The structure of Li7− and K7−

Chemical Physics ELSEVIER

Chemical Physics 206 (1996) 35-42

The structure of LiT and K7 C h a r l e s W. B a u s c h l i c h e r Jr. STC.230-3, NASA Ames Research Center, Moffett Field, CA 94035, USA

Received 11 September 1995

Abstract

The self-consistent-field (SCF) approach and density functional theory, using the B3LYP hybrid functional, yield three |ow-:ying structures for Li~'. The relative separations differ for the SCF and B3LYP approaches, however the B3LYP results are in good agreement with the coupled cluster results. For K~', only an octahedron with one face capped is found to be a minimum; this is the second most stable ,,~oJcture for Li~'. A comparison of the computed separations between the low-lying states of K7 and the photoelectron detachment spectra does not allow an unambiguous assignment of the structure of K~-.

1. Introduction

The geometry of a metal cluster is usually unrelated to the structure of the bulk metal. In general, it is not possible to determine the cluster geometry experimentally. Ab initio calculations have been very successful at determining the structure of many molecules. However, determining the optimal geometry of small metal clusters can be a very challenging task. There may be many structures with similar energy so that high levels of theory have to be used. Because analytic second derivatives, which allow one to determine if the optimized structure is a minimum and to compute the zero-point energy, are practical for only relatively low levels of theory, the geometry optimization and calculation of relative stabilities are commonly done at different levels of theory. For large clusters the problem becomes even more difficult as it might not be possible to perform calculations to compute the relative stabilities with sufficient accuracy to definitively determine the ground state from between the nearly degenerate structures. While a routine experimental determination of

structure is not possible, experimental studies of small clusters have yielded a wealth of information, such as ionization potential (IP), electron affinity (EA), absorption spectra, reactivity, .... These properties can be very different for the different isomers. Thus combining ab initio calculations with experimental results can be used to determine the ground state from among the low-lying structures; see for example the study of Cu~ by Siegbahn and co-workers [1] or the recent work by Bona~i6-Kouteck3;', Kouteck~, and co-workers [ 2-4] on Li and Na clusters. McHugh et al. [ 5 ] measured the photoelectron detachment spectra of K 7. In this technique they measure the number of detached electrons as a function of their kinetic energy. Because the detachment is a very rapid process, the geometry of the I(7 product is expected to be that of K 7. The EA can be deduced from the difference between the highest kinetic energy electrons and the photon energy. In addition to highest energy peak associated with the ground state of KT, they observed several peaks with lower electron kinetic energy, which can be interpreted as detachment to form low-lying excited states of K7. Thus this ex-

0301-0104/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved PII S030 !-0104(96)00019-5

36

C.W. Bauschlicher Jr.~Chemical Physics 206 (1996) 35-42

periment yields the separation between the ground and low-lying states of K7 at the K 7 geometry. In this work we study the geometry of Li 7 and K 7 . For Li 7 we compare the results obtained at different levels of theory. Li 7 is chosen because it is easier to treat than K 7 and there are previous theoretical results for comparison. We study the low-lying states of K7 at the K 7 geometry for comparison with the photoelectron detachment experiments.

2. Methods Three Li basis sets are used in this work. The first basis set is the 6-31 +G* set of Pople and coworkers [ 6]. The second Li set is the (9s) / [4s] set of Dunning [7], with a diffuse s (0.01) added to describe L i - . The ( 4 p ) / [ 2 p ] polarization set of Dunning and Hay [8] is also added. Relative to the second set, the third Li basis set uncontracts the p functions and adds a d function with an exponent of 0.3. Three basis sets are used for K. The all-electron basis set is derived from the ( 14s 9p) / [ 8s 5p] double zeta set of Schafer, Hora. and Ahlrichs [9]. A diffuse s (0.007) and two diffuse p (0.04 and 0.017) functions are added. This basis set is denoted AE. The second K basis set adds a 3d (0.01) polarization function to the first, which is denoted AE+d. Other basis set modifications, such as adding a second 3d function (0.03) or an additional diffuse s and p function, as well as, adding an s function to fill in the gap between the 3s and 4s orbitals and a p function to fill in between the 3p and 4p orbitals, have so little effect on the results that they were not used in any of the production calculations. The third K basis set uses the relativistic effective core potential (RECP) of Hay and Wadt [ 10], which includes the 3s and 3p orbitals in the valence space. Their orbital exponents are contracted (5s 5p) / [4s 3p]. A general contraction is used for the s space; the two outermost s functions are uncontracted. A diffuse s function (0.007) and a 3d function (0.01) are also added. The geometries are optimized at the self-consistentfield (SCF) or density functional theory (DFT) levels using analytic first and second derivatives. The second derivatives are used to confirm that the structures correspond to minima. The B3LYP-hybrid functional [ 11,12] is used in the DFT calculations. More

extensive correlation is added using the coupled cluster singles and doubles approach with a perturbationai estimate of the triples [ 13] [CCSD(T) ]. In these calculations only the eight s valence electrons are correlated. The calculations on K7, which are performed to compare with the photoelectron detachment spectra, use the state-averaged complete-active-space SCF (SA-CASSCF) approach. The K I s-3p-like orbitals are frozen in their form from an SCF calculation on K 7 due to technical limitations. (In the RECP calculations only the 3s and 3p-like orbitals are frozen.) More extensive correlation is included using the multireference configuration interaction approach (MRCI). Internal contraction [ 14] (IC) is used keep the MRCI expansions tractable. The effect of higher excitations is estimated using the multi-reference analog of the Davidson correction (denoted + Q ) . Only the seven 4s electrons are included in the correlation treatment. The choice of the active space is discussed below. The SCF geometry optimizations and vibrational frequencies were determined using Gaussian 92/DFT [ 15] or GRADSCF [ 16]. The B3LYP calculations were performed using Gaussian 92/DFT. The CCSD(T) calculations were performed using TITAN [17] interfaced into the MOLECULESWEDEN [18] program or using Molpro 94 [94]. The CASSCF/ICMRCI calculations were performed using Molpro 94 [ 19].

3. Qualitative considerations The structure of LiT is established to be a pentagonal bipyramid (Dsh symmetry) from ESR experiments [20]. The same structure is found to be the most stable in the SCF/CI calculations of Bona~i6Kouteck~, Kouteck~, and co-workers [ 21 ]. It should be noted that at the SCF level the D5h structure is not the most stable; it is only with electron correlation that it becomes the ground state. At the SCF/CI level, Boustani and Kouteck~ [22] found the lowest structure for Li 7 to be pentagonal bipyramid. As in the case of the neutral, the inclusion of correlation is required to bring the Dsh structure below two other low-lying structures. It is interesting to note that at the SCF level the D5h structure has imaginazy frequencies (for the

C. W. Bauschlicher Jr./Chemical Physics 206 (1996)35-42

Fig. I. The optimal SCF structure for the pentagonal bipyramid (Dsh symmetry) of Li7 , which is denoted Dsh in this work.

37

d

Fig. 3. The optimal SCF structure for the octahedron with one face capped (C3v symmetry) of Li7 , which is denoted Oh+ I in this work.

Fig. 2. The optimal SCF structure for the tetrahedron with three laces capped (C3v symmetry) of Li 7 , which is depoted Td+3 in this work.

[3s Ip] basis set used by Boustani and Kouteck)~). Thus the structure that is the most stable at the CI level is not even a stationary point at the SCF level. In a recent review article, Bona~,i6-Kouteck~, Fantucci, and Kouteck~ [ 23 ] state that the most stable structure is not Dsh, but a tetrahedron with three faces capped, which we denote as Td+3. The most sta~le K7 structure, at both the SCF and fourth-order perturbation theory levels is the pentagonal bipyramid [24]. One way of thinking about possible structures for X~- is to start from X6 and add an atom or start from

X8 and remove one. For Li6 there are three low-lying structures. The most stable structure at the SCF/CI level is a planar system with D3h symmetry [2]. However from a comparison with the experimental absorption spectra, it is clear that the ground state corresponds to a tetrahedron with two faces capped [2]. The most stable LiB cluster at the SCF/CI level is a tetrahedron with a capping atom on each face. The computed absorption spectra is in good agreement with that found experimentally. Unfortunately, the computed absorption spectra for the second most stable LiB cluster also agrees reasonably well with experiment. This second LiB cluster has D2d symmetry and looks like a distorted octahedron with two adjacent faces capped. Two candidates for the ground state of K 7 are the pentagonal bipyramid or Td+3 structure, which previous calculations [22,23] have been found to be the most stable structure for Li 7. This Td+3 structure would arise from the Li6 by capping an additional face or by removing one capping atom from LiB. Another candidate can be derived from the LiB D2d structure by removing one of the capping atoms, which leads an octahedron with one face capped, denoted Oh+ 1. These three structures are those found to be low-lying by Boustani and Kouteck~ [22] in their study of Li 7. These three structures are illustrated in Figs. 1-3. Starting from the pla,~ar form of Li6, which

38

C. W. Bauschlicher Jr.~Chemical Physics 206 (1996) 35-42

Fig. 4. The optimal SCF structure for the planar structure of Li 7 .

is found to be the most stable in previous SCF/CI calculations [ 2 ], another possible structure for Li 7 arises by capping an edge - see Fig. 4.

4. Results and discussion 4.1. Li 7

We first consider Li and Li2. The CCSD(T) calculations are performed using the [5s4p ld] basis set, while the B3LYP calculations use the 6-31+G* basis set. As expected, the electron affinity (EA) of Li computed at the CCSD level is 0.62 eV, which is in excellent agreement with the experimental value [25] of 0.618 eV. The B3LYP value of 0.55 eV is slightly too small. The B3LYP results for Li2 are in good agreement with experiment [ 26] (in parentheses)" re 2.726(2.673) ,/k, tOe = 341(351) cm - i , and De -0.88(1.04) eV. The computed EA of Li2 is 0.43 eV, which is in good agreement with our previously computed value [ 27] of 0.43-t-0.02 eV. Thus the results for Li and Li2 support the accuracy of the B3LYP approach for this problem. We optimize the geometry of Li7 using the [ 4s 2p ] basis set at the SCF level starting from the four geometries described in the previous section. All four correspond to local minimum and are close in energy at the SCF level - see Table 1. This differs from Boustani and Kouteck~ [22], who did not find the Dsh structure to correspond to a minimum. We attribute the difference to the larger basis set used in this work. Improving the basis set does not significantly change the separations at the SCF level. The relative ordering of the structures changes dramatically with the addi-

tion of correlation. At the CCSD(T) level, the O5h structure is the most stable and the planar structure can clearly be ruled out as a candidate for the most stable isomer. The difference in the stability of the clusters at the SCF and correlated levels indicates that it is desirable to be able to optimize the cluster geometry at the correlated level. The B3LYP approach is a practical method of performing such calculations. The relative separations at the B3LYP level of theory are in much better agreement with the CCSD(T) results than the SCF - see Table 1. The zero-point energies are very similar for all four structures, such that the relative separations between the lowest three are unchanged when it is added. The CCSD(T) calculations performed using the B3LYP geometries yield lower total energies than those performed using the SCF geometries, however the relative separations are only slightly changed. Despite the small change in the CCSD(T) relative separations, there are nontrivial changes in the B3LYP geometries relative to those obtained at the SCF level. We consider the D5h cluster geometry, because there are only two independent parameters and therefore it is practical to optimize this cluster at the CCSD(T) level of theory. From the center of the cluster, the two parameters are: ( l ) the distance to the capping atoms and (2) the distance to the five in-plane atoms. At the SCF level using the [Ss 2p] basis set these two values are 3.60 and 4.92 a0, respectively. The B3LYP results are significantly shorter, being 3.23 and 4.66 a0. The CCSD(T) results, obtained using the [5s4p ld] basis set, are very similar to those obtained using the B3LYP, namely 3.27 and 4.68 a0. Optimizing the geometry at the CCSD(T) level stabilizes the cluster by 0.105 eV, relative to using the SCF geometry. The CCSD(T) energy at the B3LYP geometry is only 0.001 eV above that using the optimal CCSD(T) geometry. On the basis of this calibration, we conclude that the B3LYP geometries are significantly better than those obtained at the SCF level. The atomization energy at the CCSD(T) level, using the SCF geometry, to 6 L i + L i - is 6.35 eV for the D5h cluster. The B3LYP result of 5.87 eV is somewhat smaller, but in reasonable agreement with the CCSD(T) result. It is also much better than the SCF result of 2.55 eV, obtained using the [5s4p ld] basis. Our CCSD(T) atomization energy is about 0.7 eV

CW. Bauschlicher Jr./Chemical Physics 206 (1996) 35-42

39

Table I Relative energetics (in eV) of the Li7 and K7 clusters. The CCSD(T) calculations are performed at both the SCF and B3LYP geometries 15s2pi

15s4p ldl

Li 7

SCF

SCF a

CCSD(T) SCF a

CCSD(T) B3LYP c

B3LYP b

Dsh Td +3 capping Oh + I capping planar

0. ! ! 0.00 0.04 0. ! 3

0.1 ! 0.00 0.05 0.15

0.00 0. I 0 0.05 0.69

0.00 0.08 0.03

0.00(0.00) 0.04(0.04) 0.02 (0.02) 0.56( 0.53 )

K7

SCF

B3LYP

AE

AE

AE+d

RECP

AE

0.00 0.02 0.05 e

0.00 0.03 e 0.04 e

0.00 0.03 e 0.03

0.00 0.06 0.03

0.00 0.06

Oh+ I capping Td+3 capping Dsh 0.09 e a b c d c

The The The The Has

6-31+G*

CCSD(T) d

SCF 15s2pl geometry is used. results including zero-point energy are given in parentheses. B3LYP/6-3 i +G* geometry is used. B3LYP/AE geometry is used. two imaginary frequencies.

larger than the CI+Q value reported by Boustani and Kouteck2L Even our B3LYP result is larger than the previous CI+Q result.

4.2. K 7

The EA of K at the B3LYP level of theory, using the AE basis set, is 0.52 eV, which is in good agreement with the experimental value [25] of 0.50 eV. The results for I(2 are also in good agreement with experiment [5,26] (in parentheses)" re 3.966(3.905) A,, We 89(92) cm - l , De 0.53(0.52) eV, and EA 0.48(0.493) eV. These results support the use of the B3LYP approach for K 7. On the basis of the Li 7 results, we exclude the planar structure and consider only the three 3-D structures. While the relative separations differ slightly, the SCF and B3LYP levels of theory yield the same order for the structures; the Oh+ 1 structure is the most stable (see Table 1 ), and corresponds to a true minimum at both levels of theory. The Dsh structure has two imaginary frequencies, and if displaced along one of these two modes, the B3LYP optimization yields the On+l structure. At the B3LYP level, the Td+3 structure, optimized with C3v symmetry, has two imagi-

nary frequencies. However, the potential is very flat in this region. If the Td+3 structure is only slightly displaced from equilibrium, the geometry of the structure with Cs symmetry is found to be converged to within the default thresholds in GAUSSIAN 92/DFT. If the threshold for the geometry optimization is tightened (to that termed "tight" in Gaussian) or the size of the displacement increased, this structure collapses to the Oh + 1 structure. At the SCF level of theory the Td +3 structure is a minimum, but two lowest modes have a frequency of only 0.6 cm-l. The B3LYP calculations were repeated using the AE+d basis set and only Oh+l structure was found to correspond to a minimum. Adding the d functions has very little effect on the geometry or even the atomization energy. For the Td+3 structure, other basis sets were tried, and all yielded imaginary frequencies at the B3LYP level. Thus the difference between K 7 and Li 7 does not appear to be an artifact of the K basis set used. We also note that unlike Li 7, the relative order of the K7 structures is the same at the SCF and B3LYP levels. The atomization energy of K 7 (2.93 eV) is only about half that found for Li 7. This is consistent with the results for the diatomics, where the binding energy of Li2 is about twice that of K2.

40

C W. Bauschlicher Jr./Chemical Physics 206 (1996) 35-42

The Oh+l structure is also the most stable at the CCSD(T) level of theory using either the RECP or AE basis sets. The CCSD(T) Oh+l-To+3 separation is larger than found at the B3LYP level, while the CCSD(T) Oh+l-Dsh separation is smaller. This leads to an inversion in the order of the Td+3 and Dsh structures. We next compute the separation between the lowlying states of K7 using the SA-CASSCF/ICMRCI approach in conjunction with the RECP basis set. For comparison, we include, in Table 2, the experimental separations deduced from the spectra of McHugh et al. [5]. The Oh+l is the first structure that we consider. Both the K 7 B3LYP and SCF geometries are used. While the system has C3v symmetry, the calculations are performed in Cs symmetry. The active space consists of the seven 4s-like orbita!s, 5a' and 2a". On the basis of preliminary calculations, four 2A~ and three 2A" states were included in the SA-CASSCF calculation. All of the configurations in the CASSCF calculation were used as references in the subsequent ICMRCI calculations. The results of these calculations are summarized in Table 2. We first note that the results are similar for the SCF and B3LYP geometries. Increasing the active space to 7 a ~and 3 a" orbitals and averaging for five 2At and four 2A" states, which is labelled big active space in Table 2, reduces the size ef the +Q correction, but does not significantly improve the agreement with experiment. Namely, there is no obvious candidate for the state at 0.34 eV. Changing the basis set from the RECP to the AE basis set tends to increase the separations very slightly, but still provides no good candidate for the state at 0.34 eV. Since a correction of only 0.2 eV would bring the computed results into agreement with experiment, one cannot exclude the possibility that an improved treatment would yield separations in agreement with experiment. The results for the Td+3 structure agree somewhat better experiment than those obtained for the Oh+l cluster, especially for the SCF geometry. However, it is not possible to definitively assign the experimental results as arising from the Td+3 cluster. The results for the Dsh structure show a much larger effect of adding correlation. Therefore, the active space was increase by four orbitals, one in each irreducible representation of the C2v point group, in which the calculations are performed. This reduces the differential effect of electron correlation and of

the Davidson correction. However, as for the other clusters there is little correspondence between the computed separations and those deduced from experiment. The disagreement between all three clusters and experiment is disconcerting. This seems to imply that we have not found the most stable structure. However, attempts to find other low-lying structures were unsuccessful. The SCF and B3LYP results are similar, suggesting that higher levels of electron correlation would not yield dramatically different geometries, and hence not yield very different separations for the K7 excited states. The B3LYP geometries are insensitive to further basis set enhancements, and the K7 separations change only slightly between the RECP and AE basis sets. With only 7 electrons correlated, it seems unlikely that we have dramatic errors in our SA-CASSCF/ICMRCI treatment of KT, which is supported by the larger active space calculations. From the large number of very low frequency vibrational modes, the cluster geometry should by quite fluxional. Thus the average geometry sampled in the photodetachment experiment may not correspond to the equilibrium geometry determined in the calculations. In addition to the choice of geometry, it is possible that all of our errors, such as neglect of zero-point energy, neglect of K inner-shell correlation, limited size of the active space, and finite basis set all work in the same direction, so that the maximum error in the SA-CASSCF/ICMRCI treatment is 0.2 eV, such that we are unable to confirm that the Oh+l structure is the true minimum.

5. Conclusions We have studied Li 7 and K 7. For Li 7 we find three low-lying structures, corresponding to a pentagonal bipyramid, a tetrahedron with three faces capped, and an octahedron with one face capped. The relative separations at the SCF level disagree with those obtained at the B3LYP or CCSD(T) levels. The B3LYP separations and pentagonal bipyramid geometry are in good agreement with the results obtained at the CCSD(T) level. For K 7, only the capped octahedron is a minimum at the B3LYP level of theory. Unfortunately, we are unable to confirm the geometry of K7 using the experimental photoelectron detachment spectra. In spite of this somewhat negative conclusion, we note

C W. Bauschlicher Jr.~Chemical Physics 206 (1996) 35-42

41

Table 2 Relative energetics (in eV) of the low-lying states of KT, at the K7 geometry. For comparison the relative separations that we deduce from the experimental spectra 151 are: 0.00, 0.19, 0.34, 0.77, and !.02 eV Oi~ + I B3LYP geometry

SCF geometry

SA-CASSCF

ICMRC!

ICMRCI+Q

SA-CASSCF

ICMRC!

ICMRCI+Q

0.00 0.00 0.26 0.54 0.52 0.78 0.62

0.00 0.01 0.26 0.66 0.76 0.92 0.83

0.00 0.02 0.22 0.63 0.85 0.87 0.90

0.03 0.00 0.26 0.50 0.45 0.73 0.58

0.00 0.01 0.25 0.58 0.69 0.86 0.77

0.00 0.03 0.22 0.57 0.8 I 0.83 0.85

B3LYP geometry AE basis set

B3LYP geometry big active space SA-CASSCF

ICMRC!

ICMRCI+Q

SA-CASSCF

ICMRCI

ICMRCI+Q

0.00 0.00 0.25 0.50 0.73 0.87 0.87 0.99

0.00 0.00 0.22 0.60 0.76 0.83 0.83 0.91

0.00 0.00 0.20 0.61 0.74 0.77 0.78 0.82

0.00 0.00 0.27 0.55 0.54 0.80 0.64

0.00 0.01 0.26 0.67 0.78 0.93 0.84

0.00 0.02 0.22 0.64 0.86 0.87 0.90

Tj+3 SCF geometry

B3LYP geometry SA-CASSCF

ICMRC!

ICMRCI+Q

SA-CASSCF

ICMRC!

ICMRCI+Q

0.00 0.09 0.12 0.54 0.73 0.68 0.81

0.00 0.13 0.16 0.67 0.92 0.95 1.04

0.00 0.15 0.19 0.62 0.92 1.07 1.08

0.00 0.22 0.22 0.53 0.66 0.61 0.72

0.00 0.25 0.29 0.67 0.85 0.83 0.94

0.00 0.25 0.31 0.64 0.84 0.90 0.95

Dsh B3LYP geometry large active space

B3LYP geometry small active space SA-CASSCF

ICMRCI

ICMRCI+Q

SA-CASSCF

ICMRCI

ICMRCI+Q

0.00 0.31 0.~ I 1.44 0.66 1.01 1.03

0.00 0.29 0.3 i ! .09 0.74 0.94 ! .03

0.00 0.25 0.28 0.62 0.69 0.73 0.88

0.00 0.33 0.33 0.93 0.66 0.93 0.98

0.00 0.28 0.29 0.88 0.7 i 0.83 0.90

0.00 0.25 0.27 0.66 0.69 0.72 0.84

42

C. W. Bauschlicher Jr./Chemical Physics 206 (1996) 35-42

that the B 3 L Y P results for Li2, Li 2 , K2, and K 2 are in good a g r e e m e n t with experiment, and the results for Li 7 are in m u c h better a g r e e m e n t with the C C S D ( T ) than are the S C F results. T h u s the B3LYP is a m u c h better starting p o i n t for the s t u d y o f these clusters than is the S C E Clearly additional w o r k to establish the structure o f K 7 and explain the photoelectron spectra is desirable.

References

Iil H. Akeby, 1. Panas, L.G.M. Pettersson, P. Siegbahn and U. Wahlgren, J. Phys. Chem. 94 (1990) 5471. 121 J. Blanc, V. Bona~i6-Kouteck~, M. Broyer, J. Chevaleyre, Ph. Dugourd, J. Kouteck~, S. Scheuch, J.P. Wolf and L. W0ste, J. Chem. Phys. 96 (1992) 1793. 131 V. Bona~i6-Kouteck~, P. Fantucci and J. Kouteck~, J. Chem. Phys. 93 (1990) 3802. 141 V. Bona~i6-Kouteck)~,J. Pittner, C. Scheuch, M.E Guest and J. Kouteck~, J. Chem. Phys. 96 (1992) 7938. 151 K.M. HcHugh, J.G. Eaton, G.H. Lee, H.W. Sarkas, L.H. Kidder, J.T. Snodgrass, M.R. Manaa and K.H. Bowen, J. Chem. Phys. 91 (1989) 3792. 161 M.J. Frisch, J.A. Pople and J.S. Binkley, J. Chem. Phys. 80 (1984) 3265. 171 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. 181 T.H. Dunni,g and P.J. Hay, in: Methods of electronic structure theory, ed. H.E Schaefer (Plenum Press, New York, 1977) pp. 1-27. 191 A. SchMer, H. Horn and R. Ahlrichs, J. Chem. Phys. 97 (1992) 257 I. P.J. Hay and W.R. Wadt, J. Chem. Phys. 82 (1985) 299. llOl I t l l EJ. Stevens, EJ. Devlin, C.E Chablowski and M.J. Frisch, J. Phys. Chem. 98 (1994) 11623. 1121 A.D. Becke, J. Chem. Phys. 98 (1993) 5648, and references therein.

[13] K. Raghavachaxi, G.W. Trucks, J.A. Pople and M. HeadGordon, Chem. Phys. Letters ! 57 (1989) 479, and references therein. !141 H.-J. Wemer and P.J. Knowles, J. Chem. Phys. 89 (1988) 5803. 1151 M.J. Frisch, G.W. Trucks, l-i.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.W. Wong, J.B. Foresman, M.A. Robb, M. Head-Gordon, E.S. Replogle, R. Gomperts, J.L. Andres, K. Raghavachari, J.S. Binkley, C. Gonzalez, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart and J.A. Pople, GAUSSIAN 92/DFr, Revision G.2 (Gaussian, Inc., Pittsburgh PA, 1993). [161 GRADSCF is a vectorized SCF first- and second-derivative code written by A. Komomicki and H. King. [171 TITAN, a set of electronic structure programs written by T.J. Lee, A.P. Rendell, and J.E. Rice. I!81 MOLECULE-SWEDEN is an electlonic structure program written by J. Alml6f, C.W. Bauschlicher, M.R.A. Blomberg, D.P. Chong, A. Heiberg, S.R. Langhoff, E-A. Malmqvist, A.P. Rendell, B.O. Roos, P.E.M. Siegbahn and P.R. Taylor. 1191 MOLPRO 94 is a package of ab initio programs written by H.-J. Werner and P.J. Knowles, with contributions from J. Alml6f, R.D. Amos, M.J.O. Deegan, S.T. Elbert, C. Hampei, W. Meyer, K. Peterson, R. Pitzer, A.J. Stone and P.R. Taylor. [20l D.A. Garland and D.M. Lindsay, J. Chem. Phys. 80 (1984) 4761. [211 I. Boustani, W. Pewestort, P. Fantucci, V. Bona~i~-Kouteck~ and J. Kouteck~, Phys. Rev. B 35 (1987) 9437. [221 I. Boustani and J. Kouteck~, J. Chem. Phys. 88 (1988) 5657. 1231 V. Bona~i~-Kouteck~, P. Fantucci and J. Kouteck2~, Chem. Rev. 91 (1991) 1035. [241 A.K. Ray and S.D. Altekar, Phys. Rev. B. 42 (1990) 1444. 1251 H. Hotop and W.C. Lineberger, J. Phys. Chem. Ref. Data 14 (1985) 731. [261 K.P. Huber and G. Herzberg, Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [27] H. Partridge, D.A. Dixon, S.P. Walch, C.W. Bauschlicher and J.L. Gole, J. Chem. Phys. 79 (1983) 1859.