Potential curves of BO and LiO calculated with the complete active space SCF (CASSCF) method

Chemical Physics57

(1981) 197-206 North-HollandPublishingCompany

POTENTIAL

CURVES

OF BO AND

LiO CALCULATED

WITH

THE

COMPLETE

ACTIVE SPACE SCF (CASSCF) METHOD Alexandr V. NEMUKHIN*, Jan ALML6F** and Anders HEIBERG Department of Chemistry, University of Oslo, Blindem, 0x10 3, Norway Received

1 September

1980: in final form 19 January 1981

The CASSCF method was applied to compute the potentialenergy curvesfor the lowest‘%‘, ‘.?I and ‘.‘A states of BO, and the lowest‘X’, ?I and ‘E- states of LiO. The curveswere obtained for internucleardistancesrangingfrom 2 to 25 au usinga medium-sizedbasis set of CGTOs. The resultswere comparedwith previousaccurateab initio calculationsand availableexperimentaldata. The changesin the wavefunctionstrucfurealong the potentialcurveswere discussed,

1. Introduction The multiconfiguration

task. Moreover, the CAS idea leads to some computational advantages over the conventional self-consistent

MC schemes (see ref. [4] for a comprehensive

field

(MCSCF) method, which allows for optimization of both the configuration mixing coefficients and of the orbitais entering these configurations,

is

one of the most powerful tools for a theoretical description of interacting molecular species. The complete active space SCF (CASSCF) method D-33, being a particular case of the MCSCF approach, is especially suited for calculations of full potential surfaces covering not only regions of stability of the whole molecular system but also the regions corresponding to different dissociation limits. In this approach the chemical effects of interest are assumed to be described by a subset of the orbital space (the active space) and a complete configuration interaction (CI) expansion in this space is constructed keeping inner electronic shells doubly occupied and the remaining orbitals empty. The difficult dilemma of selecting “important” configurations typical for other MC approaches is thereby

replaced by the problem of choosing an active space, which is conceptually a slightly easier * Permanentaddress:Departmentof Chemistry,Moscow l

State University.Moscow117234. USSR. * Author to whomcorrespondenceshould be addressed.

0301-0104/81/0000-0000/$02.50

@ North-Holland

description of the MC technique). We report here the results obtained with the CASSCF method for the potential curves of BO and LiO. We were primarily interested in examining the general picture of the full curves rather than in obtaining high spectroscopic accuracy, and therefore a medium-sized basis set of GTOs was used. The states studied in the present work were the “C’, ‘II, ‘A, ‘IX+, ‘n and “A states of BO referring to the first dissociation limit B(‘P) -I-O(~P), the ‘II and ‘H- states of LiO corresponding to Li(‘S) +O(3P) and the ‘Hi state referring to Li(‘P) t- 0(3P). All these states represent the lowest roots of the pertinent secular equations within the Ct., symmetry

group, which for computational reasons was used instead of full C,, symmetry. The active space employed for both systems consisted of the 417, 5~, 6a, lsr,, 25;;. lz,, 27rY MOs (the nuclei were located along the z-axis) and covers all valence states of the diatomics. After some experimentation we found it possible to keep the 3u orbital (correlating with the 2s orbital of oxygen) along with the la and 20 MOs within the inactive space, i.e. always doubly occupied. Publishing

Company

A. V. Nrmxkhin el al. / CASSCFporeltrial curves forBO and LiO

198

These diatomic systems present a di&cult problem for an MCSCF calculation. In the case of BO there are a large number of valence states within a comparatively narrow ener,g range, and LiO is an example of a highly polar molecule. For both species considerable changes of the wavefunction structure occur when passing along the potentia1 cm-ves. Previous ab initio investigations beyond the Hartree-Fock approximation of potential curves for these molecules include the CEPA studies of the lowest ‘xi and ‘II states for BO [5] and the accurate CI calculations of the lowest ‘II and ‘Z” states for LiO [6]. Experimental results are also available [7_ 83.

of the generators l& or of products of such generators_ Their evaluation is carried out by means of a graphical technique [lo-121. After solution of the CI problem with a trial set of molecular orbitals the s;?in independent first- and second-order reduced density matrices Dpq = 1 C,CvA,;,

Gal

Pp._vs =‘;: C, C&&

(4b)

P”

are calculated and passed on to the orbital optimization part. The starting point for the orbital optimization is the energy expressed in terms of the density matrices D and P: E = 1 &Jr,,

w

2. Theory A detai!ed description of the CASSCF method is presented in refs. [l, 21. Here we will only outline the essential points of the approach. Being a particular case of the MCSCF approximation the method is based on the general expression

for the many-etectron tiavefunction. Here the Qp are configuration state functions (CSFs) of a particuiar kind (Cielfand states :9]) constructed from an orthonormal set of one-electron spatia1 functions and the C, are coefficients to be determined by solving the CI eigenvalue problem. In Lerrns of the generators of the unitary group, E, [9], the expression for the hamiltonian operator is w

P9=

This representation leads to the following relation between the matrix elements of A and the usual one- and two-electron integrals: HClv=C_4g&+ P9

1 A,c&qIx), f4=

where the one- and two-electron coupling coefficients AZ and AEm are matrix elements

(3)

+ 1 P,,,(P~

W’S

1rs).

(5)

In order to make the energy stationary with respect to the orbital variations the orbital set is subjected to a unitary transformation U. In the present approach the exponential paramcterization of real unitary matrices is employed. Thus the matrix U is written as U = exp (X),

(6)

where X is a skew-symmetric matrix containing all independent variables defining the rotation of the orbital set. To determine these parameters the Newton-Raphson method is used, resulting in the following expression for the vector x = {X1?, X13, . . - 1: x = -G-‘g.

(7)

Here g is a vector of first derivatives of the energy with respect to the rotational parameters X,, calculated at the initial point X = 0, and G is a matrix of second derivatives which is positive definite at the desired stationary point. Expressions for the first and second derivatives in terms of molecular integrals and elements of D and P may be obtained directly from eq. (5) [2, 31. After finding the X,-quantities the rotation of the orbit& set is performed and a new CI calculation using the rotated orbitals is carried out. The procedure is repeated until convergence is achieved.

A. V. Nnnukhin er al. / CASSCFporenrial cumes forBO and LiO

The outlined scheme is valid for any MCSCF approach and so far no references have been made to the particular features of the CASSCF method. T’he basic assumptions of this method that distinguish it from other lMCSCF schemes are related to the specific selection of configurations in the CI expansion (1). The orbital space is divided into inactive, active and secondary parts. The secondary orbitals &, &br - . - are not occupied in the wavefunction, whereas the inactive orbitals &, &, . . . are kept doubly occupied in all configurations. The remaining electrons (active electrons) are distributed in all possible ways among the remaining orbitals (active orbitals &, &, . . .), which may thus be occupied (singly or doubly) in one configuration and empty in another. The particular choice of configuration space implies that the energy is stationary with respect to orbital rotations within each of the three groups of orbitais. Therefore the corresponding parameters Xii, Xab and X,, are deIeted frcm the Newton-Raphson system of linear equations. The number of independent parameters X, defining the orbital variations is thereby drastically reduced. Furthermore, only density matrix elements referring to the active orbitals are needed, which for many important applications solves the problem of storage of these matrices. With these advantages the CASSCF method is a rather simple and convenient method to use in accurate studies of small molecular systems. For example, the calculations on the ‘Z* (‘A,) state of BO (the largest calculations performed in this work) included 208 configurations, whereas the number of rotational parameters was 107 and the number of second derivatives of the energy with respect to these parameters was 5778. The number of first and second order density matrix elements stored in memory was 28 and 406, respectively.

3. Computational

details

The basis set of contracted gaussian orbitals used in the case of BO consisted of double-zeta

199

sets on boron and oxygen [13, 141, polarization 3d functions with exponents ca = 0.5 and co= 1.2 (as in basis “B” of ref. [S]) and a diffuse 2p function on oxygen with the exponent co= 0.059 [KJ. In the case of LiO, the basis for oxygen was essentially the same with the exception of the d-orbital exponent (co= 0.85 [15]). For lithium, the set of s- and p-functions (with the polarization 2p orbitals included) quoted in ref. [16] was used. The total number of CGTOs was 35 for BO and 29 for LiO. As previously menticned, the partitioning of the orbital space, which is a key step in the CASSCF method, was done as follows. The lm and 2r MOs corresponding to 1s orbitals of the constituent atoms were assigned to the inactive space. Exclusion from the active space of the 3~ MO, which was essentially equal to the 2s orbital of oxygen, leads to shifts of the order 0.2 eV in excitation or dissociation energies for the states under consideration. The orbitals 4~, 5cr, 6u, l?r,, 25r,, lag, 2z,. were finally chosen as active taking into account the approximate relation between the MOs and the orbitals of the isolated atoms [lS]. In principle, this active space allows one to obtain diatomic states dissociating to combinations of isolated atoms with almost all possible distributions of electrons among the valence orbitals. A further reduction of the active space leads to substantial errors in correlation energy estimates. For example, restricting the active space to the 4a, 5u, lsr,, lr,. orbitals in the case of the %I states of LiO we obtain only five configurations (representing the minimal set of configurations needed for properly describing the dissociation of the ground X’II state). The computed dissociation energy in this approach was 1.96 eV, compared with 2.76 eV obtained with the larger active space. Actually, the problem of selection’of active space is not so simple even for the first row atoms. For LiO we expected that the following correspondence between MOs and AOs would occur: 4u -2p,(O), 5~ - 2s(Li), 6a - 2pz(Li), (this kind of I ;;r.y - 2P,.,!O), 2TT,.Y-2p,.,(Li) relation was valid for BO). Instead, however, the procedure predicts the 6u and 25;,., MOs to

200

A. V. Nemukhin er al. / CASSCFpotentialcume3for SO andLiO

be purely oxygen orbita!s. ‘Ibis situation changes when higher excited states nre treated. In the calculations the CZy point group was employed which enabled the use of programs based on the MOLECULE package [17]. When restricted to the subgroup C?,. the representations HA and 8- of C,, turn into the symmetry species Ai and A=, whereas the II and A representations decompose into the sums B1 i Bz and Ai +A?, respectively. The distinction of the various diatomic states falling in the same irreducible representation of CZV(for example, ye and 4 in A,) presents no problem, but of course this restriction reduces the number of diatomic states that can be treated as first roots of the corresponding secular equations_ With the choice of active space described above the number of CSFs pertinent to this work is (C2, symmetry): For LiO with 5 active electrons 126 ‘Al, 116 ‘A? and 124 ‘Bi configurations, and for BO (7 active electrons) 208 ‘)Alr 196 ‘B1, 184 ‘AZ, 96 *At, 96 -Bi and 104 ‘A2 configurations_ When calculating a potential curve for a diatomic molecule, it is reasonable to move aiong the curve using converged orbitals at the neighboring internuc!ear distance R as starting orbitals for a new point. Obvious candidates for the initial points are the equilibrium position, when the corresponding SCF orbitals provide a good trial set of orbitals, or the dissociation limit, where the orbitals of isolated atoms may be tried. When starting the study of a new state, IMCSCF orbitals of previously calculated states may be useful. Normally, these approaches worked well for the systems studied here. About 8-12 iterations were usually enough to achieve convergence with orbitals of the previous geometry (the threshold for the energy difference between two consecutive iterations was established at 10d6 au, for the maximum of the first derivatives at lo-& au). However, several difficult cases were encountered which required substantial efforts for their treatment_ For the ground ‘II state of LiO it was difficult to obtain convergence with the

corresponding SCF orbitals even at the equilibrium position. Certainly, this is an indication that the SCF and MCSCF orbitals differ considerably. Other difficult situations often arise when studying regions intermediate between equilibrium and the dissociation limit or around maxima, if such occur on the curve. These regions are characterized by drastic changes of the wavefunction structure regarding both CIcoefficients and orbitals, and in these situations orbitals of a neighboring geometry may not be adequate as a start. Thus, it was necessary to perform 35 iterations to obtain convergence at the point R = 3.5 au for the ‘II state of $30, starting with converged orbitals referring to R = 3-O au. In the present version of the program the following parameters can influence the rate of convergence, Or even play a crucial role for achieving convergence at all: a threshold parameter, restricting the maximal value of the variables X,, found after solution of the Newton-Raphson equations, i.e. it damps the rotations of an intermediate step if they are too large, and a parameter shifting negative eigenvalues of the matrqx G in eq. (7) in order to make it positive d-finite [2]. Manipulations wi?h these parameters may sometimes help to obtain convergence. In some cases it turned out successful to perform one or two orbital optimization steps with CI coefficients not computed from the secular equation, but given as input. As has been found previously [2], there is no need to recompute the G matrix in every iteration in nearly convergent situations. This practically does not influence the rate of convergence, and permits the time in the integral transformation section to be reduced considerably. For the largest calculations with the ‘Al symmetry of BO (35 basis functions, 208 configurations) the full transformation required 40 s, whereas the CI part took about 15 s and the orbital optimization part 2 s per iteration on a CYBER 74 computer. With only the g vector recomputed, the time in the transformation section reduced to 9 s.

A.V. Nemukhin et al / CASSCFpotentialcurces 4.

Results and discussion

201

for BO and LiO

Table 1

Energies (in atomic units) for the lowest lying doublet states of LiO relative to -82.000000 au

Cakulations were performed for 10-12 values of the internuclear distance for each potential curve. Calculated energies and corresponding values for R are shown in tables 1 and 2. These caIculations allow one to get a general picture of the potential curves for LiO and BO. However, in order to compute accurate spectroscopic constants of the bound states more points around the equilibrium would be needed, and more sophisticated basis sets should be used. Therefore, the spectroscopic properties for LiO and BO given in tables 3 and 4 present estimates rather than accurate ab initio results. The uncertainties due to the limited number of points are estimated to be 0.005-0.01 A for R,, and lo-20 cm-’ for w.- The calculation of higher order constants such as o,x, or (Y, would hardiy be meaningful on the basis of the present data. The calculated potential curves are shown in figs. 1-3. The two lowest bound states of LiO have been carefully studied previously by Yoshimine l6] using a large basis set of Slater type functions. The lowest ‘Z- state correlating with the ground state dissociation limit Li(‘S)+ 0(3P) is compietely repulsive. The properties of the bound states presented in table 3 differ sIightIy from those calculated by Yoshimine with the CI approach. Basis set effects are

R (au)

2.0 2.5 3.0 3.5 4.0 5.0 6.0 8.0 10.0 25.0

state ‘Z’

=r-

=n

-0.121411 -0.308828 -0.347713 -0.342114 -0.324642 -0.287136 -0.256709 -0.216142 -0.187888 -0.187671

0.164957 -0.222279 -0.256572 -0.258918 -0.260190 -il.262220 -0.262663 -0.262671

-0.289307 -0.355002 -0.360704 -0.347738 -0.311006 -0.281444 -0.262992 -0.261607 -0.261421

probably responsibIe for these discrepancies. Thus, the energy difference between the dissociation limits Li(‘P) + C$P) and Li(‘S) + 0(3P) is predicted with our basis set to be 0.0737 au compared with the experimental value 0.0697 au. This explains the overestimation of T,(‘rG2Z.‘). Figs. 2 and 3 present results for the doublet and quartet states of BO. The region around the minima for the X’H’ and A211 states have been calculated previously by Botschwina [5] with the SCF and CEPA approaches using basis sets comparable with ours. All the states are strongly bound with the exception of the ‘II

Table 2

Energies (in R (au)

1.5 2.0 2.275 2.5 3.0 3.5 4.0 5.0 6.0 10.0 20.0

atomic units) for the lowest lying doublet and quartet states of BO relative to -99.000000

State “I’

Q

0.149047 -0.594095 -0.654397 -0.642393 -0.553214 -0.463527 -0.404198 -0.362245 -0.361758 -0.362225 -0.362236

0.478214 -0.393539 -0.508468 -0.536087 -0.518704 -0.474874 -0.433608 -0.387377 -0.366026 -0.362322 -0.362236

‘A

%’

-0.197034

-0.254015

-0.178177

-0.386094 -0.391526 -0.357864 -0.342000 -0.356367 -0.360840 -0.362191 -0.362235

-0.431704 -0.424526 -0.384523 -0.358528 -0.356355 -0.361395 -0.362225 -0.362236

-0.314779 -0.300560 -0.329626 -0.345458 -0.359179 -0.362236 -0.362290 -0.362236

;n

0.682751

Ad 0.706902 -0.362090 -0.403568 -0.403001 -0.370812 -0.353025 -0.357980 -0.361170 -0.362191 -0.362235

au

A. V. Nmxkhin er at. / CASSCFpotential curres for BO and LiO

202 1 - 82.10-

-82.X '; 2 B : t -0230

*

t

,

6

‘

2

*

10

R(au)

Fig. 2. Potential energy cutves for the lowest ‘8’. ‘A states of BO.

‘iI and

Fig. 1. Potential energy curves for the lowest ‘Z+, ‘il and “5- states of LiO.

state which strictiy speaking is not bound but possesses a IocaI minimum at 1.30 .k. The caIcu!ated spectroscopic characteristics of these states are collected in table 4. It is worth noting that the error in R, values for the ‘H’ and ?l states is larger than for SCF calculations using a similar basis iet [5]. As usual, the success of approximate SCF calculations for predicting bond Iengths arises from the cancellation of different errors, mainly the neglect of correlation and the incomplete basis set. In our calculations, the correlation effect on R, has been accounted for to a !arge extent by inciuding aII configurations necessary for a correct dissociation. However, we are left with the error caused by the truncation cf the LCAO basis, which is likely to make the bond too weak and therefore longer than experiment.

p-9930-

Spectroscopic

coostana

for the X%I and X’Z’

states of

LiO

X’I-I

A?z*

-2 2 3.

Table 3

State

One of the greatest advantages of the CASSCF method in studying diatomics is the possibility to adequately examine the changes of the electronic structure along the potential curve. All possible configurations with the active orbital are taken into consideration and if the active space is properly chosen there are no chances to omit important configurations at any point on the curve. Table 5 contains the CI coefficients for LiO referring to the CSFs with the largest weights at several internucIear distances. Coefficients with magnitude smaller than 0.1 are not listed. When analyzing these data the deformation of the

Ref.

this work C6l CSI (exp.) this work

161

Rb (X)

D, (eW

1.76 1.70

2.76 3.37 3.38kO.26 4.40 4.90

1.62 1.60

Z-l,

T, (cm-‘)

829 851

0 0

833 867

3018 2330

:

w

-9940-

r Fig. 3. PotentialenergyCIBVS for the lowest ‘EC, ?I and ‘A states of BO.

203

A.V_ Nemukhin et al. / CASSCFpotential tunes for BO and LiO Table 4 Spectroscopic state

‘z-

‘n

‘A

et+ ‘n a&

characteristics

for the BO states Minimum

Ref.

Maximum

R, (A,

R. (e’.‘)

WC(cm-‘)

r, (cm-‘)

this work PI [7, 81 (exp.) t5is work r5i

1.22 1.21 1.20 1.38 1.36

7.97

1909 1873 1886 1238 1289

0 0 0 25401 21882

[71 (exp.1

1.35

this this this this

I.55 1.36 1.30 1.44

work work work work

8.33 50.22 4.80

1261 0.83 2.04 1.38

1231 1279 1558 1145

active orbitals from one point to another due to orbital optimization should be taken into account as well. Nevertheless, to simplify the interpretation we can regard the 4w and lirX., orbitals as being primarily Zp, and Zp,., orbitals OR oxygen, whereas the SC+orbital tends to the 2s orbital on lithium at the dissociation limit. The only exception to this rule is the *E+ state for which one of the lrr orbitals tends to a 2p orbital on lithium whereas the 5o orbital tends to a weakly occupied s orbital on oxygen. The CSFs listed refer to natural orbitals. The changes in the electronic structure are seen Table 5 Configuration state function meficients differ in the spin-coupling scheme Stare

CSF

R,

t&

LE(R,l(cm-‘1

E-1

23959

23834 55471 46738 56668 52829

2.13 2.76 1.54 2.27

4920 1260 13678 2529

clearly. In the vicinity of the equilibrium position the ‘II and ‘E+ states describe practically ionic species, Li’O- (the population analysis gives Li2.‘40s-76 for ‘IL and Li2~“50e~75for ‘xc at R = 3.0 au). At large R values the curves should go to combinaticns of the neutral atoms. This is illustrated in fig. 4 where the variations of the dipole moment with R are shown. We note that the change from ionic to covalent structure in the case of the *Ci state occurs at a larger distance than for the ‘II state. This is due to the fact that the dissociation limit for the ‘Xi state (Li(‘Pj+O(3P)) is closer to the ionic Iimit

for the LiO states. Coetiicient

is omitted if its magnitude

R (au) 5.0

8.0

10.0

0.975 -0.119

0.707 -0.513 a.472

0.636 -0.580 0.494

0.979

0.978

0.652 0.425 -0.248 -0.567

0.860 -0.497

0.860 -0.497

0.860 -0.497

is less than 0.1. Starred

CSFs

20-I

A.V. Nemukhin et al. / CASSCFpotmtiaI

’ R (au) Fig. 4. The dipok moment for three electronic as a function of the intemuckar separation.

states of LiO

Li’(‘S)+O-(‘P) than is the limit for the ‘II state (Li(‘S) + 0(3P)). Furthermore, not so many states are involved in the complicated picture of avoided crossings. The ‘H- state describes practically non-interacting atoms, and no significant changes in the electronic structure are noticeable when moving along the potential curve. The corresponding data for BO are presented in table 6. The same approximate relation between the MOs and the AOs of the isolated atoms [i.e. 4~ -2p=(O), 5a -Zs(B). 6u2PZ (B), 12-X.,-2p,.,(O), 2rx., -2p43)l is valid at all internuclear separations with the exception of the intermediate regions around 46 au where there is a strong mixing of all active orbitals. Most of the states are characterized by the following change of dominant configuration during dissociation: 4&a ___(small R) + 4&a’ . .. (large P). Thus, at infinite internuclear separation the configurations important a.: e?nilrbrium would describe a 2s + 2p excitation on boron, i.e. the dissociation limit B(‘P)+ O(‘P). Cnly the least energy principle forces the curves to go to the ground state limit B(*P) + 0(3P). The situation is analogous to the LiO case where the contigurations being dominant at equilibrium tend to the leading configurations for the higher dissociation limit referring to the ionic pair Li’O-. This explains the maxima typical for the BO potential curves, which are certainly the results of avoided crossings. For the states ‘h, ‘Z- 2nd ‘A of BO two different self-ccnsistent solutions were found at

curzt?s for BO and LiO

distances around the potential energy maxima. The two solutions are different with regard to the symmetry equivalence of ;r, 2nd ;s, orbitals, the breakdown of which leads to a lower energy. For distances around the equilibrium the only self-consistent solution was obtained for a wavefunction with proper linear symmetry, whereas for long distances the ‘i;;, T,, equivalence does not lead to a minimum on the energy hypersurface. In caSes where two solutions were obtained, the energy difference between the two was less than 0.1 eV in all cases, and would hardly be visible on figs. 2 and 3. The spectroscopic constants of table 4 were calculated using the lower of the two potential curves. The corresponding values obtained with the upper curves differed by less than 0.005 A for R, and 6 cm-’ for in,.

5. Conclusion The CASSCF method turns out to be a useful tool for the study of diatomic molecules. It makes possible the prediction of fairly reliable numerical characteristics of the molecules 2s well 2s adequate descriptions of the potential curves over the whole range of internuclear separations. The convergence behavior of the calculations carried out is generally good. Slow convergence, whenever it occurs, seems to be connected partly with the shape of the energy hypersurface and partly with the two-step procedure (CI and orbital optimization steps) employed to reach the desired stationary point OR this surface. In particular, it should be pointed out that the coupling between the orbital and CI coefficient variations has not been explicitly taken into account in the present caIculations. For states that are not the lowest within the given symmetry representation, this may lead to inherently divergent situations [19]. In fact, calculations of potential curves for such excited states of BO were successful only within certain ranges of the internuclear distance. The CASSCF method may, however, be adjusted to treat such states as well. These questions will be discussed elsewhere [20].

A. V. Nemukhin ec aI. / CASSCFporendal Table 6 Configuration

state function ccefiicients differ in the spin-coupling scheme

for the BO states. Coefficient

State

R (au)

CSF

2.5

3.5

0.961

CWL-esfor BO and LiO

205

is omitted if its magnitude is less than 0.1. Starred CSFs

5.0

10.0

0.918 0.588 0.464

0.596

-0.120 -0.155 0.464 -0.281

-0.421 0.243

-0.120

=n

0.951

0.179

‘a

-0.155

-0.594

0.828 -0.275 -0.237 -0.122 0.140 0.162 0.237

0.601 -0.589 0.486

0.971 -0.159 0.166

0.972 -0.166 0.166

0.971 -0.159

0.972 -0.166

-0.163

-0.166

0.971 -0.154 0.171

0.972 -0.163 0.167

0.970 -0.158

0.972 -0.166

-0.164

-0.166

0.792 OS61

0.862 0.494

4-T-t

0.673

0.673

0.694

0.966 -0.104 0.194 0.681

0.681

References Cl1 B.O. Rws,

[4] A.C. Wahl and G. 3as,

in: Methods of electronic structure theory, ed. H.F. Schaefer (Plenum Press, New York, 1977).

P.R. Taylor and P.E.M. Siegbahn,

Chem.

Phjs. 48 (1980) 157. [2] P.E.M. Siegbahn, 3. Almlbf. A. Heiberg and B.O. Roes, J. Chem. Phys.. to be published. [3j P. Siegbahn. A. He&erg, B. Roes and B. Levy, Phys. Scripta 21 (1980) 323.

[51 P. Botschwina, Chem. Phys. 28 (1978) 231. [6J M. Yoshimine, J. Chem. Phys. 57 (1972) 1108. [7] K.P. Huber and G. Henberg, Constants of diatomic molecules (Van Nostrand, Princeton, 1979). [81 L. Brewer and G.M. Rosenblatt. Advan. High Temp. Chem. 2 (1969) 1.

[9] I. Paldus, in: fheorerid

[IO]

[Ill [12] [13] [14] [lj]

chemistry: advances and perspectives, Vol. 2. eds. H. Eqting and D. Henderson (Academic Press. sew York. 1976). I. Shavit!. Intern. J. Quantum Chem. S12 (1978) 5. P.E.M. Siegbahn. J. Chem. Phys. 70 (1979) 5391. P.E.?.I. Siegbahn, J. Chem. Php. 72 (1980) 1647. S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. TX. Dunning. J. Chem. Phys. 53 (1470) 2823. T.H. Dunning and P.J. Hay, in: Methods of electronic structure iheory, ed. H.F. Schaefer (Plenum Press, New York, 1977).

[16] MM-L. Chen and H.F. Schaefer. J. Chem. Phys. 72 (1980) 4376. [17J J. AImI%, USIP report 74-29. University of Stockholm (1974). [IS] A.C. Hurley, Introduction to the eIectroil theory of small molecules (Academic Press, New York, 1976). [19] D.L. Yeager and P. Jorgensen, Mol. Phys. 39 (1980) 587. [2O] J. AlmlBf. A.V. Nemukhin and A. H&berg, Intern. J. Quantum Chem.. to be published.