Manybody calculations on molecules with secondrow atoms: H2S and H2CSW. von Niessen, L. S. Cederbaum, W. Domcke, and G. H. F. Diercksen Citation: The Journal of Chemical Physics 66, 4893 (1977); doi: 10.1063/1.433827 View online: http://dx.doi.org/10.1063/1.433827 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/66/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in MANY-BODY PHYSICS IN ATOMS AND MOLECULES Am. J. Phys. 82, 269 (2014); 10.1119/1.4867377 Manybody calculations of hyperfine constants in diatomic molecules. II. Firstrow hydrides J. Chem. Phys. 84, 6336 (1986); 10.1063/1.450726 Model potential calculations for secondrow transition metal molecules within the localspindensity method J. Chem. Phys. 83, 4573 (1985); 10.1063/1.449027 Diagrammatic perturbation theory: Manybody effects in the X 1Σ+ states of firstrow and secondrow diatomichydrides J. Chem. Phys. 66, 5400 (1977); 10.1063/1.433902 Manybody perturbation theory applied to electron pair correlation energies. II. Closedshell secondrow diatomichydrides J. Chem. Phys. 64, 4578 (1976); 10.1063/1.432091

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Many-body calculations on molecules with second-row atoms: H2S and 'H2CS

W. von Niessen

Lehrstuhl fiir Theoretische Chemie. Technische Universitiit. D-8 Miinchen 2. West Germany

L. S. Cederbaum and W. Domcke

Physik Department. Technische Universitiit. D-8 Miinchen 2. West Germany

G. H. F. Diercksen

Max-Planck-Institut fiir Physik und Astrophysik. D-8 Miinchen 40. West Germany (Received 22 June 1976)

The ionization potentials of H2S and H2CS are calculated by a many-body Green's function method. Correlation energy changes in the diffuse part of the charge cloud require an extension of the polarization function basis compared to the first row atoms. When f-type functions are included all calculated ionization potentials of H2S agree with the experimental values to within 0.1 eV. The ionization potentials of H2CS agree satisfactorily with the recently determined experimental values except for the fourth band where possibly predissociation occurs. The ordering of the ionic states is the same for H2CS as for H2CO. The vibrational structure in the photoelectron spectrum of H2CS is computed.

I. INTRODUCTION

In this article, the results of a theoretical investiga­tion on the ionization potentials (I. P. 's) of the molecules HzS and HzCS including the vibrational structure in the photoelectron spectrum (PES) of HzCS are presented. Based on a Hartree-Fock wavefunction, the I. P. 's are calculated directly by a many-body Green's function method, which includes the effect of electron correlation and reorganization. 1 The PES of H~ has been carefully investigated experimentally (Refs. 2,3 and references given therein) and the asSignment of the bands presents no problem.

The I. P. 's of HzS have been computed by Chong et al. by a Rayleigh-8chrMinger perturbation theory taking into account electronic correlation and reorganization, 4

by Raimondi e tal. using a valence bond method, 5 and by Guest and Saunders by the ~c F method, which takes into account the electronic reorganization. 6

The HzCS molecule is less well characterized than HzS. Increased interest in this molecule has arisen be­cause of its occurrence in interstellar clouds. 7 The geometry of HzCS is known from microwave measure­ments. 8 The infrared spectrum9 and the rotational Zeeman effectlO have been investigated. Excited states have to some extent been studied and theoretically cal­culated. The paper of Peyerimhoff and co-workers con­

. tains the relevant information. 11 Kroto and Suffolk were able to identify only the first two bands in the spectrum of H2CS, because of contamination by bands due to other molecules which were formed in the pyrolysis yielding HZCS. 1Z Very recently the PES of pure H2CS has been recorded and the I. P. 's calculated by a configuration interaction method (CO. 13

On the example of the HzS molecule, it will be investi­gated as to what must be done to accurately calculate I. P. 's for molecules containing second row atoms. The many-body Green's function method used in this work is equally applicable to molecules containing atoms from any row of the periodiC table. In calculations on the

The Journal of Chemical Physics, Vol. 66, No. 11, 1 June 1977

molecules phosphoridine, pyridine,14 thiophene, and furan,15 it was found, however, that some of the com­puted I. P. 's in phosphoridine and thiophene were con­siderably in error (by about 0.5 to 0.6 eV) when com­pared with the corresponding results for pyridine and furan which were in error by 0.2 to 0.3 eV. The basis sets employed were of double- b quality with ad-type function on the second row atom, and are generally con­sidered to describe first and second row atoms with equal quality. The error occurred for the lowest I. P. 's, where the associated molecular orbitals (MO's) were ap­preciably localized on the second row atom and the cal­culated I. P. 's were smaller than the experimental val­ues. This strongly suggests that the reorganization which lowers the value of the I. P. 's is described cor­rectly, but that there is a considerable deficiency in the description of the electronic correlation.

The investigations on the H~ molecule which serve to establish the basis sets required to calculate the I. P. 's to within 0.2 to 0.3 eV of the experimental value are presented in Sec. II. This information is used to calcu­late the I. P. 's of HzCS in Sec. III. The vibrational structure in the PES of HzCS is discussed in Sec. N.

II. IONIZATION POTENTIALS OF H2S

The calculations on the H:aS molecule in its experi­mental geometry (r = 1. 328 'A, () = 92. 2 0 )16 were per­formed with the program system MUNICH, 17 using a num­ber of different basis sets of Cartesian Gaussian func­tions. For the H atom, either the 4s basis set of Huzinaga was used contracted to 2S18 or the 6s basis set of Salez and Veillard contracted to 3s. 19 For the S atom, the 12s/9p basis set of Veillard was employed contracted to 6s/4p or 8s/6p.20 In addition, a larger basis set consisting of 15 s-type and 12 p-type functions was determined by optimization on the S<: p) atom using a program of Roos et al. 21 Care was taken to obtain a basis set flexible in the valence region. This basis set and the atomic orbital expansion coefficients are listed in Table I.

Copyright © 1977 American Institute of Physics 4893

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4894 von Niessen et al.: Many-body calculations on molecules

TABLE 1. Exponential parameters of the 15S/12p basis set and atomic expansion coefficients for S{3p ). a

Q s CL«f~-92.0036) C",«--9.00427) C 3s «~-0.k7952)

153862.0 0.1728":! D-U:J -0.477941 D-04 0.138742 D-U'l 19696.7 0.172317 D-02 -0.47,,936 D-03 0.138164 D-03 3815.95 0.107358 D-01 -0.299179 D-02 0.869044 D-O:! 1007.68 0.440704 D-01 -0.124561 D-01 0.362396 D - 02

330.969 0.13:!921 -0.-101631 D-01 0.llH28 D-01 122.195 0.302303 -0.101595 0.300446 D-01

-19.142 O.~06 GOO -0.186577 0.567,,46 D -01 20.900 G 0.228911 -0.H4589 0.461976 D - 01 7.74014 0.256979 D-01 0.244 522 -0.881366 D-01 3. 56:119 -0.33H75 D-02 0.580466 -0.277261 1.65210 0.250989 D -08 U.315794 -0.314000 0.862488 -0.142066 D-02 0.151799 D-01 0.186741 O. :366 050 +- O. H39 ·127 D -0:3 f (J.l:11 902 D -01 0.018 :;18

0.220000 -0053-1085 D-03 -0.680839 D-02 0.324219 0.119965 0.109029 D-03 0.171385 D-02 0.239985

O!p C2P«~ -6.G82U) C3p«=-0.43735)

608.130 0.222828 D-02 - 0.568 254 D -03 142.587 0.176402 D-01 -0."53117 D-02

47.1583 0.739413 D-01 -0.192941 D-01 18.7351 0.196791 -0.529700 D-01 8.10771 0.344979 -0.954345 D-01 3.67612 0.367687 -0.110126 1.74049 0.179705 -0.257910 D-01 0.837644 0.270735 D-01 0.193059 0.406083 0.190342 D-02 0.368434 0.201329 0.373308 D-03 0.359708 0.101289 0.27H91 D-03 0.200311 0.050000 0.818561 D-05 0.390790 D-01

aJi3CF = _ 397 .5006437 a. u., virial ratio = - 2.0000376.

All SCF calculations performed on the HzS molecule are given in Table II, were the exponential parameters of the polarization functions are listed. This table con­tains a number of wavefunctions which give a total ener­gy lower than the best value reported in the literature. E;;f = - 398.68624 a. u. zz The best value computed in the present work, E~;r = - 398.70828 a. u., is within 0.011 a. u. of the estimated Hartree-Fock limit. 23

In Table III, the orbital energies and the I. P. 's calcu­lated with the Green's function method are listed. The basis sets are identified by their contraction scheme. The next column lists the number of virtual orbitals taken into account in addition to the occupied valence orbitals in the many-body calculation. In a few cases, several calculations have been performed with different sets of virtual orbitals, sometimes completely exhaust-

ing the basis. The extension of the virtual orbital basis generally has fairly little infll:lence. The orbital ener­gies are seen to be relatively insensitive to the basis sets. The energies of the bb ab bz orbitals are approx­imately -10.45, -13.65, -16.10 eV for basis sets con­taining d-type functions, and approximately - 10.60, -13.45, -16.25 eV for basis sets without polarization functions. Using a basis set without polarization func­tions gives the following values for the I. P. 's in the many-body calculation: b1: 9.91 eV, a1: 12.76 eV, bz: 15.72 eV. The adiabatic experimental values are 10.48, 12.76, and 14.56 eV, z and the maxima reported by Turner et al. are 10.48, 13.25, and 15.35 eV. Price e tal. give somewhat different values: for the adiabatic I.P. 's 10.47, 12.78, and 14.78 eV and for the maxima 10.47, 13.33, and15.47eV. 3 The centroids of the photoelectron bands with which our calculated vertical I. P. 's have to be compared can be estimated to be 10.48,13.4, and 15.5 eV from the published spectra. 2 It is thus seen that the first two calculated I. P. 's are about 0.6 eV too small, whereas the b2 I. P. is within 0.25 eV of the experimental value. The same effect as found for phosphoridine and thiophene is thus found again. For all basis sets, the bz I. P. remains within 0.2 eV of the experimental value and will thus not be considered in detail.

The a1 and the b1 I. P. behave similarly. A less con­tracted basis set (calculation 2) gives no change com­pared to calculation 1. Adding polarization functions­either one d-type function or two d-type functions with standard exponential paramete.rs taken from the litera­tureZ4 has practically no effect on the I. P. For mole­cules of first row atoms basis sets of this quality have led to computed I. P. 's that agree with the experimental I. P. 's to within 0.2 or 0.3 eV for a wide range of ex­amples. Z5 Thus something must be different in the de­scription of the S atom. Subsequently, a larger basis set of s- and p-type functions was used on the S atom but without giving an improvement. The 12s/9p basis set of Veillard is thus not the cause of errors but the polarization functions. When smaller exponential pa­rameters are chosen for the d-type functions a signifi­cant improvement in the computed I. P. results. In the subsequent calculation, the smaller s- p basis set was

TABLE II. Basis sets, polarization functions, and total SCF energies for H2S.

Basis

(12/9 + 4) /[6/4 + 2] {12/9 +4)/[8/6 + 2] {12/9/1 +4/1)/[6/4/1 + 2/1] {12/9/2 +4/1)/[6/4/2 + 2/1] {15/12 +4)/[9/6+2] {15/12/1 + 6/1)/[10/6/1 + 3/1] {15/12/2 + 6/2)/[10/6/2 +3/2] (12/9/1 + 4/1) /[6/4/1 + 2/1] (12/9/2 +4/1)/[6/4/2 + 2/1] (a) (12/9/2 +4/1)/[6/4/2 +2/1] (b) (12/9/3 +6/2)/[8/6/3 +3/2] {12/9/3 + 4/1)/[6/4/3 + 2/1] (12/9/3 +4/1)/[8/6/3 +2/1] (12/9/2/1 + 4/1)/[6/4/2/1 +2/1]

0.54 0.542.0

0.54 0.3 0.9 0.1 0.10.3 0.150.5 0.150.451.35 0.150.30.6 0.15 0.3 0.6 0.150.5 0.35

0.75 0.75

0.75 0.250.75 0.4 0.4 0.4 0.3 0.9 0.4 0.4 0.4

Efot(a.u.) -398.497904 - 398.654769 -398.547079 -398.556139 -398.563483 -398.696126 -398.701557 -398.529286 -398.535561 -398.545484 -398.708277 -398.549191 -398.701794 - 398.546697

J. Chern. Phys., Vol. 66, No. 11, 1 June 1977

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von Niessen et al.: Many·body calculations on molecules 4895

TABLE III. Orbital energies and 1. P. 's (in eV) of H2S as a function of basis set.

No. of vir- Computation No. Basis tual MO's -E(bl) I. P. (b l ) -E(al) I. P. (al) -E(b2) 1. P. (b2) time per 1. P. a

1 [6/4+2] 11 10.66 9.91 13.43 12.76 16.24 15.72 10 sec. 2 [8/6+2] 15 10.62 9.87 13.43 12.77 16.20 15.67 20 sec. 3 [6/4/1 + 2/1] 13(23) 10.45 9.96(10.01) 13.61 13.06(13.12) 16.15 15.46(15.55) 15 sec. (1-1. 5 min) 4 [6/4/2 +2/1] 20 10.45 10.01 13.63 13.12 16.19 15.54 40-50 sec. 5 [9/6 +2] 15 10.67 9.94 13.47 12.83 16.25 15.72 20 sec. 6 [10/6/1 +3/1] 23 10.46 9.87 13.67 13.14 16.18 15.52 1-1. 5 min. 7 [10/6/2 + 3/2] 13 (26) 10.46 10.00(10.11) 13.68 13.19(13.24) 16.14 15.45(15.45) 10 sec. (1. 5-2 min) 8 [6/4/1+2/1] 20 10.60 10.14 13.54 13.19 16.13 15.67 40-50 sec. 9 [6/4/2 + 2/1] (a) 23 10.53 10.25 13.60

10 [6/4/2 + 2/1] (b) 21(29) 10.48 10.24(10.23) 13.63 11 [8/6/3 +3/2] 25 10.43 10.17 13.66 12 [6/4/3 +2/1] 23 10.48 10.24 13.65 13 [6/4/2/1 + 2/1] 24(39) 10.51 10.31(10.38) 13.62 Experiment (band centroid) 10.48

~he computation times refer to an IBM 360/91 computer.

used again and the number of d-type functions and their exponential parameters were varied. Up to three d-type functions have been used on the S atom. The results of the I. P. 's appear to be stable, when two d-type functions are employed with small exponential parameters for which the values a,,(S) = 0.15 and 0.5 are recommended. The a1 I. P. is with these basis sets within 0.1 eV of the experimental value, but the b1 I. P. has still an error of 0.24 ev, which apparently cannot be decreased by adding and varying d-type functions. The next step is to add I-type functions which has been done in the last calculation. The exponential parameter is a guess (hopefully an educated one). The b1 I. P. is now within O. 1 e V of the experiment and the othe r I. P. ' s experi­ence little change.

The S atom (and presumably any other atom from the second row) thus requires a different description (basis set) than its first row homolog, the 0 atom (or the other atoms from the first row). For a proper description of the changes in correlation energy upon ionization, two rather diffuse d-type functions are required for mole­cules with second row atoms, whereas one d-type func­tion (or even no polarization functions at all) with con­siderably larger exponential parameter [a,,(O) = 0.8-1. 0] suffices for molecules containing only first row atoms. The reason is that the S atom is much larger than the 0 atom and has a more diffuse charge cloud. Similar problems as encountered for the S atom can thus be ex-

13.28 16.08 15.59 1-1.5 min. 13.25(13.28) 16.12 15.54(15.57) 1(1.5-2.5) min 13.29 16.12 15.50 1.5-2 min. 13.32 16.12 15.57 1.5-2 min. 13.32(13.36) 16.09 15.52(15.58) 1.5-2(2.5-4.5) min 13.4 15.5

pected to occur for IT-electron systems, this in fact has been found to be true. a6

The final results obtained with the basis set contain­ing/-type functions are listed in Table IV, which gives the following information: I. P. ' s in the Hartree- Fock approximation (Koopmans' theorem)a7 in second and third order of the many-body perturbatiorr-expansion, and the renormalized final results together with the corresponding pole strengths. The I. P. 's theoretically calculated by Chong et al., 4 Raimondi et al., 5 and Guest and Saunders6 are listed as well as the centroids of the bands estimated from the experimental spectrum. a All the present results are within 0.1 eVof the experi­mental centroids.

III. IONIZATION POTENTIALS OF H 2CS

USing the information obtained in the investigation on the HaS molecule, the I. P.'s of HaCS have been calcu­lated. The geometry of Johnson et al. 8 was employed. The following basis set was used: S: (12s!9p!2d)! [6s4p2d] with a,,(S)=0.15 and 0.5,0: (9s!5p!ld)! [4s!2p!ld] with a,,(C)=0.8, and H: (4s!lp)/[2s!lp] with ap(H) =0.75. The parameters of the s- and p-type func­tions were taken from the work of Veillard20 for the S atom and from the work of Huzinaga18 for the C and H atoms. The total SCF energy of HaCS is calculated to be E:~ = - 436.3780652 a. u. This basis set is expected

TABLE IV. Final results for the valence I.P.'s (in eV) and pole strengths, P, of H2S calcu­lated with the basis set including/-type functions. a

Symmetry _4CF 1. P. (2) I. P. (3) I. P. (R) p(R) 1. p.8JPb I.P. " I.p.d I.P.-

bl(7T) 10.51 10.15 10.37 10.38 0.92 10.48 10.25 10.83 9.7 al 13.62 13.08 13.35 13.36 0.92 13.4 12.99 13.55 12.9 b2 16.09 15.46 15.68 15.58 0.93 15.5 16.06 16.30 15.5

"I. P. (I) i= 2, 3, I. P. in second and third order of the perturbation expansion; I. P. (R), p(R),

"renormalized" result for I.P.'s and pole strengths. bCentroid of the bands, estimated from the spectrum in Ref. 2. "Reference 4. <iJreference 5. "Reference 6.

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4896 von Niessen et at.: Many-body calculations on molecules

TABLE V. Final results for the valence I. P. 's (in eV) and pole strengths of H2CSo a (The experimen-tal values are band maxima, whereas the calculated values should be compared to the centroids of the bands.)

Symmetry _~CF 1. P. (2) I. P. (3) 1.P. (R) p(R) 1.p.b(exp) I.P. C(exp) I.p.C(theor)

2b2 9.57 8.74 9.18 9.12 0.91 Ib j (rr) 11.41 11.47 11.67 11.63 0.91 3aj 14.72 13.49 13.95 13.82 0.90 Ib 2 17.52 15.34 16.13 15.80 0.84 2aj 21.98 19.09 19.65 19.45 0.79

~he consecutive numbering in the symmetry labels start with the first valence I. Po

9.34 11.89

to give all I. Po's to within 0.2 to 0.3 eV of the experi­mental values. The results of the Green's function cal­culations are listed in Table V which contains orbital energies, I. Po's obtained in second and third order of the perturbation expansion, and the final results ob­tained by applying the renormalization procedure, j the theoretical values obtained by a configuration interac­tion (cI) calculation, 13 and the experimental I. P. 's cor­responding to the band maxima. 13 For the flrst three I. p. 's, the agreement is as expected. The maximum error occurs for the 2b2 I. P. whose associated MO is located on the S atom (0.26 eV). But there seems to be an unexpectedly large discrepancy of about 0.6 eV in the case of the fourth I. P. (lb2). With the method and basis sets used here, all computed I. P. 's for a number of molecules have been found to agree with the experimen­tal values to within 0.2 or 0.3 eV,25 except in the case where satellite lines are lying in the same energy range. In such cases, the renormalization procedurel does not apply and the more general procedure28 should be used. This is, however, not the case for the fourth band. In addition, the value obtained by a CI calculation agrees closely with the present value. A possible explanation for the discrepancy with experiment will be given below. The fifth I.P. (2al ) is smaller than the experimental val­ue by 0.45 e V. Satellite lines are found in this energy range (see below for more detail).

IV. VIBRATIONAL STRUCTURE IN THE PES OF H2CS

Since the vibrational structure of the photoelectron bands of HzCS has not been investigated theoretically yet, it is calculated as well. The method used is de­scribed in detail in Ref. 29. The first order (linear) cou­pling constants which are to be computed are given by

. I (~) Kk(t) = -..f2 aQk 0 0

(la)

The E j are the poles of the Green's function (they are the negatives of the I. P. 's). If many-body effects are not included, the E j have to be replaced by the orbital energies €h which are the poles of the Hartree-Fock Green's function. The Qk are the totally symmetric di­mensionless normal coordinates. The derivatives are evaluated at the ground state equilibrium geometry. The second order (quadratic) coupling constants

( 0)_ l( azE j ) 'Ykk' t --"4 6Qk aQk' 0 ' (lb)

are neglected in the following. For the totally symmet­ric normal modes, they have only a minor influence on

9.38 11. 76 13.85 15.20 19.9

bReference 12. cReference 13.

9.08 11. 49 13.75 15.78 20.22

the FranCk-Condon envelope ° In particular, their in­fluence is largely compensated by anharmonic effects.a9

The nontotally symmetric vibrations can only couple to the electronic motion through second or higher order coupling constants and are thus not considered °

The quantities which can directly be obtained from a calculation of the I. P. 's at various geometries are the derivatives of E j with respect to the internal symmetry coordinates S" (aEt/aS,)o' In terms of these deriva­tives the first order coupling constants are given by

( .)_ 1 -liz,,", L (aEi) Kk t -- ..f2Wk Lr Ik as, 0 '

(2)

where (L'k are the ground state frequencies and L'k are the elements of the L matrix which transforms the nor­mal coordinates into the internal symmetry coordinates

S=Lw-l/a Q . (3)

For HaCS the internal symmetry coordinates are given by

51 =AR ,

Sa = (1/v'2) (Arl + Ara) ,

5 1 = VI ro Acp ,

(4)

where R is the C-S bond-length, r l and ra the two C-H bond lengths whose equilibrium value is ro, and cp the HCH angle. The derivatives (aE j /a5,)o are given in Ta­ble VI. To calculate the coupling constants Kk(i) from Eq. (2), the L matrix has to be obtained from the force field of HzCS in its ground state by the standard method described by Wilson, Decius, and Cross. 30

The force field of HaCS, however, is unknown. We therefore have to piece it together using as information the force field of HaCO and the known frequencies of HaCS.9 The force field of HaCO has been carefully in-

TABLE VI. Renormalized derivatives with respect to internal symmetry coordinates of the poles of the Green's function (in a.u.).

(:~)o (:~)o (:~)o 2b2 -0.024 0.0053 0.0016 Ib j (1T) 0.094 0.0061 -1. 2x 10-4

3aj 0.036 0.014 0.012 1b2 0.039 0.090 -0.024 2aj -0.024 0.083 0.016

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von Niessen et al.: Many·body calculations on molecules 4897

TABLE VII. Renormalized coupling parameters ak for H2CS.

~k 2b2 Ib1(1T)

3a1 Ib2 2a1

a1

3.87 x 10-3

4.14 x 10-5

2.75/10-3

1.15 0.66

a2 a3

0.035 0.054 0.053 1. 72 0.55 1. 58 1. 56 0.52 1. 90 0.54

vestigated by a number of authors. 31-36 Among the non­diagonal force constants, F12 is large and positive (0.3 to 1. 0 mdyn/A), while F 13 is negative according to the majority of the authors (- O. 3 to - O. 4 mdyn/A). F23

is relatively small and its sign uncertain. Peyerimhoff and co-workers have calculated Fll for H2CS and ob­tained F 11 = 6. 5 mdyn/ A. 11 They cite as the probable experimental value, the range 6.0-6.8 mdyn/A. Com­pared to H2CO, where Fu = 12-13 mdyn/A, one expects a strong reduction of F11 because of the large C-S dis­tance. To reproduce the C-S stretching frequency W3

of 1150 cm-1, it appears to be necessary to use for Fll a value larger than 6.8 mdyn/ A and a large and negative F 13 • F22 is chosen to be somewhat larger than in H2CO since the C - H distance is shorter. F 12 , F 23 , and F33

are taken approximately as in H2CO. The force field used to calculate the vibrational structure in the PES of H2CS is (in mdyn/A): Fu = 9.37, F22 =4. 89, F33 =0.48, Fl2 = O. 8, Fl3 = -1. 0, F 23 = - O. 3. The computed frequen­cies are (the values given by Johns and Olson are paren­thesis)9: w1 = 2971 cm-1 (2971), w2 = 1570 cm-1 (> 1550), w3=1152 cm-1 (-1150). A basis set without polarization functions has been used in the calculations of the deriv­atives (BE,! as, )0. If a better force field is determined in the future, one can easily compute from Table VI the coupling constants according to Eq. (2) and repeat the ensuing calculations.

The vibrational intensity distribution of the ith band is determined by the coupling parameters ak(i) =[Kk(i)/ wk ]2.37 These coupling parameters are listed in Table VII and will be discussed together with the vibrational structure of the bands below. The calculated spectrum is drawn in Fig. 1, together with the experimental spectrum of Solouki, Rosmus, and Bock. 13 The individ­ual lines have been drawn as Lorentzians (FWHM =0.038 eV). The bands are drawn with equal total in­tensity.

The following vibrational spacings have been used: First band: Because of the minimal coupling no strong changes in the vibrational frequencies compal'ed to their ground state values are expected. The ground state frequencies are thus used w1 = 2971 cm -1, W2 = 1570 cm-I ,

W3 = 1152 cm -1. Second band: V3 couples strongly. Cor­respondingly a strong reduction in frequency is ex­pected. From the experimental spectrum one deduces W3 = 840 cm-1

• The other frequencies are taken as in the ground state. Third band: A progression in 930 cm-I

is observed in the experimental spectrum. We choose w1 = 2971 cm-1, w2 = 1570 cm -1 (both are the ground state values) and Ws = 930 cm-I • Fourth band: We estimate the values: ~1 = 2300 cm-I, w2 = 900 cm-l , Ws = 1050 cm-l

and Fifth band: WI = 2600 cm-t, w2 = 750 cm-t, and Ws

=1050 cm-1 •

Our calculations show that the first two bands are rel­atively insensitive to variations (even to larger ones) in the force field. In fact, we obtain for these bands good agreement with experiment. The other three bands, however, are very sensitive to details of the force field, because here two or even all three derivatives with re­spect to symmetry coordinates are large. Besides the first two bands, only the third one exhibits resolved vi­brational structure in the experimental spectrum. For this band, we obtain qualitative although not quantitative agreement with experiment. According to the calcula­tion, both /)2 and V3 couple, with V3 dominating the struc­ture. Because of this mixture, the third and the higher vibrational lines are broadened. This is also observed in the experimental spectrum, but it appears that the mixture of v2 and V3 is even stronger than obtained from the calculation. It is possible that also nontotally sym­metric vibrations are excited in this as well as in the other bands at higher energy (e. g., out-of-plane vibra­tions in the case that H2CS becomes nonplanar).

For the fourth and fifth band, the calculations yield a very strong coupling, which is in agreement with the general structure in the experimental spectrum. For the first time VI couples to a nonnegligible extent. The details of the structure are without relevance because of their sensitivity to details of the force field. An ac­curate force field is required before predictions can be made. The calculations give the fourth band broader than found in the experimental spectrum. It is possible that predissociation occurs in this ionic state. This would result in a cutoff of the vibrational structure in the high energy region of the band. The corresponding intensity would then be shifted to higher energies and disappear in the background. This appears to be the most reasonable explanation for the surprisingly large discrepancy between the computed vertical!. P. 's and

~~~ 93 9S 11! 118 UD '3.7 139 141 143

8

30,

18

Q)

20 eV

2Q, b)

FIG. 1. The photoelectron spectrum of H2CS; (a) The experi­mental spectrum. 13 Insert shows the first three bands on an expanded energy scale. (b) The calculated spectrum.

J. Chern. Phys., Vol. 66, No. II, 1 June 1977

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4898 von Niessen et al.: Many·body calculations on molecules

the apparent center of gravity of the fourth band. A similar situation occurs possibly in the fourth band of HzCO. z The fifth band is found to be narrower in the calculation than appears from experiment. But neither the experiment (because of the low intensity of the band, the possible occurrence of satellite lines, and the close­ness to the He! threshold) nor the calculations (because of the uncertainty in the force field) admit detailed state­ments.

A comparison with the PES of the CS molecule is of interest here. The He! PES of CS shows four bands. 38.39 In contrast to previous interpretations of the spec­trum38 .39 it is shown in Ref. 40 that the third band has to be assigned to a satellite line and the fourth to the B z~+ state. The satellite line and the fourth band are only 2 eV apart. If it is also the case for H2CS that an intense satellite line is located near to the 2a1 band, then the renormalization procedure used here does not apply for this I. P . and only the third order value in Table V has some relevance. In such cases the more general renormalization procedure described in Ref. 28 should be used in order to obtain accurate results. From the somewhat poor agreement of our calculated 2a1 I. P. with the experimental centroid of the fifth band in the PES of HzCS, we conclude that an intense satellite line is situated at about 20 eV binding energy.

V. CONCLUSIONS

The many-body Green's function method has been ap­plied to calculate the outer valence I. P. 's of the mole­cules HzS and HzCS. To accurately describe the changes in correlation energy and the reorganization occurring in the process of ionization and thus to obtain theoretical 1. P. 's to within 0.2 or 0.3 eV of the experimental val­ues requires more computational expense for second row atoms such as S than for the first row atoms. If two diffuse d-type functions are used for the S atom, all 1. P. 's of H2S agree with experiment to within 0.25 eV. A further improvement is obtained by addingj-type functions. Then all I. P. 's agree with experiment to within 0.1 eV. The larger size of, say, the S atom com­pared to the 0 atom and the more diffuse character of its charge cloud are the reasons for the different be­havior. The I. P. 's calculated for H2CS show satisfac­tory agreement with the recently performed measure­ments and the calculations of Solouki, Rosmus, and Bock13 for the first three I. P. 'so The maximum error is O. 26 e V . The discrepancy of about 0.6 e V between the experimental and the theoretical values for the 1b2

I. P. is attributed to predissociation which cuts off the high energy part of the band. Intense satellites are ex­pected in the energy range of the fifth band. In this case, the application of the present renormalization method is hardly justified.

The vibrational structure in the PES of H2CS has also been calculated by a many-body approach. Since the force field is unknown, we had to guess it using as in­formation the force field of HzCO and the experimental frequencies. The first two bands have been found to be insensitive to variations in the force field, whereas the higher bands prove to be very sensitive. Good agree-

ment is indeed obtained for the vibrational structure of the first two bands, whereas only qualitative agreement can be expected for the other ones. An accurate force field of H2CS is required before detailed interpretation and predictions can be made for these higher bands.

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The PES of thioformaldehyde has been kindly provided by Professor Bock (University of Frankfurt) prior to publica­tion.

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