIV DATA DD/0.90090./. FORMAT (2X%92F15.8) FORMAT (3Xe9SULFUR 3P) FORMAT (2F1508) FORMAT (13) FORMAT (F7.4) FORMAT (* ATOMIC NUMBER =*, F4,1) FORMAT (SOXe"L. =%oT39%. NUM =%974) FORMAT (* ENERGY(FV) PolIXs*RADEINTS L=0%s14X»tL=2t) FORMAT (FISe 79ISX oF 15.7 o2XsEl5e79I0XeTHTNORM =59E15.7) FORMAT (371528) FORMAT ( 1Xo*R=teELBe 7a V(1L)=9oF1Se 7s "DV C1) 29 sEL4e7) FORMAT ( 21X9*Y¥(2)="9E15 67s "DY (A) =F E1407) FORMAT ( "ERROR OCCURRED® K=-le NIN NOT CONVERGE?) N=2 RSTART=0,003 c=1.0 RMAX=15,9 DR=0.02 E°S=0.0 WRITE (60499) WRITE (759499) REAN (59502) Z WRITE (69593) 2 READ (59591) NPOT WRITE (60591) NPOT . READ (59500) ((RR(T) oVVCI)) s ISL eNPOT) WITTE (69500) CERR(T) sVV(I)) » I=] eNPOT) REAM (5¢401) ( (ACOEF (J) sAEXP(U)) e198) WRITE (60401) ((ACOEF (J) sAEXP(5)) oU=198) NO 148 J=1.8 IF (Je6T.1) GO TO 150 : ACORF (J)= ACOEF (J) #2.#SQRT(AEXP (J) 1 ##5/SQRT (32) 60 TO 148A ACOEF (J) =ACOEF (J) #001126872348S50RT (AEXP (J) ) #49 CONTINUF WPTTE (6¢597) R=RSTART 7 VO=(YNTERP(RReVVeReNPOTs3)—=7) /R READ (55501). NENG DO 942 JJ=1eNENG REAN (56592) Eev E = EFV/27.18 aan. aaqanan 18 20 39. i= CONT INUF FINAL WAVFFUNCTTON R=PSTART LPLUSI=FLOATI(L4]) ALPHA=(7##2-LPLUS1 9 (VO4E) ) / ( (241.42) #LPLUSI) RETAS7# (7442— (34144) #(VOFE)) / (AFL PLUS] #(L42) (20143) ) Y CL) SC#R SFL PLUST# (1 6 (Z#R/LPLUS1) + ALPHARRO#2—RET ASR 43) Y(2)=C#R#*L PLUS] #(=Z2/LPLUS14+29AL PHAtR=34BETARRH42) K=1 I=] CALL MONDEN (DERIVeks2oRoVoDY gDReEPS) IF (K .LT.0) GO TO 100 PHI=Y(1)/R XPLT(I)=R YPLT (1) =PHTI I=I¢1 IF (R otLFe RMAX) GO TO 20 NUM=T=} WRITE (65504) LeNuUM INITIAL WAVEFUNCTION SUM OVER BASTS SET» PHINIT(T)=VALUF OF INITIAL WAVEFUNCTION AT R(T) 152 151 153 154 176 175 IF (JJeGTe1) BO TO 154 IF (LeEQ.2) 6O TO 154 -TES=0.0 TNORM = 0.0 DO. 153 T=1>+NUM PHINIT (1) 20,0 DO 151 J=leR S=-AEXP(J)@XPLT(T) 2 IF (S.tT.-20.0) 60 TO 151 IF (Je6T.21) GO TO 152 - PHINIT(T)=PHINIT(T) +XPL T(E) @ACOFF (J) FE XP OS) 60 TO 151 PHINIT(L) =PHINIT(T) +XPLT (1) ##3@ACNEF (J) EXP (S) CONTINUE - TNORM=AMAX1 (TNORMsPHINIT (YI) ) CONTINUE CONTINUF SUM=0,.0 IF(LeEQ.9) GO TN 175 D0 176 T=1sNUM TNORM=AMAX1 (TNORMo YPLT (IT) ) S=YPLT (1) *PHINIT(T) #XPLT(I) ##38NR SUM? (T)=S SA=ABS(S) TES=AMAX) (TFS #SA) SUM=SUM+5 CONTINUE 6N TO 39 00 178 Tt=leNUM TNORM=AMAX1 (TNORMeYPLT(T)) StYPLT (1) @PHINIT (CT) ®XPLT(1) ##3¢0R fozeake) SUM1(T)=S - SAZARS(S) ~ TESZAMAXL COTES 9SA) 178 39 77 941 100 30 SUM=SIIM4S : YPT(T)=¥PLT (1) CONTINUE RADINT (1) = SUM L=2 GO TO 18 RADINT (2) = SUM WITTE (6050A) EEVeRADINT (1) sRANINT (2) os TNORM WRITE(79509) EEVs RADINTC1)» RANTNT (2) D9 377 Y=1eNUM SUM1 (7) =SUM1 (1) /TES SUM2 (1) =SUM2 (1) /TES PHINIT(TI) SPHINIT (TL) /TNORM YPLT(T) =YPLT(T) 7TNORM YPT (IT) =¥PT (1) 7TNORM CONTINUE CALL VLAREL (9.54, 90.915,015e915e9R (BOHR) * 980099 (F 3,0) 852) CALL LABFL (06 98.5900015e91569—15, *PRODUCT' 9750) CALL LAREL (Me Og s—leshoolOoem=1et*Ctofied) CALL -PRTNUM(10.91.590.2] sEEVe* CAHFNERGY @oF7,993H EV) * 90.) CALL XYPLOT (NUM es XPLT sPHINITs 00915. 9-OeAole290099) CALL XYPLOT (NUMoXPLT9SUM1 9049156 9—-8.591 6590059} CALL XYPLOT (NUM, XPLTs YPT 906015 es=OeAele2s0e0} CALL XYPLOT (NUM > XPLT 9SUM2 0009156 9-80.59) 059DD90) CALL XYPLOTINUMeXPLTsYPLT s0.015.9-0.Rele2enDel) CONTINUE GO TO 3n CONTINUE WRITE (60530) ReVCl) sOYCI) WRITE (62540) Y(2)sDY (2) WRITE (69559) — CONTINUE STOP END ; SUBROUTINE DERIV (NeReVoDY) REAL LPLUSI COMMON Fel PLUS] oZok COMMON/VAL/RReVVeNPOT DIMENSION ¥(2) OY (2) oRR (150) svV (150), DY(1I=7 (7) 47 CL) #LPLUSI/ZR INTERPOLATE V FROM TABLE V=~YNTERP (RR eVVeReNPOT 23) /R ; DY (2)=2.0%(V~E) #¥ (1) -¥ (2) *LPLUSI/R RETURN END 7/DATA §=90 * 16. SULFUR 42 SULFUR POTENTIAL | 0.90031514 15298761569 0.90066715_ 15.97368449 0.90124639 15.95061087 0.99263862 1589453639 O.49N831LRPTA 15,87261337 0.90408677 0290524752 0.90673795 0.90865179 0.01340001 0.92075434 003214495 0.04677062 006805085 0.09301449 0.1353352A 0418498140 0,20961139 0.25283960 0.30498277 0.346559075 039160563 0467236655 0.53526143 0.60653066 0.68728928 0.82902912 1.00000000 1.13314A65 1.54893030 1,75505466 1286824596 1.98873747 2011790002. 2.25353479 2239887529 2.55358946 2271828183 - 2.99359594. 3.08021685 9.0 31.0 2P 0.15546 4P =-0,90995 4P 0.01287 4P =0,09992 4P 0.05174 4p -0,55282 4p 9,50A96 4p 002556 1 10.0 4/ 15.83543696 15. 78759937 15.72563318 15.64537966 15644403079 15.13926175 14.65118778 14,.06279503 13.27977097 12243093849 11.19515295 1, 014A5A46 9.51694911 A. 74027206 7293620A% 7239296651 6285242783 6205961474 554651371 5.N4499272 4055597903 3.86922589 3.28670529 2294989331 2017381205 1.88692895 1.75148796 1.62224996 1.49969548 1.383735A5 1.274647.92 1.17232375 1.97660S5A2 1.91137375 er) of ef 20 8.0000 14.12590 11.57180 7.91710 5.50390 2.89060 1.42750 0.96650 41. AMAA 42. Program 3. Z7ANGSUL JOB (9AN52eWHWeCHE) 9S. SHFARN* +s TIMES (1959) 7*#JOBPARM 1NC=406L INES=4 »PLT=480 4. 7/SYSPLTON DD SYSOUT=N EXEC. FORTG //FORT i]0) * AAANA AN ANAADANDAAANDGANAAAAA 102 103 104 105 106 107 108 109 110 11 113 508 INPUT CARDS 1, CORFFICIENTS OF INITIAL STATE P FUNCTIONS Y(ls—-1)5 Y¥(190)5 AND Y(lo+1) IN COMPLEX FORM, (YINM(3)5 FORMAT 6F 10.3} 2. NUMRER OF RADIAL INTEGRALS SUPPLIFDO FROM RANTAL PROGRAMS. (NRAD, FORMAT 13) 3.. ENERGIES ANN RADIAL INTEGRALS (CENERGY(T)s SOUM(T)»s AND. KDUM(T)» FORMAT 3E£15.4) THERE MUST BE EXACTLY NRAD OF THESE CARDS. 4. PLOTTING INFORMATION? 1 CARD FOR NUMBER OF PLOTS PER ENERGY AND POLARIZATIONs UP. TO 10.. (NPLOTs FORMAT 13) S. FOR GACH PLOT? WHICH. PLOT (LPLOTs AS DESCRIBED BELOW)» STEP SIZE OF VARYING ANGLE (ANSTEP) »VALUE FOR FIXED ANGLE (ANGFIX) > AND IF LPLOT4 OR 8» MINIMUM AND MAXTMUM OF VARYING ANGLE (ANGMIN ANO ANGMAX)s (FORMAT I3s4F10.7) LPLOT=le THETA = 0 TO 90 LPLOT=5» PHI = 0 TO 90 LPLOT=25 THETA = 0. TO 180 LPLOT=6>» PHI = 9 TO 180 LPLOT=35 THETA = 0 TO 360 LPLOT=7, PHI = 9 TO 360 LPLOT=#4.. THETA = OTHER RANGE LPLOT=85 PHI = OTHER RANGE 6.. NUMAER OF ENERGIES AT WHICH THE DISTRIBUTIONS WILL BE CALCULATED (NENGs FORMAT 13) 7. FYNAL STATE ELECTRON KINETIC ENERGIESs IN ELECTRON VOLTS. (EEVs FORMAT F1003) THERE MUST BE AT LEAST NENG MF THFSF CARDS, ONLY THE FIPST NENG OF THEM WILL BE READ ; DIMENSION ENG (1102) sDETRI11+15+2) DIMENSION A(3) sRADINT (2) DIMENSION CUR (89) sPANGLE (80) »s AANCLE (80) DIMENSION X(80)¥(80) : DIMENSION XA(805245) oP (8005) 09 (80-5) DIMENSION NP (5S) DIMENSION OPLT(11) sEPLT(11} DIMENSION 0n (3) DIMENSION LPLOT (5S) sANSTEP ( 5) sANGEIX (5) sANGMAX (5) sANGMIN(S) DIMENSION ENERGY (30) sSDUM(30) REAL KDUM(3N) COMPLEX YTNM (3) sCYLM(6) sANGINT (34395) CI sPSIF »YLM(6) COMMON /RADT/ENERGY ¢eSDUMsKDIIM DATA DDO/0.9%090./ FORMAT (4F 10.3) FORMAT (3X«e*UNPOLARIZED LIGHT TRAVFLTNG IN Z DIRECTION’) FORMAT (1394F10.7) ; FORMAT (14) ; FORMAT ( * PLOT# LPLOT ANGFIX ANGMIN ANGMAX ANSTEP *%) FORMAT (J3491794F10.3) FORMAT (3615.8) FORMAT (3Xs*FNERGY =*9F 8,39" EV!) FORMAT (3X%9*X POLARIZATION®) FORMAT (3X9*Z POLARIZATION®) FORMAT (3X93F15.A) FORMAT. (2°X¢*POLAR ANGLE * 9 17X94 A7TMUTHAL ANGLE '914Xe R'NETECTOR INTENSITY®) ioiekeke ke) 43. 509 FORMAT (25X et THETA’ 9 26X 9 "PHT! Pang tT (THE TASPHT) #7) 510 FORMAT. (15XeF 15.1 915X9F15e1019X0E15.7) PT=3.141593 YI=SART(1.0/7(4.08PT)) YPSSORT (45./196, PT) ) Y3=SART (45./(24,#PI)) Y4=SORT(S.0/(4.9#PT)Y RAD=PT/189.0 CT=(0.0+1.0) GENERATE ANGINT TENSOR ANGULAR INTEGRALS FROM CLEBSCH=GORDAN COEFFICIENTS ANGINT(lelsl)=( ~0.4082482, 0,0 ) ANGINT (Pelel) = 0.4 ’ 0.4082482) ANGINT (39191) =( ’ ) ANGINT (lo2sh}=( * ANGINT (292¢))=( ANGINT (30201)=¢ ANGINT (1e3e1)=¢ ANGINT(29301)=( ANGINT (30351)=(¢ ANGINT(1lels2)=( = ANGINT (29192) =( ANGINT (39192) =( ANGINT (19202) =( ANGINT (29292) =( ANGINT (39292) =( ANGINT (19352)=(¢ ANGINT (29382) =( ANGINT (39392)=( ANGINT (19193) =( ANGINT (29193) =( ANGINT (39193)=( ANGINT(le2@e3)=(. = ANGINT (29293)=( - ANGINT (39293) =( ANGINT (15353)=¢ ANGINT (20493) =( ANGINT (39393)=( ANGINT (10104) =( ANGINT (25194) =( ANGINT (39164) =¢ ANGINT (Le2e4)=¢ ANGINT (29244) =( ANGINT (3¢204)=( ANGINT(1e394)2=(.. = ANGINT (22304) 24 ANGINT (20394)=(¢ ANGINT (Lol sS5) = ANGINT (20195) =( ANGINT (39155) =( ANGINT (19295) =4 ANGINT (29295) =( ANGINT (39295)=( ANGINT(19395)=( ANGINT (2s3e5)=( ’ 7735035 0824826 ’ 982482) we s 472136» ’ 472136) 4 eee oe ee eee wesw eevee oe 472136» 16227R, ’ 162278 82574 nN eoweveoweowe eooveee VPoowoVIoOO SKY GVO eEVFKMOO F989 SCWTOO0C9 C9 F9SC C8 CfP OO FB O000 R25742 vsrooosoooos F9eVrF 990909 29903990 GC2VPI79I9I9N00 T9999 ° ’ 516397A, R2ST4A2 4 ’ ' R25742 wwewe 1 162778 es9o0oo0oreo0o0o0oneo 70 GCNPNO Gee V9 eos SoOeOVO Sees oC CVO S890 DBO e099 ° eneoevaot®eeeevevenvreeveaeeveeaeeanevee see wemaeane eee eveae ee SPoeoervWver2o0 Co CcSorw NFB FF OKFP FSC S9 SG FWHFQOO9D Ses oes roocFrF NDE So ee ee er en er er ener ore re) a» nN nN =~ 2» e 272 en 899999 Oana aaaa Aaaaa an aaa 44, ANGINT(39305)=( 006 ANGINT () 0196) =( 9.0 ANGINT (29106) =( 0.0 ANGINT (2¢] 56) =( 0.0 ANGINT(19296)=( 9.0 ANGINT (29296) =¢ 0.0 ANGINT (342e6)=( 0.0 ANGINT (1¢396)=( 0.4 ANGINT (26356) =( 0.0 ANGINT (30396) =(¢ 9.9 472136 2299099 393293939 era 2og 090909 ee ee ee wee ww READ(Se102) YINM READ (54105) NPLOT 00 19 TJ=lsNPLOT REAN (59104) LPLOT(IU) sANSTEP (TU) » ANGFIX (TU) sANGMIN(TU) pANGMAX (TJ) 19 CONTINUE : ENERGY LOOP 690 691 READ (59105) NENG DO 771. TENG=1sNENG READ. (59108) EEVsRADINT (1) sRADINT (2) WRITE (60113) EEVeRADINT (1) sPANINT (2) POLARIZATION LONP FOR UNPOLARIZED LIGHT A(2)=(0099.) DO 772 IPOL=193 GO TO (69056919693) sIPOL ACL) =(1.90.} A(3)=(0.90.) WRITE (62110) 60 T0692 A(3)=(1e.00.) WRITE (60111) 692 CONTINUE CALCULATE THE SUM OVER INITIAL STATES OF THE ANGULAR INTEGRAL D0 20 J=1+6 . CYLM (JS) =(02090.0) 00 15 IX=1e3 DO 15 IR=1ls3 15 CYLM(J) =CYLM (J) +A (IX) MANGINT (1X eT Re J) HY ENM (IB) MULTIPLY BY RADINT, TOTAL DETECTOR INTENSTTY IS PROPORTIONAL TO 20 THIS RESULT TIMES THE SPHERTCAL HARMONTCS Y (LoM) EVALUATEN AT THETA AND PHI. CYLM (J) SRADINT (2) #CYLM( 9) TF (JeEQ.1) CYLM(J)=RADINT (1) #C¥tM (J) ZRADINT (2) CONT TINUE EVALUATE Y(LoM) AT ANGLES FOR PLOT 693 CONTINUF aan aaAaaND oan AaNIGBaa aan 45, DO_55 IJ=leNPLOT LPT=LPLOT (IW) : GO TO (19791226125) sIPOL 122 CONTINUE GO TO. (21922923 624921 92? 922024) eL PT 21 ANGMIN(T J) =060 ANGMAX (1.)).=9020 GA TO 26 22 ANGMIN(TJ) =000 ANGMAX(TJ)=180, 60 TO 24 23 ANGMIN(TJ)=06 ANGMAX (TJ) =360. 24 CONTINUF WRITE (69106) WRITE (69107) ISoL PLOT (IU) sANGFIX (TI) pANGMINIT I) » & ANGMAX (TJ) sANSTEP (TU) WRITE (65508) WRITE (69599) GO TO (259259254925 926926926 926) oL PT 25. CONTINUF SWEEP THETA WHILE MOLDING PHI FIXED ZERO ANGLES PO=ANGMIN(IJ)-ANSTEP (IJ) AZ=ANGFIX(T J) . 120 37 CONTINUF LOOP OVER THETA PO=PO+ANSTEP (IJ) I=I+). ; ANGVAR=P0 60 TO 39 26 CONTINUF SWEEP PHI WHILE HOLDING THETA FIXED 7ERO ANGLES PO=ANGFIX (TJ) AZ=ANGMIN(TJ)“ANSTEP (IJ) 1=0 38 CONTINUF LOOP OVER PHI AZT=AZ+ANSTEP (TJ) YsI4] ANGVAR=A7 a9 CONTINUE PSTF=(0.0+60,0) CALCULATE VALUFS OF THE SPHERICAL HARMONICS AT THETA.AND PHI ; JEl3 -LoM=08 Js? TO 62 Lats Mzm?2 TO 2? a aan omens) aan anon 46. CPO=CNS (PN#RAD) SPO=STN(POFRAD) CAZ=COS (A7#RAD) SAZ=SIN(AZ#RAD) C2A7Z=COS (2, #AZSRAN) S2AZ=SIN( 7, @AZ#RAD) YLM(1)=71 YLM (2) =¥24SPO#H2% (C2AZ=—CIeS247) YLM (3) =¥3#SPO#CPN® (CAZ—“CI#SAZ) YLM (46) =Y4% (1 oS #CPNH#2-0,5) YLM (5) s=¥ 3#SPO8CPNF (CAZ4+C 1 #SAZ) YLM (6) =¥2#SPO##2# (CZAZ+CI *S2A7) DO 48 J=196 PSIF=PSTF4+CYLM (J) FYLM (J) WRITE (8o201) YEMePSIF sJsCYLM (J) : 201 FORMAT (* YLM=t. ¢2F12659° PSIFs! 92F12059" CYLM(', RI19%)=09P?F 12.5) 48 CONTINUE DETECTOR INTENSITY = "CUR? CUR (1) =PEAL (PSIF#CONUG(PSIF)) XACT sIPOL oT) SCHR (1) 50 CONTINUE PANGLE {T) =PO AANGLE (1) 2AZ P(19TJ)=PANGLE (1) QUTsIU)=AANGLE(T) oo, WRITE (64510) PANGLE (7) sAANGLE (7) sCUR(T) IF CANGVAR.GE.ANGMAX(TU)) 69 TO: 53 RETURN FOR NEXT ANGLE _, GO TO (37037937 93793863Rs3As3A8) LPT 53 CONTINUF NPOTN=ST - NP (TJ) =NPOIN FIND MAXIMUM DETECTOR CURRENT TEST=CUR (1) DO 60 K=?eNPOIN . XA (Ke TPOL e TJ) SCUR CK) TF (CURC(K)=TEST) 6996005A 58 TFEST=CUR IK) 60 CONTINUE 60 TO 120 FOR UNPOLARIZFD LIGHT 125 CONTINOF NPOITN=NP (TJ) 12) TEST=0,9 WRITE (60103) WOILTE (60106) WRITE (60107) FUoL PLOT (TS) sANGFIX (TJ) SANGMINGT J) sANGMAX (TU) » & ANSTEP (TU) WRITE (69599) WRITE (69599) aaa aan Qaa aaa 47. DN 123 K=) eNPOIN XACK og be TNSXA(K oe Le TUP XA (Ke Po DS) PANGLE (K) =P (KeTJ) AANGLE (K=O (Ke TJ) CUR (K)=XA(Koelely) WRITE (64510) PANGLE (K) sAANGLE (K) 9 XA (Kel 9 TU) IF CXA(K eo Le IU) “TESTI 123912301246 124 TEST=XA(KelgIJ) 123 CONTINUE 120 CONTINUF NORMALIZE CURRFNT DN 62 K=1eNPOIN 62 CUR(KI=CUR(K)/TEST PLOT CALL PRTNUM (10.926e0e2)9EEVe! (RHENFRGY SFT. 393H Ev) 's0.) IFCTPOL=2) 10011912 10 CALL SYSSYM(10.91.590e219°X POLARTZATTION' 91490.) 60 TO 14 11 CALL SYSSYM(10691,50002101Z POLARTZATION* » 1490.) 69 TO 13 12 CALL SYSSYM(10091.590e219"UNPOLARTZED LIGHT’ 51790.) 13 CONTINUF GO TO (63+64965 565 966067 068 +6R) of PT LPLOT = 1 63 CONTINUE eos DO 81 K=1eNPOIN X(K)=CUR (K) @SIN(PANGLE (K) #RAD) ¥ (K) SCUR (K) #COS (PANGLE (K) #RAD) 81 CONTINUE CALL VLAREL (3o93090e0le0Seols*CUPDENT! 97509! (F2. 0)¢,1) CALL VLAREL (30930 90eolesSevle "CURRENT oTole" (F2, 0)%,1) CALL. PRTNUM(10.0598 2090.21 sAANGLF og *(SHPHY =9F54158H DEGREES) 4500) CALL SYSSYM(10e59ReSeOe21e* THETA = 0 TO 90 DEGREES' +2390.) CALL XYPLOT(NPOINe Xe Ve=0ebe2ede=N,fole4oMNel) 60 TO 54 LPLOT = 2 64 CONTINUE NO A2 K=leNPOIN X(K)=CUR(K) #SIN(PANGLE (K) #RAD) Y¥(K)=CUR (K) #COS (RANGLE (K) #RAD) R2 CONTINUF . Loe . CALL VLAREL (20 AesDeslesZeoletCHIRPENT! 7909" (F220) %51) CALL LARFL (Ze ebgemlesles 4eem2e *CURPENT #9091) CALL PRTNUIM(10.19R 0090621 9AANGLE e *(SHPHI =9F5.198H NEGREES) toe) CALL SYSSYM(10098.590e219*THETA = 0 TO 180 DEGREES*,24e0.) CALL XYPLOT (NPOTNOXsVomleoheSenmGy ole shed) 60 TN 54 Cc c aaa aan aan 48. LPLOT = 3 08 4 DO A3 K=)eNPOIN: X(K)=CUR(K) #SIN(PANGLE (K) #RAD) Y (KY =CUR (K) #COS (PANGLE (K) #RAD) 83 CONTINUE CALL LARFL (Le sSeemlovlesl0es=20 tCLRRENT! 99590) CALL LAREL (Fes DO omleslaslOsem2sCIIRRENT* 9091) CALL PRTNUM(10.09R 0099.2] sAANGLE «6 # (SHPHT =9F5.198H DEGREES) ' 90.) IF (LPLOT(TU) -EQ9.4) GO TO 51 CALL SYSSYM(1069A.590e2199THETA = 0 TO 360 DEGREES*.2499.) 51 CONTINUE CALL XYPLOT (NPOTNsXoVo—le2eleAs—] oMel.09NDs1) GO TO 54 . LPLOT = 5 66 CONTINUE DO 84 K=]eNPOIN Y(K) =CUR (K) SIN (AANGLE (K) #R 4D) X(K)=CUR (K) #COS (AANGLE (K) #RAD) A& CONTINUE CALL. VLARFL (3093.90 0lesSests SCUPPENT! 97909" (F2,0) % yl) CALL VLAREL (30930 90eelesSeels*ClIOPENT? o7elo"(F2.0) %ol) CALL PRTNUM(10.98.0090.2] sPANGLE 9 *(7HTHETA =9F5e19BH DEGREES) § 900) CALL SYSSYM(10.00R.590.21s*PHT = 9 TO 90 NEGREES' +21 90,) caLL xyator (NPOINOX 9 Vo~e6e2ebomehnledeNDel) LPLOT = 6 67 CONTINUE DO 85 K=1leNPOIN : Y(K)=CUR(K) ®SIN(AANGLE (K) @RAD) X(K}=CUR(K) #COS (AANGLE (K) #RAD) 85 CONTINUF CALL LABEL (6093e00eeleoSaeml es (CURRENT? +091) CALL PRTNUM(10.9R.0090.21 2PANGLFs*(7HTHETA =9F5o198H DFGREFS) M0) CALL SYSSYM(10.099.590.219*PHI = 0 TO 180 DEGREFS* 92290.) CALL XYPLOTINPOINe XeVorle2eleAemeheledsNDeol) 60 TO 54 LPLOT = 7 OR A 68. CONTINUE DO 86 K=1]eNPOIN Y(K)=SCUR (K) SIN (CAANGLE (K) #RAD) X(K) =CUR (K) #COS (CAANGLE (K) #RAD) 86 CONTINUF CALL LARFL (Nes3eemlesleos S4eam2e!ClIRPENT #9090) CALL LARFL (2esloemlevles 400-26 "CURRENT! 9091) CALL, PRINIM(10. 63.50 90e21 sPANGLE e * (7HTHFTA =5F5.198H DEGREFS) #900) IF (LPLOT(TY) -FQ.8) GO TO S? CALI. SYSSYM(10.0046000.21e*PHT = © TO. 3260 DEGREFS's22¢0,) S2 CONTINUF CALL XYPLOT (NPOINeXsVoml eM e6e5emtoSe te 9NDel) 49. 54 CONTINUE 55 CONTINUF 772 CONTINUE 771 CONTINUE 200 CONTINUE 100 STOP ‘ END SUBROUTINE RADIAL (EF SeFK sNRAD) DIMENSION ENERGY (30) »SDUM(30) RFAL KDUM(39) COMMON /RADI/ ENERGY »SDUM»sKDUM FS=YNTERP (ENERGY »SDUM sE »NRAD 9 3) FK=YNTERP (ENERGY eKDUM2E sNRAD 2 3) RF TURN END 7/0ATA 00 2 SULFUR @ §,0 7 5.0 Ss 0.1000NN00E 0.1 000N000E 0.2>5000000E 0,5000Q0000€ 0.10000000EF 4/ « 1.9 0,0 10.0 01-0,2214237S5SE 00 O0,11007625F 00 02-0,10206109F 00 0,144656525F=-0) 02-0,42851407E-01-0,29119879E=02 02-9 .28469468E=-01-0.29701153E=02 03-9 .67723021E-02~0,12967903E-02 50. Program 4. Z/RADTAALE JOS (98057 eWHWsCHE) sSHEARN eT IMFH (1040) 7*#JOBPARM T0C=15 4/ EXEC. FORTG 7/SYSPLTON DO SYSOUT=N S/FORT AAQGAIANAAANNANAAARAAIAAEANAH aOaaan0nnanananaaaa bpd) * THIS PROGRAM LISTS AND PLOTS RANTAL INTEGRALS VS. ENERGY FOR -1P TO 8 ATOMSsUP TO 5N ENFeERIES PFR ATOM. THERE ARE 4 ATOMS TO A GRAPH SHEET» WITH THF TOP GRAPH RFING RADINT(1)+ THE MIDDLE BEING RANTNT(2)»9 AND THE BOTTOM RETNG RADINT(2)/RADINT(1)+ MAKING & TOTAL OF 12 GPAPHS PER SHEFT. NATM TS THE NUMBER OF ATOMS, ft INDEXES THEMs AND YATM INDEXES THE ATOMS ON ANY ONE GRAPH SHEETe NENG (= NE(T)) TS THE NUMBER OF ENERGIES FOR WHICH THERE ARE RADIAL INTEGRALS FOR ATOM I INPUT NATMs THEN NENG FOR EACH ATOM (FORMAT(I3) »ONE CARD EACH)» THEN INPUT ENERGIES AND RADIAL INTEGRALS FROM RADINTS PROGRAM (PUNCHEN CARDS) «+ FORMAT. (3E£15.8) THE ENERGIES ARE ASSUMED TO CoveR 0 TO 100 EV EPLT IS THE X=AXIS MATRIX AND NPLT TS THE Y=AXIS MATRIX DIMENSTON ENERGY (8956) ¢RAD) (Ae5%) ePAN2 (8950) RADI (9950) DIMENSION NE (8) ws DIMENSTON YMP (3) sYPP (3) DIMENSION X(4) 9¥ (3) sDPLT(50) sEPt T (50) DIMENSION XMINP (4) oXMAXP (4) » YMINP (3) 9 YMAXP (3) . DIMENSION 00 (3) INTEGER TEST DATA X/2 095 e9Be elles a DATA Y/7.e4e9]1e/ , DATA XMINP/=100, 9-250 69-4000 9-5506/ DATA XMAXP/650 095006 9350092006/ DATA YMINP/=0.350-0,209=0.5/ DATA YMAXP/0.15590.3094,5/ DATA YPP/0.1090019120/ DATA YMP/0.090.190.0/ DATA OD /0.050.050.0/ X(T) AND Y(J) ARE THE AXIS STARTING POINTS FOR THE GRAPH IN THE. ITH COLUMN AND THE JTH ROW XMINP(T) AND YMINP(T) ARF THE VALUES OF THE LEFT AND RIGHT HAND EDGES OF THE PLOTTING AREAS FOR THE ITH COLUMN OF GRAPHS ON THE PAGE YMINP(T) AND YMAXP(1) ARE THE VALUES OF THE TOP AND BOTTOM OF THE PAGE FOR. THE ITH ROW OF GRAPHS. THESE MUST BE ANJUSTED EACH TIME you CHANGE VERTICAL SCALES ON THE GOAPH AXESe YMP(T) AND YPP(T) ARE THE UPPFR AND LOWER VALUES OF THE AXES FOR THE ITH ROW OF GRAPHS. CHANGING THFSF CHANGES THE ORIGIN AND THE VERTICAL SCALE. THEY AFFECT ONLY THE AXTS PLOTTINGs NOT THE LINE PLOTTING. enh ene 203 en4 205 206 en7 208 209 210 161 130 ol. FORMAT (3F15¢8) FORMAT(1T3) FORMAT (* NATM =%913) FORMAT (21X et ATOM NOS* oT 29S59X et ATOM NOS IZ) FORMAT (6X e SENG4 RX "RAD #4 8Xs PRANZE AX s *RAD39929X 9 ENG? BK KR TRADI ® 6 AX, *RANZ# SAXs "PAN3*) FORMAT (F12.6°3F1205920X 9F 12069 3F 1205) FORMAT (21X%9*#ATOM NOw* oT 2) FORMAT (6X0 *ENG 86 8Xs *RADI# 9AXs FRAN? GAXs PRADZ#) FORMAT (F12.693F12,5) FORMAT(* NENG ='413) READ (5.292) NATM WRITE (69203) NATM- TEST=0,0 DO 161 T=leNATM READ (59202) NENG WRITE (63210) NENG NE (1) =NFENG TEST=MAX0 (TEST » NENG) CONTINUF 00 160 [=1yNATM NENG=NE (1) DO 130 J=1 NENG . _ . REAN (59201) ENERGY (TI oJ) sRADI (19d) eRAD2(T oJ) RAD3(T oJ) =RAD2(T oJ) /RADI (IT oJ) CONTINUE _ KSNENG#4) 140 160 150 180 155 IF (NENG.FQ.TEST) GO. TO 160 DO 140 JekeTEST ENERGY (19J)=0.0 RADL(TsJ)=1. RAN2(Toy)=1. RADZ(Ts JH]. CONTINUE CONTINUE 00 190 JJ=1.20 TE=(NATM41) 72 IF (JJeGT.TFE} GO TO 190 IN=NATM/2 - IF (JJeGT.ID) GO TO 180 JK=JJ#2 : JI=JK=1 WRITE (65204) JIoJK WRITE (69205) DO 150 J=1sTEST WRITE (49206) ENERGY (JI oJ) sRANL (UT oJ) eRAD2 (ST oJ) SRANS (UT oJ) > & ENERGY (JK oJ) sRAD] (UK oJ) pRAN2 (UK 9.1) sRANS (UK 9) CONTINUE 69 TN 198 CONTINUF JT =JJF2=1 WRITE (60707) JT WRITE (45208) DN 155 JaleTEST WRITE (602799) ENERGY (JT oJ) oRADL (UT oJ) eRANZ (ST 9 J) sRADZ (JT 9 JU) CONT TINUF 190 CONTINUE no. 301 311 aan 304 aaa 305 303 306 309 307 310 Nan 302 30] 7/OATA o2. T=] oNATM LABL=9 TATM=T TF (LeAT.A) LATMay-3 IF (LATM.GT.V) 60 TO 311 CALL SYSSYM (90,2599, 39 e021 et ATOME G4e0,) CALL SYSSYM(0,25 Rel sMe2l ey LE0' 530%.) CALL SYSSYM(U,25 95010002] 9*L=2 453000) CALL SYSSYM(0 625 eAeloMe2l ea PATIO 5900) CALA SYSSYM(2.99 Fe39e0.2botSTt e200.) CONTINUE FOR EACH ATOM: STARTX=(FLOAT (TATM) #3.) 0,05 STARTX=X¢IATM) . NENG=NE (1) 00. 304 3 KC=) »NENG . EPLT (KC) sENERGY (I 9 KC) CONTINUE XM=XMINP (TATM) XX=XMAXP (LATM) FOR EACH GRAPH RELATED TO THAT ATOM? DO 302 IPLT=193 GO TO (30553069307) sIPLT DO 303 KC=l1»NENG DPLT (KCI =RANL CT KC) 60 TO 308 |. 00 309 KC=1+NENG DPLT (KC) =RAN2 (1 KC) GO TO 30R DO 310 KC=l1sNENG DPLT (KC) =RAN3 (1 eKC) IF CTATMSFEQ.3) LABL==1 THIS. MEANS THE LAST GRAPH ON THE SHEET CONTINUE STARTY=Y(IPLT) YM=YMINP (TPLT) YX=YMAXP (IPLT) YMN=YMP (TPLT?} YMX=YPP(TPLT) STRTY=YC(TPLT) CALL. VLABEL (STARTX sSTARTY oO o510M.e2e929 "ENERGY (EV) 4011905 & (F520) 494) CALL VLAREL (STARTXeSTARTY sYMNGYMX o2ea2e tC Ogle! (F6,3) 146) CALL XYPLOT (NENG cEPL T oDPLT oXMe xh op ¥Me YY SND sLABL) CONTINUE CONTINUF. SsTne END 90 * 53. APPENDIX 2 Laser-Induced Chemisorption Studies This work occupied much of the actual time of my master's work. The project consisted mainly of constructing the vacuum system used. However, since no experiments have been completed as-‘of this time, this work is included only as an appendix. The work will be completed by others in the future. 54. Surface studies attempt to understand the nature of bonding between solids and the species adsorbed on them. An important area of investi- gation is the characteristics of the activation energy barrier to ad- sorption of saturated hydrocarbons on transition metals. This research has important implications: in the field of industrial catalysis. One would like to know what type of energy is needed when a molecule sticks to a surface and dissociates: is it translational, rotational, or vibrational energy of the gas molecule or vibrational energy of the surface? The answers to these questions are well developed for the specific system of methane on rhodium. In a novel series of careful experiments, Charjies Stewart and Gert Ehrlich? used a heated molecular beam of methane directed at a rhodium field emission tip to measure adsorption rates for three isotopes of methane, CH)» CHAD,» and CD). Analysis of the isotope data ruled out translational energy, surface residence lifetime, electronic or rotational. energies as determining adsorption probabilities. “Two methods of vibrational analysis were then applied to the data to determine which of the vibrational modes was most likely to be responsible. Of these methods, the better accepted RRKM activated complex theory made predictions that were in poor agreement with the results, although rates calculated for the V3 asymmetric stretch (see Fig. 1) agreed least poorly. Slater's dynamical model of unimolecular reactions, which uses bond extension due to a random superposition of normal vibrational modes as the critical factor for dissociation, appeared to agree much more closely with the results. From this analysis, Stewart and Ehrlich concluded that the active vibrational mode is Vas an asymmetric bend in which the mean square 55. displacement of the hydrogen-hydrogen distances is a maximum. Such a mode would show appreciable isotope effects since the heavier deuterium atoms would not vibrate so far. Our experiment was designed to test whether the V3 OF vy vibrationally excited mode of methane is more likely to induce the dissociative chemical 2 Both of these modes are infrared-active modes, adsorption of methane. and can thus be excited by an infrared laser. The energy levels, lifetimes and. decay characteristics of the methane vibrational modes have been extensively studied by C. B. Moore and coworkers” using a helium-neon infrared laser line which happens to overlap-a vibrational level of. the V3 state. Our experiment was designed to use the same technique to excite methane molecules in the vicinity of a rhodium surface into the V3 excited state. The resulting adsorption would be compared to that of unexcited (room temperature) methane. Then by deactivating this mode to the lowest excited vibrational level, Vy» we can measure adsorption from this excited state. The deactivation would be accomplished by a high pressure (~1 torr) of helium gas, which would increase the molecular ‘collision probability by several orders of magnitude. The experiment would then proceed as follows: A. Low Pressure (v.,) Test The methane will be admitted through a capillary doser to the laser cell. It will then be excited by a helium-neon infrared laser line at 3.39 H. The configuration of the laser cell will allow the radiation to be reflected up to ten times in front of the crystal. After a variable time of excitation, the cell will be evacuated, and the thermal desorption spectrum of species left on the rhodium surface will be measured using a 56. quadrupole mass spectrometer. Hydrogen will be measured primarily, as it is the expected product of CH, dissociation. B. High Pressure (v,) Test: This test will then be performed to compare with the former results. In addition to methane, approximately one torr of helium will be admitted to the laser cell prior to excitation. The helium pressure can be measured accurately with a capacitance manometer. The helium, 99.9999% pure to begin with, will have been purified to less than one part in 10” hydrogen impurities by diffusing through a heated quartz tube. The methane will be excited as in the first experiment by shining the laser on the cel] in the same multipass configuration. Because of the large number of collisions that the methane molecules undergo before _ reaching the crystal, they will have deactivated to the Vy vibrational mode. The cell will be evacuated and the flash desorption hydrogen signal will be recorded along with adequate controls, such as moving the crystal away from the region of laser excitation, performing the experiment without laser excitation, and performing the experiment without the crystal (to test for effects due to the crystal support wires). These two experiments _ should be sufficient to determine whether V3 or Vy is involved with over- coming the activation energy barrier to the dissociative chemisorption of methane on a cold rhodium surface. The excitation energy of the V4 mode of methane is 1306 wave numbers which translates to 3.9 kcal/mole. Since the activation energy of chemi- sorption is measured to be ~ 7 + 1 kcal/mole, it is expected that translational energy is important to the process. If no adsorption is seen from either the V3 or the V4 mode, this will probably be the reason. o7. In that case we will heat the methane first to a temperature of about 400 - 500 K, at which temperature Stewart and Ehrlich still. saw no dissociative adsorption, but which Should be high enough to provide the necessary translational energy. Construction of the System: Fig. 2 shows a schematic diagram of the final system. Two nickel getters wil] be used to purify the methane and one titanium getter will be used on the helium. Oxygen and hydrogen will be used for cleaning and calibration purposes. The system construction involved an inordinately large number of technical problems. Several bakeout cracks forced the installation of glass bellows at every possible stress point. The system was initially designed with doubly trapped mercury diffusion pumps but. one of these cracked off the system and contaminated the mass spectrometer with mercury, so it has been replaced by an ion pump. Because of the large number of setbacks, the experiments have yet to be performed. Preliminary data included the determination of the extinction coefficient of methane using our laser. Fig. 3 shows the data and a weighted least-squares regression line, giving an extinction coefficient of ¢ = 0.13 torr” tcm™! Venn? compared: with literature values of ¢ = 0.12 and c = 0.18 torr” Construction of the system and performance of the experiments will be continued by Dr. John T. Yates, dr. and Jenna J. Zinck, both at Caltech and in Washington, D.C. at the National Bureau of Standards. 58. References: 1. Stewart, C. N., and Ehrlich, G., J. Chem. Phys. 62 (1975) 4672-4682. 2. The reader is referred to my Ph.D. Candidacy report (March 1978) for a detailed description. 3. Moore, C. B., Acc. Chem. Res: 2 (1969) 103-109, and references ‘therein. o9. v 1 symmetric stretch 3 y | asymmetric stretch bend Figure 1. Vibrational modes of methane. dapurpA wnt ay OL “(2b ZewayIS) waqyshs wnndeA *zZ aunbr4 dung aL qeaAow 3 OL ouL[au04 Of WNL LSH 4O4 Ging 493309 wn ueyt | dun * wing pue JAszououey sdeul oy yea] WODAA aouRz Lede) uduzeuRg AXLALZLSUaS YBLH L = i C) : t &y) abeny 3 ] e uoLzezLuoy | fy S 2 # quad [y-puekeg © wing pue = | sdeszy of — hae 1 {ides -) aueyu.ay wo} Sqing 493309 [B4DLN | / 40d 4 id) ty aueyzay qpUoUe | Ly abeny ysety Ss uotyezLuo] : quad Ly-puekeg wan) D) (S) LL99 uaseq 4 C AaTswouep AazaAMWMOUZIAdS aouey 12ede9 SSPy aLodnupenh UO. eUeg 2.0 3.0 1.0 LOG INTENSITY 0.0 -0 -1 -2.0 61. | | J 0.0 1.0 2.0 3.0 “0 METHANE PRESSURE ¢ TORR? EPSILON = 0.132 /TORR-CM Figure 3. Extinction coefficient of methane. 62. APPENDIX 3 Propositions 1. Matrix Isolation of Metal-Silica Models. 2. A Hartree-Fock Model of Physical Adsorption. 3. Metal Carbonyl Complexes on Surfaces. 63. Proposition I Metal catalysts are frequently dispersed as small particles in order to increase their surface areas and their per-atom efficiencies. For proper handling and recovery of the particles the catalysts must be attached to some sort of support material. Silica gel and other silicates are often used for this purpose because. of their inertness to catalytic reagents, their ability to hold the catalyst, and often their ability to enhance the catalytic activity.2? 2 The importance of supported meta) catalysts is an obvious motivation for research into the nature of the metal-silica bonding in supported heterogeneous catalysts. Much is known about the properties of the metal- silica bond in homogeneous systems. (one or two metal atoms)° but corres- ponding information on larger systems such as clusters and particles does “not exist.4 It is therefore proposed that a study be undertaken of the bonding between metal clusters and silica or silicates. The aims of such a study would be to describe as many different bonding parameters as possible, such aS the geometry of such complexes (bond distances, and which atoms are involved in bonding), the effect on the bonding of other atoms bound to silicon, the relative numbers of support and metal atoms, and the effect of the support on adsorption (ligand-binding) properties of the metal. Of course no one technique can definitively answer all of these questions for even one system, but a technique which promises to give clues to some of them is cryophotoclustering or matrix isolation. In matrix isolation experiments, the atoms whose properties are to be 64, Studied are co-condensed in a matrix of inert gas atoms at low temperatures (10 - 77 k).° —8 Gas phase spectra of the atoms are compared to the ultraviolet and optical, or infrared absorption spectra of the species in the matrix. As irradiation continues the matrix begins to heat up. Care- ful control of the wavelengths and intensities of irradiation allows one to follow new species as they are formed by the thermal diffusion of species inside the matrix. In this manner one can selectively study small clusters 7 6 or two”? co-condensed species. (one to about six atoms) of one When choosing a system one must keep several questions in mind. Is the system relevant to current scientific studies? Will it be simple enough to understand, and is it interesting? A system which seems to fit these criteria is rhodium atoms and Si0 molecules in an argon matrix. 1,2,9-12 and 13 — 16 Rhodium is selected because of its catalytic importance because homogeneous solutions of complexed rhodium, both atoms and clusters 4,13, 17, 18 are well known. Optical spectra of the clusters, if they are not now available, should at least be fairly easy to obtain. Silicon monoxide, Si0, is chosen as the model support species. because it is intermediate between true support materials such as zeolites, which are too large to understand or possibly even to incorporate in such matrices, and silicon atoms, which would be simpler to study but are far different from support materials. Furthermore, Si0 contains oxygen, which is fre- quently indicated as the atom through which metals bind to supports.°* 29 It will be possible to study such interactions in this experiment. Matrix trapping of the two species will be performed by heating them in separate furnaces. Rhodium wire will be wound around high melting tungsten wire and evaporated onto spectral plates such as NaCl (for optical 65. and ir. experiments) and CsI (for ir studies) .° Simultaneously, SiO wil} be formed by heating Si0, and Si to 1480 ~ 1680 K in an argon atmosphere. /? All three gases will be deposited onto the cold plates. Concentration ratios will be varied over several orders of magnitude using the more dilute samples to study atomic absorptions and the more concentrated samples for clusters. During the deposition process total amounts deposited can be monitored using a quartz crystal microbalance.® This consists of a5 MHz quartz crystal oscillator which is exposed to the same deposition current as the spectral plates.® The rate of frequency change of the crystal is propor- tional to. the added mass. By correcting for geometrical factors one can calculate the mass added to the spectral plate. With the variables of concentration ratios, irradiation frequency, and irradiation time one can perform a large number of useful experiments. Some of the more straightforward types are listed in Table 1. One thing to bear in mind is that taking an absorption spectrum causes heating of the matrix in localized spots around the irradiated species. That species can then diffuse through the matrix if the heating is intense enough. By varying the time of irradiation.one can control this effect. As the species diffuse through the matrix they will form various types of chemical bonds. Since Rh-Rh bonds, Rh-Si0 bonds and Si0-Si0 bonds are stronger than any bonds to argon, one can expect to build larger clusters of these types as irradiation continues. By selective frradiation, say of rhodium atomic lines, one can heat the matrix around a rhodium atom until the atom moves to another Rh or Si0. The bonding will change the absorption frequency so that the newly formed cluster will no longer be able to diffuse. Thus the absorption due to rhodium atoms will decrease and absorptions due to clusters will increase with time. Note that the former lines can be 1. 2. 3. 4, 12. 13. 14. 15. 16. 17. 18, 19, 20. Concentrations Ar:Rh + Si0 Rh:Sid Irradiate 1000:1 1:0 Rh 100:1 1:0 Rh . 10:1 1:0 Rh 10:1 1:0 Rhy after Rh 10:1 1:0 Rh, after Rh 1000:1 0:1 Si stretch ~ 1000:1 0:1 S10 opt- cal bands same as O:1- Sid vi- 2-5 brations 1000:1 10:1- Rh 100:1 1:1 ~~- Rh 10:1 1:1 Rh 1000:1 l:l- Sid : 1:10 100:1 1:1 $id 10:1 1:1 $i0 100:1 10:1- Rh then 1:10 $id 10:1 10:1- S10 then 1:10 Rh 100:1 10:1- —- Rh-S10 or 10:1 1:10 66. Table 1 Rh atomic absorptions Rh atoms plus some Rho Rh atoms, some Rho and higher clusters Rho disappearing Rh, Rh, n> 2 growth Rh, disappearing, others growing Si0 vibrations $10 electronic excitation $i,0, n= 1 and up clusters growing May see some Rh-Si0 perturbations Probably will see Rh-Si0 perturbations; RhoSi0, maybe. : Probably will see Rh-Si0 perturbations, Rh, S10 clusters growing May see Rh-Si0 perturbations Probably see Rh(Si0), maybe Rh(S10)4 Probably see Rh(S10),, See Rh, Si0 » Rh, (S10), See Rh(S40),, > Rh, (S10), Study M-Si0 bond stretch, determine if M-Si0 or M-0 67. continuously monitored while the latter absorptions must be measured in Short pulses, or one would heat the clusters as well. 6,7 and lasers® have been used as light Both conventional lamps sources in cryophotochemical experiments. In addition to tr and uv/ optical studies, laser Raman experiments have proved useful.® Raman spectroscopy is advantageous in that different selection rules allow transitions which are symmetry-forbidden in infrared absorption exper- iments. This allows a more complete characterization of the vibrational bands of the species formed. Since Raman experiments are technically more complicated than absorption experiments, it is recommended that these be postponed until after the corresponding ir and ug@ptical data is under- stood. It-is expected that the proposed experiments will be able to answer a number of questions about the Rh-Si0 system. Rates of formation of various rhodium clusters will be calculated, giving an indication of the activation energies involved. Thermodynamic stabilities can be inferred from the products obtained after extensive irradiation. Vibrational frequencies can be measured, and they will. help determine the structures of the Rh, (Si0),, complexes, indicating how the Si0 molecules are bound to each other and to rhodium. — Other metals can be studied using cryophotoclustering, as can models for other support materials such as alumina. This technique promises a wealth of basic information about bonds characteristic of supported metal catalysts. It is hoped that in the long run this information will help predict more efficient and useful catalytic procedures. 68, References 1. 11. 12. 13. 14. 15. 16. “17. 18. 19. 20. Takahashi, N., Okura, I., and Keli, T., 3. Mol. Catal. 3 (1977) 277 - 283. Anderson, J. R., Structure of Metallic Catalysts. Academic, San Francisco, 1975. Ellis, J. J., Organometat. Chem. 86 (1975) 1 - 56. Smith, A. K.; and Basset, J. M., J. Mol. Catal. 2 (1977) 229 - 241. Ozin, G. A., Power, W. J., Upton, T. H., and Goddard, W. A., III, to be published. Kiotzbiicher, W. E., Mitchell, S. A., and Ozin, G. A., Inorg. Chem. 16 (1977) 3063 - 3070. Ozin, G. A., Huber, H., and McIntosh, D., Inorg. Chem. 16 (1977) 3070 - 3078. Kiindig, P., Moskovitz, M., and Ozin, G. A., J. Mol. Structure 14 (1972) 137: - 144, Brinen, J. S., Acc. Chem. Res. 9 (1976) 86 - 92. conan, J., Bartholin, M., and Guyot, A., J. Mol. Catal. 1 (1976) 375 - 82. Bartholin, M., Graillat, C. H., Guyot, A., Coudurier, G., Bandiera, J., and Naccache, C., J. Mol. Catal. 3 (1977) 17 - 32. Michalska, Z. M., J. Mol. Catal. 3 (1977) 125 - 134. Baird, M. C., Progr. Inorg. Chem. 9 (1968) 1 = 159. Cotton, F. A., Chem. Soc. Revs. 4 (1975) 27 - 53. Dror, Yi, and Manassen, J., J. Mol. Catal. 2 (1977) 219 - 222. Spek, T. G., and Scholten, J. J. F., J. Mol. Catal. 3 (1977) 81 - 100. Chini, P., Longoni, G., and Albano, V. G., Adv. Organometal. Chem. 14 (1976) 285 - 344. Anderson, J. R., Elmes, P. S., Howe, R. F., and Mainwaring, D. E., J. Catal. 50 (1977) 508 - 518. Morrow, B. A., J. Phys. Chem. 81 (1977) 2663 - 2666. Kozhevnikov, G. N., Izv. Akad. Nauk., SSSR, Metal. 4 (1972) 82 - 85. 69. Proposition 2 Physical adsorption of a molecule on a metal surface is due to a number of interactions, such as van der Waals, fonic and dipole attractions, in which the interaction energies are very Tow.? Typical physisorption bond strengths are less than about 5 kcal/mole. It is instructive to study physical adsorption theoretically in order to understand the changes in the electronic structure of the surface and the admolecule which lead to a stab- ilization energy of that magnitude. Many theoretical models of physical adsorption have been studied. In the jellium model the metal surface is treated as a sea of electrons in a neutralizing background, and the effects of the field due to this surface ~ on the orbitals of nearby adparticles are calculated. This method is not applicable to stronger bonds which involve mixing of surface and admolecute orbitals because the metal is treated in a highly approximate manner. In amore exact manner one can treat the surface using Green's function expressions for the metal density of states.” Limitations in computational facilities then force a more approximate description of the admolecule; the reference cited uses one nondegenerate energy level per adatom. I would like to propose.a type of physisorption calculation in which metal orbitals and admolecule orbitals are treated with more nearly equal degrees of accuracy. Furthermore this scheme allows a detailed description of the bond character in that it separates out the interactions due to inter- molecular charge transfer, intramolecular orbital polarization, purely electrostatic contributions, and quantum mechanical effects such electronic exchange and coupling. As the molecule moves closer to the surface, changes 70. in its structure can be quantitatively described and broken down into the several types of contributions. Insight of this sort can lead to paradigms for the electronic properties of physisorbed states and hence will contribute to a basic. understanding of adsorption bonds. The Energy Decomposition Method The technique has been outlined in a series of papers by Morokuma and co-workers .° —§ They describe a method which uses the Hartree-Fock LCAO-M0 description of two isolated systems as a starting point and calculates the interaction of these two systems using Hartree-Fock theory. By isolating | different terms in the interactional Fock matrix, they optimize the energy of the complex while allowing only one type of interaction to occur at a time. For example, a system of two isolated molecules which are allowed to approach each other is represented by A+ 8 -AB. To calculate the electro- static interaction energy a Hartree-Fock calculation is performed in which no orbitals of A are allowed to mix with B orbitals (in other words, all overlap matrix elements of this sort are set to zero during the calculation). The energy calculated is the energy of the two subsystems at infinite separation (zero-order energies) plus internuclear repulsion and the attraction of each nucleus to the (zero-order) electron density of the other molecule. Nuclear repulsion and electron-nuclear attraction are calculated at some finite distance, usually the equilibrium bond distance. Polarization energy can be calculated by letting the occupied and vacant orbitals of each subsystem mix, but not allowing inter-system mixing. Such an energy contains zero order, electrostatic, and polarization terms; polarization energy is obtained by subtraction. Similarly charge transfer is calculated allowing the occupied 71. orbitals on each system to mix witn the vacant orbitals on the other. Exchange energies are calculated by allowing occupied or vacant orbitals on. each system to mix with the occupied or vacant orbitals, respectively, on the other system. Finally, the coupling error among these different contributions is evaluated by performing a calculation which allows all terms, and subtracting the sum of the individual interactions energies from this total interaction energy. Figures 1 and 2 diagram these calculations. The coupling term serves the useful purpose of indicating the reli- ability of the calculations. For strong bonds, such as chemisorption bonds, this coupling term will generally be very large, indicating that tnis energy decomposition scheme is not valid.® This will happen when the bond distorts the initial subsystems so greatly that their zero-order states become meaningless tn the complex. Because physical adsorption is a weak interaction, comparable in magnitude to others which have been studied by 5 3 this technique (such as hydrogen bonds~ and electron donor/acceptor complexes, it is expected that the technique will work similarly well for physisorption. Proposed System A proper model for a physisorption system must include a metal and an admolecule. However, these subsystems are constrained to have smal? sizes 4 where n is the total because the computing facilities required grow as n number of orbitals involved. Two choices are possible for the metal: use a small metal atom such as beryllium, or approximate the orbital structure of a large one such as nickel using pseudopotentials to represent core orbitals. (In both cases the total number of atoms must be limited to three or four.) Although. the second approach is quite useful much work has already been done 7 in the area’ and I shall focus on the first approach. 72. Energy = 0 A B EQ Ey isolated A molecule { isolated molecule Ey s ef + BB zero-order Hartree-Fock energy for the system. ES = energy stabilization due to electrostatic interactions. PL = polarization stabilization energy. CT © charge transfer stabilization energy EX = stabilization due to electron exchange. MIX = coupling term. . Total . Energy E = ES + PL + CT + EX + MIX + ED total (Sign convention: + for stabilization) Figure 1. Zero-order energies. 73. Energy = 0 A 21 © q © ® 10 Oo |” 0 q Eo. Ee, + ES + EX 0S gp + ES + PLS | | “total ty Ep t ES + Cr pn ALES E + ES + EX + PL + CT (added together) Calculation. Allows QO © © © OOO Hartree-Fock convergence on isolated system A. Hartree-Fock convergence on isolated system B. Addition to sum of 1 and 2 of electrostatic interactions (calculated classically). Hartree-Fock convergence allowing Aoce 7? Avac and Boce —_— Bac mixing. Hartree-Fock convergence allowing A —. occ Boce and Avac _— Bac mixing. Hartree-Fock convergence allowing Avce - Byac and Avac <_ Boce mixing. Hartree-Fock convergence on system AB (allows all types of mixing). Figure 2. Energy Decomposition 74. One must select a system that represents a true physisorption situation as closely as possible, under the constraints of the model. For this reason I choose to study the interaction of Be, with CO and 05. The surface chemistry of beryllium has not been studied much, presumably because of the metal's cost as well as its low catalytic activity; but some useful gener- alizations should be available from the study of its interaction with typical ligands. CO is probably the most studied admolecule of all (with the possible exception of hydrogen) and 0, is not far behind. One should be able to compare the isolated structures of these ligands with their interacting structures using the energy decomposition method. Several geometries must be studied, not only to determine which configur- ation is energetically most stable but also what changes occur in the molecule to make that geometry most favored. Figure 3 shows some reasonable configur- ations to start with. In all cases the bond lengths between the metal and the ligands can be optimized for the most stabilization. It would also be inter- esting to calculate the most stable C-0 and 0-0 bond lengths in the most favored geometries. All these energies are total interaction energies. Once the best geometry is calculated one then “decomposes” the total energy by repeating the calculation with selected mixing terms set to zero (not allowed). By performing calculations numbered 3 through 6 in Figure 2, (1, 2 and 7 were done previously) one subtracts to get the decomposed energies. Perhaps the most useful presentation of the data in these calculations entails electron density maps. In these maps one plots the change in electron density which occurs in the two subsystems as they approach each other. Such maps are conmonly made for the total energy contribution, but this scheme allows one also to plot changes incurred by individual energy contributions. It is computationally easy to obtain these maps since each Hartree-Fock 75. Be — Be Be — Be Be — Be Be — Be Be — Be C-0 0-0 ¢ 0 0 ] ! 0 C 0 Be — Be Be — Be Be — Be ---—C—0 \e Wye Be -Be ---0-—C Be - Be --- 0-0 (pseudo-tetrahedral ) Figure 3. Bonding geometries to test. 76. calculation derives molecular orbitals as well as energies. The electron density at any point is simply the sum of probabilities that an electron will be at that point for each orbital. Some electron density maps are shown in Figure 4,3 These came from a calculation of the interaction of CO with BH,. As an example of the chemical usefulness of these calculations, I show the charge transfer maps for both forward (HB —~+ CO) and backward (CO — BH) electron donations. Note that in the second case CO loses w-electron density (dotted lines) which BH, gains in the vacant boron lone pair orbital. It would be very interesting to see similar maps for ligand-to-metal interactions, both for the proposed model and for heavier metals (using pseudopotentials). I strongly suggest such calculations be performed. 77. { CTy,8*C0) ur7k 4 CTn,aeco) irk “10 00 io0-|-~— 20 30 Figure 4, Density maps of forward and back charge-transfer components for the complex OC-BHy, at R(C~B) = 1.77 A in the Cy, approach. Contours indicated are, successively, 40.5, +2.0, #3., 46.0 X 10 7 buh’, Figure 4. From Reference 3. 78. References 1. 1970} A., The Theory of Adsorption and Catalysis. Academic, New York, 2. to, W., Cunningham, S. L., and Weinberg, W. H., Surf. Sci. 62 (1977) 662 - 674. 3. Morokuma, K., Acc. Chem. Res. 10 (1977) 294 - 300, and references therein. 4. Kitaura, K., Morokuma, K., Int. J. Quant. Chem. 10 (1976) 325 - 340. 5. Morokuma, K., J. Chem. Phys. 55 (1971) 1236 - 1244. . 6. K. Morokuma, personal communication. 7. Goddard, W. A., III, to be published. 79. Proposition 3 ABSTRACT Metal carbonyl? complexes, specifically Ni(CO), and later large meta] cluster complexes, are to be adsorbed on single crystal surfaces, specifically copper. Ultraviolet photoemission spectroscopy wil) be used to monitor the dissociation of carbony] groups from the nickel atoms. Tip angles of the carbonyls still attached to nickel can be measured using angle-resol ved photoemisston spectroscopy. From the change in these angles with time and temperature, one can calculate ligand repulsions and dissociation activation energies. Other techniques also allow the calculation of relative adsorption bond strengths between different adsorption sites. Introduction Homogeneous and heterogeneous catalysis are more closely related than 1s usually acknowledged. Homogeneous catalysts are easier to characterize and to modify by means of exchanging ligand groups. They are also efficient in that every metal atom is available to the reagents. Heterogeneous catalysts are prepared so as to maximize the proportion of active, surface atoms while retaining the advantages of separability and stability. It is important to investigate both types as well as to create and study new forms of catalysts. Recently, transition metal cluster compounds have shown increasing industrial applicability.2 It is important ‘to research such cluster compounds 80. as intermediates between homogeneous and heterogeneous catalysts. In particular, it 1s desirable to study the changes that occur in atomic properties of meta] atoms as the compounds range from mononuclear complexes through small? and large” -4 clusters to particles and single crystal surfaces. At present, there are few techniques which can be suitably applied to all catalyst forms. Homogeneous catalysts are most frequently characterized by their infrared absorptions and Raman spectra » certain electrochemical redox reactions 6 7-9 and other spectroscopies. These techniques can give information about the geometries, vibrational force constants, electronic structures and chemical reactivities of the ‘complexes. Many of these techniques cannot, however, be applied to single- crystal surface studies, which are carried out in ultrahigh vacuum con- ditions. ‘Using surface techniques one can deduce similar information about heterogeneous catalytic species: electron diffraction indicates surface ordering which can imply adatom interaction; photoemission and work function studies can describe surface-induced electronic structure and admolecule geometries; vibrational spectroscopies can determine structures and force constants; and thermodynamic and kinetic stabilities and chemical reactivities can be calculated from temperature-dependent effects. It is proposed that these surface techniques be used to study the gradual change. in many molecular properties as one examines homogeneous catalysts, cluster compounds, meta? particle and single crystal surfaces. These properties include preferred binding sites of ligands (adatoms), such as bridged, terminal or low-symmetry; the optimum number of ligands per metal atom, and bond lengths and angles between them; bond energies; internal structural changes in the ligands; and electronic structural changes in the metal atoms. For example, how do the electronic and 81. ‘vibrational structures of carbon monoxide vary between gaseous molecules, physically and chemically adsorbed molecules and cluster and mononuclear metal complex carbony]s? Proposed Experiment I propose that the change in molecular properties of metal carbonyls be followed as additional metal atoms are added to a mononuclear complex. One can adsorb metal carbonyls on well-oriented, single-crystal surfaces and study the Subsequent behavior of the carbonyl] moieties with ultraviolet photoemission spectroscopy and other surface techniques. The first experiment of this type should employ a volatile, easily obtainably metal carbonyl. and a compatible surface. Nickel carbony] (Ni(CO),), with a boiling point at atmospheric pressure of 316 K , is a suitable admolecule. A logical first choice for the substrate would be a single crystal plane of nickel. If such a surface were ideal, with no imperfections or steps, this substrate would give useful information. In this case a nickel atom would bind to the nickel substrate through d-electron interactions, pushing the carbonyls away from the substrate nickel atoms. If more than one carbonyl remained attached to the original nickel atom, the carbonyls would not be oriented perpendicular to the surface, and the angular distri- bution of ultraviolet photoelectrons might be able to measure this "tipping" effect. Heating the crystal would initiate surface diffusion of these carbonyls and the photoemission patterns would revert to those of CO on a flat nickel surface. However, it is not currently possible to prepare such an atomically flat surface. In the case of an imperfect surface the smal] number of 82. added nickel atoms with attached carbonyls 1s likely to be insignificant relative to. the number of substrate nickel atoms which adsorb carbonyls from other sources (such as those lost when a Ni(CO), molecule dissoci- atively adsorbs elsewhere). It would be better to choose a surface whose effects would be separable from those of the added molecule. Copper binds CO in an entirely different ‘manner from nickel: the adsorption bond strength is about one-third lower, so. at room temperature less CO remains adsorbed on copper than nickel. 2! Furthennore, since the work functions of clean copper and nickel differ by as much as 3 ~ 4 ev,22 the photo- emission spectrum of the surface should show distinct peaks for CO on copper and CO on nickel. One could moniter the nickel CO peak intensity to measure how much CO remains on nickel as a function of time and temper- ature. At low temperatures the CO that leaves will move to other sites on the copper surface, whereas at higher temperatures (around 300 K'and above) the carbonyl is likely to desorb altogether. One can monitor total CO photoemission intensities to distinguish which effect is occurring under any given conditions. Copper is a good substrate for another reason: bimetallic clusters of copper and nickel have been studied by chemisorption and activity. measure- 13 providing a basic understanding of the interactions between the ments, two metals. The proposed experiment will both utilize and help clarify these conclusions about the chemisorption properties of nickel atoms on copper. The experiment will proceed as follows: 1. A clean copper single-crystal surface will be prepared using Standard techniques. 2. Ni(CO), will be admitted to the vacuum chamber for specific exposure times. 83. 3. Ultraviolet photoemission spectra, both angle-integrated and angle-resolved, will be taken of the surface as a function of time at constant temperature for severa] temperatures in the range 90 - 300 K. 4. Low-energy electron diffraction (LEED) patterns and their intensity vs. voltage curves will be measured under several conditions. 5. After each experiment the crystal will be flashed to high temper- atures and the thermal desorption of surface species will be measured using a quadrupole mass spectrometer. From the temperature dependence of angle integrated photoemission intensities due to CO on nickel, one can calculate activation energies of dissociation of the surface nickel carbony1 species. The angular distri- butions of these photoelectrons can be compared to distributions calculated theoretically!4 for CO molecules tipped away from vertical at various average tip angies. The best fit will determine the equilibrium positions of the carbony] ligands relative to the other ligands and to the surface. As the temperature 1s raised more CO's are expected to move away from the nickel atoms and the remaining ones will stand up more vertically. 2° Desorption energies calculated from thermal desorption peak temperatures will suggest models for the surface species. Finally, LEED data will be studied to determine surface ordering of Ni(CO) 4 and CO, especially as carbony!s dissociate from the nickel atoms. This study as outlined above will determine the feasibility of studying metal carbony] complexes on a surface. With Juck one will then be’ able to adsorb larger clusters on metal surfaces. With clusters such as Rhg(CO) 16 one may be able to distinguish carbonyls attached to the cluster even when the substrate used is also rhodium. This would be a useful way 84. to study chemisorption properties of metal edges,. corners and imperfections, represented by the metal clusters, relative to flat surfaces, represented by the substrate. In the meantime 1t would be quite interesting to study smalt metal complexes on different surfaces. 85. References 1. 2. 3. 4. 10. 11. 12. ‘13, 14, 15. Smith, A. K. and Basset, J. M., J. Mol. Catal. 2 (1977) 229-241. ‘Chini, P., Longoni, G., and Albano, V. G., Adv. Organometal. Chem. 14 (1976) 285-344. Anderson, J. &., Elmes, P. S., Howe, R. F., and Mainwaring, D. E., J. Catal. 50 (1977) 508-518. Primet, M., Basset, J. M., Garbowski, E., and Mathieu, M. V., d. Am, Chem. Soc. 97 (1975) 3655-3659. Ozin, G. A., Acc. Chem. Res. 10 (1977) 21 - 26. Ferguson, J. A., and Meyer, T. J.. J. Am. Chem. Soc. 94 (1972) 3409 - 3412. Frankel, R. B., Ruff, W. M., Meyer, T. d., and Cramer, J. L., Inorg. Chem. 13 (1974) 2515 - 2517. Barbers M., Vickerman, J. C., and Wolstenholme, J., J. Catal. 42 (1976) 48 - 53. Thomas, M. G., Pretzer, W. R., Beier, B. F., Hirsekovn, F. J., and Muetterties, E. L.,.d. Am. Chem. Soc. 99 (1977) 743 - 748. Cotton, F. A., and Wilkinson, G., Advanced Inorganic Chemistry, 3rd ed., Interscience, New York, 1972. There exists some controversy over the nature of CO adsorption on copper at room temperature. See Isa, 5. A., Joyner, R. W., and Roberts, M. W. 0. Chem. Soc,: Faraday Trans. I, 74 (1978) 546-554; Chesters, M..A., and Pritchard, J., Surf. Sci. 28 (1971) 460-468; Tracy, J. C., J. Chem. Phys. "66 (1972) 2748-2754; and Joyner, R. W., McKee, C. S., and Roberts, M. We, Surf. Sci. 26 (1971) 303-309. Cu: 8-9 eV; Kittel, C. Introduction to Solid State Physics 5th edition, Wiley, New York (1976) p, 154. : 4-5 eV, Chemical Rubber Company handbook, 53rd edition, CRC press: Cleveland (1973) p. E 70. Sinfelt, J.-H., Acc. Chem. Res. 10 (1977) 15-20. Davenport, J. W., Phys. Rev. Lett. 36 (1976) 945-949, Andersson, $., Solid State Commun. 21:(1977) 75-81,