All-electron CI calculations for core-ionized, core-valence excited and shake-up states of N2

Volume 52, number 3

CHEMICAL PHYSICS LETTERS

ALL-ELECTRON CI CALCULATIONS CORE-VALENCE Werner BUTSCHER,

15 December 1977

FOR CORE-IONIZED,

EXClTED AND SHAKE-UP STATES OF N2 Robert J. BUENKER* and Sigrid D. PEYERIMHOFF Chemie and Institut ftirPftysikalische Chemie, Universitat Bonn.

Lehrstuhl fiir Theoretische D-5300 Bonn. Germany

Received 25 August 1977

All-electron MRD CI calculations are reported for various core-ionized and core-valence excited states of N2 using a diagonalization technique which is capable of extracting arbitrarily high roots of large secular equations. The computations arc carried out in a delocalized framework so that gerade and ungerade electronic states cdn be distinguished just as they are in the limit of a full CI treatment. Configuration spaces of order up to 300000 arc considered. In the most flexible A0 basis studied (69 AO’s) a 1s IP of 410.01 eV is obtained in good agreement with the corresponding experimental value of 409.9 eV; the energy difference between the log-l ion and its lowest shdkc-up state is calculated to be 9.39 CV versus 9.3 eV experimentally.

1. Introduction The rapid development of experimental techniques in the field of photoelectron spectroscopy in the X-ray energy region during the last decade [I] has stimulated interest in describing such inner-shell excitation processes theoretically on a quantitative basis. Calculations at the SCF or Hartree-Fock level have been shown to be quite useful for the assignment of the major features of these highenergy spectra [2] but experience with related studies of electronic transitions within the valence shell [3] suggest very strongly that the method of configuration interaction should be more appropriate for providing the more accurate prediction of such experimental findings and at the same time should enable a wider range of applicability, covering corevalence excitations and so-called shake-up processes as well as simple ionizations. In spite of the fact that the CI method is completely general in principle it is clear that a number of special technical problems arise in connection with the study of highly excited states which are not encountered in conventional treatments of valence-shell spectra. * Present address: Gesamthochschule WuppertaI, Lehrstuhl Theoretische Chemie, D-5600 Wuppertal-Elberfeld,

Germany.

Because of the partial occupation of inner shells in these species it would appear necessary to employ basis functions which are capable of accounting for correlation effects in such highly contracted shells, at least as long as the proper balance with the corresponding molecular ground state is sought- In addition the possibility that core-excited states may require a generally more compact set of A0 basis functions than valence excited species should be investigated_ Furthermore the CI procedure itself needs to be altered in at least two important respects, namely so as to enable the correlation of all of the electrons of the system and also to allow for the extraction of relatively highenergy roots from the associated secular equations. In order to consider these various complicating factors in a systematic fashion it is clearly important to employ a test system for which a wide range of experimental data is available and for this reason the N, molecule seems to be especially appropriate for the present study. In addition to the 1s II?% themselves the location of a good number of the related satellite lines for the shake-up states of this system are known from the work of Celius [4] and a highly resolved spectrum for the corresponding core-valence transitions in N, has been reported by Nakamura et al. [S] and Wight et al. [B] . It is hoped then that by looking at a broad range of core449

Volume 52, number 3

CHEMZCALPHYSICS LETTERS

excitation processes, running the gamut from ionizations with and without simultaneous change in valence occupation to simple electronic transitions between core and valence MO’s, a more gencralIy applicable calculative procedure for the description of such phenomena in molecules will emerge than has heretofore been available.

2. Details of the CI calculations The CI method employed for the present study of core-excited states paraIIeIs in so far as possible that used for ground and valence-shell species (including Rydberg states) [7] _The configuration spaces are generated by taking aII singIe and double excitations with respect to a series of the most important terms (main or reference configurations) in the Cl expansion of a given state (multi-reference, MRD CI), Although the resulting secular equations arc generally too large to be solved by direct matrix diagonalization the use of an appropriate energy extrapolation procedure [7,8] makes it still possible to obtain the desired eigenvalue to suitable accuracy. The secular equations for subsets of the total Cl space require explicit solution and for this purpose a special diagonalization techni;;::: (roothoming procedure [9 J ) is empIoyed which is capable of extracting roots strictly on the basis of t!re appearance of the corresponding eigenfunction; thus the’root which contains the largest contribution from the desired corevalence excited configuration (or configurations) can thereby be obtained regardless of the position of its eigenvalue in an energy-ordered set *. The orbital basis set used to construct the various configurations consists of either the SCF MO’s of the parent state (or some closely related species) or else of * Note that by employing

a standard MRD CI trcdtmedt, in which all sir-@ rend doubIc excitations relative to each of the reference configurations are taken into account, low-energy species in which the inner shells arc complctcly filled arc also considered (and are found to bc non-negligible). As a result the dcsircd Is-cucited species do not correspond to the lowest roots of the CI secular equations, thereby necessitating the type of generally Jpphwble dia,vonalization procedure employed in Otis study- rrom the standpoint of perturbation theory one expects the arbitrary exclusion of such low-energy sccondxy configurations to result in an overcstimntion of the stability of the Is-excited states relative to what would result in a more extensive CL procedure.

450

15 December 1977

a corresponding set of natural orbitals generated specifically for the state under consideration. In this instance the NO’s are obtained in a single-iteration procedure simpIy by diagonalizing the first-order density matrix of a CI wavefunction from the MRD CI treatment undertaken at a relatively high selection threshold [IO]. The caIcuIations are carried out in the full point group symmetry of the hamiltonian (D_IJ and they therefore distinguish between gerade and ungerade electronic states. At the SCF or single-configuration level a symmetry breakdown caused by the near degeneracy of the Is levels of a homonuclear diatomic makes it often desirable to ignore the inversion symmetry in such treatments [2,1 I] but in a large-scale Cl procedure such as is considered in the present work there Is no comparable advantage to be gained in working with a lower-order point group than that of the Schrodinger equation itself [12,23]. 3. Simple Is ionization T’be role of the A0 basis in the description of coreexcited states is first tested by means of a series of calculations on the (I cil) ‘.Zi ion of N, _The smallest basis thereby considered consists of the N(9s, Sp) Huzinaga Cartesian gaussian set Cl43 in the standard [Ss, 3p] Dunning contraction (15 J augmented with two sets of s and p bond polarization functions whose exponents are given in table I ; a very similar basis (aIthough augmented with Rydberg-type species) has also been used effectiveiy to study the low-energy electronic spectrum of N, [16]. The calculated (I 0;‘) II? vahres thereby obtained at the SCF and CI levels of treatment are compared in tabte 1 with previous theoretical results 117,181 as well as with experiment 611. The present SCF value for this H’ obtained with delocalized orbitaIs lies over 8 eV above the corresponding result obtained in a localized MO framework with a simiIar A0 basis [ I71 but the CI treatment eliminates this discrepancy quite effectively so that the resulting IP value now is actually in somewhat better agreement with experiment than the triple-zeta SCF result reported in ref. [I g] _ The use of natural orbitals in the CI treatment leads to an energy Iowering of roughly 0.006 hartree for both the I$ ground state and the core-ionized species relative to the results using SCF MO’s but the II? value itself is decreased by only 0.06 eV as a result of the refinement in the theoretica treatment.

Volume52, number3

15 December I977

CHEMICAL PHYSICS LETTERS

Table 1

fonizationpotential for K shelf ionizationin N2 obtainedfrom different treatments Treatment

A0 basis

IP(eV)

SCF for % ItX2 and (1~~)~~ 2 ;Sg

DZ plusa) bond s and p (36 contracted functions) smne 3s above

419.77

same as above DZ, Slater AO’s s3me as above tripte-zeta

410.15 b)

MKD Cl, MO’s of botil states 4 mains/3 mains MRD CI, NO’s for both states SCF, localized hole, ref. [ 171 Roopmans' theorem, ref. [ 17 1 SCF, localized hole, ref. f 181

experimentaif 11 ESCA

410.51 W

411.6

428 410.8

409.92

af The exponentsusedfor the bond speciesare: utfs} = LOS, as(s) = 0.09, cZr(pTr) = 0.7, aa = O-OS. b) me total energies for the corresponding % 1 2; N2 @ound state is - 109.1588 hartree in the MO basis and - 109.2646 hartree if NO’s are empioycd.[n this case a set of four (formally D,h) main (or referenceconfigurations)b taken for 2 r Xg while three suchspecies are used for the ionized state; the secular equations actually solvea are of orders 14 19 anb 2118 respfctiveIy for the sound

‘i?csX$

state and the ion while the extrapolated total energy corresponds to tbc generated configuration spaces of order 23121 and 3 1694 (2P;>).

The next series of calcuIations undertaken involves the addition and subsequent exponent optimization of a compact p-type function in order to better describe inner-shell correlation effects (table 2). ate energy of the I$ ground state is thereby lowered by 0.041 hartree Table 2 CI enegies of tfie N2 ground state and the Nz (lo$) ion as a function of orbital exponent for functions of p type fa) and d type (b) added one set at a time to the original basis Set of table f (the SCF MO’sare used in the C1 basis) Exponent

S?cg* (haftree)~)

(la-r)z+ (in&e) -Y a

IP (eV)

(a) cY@)= 4.0

- 109.2662 -109.2767 -109.2840 -109.2896 -109.2977 -109.2PPl -109.2999

-34.1832 -94.1927 -94.2039 -94.2016 -94.2011 -94.2007 -94.20110

4IO.42 4 10.45 410.34 410.56 410.79 410.84 4 10.88

(b) a
-109.2877 -109.2954 -109,3072 -109.3188 -109.3351 -109.3277

-94.2122 -94.2302 -94.2472 -94.2488 -94.2650 -94.2425

410.21 409.94 409.7% 410.07 410.07 410.48

6.0 8.0 10.0 15.0 17.0 20.0 5.0 4.0 3.0 2.0 1.0 0.7

a) The scular equations solved for % t L‘a are in the order of 2000 while extrapolation is carried out for a spsce of appboximately SO000 configurations. Corresponding data for the ion are 3000 and 64000 respectively.

causing the (10,-I) IP to increase to about 410.8 eV The ionic species is found to prefer a somewhat smaller value for the p-type orbital exponent; the energy resuits for at(p) = 15 are seen to be close to minimal for both states. E%ycontrast addition of a d-type function to the original double-zeta plus bond basis ieads to a distinct lowering of the IP value to 4 10.07 eV for both o(d) = 2.0 and 1.0 (table 2b); in this case the ground state energy is lowered by nearly 0.08 hartree through the addition of this type of polarization function. On the basis of the results of table 2 a larger basis set is constructed for further investigations which included a total of 69 AO’s: two d-type functions with exponents Q = 2.0 and c)r= 0.65 respectively are thereby added to the originat double-zeta plus bond set along with a compact p A0 with fy = 15.0. En addition the is-type group flS] with exponent 199981 and 1.68 is decomposed to give an N atom basis of (9,6,2) contracted to (6,4,2). Finaliy, diffuse s and p-type functions with exponents 0.025 and 0.020 respecti~iy are added at the inversion center to allow for the treatment of Rydberg states in the N, core-excited spectrum. The energy results for the ground state and the 1UC’ ion obtained in this basis are collected in table 3. The ground state energy obtained in an MRD Cl treatment with four reference configurations employing NW is -109.4Cl98 hartree, and as such is lower than any previously reported value for this qu~ti~y in the Iiterature. The analopus C1 with a closed K shell 451

CHEMICAL PHYSICS LETTERS

Volume 52, number 3

15 December 1977

Table 3 Total energies (in hartree) for the NZ ground and Is hole state and corresponding ionization potential (in eV) obtained usingthe large A0 basis (69 functions). The order m of the secular equations actually solved and the generated configuration spaces n to which extrapolation is carried out are also given in parentheses as (m/n) Treatment

zlz+ g -108.9813 -109.3251 (274 i/53300)

(I@%g+

IP

-93.5673

419.42

MRD Ci, SCF MO’s=d

-109.4032

MRD CI, NO’s

-94.3188 (2941/161581) -94.3418 (3706/161581)

410.45

(3864/l 18306) -109.4098 (3687/l 18306)

SW MRD CI witi_ core {4 mains, X SCF MO‘s)

a) in this case the % ’ XiSCF MO’s are employed for the ground state calculation hole state SCF MO’s arc used for *xi (3 reference configurations).

(IO valence electrons) yields an energy which is 0.0847 hartree (2.30 eV) higher. When a comparabie all-electron CI is carried out with the same A0 basis for the (log’) 2Xg ion an IP value of 410.01 eV results, in good agreement with the corresponding experimental valuet. The use of NO’s is seen to be more important in the larger basis, causing a lowering of 0.62 eV in the 2Xz energy compared to what is obtained with the SCF MO basis; the IP vafue obtained in the MRD CI using SCF MO’s for both states is 0.44 eV higher than in the present best CI treatment.

4. Core-valence

excitation

and shake-up states

The results of the last section show that use of standdard CI methods for a reasonably flexible A0 basis leads to good agreement with experiment for N2 core ionization and hence raises the question as to how accurately other coreexcitation processes can be described in such calculations. To this end analogous MRD CI treatments are carried out for a number of core-vafence excited states of N2 employing the largest basis t Based on results for a two-electron system with 2 = 7 it can be estimated [ 191 that the non-relativistic IP value is 0.24 eV lower than the corresponding relativistic result On the other hand from the ~yrnrne~y of the N-Is ionization peak in NZ Celius has suggested (41 that a vibrational correction of roughly the same magnitude but opposite sign needs to be applied to the relativistic energy difference to predict the ~rr~pondin~ location of the intensity maximum. Hence for ati practical purposes it may be assumed that the exacr non-rctativistic energy difference coincides with the measured peak location of 409.9 eV. 452

_

410.01

(4 main or reference configurations)

while the

discussed above, and the resuIts are given in table 4. Comparison of the resulting transition energies with the corresponding experimental data indicates that the calculations overestimate such quantities by about I.4 eV in all foyr cases considered (table 4), despite the fact that the results for the II’ itself in the analogous treatment are in almost quantitative agreement with experiment. The error margin for -rhe core-valence transition energies is very nearly constant, however, so that the energy differences among such excited states themselves are seen to be in exceIIent agreement with the mea~~ments (table 4). As a result the present calculations are seen to support quite well the tentative assignment of this core-excited spectrum given by Nakamura et al. [5], which was based on a comparison with the low-energy spectrum of NO. In other words the CI calculations are able to order the core-valence excited states quite accurately just as is generally found to be the case for intra-vaIence species [3] in most conventional treatments of iowenergy spectra but the position of the corresponding ionized species is fess satisfactorily predicted. Since Is valence states appear too high in the calculations compared to the N, ground state the question arises as to whether the use of AQ basis sets optimized for the neutral nitrogen atom is appropriate for such highly excited species, especially since it is well known that removal of Is-type electrons is tantamount to increasing the atomic number of the corresponding nucleus [ISJO] _ To investigate this point more quantitatively additional CI calculations are carried out for the ground and 1 ug + ng excited states of N2 and the 10;” ionic species using different scale factors for the func-

CHEMICAL PHYSICS LETTERS

Volume 52, number 3

1.5 December 1977

Wble 4 Total energies E and transition energies ti relative to the N2 ground state for various core-valence excited states obtained from an MRD CI treatment (69 A0 basis set), and comparison with corresponding experimental quantities. The relative locations of the opticauy abowed transition with respect to the tDu(lou -) 33 state and the configuration spaces (see table 3) are also given State

E(hartree)

Vx; “Wl(l~”

“Tg)

‘z:(lou

+ 3s)

-109.4098 (3687/11830) -94.6279 126781198070) -94.4506

(3336~247328) lnu(IQg-

SPXU)

-94.4 I 19 (28431308238)

-94.6359

tDg(lag*ng)

Excitationa) exptt.

aE(eV)

(eV)

Relative excitation (e-V) G&C_

exp

0.0

0.0

402.23

400.84

0.0

0.0

407.05

405.59

4.82

4.75

408.11

406.Sob)

5.88

5.66

406.72 =400=)

402.01

tl.

-

5.88

-0.22

-c)

(3354/199551)

3) Refs. 15,6]. b, These peaks have been assigned in ref. [5] as lag -+ 3pa, and lug -c 3~0~. c, According to ref. f2lj this feature contains twa unresolved peaks lying very close to one another which 3re expected to correspond to lag -+ ag and lau - 3g excitations (1.8-2.4 eV resolution).

Table 5

Total energies (ii hartree) and corresponding IP and excitation energies for various N3 and I$ states obtained from the MRD CL treatment (D2 plus bond function A0 basis. 3s in table 1) 3s 3 function of the scaling factor for valence (qval) and inner shell (n&

functionsa) ‘)I5

OV3l

%‘z+

1.0

1.0

-109.2588

-94.L725

1.135 1.27 - 1.00

-io9.2554* -109.2648 -109.2578

-94.1956 -94.2032 -94.1739

LO 1.0 1.02

i?

(if&

%;

1.02 1.02 1.02 LO.02 1.02 1.04 1.04 1.04 1.04 1.04 1.04 1.09

1.135 1.27 1.3 13S 1.4 0.90 0.95 1.0 1.15 1.135 1.27 1.00

-109.2641 -109.2633 -i09.2627 -109.2612 -109.2596 -109.2429 -109.2502 -109.2554 -109.2619 -109.2616 -109.2609 -109.2432

-94.1962 -94.2034 -94.2034

1.09 1.09

1.135 1.27

-109.2501 -109.2502

-94.1921 -94.1921

(log

* +

*

ng)‘ng

IP (eV)

(10~ + +3E

-94.4670 -94.4918 -94.4982

41OSL 410.06 400.84

402.50 402.00 401.81

-94.4925 -94.4983

4 10.45 410.01 409.79 409.78

401.95 401.77

-94.4980 -94.2027 -94.1416 -94.1603 -94.1744 -94-1973 -94.1961 -94.2029 -34.t715

(lo?‘)

*

401.72

402.50 402.28 401.88

-94.4979

409.7 1 410.92 410.61 410.37 409.92 409.95 409.74

-94.4689 -94.4887 -94.4937

410.12 409.74 409.75

402.02 401.67 401.54

-94.4583 -94.4717 -94.4930

(eV)

401.72

a) The enere~ minimum b each column is indicated bY 3rt asterisk. The secular equ3tions actually soabed are in the order of 2800 for the ion and 3000 for ’ fXgwhite extrapolation is carried out for con~guration spaces of 3pprox~ate~y 32000 for the ion 3nd 37000 for the excited species.

453

Volame 52, number 3

X5 December 1977

CHEMICAL PHYSICS LFTRXRS

tions in the original double-zeta plus bond basis set. When separate scale factors are used for functions of core and valence type respectively it is found (tabIe 5) that all three states benefit from such exponent variation, but by far the largest energy lowerings occur for the I s-excited species. A&r optimization the f 0% IP value changes from 410.51 to 409.54 eV while the corresponding Iag + 7~~transitibn energy goes from 402SO to 401.82 eV, a drop of nearly 0.7 eV in both cases, This result does not alter the separation between the two core-excited species but it does indicate that if scaling were carried out far the large basis of tables 3 and 4 the I og IP would become too low compared to experiment (by about OS eV) while the 1og + Z~ transition would then be in significantly better agreement with the observed results. Furthermore, on the basis of the expansion coefficients for the various wavefunctions being considered (table 6) it seems quite likely that as the CI treatment is expanded toward the full CI limit the trend would be reinforced, since the reference configurations for the is-excited state are seen to account only for about 89% of the tatal density while the corresponding value for the Nz ground state is 94%?, None of these considerations appear to alter the spacing between the IO;’ ion f E&XXX perturbalion

theory it can be estimated that the fuitLI CI transition energy results are about 0.S eV lower For both the lug - srg and 1 og ionization process than the values obtained with the MRD CI calculations carried oat explicitly in the present work.

and the corresponding core-valence excited states, which continues to be underestimated in the calcufations by a~pro~matel~ I ,I eV (i.e. by taking the experimental 1 c.r93 7fg transition energy to be exactly equal to the lo, --f Z~ value of400,8 eV f5,6,21])As a result it is concluded that the earrelation energy of such core-valertce excited states is at least 1.O eV greater than for the corresponding core-ionized species, as seems reasonable on qualitative grounds. I?inally a calculation for the lowest shake-up state of N2 is carried out using the basis set with only one d-type function considered in table 2, A value of cr(d) = 4.0 is chosen since it provided for an accurate 1~;” IP value once I?Ws are employed in the CI (409.83 ev). S mce the spacings among core-valence excited states are represented quite well in the CI treatment it seems quite reasonable to expect that the same situation should hold for ionic species with differ only in their valence-shell occupation. To this end the (lO,f )fl” + ng 2X:” shake-up state is obtained using a treatment with its s!x most important configurations as reference species (table 7). The fact that a number of the secondary configurations thereby found to be important (at least 10% on a c2 basis) are related by a doubI~~xcitation to the leading term indicates quite strongly that a single-excitation Ci is not adequate for an accurate representation of this electronic state. If the NO’s of the I o;” ion are employed in the MRD CE treatment an excitation energy of 10.07 eV results, in moderately good agreement with the corresponding ex-

Tablr 6

Dominant terms in the CI expansion for tk?e representative N2 states employingthe large (69 AO) basis State


n%g

09021 0.0367a) sum

SUIIl

o.1279b) sum

0.8964

a) Formally three D& co~~~r~tjoo5. b) Form&& two I&h co~~gurat~ous.

454

%i

2uP,

3%

30 g

%a

=g_

2 2

2

2

2

2

2 2

2

2

4 2

2

f 2

2 L

2 2

2 2

2 2

4

1

2

2 2

2 2

2

2

4

3

I

0.8859 0.7685

UQg’gl*Qh&

%

093SB G-7733 0.1126b)

flog:-12) rg+

Configuration (occupation numbers)

2

I

3

1

2

CHEMICAL PHYSICS LETTERS

Volume 52, number 3

Table 7 Dominant terms in the CI expansion for the (lo;*)

(Coefficient)* 0.7645b) 0.0777 0.0244 0.02 1 I b) sum

x,, + mg ‘Xi

15 December 1977

shake-up state of N2 (the A0 basis contains only one d functiona)

Configuration (occupation numbers)

(lo,’

ion)

‘*g

hl

2%

2%

3ag

=ll

2 1 1 1

1 2 2 2

2 2 2 2

2 2 2 2

2 2 2

3 2 4 3

1

=g

3%

1

2 1

1

0.8877

a) The total energy is -93.8702 hartree of the NO’s if the (la,‘) . 6. used (255342~ionfigurations generated). bJ Formally two D,h configurations.

perimental value of 9.3 eV [4] _ Use of its own NO’s

lowers the energy of the shake-up state considerably, however, reducing this separation to the more acceptable value of 9.39 eV. In recent CI calculations in a localized representation Rodwell et al. [22] report a separation of 9.55 eV and also find similar (albeit somewhzt worse in two of the cases) agreement with experimental values for three other satellite lines of higher energy.

5. Conclusion The calculations described above indicate that the energy separation between various core-valence excited states can be determined quite accurately through the use of conventional CI methods provided a moder-

ately flexible A0 basis set is employed;energy relationships between core-ionized species including shakeup

ion are employed while it is -93.8951

har:ree if its own NO’s are

tive to the ground state; as a result this procedure leads to an underestimation of the Is hoIe II? by from OS0.7 eV, but at the same time it causes a desirabIe reduction in the absolute value of the core-valence transition energies. Since neither of the above considerations affects the relative position of the core-excited states among one another it appears very likely- that Cl calculations carried out with normal-sized A0 basis sets can be used quite effectively in assigning features in inner-shell spectra without restorting to measures normally not required in analogous treatments of valence-shell (and Rydberg) excited states.

Acknowledgement

The services and computer time made availabie by the University of Bonn Computer Center have been essential to this study and are gratefully acknowledged_

states also appear to be quite faithfully predicted in such theoretical treatments. If a normal unscaled A0 basis

is used with various types of polarization functions for both ground and excited states it is found that the core ionization potential of N2 itself is quite accurately predicted, but with the same basis the core-valence transition energies (Is + ng, 1s 4 3s etc.) are (uniformly) overestimated by about 1.4 eV; this result indicates quite strongly that the correlation energy in such corevalence excited states is considerably higher than for their core-ionized counterparts. Separate scaling of the valence and core basis functions has virtually no effect on the relative energies of any of the core-excited and core-ionized species but it does lead to a uniform reduction in the transition and ionization energies rela-

References

[II K. Siegbahn,

PI

C. Nordling, G. Johansson, I. Hedman, P-F. HedCn, K. Hamrin, U. Gelius, T. Bergmark, L-0. Werme, R. Manne and Y_ Baer. ESCA applied to free molecules (North-Holland, Amsterdam, 1969). P.S. Bagus and H.F. Schaefer III, J. Chem. Phys. 55 (1971) 1474; D-T. Clark, in: Applications of MO theory in organic chemistry, Progress in theoretical organic chemistry,

Vol. 2, ed. LG. Csizmadia (Elsevier, Amsterdam, 1977). I31 SD. Peyerimhoff and R.J. Buenker, in: Advances in quantum chemistry, Vol. 9, ed. P.-O. Lawdin (Academic Press, New York, 1975) p. 69. [41 LJ. CMius, J. Electron Spectry. 5 (1974) 985.

Volume

52. number

3

CHEMICAL

PHYSICS

[5] M. Nakamura. M. Srsanuma, S. Sato, M. Watanabe, H. Yamashita, Y. Iguchi, A_ Ejiri, S. Nakai, S. Yamaguchi, T. Sagawa. Y.-Nakai and T. Oshio, Phys. Rev_ 178 (1969) 80. (61 G.R. Wight, C-E. &ion and M-J. van der Wick, J. Eiectron Spectry. 1 (1972/73) 457. [7] R.J. Buenker and S.D. Peycrimhoff, Thcoret. Chim. Acta 35 (1974) 33; RJ. Buenker, SD. Pcycrimhoff and W. Butscher, submitted for publication. f S] R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 39 (1975) 217. . [9] W. Butscher and W.E. Kammer, J. Comp. Phys. 20 (1976) 313. [lo] K.H. Thunemann, R. Rdmelt, SD. Peyerirnhoff and R.J. Buenker, Intern. J. Quantum Chem. 1 I (1977) 743. [ 11] P.S. Bagus and H-F. Schaefer III, J_ Chem. Phys. 56 (1972) 224.

456

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15 December

1977

[ 121 A. Denis, J. Langlet and J-P. hlalrieu, Theoret. Chim. Acta 38 (1975) 49. f 131 L-S. Cederbaum and W. Domcke, J- Chem- Phys 66 (1977) 5084. 1141 S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [ 151 T.H. Dunning Jr., J. Chem. Phys. 53 (1970) 2823. [16] W. Butscher, S. Shih, R.J. Buenker and S-D. Peyerimhoff, to be published. 1171 LJ. Aarons, M-F. Guest, M-B. Hall and 1-H. Hillier, J. Chem. Sot. Faraday Trans. 69 (1973) 563. [ 181 D-T. Clark and J. Mufler, Theoret. Chim. Acta 41 (1976) 193. 1191 C.L. Pekeris, Phys. Rev. 112 (1958) 1649. [ 201 W.H.E. Schwarz and R-J. Buenker, Chem. Phys. 13 (1976) 153. 1211 J.S. Lee,T.C. Wong, R-A. Bonham and R.E. Kennerly, preprint. 1221 W-R. RodweIi, M.F. Guest, T. Darko, 1-H. Hillier and J. Kendrick, Chem. Phys. 22 (1977) 467.