A Discrete Basis Set Technique for Calculating Photoionization Cross Sections Utilizing Moment Theory Citation Gerwer, Alexander Sherwin (1980) A Discrete Basis Set Technique for Calculating Photoionization Cross Sections Utilizing Moment Theory. Master's thesis, California Institute of Technology. doi:10.7907/E1ZA-6224. https://resolver.caltech.edu/CaltechTHESIS:05042010-123305037 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): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1980 Record Number: CaltechTHESIS:05042010-123305037 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05042010-123305037 DOI: 10.7907/E1ZA-6224 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5769 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 09 Jul 2010 21:42 Last Modified: 21 Dec 2019 02:16 Thesis Files Preview PDF - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

A Discrete Basis Set Technique for Calculating Photoionization Cross Sections Utilizing Moment Theory Alexander S. Gerwer Submitted in partial fulfillment of the requirements for the degree Master of Science in Chemistry 978-190 California Institute of Technology Pasadena, California Contents (I). Introduction..... comme eee eee ewer eee ne enaseesese 1 (II). Theory A) The Complex Polarizability............ wees 5 -B) Quadrature Approximation to the Spectral Resolution................ 4.10 C) Photoionization: Equivalent Quadrature Unknown... ... ccc cece cence en ee 13 D) Extraction of o(E) from its MOMENTS... . cc cece ee ee eee eee eens rrr 16 EB) Principal Pseudostates..........eeceeee eee 20 F) Linearization of the Moment Problem......... oe eeh eee Roce e ee eee eee eceee 22 G) Computational ASPpeCts.... cece reccccceccecse 26 H) Illustrative Example.........cc cece ccccsoes 29 (III). Current Work... .. ee cc cc cee ee cere cree ee ewe ew enee 33 References and NOteS.....cecceccsecccscccscccees 38 ‘Appendix A: Photoexcitation and Ionization in Molecular Oxygen: Theoretical Studies of Electronic Transitions in the Discrete and Continuous Spectral Intervals. (J. Chem. Phys.: 72, 713 (1980)) Appendix B: On m-Shell Photoionization in Molecular : Nitrogen. (Chem. Phys. Lett.: 66, 116 (1979)) (I) INTRODUCTION Techniques for measuring the photoionization cross sections of diatomic molecules have improved significantly ‘in the past twenty years. )>? Precise ab initio photo- ionization calculations in the semiclassical approximation, however, have been limited to the simplest of diatomic systems.” This is primarily due to difficulties that arise in the construction of many-electron continuum functions appropriate for the unbound motion of an ejected electron in the presence of a nonspherical ionic molecular core. Consequently, ad hoc computational approximations, including the additive atomic approximation,’ one-center many - body calculations,” the use of plane-wave final states,° the scattered xy method,’ and the use of one--and two- center Coulomb wavefunctions, ® have been employed in semi- classical photoionization calculations. These techniques - require lengthy calculations, are of unknown accuracy and are difficult to systematically improve. Because computation of many-electron continuum eigen- functions for diatomic molecules is so difficult, various procedures have been suggested recently for constructing photoionization profiles utilizing the discrete oscillator strengths and transition frequencies which are obtained 9-11 from variational calculations in Hilbert space, These techniques have long been useful in conventional bound- state configuration-interaction calculations. !4 In particular, the Stieltjes-Tchebyscheff technique, 911 based on the theory of moments, => can accomodate different ‘types of basis functions and can provide highly reliable photoabsorption cross sections, Stieltjes-Tchebyscheff profiles have been reported for the negative hydrogen, !1> lithium, sodium, and potassium ions, !4 atomic helium, /0¢»11b 15 16 18 molecular hydrogen, +” formaldehyde, 20 lithiun, boron, molecular nitrogen,” and carbon monoxide, In the present report, the Stieltjes-Tchebyscheff technique is employed in calculations of total photoionization cross sections. This procedure can be outlined as follows: Moment theory is used in computing the cummulative oscillator strength distribution of the molecule in question. This- is done by employing the static-exchange approximation for the Hartree-Fock ionic core. Various sets of virtual orbitals are created for the electron which is photoejected into the molecular continuum. It is assumed that the various physically distinct excitations, or channels, of the molecule are independent of each other. This approximation greatly simplifies the computations and avoids possible spurious results caused by weak channel interactions. Each “channel is separated into its individual components as defined by the various allowed final states in which the molecule could be left by excitation. An appropriate set of virtual orbitals is generated using a basis set of normalizable Gaussian functions. The oscillator strength distribution can be computed from the dipole-allowed transitions between ‘the ground state and the virtual orbitals.?? The energies of the virtuals and the corresponding oscillator strengths serve as points and weights, respectively, of a histogram approximation to the true cross section. The total cross section is taken as the sum of each individual spectrum computed in this manner. In the case of 02 we looked at the three main ionizations that contribute to the total cross section; °Z; + hv > X71 + e , “Ey + hv > (a°, + A‘lL) +e, and “Ey + hv > (b*Z, + Ber) + e in the case of valence shell photoionization. There are eight possible allowed final states connecting - these ionizations to the 02 ground state. A partial cross section was computed for each channel by taking the sum of the components which describe that channel. These partial cross sections showed considerable agreement with the — available experimental data. The sum of these three channels yielded the total cross section, in good agreement with experiments. We were also able to show that the Stieltjes method provided reliable results for inner shell photoionization cross sections. We computed the cross section for ionization out of the K-shell in the oxygen molecule. Again, the results obtained were consistent with experimental data. In summary, the separated-channel, static-exchange calculations employing the Stieltjes-Tchebyscheff procedure yielded excellent first approximations to photoionization ‘processes in molecular oxygen, a complex, open shell system, It should be possible to obtain cross sections for a variety of complex molecules using this technique, and hence avoid the involved and lengthy computations that have previously been required to construct the spectra of complex molecules. Future work could also involve calculation of angular distributions of photoejected electrons using integrated partial cross sections in conjunction with the scattering theory of asymptotic phases of the continuum orbital wave- function. The Stieltjes-Tchebyscheff technique is detailed in- Section II. Section III provides a look at the direction in which future work will procede. The Appendix consists of two papers (in preprint form) in which we have applied the Stieltjes-Tchebyscheff technique to molecular oxygen, (IT) THEORY This section starts with the concept of a complex dynamic dipole polarizability and procedes to develope the Stieltjes imaging technique. In the last subsection, a simple case is worked out as an example. A) The Complex Polarizability The interaction of radiation with matter is described by the time-dependent Schrodinger equation?! (HO (ry) + HO r,t) - i(a/at)]u(r,t) = 0, (1) where HO) Cr) is the field-freé Hamiltonian and, in the semiclassical approximation, 77 af) (r,t) = u(r)coswt (2a) N with u(r) = -) arr, (2b) i=1 being the component of the electric dipole moment operator for an N-electron system in the direction of polarization of the (unit) electric intensity a. The latter is chosen along a principal axis of the polarizability tensor for simplicity. The symbol r represents all appropriate spin and space. coordinates. The wavefunction W(r,t) can be expanded in a perturbation * series up to first order vir t) = vO at) + ~Pat + ..., (3) where wy") (r,t). = 6) (ryexp(-iE lt) (4) (60) (ryexp(iwt) + 9? xtexp (-iE() £) (5) il y 9 (r,t) (r)exp(-iwt)] . where the function 96) Cr) is an eigenstate of the unper- turbed Hamiltonian and gO) is the particular solution of OM) - BO + oo, ~ +um@meM@ =0 (6) that is orthogonal to or), Expanding the solution of eqn. (6) in terms of the complete set of discrete and continuum eigenfunctions (6,0 qm, 6,69 (2)] of HO") (ry, o <6, 6 ufoO)> 6, = - 6, itm pay €. = Ww on 6°) (rae, (7) 1 : where e; = po) - ae ) are the discrete and ¢ = E - pO) the continuum transition frequencies, with the continuum eigenfunctions 6,6) delta-function normalized in 23 energy. The induced electric dipole moment along the direction of the unit electric field intensity a is D(w,t) = yO fu jyOds + cv) jypyl)s Substituting for yl) and yo) from eqns. (4) and (5), respectively, D(w,t) = a(w)coswt, - (8) where the frequency-dependent dipole polarizability is given by a() = <6) [ufo I> + <g [npg l> If we use the expression for ¢ Ver) given in eqn. (7) along with the orthonormality of the set om, o CI ery], we may write E o(w) = Joe z+] x42 de, (9) i=1l°i ~ ® wo : t with Wee the photoionization threshold frequency and th I = 20, 1<0, 0 [ufo >]? (10a) 2e [<op 0) full]? | (10b) g(e) the oscillator strengths and density for discrete and continuum transitions, respectively. Equation (9) can be formulated as a Riemann-Stieltjes integral, oz) = [. eee, (11) where the result has been continued to the entire complex frequency plane, €; is the resonance frequency, and € Ep @ . ce "Lie (e, - e*) + s(e)| de~ £(e) | (12) is the cummulative oscillator strength. The dynamic complex polarizability is analytic throughout the complex Z-plane except for an infinite number of simple poles on the real axis at z = te and a branch cut along the real axis in the photoionization interval 0, <|o]<e. The behavior of a(Z) across the real axis in the photo- ionization interval can be shown from its behavior in the neighborhood of z =w ti6 as 670. a(w tio) = Rea (w) + iIma(#), | (13) wtiere Rea(w) = pL agile), | | (14b) and Imo(w) . = ™g(w)/2w, (14b) 24 _ with P implying the Cauchy principal value, Both the photoabsorption and dispersion profiles are contained in a(Z), since, under appropriate conditions, the photo- * . . 25 absorption cross section is o(u) = 22 g(w) o(w) = 4mm (w) | (15a) and the refractive index is given by7® n(w) - 1 = 27 NjRea(#) | (15b) with c the speed of light and N, the number of molecules. 10 B) Quadrature Approximation to the Spectral Resolution The completeness of the set of electronic eigenfunctions of the Born-Oppenheimer Hamiltonian, H, allows an eigen- function expansion or spectral resolution for H H = 1165>E,<o,| + [aE] 6(E) >E<0(B) |, (16) ; 1 where the 1o5> are bound state square-integrable eigenfunctions H]o>= E,|o5>, (17) with the orthonormality condition <oj1o5> = Sigs (18) and the |@(E)> are continuum eigenfunctions HJO(E)> = BlO(E)>, | (19) which are delta-function normalized <o(E)|o(E*)> = S(E-E*) . | (20) Only a subspace defined by a single non-degenerate symmetry is considered in eqn. (16) 27 Thus, for each symmetry, 11 the complete set of eigenfunctions may be expressed as a sum over bound states and an integral over the continuum whose origin has been chosen at E = 0. | In an improved virtual orbital calculation, 7° a finite basis of square-integrable functions (e.g. Slater determinants composed of molecular orbitals, which are in turn expanded in terms of Gaussian orbitals) is employed. The matrix representation of the Hamiltonian, N, is diagonalized to give a set of approximate eigenfunctions, ~ Ix3>s and eigenvalues E,, ~ H| x5> = E,1x4?> (21) appropriately normalized <x 4 1X5? = ij: (22) Within the subspace defined by the choice of basis, the spectral resolution of H can be expressed as H = .> E.< | + ->E.< ele (23 0! *4 46%! bdo. poxyh ) Here the sum over approximate eigenfunctions has been divided into a portion over negative eigenvalues which 12 ‘intuitively corresponds to the bound state sum of eqn. (16), and a portion over positive eigenvalues which is expected to approximate the continuum. It has been shown? that 1 Ixj>Bj<xgl = fazlece > zoe) | (24) E200 J J 3 ° : when properly interpreted as a numerical quadrature. Specifically, to evaluate a matrix element such as u <b[H]v> = J<vlo,>B;<o,|v> + | dE<W|$(E)>E<o(E) |v>, (25) 1 ° where |) > is an arbitrary square-integrable function, one might, in a purely numerical sense, evaluate the continuum part of the matrix element as [ae<y| 6(E)>E<6 (B) |v> ~ Yuj<vlo(E5)>B;<6(E5) 1¥>, (26) ° J where the E, and Ws are appropriate numerical quadrature abscissas and weights, respectively, This is just the standard numerical technique of replacing an integral by a quadrature sum. 39 Following the suggestion of eqn. (24), the quadrature approximation of eqn. (16) should be related to the approximation of the continuum part of <wlH|y> as | dE<p|o(E)>E<o(E) |v> = 2 <wlxs>B <x. [vr (27) E.0 JI jJo3 ° 13 In both eqns, (24) and (26) an integral is to be replaced by a sum. In eqn, (26) the sum is generated by choosing a specific numerical quadrature, while in eqn. (24) the sum arises directly from the diagonalization of H. In certain cases??? 31 the seemingly distinct approximations of eqns. (24) and (26) not only yield equivalent results for <p|H|w> but the sums are identical term by term. Thus, the square-integrable functions, Ix5? [E,>0], are proportional to the continuum functions, OE ), evaluated at the same energy. When this is the case, photoionization information (at energy E5) may be easily extracted directly from the Ix5?- Interpreted more generally, the square-integrable eigenfunctions arising from discretization of an arbitrary electronic Hamiltonian in an arbitrary square-integrable basis give a quadrature representation of the actual spectrum of eigenvalues and eigenfunctions of the Hamiltonian itself. This implicit quadrature is refered to as the equivalent quadrature-” generated by the Hamiltonian and the basis. C) Photoionization: Equivalent Quadrature Unknown Equation (23) is interpreted as a quadrature represent - ation. Thus Ix,> = (02% *leGa> EPO), (28) 14 ~ where o(E;) is the actual electronic continuum state at energy E and w F9 is the equivalent quadrature weight implicit in the choice of basis. Equation (28) holds over a limited range of coordinate space. When this range spans the range of ul o> where @ is the N-electron ground state wave function, the transition amplitude can be computed. <o(E.)[plo> = (w.F4 Fey, Ju for (29). _ provided that the equivalent quadrature weights are independ- ently known. For an arbitrary N-electron system, however, the equivalent quadrature implicit in the basis discretization will not be known. Thus photoionization information cannot be extracted directly from the states IX>- Nevertheless, because a quadrature is implicit in the discretized spectral resolution, matrix elements of the form ok =<o|uHXule> = col eget + | ss] oce)>eco ey ul | ° = <olul|y cE. S<x,Julo> =k = 1,2,3,-5. (30) “joj 2 9% 15 which involve summation and integration over the whole spectrum of the operator can be approximated. Regrouping terms in eqn. (30) <o|uHSy|o> = Jo,B,* + foceEkaE, (31) : 1 0 where | o(E) = |<|ulo(E)>|? (32a) . _ 2 and 0; 5 I<olulo.>| (32b) are proportional to the photoionization cross section and bound-bound oscillator strengths, respectively. Thus <o| uHSy | o> gives the yth moment of the photoabsorption cross section which, according to eqn. (30) is approximated, in a quadrature sense, by oX = yo.E.* | (33) Q2 il where <olulxs>l? (34) Thus, although for an arbitrary basis o(E) cannot be calculated directly at the energies E, because of a lack of knowledge of the equivalent quadrature weights, the moments of the cross section can be calculated. Each of the moments ok represents a smoothing of the distribution 16 ~ of transition moments 0; calculated in the square-integrable basis set. When the basis is enlarged, the converged low order moments represent the useful information contained ‘in the improved virtual orbital calculation. _ Dd) Extraction of o(E) from its Moments The method for constructing o(E) from its moments ok can be developed from the theory of Gaussian integration.°° To approximate b [oc hexddx , | (35) a a sum of the form n is constructed. p(x) is a known weight function and h(x) is the well-behaved function to be integrated. First, the monomials x", n = 0,1,2,... are integrated; that is, the moments . | M(k) = Jo (x)axKax | (37) a are determined. When quadrature approximations are used for the moments, eqn. (37) leads to 17 b n M(o) = | o(x)dx = J w Px > (38a) 3 é& rb n 1 M(1) = | p(x)xdx = J w,°x, (38b) “a ; i=l rb . n M(2n-1) = o(x)x?2 ax = } w Px, 7nt . (38c) Ja i=l The solution of this set of 2n non-linear equations for the 2n unknowns {w5°,x,} yields a set of quadrature weights and abscissas such that the approximation b n - | o(x)h(x)dx = JY w.°h(x.) (39) a i=1 * te + is exact if h(x) is a polynomial of degree 2n-1 or less, ‘Thus, given the moments of, a quadrature can be constructed to evaluate b | o(E) h(E) dE a for arbitrary h(E). An approximation to o(E) itself, however, is needed. Here the idea of a Stieltjes approximation and Stieltjes imaging! © enters. Stieltjes** suggested that if b n | o(x)dx = ») w.?, | (40a) 18 y nt then | o(x)dx * } w, (40b) a 1=1 where a < y < b and the integer n* is chosen such that x ne < YS Xe] This gives a histogram representation of _the integral, with the rigorous Tchebyscheff bounding 32 property. nt Xa n¥4+1 a < | p(xjdx < Jw,” (41) i= a i=l More importantly the midpoint of each "rise", i.e. * 9 ie P ) Ws + J W- i=1 7 i=1 7 1 “2 ’ gives a remarkable approximation to the integral at the points of increase, x nx lnis being the case, these mid- points can be interpolated with a "smooth" analytic fit which can be differentiated. ae (xJdx = p(y) (42) aa] PlxX)ax = ply dy a , to give the weight function, 0, itself. For photoionization the quadrature weights and abscissas fo} (wo. >? E;) can be obtained from the moments o* by solving a set of moment equations analogous to those of eqns. (38). Thus, the Stieltjes approximation to 19 [ooce yar (43) fe) can be found and finally differentiated to form o(E) itself. Note that the weights w 5° derived via this process are the quadrature weights generated by the positive definite distribution o(E), rather than the equivalent quadrature weights w 9 discussed in Section IIC, which are generated directly by diagonalization of H. In practice, rather than calculating approximate _tmoments of the photoabsorption cross section, it is convenient to work with the oscillator strength distribution of eqn. (12) of Section IIA. According to eqn. (15a), the oscillator strength at each energy differs from the absorption cross section by a known factor. Thus Stieltjes imaging of the negative moments S(-k) = | © Kagce) (44) €. 1 of the oscillator strength distribution will immediately allow construction of the photoionization cross section. In approximate calculations the moments are calculated as (45) 20 ~ where es is the computed transition energy (E;-E) and the pseudo-oscillator strengths are defined as f. = 2e,|<Olulxg>l? . : (46) The f; are easily. accessible from the eigenvectors of a normal improved virtual orbital calculation. E) Principal Pseudostates The pseudostates, xX, (2), i = 1,2,...,n, can be chosen so that 2n sequential spectral moments S({-k 1) (cf eqn. (44)) for k = 1,2,...,2n are reproduced exactly by eqn. (45), in which case they are designated as principal pseudostates. To show this, consider the linear combination, n where the biG are chosen to satisfy eqns. (21) and (22). The 6, (2) are given by? ete) = (2) KH xy BO uo @w, (48a) k = 1,2,...,2n , where 96) (x) is an eigenfunction of the field-free Hamiltonian, HO) cry, with eigenvalue pO), Equation (48a) may be rewritten in the form of a recursion relation (HO) (ry-B © yor cry + u(ryo6) Ce) = 0, (48b) 21 pHO°) (ry-E0 je, (x) + e, 4 (x) = 0 (48c) . 2<k< 2n. Thus it can be shown that the expansions - & k ; | 6m = J en te @u*? (49) ~ k=1 , satisfy eqn. (6) by direct substitution followed by application of the identity in eqn. (48c). From eqns. (44) and (48)°°?°4 | S(-k+1) = (-1)*2<66 |plo,>, k = 1,2;...,2n, (50) showing that 6, (2) determines the spectral sum S(-k+l1). Noting that eqns. (48) are similar to the equations of static perturbation theory, it is evident that a 2n-1 35 theorem” exists such that if the function 6,(r) is known, the sum S(-2k+1) is also determined. Therefore, if the n functions 8, (4) for 1 < k < n are employed as primitive basis functions in the construction of the xX; () (cf eqn. (47)), the variational development will contain the correct wavefunction to o(w2 ty (cf eqn. (49)), and the 2n sums S(-k+1) for 1 < k < 2n will be reproduced exactly by eqn. (45).°° The discrete transition frequencies and i =1,...,n] that reproduce oscillator strengths [e;> f., 2n of the correct spectral sums are an optimal, or. principal, 22 choice with reference to imaging a(z) for all values of complex frequency. Consequently, the primitive basis functions 8, (2) of eqns. (48) and (49) and the linear ‘combinations results in the pseudostates X; (1) of eqns. (21), (22), and (47), are designated as optimal or principal basis functions and pseudostates, respectively, in that they reproduce the principal c and f,. From an arbitrary larger basis set of functions, n principal pseudostates can always be extracted and the principal ce, and f values can be constructed. These latter values alone, out of all possible effective transition frequencies and oscillators constructed from a basis set of n terms, exhibit the useful properties described in Section IID, hence motivating their calculation, >” F) Linearization of the Moment Problem The moment problem of Section IID is linearized through the introduction of Padé approximants>°® P-162)/Q (2)> | (51a) [n,n-1](z) = P (z) = mtg, (m) gi (51b) n-1 izo , Q(z) = 1+ fb, Mai (Sic) n 24,074 Po i=l 23 to the continued fration”” Bo A(z). = 1 . Bi Ba associated with the Stieltjes integral exnanded in powers of z " i178 n cn 1 x nd N a (53) up to order 2n-1l [Injn-1](2) = 4a(z) = 8(2) = 0(22974), (54) Equation (54) provides explicit algorithms for determining the a,(™), p, and a 0 » &. from the moments S(-k).* i n n When large numbers of moments are employed it is convenient . 13 to use the recurrence relations Q,(2) = (1-0, 29Q,.1) - 278,79), Q.,(2) = 0 , Qo(4) = 1, (55a) 24 P-1 7) = (1- n2)Py-2 64) 7 2 n-1Pn-3 67) > P_ (4) = 0 , -Po(2) = Bo, (55b) in constructing the polynomials Q (2), P (Z) orthogonal 7 n n-1 1 with respect to the distribution £(e).* The om and Bo appearing in eqns. (52) and (55) are determined from the moments in the forms, “2 _ Yn-1,n Yn-2,n-1 = 0 On ¥ " ¥ 705 7% n-l,n-l n-2,n-2 . S(-1) Yh n | = os = = * BY Y 7 By 0, By S$ (0) (56b) n-l,n-1l where the matrix elements - {-.-%-n Yn, = J Q,, Ce) d£ Ce) (57a) satisfy Yny2 = Yn-2, 241 - On n-1,2 - By Wn-2,0 , (975) Yn,2 = 0 ,> &’<n, | (57c) Yon = S(-&), | | (57d) Yn,-2 = Yiise = 0, (57e) 25 The recurrence relation (57b) is obtained directly from _ eqn. (55a) and the definition of eqn (57a). Equations (56) can be written in the alternative forms 1 (yy 2n-1 ; : “n ” BoBi-- Bay] le) Q, (2) *d£(e) , (58a) 1 (4) 2n-1 , Pn BoBi---By E] Qe) dle), (58b) ie) which are particularly useful when the recurrence coefficients are to be calculated from variationally determined pseudospectra 5 and f;, i= 1,...,N. In this case, eqns. (58) become ~ N. 2n-1 : _ 1 & ~ \ oz Oo. = =< = io Q (e.)*£., (59a) n ByB, **Bn-1 LlE n-l°"i i N 2n-1 o~ ~ * 1 1 2 al a nes ) [4 Q Ce.) °f.. (59b) n Bobi q=1 84 n-l‘ “i 1 Here the values Q,-1 (4) and Qe) are obtained from the recurrence relations of eqns (55) using O and Ba values ~ of lower order than the a, and 8 being calculated. Once the = and BL are determined from eqns (55) and (59) and variationally determined pseudospectra, eqns. (55) provide the polynomials Q,(2) and Pi. 62) - The e; (n) and £,(n) are obtained from the roots and residue of [n,n-1](Z) (cf. eqns. (51) - (54)) according to 26 Q(€;(n)) = 0, | (60a) P(e. (n)) | _ n-1*"i 7M = SMUT - Coo) Finally, then we have generated a set of smoothed points and weights which determine the calculated photoionization cross section. G) Computational Aspects We have developed an approximate procedure for applying the Stieltjes imaging method to molecules which avoids the potentially spurious results that arise from weakly interacting spectra and also the need for extensive CI- calculations to generate the pseudospectra. These approximations have been successfully tested on molecular nitrogen’? and offer quantitatively significant simplifications in the application of the Stieltjes imaging method to molecules. In order to avoid the potentially spurious results arising from weakly interacting channels, an approximate solution of the coupled-channel problem is obtained by separating out the various physically-distinct, low-lying states of the molecular ion. Wavefunctions are constructed for these states and using these wavefunctions the ionic molecular fields in which the free electron is scattered can be defined. Each molecular ionic wave- 27 function determines an effective one-particle potential which reflects the nonspherical nature of the molecular core and also contains. exchange effects. A set of square- ‘integrable one-electron eigenfunctions are obtained using this potential. These eigenfunctions are then coupled to the orbitals of the core to produce dipole-allowed total molecular states which provide the discrete excitation energies and oscillator strengths, (e,,£,), that are used to construct approximate spectral moments, S(-k), and | hence pseudospectra. Coupling between different ionic channels is ignored. This is equivalent to using static- exchange potentials for each ionic channel. The total photoionization cross section is just the sum of the contributions for all such channels. In this model a set of N orbitals, 4 and hence N discrete excitation energies and oscillator'strengths, is obtained from a single diagonalization of the appropriate one-particle Hamiltonian associated withthe residual ion core. A final state wavefunction is of the form where A is the antisymmetrizer, ®cornR(N-D) represents the core, and oy is a one-particle eigenfunction of the potential produced by the core. More specifically, the oy 28 satisfy <0; |h+V(N-1) loy? = EK ojk (62) where V(N-1) = }(2a;54-biks) . (63) Here h is the kinetic energy plus nuclear attraction terms while J; and Ks are the usual Coulomb and exchange operators. The summation goes over the orbitals of the core. The coefficients as and b; are needed to specify the Coulomb and exchange potentials of the core orbitals. For closed- shell contributions or for spatially non-degenerate orbitals in unfilled shells these coefficients are simple. The proper construction of these V(N-1) potentials is important. According to eqn. (63), the {$3 are equivalent to the virtual orbitals of the V(N-1) potential. These so-called improved virtual orbitals are frequently used in bound- state molecular applications. Our approach allows construction of the discrete pseudospectra required for the Stieltjes moment analysis from diagonalization of relatively small matrices: (cf. eqn. (62)) occuring in computations with SCF Hamiltonians. In addition, there is no need to use the same set of basis functions to expand both the occupied core orbitals and the virtual orbitals OL A modest number of valence- like basis functions with optimized orbital exponents is employed in solving for the SCF occupied orbitals. The 29 Hamiltonian of eqn. (62), however, is diagonalized over a much larger set of basis functions. This provides the rich spectrum required for the moment analysis. Since the augmented basis set is much larger than the basis set originally employed in the SCF calculations, only the necessary number of two-electron integrals is evaluated. If M functions are used in solving the SCF equations and N functions, are used to expand the virtual orbitals, the number of two-electron integrals needed goes as N? x M?. In contrast, a CI calculation in this basis requires the full number of two-electron integrals (~N*) anda trans- formation of these integrals which goes like N°. H) Illustrative Example The energies and oscillator strengths calculated from a set of discrete pseudostates can be used to crudely approximate smooth photoabsorption cross sections without even resorting to the full machinary of the Stieltjes imaging method. We will show this to be the case for helium. The pseudostates needed for this calculation are obtained from the standard IVO computation detailed in Section tir, 79 Table I provides a set of pseudostate energies, ei, ™ and oscillator strengths, f for He. The ej and f. i? values furnish a histogram approximation to the cumulative 30 oscillator strength distribution in the form f(e) = 0, O<e< €) f(e) = df ’ e; < oe < E541 (64) ne ~ f(e) = fi ; e,<e This histogram approximation is given in Table II and Figure 1. The differential oscillator-strength distribution, g(e), is the Stieltjes derivative of eqn. (64) df . Af (65) g(e) = ge * ter ©? &1 according to eqn. (12). To evaluate g(e), consider the interval between ey and e541 in Figure 1. The midpoint and the width of the interval is ~ he = (€5 44765) (67) 31 The change in thecummulative oscillator strength has a , different value at opposite ends of the interval Af(e544) = fied (68a) Af(e.) = £. (es) j (68b) In order to evaluate the quantity on the right side of eqn. (65) at Ws > we take the average of the values for Af obtained at the right- and left-hand limits of the energy interval 1 7 ~ Af = zlafles 1) + Af(e;)] (69) = ire ¢ | = ZtEj +151. The approximate expression for (5) is therefore — (£.+£. ) 1 gla) =p (70) (e5447&5) Applying this result to the pseudostate energies and oscillator strengths listed in Table I gives the values for the differential oscillator strength in Table III. The photoionization cross section is obtained from the differential oscillator strength by utilizing the relation- ship of eqn. (15a) and including a factor of three to account for the spatial degeneracy of the p-wave representing 32 the photoejected electron. The values for the cross section are plotted in Figure 2 along with the experimental 44 The results of even this crude calculation data of Samson. -turn out to be in agreement with experiment. to better than 15%. 33 (III) CURRENT WORK It was found that core relaxation and basis set effects caused insignificant differences in the photo- absorption cross section of N2 calculated using the - Stieltjes imaging technique.!? The apparent insensitivity of the Stieltjes calculation to the precise nature of the required virtual orbitals led us to explore the possibility of using V(N) rather than V(N-1) potentials in computing photoionization cross sections. The motivation is . convenience as well as insight into what features of the core potential have an important effect on photoionization phenomena. In general, the construction of the V(N-1) potential is considerably more-involved than the construction of the V(N) potential for a particular system. Also, in “obtaining the necessary virtual orbitals, the same potential is employed for all of the separate channels when dealing with the V(N) potential. When using the V(N-1) potential, however, a different potential must be constructed for each channel. | The computational procedure using the V(N) potential is similar to the procedure employed with the V(N-1) potential. An SCF calculation is performed to get the target wavefunctions. The system is then partitioned into its physically distinct channels. For each channel the original SCF basis is augmented with orbitals of the 34 appropriate symmetry. A virtual orbital calculation is performed using the augmented basis with the Hamiltonian chosen to represent the photoejected electron in the ‘presence of the full V(N) potential. Care is taken to insure that the virtual orbitals constructed are orthogonal to the target orbitals. The excitation energies must be determined for the various channel components. For example, in the case of singlet ground and final states, for an excitation between orbitals 5 and a (corresponding to a transition between states ‘6, and *®. 4) with orbital energies €, and Ea? repectively, it can be shown’? that 1 EC Pia ) - E(’4,)) = e, 7&4 - (J, ,-2K, 4) (71) It is noteworthy that the average excitation energy of the singlet is not what would be expected offhand, namely, Ee, 7 &4° The reason for this is that es and €, are both eigenvalues of the SCF Hamiltonian of the ground state; for the excited state the quantity €, can therefore not _ be expected to have the same meaning as the orbital energies Ee; of the ground state molecular orbitals. This is known as the single-transition approximation. The matrix elements [J,,72Kj,] are conveniently obtained by constructing the matrix representation of the operator 35 [J;-2K, ] over the basis of atomic orbitals, transforming ~ this representation to the basis of molecular orbitals, and extracting the ath diagonal element of the latter matrix representation of [J;-2K,] to get the element [J;,72k,]- It should be noted that the two-electron integrals required for this procedure are among those constructed for use in the virtual orbital calculation. To get values of E(*9,,) from eqn. (71), E(‘'%)) is chosen so that the lowest value of E(*®, .) corresponds to the appropriate experimental first Rydberg energy. Having found the appropriate excitation energies, the transition moments, u between orbitals 5 and do ia’ are calculated. The transition moments, u and the ia’ corresponding excitation energies E(*9,,), are then used to compute a set of oscillator strengths. The excitation energies and corresponding oscillator strengths serve as points and weights, respectively, for a Stieltjes moment analysis. This finally provides the partial photoionization cross section for a single channel. The sum of all of the partial cross sections yields the total photoionization cross section. We have teseted our new approach to photoionization calculations on helium. An SCF wavefunction for the ground 'S, state of He is obtained in the basis of 10 s 46 Gaussian functions given by Martin and McKoy. For the 36 1s + kp transitions, 8p functions are added to the original 10s basis. The photoionization calculation was carried out using both the V(N-1) and the V(N) potential techniques. ‘The pseudospectra computed using the V(N-1) potential and the V(N) potential are given in Table IV. Below threshold (0.90357 au) both sets of results are considered unreliable. Above threshold the V(N) potential oscillator strengths are larger than the corresponding oscillator | strengths for the V(N-1) potential but the two sets of results converge to the same values as one goes to higher energies. This behavior should be reflected in the resulting photoionization cross sections. The photoionization cross sections obtained from the Stieltjes-Tchebyscheff moment analysis of the pseudospectra of Table IV are shown in Figures 3 and 4. The cross section calculated: using the V(N) potential lies consistently above the cross section computed using the V(N-1) potential. Both calculations yield results in good qualitative agreement with the experimental work of Samson; "+ quantitatively, the cross section obtained using the V(N-1) potential is closer to experimentally determined values, | | What we have concluded thus far is that photoionization cross sections in good qualitative agreement with experiment may be conveniently constructed using the 37 Stieltjes imaging technique where the potential in which: the photoejected electron recedes is the full V(N) potential of the system. Apparently, then, the structure ‘of the cross section is not critically dependent on the precise nature of the V(N-1) potential. We are left with the question: Why is the cross section calculated using the V(N) potential generally larger than the cross section calculated using the V(N-1) potential and the experimental cross section? We hope to be able to answer this question and apply our new approach to molecular nitrogen in the near future. 38 REFERENCES AND NOTES R. D. Hudson, editor, "Critical Review of Ultraviolet Photoabsorption Cross Sections for Molecules of Astrophysical and Aeronomic Interest", NSRDS-NBS 38 (U.S. Govt. Printing Office, Washington, 1971). F, J. Wuilleumier, editor, Photoionization and other Probes of Many-Electron Interactions (Plenum, New York, 1976). a) D. R. Bates; J. Chem. Phys., 19, 1122 (1951); b) D. R. Bates, V. Opik, and G. Poots, Proc. Phys. Soc. A, 66, 1113 (1953). E. J. McGuire, Phys. Rev., 175, 20 (1968). H. P. Kelly, Chem. Phys. Lett., 20, 547 (1973). . O. Ellison, J. Chem. Phys., 61, 507 (1974); . W. Rabalais, T. P. Debies, J. L. Berkosky, it) Net ey Qy Las ae) q4y . J. Huang, and F. O. Ellison, J. Chem. Phys., 61, 516 (1974). a) J. W. Davenport, Phys. Rev. Lett., 36, 945 (1976); b) D. Dill and J. L. Dehmer, J. Chem. Phys., 61, 692 (1974); c) J. L. Dehmer and D. Dill, Phys. Rev. Lett., 35, 213 (1975). a) B. Schneider and R. S. Berry, Phys. Rev., 182, 141 (1969); b) H. C. Tuckwell, J. Phys. B, 3, 293 (1970); 10. 11. 39 c) S. Iwata and S. Nagukura, Mol. Phys., 27, 425 (1974); d) C. Duzy and R. S. Berry, J. Chem. Phys., 64, 2421 (1976). a) Green's function methods - M. Oppenheimer and H. Doyle, Phys. Rev. A, 13, 665 (1976); b) Coordinate rotation methods - T. N. Rescigno, C. W. McCurdy and V. McKoy, J. Chem. Phys., 64, 477 (1976); c) Equivalent-quadrature methods - H. A. Yamaui and W. P. Reinhardt, Phys. Rev. A, 11, 1144 (1975); d) J-matrix methods - J. T. Broad and W. P. Reinhardt, J. Phys. B, 9, 1491 (1976). e) Analytic continuation methods - M. A. C. Naseimento and W. A. Goddard, III, Phys. Rev. A, 16, 1559 (1977). a) P. W. Langhoff, Chem. Phys. Lett., 22, 60 (1973); b) P. W. Langhoff and C. T. Corcoran, J. Chem. Phys., 61, 146 (1974); c) P. W. Langhoff, J. Sims, and C. T. Corcoran, Phys. Rev. A, 10, 829 (1974). a) P. W. Langhoff and C. T. Corcoran, Chem. Phys. Lett.,. 40, 365 (1976); b) P. W. Langhoff, C. C. T. Corcoran, J. S. Sims, F. Weinhold, and R. M. Glover, Phys. Rev. A., 14, (1976); c) C. T. Corcoran and P. W. Langhoff, J. Math. Phys., 18, 651 (1977). 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 40 H. F. Schaefer III, The Electronic Structure of Atoms and Molecules (Addison-Wesley, Reading, 1972), p. 306. J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Mathematical Surveys; 1 (American Mathematical Society, Providence, 1943). R. F. Stewart, C. Laughlin, and G. A. Victor, Chen. Phys. Lett., 29, 353 (1974). P. W. Langhoff, C. T. Corcoran, and J. S. Sims, Phys. Rev. A, 16, 1513 (1977). | R. K. Nesbet, Phys. Rev. A, 14, 1065 (1976). S. V. O'Neil and W. P. Reinhardt (submitted for publication, J. Chem, Phys.) P. W. Langhoff, S. R. Langhoff, and C. T. Corcoran, J. Chem. Phys., 67, 1722 (1977). T. N. Rescigno, C. F. Bender, B. V. McKoy, and P. W. Langhoff, J. Chem. Phys., 68, 970 (1978). T. N. Rescigno, B. V. McKoy, and P. W. Langhoff (unpublished work) P. W. Langhoff, S. T. Epstein, and M. Karplus, Rev. Mod. Phys., 44, 602 (1972). a) H. Hameka, Advanced Quantum Mechanics (Addison- Wesley, Reading, 1965); | b) G. Wentzel, Z. Phys., 41, 828 (1927); c) 0. Klein, Z. Phys., 41, 407 (1927). 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 41 In the development of eqn,s (1) - (7), implicit use- has been made of an adiabatic switching function to turn on the perturbation of eqns. (2) sufficiently . slowly to ensure that eqns. (5) and (6) appropriately describe the steady state response. See Ref. 21. P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 368. G. V. Marr, Photoionization Processes in Gases (Academic, New York, 1967). S. A. Korff€ and G. Breit, Rev. Mod. Phys., 4, 471 (1932). | By specifying a single non-degenerate symmetry we are excluding cases in which the same overall electronic symmetry can be obtained by several different couplings of photoelectron and residual ion angular moments. H. F. Schaefer, Electronic Structure of Atoms and Molecules (Addison-Wesley, Reading, 1972). . E. J. Heller, W. P. Reinhardt and H.’ A. Yamani, J. Comp. Phys., 13, 536 (1973). A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965). H. A. Yamani and W. P. Reinhardt, Phys. Rev. A., ll, 1144 (1975). N. I. Akhieser, The Classical Moment Problem (Hafner, | New York, 1965). 33. 34, 35. 36. 37. 38. 39. 40. 42 P. W. Langhoff and M. Karplus, J. Chem. Phys., 52, 1435 (1970). a) P. W. Lanhoff, Chem. Phys. Lett., 9, 89 (1971); b) P. W. Langhoff, J. Chem. Phys., 57, 2604 (1972). . A. Dalgarno and A. L. Stewart, Proc. Roy. Soc. A, 238, 269 (1956). That is, since a trial function in the form of eqn. (49) with the 6, (4) given by eqn. (48) can reproduce the correct wavefunction to O(we ty, the variational principle ensures that it will do so. The 2n+1 theorem for time-dependent problems ensures that the resulting polarizability will be accurate to 0(w23) and that the associated sum rules will be accurate for k < 2n, See S. T. Epstein, J. Chem. Phys., 48, 4716 (1968). The 6, (2) of eqns (48) can only be determined in closed form for one-electron and model systems. G. A. Baker, Jr., Essentials of Padé Approximants (Academic, New York, 1975). H. S. Well, Analytic Theory of Continued Fractions — (Van Nostrand, New York, 1948). a) R. G. Gordon, J. Math. Phys., 9, 665 (1968). b) P. W. Langhoff and M. Karplus, Phys. Rev. Lett., 19, 1461 (1967). c) P. W. Lanhoff and M. Karplus, J. Chem. Phys., 52, 1435 (1970). 4l. 42. 43. 44, 45. 46. 43 It is convenient to let x = 1/e and define new poly: . . _~ h - _{n-1l nominals g, (x) = x Q, GA/x) and pO) = xf pp 0/9). See Ref. 13. The Stieltjes moment theory has been used previously in the study of the dynamics of crystals lattices. See: a) J. Deltour, Physice (Utr.), 39, 413 (1968); b) C. Blumstein and J. C. Wheeler, Phys. Rev. B, 8, 1764 (1973). A. Gerwer and V. McKoy (unpublished work) J. A. R. Samson, Adv. Atomic and Molecular Physics, 2, 177 (1966). T. H. Dunning and V. McKoy, J. Chem. Phys., 47, 1735 (1967). * P. H. S. Martin, W. H. Henneker, and V. McKoy, J. Chem. Phys., 62, 69 (1975). TABLE I. Pseudostate Energies and Oscillator Strengths (in atomic units) for He. 7 i 0.87653092 0.90567709 x 10° 0.92263598 i 0.23140472 x 107? 1.07249611 0.68793441 x 107° 1.65871589 0.12857756 x 10° 4.19594832 0.702062057 x 10- TABLE II. Histogram Approximation to the Cummulative Oscillator Strength for He e(a.u.) é (eV) YF(a.u.) 0.87653 - 0.92264 23.852- 25.106 0.90567709 x 10° 0.92264 - 1.07250 25.106 - 29.184 0.32197243 x 10° 1.07250 - 1.64872 29.184 - 40.468 0.10099068 x 10° 1.64872 - 4.19595 40.468-114.179 0.22956824 x 10° TABLE III. Interpolated Oscillator Strength Densities and Cross Sections for He w;(a.u.) a; (eV) g(a.u.) o (Mb) 0.89958345 24.479 .34917255 8.448 0.99756605 27.145 . 30673239 7.422 1.34060600 37.024 .17126364 4.143 2.92233211 79.522 .03901956 0.945 TABLE IV. Pseudospectra of Transition Frequencies and Oscillator Strengths (in atomic units) for Each Channel of (1s)'S) + (np)'P: of He. ° V(N-1) V(N) “on fon/? *on fon! ? 0.782824 0.425334 x 10°? 0.779723 0.900887 x 10° 0.849298 0.120127 x 10° 0.783786 0.761726 x 10°. 0.876531 0.905677 10° 0.791455 0.548438 x 10° 0.922636 0.231405 x 10° 0.830775 0.272770 x 107? 1.072496 0.687934 x 107’ 0.980027 0.984244 x 107° 1.648716 0.128578 x 10° 1.568742 0.162620 x 10° 4.195948 0.702062 x 10- 4.124936 0.728655 x 10° 13.693166 0.412790 x 10° | 13.622746 0.383364 x 10° FIGURE CAPTIONS Y Figure 1. Histogram approximation to the cummulative oscillator strength for He. Figure 2. Photoionization cross section of He from inter- polated values of the oscillator strength density. x values from Stieltjes theory + experimental values of Samson ‘Figure 3. He photoionization cross section calculated using V(N-1) potential; plus signs are calculated points; curve is fitted to experimental data of Samson. ** Figure 4. He photoionization cross section calculated using V(N) potential; plus signs are calculated points; curve is fitted to experimental data of Samson. “* 8S vo 0s ov ‘zy ltl se ‘ye! ocl — ozi-l, zz | | | | Poy | ne A I _ I A ! _ 1 A ” 4 a“ eee second Ath, a _— — — y vo'o 80°0 ZL°0 9L'O FIG.1- c8 82 vl OL 99 co } ‘9 Om ob oe — 09! 4 06: —+ OZL — SEL — 0S. 2 FIG.2 ‘AP. Zp se VE, of ce. GL! Gy 09 06 GOL FIG.3. 6.0 [- aN ABE (3.0 -- 15 30. (34 38. 42: - 46 50 54 58 eV. Appendix A -Al- Photoexcitation and Ionization in Molecular exygen RNAV DVN NNN NNN NID NNN NN NN RN NR RN NR DANN VRAIN NNW SW NNN MANN NNN NN NNN ND nw NNN NNN ID NNN NNN IND NNN NNN VN NNN NNN A. Gerwer, €. Asaro, and B. V. McKoy A. A.. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125 and P. W. Langhoff Department of Chemistry Indiana University, Bloomington, Indiana 47401 Abstract Theoretical studies of valence-electron (11, 1m, 30.) photoexcitation and ionization cross sections in molecular oxygen are reported employing separated-channel static- exchange calculations and the Stieltjes-Tchebycheff moment- _ theory technique. As in previously reported investigations of photoexcitation and ionization in small molecules following this approach, canonical Hartree-Fock orbitals, large Gaussian basis sets, and many-electron eigenstates of correct symmetry are used in defining appropriate noncentral static- .exchange potentials and in computations of the appropriate -A2- discrete and continuum transition strengths. It is parti- cularly important in molecular oxygen to incorporate the appropriate ionic parentages of the various photoionization multiplet states in order to obtain the correct partial-chan- nel cross sections. The calculated discrete series associa- ted with Im, excitation are found to be in good agreement with available experimental assignments and previously reported theoretical studies, and the predicted states associated with It, and 355 excitations are in general accord with assignments for the higher series based on spectral and quantum-defect analysis. Although the observed photoelectron spectra and photoionization cross sections are relatively complex, the calculated total vertical electronic photoabsorption cross section and the partial-channel photoionization cross sections for production of x1, aI» An 27m, 3° biz. and BoE, ionic states are found to be in good accord with recent synchrotron radiation, line-source, electron-impact, and (e, 2e) dipole oscillator-strength measurements when proper account is taken of the parentages of the various multiplet states, The partial-channel cross sections exhibit resonance- like structures that can be attributed to contributions from diabatic valencelike virtual states that appear in the appro- priate photoionization continua, rather that in the corres- ponding discrete spectral intervals. These features in the dipole spectrum of molecular oxygen are discussed and are con- trasted and compared with the results of previously reported related studies in molecular nitrogen and carbon monoxide. -A3- I. INTRODUCTION Recently reported theoretical! ~® and experimental’ 14 photoexcitation and ionization studies reveal the presence of resonancelike features in certain of the partial-channel photoionization cross sections of diatomic (N2, CO) and poly- atomic (120 ,H2CO) molecules. Elementary qualitative arguments suggest these resonance features can be attributed to contri- butions to the cross sections from o* and other valencelike virtual orbitals that appear in the appropriate photoioni- zation continua, rather than in corresponding discrete spec- ‘tral intervals. To achieve quantitative agreement with the measured outervalence-shell partial-channel cross sections, it is apparently sufficient to employ the static-exchange approximation and noncentral molecular potentials in the calculations. Appropriately chosen local noncentral poten- tials can also reproduce some of the general features of 15,16 although Simpler approximations involving, 17 18,19 the spectra, for example, plane-wave and one- and two-center Coulombic final states generally lead to qualitatively in- correct cross sections that can be an order of magnitude or more in error. In view of the fundamental importance of resonances in molecular photoionization continua to our understanding of electronic excitation/ionization spectra and their molecular-orbital interpretations, and in light of continuing line-source, synchrotron and bremsstrahlung -A4- radiation, electron-impact, and (e, 2e)/(e, e+tion) coinci- dence studies of dipole processes in small molecules, further quantitatively reliable theoretical investigations of appro- priately chosen systems are clearly in order. The photoabsorption spectrum of molecular oxygen has been the subject of continuing experimental and theoretical interest, as a partial consequence of its importance in atmos- 20 pheric and related photochemistry. In spite of a great many previous theoretical and experimental investigations,“ the general features of the oxygen continuous spectrum have been experimentally established comparatively recently,*2 28 and the reported corresponding theoretical studies do not particularly clarify the observed results. In the present study, outer-valence-shell electronic photoexcitation and ionization cross sections in molecular oxygen are obtained in the separated-channel static-exchange approximation employing the previously devised Stieltjes-Tchebycheff (S-T) moment-theory technique and L? variational methods . 297 32 As in previously reported studies of photoexcitation and ionization employing this approach, /~® Hartree-Fock canoni- cal orbitals and Gaussian basis functions are used in mole- cular oxygen in calculating one-electron discrete excitation series and continuum pseudostate spectra, and many-electron eigenstates of correct symmetry are employed in calculations of appropriate transition strengths. Particular care must be exercised in properly incorporating the ionic parentages -A5S-~ ‘ of the various multiplet photoionization states in order to obtain the correct partial-channel cross sections in molecular oxygen. Excitations of the It» lt and 355 outer valence orbitals are considered, whereas studies of the 205, zo, inner-valence and lo,» lo ,K-shell orbitals are described in separate reports. The calculated discrete series converging on the ground (1m) XT, molecular ionic state are found to be in generally good agreement with experimental assignments and previously reported theoretical values, in spite of the use of a frozen core and the single-configurational approxi- mation. Although the predicted positions of the higher 1t> 30, excitation series are in general accord with assignments made on basis of spectral and quantum-defect analysis, defini- tive studies in these regions will require appropriate auto- ionizing line-shape calculations not reported here. The calculated partial-channel cross sections for the production g g & are found to be in good accord with recent line-source 28 of X7I_, a“, A*lL 2°TL 370, DYE and B*x_ ionic states 25 and (e, 2e) dipole oscillator-strength measurements. Similarly, the calculated total photoabsorption/ionization cross section is found to be in good accord with the results of recent electron impact, 72728 23-26 continuum and line source, and (e, e+ion) 28 dipole oscillator-strength measurements. Although the measured photoelectron spectra and corresponding partial-channel and total photosbsorption/ionization cross sections in molecular oxygen are relatively complex, the 4 -A6- static-exchange approximation is seen to account satisfactor- ily for the observed results, provided proper cognizance is taken of the parentages of the various multiplet states. Moreover, a previously unassigned band in the photoelectron/ ionization spectrum is assigned on the basis of the present investigation. Resonance features appearing in certain of the individual channel.components are attributed to contri- butions to the cross sections from valencelike diabatic states appearing in the corresponding photoionization continua. Specifically, compact 0,2p,7,,3p,7, 3d, ,5d, and 6 3d orbitals > ; In, and 3057 valence photoionization channels, rather than to the corresponding apparently contribute to the lt discrete spectral intervals. These and other features of the molecular oxygen spectrum are discussed, and are contrasted and compared with the results of related previously reported theoretical investigations of dipole excitation/ionization . . : 1-3 spectra in molecular nitrogen and carbon monoxide. A brief description of the Stieltjes-Tchebycheff tech- nique in the separated-channel static-exchange approximation is given in Sec. II, the calculation required in molecular oxygen are summarized in Sec. III, and the resulting photo- excitation and ionization cross section are reported in Sec. Iv. A summary discussion of the resonance features is pro- vided in Sec. V and concluding remarks are made in Sec. VI. II. THEORETICAL APPROACH The Stieltjes-Tchebycheff procedure and the separated- 4 -A7- channel static-exchange approximation are described in con- 1-6,29-32 Gonee- siderable detwil in previous publications. quently, it is sufficient here to provide a brief heuristic description of the general scheme, and to refer the reader to earlier reports for further details. A ground-state Hartree-Fock function is first construct- ed near the equilibrium molecular geometry employing Gaussi- an basis orbitals and appropriate computational methods .>°?>4 Many~electron eigenstates of correct parentage and symmetry are employed in constructing noncentral static-exchange po- tentials for each orbital excitation considered from the ‘occupied canonical orbitals, and corresponding one-electron Schrodinger equations are formed for the calculation of exci- tation spectra. The one-electron equations are solved vari- ationally employing significantly larger basis sets than | that used in constructing the ground-state Hartree-Fock func- tion. A considerable reduction in computational effort is achieved by noting that only a subset of two-electron inte- grals, involving two basis orbitals from the occupied valence space and two from the supplemental or virtual space, is re- 1,2 The one- quired in constructing variational solutions. electron eigenvalues obtained that are below the occupied- orbital ionization potential, and the corresponding one-elec- tron orbitals, provide approximations to the appropriate dis- crete or autoionizing valence and Rydberg states, Those eigenvalues above the ionization potential, and their corres- ponding orbitals, provide pseudospectra for the Stieltjes- Tchebycheff moment analysis, from which the appropriate photo- ionization cross sections are obtained. The necessary exci- tation and ionization transition strengths are constructed from many-electron eigenstates of correct parentage and sym- metry. It is important to note that in order for the S-T an- alysis to be satisfactory, and to provide the correct static- exchange photoionization cross sections, it is necessary that the calculated pseudospectra correctly describe the appro- priate continuum eigenstates over the finite regions of space. The S-T approach employs a pseudospectrum of discrete transition frequencies and oscillator strengths, calculated in the present study as indicated above, in a smoothing pro- | cedure in which a continuous approximation to the underlying. photoionization cross section is obtained. Conventional spec- 2 pseudospec- tra power moments, which are convergent in the L trum of energies and strengths,” are generally regarded as. intermediaries in the smoothing procedure, providing a basis for establishing the convergence of the overall procedure. In actuai calculations, however, the power moments are avoid- ed, and a highly stable computational algorithm is employed in constructing so-called recurrence coefficients, and moment- theory spectra of (principal) frequency points and weights, directly from the quantum-mechanically determined pseudospec- trum, 2974? The moment-theory spectra exhibit useful proper- ties which the original pseudospectrum does not exhibit, and, in particular, provide bounds on the cumulative oscillator- strength distribution. The latter can be differentiated in -A9- a variety of ways to provide a final expression for the photo- ionization cross section. Stieltjes derivatives of various orders are generally employed to obtain a first approximation :to the continuous cross section, and to establish the range of convergence of the variationally calculated recurrence co- efficients. Tchebycheff derivatives and appropriate recur- rence-coefficient extention procedures are then used to con- struct a continuous approximation to the cross section. Al-. ternatively, specific analytic forms fit to the cumulative Stieltjes histograms provide expressions convenient for | differentiation. Previous investigations indicate the method is highly satisfactory, provided that the quantum-mechanical © pseudostate calculation contains sufficient information per- taining to the underlying continuous spectrum. 17° It is perhaps appropriate to note that the S-T approach to photoionization is not limited to the separated-channel static-exchange approximation employed here. Coupled-channel static-exchange calculations, for example, can also be per- formed employing generally available computer codes. Since a number of static ionic cores are employed ‘in this approach, well-defined total and partial-channel cross sections are ob- tained directly from the calculations. By contrast, when correlated ground-state functions and polarized or relaxed final-state core functions are employed, special projection or related procedures--which have yet to be clarified fully --are required in determining the pseudospectra corresponding | to specific photoionization channels. Moreover, the necessary quantum-mechanical calculations, of course, become somewhat -A10- more elaborate when correlated ground-state and ionic core functions are employed. In order to gain further experience with the S-T approach to molecular photoionization, and to ‘clarify the general natures of molecular photoionization cross sections, it is. perhaps appropriate to investigate mem- bers of the large class of molecules for which the static- exchange approximation is suitable. This class should include those molecules in which photoelectron spectra are well des- cribed by the orbital approximation and Koopmans' theorem, and in which the effects of shake up and rearrangement are 36-38 small. Although the observed valence photoelectron and 25-28 spectra in molecular oxygen are relative- photoionization ly complex, it will be seen that the static-exchange approxi- mation accounts satisfactorily for the measured results pro- vided proper cognizance is taken of the ionic parentages of the various multiplet states. Relaxation and related effects are presumably of more consequence in the inner-valence and core spectra, however. “2724 III. MOLECULAR OXYGEN CALCULATIONS The electronic ground state of molecular oxygen is written in the Hartree-Fock approximation in the symbolic form. (Lop1oj20020 (301m Ine) XE, (1) Hog g ° where the indicated canonical orbitals are obtained in the conven- ° -All- tional manner.°9?>4 In the present calculations a (10s,5p,14)/ [3s, 2p, 1d] -sontracted Gaussian basis set is employed at R, = 2.282a, » giving a total SCF energy of -149.634 a.u.,°4 e compared with the H-F limit of -149.666 a.u.?” The types of one-electron excitation and ionization series considered here can be written in the symbolic forms “J 3 -1 3_- “1. 3 Ong Ko )Ty ’ (In, kr) Dare (In, ké ) Ty? (2a) +1 3 -1 3_- -1 3 (11. ko 5) qi? (10, kn 5) Cy? (1m, ké I? (2b) -1 3 -l 3 / (30, ko) ra? (30, kn JD, ’ (2c) where excitations of the 20g 20, and logs lo orbitals, which remain doubly occupied, ate not considered. There are, consequently, eight distinct individual channels for excita- tion of the 3c, 51m, and In, orbitals, three corresponding to om symmetry, and five to + symmetry. However, because of the open In, shell in 0. some of the eight channels are com- prised of two or more subchannels associated with the forma- tion of various distinct parent molecular 0, ions. Appropri- ate many-electron pseudospectral wave functions and correspon- - ding static-exchange potentials must be constructed for each of these subchannels. In the case of excitation and ionization of the outer- most In, orbital [Eq. (2a)], the resulting core configuration gives rise to only the ground x71 molecular 05 ionic state. Consequently, in this case three appropriate sets of many-electron pseudo- spectral functiens and three corresponding static-exchange potentials must be constructed to describe the excitation and ionization of Ing electrons into Ko Krys and ks, orbitals, respectively. In the case of excitation and ionization of the Any orbital [Eq. (2b)], however, the re- maining Inrglirg core configuration gives rise to five separate electron- “ic molecular ionic states. Of these, two [a"a,» 4] arise from the ground-state (Ing) “pg Parent, whereas three [*o “i (2), “I (3)] arise from the excited-state (Ine) Ee g | of the frozen-core approximation inherent in the static-exchange approach, and (ne) by parents. As a consequence only the ail, and (1) ionic states contribute to dipole transitions in molecular oxygen, since these arise from the (Ine) rg parent. However, 2. 2 2a at 121» and 3°Tl 0, 2 approximations to the spectroscopically observed A states? 2 are formed from linear combinations of the 1,Q), “n,,(2)5 and HQ) static-exchange states in the present development. Consequently, four parent 05 ionic states [a"n,, A’t, 20, 370, 1 can be produced on basis of the static-exchange approach upon excitation and ionization of a In, electron. Strictly speaking, it is necessary to construct distinct Iq, > Koy kitg» KS, orbital pseudospectra corresponding to each of these four separate ionic states. However, in accordance with the approximations inherent in multiplet theory, virtual orbitals are chosen here in a demo- cratic fashion to be appropriate for all the states arising from lnglngkoys Inglgkty and Ind k8 g configurations. In this way only three distinct or- bital pseudospectra are required in forming the 12 many-electron pseudo- spectra associated with the four parent 0; ionic states formed upon exci- tation and ionization of a In, electron.. In the case of 30g excitations 4 257 2 [Eq. (2c)], the 30,11 configuration gives rise to b g LY - + B Z_, and Uy? g’ ~A13- Ca final 05 ionic states. Of these, only BE, and BED, which arise from the (tne) "rg ground-state configuration, contribute to the molecu- lar oxygen excitation and ionization spectra in the static-exchange ap- proximation. As in the case of the In, orbital, democratically chosen ko,, and kn, orbitals appropriate for both ionic states are employed in constructing many-electron wave functions for 30g excitation and ioni- zation. More refined calculations in which orbitals associated with specific. final ionic states ‘are employed can, of course, also be per- formed. It is found, however, that the pseudospectral orbitals are rel- atively insensitive to these refinements, and, consequently, democratic- ally chosen orbitals are employed here. In each of the eight individual cases [Eqs. (2)], the appropriate one-electron Schrodinger equations are written as - e)o, _ = 0 a 3 where €« corresponds to continuum or discrete energies, and hy, =T+ V+ Vp so (4) is the static-exchange Hamiltonian, with T and V the kinetic, and nucle- ar-framework-potential energy operators, and Vp the appropriate nonlocal channel potential. The nonlocal molecular channel potentials V, are all. written in the forms = r r . . i -Al4- where J; and K; are the familiar orbital Coulomb and exchange operators,*> and the coefficients ai, bi must be determined in each case. Explicit expressions for the latter, based on a previously reported analysis of open shell SCF energy expressions , are given in Table I. It is per- haps helpful to note in this connection that the static-exchange calcu- lations give results identical with dipole spectra obtained from appro- priate single-excitation configuration-interaction calculations in which Hartree-Fock ground-state canonical orbitals are used to describe the frozen final-state core configurations .47 Consequently, the open-shell static-exchange potentials of Table I are obtained simply by including in the appropriate energy functionals multiplicity weighted contributions from all linearly independent eigenstates of correct symmetry that can be formed by coupling the photo-excited final-state orbitals with the appropriate static core states indicated above. In order to obtain useful variational solutions of Eq. (3) in each case, it is necessary to use relatively large basis sets of Gaussian or- bitals. This large-basis-set requirement is a partial consequence of the structure of Cartesian Gaussian functions, which exhibit only the mini- mum number of nodes for each orbital angular momentum considered. For- tunately, it is not necessary that the variational solutions reproduce in detail the oscillatory behaviors of scattering functions over large regions of space in order to obtain reliable photoionization cross secz tions, since the product of dipole moment operator and ground state or- bital provides a finite box in which the final-state function must’ be determined. °” Even with this helpful simplification, however, it is generally necessary to expand the basis sets employed to the limit of linear dependence in order to obtain fully converged results. -A15- In Table II are shown the supplemental basis functions employed in the present study in addition to the valence basis used in construction of the ground-state Hartree-Fock function.°* The variational solutions of Eq. (3) obtained in this extended basis correspond to pseudospectra of 230» 230,,> 16m 5» 16r,,» 136 5» and 9S orbitals. As a convenience in evaluating the appropriate transition integrals, the solutions of Eq. (3) are explicitly orthogonalized to the occupied canonical Fock orbitals. Past experience indicates this generally has small. effect on the calcu- lated cross sections relative to results obtained without the explicit orthogonality constraint. In the case of molecular oxygen, however , there is a somewhat more significant effect in that the In, 7 In, trans- ition is excluded from the calculation by the orthogonality constraint. Since this excitation in the static-exchange approximation corresponds 3 3 to the well-known X Ip? B By Schmann-Runge bands ,“? however, it is not necessary to explicitly add the appropriate vertical energy and £ number to the spectrum in this case. 1-3 As in the previously reported studies in this series, one-elec- tron transition energies and oscillator strengths See tey > : (6a) qe Ty Th 20 are formed using experimental, rather than Koopmans, adiabatic ioniza- 36-38 tion potentials En thereby insuring convergence to the appropriate ? series limits, and allowing in some measures for the effects of core re- laxation, When the one-electron pseudospectra of Eqs. (6) are determined for the eight individual channels of Eqs. (2) to (5) and Table I, it only remains to determine the total line strengths associated with each of the many-electron final states./>7"4 These provide the many-electron pseudospectra of transition frequencies and multiplet oscillator strengths associated with the production of specific parent molecular 05 ionic states. The foregoing analysis indicates that the static-exchange approach to valence-electron (ln Im» 304) photoionization in molecular oxygen g? predicts the appearance of seven distinct ionic states [a’n. a7, , b'Es xt, B Tg» 2s 3m] in the photoelectron spectra. These are all observed in the (e, 2e) experiments 7° although some of the states are not separ- ately resolved. Higher resolution lower-energy line-source measurements 25 36-38 are helpful in these cases in separating the unresolved (e, 2e) data. The additional ionic states observed in the (e, 2e) spectra 2.- 2,4 [cin Dy? Bgl] can be attributed to 20, and 205 photoionization not treated here, with the exception of the ct Cy state, which is a well-known two-electron excitation.”> The latter contributes weakly to the photo- electron and ionization spectra, however, and, consequently, the static- exchange approximation apparently accounts for all the important features of the valence-shell measurements. IV. PHOTOEXCITATION AND IONIZATION CROSS SECTIONS IN MOLECULAR OXYGEN In Tables III-V and Figs. 1-4 are shown the discrete excitation series and photoionization cross sections, respectively, obtained from the present calculations. These results are discussed in detail in the following subsections. In order to obtain some preliminary qualitative indication of the natures of the spectra, it is perhaps helpful to re- -Al7- e call that both compact and diffuse virtual molecular orbitals are antici- pated in the excitation spectrum of molecular oxygen on basis of elemen- 46,47 ° ° ° s . * ° tary considerations. Specifically, a valencelike 0,,2P oro virtu- al orbital is expected to participate in strong N+ Vy excitdtions in the molecular oxygen spectrum, and strong. N+ Vv, transitions into the val- encelike In, or 1. orbital are also expected on this basis, since this shell is partially open. In addition to the presence of these predomin- antly valencelike o and 1. orbitals, previous experience suggests the appearance of certain partially valence or pre-Rydberg 555d, 63d, T)°P » 48 and appropriate discrete and continuum Rydberglike 1-6 and Tp orbitals, orbitals, in the molecular oxygen excitation/ionization spectra. Although the foregoing qualitative concepts are suggestive, the quantitative details of photoexcitation and ionization cross sections must, of course, be determined by appropriate calculations. In particu- lar, the contributions of strong intravalence transitions to the discrete or continuous spectra are expected to be quite sensitive to the details of the calculations and, consequently, relatively large basis sets are required to obtain the correct separated-channel static-exchange results. Previous experience indicates that the valencelike orbitals indicated above can contribute substantially to the continuous spectra in first row diatomics .1~© A, Ing spectra (I.P, = 12.1 eV) In Table III and Fig. 1 are shown the results of the present calcu- lations for In, excitation and ionization, respectively. Referring to Table II, it is seen that 230)» 16n,,> and 96) pseudospectra are used in calculating ln, excitation and ionization cross sections. As indicated g 4 ally ~0.6 to 0.8. -Al18- in the preceding section, the various Rydberg series are made to converge on the experimentally observed x" 05 adiabatic ionization potential. 36-38 The predicted Ing> Ty excitation series is evidently in good agreement with the positions and intensities of the observed npr, series, 21,49-51 and the first three members of the series are essentially iden- 50 tical with the earlier IVO calculations. The position of the n = 3 -member of this series, corresponding to excitation into a 3pm Rydberg * orbital (n ~ 2.2, § ~ 0.8), is taken from the recent electron impact- 50 excitation measurements. The effective principle quantum number (2.2) of the resonance transition substantiates the 3pm, assignment, since Rydberg p-orbital quantum defects for the first row diatomics are gener- > Of course the present calculation neglects the Ryd- -1 1 & . figurations, which apparently results in a shift of ~0.4 eV in the posi- tion of the first Rydberg state.77>4 berg-valence interaction between the (17 3pm.) > and (1m, Ing)" con- In addition to the discrete Rydberg npr, States in 0, indicated above, relatively compact pre-Rydberg atomic oxygen 3p and 3d orbitals (Table II) can also contribute to the In, > kn, photoabsorption spectrum. “8 Referring to Fig. l(a), a very broad and strong resonancelike 7,34 fea- ture is evidently present in the lr > kn, photoionization cross section, g having a maximum at ~28 eV, An oxygen d-orbital interpretation of this feature is consistent with earlier theoretical studies, which indicate that compact 9d oxygen orbitals do not contribute Significantly to the discrete In, > On, excitations in molecular oxygen.” However, since a very large Gaussian basis set is employed in the present calculation, it is not possible to unambiguously assign the qualitative nature of this resonancelike feature by inspection of the appropriate IVO orbital vec- -A19- tors. Single-center expansions or computer plots of the re- sonancelike orbitals, which are not presented here, would pro- vide useful information and are the subject of subsequent ‘in- vestigations. Moreover, diabatic calculation on configura- tional states of the type (1n5 tm 34) 25 and (Ing *n,3p) 725 would also be helpful in this connection. The Ing > Noy, excitation series shown in Table III is evidently in good agreement with the experimental assignments 49,50 and previous IVO calculations. Since the low-lying state at ~8.5 eV has such a small f number it is not observed ex- perimentally, The nature of this state, and of the entire In, > no,,/ko, spectrum, is clarified by inspection of the appropriate potential curves,>" which indicate a diabatic mo 2p) “ty state crosses the (In, almost vertically in the neighborhood of the xPEs equilibrium 3 . (1 npo,,) “I, Rydberg series u internuclear separation. Rydberg-valence interaction results in the weak low-lying transition at ~8.5 eV, with the diaba- tic intensity spreading over the higher Rydberg states and into the photoionization continuum. Indeed, there is evidence of a weak resonancelike In, > 5y2P contribution just above threshold in the In, > ko, photoionization cross section of Fig. 1(a). This interpretation is also consistent with re- cent valence-basis CI calculations in 0,, which locate the wn state at ~ll eV,” although inspection of the appropriate vector is required to confirm this assignment. Interaction of the diabatic valence state with diffuse basis functions, not included in the CI calculations,” presumably gives rise -A20- to the small shift to higher energy obtained from the present study. The second state in the It, > NPo, series is perturbed but clearly Rydberg in nature with principle quantum n = 3 and quantum defect § = 0.5. Evidently, the calculated f num- bers for the series are not entirely in accord with the approx- imate experimental intensity designation, a discrepancy that can be attributed to the sensitivity of transition strengths to Rydberg-valence interactions. | The ing > ng, series of Table II is evidently weak, ac- counting for the lack of an experimental assignment in this case. ‘Those values listed under "vrevious calculation" in Table III in this case are obtained from the Rydberg formula’ ot as is for n = 3 and 4 employing a quantum defect of 0.1, indicated in the table. The corresponding effective princi- ple quantum numbers of 2.9 and 3.9 are in poor accord with the calculated IVO values of 3.55 and 4.93, suggesting that a compact 6,34 orbital contributes only weakly to the dis- excitation series. However, referring to crete In, + ns, Fig. l(a), a 3d pre-Rydberg atomic oxygen orbital can be re- garded as contributing strongly to the In, > kéy photoioniza- tion continuum at ~28 eV, since Table II indicates that the Koy final-state resonance feature can only be due to atomic oxygen d orbitals. It is of particular interest to note that the peak in the cross section in this case is identical with that in the lq +kny profile, substantiating in some measure g the 3d atomic oxygen character of both features (Table II ). ' Diabatic calculations on the (1n57 6,34) “ny configurational - -A21- - state would be helpful in this connection. Since the potential energy curve of the x71.,05 State is deep and situated approximately vertically above the x35 70 u2 40 the Franck-Condon factors favor only a few ground state, low-lying vibrational states,°° and the sum of the three curves of Fig. 1(a), which incorporate appropriate line- strength factors,”° provides a vertical-electronic approxima- tion to the vibrationally averaged result that can be compared directly with corresponding cross-sectional measurements.2> 2° In Fig. 1(b) are shown the partial-channel IVO photoioniza- tion cross section for production of x71,09 parent ions and corresponding results obtained from recent line-source” and (e, 2e) dipole oscillator-strength measurements .°° Although = there is considerable structure in the line-source measure- ments at lower energy, which is.presumably due to the presence of autoionizing Rydberg series converging on higher ionic states and to vibrational structures, °° and there is a small quantitative discrepancy at intermediate energy, it is quite clear that the experimental and theoretical values are in good agreement. The small quantitative discrepancy near the maxi- mum at ~28 eV in Fig. 1(b) can perhaps be attributed to the relatively small 98 basis employed (Table II), presumably making the description of the In, + ké,3d resonance feature in Fig. 1(a) somewhat imprecise. It is particularly satis- fying that the calculations for this channel are in good a- greement with experiment at higher energies, where the reson- ance feature does not contribute and a reliable description gn Noe ° -A22- of the scattering functions can be expected in the basis employed. The experimental data of Fig. 1(b) evidently pro- vides substantial support for the presence of the In ,>kn- 3d and k6,,3d resonances in the (1m, - "XT, cross-section components of Fig. l(a). &B. dn, spectra (I.P.=16.leV) The calculated In, excitation series are compared with available spectral assignments and quantum-defect estimates in Table IV. Referring to Table II, it is seen that 230_, g lom,; and 136 _ pseudospectra are employed in this case. g The indicated adiabatic ionization potential refers to the a‘ Il, ionic state, and consequently, the calculated excita- tion series are made to converge on this value [Eq. (6a)]. Moreover, the calculated strengths cited refer only to the a*tL, contributions, which correspond to 2/3 of the entire transition array.>° Corresponding experimentally assigned series converging on the A’, ionic state are generally within ~ 0.5 eV of the values cited,°! although the higher- lying series converging on the 221 and 3° 11,05 states have apparently not been studied. The 2° TL, state in oo is re-- ported at ~24eV, and the unassigned band at ~33eV in the photoelectron spectra is attributed here to the 3°I, state. 78 As indicated in the preceding section, the static-exchange approach employed here predicts the appearance of four parent 0} ionic state [a*m,,A?1,,270,,,371,], and correspond- ing partial-channel photoionization cross sections, upon -A23- removal of a 1n,, electron. Since the discrete states of Table IV lie above the 12.1 eV In, ionization potential, they can autoionize into the underlying continua. Although autoionization linewidth calculations are not reported here, the calculated positions should be relatively insensitive to the appropriate interaction with the background continua. The calculated Im >nd , series is evidently in general agreement with the assigned So, series leading to the | a"Tl ionic state, and when shifted by ~ 0.5eV with the cor- responding series at slightly higher energy converging on 1 Moreover, both IVO and the A’IL, ionic state (not shown) .> experimental values are in general agreement with the positons obtained from the Rydberg formula and appropriate quantum defects, also listed in the table. The n=3 resonance in the series is evidently 380, Rydberg in character (n*- 2, 6 ~1), and there is no evidence of substantial valence- orbital contribution to this particular subchannel, since the corresponding photoionization profile [Fig. 2(a)] is monotonically decreasing. These results are in general accord with expectations and with previously reported studies of the In, ono ,/ko and 1lnm-no/ko cross sections in N2 and CO, respectively.*?* g in the present calculations due to the orthogonality con- There is no evidence of a strong In ln, transition straint imposed, as indicated above. Since this excitation corresponds to the familiar XeE BL, Schumann- Runge 1 -A24- 92,93 it is not necessary to perform a separate in- system, vestigation of its position and intensity here. The experi- mentally assigned first member of the ndq , series is in.- some accord with the present calculations, although the theoretical position is higher, perhaps as a consequence of. the orthogonality constraint, and the predicted intensity is apparently weak. The IVO In >nm calculations are in reasonable agreement with the positions obtained from the ot also. Rydberg formula and appropriate quantum defects, shown in the table. Referring to Fig. 2(a), the correspond- ing In ok, photoionization cross section is quite weak, al- though there is evidence of a very broad resonance like fea- 1 effective potential sup- ture at ~30eV. Apparently the In, ports only one valencelike T, orbital, which appears in the discrete spectrum in this case (Schumann-Runge transition). The Ln okt, cross section in O02 of Fig. 2(a) is apparently dissimilar in appearance from the previously reported corresponding Iny*kn and ln>kn cross sections in Nz and co, respectively, which exhibit large peaks just above threshold. “> Reexamination of the previously reported static-exchange calculations of ln, and lv excitation and ionization spectra in N, and CO, however, indicates that In okn, and 1In-kn photoionization cross sections similar to that of Fig. 2(a) are obtained when appropriate pro- jection techniques are used to isolate the strong 1>1* . oe . 5 transitions in each case. 7 ° -A25- 1 The experimentally assigned first member of the ‘tT u ondé , series is in some accord with the present calculations, although the IVO resonance transition apparently contains only weak contributions from 3d orbitals, since n*=3.2 in this case. Indeed, much of the intensity in this IVO channel is concentrated in the strong resonance in the Im >k6, ‘photoionization cross section of Fig. 2(a) at ~ 22eV. This feature can be attributed to the presence of compact 6, 3d molecular orbitals in final state symmetry, similar to the case of the In oko, channel [Fig. 1(a)]. Moreover, the 1m 7k6, profile of Fig. 2(a) is highly similar to the pre- viously reported corresponding Im oko, and Ln>ké photo- ionization cross sections in Nz and CO respectively.2?9 As in these latter cases, diabatic calculations on the (1m, °6,3d) *, state in 0. would be Helpful in clarifying the nature of the resonancelike feature in this channel. The aI, and A*TL 0 ionic state potential curves are not as deep as is that of the I, ionic state, and their minima are shifted to somewhat larger internuclear separa- tions (R~ 2-840). Moreover, the 2° TL and 37 TL state curves are most likely repulsive.*° Nevertheless, the Franck-Condon regions are still expected to be relatively 38 and the sum of the three vertical limited in these cases, photoionization curves of Fig. 2(a) should provide a vir- brationally averaged cross section suitable for comparison with appropriately interpreted measured data. In Fig. 2(b) -AZ6- the theoretically determined vertical electronic cross section so obtained for formation of a"ts An 27m and 37m pa- +. . . rent 0, ionic states is compared with the results of recent 28 line-source?” and (e, 2e) dipole oscillator-strength mea- surements. The an, and An, contributions are obtained from ‘the (e, 2e) partial-channel data by subtracting the by g channel contribution from the measured values corresponding to the sum of the a‘m,, an and bir channels.**® Since the boa, cross section is not resolved by the (e, 2e) measure- ments, an approximation to its contribution is obtained using twice the measured BES cross section (which is experimental- ly resolved), 7° employing the theoretically predicted ratio b’ 2, /BZ, = 2,76 Support for this theoretically determined ratio is provided by the line-source branching-ratio measure- 4 2 25 ments of the b Zo and B Bo contributions, which are precise- ly in the ratio 2:1 at the highest energy reported. To the sum of aT, and a the 27T, contribution and the separately resolved contribu- Thy partial-channel cross sections are added tion appearing at 33 eV in the photoelectron spectra, 7° which 2 + - 2 1,02 parent ionic state. The 2 Ty 4 is here assigned as the 3 cross section is separated from the c Ey channel with which it is degenerate by making use of the corresponding line- 25 which indicates the former contributes ~ 70% ot bands, In adding the 27m, 4 source data, to the combined 27m, and c and 37m, contributions. to thea 1, and An cross sections 4 it is, of course, necessary to refer the former to the a Ty threshold at ~ 16 eV. Although this procedure perhaps takes -A27- e some liberties with the experimental data, it is seen from Fig. 2(b) that generally good agreement between theory and experiment is obtained. The structure in the measurements ‘at lower energy is presumably a consequence of autoionizing states and vibrational features. Since the calculated ln, > k6 3d resonance appears in this spectral region, the measure- ments do not provide an unambiguous experimental verification in this case, although the experimental results are compat- 2 2 . IL, and 3°, partial- ible with its presence. Moreover, the 2 channel cross sections separately have maxima ~ 5 eV above their thresholds, in general support of the In, > k6 3d fea- ture of Fig. 2(a). C. 355 spectra (I.P. = 18.2 eV) The calculated 30, excitation series and ionization cross sections are shown in Table V and Fig. 3, respectively. Table II indicates that 2305 16m,» and 98, pseudospectra are employed in calculating the 305 excitation and ionization cross sections. The calculated energies are made to converge on the experimentally observed b’ ae shown correspond to contributions from series converging on 05 state, and the strengths this state only, which accounts for Z of the entire transi- 56 tion array. . Evidently the 30, > no, IVO series is in.very 51,58 good agreement with the assigned nPo,, series, and with estimates obtained from quantum defects. Of course, the ef- -A28- fects of autoionization must be included in order to obtain _ a quantitative comparison of intensities. The n = 3 reson- * . ance transition in the series is essentially Rydberg (n ~2.2, a $8 ~ 0.8) in character, and there is no evidence of a strong 30g +o,,2p intravalence excitation present in the discrete _ spectrum. However, referring to Fig. 3(a), the 30, 7 ko, Ivo photoionization channel has a prominent resonancelike feature at ~20 eV, which can be attributed to the presence of valencelike 9,,2P orbitals in the ko y photoionization con- tinuum. Similar resonances appear in the kg, and ko continua of Ny and CO,respectively,’° sr, 3°55 states corresponding to b and presumably related valence- like diabatic 2 45° and g B2 g parentages, respectively, are obtained at 18.4 and 20.9 eV from recent CI calculations.°> The calculated 30 5 + nT, excitations are evidently in very good agreement with the experimental positions and quan- tum defect estimated for the mpT,, series, although the strong intensity assignment is clearly at variance with the theore- tical results. Evidently, the corresponding 35,7 kt, photo- ionization cross section [Fig. 3(a)l is very weak, although there is some evidence of a broad high-energy maximum at ~30 to 40 eV, presumably corresponding to the associated reso- nance at ~30 eV in the 1m, + kt, cross section [ Fig. 1(a)]. The Franck-Condon factors for production of bors and Ber ionic states evidently favor a narrow (~0.5 eV) band of g low-lying vibrational states, and do not extend into the dis- 38,40 sociative photoionization continuum. Consequently, the two subchannel components for vertical photoionization of Fig. 3(a) are summed and compared in Fig. 3(b) with the re- 25 sults of recent line-source and (e, 2e) 28 dipole oscilla- tor-strength measurement. The (e, 2e) results shown .are “simply three times the measured BYE | channel cross section, making use of the theoretically predicted ratio bor /BoEs = 2 indicated previously.°° Although the theoretical and experi- mental results are evidently in general agreement, it is not entirely clear if the predicted 30, + ko, resonance feature is experimentally verified, since it appears near threshold in the region of possible vibrational and autoionizing struc- ture. Nevertheless, the presence of a resonance-like feature just above threshold in the 30. + ko, cross section is con- & sistent with the experimental data of Fig. 3(b), with the absence of a strong intravalence transition in the corres- ponding discrete spectrum (Table V), with the appearance of a similar weak resonancelike feature in the lr + ko, cross g section of Fig. l(a), and with the recent valence-basis cal- culations for the 2°" and 375 diabatic states.” Moreover, it is of considerable interest to note that measurements of the K-edge photoabsorption cross section in 05 indicate the presence of a resonancelike feature at or just above the - 22-24 This feature presumably can photoionization threshold. be assigned to the 1 % > ka, channel, and corresponds to the related 30 5 + ko, resonance of Fig. 3(a), providing further indirect support for the presence of a valencelike g,2P con- tribution to the ko, continuum in 0,- -A30- D. Valence-shell photoabsorption cross section A further comparison between theory and experiment is provided by the results of recent synchrotron radiation, 7° and electron-impact and (e, e + ion) coincidence studies “8 of total absorption and 05 and O- photoproduction in 0, in the 18 to 70 eV energy interval. In order to obtain a mean- ingful comparison between theory and experiment in this case 2 2 it is necessary to refer the 2 qh and 3 TL partial-channel contributions to the appropriate thresholds (24 and 33 eV, respectively), rather than to the at qs threshold, as in Fig. 2(b). When this procedure is followed, the In) con- tribution will exhibit three distinct thresholds, associated with the a‘n and A*t, 27m and 37m states, respectively. In the absence of appropriate line shape calculations, the contributions to the total cross section from autoionizing states converging on the 271 and 37m thresholds are neglec- ted, resulting in small discontinuities in these cases. Al- ternatively, if the states below these two thresholds are included in the S-T procedure, corresponding to introduction of very broad autoionization widths, the discontinuities are no longer present and the cross section increases somewhat in ~~ regions. ee the relative contributions of the Ae qT. 27 w and 32 qh partial-channel cross sections are ob- tained from a simple three-term 0; configuration-interaction calculation in which the experimentally observed thresholds are introduced as constraints. The results of this semiempir- ical analysis, which is in the spirit of the use of experi- -A31- mental ionization thresholds in Eqs. (6), indicate that the Mn cross section is very weak, whereas the 27n and 37n contributions are in the ratio ~2:1. These predictions - are in accord with the measured (e, 2e)2°m, and 37m ‘cross 28 2 sections, and with the apparent weak contribution the A Ty state makes to the high-resolution photoelectron spectra.°° 59 although not with previously reported theoretical studies. In Fig. 4 the results of the present IVO calculations for photoionization of the 1th, In 3 constructed as indicated above, are compared with measured and 30 _ orbitals in 0 g 2 cross sectional values. Evidently, the theoretical results are in good agreement with the measured photoabsorption cross section at energies s 30 eV, whereas the measured photoabsorp- tion cross section is greater than the theoretical result above ~30 eV, presumably as a consequence of dissociative (20 20,4) and two-electron photoionization. Moreover, the g? inflection in the data at ~40 eV can be attributed to an increase in 0° production at this energy.2® . At higher ener- gies the calculations approach the measured photoionization cross section for 05 production, and consequently are in ac- cord with expectations. Although the total cross sectional measurements do not provide direct evidence regarding reson- ance features, the presence of a maximum just above threshold in the measured values is not incompatible with the calcula- tions, providing additional indirect support for the 30, > g ko, resonance. -A32- V. Resonance Features The various resonance features appearing in the partial- channel photoabsorption cross sections in molecular oxygen warrant further summarizing discussion. Table VI provides a summary of the theoretical assignments and positions, and of the corresponding experimental estimates. The energies given in the first column are taken from recent valence-basis 55 which refer to dia- configuration-interaction calculations, batic states, rather than spectroscopic states. Evidently, the predicted IVO and CI positions are in general agreement for the lowest-lying states of each symmetry. This suggests, as indicated above and discussed further below, that the po- Sitions of resonances in photoionization cross sections are determined in large measure by the diabatic approximation, although the shapes of the resonance cross sections are sen- sitive to the details of Rydberg-vaience interaction in the continuum. Table VI emphasizes the very good agreement ob- tained between the separated-channel static-exchange results and the estimated positions of the measured photoionization resonances. Although the results of Table VI are satisfactory, it is desirable to provide a somewhat more detailed characteri- zation of the resonance features in the photoionization -pro- files in molecular oxygen. Referring to Fig. l(a), the lot > Kn, and In | g g . foregoing discussion (Sec. IV.A) to contributions to the > k6,, resonances are attributed in the photoionization continua from relatively compact pre-Rydberg -A33- 3d atomic oxygen orbitals (Table II). This view is in accord with the observation that 3d oxygen orbitals do not contri- bute significantly to the discrete In, + nt, excitation series, °° and with the evident highly similar natures of the It, > kt, and In, > ks, profiles. Nevertheless, it is only possible to make an unambiguous assignment in the case of the In? ko, profile, since the supplenental basis func- tions employed are all of d, type in this case (Table II). Y For the In? kt profile it is possible that the relatively compact portion of the oxygen and CM Py basis (Table II) can contribute significantly to the resonance feature. Of course, the presence of a well-defined n = 3 member in the discrete npt, series (Table III) makes this possibility somewhat less likely. Configuration-interaction calculations that include the (in 576,34) 1, and (m5 m,34) 725 configurational states, which are apparently unavailable at present, would be help- ful in this connection, as indicated above. Presumably the appropriate diabatic curves, which should correlate with the dipole allowed (2p 13a) *p° atomic oxygen state in the sepa- rated-atom limit, cross the outer limbs of the Rydberg series converging on the x71 ionic state (Table III), and become photoionization resonances at the ground-state equilibrium internuclear separation. By contrast, the (in 5) ,3P) 725 diabatic state presumably correlates with the (2p13p) *pe atomic oxygen state in the separated-atom limit, and, conse- quently, is not dipole connected to the ground state in this ‘limit. It is not possible to provide an unambiguous molecu- -A34- lar-orbital characterization of the small but distinct fea- ture near threshold in the 1, > ko, channel [Fig. 1(a)] on the basis of the present calculations. However the absence of a strong valencelike member in the calculated and observed npo,, excitation series (Table III), as well as the results of 54,55 which indicate the previously reported CI calculations, presence of a diabatic valence state crossing the Rydberg series, suggests that this small feature can be attributed to a compact o.,2P orbital lying just above the threshold. Indeed, referring to Table VI, the position of the diabatic (1n,” o,2P) °1, state is seen to be in rather good agreement with the present IVO result. Since only the x74 05 ionic state is produced upon removal of a In, electron, the calcu- lated orbital photoionization cross sections [Fig. 1(a)] are simply related to the experimental results [Fig. 1(b) ], which later provide considerable support for the existence of the predicted 13d and 6 3d resonance features. Referring now to the In, photoionization profiles of Fig. 2(a), the prominent feature in the 1,7 ks, subchannel can be attributed to contributions from relatively compact pre-Rydberg 3d atomic oxygen orbitals (Table II), in accor- dance with the observation that the discrete resonance in the ndé, series (Table IV) has an effective principal quan- tum number of ~ 3.2. Evidently, the opening of the In,, shell provides a somewhat more attractive potential for the k 6, spectrum than does the Im, shell for the ké, spectrum, since the resonance in the former appears at significantly lower -A35- energy ( ~5 eV above threshold) than does the latter ( ~16 eV above threshold). Recall that a prominent In > ks, re- sonance feature also appears in the previously reported ‘corresponding cross section in No? The appropriate diaba- tic state calculations for these configurations in 0, and Ny have, unfortunately, not been reported. Nevertheless, it can be expected that the (mn 576 34), diabatic state, which should correlate with the dipole allowed (2p43a)*p® atomic oxygen state, crosses the outer limbs of the nd 6, Rydberg series (Table IV) and becomes a photoionization resonance at the ground-state equilibrium internuclear separation. — The 1m > kg profile is evidently monotonic, and, of course, joins smoothly to the Stieltjes histogram in the discrete 35 spectral interval, which is dominated by the strong ln > 38 9, Rydberg transition (Table IV). Again, a similar beha-_ vior is seen in the corresponding profile in Ny. By con- trast, the 11,7 kt, profile in No shows a strong peak near threshold, whereas the present ny > kT, cross section in 05 [Fig. 2(a) | is flat and broad, with no evidence of a large contribution near threshold. As indicated above, re-examina- tion of the No calculations suggests that the large threshold value is a consequence of am> 1 contribution. When appro- priate projection procedures are employed in Ny to localize the 7+ m* contribution, a 1m, > kr, photoionization profile similar to that of Fig. 2(a) is obtained.” Recall that the strong 1n,+ In, intravalence transition is not included in § the present calculations, thereby eliminating its possible " -A36- contamination of the continuum. Because four parent 0, ionic states [an aT 27a, 37m] are formed upon removal of a 11, electron, comparison with experiment must be made with eare. It is found that when the measured a7I and a qt con- tributions, which are not separately resolved, are combined with the measured 270 and 370 contributions, the latter ‘being shifted to the aT, threshold, the theoretical and ex- perimental results are in good agreement . Moreover, the three separately measured cross sections [a‘m, + At, 241, 3° 0] exhibit peaks ~5 eV above their respective thresholds, © providing substantial support for the presence of the predic« ted In ,> k 43d resonance feature. The appearance of a prominent resonancelike 0,,2P feature in the 30g > ko, cross section of Fig. 3(a) is in complete accord with the resuits of the previous investigations in 1-3 No with the In, > ko, channel, and with the observation that the and CO, with the foregoing remarks made in connection discrete 30,7 no,, series (Table V) does not include an in- travalence contribution. Moreover, the valence-basis CI cal- culations (Table VI) predict the presence of two diabatic states, corresponding to b's and BES parentages, at the appropriate resonance position. Finally, the presence of a prominent feature in the 0,K-edge cross section just above threshold, which can be given a lo, > ko, assignment, pro- vides additional support for the appearance of a 0,,2P con- tribution in the photoionization continuum, 24 Although the experimentally determined partial-channel -A37- cross sections for Im 5» Im > and 30 photoionization in 0. do not provide unambiguous verification of the calculated resonance features, the measured values are certainly compa- tible with the theoretical results and with the corresponding interpretations. Moreover, the independent measurements of the total photoabsorption and ionization cross sections shown in Fig. 4 are also compatible with the present calculations, and perhaps also provide some support for the 50, > ko, 2p : resonancelike feature slightly above the corresponding threshold. °° VI. CONCLUDING REMARKS The qualitative discussions of electronic excitation and ionization processes presented here, and the quantita- tive separated-channel, static-exchange calculations upon which they are based, attempt to provide an overall elemen- tary account of both the discrete and continuous portions of the valence spectrum in molecular oxygen: Use of the Stieltjes- Tchebycheff technique and L? calculations allows rather straightforward application of familiar molecular-orbital concepts to the photoionization continua corresponding to various occupied orbitals, and thereby provides a basis for discussing the discrete and continuous excitations from.a common perspective. Most important, the concepts of intense intravalencelike transitions on the one hand, and generally _ weaker Rydberg-like transitions on the other, are seen to carry over in a natural way into the photoionization continua, -A38- In particular, the 1M, photoionization continuum in oxygen is seen to bédominated by features that are conveniently regarded as arising from the presence of 7 3d, 6 3d, and 5,,2P final-state resonancelike orbitals in the kt > kS> and ko,, channels, respectively, Similarly, the lt, and 35, cross sec- tions contain 6,34 and o,2P resonancelike features, respective- ly. These final-state valence orbitals do not contribute to. the corresponding discrete spectral regions, but rather are associated with diabatic states that cross the outer limbs of the vari- ous Rydberg series as functions of internuclear distance, be-: coming photoionization resonance above the appropriate ioniza- . _tion potentials at the ground-state equilibrium separation. | In addition to providing an overall descriptive account of the electronic dipole excitation spectrum in molecular oxygen, the calculations reported here are seen to be in generally good quantitative accord with very recent corres- ponding experimental studies. Some care is required in making comparisons* between theory and experiment in molecu- lar oxygen, particularly since there are four final ionic states and corresponding cross sections associated with the removal of a In, electron. Although the measured photoelec- tron spectra and partial-channel cross sections are relative- ly complex, good agreement between theory and experiment ob- tains when proper cognizance is taken of the ionic parentages of the various multiplet states. Although further computa- tional refinements, including coupled-channel, static- exchange calculations, allowance for core polarization and -A39-— relaxation, explicit treatments of vibrational degrees of freedom, and “df aspects of autoionizing states, will im- prove agreement with experiment, it is clear that the gener- ,al approach employed here should continue to provide useful first approximations to discrete and continuous dipole exci- tation spectra in small molecules. Further computational - applications are consequently in progress and will be re- ported subsequently. ~ ACKNOWLEDGEMENTS The authors thank C. E. Brion and.M. J. Van der Wiel for kindly making the results of their experimental studies available prior to publication, and for helpful correspon- dence. We also thank R. L. Blake for permission to cite his experimental results prior to publication, and A. U. Hazi for his help in clarifying the origins of the four In, cross sections. Achknowledgement is made to the National Science Foundation for support provided to B.V.M., and to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the National Research Council, for providing support to P.W.L. The kind hospital- ity of J. 0. Arnold and S. R. Langhoff of the NASA-Ames Research Center Computational Chemistry Group, and of D. Bershader of the Department of Aeronautics and Astronautics, Stanford University, to P.W.L. is also gratefully acknow- ledged. n 10, lly P. ° -A40- REFERENCES AND NOTES a . N. Rescigno and P. W. Langhoff, Chem. Phys. Lett. 51, 65 (1977). . N. Rescigno, C. F. Bender, B. ¥. McKoy, and P. 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Schneider (Plenum, New York, 1979), pp. 183-224. 33 34q 35 36 37 385 39 40 415 42 - -A43- F. Schaefer III, The Electronic Structure of Atoms and Molecules (Addison-Wesley, Reading, MA, 1972); Ann. Rev. Phys. Chem. 27, 261 (1976). . H. Dunning and P. J. Hay, in Modern Theoretical Chemts- try, edited by H. F. Schaefer, III (Plenum, New York, 1976). Vol. 3, Chap. 1. | . W. Langhoff and C. T. Corcoran, J. Chem. Phys. 61, 146 (1974). . W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, Molecular Photoelectron Spectroscoy (Wiley, New York, 1970). . Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. F. Hedén, K. Hamrin, U. Gelius, T. Bergmark, L. 0. Werme, \ R. Manne, and Y. Bauer, ESCA Applied to Free Molecules (North-Holland, Amsterdam, 1969). . Raffery and W. G. Richards, Int. J. Mass Spectrom. Ion Phys. 6, 269 (1971); 0. Edqvist, E. Lindholm, L. E. ‘Selin, and L. Asbrink, Phys. Scr. 1, 25 (1970). E. Cade and A. C. Wahl, At. Data Nucl. Data Tables 13, 339 (1974). R. Gilmore, J. Quant. Spectrosc. Radiat. Transfer. 5, 369 (1965). B. Rose and V. McKoy, J. Chem. Phys. 55, 5435 (1971). Lefebvre-Brion, C. Moser, and R. K. Nesbet, J. Mol. Spectrosc. 13, 418 (1964). 435. 44 45x 46p 47 485. 49 50 ol, 52. 535. 4p -A44- U. Condon and G. H. Shortley, The Theory of Atomte Spectra (Cambridge U. P., London, 1963), p. 98. Herzberg, Speetra of Diatomie Molecules (Van Nostrand, New York, 1950), 2nd ed. . Tanaka and M. Yoshimine, J. Chem. Phys. 70, 1626 (1979). S. Mulliken and C. A. Rieke, Rep. Prog. Phys. 8, 321 (1941). . S. Mulliken and W. C. Ermler, Diatomie Molecules (Acade- mic, New York, 1977). Sinanoglu, "Theory of Intravalency and Rydberg Transitions in Mdlecules," in Chemical Spectroscopy and Photochemistry tn the Vacuum Ultravtolet, edited by C. Sandorfy, P. J. Ausloss, and M. B. Robin (Reidel, Boston, MA, 1974), pp. 337-384. | | . Tanaka and T. Takamine, Tokyo Inst. Phys. Chem. Res. 39, 437 (1942). C. Cartwright, W. J. Hunt, W. Williams, S. Trajmar, and W. A. Goddard, J. Chem. Phys. 8, 2436 (1973). Lindholm, Ark. Fys. 40, 117 (1969). Morokuma and H. Konishi, J. Chem. Phys. 55, 402 (1971). F, Schaefer, III and W. H. Miller, J. Chem. Phys. 55, 4107 (1971). . J. Buenker and S. D. Peyerimhoff, Chem. Phys. 8, 324 (1975); Chem. Phys. Lett. 34, 225 (1975). 55 56 57 58 59 60 R. P, D. -A45-— P. Saxon and B. Liu, J. Chem. Phys. 67, 5432 (1977). W. Langhoff, A. Gerwer, C. Asaro, and B. V. McKoy, Int. J. Quantum Chem. Symp. (to be published). . N. Rescigno, A. Gerwer, B. V. McKoy, and P. W. Langhoff, Chem. Phys. Lett. 66, 116 (1979). Leclercq, Ann. Astrophys. 30, 93 (1967). N. Dixon and S. E. Hull, Chem. Phys. Lett. 3, 367 (1969). Gustafsson (private communication) has recently completed high-resolution PES and partial-channel photoionization cross section studies in 0, the results of which are in excellent accord with the present calculations, and which, in particular, verify the existence of the 35, > ko, =p resonance just above threshold. We thank Dr. Gustafsson for making his results available prior to publication. / -A46- c/t r-/t- 1/T T/T T/T cash + t L — ~ /ru T/T Tay c/¢ 1-/t T/T. 1/1 / xt ! 3 L/L L/L L/t Moy « Sut t/t +/ 4 T/T vt / x T/T T/T t/¢ T/¢ Ax 3 aft /z S/T ae [% I/T td + z/z z/z z/h Zz fv Boy « Mut e/ § t/t tls T/¢ VT t zz Zz /2 _ [2 May t/t t+ T/T T/T t /* x Zz /Zz Zz /Z — /% “oy « “og ffs - 4 T/T T/T t- /5 - 3 n n 3 AAT xt KUT xt o¢ T ‘I @IdVL q‘e ‘ue8hxo eTNIeTOW ur sTetzusezod TTeys-uedo o8ueyoxXe-5T3e1S -A47- TABLE I. - FOOTNOTES r i appearing in the potential values of the coefficients a,/d of Eq. (5), determined employing the procedures of J. B. Rose and V. McKoy, J. Chem. Phys. 55, 5435 (1971). b The lo,, lo, and 20 5» 20, orbitals remain doubly occupied and, consequently, ai = 1, by = 1 in these cases. ce A kx, ,h 500°0-$'0/T°0-S°0 6/S *p/**p 0/0 n._,3 Axrzowuuks ( 9/9) . ‘X,,ZX ; , ur o/°w’s 100°0-$Z0°0/SZ0"0-S"0 9/9 dj Pp moseurs 90°0-L°0/T"0-L°0 v/t p/P | 0/0 90°0-4°0/9°0-L°0 v/t *a/*d 0/0 | n 58 Azzowuks ( u/ wu) n . a t Zz Z$000°0-2°0/Z2S000°0-2'0 Z1/ZT d/s tuo / ured 69T00°0-££90°0/ZT'0-0°2 v/¥ “a/7d . 0/0 vZ00°0-2980°0/ZT°0-0°2 | v/v © s/s . — 0/0 | n._,3 Aiqawurss (0/0) esueir .UsUOdxXY Loqunn . edky uot }Bed07 q 2 “SUOTJE[NITeD useskXO AV[NISTOW OF STseq ueTSsNeD Tequewstddng “Il AIAVL -A49- TABLE II. - FOOTNOTES 4supplemental basis functions employed, in addition to the (10s, 5p, 1d)/[3s, 2p, ld] valence basis, in solution of Eq. (3). Ratios of successive exponents vary geometrically. -A50- Tv tt “ee “°° /PL TT "°° /89°OT . . i $T000°0/22°TT n n ‘3 3 Ie C gu) + SoC ut) wee eee 798° TT oa Z9° TT 66000°0/SL°TT ee — pS Tl 09T00°0/£9° TT 10Z200°0/2E° TT | 9¢°TT $8Z00°0/Tr' IT 94£00°0/96°0T 86° OT 67S00°0/00°TT €£800°0/86°6 8U013S//16°6 S$£800°0/9T"OT "°° /66°L ue — 00000°0/sS°8 "yu, Modu) + ®t) a Z9° TT "*/OL° TT mee ps‘ tt 91000°0/6S°TT €£000°0/ZE°TT 9f° TT - 2¢€000°0/rE* TT £L000°0/S8°0T | ov'OT — | 91000°0/Z8°0T §1200°0/rr'6 | yeom/TS "6 a 1LZ00°0/Zv'6 My (Mudu) + 3z_ (Fut) ~zequnn 5/(A9) AB190Ug Aqytsuaequy/(Aja) AS10Ug IequnN F/(A9) ABOUT 5 SUoTyeTNTeD snoTAeld gSontea Tequoursrodxg potinset queso - Cg ut e1qdeds uotie2TOXe But “III ATaVL -A51- TABLE III. - FOOTNOTES 4Values obtained from Eqs. (3) to (6) and appropriate multi- plet line-strength factors employing the basis sets described in Sec. III and Table II. The indicated series are made to converge on the experimentally determined x1, ionic state, as discussed in the text. Dspectral assignments and intensities taken from Y. Tanaka and T. Takamine, Tokyo Inst. Phys. Chem. Res. 39, 437 (1942); E. Lindholm, Ark. Fys. 40, 117 (1969). See also the tabula- tions of P. H. Krupenie, J. Phys. Chem. Ref. Data 1, 423 (1972), and references cited therein. “Previously reported IVO calculations of D. C. Cartwright, W. J. Hunt, W. Williams, S. Trajmar, and W. A. Goddard, J. Chem. Phys. 8, 2436 (1973). Additional theoretical calcula- tions are reported by J. LeClereq, Ann. Astrophys. 30, 93 (1967); T. Betts and V. McKoy, J. Chem. Phys. 54, 113 (1971); H. F. Schaefer and W. H. Miller, J. Chem. Phys. 55, 4107 (1971); K. Morokuma and H. Konishi, J. Chem. Phys. 55, 402 (1971); R. J. Buenker and S. D. Peyerimhoff, Chem. Phys. 8, 324 (1975); Chem. Phys. Lett. 34, 225 (1975); R. P. Saxon and B. Liu, J. Chem. Phys. 67, 5432 (1977). The values cited for the In, ~né, series are obtained from the Rydberg formula for n = 3 and 4 employing a quantum defect of 0.1. Estimates for f numbers obtained from the Coulomb approximation Ref. 35). GL70°O/T2Z°ST 6990°0/8t' tT STIO0°O/TZ°ST 64200°0/8t'° VT 2£00°0/Z8°ST 0S00°O/TL°ST £800°0/9S°ST OLTO0°O/SZ°ST £0v0°0/6S" bT TOST*O/OL°2T WNT pew /8s* PT "yu Copu) « 23, ("ut) UuNnTpeu/sp' rt 3 3 "6 ( upu) + "gz, ("uT) 6Z°ST ES‘ BU01YS/SE*ZT "yo (osu) + Ba, (ut) S9TO°O/ES°ST 961T0‘'0/08'rT £6000°0/88' tT 0v00°0/£0°9T 7v00°0/S6°ST vL00°0/9L°ST Tvpl0°O0/2v'°St pIS0°O/0L* rT $260°0/6S‘°2T a) zaqunu 3 /(A9) AB19Ug SOJEWTISS JIdFOp-unUeNhH Aqytsusquy/(A9) A819Uug SjuoWwUusTSse Te.UoWTIedxg requnn 5/(Ae) ASa29ug pstiInser jUsseid (A® T°9T = ‘d‘I) fo UT BIQZDOdS UOTIEYTOXO UT a "AL ATAVL - -A53- TABLE IV. - FOOTNOTES “as in Table III. The calculated energies are made to con- ¥ verge on the am 05 ionic state, and the strengths refer to contributions from the series converging on this state only, as is discussed further in the text. bvalues taken from the tabulations of E. Lindholm, Ark. Fys. 40, 117 (1969). The entries refer to Rydberg series con- verging on the at parent 05 ionic state. Additional series converging on the A*n, ionic state are also reported at en- ergies generally within ~ 0.5 eV of the values cited, although the series converging on the higher-lying 27m (24 eV) and 2 5° Ty (33 eV) ionic states also formed upon removal of a ln, electron are apparently not reported. As indicated in “re > B° is not included in the present calculations. the text, the familiar X ae Schumann-Runge transition Cvalues obtained from the Rydberg formula and the Coulomb limit (Ref. 35) for n = 3, 4, ... using quantum defects of 1.0 and 0.1 for the ns and nd series, respectively. T£000°0/98°LT Z$000°O/ZL°LT °°" /06°LT 6v000°O/ZL°LT -A54- £6000°0/9b° LT | SPL Z26000°O/Lv°LT v1Z00°0/S6°9T £6°9T 90200°0/96°9T TE900°0/£9°ST BU0LIS/SS°ST Z22900°0/T9°ST “ug (udu) + ®2 Pos) £500°O/v6°LT T6°LT °°" /06°LT TS$00°0/98°LT : - €8'°Lt 6v00°0/S8°LT S$800°0/Z2L°LT | 69° LT T800°0/TL°LT 6S10°0/9b'° LT po Lt Lvt0°O/Sv°LT 2S60°0/S6°9T 66°9T L0£0°0/T6°9T © BE0T°O/E9°ST UNTPeW/gzZ "ST §780°0/Sb'ST Mz, (Modu) + 2, (Fog) requnn 3/(Ae) AB19uq pStInseil JUsseid requnn ¥/(A9) AB19OUg SOJBUTISS JOOFOp-wnzUeNh Aytsuequy/(Ae) AB190ug sjuswUstssy Tequoulzedxg 2 q = “(A@ z°8 = ‘d°I) 20 ut vaq00ds uotaeazoxe Fog +y aTAVL -A55- TABLE V. - FOOTNOTES © #as in Table III. The calculated energies are made to con- 4 ze parent 05 ionic state, and the strengths * verge on the-b refer to contributions from series converging on this state only, as is discussed further in the text. Ospectral assignments taken from tabulations of J. Leclercq, Ann. Astrophys. 30, 93 (1967). See also the references cited in Footnote b of Table III. The entries refer to © Rydberg series converging on the parent 05 ionic bt 2 E_ state. g De ionic states are also reported at energies generally with ~ 2 eV of the Additional series converging on the “a, and B values cited. Sas in Table IV, employing 6 = 0.7 for np series. -A56- 97~ 02~ 97~ TZ~ sz/psoy « Fut zz/pssoy « “ut 3 zt/dz"oy + Put S3IDUBUOSOL "Wy 3 8Z/PS uy « “ut 12/dzoy + Fos edz 2 ut «ut n soouruosel 7 S*O/"Ugt 6°02 SP 8T/ ee 62 £0 6/ 3,4 (A2) AB1IOUT 5 tteut redxg (Ae) ASL9Ug/UOTITSUeL] qStoraernoTes asueyoxa-dTIeIS (Ae) AB10ug/92"eIS pouotyetnotes 19D STSEQ-ddUSTBA “usSAkXO IBTNIPTOW UT sjUSWUSTSse soUBUOSET UOTIeZTUOTOJOYG “IA ATAVL — -A57- TABLE VI. - FOOTNOTES *R. P. Saxon and B. Liu, J. Chem. Phys. 67, 5432 (1977). The vindicated values refer to vertical excitation energies at R = 2.3ap.. See also R. J. Buenker and S. D. Peyerimhoff, Chem. Phys. 8, 324 (1975); Chem. Phys. Lett. 34, 225 (1975). bvalues obtained from Tables III to V and Figs. 1 to 4. “Estimates obtained form the references cited and data appear- ing in Figs. 1 to 4. Fig. 1. ‘Fig. 2. Fig. 3. Fig. 4. - -A58- FIGURE CAPTIONS (a) Vertical electronic S-T oscillator-strength densities obtained from Eqs. (6) and appropriate multiplet factors for It, > ko.» kr» and ké, ion- ization in 0,. (b) Partial-channel photoionization cross section for production of (1m 57) x71 05 ions; (—-) vertical electronic S-T cross section; @, line- source measurements (Ref. 25);4, (e, 2e) coincidence measurements (Ref. 28). 1 Mb = 10°28 cm?. (a) As in Fig. l(a) for In, > Koy» kn and ks, photoionization. (b) As in Fig. 1(b) for the pro- 4 2 2 2 +. . Tye A Ty? 2 1,» and 3 11,95 ions. The duction of a (e, 2e) results are constructed as discussed in the text. (a) As in Fig. l(a) for 30 > ko, and kr photoion- ization. (b) As in Fig. 1(b) for the production of 4 - » b and B°E (0) ions. The (e, 2e) results are con- structed as discussed in the text. Valence-shell photoabsorption and ionization cross sections in 0.- S-T results as indicated; Experi- mental results; ®, total absorption cross section from electron impact dipole oscillator-strength stu-. dies (Ref. 28); photoabsorption measurements of ° -A59- Fig. 4 (cont.). ny: . + the total cross section (Refs. 25 and 26); @, 0, . photoionization cross section from (e, e + ion) dipole oscillator-strength studies (Ref. 28). 1 Mb = 10°18 cn? 4 Strength (df/d€) x 10° (eV~') Cross Section O(Mb) w q yp: T _ ] Iq (a) 10-- 10 20 30° 40 50 Excitation Energy € (eV) TO FIG. . Strength (df/d€) x 107 (ev"") Cross Section O(Mb) in, > kdg (x1/2) (a) (b) 10 Excitation Energy € (eV) FIG.2 Strength (df/dé) x 107 (ev™') Cross Section O(Mb) — -.y Ny 3 @ o Bs 14 12 10 (a) 10 20 Cross Section O (Mb) 35 30 15-- 10; 5- ! J ! L 1 10 20 30 40 50 60. Excitation Energy € (eV) . Appendix B | -B1- POR RRR RR NNN NOI T. N. Rescigno Theoretical Atomic and Molecular Physics Group University of California, Lawrence Livermore Laboratory Livermore, California 94550 A. Gerwer and B. V. McKoy A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125 _ and P. W. Langhoff Department of Chemistry Indiana University Bloomington, Indiana 47405 Abstract Static-exchange calculations of 1-shell photoionization in molecular nitrogen yield a physically incorrect description of the In, > kn, partial- channel cross section; difficulties are attributed to spurious perturbations arising from the intense 7 + 1* intravalence transition. A simple projec- tion procedure is devised that retains the computational simplicity of the frozen-core approximation and gives cross sections in good accord with ’ measured values. Introduction: ay A recent series of articles reports static-exchange calculations of partial-channel photoionization cross sections in the diatomic “molecules No 5 CO, Q -and F, [1-6]. Although the results of these calculations are in generally good agreement with corresponding line-source [7- 10], synchrotron radiation [11-13], and (e,2e) [14- 16] measurements, discernable quantitative discrepancies between theory and experiment. are evident in the cases of the (In, *) and (1n"!) channels ~ in No and CO respectively. Because the separated-channel static-ex- change approximation has proved to be otherwise satisfactory in studies of the outer-valence-shell photoelectron spectra [17] and photoionization cross sections of light diatomic [1-6] and polyatomic molecules [18-21], it is important to identify the origins of the dis- crepancy between theory and experiment in the (1m, *)Ne and (17 ')CO channels. In the present letter, the discrepancy between our previous results [2] and experiment in the case of the In, channel in N2 is investi- gated, and attributed to the presence of the strong m>m* intravalence transition in the calculated spectrum. A simple procedure for avoid- ing the difficulties that can arise in this connection is suggested, and detailed applications are presented. In order to understand the origins of the difficulty, it is helpful to recall that strong mixing occurs between the (r*)V__ and (o0*)V, configurational states in Ne [22-24], resulting in low-lying and high-lying intravalence states in this case. The former corresponds to the bE state in Ny with a vertical excitation energy of 14.4 eV [25], whereas the latter -B3- « corresponds to a strong photoionization resonance in the (30 ,>ko, XE partial-channel cross section centered at ~ 28 eV [2]. Because the static-exchange model in Np does not include o-n correlation or ,core relaxation, the m>n* transition is uncorrectly located in the spectrum, producing a significant spurious perturbation of the corre- sponding Rydberg series and photoionization continuum. Moreover, the position of the Va state is very sensitive to.the specific form of the potential employed in the calculation, and it is shown here that the transition can be made to fall above or below the ionization — threshold upon appropriate choice of potential. Consequently, caution must be exercised in applications of local exchange potential and re- lated approximations to studies of molecular excitation/ionization spectra that include intravalence transitions. To obtain a reliable approximation to the In kre photoionization cross section in N, in the presence of the strong N-V_ intravalence transition would require an extensive configuration-interaction study on states of 1y symmetry in he, including the necessary o-1 inter- actions. In this letter, these important physical interactions are modelled by employing an approximate P-Q projection procedure within the framework of the static-exchange approximation. The procedure is similar to previous studies of autoionizing states in which a back- ground continuous spectrum orthogonal to a zeroth-order compact lo- calized state is constructed and employed in shift and width calcula- tions [26]. When this approach is employed in Nz, using a In, orbital obtained from a static-exchange (In *4n.) °) state calculation, the resulting In, kr, photoionization cross section.is found to be relative- 4 -B4- ly insensitive to the potential employed and similar to the correspond - ing cross sections in Oz and F, [4-6]. Moreover, the resulting partial-channel cross section for formation of (In, AT parent Nz ions is found to be in excellent agreement with the measured data [7,9,11,12,14,15] when a factor of two error in the previously re- ‘ported Im ks, component [2] is also corrected. The static-exchange calculations in N, are reported in section 2, comparisons are made with experiment in section 3, and some concluding remarks are made in section 4. 2. Static-exchange calculations in N:: In this section static-exchange calculations in N. are reported employing (i) a potential appropriate for closed-shell systems with non-degenerate orbitals, (ii) a properly equivalence-restricted po- tential appropriate for a closed In, shell, and (iii) an equivalence- restricted potential with a background spectrum orthogonal to the 1 i . Ts orbital The previously reported static-exchange calculations in N, involve solutions of the one-electron equation [1,2] T+V+V, - 96,70 (1) Here, T and V are the kinetic-and nuclear-potential-energy operators, and Vo = £(2.-K.)+I. +K + WJ -XK . (2) Tux it ¥ Tx Tox Tuy "uy is the conventional static-exchange potential for excitation of a non- -BS- degenerate orbital in a closed-shell system [27,28]. The sum over i includes all the doubly occupied canonical o orbitals in NM, but not the 1n,, shell in this case. The energy variable e refers to discrete “states below the ionization threshold, as well as photoionization states above the threshold. In Fig. 1 curve (1) shows the partial-channel In, pam o/kn photo- excitation/ionization cross section in N) obtained from solution of eq. (1) employing a 27-term Ty, basis set and the Stieltjec-Tcheby- cheff moment- theory technique [2,29]. The discrete spectrum is pre- sented in the so-called Stieltjec sense [30], so that it joins on to the photoionization cross section at threshold, and the two can be - treated on an equal footing. There is clearly a very strong 1>1* transition in the spectrum~ 2.4eV below the ionization threshold of 16.7 eV [31]. In addition to this state, the compact valence portion of the basis set also contributes to the associated Rydberg series, ‘the intensities of which are strongly perturbed, and to the cor- responding photoionization cross section, the latter taking on a spuriously large value at threshold. What is needed to obtain a profile more in accord with reality is configuration mixing with the (o+0*) states, which will reduce the spuriously large f number of the transition, but leave the position of the Rydberg series unchanged, - thereby removing its perturbation and that of the continuum by the valence Vi state. It is interesting to determine whether the spurious result for the In, +kr cross section can be attributed to an incorrect choice of potential in eq. (2). Indeed, the potential of eq. (2) is inappropri- -B6- ate, since it. involves only Un, 7M gy / Ie 9 excitation/ionization, and, consequentiy, does not take proper cognizance of the orbital- equivalence restriction [32]. An equivalence-restricted static-ex- change potential is obtained from a wavefunction and energy functional involving both 7..»1_ and 7 > symmetry components in the form ux” gx uy gy [33] Vo = E(25.-K.) +l. + 3K Tox i 1 21 Tox Tox + 5/23. - 5K, . (3) "uy "uy where the sum over i includes all the o orbitals inN. Of course, . other symmetry restricted potentials can be devised that differ from that of eq. (3), although the latter gives results identical with those obtained from single-excitation configuration-interaction studies [34]. The In, oom /kr photoexcitation/ionization cross section obtained from eqs. (1) and (3), employing the 27-term 7, basis set indicated. g above, is shown as curve (2) in fig. 1. There is a dramatic difference between this result and that of eqs. (1) and (2) in that the mn* transition now appears ~ 2eV above threshold in the photoionization continum. It is highly unlikely that the use of a still larger basis set will significantly alter the photoionization cross section obtained from eqs. (1) and (3). Consequently, the position of the m-n* transition in the spectrum is seen to be rather sensitive to: the static-exchange potential [eqs. (2) or (3)], and an apparent "improvement" in the potential is seen to result in a highly unphysi- cal cross section. -B7- Since it is known that the bor state in N, falls below the (In, ATL, ionization threshold, and it is clear that the transition significantly perturbs the static-exchange calculations, a projection procedure in which the In, valence orbital is removed would seem to be appropriate. In this way the complexities of a configuration- interaction calculation can be avoided, the simplicity of the static- exchange approximation can be retained, and the usefulness of the pro- jection procedure for describing the In km, photoionization cross section can be assessed. A valence-1like In, orbital is obtained in the present development — from calculations of the (In, *4m,) °2y state in the 27-term Ts basis indicated above. This orbital constitutes the P space, and the ap- propriate Q-1-P complement is obtained by direct orthogonalization of the basis to the In, orbital, giving a 26-term T, background basis set. Interest here is focused primarily on the photoionization continuum, which is expected to be less sensitive to the specific choice of In, orbital than in corresponding 13h (ron) state. Solutions of eq. (1) using both the eq. (2) and eq. (3) potentials are construct- ed in this orthogonalized basis. The two results are found to be virtually identical and are shown as curve (3) in fig. 1 The profile is seen to be quite weak compared to the two other results also shown in the figure, which latter contain significant contributions. from the In, valence orbital, as indicated above. By contrast, the In ok, photoionization cross section obtained in the absence of the In, valence orbital in No is very similar to the corresponding results in Q and F, [4-6], in which cases the appropriate In, valence orbitals -B8- do not appear explicitly. Although the static-exchange results with projection are seen to be sensible, of course the ultimate test of the quantitative reliability of the calculated cross section is to be found in comparison with measurements. 3. Comparison with experiment: The "projected" static-exchange calculations of the In, ok, cross section in N, described in the preceding section are combined with the previously. reported In, ko and KS, components [2] and presented g in fig. 2. It is seen that the In, >ko, contribution to the 1m," partial-channel cross section is relatively small, whereas the In k6, component provides the dominant contribution. As indicated above, , the Ln, rks, cross section of fig. 2 differs from that previously reported by a factor of two [2]. Evidently, the results so obtained are in very good agreement with the line-source [9], synchrotron- radiation [11,12], and (e,2e) [14] measurements also shown in the ‘figure. By contrast, if either of the two In kn, cross sections of fig. 1 [(1) and (2)] containing the In, contribution are employed in fig. 2, the results so obtained for the (im, *) cross section are in significant disagreement with experiment. In order to further clarify the range of validity if static-exchange calculations of photoionization cross sections in N,, the partial- — channel cross sections for production of (50,°*)22%, (Im, JAtML,, and (20,7) Ba, parent Ne ions are compared with experiment in fig.3. The (In *)A'T channel shown is taken from fig. 2, whereas the other two channels are taken from the previously reported studies [2]. It -is seen that the three calculated results are in correspondingly -B9- good agreement with the measured values. Moreover, the sum of the three results is also in good agreement with the independent (e, e + ion) measurements of the N5 cross section [15]. 4. Concluding remarks ¥ In the present letter, discrepancies between theory and experiment in the In, partial-channel photoionization cross section in N, reported previous- ly are discussed and clarified [2]. These discrepancies are attributed, on the basis of the present study, to complications that can arise in static- -exchange cross section calculations in the presence of strong intravalence tran- “sitions. It is shown that the static-exchange approximation to the In, > In, transition in N, is rather sensitive to the particular noncentral, nonlocal potential employed, and that its presence in the calculated spec- _ trum can strongly perturb the corresponding 1m, > km, photoionization cross section in an entirely spurious fashion. Because an incorrect multiplici- ty factor was employed for the In, > Ké, component in the anya’ i partial-channel cross section, the previously reported results (ref. [2], figs. 2 and 6) do not appear to be in significant disagrement at threshold with the measured values. | A simple projection procedure is devised for calculation of the In, > kn, photoionization cross section in N, in the presence of the strong n> In, intravalence transition. A valence-like It, orbital obtained from static-exchange calculations of the (1ny1n,) "2 state is used in construc- tion of a background TW, space for determination of the appropriate cross section. Although the ty (mn > 1) state properties are expected to be high- ly sensitive to the choice of In, orbital in this procedure, the corres- ponding background photoionization continuum should be less sensitive to orbital choice. The 1m, * kt, cross section so obtained is found to be in- -B10- sensitive to the specific ‘static-exchange potential employed, is in ex- cellent accord with measured values, and is consistent with the corres- ponding recently reported m-shell cross sections in 0, and F, [4-6]. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] T. -Bli- REFERENCES N. Rescigno and P. W. Langhoff, Chem. Phys. Letters 51 (1977) 65. . N. Rescigno, C. F. Bender, B. V. McKoy and P. W. Langhoff, J. Chem. Phys. 68 (1978) 970. . Padial, G. Csanak, B. V. McKoy and P. W. Langhoff, J. Chem. Phys. 69 (1978) 2992. . Gerwer, C. Asaro, B. V. McKoy and P. W. Langhoff, J. Chem. Phys. (1979), to be published.: . W. Langhoff, A. Gerwer, C. Asaro and B. V. McKoy, Intern. J. Quantum Chem. S13 (1979), to be published. E. Orel, T. N. Rescigno, B. V. McKoy and P. W. Langhoff, J. Chem. Phys. (1979), to be published. L. Blake and J. H. Carver, J. Chem. Phys. 47 (1967). 1038. | R. Samson and J. L. Gardner, J. Electron Spectry. (1976) 35. 10 (1977) 1749. A 8 | | . A. R. Samson, G. N. Haddad and J. L. Gardner, J. Phys. ; A R. Samson, J. L. Gardner and G. N. Haddad, J. Electron Spectry. 12 (1977) 281. R. Woodruff and G. V. Marr, Proc. Roy. Soc. A 358 (1977) 87. | . W. Plummer, T. Gustafsson, W. Gudat and D. E. East- man, Phys. Rev. A 15 (1977) 2339. | Gustafsson, private communication. [14] [15] [16] [17] [18 J [19] [20] [21] [22] [23] 124) [25] [26] [27] [28] -B12- A. Hamnett, W. Stoll and C. E. Brion, J. Electron Spectry. 8 (1976) 567. | G..R. Wright, M. J. van der Wiel and C. E. Brion, J.’ Phys. B 9 (1976) 675. C. E. Brion, K. H. Tan, M. J. van der Wiel and Ph. E. ‘van der Leeuw, J. Electron Spectry. (1979), to be published. | L. S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1978) 205. P. W. Langhoff, A. E. Orel, T. N. Rescigno and B. V. McKoy, J. Chem. Phys. 69 (1978) 4689. | G. R. J. Williams and P. W. Langhoff, Chem. Phys. Letters 60 (1978) 201. | N. Padial, G. Csanak, B. V. McKoy and P. W. Langhoff, J. Chem. Phys., to be published. | N. Padial, G. Csanak, B. V. McKoy and P. W. Langhoff, J. Chem. Phys., to be published. R. S. Mulliken, Chem. Phys. Letters 25 (1974) 305; 46 (1977) 197. | R. S. Mulliken, Accounts Chem. Res. 9 (1976) 7. R. S. Mulliken and W. C. Ermler, Diatomic molecules (Academic Press, New York 1977). J. Geiger and B. Schroder, J. Chem. Phys. 50 (1969) 7. U. Fano, Phys. Rev. 124 (1961) 1866. | P. W. Langhoff, M. Karplus and R. P. Hurst, J. Chem. Phys. 44 (1966) 505. | W. J. Hunt and W. A. Goddard, Chem. Phys. Letters 3 (1969) 414. [29] + [30] [31] [32] [33] [34] -B13- e . W. Langhoff, in: Electron-molecule and photon-mole- cule collisions, ed. T. Rescigno, V. McKoy and B. Schneider (Plenum Press, New York, 1979) pp.183-224. . W. Langhoff and C. T. Corcoran, J. Chem. Phys. 61 (1974) 146. . W. Turner, C. Baker, A. D. Baker and C. R. Brundle, Molecular photoelectron spectroscopy (Wiley, New York, 1970). . C. J. Roothaan, Rev. Mod. Phys. 32 (1960) 179. . B. Rose and V. McKoy, J. Chem. Phys. 55 (1971) 5435. . Lefebrve-Brion, C. Moser and R. K. Nesbet, J. Mol. Spectry. 13 (1964) 418. Fig. 1. Fig. 2. Fig. 3. -B14- FIGURE CAPTIONS Static-exchange calculations of In, > nt of kn photoexcitation/ ionization cross section in N,: (1) values obtained using eqs. (1) and (2) without In, projection; (2) values obtained using eqs. (1) and (3) without In, projection; (3) values obtained using eqs. (1) and (2) or (3) with In, projection, as discussed in the text. 1 Mb = 10°28 cn’, - -] Partial~-channel In? Ko 5» kn» KS, components and (in, A’n,, photoionization cross section in N,: (-——) static-exchange cal- culation, including In, projection in the kn, component and cor- recting a factor of two error in the ko, component [2]; (e) line source measurements [9]; (a) (e, 2e) measurements [14]; (e) syn- _ chrotron radiation measurements [11, 12]. Total and partial-channel photoionization cross sections in N, + g N5 ions: (-——) static-exchange calculations, .including Ir for production of (3057 )x"r , an ya’n , and (20; 1)B"z" parent g Pro- jection in the kr g channel; (@,@) as in fig. 25 (@) total cross section for N5 photoproduction [15]. Cross section o(e), Mb 30 N ol N ° | ob oi 1 3° T (64.1) (1) =) 15 20 25 30 35 Excitation energy €, eV FIG.1 Cross section o(€), Mb 20 Cross section o(e), Mb 30 NO ol N © od oT ond foo) q q q q No + hy > N35 (X, A, B) + e7 Excitation energy ¢, eV FIG.3