J. Phys. Chem. 1995,99, 8562-8566
8562
A Non-Atom-Based Orbital Basis Set for Small Sodium Clusters Melissa R. Prince and Randall W. Hall* Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803-1804 Received: November 9, 1994; In Final Form: March 8, 1995@
The electronic structure of small sodium clusters is investigated using a basis set motivated by path integral Monte Carlo calculations. The goal of this study is to develop a small basis set that can be used to study large sodium clusters. Past and present path integral calculations suggest that the correlated electronic wave function is characterized by electron density located between several atoms, rather than on the atoms. This is in agreement with previous quantum chemical and density functional calculations, which find the total electron density has maxima in the same locations. A basis set used in previous quantum chemical calculations is modified by moving valence orbitals from atoms to locations between atoms to create a non-atom-based basis set. This results in a smaller basis set, as there are fewer non-atom-based sites than atoms; this has the advantage of allowing the study of larger clusters. The vertical and adiabatic ionization potentials are evaluated using both the old and the new basis sets and are compared with experimental results. In addition, the electronic absorption spectrum is calculated and compared with both theoretical and experimental values. It is found that the non-atom-based basis set gives agreement with experimental values as well or better than the relatively larger atom-based basis set, indicating that the new basis set provides the opportunity to study larger clusters.
I. Introduction
Alkali metal clusters have attracted attention from both experimentalists and theorists for a variety of fundamental reasons.’ Since an insulator to metal transition will occur as the cluster size is increased, the study of properties related to size can lead to insights into design and control of new materials. The interplay between electronic and geometric structure can also be examined, particularly since these clusters display bonding pattems uncharacteristic of covalently bonded materials. Finally, the evolution of the electronic properties can aid our understanding of the Mott transition and related phenomena. Previous experimental and theoretical work has led to a general understanding of the electronic structure of small clusters and to models for the electronic structure of larger clusters that give reasonable agreement with experiment. It is found that the ionization potentials, excitation spectra, and photoelectron detachment spectra can be predicted qualitatively using a model in which the electrons are treated as being bound to the cluster, rather than individual atoms, suggesting that the electronic structure of even small clusters is similar to the bulk metal. In fact, good agreement with experiment can be obtained by simply modeling the electrons as noninteracting particles localized in a 3-D harmonic well. All the simple models lead to a description of the electronic structure using hydrogen-like atomic orbitals (s, p, d, etc., often called “shell orbitals”). Detailed quantum chemical, density functional, and path integral calculations have been performed on small clusters in order to investigate the relative simplicity of the electronic structure.’ Of particular interest is the type and location of the shell orbitals and their evolution as a function of cluster size. Previous quantum chemical and density functional calculations have focused primarily on the geometry and electronic properties of these clusters and less on the type and location of the shell Previous approximate path integral calculations on N a - N a have suggested that the correlated electronic structure is typified by electron density located in the centers of triangles
* To whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, April 15, 1995.
formed by triplets of atoms.3 This result is not in disagreement with quantum chemical or density functional results which, while examining the correlated one-electron density or the molecular orbitals, find an accumulation of electron density at the same locations as the path integral calculations. It is therefore of great interest to follow the evolution of the electronic structure by performing accurate calculations on larger clusters. Studies of larger clusters are hindered by both the number of basis functions required and the need to investigate the relative stabilities of different isomers of a given cluster size. Since the calculation time is typically proportional to N4 or higher; a reduction in the number of basis functions by a factor of 2 decreases the time required to perform a calculation by a factor of 16 or, more imoortantlv, increases the size of cluster that can be studied by a-factor of 2. The purpose of this paper is to investigate the possibility of using relatively small, non-atombased basis sets to study sodium clusters. In this work, we extend our path integral studies to non-planar geometries by studying Nag, noting the locations of the electrons. On the basis of these studies, we postulate a new basis set for studying alkali clusters that uses non-atom-centered orbitals to represent the locations of the electrons in the path integral calculations. We compare the calculations using this new basis with results from previous quantum chemical ~ t u d i e s . ~The -~ comparison suggests that non-atom-based orbitals can indeed be used to describe the electronic structure and can be used to study larger clusters. The advantage of using a smaller basis set that still gives reliable agreement with experiment is the ability to study larger clusters. 11. Path Integral Calculations Our previous calculations3 suggest that the dominant correlated electronic configurations in the small alkali clusters consist of electrons centered in triangles, as shown in Figure 1. To see if this picture persists in non-planar clusters, we studied Nag at the geometry of Bonacic-Koutecky et aL7 We used our exact implementation of Feynman’s path integral formulation of quantum mechanics, as described in a previous paper.’ For Nag, we performed calculations with p = l / k ~ T = 250 au, P = 512, and k,,,/P = 0.95, 0.90, and 0.85 (for explanations of
0022-3654/95/2099-8562$09.00/0 0 1995 American Chemical Society
Orbital Basis Set for Small Sodium Clusters
J. Phys. Chem., Vol. 99, No. 21, 1995 8563 TABLE 1: QCISD(T) Energies (au) for Nan Optimized Neutral and Cationic Geometries with Different Basis Sets (Nvaland N,,, Are the Number of Primitive Valence Orbitals and &e Number of NAB Groups, Respectively)
n 4 5 6
8
Figure 1. Dominant valence electron configurations from path integral calculations.
symbols, see ref 8). Both the energy and the electronic structure had converged by this value of k,,,,,lP. The geometry of the Nag cluster, as shown in Figure 1, consists of five tetrahedra, suggesting that the centers of these should be taken as the locations of possible shell orbitals. For each of our three simulations, We found the dominant electronic configurations by assigning each of the eight electrons from our path integral Monte Carlo calculations to one of the five sites and determining the fraction of the electronic configurations with a specific occupancy in a manner similar to our previous work.3 The dominant electronic configuration is one with spin-paired electrons in the outer four tetrahedra, as one would expect from simple electrostatic considerations. Thus, we are led to suggest that the shell orbitals are constructed from a basis set consisting of non-atom-based orbitals located at the centers of simple geometric structures (triangles in 2-D and tetrahedra in 3-D). The next part of this paper takes this suggestion literally and uses such a basis set for precise quantum chemical calculations.
4 5 6 8
QCISD(T) NAB basis
Nval QCISD(T) (Ngroup) B2 basis
N,,,
-647.4389 -809.3027 -971.1779 -971.1820 -1294.9266 - 1294.9325
Neutral 28 (2) -647.4342 41 (3) -809.2991 42(3) -971.1737 54 (4) 56(4) -1294.9185 68 (5)
-647.2914 -809.1690 -971.0308 -1294.7838
Cationic 28 (2) -647.2865 29(2) -809.1634 42(3) -971.0233 56 (4) -1294.7725
energy difference QCISD(T) (kcaumol) 36 45 54 72
36 45 54 72
2.89 2.26 2.64 5.21 5.08 8.79 3.07 3.51 4.71 7.09
TABLE 2: Vertical and Adiabatic Ionization Potentials (eV) for Naa. Nas. Nar, and Nau vertical IP vertical IP adiabatic IP adiabatic IP experimental NAB basis B2 basis NAB basis B2 basis IP
Na Nas N% Nag
4.07 3.87 4.29 4.13
4.08 3.98 4.32 4.18
4.01 3.64 4.00 3.89
4.02 3.69 4.09 3.97
4.24 3.99 4.23 4.22
111. Quantum Chemistry Studies
We now describe studies using a basis set with non-atombased orbitals located at sites suggested by the path integral calculation^.^ We choose to study Na4, Na5, N&, and Na8 for which several previous quantum chemical studies are available' (in addition to existing path integral results). In this study, we have used the most accurate possible implementations of standard quantum chemistry packages and performed all-electron calculations. Two basis sets were employed in our calculations, both were adapted from the basis set B (13s8pld) of Bonacic-Koutecky et aL7 This basis set will be referred to as the B1 basis set. Inspection of the energies obtained with this basis set suggested that the d orbitals were necessary only for quantitative comparison with experiment, so we have removed the d orbitals from the B1 basis set in order to speed up our calculations. The B1 basis set without the d orbitals will be called the B2 basis set. To assess the physical significance of non-atom-based orbitals, we modified the B2 basis set by moving some valence orbitals from atoms to non-atom-based locations. Specifically, the valence s and p orbitals (s(0.024617), s(O.OlO), p(0.060), and ~(0.020))were removed from the atoms and used as nonatom-based orbitals. A collection of orbitals consisting of the set of valence s and p orbitals removed from an atom plus an additional diffuse s and p orbital are placed at a single site in space and will be referred to as a NAB group. NAB groups
Figure 2. Geometries of the Naa, Na5, N a , and Nag neutral clusters optimized at the SCF level using the B2 basis set. The NAB groups are denoted by x . Distances are given in angstroms.
were placed at different spatial locations in our calculations, typically located at sites suggested by path integral calculations. One valence s orbital, s(0.058065), was left on each atom to account for electron density remaining on the atom. The resulting molecular orbitals have minor contributions from this orbital. For example, the largest contribution from any atombased s orbital to the two highest occupied molecular orbitals in NQ is 7.0%. We will refer to this new basis set as the nonatom-based (NAB) basis set. The exponents on the extra s and p orbitals in the NAB basis were optimized using GAMES9 for the sodium tetramer and pentamer as described below and were found to be ~(0.065)and ~(0.040).The NAB basis set is typically smaller than the B2 basis set as seen in Table 1. To optimize the exponents and geometries of the neutral fourand five-atom clusters, we used the following procedure. Geometries were optimized at the SCF level using the B2 basis set with GAMESS. The optimizations were done using the B2 basis set; test calculations on the smaller clusters using the NAB
Prince and Hall
8564 J. Phys. Chem., Vol. 99, No. 21, 1995
m @ -1.41
4.88
Na (4.05)
4.81
(-0.06) -1.18
@==3.43 &A
Na
4.97
(4.23)
Figure 3. Geometries of the NQ, Na5, N%, and Nag cationic clusters optimized at the SCF level using the B2 basis set. The NAB groups are denoted by x . Distances are given in angstroms.
Na
@
@-IS8
Na (4.06)
-1.41
Na
Na
Na
4.89
4.75
(-0.15)
(4.11)
c0.72
Figure 4. Lowdin electron densities of the atoms and the shell orbitals are shown for the NQ, Na5, N%, and Nag neutral clusters. The numbers in parentheses are the Lowdin electron densities for the sodium atoms only using the B2 basis.
basis to optimize the geometries led to similar structures. Initial NAB group positions were chosen to be the centers of triangles, based on previous path integral calculation^.^ The exponents of the added s and p orbitals were optimized using the TRUDGE algorithm included with the GAMESS quantum chemistry package, Next the positions of the NAB groups were optimized at the SCF level using the scan option in Gaussian 9O.Io The collection of NAB orbitals at each spatial location (a single NAB group) was moved together. The SCF energy was insensitive to small changes in the positions of the NAB groups, so further optimization was performed at the QCISD(T) level. NAB group positions changed by 0.57 au away from the center of the cluster for NQ as a result of this higher level of optimization. While this is a large difference for nuclei to move in an atom-based orbital optimization, this difference is relatively small on the scale of the bond lengths and may just be a reflection of not optimizing all the exponents of the NAB groups but just the extra s- and p-orbital exponents (see below). As a check for consistency, the exponents were again optimized at the SCF level using the TRUDGE algorithm in GAMESS but only a very small improvement in energy resulted. A full geometry optimization including sodium atoms and NAB groups at the
4.73
4.90 (+0.15)
Figure 5. Lowdin electron densities of the atoms and the shell orbitals are shown for the Naa, NaS, N%,and Nag cationic clusters. The numbers in parentheses are the Lowdin electron densities for the sodium atoms only using the B2 basis.
SCF level for NQ with the NAB basis led to insignificant changes in atom positions. A similar procedure was followed for the sodium pentamer with three NAB groups, each located at the center of one of the three triangles shown in Figure 2. The optimized s and p exponents of the extra s and p orbitals of the NAB groups for the Na4 and Na5 CI optimized cluster geometries were similar so we set them to ~(0.065)and p(0.040) for all calculations using the NAB basis set. We used these optimized exponents and the optimized geometries obtained with the B2 basis in our further studies of N a , Nas, NQ, and Nag. The NAB group positions were optimized at the SCF level with the scan procedure in Gaussian 90 for N&, Nag, Nq', Na5+, NQ+, and Nag+. The NAB orbital positions were varied by hand to determine the minimum QCISD(T) energy. We note that the manual determination of NAB group positions is not essential for studies of larger clusters. In the larger clusters it will be necessary to use methods from statistical mechanics (such as Monte Carlo or molecular dynamics) to properly locate and statistically weight the different isomers that are possible with increasing cluster size. The same statistical procedures can be used to locate the NAB positions without significant additional effort (this is discussed in more detail in the conclusions). Figures 2 and 3 show the optimized locations of the NAB groups. We judge the two basis sets by comparing the total energies and the ionization potentials of all the clusters as well as the electronic absorption spectrum for the four-atom cluster. If the smaller NAB basis is capable of reproducing experimental results as well as the larger B2 basis, this suggests the NAB orbitals can be useful in studying sodium clusters. Vertical and adiabatic ionization potentials for N q , Na5, NQ, and Nag were calculated with Gaussian 90, and the absorption spectrum for N q was calculated with M0LCAS.I' Further analysis of the basis sets includes an examination of the molecular orbital coefficients and the Lowdin density; these provide a way to determine the amount of electron density residing on the NAB groups and, thereby, the physical relevance of these orbitals.
IV. Results The QCISD(T) energies are shown in Table 1. Energies calculated using the NAB basis set are more stable than those calculated using the B2 basis by 2-8 kcal/mol. This is despite
J. Phys. Chem., Vol. 99, No. 21, 1995 8565
Orbital Basis Set for Small Sodium Clusters
TABLE 3: Ground- and Excited-State Energies of NQ for the Optimized Ground-State Neutral Geometry from MOLCAS (Also Shown Are the Coefficients of the Major Contributing Molecular Orbitals Configurations"
state
E(MRC1) (au) NAB basis
excitation energy (eV)
3B~, 3B3u 'Big 3B2u
-647.440 -647.419 -647.406 -647.404 -647.397
38 17 90 83 87
0.577 0.91 1 0.967 1.157
3B~, 3B~u
-647.396 64 -647.380 79
1.190 1.621
"4,
-647.380 69
1.624
lB2"
-647.379 87
1.646
'B3" 'B3u
-647.372 87 -647.366 26
1.837 2.017
IBI"
-647.358 56
2.226
'B2u
-647.345 83
2.573
'BI,
-647.329 90
3.006
major configurations and coefficients 0.904(a2b3,2) 0.877(a,2b3,1b2,1) 0.817(a,2a,lb3,]) 0.876(a,2b3,Ib2,') 0.672(a,'b3,2b2,1) +0.524(a,2b3,'b1,') 0.858(a,2b~,'b1,~) 0.672(a,2a,'bzU1) -0.472(a,2b3,'blg1)
-647.435 -647.419 -647.404 -647.406 -647.395
0.532(ag2b3u1bj,*') +0.504(ag1a,'bj,2) -0.442(a,2b3,'b3,') 0.583(a:ag1 b2,I) -0.423(a,2b3,'bl,') -0.326(a,'b~,2b2,') 0.863(a,2ag'b3,') 0.765(a,2ag*'b3,I) -0.527(a,2ag'b~,') +0.527(a,2b3,'b2,') -0.692(a,'b3,2b2,') +0.472(a:b3,' bl,]) -0.61 5(a,2b3,'b2,') 0.569(ag1b3,2b~,l)
* refers to the third orbital of that particular symmetry.
NAB basis
B2 basis
B1 basis
IBzu IB3, 'B3, 'BI, 'B2, 'BI,
1.65 1.84 2.02 2.23 2.57 3.01
1.45 1.66 1.69 1.71 2.50 2.85
1.57 1.74 1.92 2.12 2.52 2.80
P1 basis 1.60 1.86 2.00 2.27 2.57 2.91
excitation energy (eV)
79 90 62 81 73
0.432 0.848 0.788 1.090
-647.396 96 -647.381 99
1.057 1.464
-647.382 52
1.450
-647.382 60
1.447
-647.374 70 -647.373 55
1.662 1.694
-647.373 04
1.707
-641.343 90
2.500
-647.331 11
2.848
major configurations and coefficients 0.905(a,2b3?) 0.838(a,2b3,'b2,') 0.779(a2ag1b3,') 0.833(a2b3,'b2,') 0.636(ag'b3?b~,') +0.526(a~b3,'b1,') 0.884(a,2b~,'b1,') -0.5 17(ala,' b2"') +0.492(a,2b3,'b1,') +0.304(a~a,#'b2,1) -0.50 1(a,'a,1b3,2) +0.482(a2b3,'b3,*1) -0.443(a,2b3,'b3,*I) O.528(ag2a,'b~,') -0.398(a2b3,Ib1,') +O.343(ag1b3,2b2,') 0.833(a,2ag1b3,l) 0.688(a~a,''b~,') -383(ag1a,2b3,1) -0.262(ag2b2,Ib~,') 0.521(a,2a,'blU') +0.495(a,2b3,'b~,') 0.679(a,2b3,1b~,1) +0.525(ag'b3?bz,') -0.501(a,2a,'b1,~) +0.474(ag'b3,2b~,') -0.233(a,2a,*lb1,')
# refers to the fourth orbital of that particular symmetry.
TABLE 4: Comparison of the Energies (eV) of the Visible Excited States That Comprise the Vertical Spectrum for N q (Number of Valence Basis Functions Is 28 (NAB), 36 (B2), 56 (BO, and 44 (Pl)) state
E(MRC1) (au) B2 basis
experimental 1.63 1.80 1.93 2.18 2.50 2.77
having fewer orbitals in the NAB basis than in the B2 basis. For comparison, we note that the original B1 basis used approximately 55% more basis functions than the B2 basis to gain 6-15 kcdmol in energy' relative to the B2 basis. The NAB basis set uses up to 35% fewer basis functions, yet still gains 2-8 kcal/mol in energy relative to the B2 basis. This suggests that the NAB groups provide a representation of the exact electronic structure similar to the larger atom-based B 1 and B2 bases. It is also possible that the NAB groups provide some effective d-character to the molecular orbitals as they are not located on the atoms. The difference in energies of the B2 and NAB basis set is not simply a unitary transformation of molecular orbitals; if it were, the energies and other physical properties would be identical when calculated using both basis sets. Note that placing an extra NAB group in the center of each of the N% and Nag clusters provides increased stability due to the larger number of basis functions; still, it is an easy calculation to do, and there are the same or fewer basis functions than with the B2 basis set. We now examine whether the NAB orbitals are contributing significantly to the total electron density. The results of this analysis are shown in Figures 4 and 5, where the total valence electron charge density (via Lowdin analysis) is shown for both atoms and NAB groups. As can be seen, a significant amount of the electron density of the valence electrons has moved from
the sodium atoms onto the NAB groups. The fraction of the electron density that can be accounted for using the NAB groups ranges from 70-86%, clearly demonstrating the importance of the NAB orbitals. The structures of the cations are shown in Figure 3, and the energies from the different basis sets are shown in Table 1. Note the extra stabilization obtained using the NAB basis set. The vertical and adiabatic ionization potentials are shown in Table 2. The ionization potentials obtained with the NAB basis set are similar to those obtained with the B2 basis set. Both the B2 and the NAB basis sets agree qualitatively with experiment,'* neither demonstrating a better fit. The addition of a fourth or fifth NAB group to the N% or Nag cluster, respectively (to the inner triangle in N% and the inner tetrahedron in Nag), causes the vertical ionization potential to change by only 0.01 eV. A final test for the two basis sets is their ability to predict the absorption spectrum for a given cluster. We have checked this for N q using MOLCAS." Shown in Table 3 are the ground and excited states of the four-atom cluster using both basis sets, along with their excitation energies. The excited states found with the NAB basis are purer than those found with the B2 basis at least for the configuration with the largest coefficient. A comparison with experiment and previous results using the B1 basis and those of Poteau et al., who used a large non-atom-based basis, centered at the origin of the c l u ~ t e r ,is' ~ given in Table 4 for the visible states. We include their results for the 4s, 4p, 4d 2s/atom basis (Pl). Unfortunately, the standard quantum chemical packages we used do not allow us to calculate transition moments for the experimentally visible transitions. Even so, it is clear that the NAB basis set is far superior to the B2 basis set, both in absolute absorption frequency and in relative spacing of frequencies. In addition, with the exception of the highest energy excitation ('BI"), the NAB basis set performs about as well as the B1 and P1 basis sets, despite having only half as many valence orbitals. It is
+
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8566 J. Phys. Chem., Vol. 99, No. 21, 1995
thus clear that the much smaller NAB basis can be used to predict accurately the properties of these small clusters.
V. Conclusions A non-atom-based orbital scheme was used to study the electronic structure of small sodium clusters. The results of previous path integral calculations motivated this study. Using NAB groups located near the centers of simple geometric structures, we obtain excellent agreement with experiment and previous quantum chemical studies which used much larger basis sets. Our procedure has some similarities with the calculations of Poteau and S~iege1mann.I~ Studies of larger clusters are hindered by the need to identify and properly statistically weight the different possible isomers. As the cluster size is increased, the number of possible local energy minima dramatically increases, and it becomes necessary to use statistical mechanics methods to efficiently sample the available phase space. Thus Monte Carlo or molecular dynamics procedures are commonly used to study large clusters. This is relevant when contemplating using and optimizing the positions of NAB groups. In a Monte Carlo calculation, for instance, the positions of the nuclei would be varied simultaneously with the NAB group positions, allowing the determination of overall structure (nuclei NAB groups) in a manner that does not require manual variations of the NAB groups and clever guesses about the locations of the NAB groups. By using a small set of NAB groups and pseudopotentials for core electrons, a Monte Carlo or molecular dynamics procedure can be used to determine geometries at the SCF level. For instance, a set of orbitals consisting of one s orbital and one p orbital per NAB group, no valence orbitals on the atoms, and the use of a simple pseudopotential for the core electrons, gives the correct geometries for Na4-Nas.'5 Since the determination of the lowest energy isomer is typically done at the Hartree-Fock level and, thus, the calculation time scales at least as N4, this type of basis set could be quite useful in studying very large clusters. We are currently extending our preliminary work using a Monte Carlo procedure with s and p orbitals in an effort to treat larger clusters.
+
The results of our calculations suggest that the electronic structure of alkali clusters may be viewed as electron density located between atoms rather than on them as is the case for conventional molecules. This is what one should expect given the low ionization potentials of these atoms and the fact that they exist as metals in the bulk. Acknowledgment. We gratefully acknowledge useful discussions with Dr. Jaime Combariza and Professor Neil Kestner. This work was supported by NSF Grant CHE-9001362. References and Notes (1) Bonacic-Koutecky, V.; Fantucci, P.; Koutecky, J. Chem. Rev. 1991, 91, 1035. (2) Rothlisberger,U; Andreoni, W. J. Chem. Phys. 1991, 94, 8129. (3) Hall, R. W. J. Chem. Phys. 1990, 93, 8211. (4) Bonacic-Koutecky, V; Fantucci, P; Koutecky, J. Phys. Rev. E 1988, 37, 4369. (5) Bonacic-Koutecky, V; Boustani, I.; Guest, M.; Koutecky, J. J. Chem. Phys. 1988, 89, 4861. (6) Bonacic-Kouteckv. V.: Fantucci. P.: Kouteckv. J. Chem. Phvs. Lett. 1990, 166, 32. (7) Bonacic-Kouteckv, V; Fantucci. P.: Kouteckv, J. J. Chem. Phvs. 1990, 93, 3802. (8) Hall, R. W. J. Chem. Phys. 1992, 97, 6481. (9) Gamess. Original program assembled by the staff of the NRCC: Dupuis, M; Spangler, D.; Wendoloski, J. J. National Resource for Computations in Chemistry Software Catalog, University of California, Berkeley, CA, 1980; Program QGO1. This version of GAMESS is described in the Quantum Chemistry Program Exchange Newsletter: Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Jensen, J. H.; Koseki, S.; Gordon, M. S.; Nyguyen, K. A.; Windus, T. L.; Elbert, S . T. QCPEEull. 1990, IO, 52. (10) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M. A,; Binkley, J. S.; Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A,; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. GAUSSIAN 90, Gaussian Inc., Pittsburgh, PA, 1990). (11) MOLCAS version 2, Andersson, K.; Fiilscher, M. P.; Lindh, R.; Malmqvist, P.-A.; Olsen, J.; Roos, B. 0.;Sadlej, A. J. University of Lund, Sweden, and Widmark, P.-0. IBM Sweden, 1991. (12) Kappes, M. M.; Schir, M.; Rothlisberger, U.; Yereztzian, C.; Schumacher, E. Chem. Phys. Lett. 1988, 143, 251. (13) Poteau, R.; Maynau, D.; Spiegelmann, F. Chem. Phys. 1993, 175, 289. (14) Poteau, R.; Spiegelmann, F. J. Chem. Phys. 1993, 98, 6540. (15) Prince, M. R.; Hall, R. W., unpublished results. JP942709W
