Ab initio configuration interaction study of the electronic spectrum of SiH+

12May 1995

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 237 (1995) 204-211

Ab initio configuration interaction study of the electronic spectrum of Sill ÷ Abani B. Sannigrahi 1, Robert J. Buenker, Gerhard Hirsch, Jian-ping Gu

2

Bergische Universitiit - Gesamthochschule Wuppertal, Fachbereich 9 - Theoretische Chemie, Gauflstrasse 20, D-42097 Wuppertal, Germany

Received 13 February 1995; in final form 28 February 1995

Abstract Ab initio configuration interaction calculations have been carried out for Sill + using a large Gaussian basis set. Calculated spectroscopic constants of the X aX ÷ and A llI states as well as the excitation energy and the f0o value of the A III ~ X 1X ÷ transition and the lifetime z o of the A1H state are obtained in good agreement with experiment. In addition, the energies of the various asymptotes (up to 13 eV above the X 1~+ minimum) computed for large internuclear separations are found to agree with observed atomic values to within 0.3 eV in each case. Spectroscopic constants, excitation energy, lifetime and other quantities are predicted for a large number of higher excited states. Many other bound states have been indicated in the present calculations and their computed spectroscopic constants should provide a basis for further experimental investigations of the spectrum of this ion. Special emphasis is placed on the long-range characteristics of the Sill ÷ potential curves on the way to their respective atomic asymptotes.

I. Introduction The S i l l ÷ ion was first observed in the laboratory [1] by Douglas and Lutz in 1970. It was detected [2] in the solar photospheric spectrum in the same year. In view of its astrophysical importance [2-4], the S i l l ÷ ion has been the subject of several spectroscopic [1,5] and theoretical [6-8] investigations. The spectroscopic constants of the ground (X iF,÷) and

1 Permanent address: Department of Chemistry, Indian Institute of Technology, Kharagpur 721 302, India. 2 Permanent address: Department of Modern Chemistry, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China.

A1H states of S i l l + and the Te value for the A I H X 1 E + transition have been determined experimentally [4,5,9]. Bruna and Peyerimhoff [6] have computed potential energy (PE) curves of the ground and ten excited states of S i l l +. They tabulated the vertical excitation energy of the four lowest-lying excited states, but did not report values for the spectroscopic constants of the bound states. The details of their method of calculation were also not given. The PE curves of the four states (X 1• +, a 3 I~, A i l-I and 3 • + ), all dissociating into Si+(3s23p; 2p,) and H ( l s ; eSg) were calculated by Hirst [7] and his results for the spectroscopic constants of the ground state are in good accord with experiment. For the A l lq state Hirst only reported a computed D e value which is only in

0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 3 0 7 - X

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

qualitative agreement with experiment. No experimental values are available with which to assess the accuracy of his results for the a 31-1 state while the 3E+ potential curve was simply found to be repulsive. Using a large G T O / C G T O basis set and the coupled pair functional (CPF) method of Ahlrichs et al. [10], Matos et al. [8] calculated the spectroscopic constants and the lifetime of the A 11-1 state with high accuracy. For the ground state they used the complete active space SCF (CASSCF) approach [11] and also obtained quite accurate values for its spectroscopic constants. In order to aid experimentalists in further investigations of the Sill ÷ system it is clearly necessary to have much more information about the potential curves of its excited states than is currently available in the literature [9,12]. We have therefore carried out ab initio configuration interaction (CI) calculations on the ground and a large number of excited states (two 1A, three ~+, four 1II, three 3£+, two 3 ~ - , five 31"I, one 3A, one 31-1 and one 3 £ - ) of this system. Comparison will be made with previous calculated results for the lowest-lying X 1E÷, a 31-I and A IlI, as well as with available experimental data for the two singlet states, in order to assess the level of accuracy of the present theoretical treatment. In addition, the computed results for large internuclear separation can be compared with observed atomic data to obtain an independent assessment of the overall reliability of the present calculations.

2. Method of calculation The basis set employed in the present investigation consists of (16sl0p5d3f/12s7p4d2f) and (12s6p4d/8s4p2d) Cartesian Gaussian functions centered on Si and H respectively. It is the larger of the two basis sets employed by Matos et al. [8] in their study of Sill ÷ and contains 109 G T O / C G T O functions including diffuse s-, p- and d-type functions for a description of the 4s, 4p and 3d Rydberg orbitals required in the energy region studied. Calculations of the multireference single- and double-excitation CI (MRD-CI) type [13-15] have been carried out using the C2v point group, a subgroup of C~ v to which Sill ÷ belongs. The SCF MOs of the ground state (lo'22cr23trz4o'25cr21"rr4) have been employed to

205

generate configurations for the description of the ground as well as various excited states. For this purpose only four valence electrons are included in the active space. Five MOs corresponding to the Si ls, 2s and 2p AOs (three from a I and one each from the b 1 and b 2 irreducible representations of C2v) are kept doubly occupied in all configurations, while eight virtual MOs with the highest orbital energies (six from a I and one each from b I and b 2) are excluded entirely from consideration. Thus the four valence electrons are correlated among 109 - 1 3 = 96 MOs. A value of T = 5.0 × 10 - 6 E h has been used for the configuration selection threshold. The generalized version [16,17] of the Langhoff-Davidson formula [18] is employed for estimating energies corresponding to the full-CI treatment. Only these energy values have been used to calculate transition energies and spectroscopic constants. Other technical details of the CI calculations will be given in the next section. Results are obtained for a series of intemuclear distances in the range: 2.0 a 0 ~< r ~< 16.0 a 0. The calculated energies are fitted to polynomials using standard numerical techniques [19-21] to determine the spectroscopic constants of the bound states. The energy values at r e and r = 16.0 a 0 have been used for the calculation of dissociation energies. In addition to the energy, the electric dipole matrix elements have been calculated for all pair of states for which these quantities do not vanish because of symmetry. For these calculations only the variational wavefunctions are employed, however. The oscillator strengths of the allowed transitions and the lifetimes of excited states are then calculated from the transition moments and the vibrational wavefunctions.

3. Results and discussion The leading electronic configurations of the excited states along with the technical details of the CI calculations undertaken are given in Table 1. These data refer to r = 2.8 a0, which is close to the r e value of the X a£+ state. For larger internuclear distances several configurations become important in most cases. Over the entire range of r and for all the states of each symmetry the reference configuration sum value ~]p Cp2 varies within 0.94 and 0.98, indi-

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

206

cating that the selected configurations are sufficiently representative o f the entire CI space generated.

3.1. Potential energy curves and spectroscopic constants The calculated PE curves o f the ground and t w e n t y - o n e excited states o f S i l l + are displayed in Fig. 1. W i t h the exception o f the 1 3E+, 2 3 E - , 3 31I and 4 311 states all other excited states are predicted to be bound or at least h a v e a barrier to dissociation. The spectroscopic constants o f the latter states are g i v e n in Tables 2 and 3. In Table 2 only the X 1E+, A 1ii and a 3 i i states are included.

Results of both experimental and earlier theoretical studies are available for these states w h i c h can be used to assess the accuracy o f the present theoretical treatment. A s can be seen from Table 2, the spectroscopic constants o f the X 1 ~ + and A t YI states are obtained in g o o d a g r e e m e n t with experiment and the best previous theoretical estimates [8]. Closer scrutiny s h o w s that our results for the A 1 H state are marginally better than other calculated values. In this context w e w o u l d like to point out that the present CI treatment also yields slightly l o w e r total energies for both X 1~+ ( A E - - - 0 . 0 0 7 g h) and A l 1 1 ( A E = - 0 . 0 0 1 E h) states than obtained by Matos et al.

[8]. The asymptotic energies for the present potential

Table 1 The leading electronic configurations giving rise to various excited states and the technical details of the CI calculations at r = 2.8 a 0 Symmetry of the states C2v notation C=~ notation

Leading transition a

pMqR b

SAFs generated/selected

T.p cp2c

1 IA1 2 1A1 3 1A1 4 ]A]

X 1~+ 1A 2 1E+ 3 ]E+

_

123M4R

261723/4700

2"n2 - 2~r~ 50. 2 ~ 2~r2 + 2"n'~ 50. ~ 60.

0.968 0.963 0.953 0.959

1 1B1.2 2 1B1,2

A 11] 2 111

l14M4R

221748/4165

3 1B1,2 4 ]B1,2

3 lII 4 1n

0.969 0.952 0.951 0.959

1 1A2

93M3R

221266/3746

2 1A2 3 1A2

1 1A 2 1A 1 1E-

50"---) 13 40.5(r ~ 2~xx21ry

0.966 0.949 0.946

1 3A1 2 3A1 3 3A1 4 3A1

1 3~+ 2 3E+ 1 3A 3 3E÷

50. --~ 70. 50. ~ 60. 40.50. ~ 2";rx2"rry 50. --* 80.

121M4R

388586/5378

0.951 0.961 0.944 0.947

1 3B1,2 2 3B1,2

50. -~ 21rx,y 40. --) 2"rtx,y

108M4R

311172/5068

3 3B1,2

a 311 2 31] 3 3FI

4 3B1, 2

4 311

50. ~ 4'n'x,y 50. ~ 3~x,y

0.975 0.962 0.952

3A2

2 3A2 3 3A2 4 3A2

1 3E1 3A 2 SE2 3A

50. 2 -~ 2~rx2axy 40.50. -~ 2~rx2~y 40.50. ~ 2~x2~y 50" ---~13

1 5B1,2

1 51]

40.50. --) 70.2~'x,y

60M1R

111950/838

0.959

1 5A2

1 5~-

40.50. ~ 2~x2~ r

87M2R

141319/1440

0.975

1

50 .2 ~

50. --~ 2"ffx,y 40. ~ 2Xrx,y 50. ~ 41rx,y 50. ~

3~x,y

50" 2-') 2~x2~y

0.954

107M4R

365047/3676

a The transitions are with respect to the ground state electronic configuration (10.220.23tr24(r250.21"tr4).

b pMqR denotes the number of reference or main configurations and the number of roots treated. c ~p c2 is the sum of the square of the coefficients of the reference configurations in the selected CI space.

0.972 0.963 0.954 0.955

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

207

with only a few minor exceptions, indicating that the full-CI limit has been approached fairly closely in the present work. For infinite separations all such atomic and molecular full-CI energies corresponding to a given asymptote must be exactly equal, regardless of the AO basis chosen. The errors in the calculated atomic term values are thus seen to be caused primarily by deficiencies in the present AO basis. The largest discrepancy in Table 4 is for the Si+(4Pg) + H(2Sg) term value, which is underesti-

energy curves correspond alternately to various Si + H + and Si++ H atomic limits. All molecular states corresponding to the first five such asymptotes have been calculated and their corresponding energies at r = 16 a o are compared in Table 4 with relevant experimental and computed atomic term values. It can be seen that there is generally good agreement between the various long-range molecular energies and the corresponding calculated atomic value. Discrepancies generally lie in the 200-300 cm -1 range

E/E h

-0.8-

411"! 1H

-0.9-

-I.0-

13~ + -I.i-

-1.2-

3'.0

4'.o

s'.o

6'.0

71o

rlao

Fig. 1. Potential energy curves of the lowest-lying states of S i l l +. All energy values are given with respect to - 288.0 E h.

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

208

Table 2 Comparison of calculated and experimental spectroscopic constants (Te and D e in eV, r e in ,~ and toe in c m - ' ) of the X IX +, A 1II and a 31-I states ~ of Sill + State

Constant

Calculated this work

X 1X+

Te

Experimental

Ref. [8]

Ref. [7]

0.00

De re toe

3.40 1.505 2172

a 3II

T, De re toc

2.29 1.14 1.538 1793

A 1II

Te De re toe

3.24 0.16 1.865 451

Ref. [5]

Ref. [9]

0.00 3.43 1.505 2161

Ref. [4]

0.00

3.23 1.507 2155

3.33 1.499 2157

3.30 1.504 2157

2.25 b 0.98 1.540 1730 0.18 1.888 438

3.14 b 0.09

3.21 0.18

3.20 0.10 1.878 390

448

0.13 1.871 469

a All the three states dissociate into Si÷(3s23p; 2Pu) and H(ls; 2Sg). b Computed from tile tabulated potential energy points of Ref. [7].

mated by 2796 cm -1. One expects the correlation energy error to be least for a system with completely unpaired electrons and parallel spins and this is the case for this combination of atomic states. The other four asymptotes included in Table 4 have at least one closed shell or two 2p electrons with opposite spins. The next largest error is for the S i + ( 2 D g ) + H case (1345 cm-1). The other two term values mentioned correspond to neutral silicon states, with 2s22p 2 configurations. Their correlation energy errors are much closer to that of the Si+(2P u) + H(2Sg) ground state. The Si(3pg) term value is overestimated by only 245 cm -1, while the Si(1Dg) counterpart is too low by 446 cm -1. It is almost invariably found that correlation errors are smaller for molecular states in the bonding region than for their respective atomic limits, and therefore these results suggest that the present calculated To values should not be in error by more than 0.1-0.2 eV. Neither experimental values nor the results of any previous theoretical calculation are available for the higher-lying bound states of Sill + included in Table 3. The dissociation energy ( D c) and the equilibrium vibrational frequency (toe) of the three states of A~ symmetry decrease in the order: 1A > 2 1 ~ + > 3 1~ ÷, while their equilibrium internuclear distance follows a reverse trend. The 1A and 2 1~ + states have comparable re values since both result from a

transition to the same orbitals (2q'rx a n d 2q'ry). These MOs consist of diffuse Si p orbitals and are thus nonbonding. The slight increase in the re values of the 1A and 1~+ states with respect to that of the ground state is due to partial depopulation of the bonding 5tr MO. In contrast, the 3 iX+ state results from the 5tr ~ 6cr transition. The latter MO is of pure Rydberg type (Si 4s). It is worthwhile to note in this context that the C 1~+ state of the isoelectronic AIH molecule is characterized [22] by a doubleminimum potential. The D e (0.76 eV) and re (1.614 Table 3 Calculated spectroscopic constants (Te and D e in eV, rc in A and toc in c m - 1 ) of other bound (or quasi-bound) states of Sill ÷ State

Te

Dc

re

toe

Dissociation products

1 3X2 317 1 IA 15X2 iX+ 2 1ii 3 iX+ 1 3A 2 3X+ 3 3X+ 2 1A 3 11I 2 3A 4 11i

6.16 6.91 7.10 7.93 8.26 8.90 9.22 9.62 10.20 11.02 11.14 11.81 12.11 12.28

2.36 1.63 2.57 0.59 1.46 0.80 0.87 0.49

1.592 1.873 1.566 1.994 1.598 2.040 2.338 1.957 1.684 2.070 1.633 1.629 1.664 1.863

1657 1330 1758 941 1639 763 640

Si+ (4Pg) + H(2Sg) Si(3pg)+ H + Si(1D,)+ H + Si+ (g'Pg) + H(2Sg) Si(1Dg)+ H + Si(1Dg)+ H + Si+ (XE~)+H(2Sg) Si + (213 g) + H(2Sg) Si+ (213 ~) + H(2Sg) Si+ (25 ,)+ a(ZSg) Si + (2]] 'g)+ H(2Sg) Si+ (213 ~)+ H(Esg) Si + (2]] ~)+ H(2Sg)

1.04

1588

Si + (2D ~) + H(2Sg)

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

,~) values with respect to the inner AIH minimum, which also corresponds to a 5tr ~ 6tr transition, compare quite favorably with the corresponding values for the 3 1~+ state of Sill ÷ (Table 3). The equilibrium internuclear separation in the 2 11/ and 2 3I-I states of Sill + is greater than that in the ground state and the respective lower state of the same symmetry because of the double occupancy of two nonbonding MOs (4~r and 2"rr). The results of Fig. 1 show that there is an avoided crossing between the 3 1H and 4 l l-I states at r ~ 4.0 a 0. The lower of these two states oPossesses a potential energy minimum at 1.629 A, which is an unusually short r e value when one considers that this state partially occupies an antibonding (470 MO. It dissociates to t h e S i + ( 2 O g ) + n(ESg) atomic limit, which lies 10.18 eV above the X 1~+ energy minimum according to experimental data [5,23]. The present calculations for r = 16.0 a 0 yield a value which is only 0.07 eV lower than the measured result (10.11 eV). The computed Te value for the 3 1H state is

209

11.78 eV (Table 3), so the state is actually only quasi-bound. The corresponding barrier to dissociation is 1.57 eV. The 4 11/ is bound by 1.04 eV, and again the long-range energy limit computed for this state is in good agreement (0.10 eV) with the measured atomic data. The 2 1A ( T e = 11.14 eV) is computed to lie somewhat below 3 11/ and thus to be only quasi-bound (it correlates with the same atomic limit as 3 lII). The first 3 ~ + state is unbound (Fig. 1) and correlates with the same atomic limit as X 1E+, a 31-I and A lII. The lowest 3 ~ - s t a t e has a Si+(4pg) asymptote, however, and is com~uted to have a D e value of 2.36 eV (r e = 1.592 A). The difference between its Te value and that of a all is only 3.87 eV. Close by is the 2 31/state, with a notably larger r e value of 1.873 ,~,. Neither the 3 3H nor 4 3II states are bound, despite the fact that their electronic configurations are similar to those of the quasi-bound 3 11-I and the bound 4 l I-I species. Otherwise several quasi-bound 3E + and 3A states are computed with Te

Table 4 Calculated and experimental energies (in cm -1) of various dissociation limits of Sill +. The calculations are carried out for both the molecule at r = 16 a 0 and the separated atomic fragments

AE(r = 16 a 0)

Asymptote

Molecular states

Si+(2Pu) + H(2Sg)

X 1•+ Al11 3X+ a 311

Si(3pg) + H+(1Sg)

3 31I 2 32-

44187 44436

44181

43936

Si+(4P s) + H(2Sg)

2 31-1 3~sE511

41422 41489 41528 41558

41284

44080

Si(1Dg) + H+(1Sg)

2 1E+ 2 1H 1A(A2) 1A(A 1)

50577 50721 50896 51044

50681

50235

Si+(2Dg) + H(2Sg)

3 1~+ 2 37~+ 3 111 3A(A 2) 2 1A(A 2) 4 311 3A(A 1)

53961 54129 54164 54232 54297 54377 54609

53959

55304

0a 89 125 208

a The calculated energy for X 1~+ at r = 16 a o is -289.132943 E h. b The calculated energy for Si+(2P u) + H(2Sg) is -289.132539 E h.

Atomic calc.

Expt. [23]

0b

0

210

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 2 0 4 - 2 1 1

Table 5 Excitation energy Tc (in eV) and oscillator strength f for the transitions from the ground state (X 1~ + ), and lifetime -r (in ns) of various excited states of S i l l + State

T~

v'

f a

~.

v" = 0

-

v" = 1

2) 2) 3) 4)

0.56( 0.72( 0.24( 0.45(

-

v" = 2

3) 2) 2) 3)

0.57( 0.13( 0.52( 0.30(

-

v" = 3

2 1E ÷

8.26

0 1 2 3

0.91( 0.14( 0.17( 0.18(

4) 2) 2) 2)

0.32( 0.19( 0.21( 0.38(

-

5) 3) 2) 2)

3 1~+

9.22

0 1 2 3

0.13(-4) 0.12( - 2) 0.60( - 3) 0.20( - 2)

0.18(-3) 0.14( - 2) 0.57( - 2) 0.15( - 1)

0.13(-2) 0.79( - 2) 0.25( - 1) 0.50( - 1)

A 1H

3.24

0 1 2

0.80(-3) b 0.10( - 2) 0.15( - 2)

0.11(-2) 0.10( - 2) 0.10( - 2)

0.92(-2) 0.58( - 3) 0.35( - 3)

2 lII

8.90

0 1 2 3

0.52(-3) 0.17( - 2) 0.30( - 2) 0.42( - 2)

0.30(-2) 0.69( - 2) 0.92( - 2) 0.96( - 2)

0.77(-2) 0.12( - 1) 0.94( - 2) 0.55( - 2)

0.12(0.92( 0.24( 0.18( -

3 111

11.81

0 1

0.95 0.37

0.48 0.27

0.11 0.42

0 . 1 3 ( - 1) 0.27

0.59(-2) 0.29( - 1) 0.66( - 1) 0.93( - 1)

47.6 46.6 48.0 63.7 44.5 8.3 3.21 1.89 1737 c 1874 1711

1) 2) 2) 4)

29 22.3 25.7 30.6 0.22 0.23

a The numbers in parentheses denote powers of 10. b The experimental foo values are (2.4 + 1) × 10 -3 (laboratory value [5]) and 0.5 X 10 -3 (solar spectrum value [2]). c The experimental value [8] of r o is 1025 + 80 ns.

values in the 10-12 eV region. In all cases good agreement is found ( + 0 . 1 0 eV) between the computed and measured atomic limits, indicating that the level of theoretical treatment employed is sufficiently accurate.

and that obtained from solar measurements [2] Matos et al. [8] have reported a value of 1.2 × 1 0 - 3 for this quantity. The agreement between the present calculated (1737 ns) and the experimental value (1025 _+ 80 ns) of % is reasonably good. 10 -3

(0.5 X 10-3).

3.2. Excitation energy, lifetime and oscillator strength The excitation energy and lifetime of the excited states and the oscillator strength of all the symmetry-allowed transitions from the ground state to various excited singlet bound states are given in Table 5. These transitions are associated with high excitation energy (= 8-12 eV) and are predicted to be rather strong. The 3 1H-~ X 1•+ transition has an oscillator strength of the order of unity and a short lifetime. None of these states has been observed in the. absorption spectrum. The only singlet-singlet transition observed to date in Sill ÷ is A III ~ X 12+. For this transition our calculated T¢ value is in excellent agreement with experiment. The computed f00 value lies between the estimated laboratory value [5] (2.4 ± 1) ×

4. Summary We have carried out ab inito CI calculations on the ground and several excited states of Sill ÷. Our results for the spectroscopic constants as well as the spectral data corresponding to the A l II ~ X 12+ transition are in excellent agreement with experiment. In addition the results for the energies of atomic limits obtained from calculations for large internuclear separations are quite consistent with the experimental data. This indicates that the level of theory employed in the present investigation is suitably reliable. Thus our calculated values for the spectroscopic constants, transition energies and other

A.B. Sannigrahi et al. / Chemical Physics Letters 237 (1995) 204-211

properties of the higher-lying states should serve as a guide to future experiments on the S i l l + ion.

Acknowledgement O n e of us ( A B S ) wishes to thank the Deutsche F o r s c h u n g s g e m e i n s c h a f t for support e n a b l i n g him to visit the University of W u p p e n a l for a period of three m o n t h s w i t h i n the Forschergruppe " E r z e u gung, Nachweis und Eigenschaften reaktiver Moleki~le". The financial support of the Fonds der C h e m i s c h e n Industrie is also hereby gratefully acknowledged.

References [1] A.E. Douglas and B.L. Lutz, Can. J. Phys. 48 (1970) 247. [2] N. Grevesse and A. Sauval, Astron. Astrophys. 9 (1970) 232. [3] N. Grevesse and A. Sauval, J. Quant. Spectrosc. Radiation Transfer 11 (1971) 4231. [4] A.A. de Almeida and P.D. Sing, Astrophys. Space Sci. 56 (1978) 415; P.D. Sing and F.G. Vanlandingham, Astron. Astrophys. 66 (1978) 87. [5] T.A. Carlson, J. Copley, N. Duri, N. Elander, P. Erman, M. Larsson and M. Lyra, Astron. Astrophys. 83 (1980) 238. [6] P.J. Bruna and S.D. Peyerimhoff, Bull. Soc. Chim. Belg. 92 (1983) 525. [7] D.M. Hirst, Chem. Phys. Letters 128 (1986) 504. [8] J.M.O. Matos, V. Kello, B.O. Roos and A.J. Sadlej, J. Chem. Phys. 89 (1988) 423.

211

[9] K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [10] R. Ahlrichs, P. Scharf and C. Erhardt, J. Chem. Phys. 82 (1985) 890. [11] B.O. Roos, Advan. Chem. Phys 69 (1987) 399. [12] G. Herzberg, Molecular spectra and molecular structure, Vol. 1. Spectra of diatomic molecules (Van Nostrand Reinhold, New York, 1950) pp. 266-267. [13] R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 35 (1974) 33; 39 (1975) 217. [14] R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771. [15] R.J. Buenker, in: Proceedings of the Workshop on Quantum Chemistry and Molecular Physics, Wollongong, Australia, ed. P. Burton (University Press, Wollongong, 1980); in: Studies in physical and theoretical chemistry, Vol. 21. Current aspects of quantum chemistry, ed. R. Carbo (Elsevier, Amsterdam, 1981) p. 17; R.J. Buenker and R.A. Phillips, J. Mol. Struct. THEOCHEM 123 (1985) 291. [16] R.J. Buenker, S.D. Peyerimhoff and P.J. Bruna, in: Computational organic chemistry, eds. I.G. Csizmadia and R. Daude| (Reidel, Dordrecht, 1981) p. 55. [17] G. Hirsch, P.J. Bruna, S.D. Peyerimhoff and R.J. Buenker, Chem. Phys. Letters 52 (1977) 442. [18] E.R. Davidson, in: The world of quantum chemistry, eds. R. Daudel and B. Pullman (Reidel, Dordrecht, 1974) p. 17. [19] J.W. Cooley, Math. Comput. 15 (1961) 363. [20] R.J. Buenker, S.D. Peyerimhoff and M. Perid, Chem. Phys. Letters 42 (1976) 383. [21] M. Peri6, R. Runau, J. R6melt, S.D. Peyerimhoff and R.J. Buenker, J. Mol. Spectry. 78 (1979) 309. [22] J.M.O. Matos, P.-A. Malmqvist and B.O. Roos, J. Chem. Phys. 86 (1987) 5032. [23] C.E. Moore, Arch. US Natl. Bur. Stand. No. 467, Vol. 1 (1971).