J. Phys. Chem. 1996, 100, 10055-10061
10055
Theoretical Study of the Electronic States of Si2C A. Spielfiedel,* S. Carter, and N. Feautrier ObserVatoire de Paris, DAMAp, F-92195 Meudon, France
G. Chambaud and P. Rosmus Laboratoire de Chimie The´ orique, UniVersite´ de Marne-la-Valle´ e, F-93166 Noisy-le-Grand, France ReceiVed: January 22, 1996; In Final Form: March 25, 1996X
The ground state and electronically excited singlet and triplet states of Si2C have been investigated by ab initio calculations. For the electronic ground state, rovibrational energy levels have been evaluated by a variational approach using several different potential energy functions. The equilibrium geometry, the centrifugal distortion constants, and the vibrational levels up to 1500 cm-1 are given. The best function yielded an electronic barrier to linearity of 859 cm-1. The changes in the Ka level structures of the rovibrational levels caused by this barrier are predicted to appear between the V2 ) 4 and V2 ) 6 vibrational states. The results are compared with available experiments and with those of other theoretical studies. The electronically excited singlet and triplet states have been characterized by their potential energy functions. The avoided crossings and conical intersections in the excited states have been located and their influence on the electronic spectra is discussed.
I. Introduction The Si2C molecule has not yet been detected in outer space, even though it can be expected to be a constituent of carbon star atmospheres and molecular clouds. The related SiC and SiC2 molecules have been already identified in the millimeter wavelength region in astronomical sources.1-4 To date, little has been known about Si2C. Only three vibrational transitions have been observed in cold matrices. Presilla-Marquez and Graham5 confirmed a previously observed transition at 1188.4 cm-1 by Kafafi et al.6 and assigned it to the antisymmetric stretching mode. They also identified a new transition at 839.5 cm-1 as the symmetric stretching mode and the (ν3 + ν2) combination mode at 1354.8 cm-1. There have been several theoretical studies of the Si2C molecule.7-12 The ground state has been found to have a bent equilibrium structure with a shallow bending part of the potential energy function (PEF). The assignments made in ref 5 have been confirmed by the calculations of the harmonic fundamental vibrational wavenumbers and intensities by Rittby.10 Previously, we have reported the anharmonic vibrational wavenumbers, vibrational band intensities, and absolute line strengths.12 A detailed study of the rovibrational problem including the anharmonic effects has also been published by Barone, Jensen, and Minichino11 based on a quartic MP2 PEF. Using secondorder polarization propagator calculations, Sabin, Oddershede, Diercksen, and Gru¨ner9 reported the vertical valence excitation energies and oscillator strengths for the singlet and triplet states of Si2C. As in many other silicon-containing compounds, the shallow part of the PEF for the electronic ground state of Si2C is difficult to calculate accurately. One of the goals of the present study is to improve the accuracy of previous PEF’s, in order to obtain a more accurate value for the barrier to linearity. We have used two different basis sets and three different approaches accounting for high portion of the electron correlation and generated five different three-dimensional PEF’s for the electronic ground state. The anharmonic effects, and the Ka level pattern close to the X
Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(96)00191-8 CCC: $12.00
barrier to linearity, are discussed. Also, the electronically excited singlet and triplet states have been characterized by their PEF’s. Very close to the Franck-Condon region of the absorption spectra avoided crossings and conical intersections have been found, which will play an important role in the interpretation of the electronic spectra of this species. II. Computational Details Two different atomic orbital basis sets have been used. The basis set A (used also in our previous work) comprised the Si (12s, 9p) basis set of Huzinaga14 augmented by one diffuse s (exponent 0.025) and p (exp 0.0378) and two d (exp 0.65, 0.19) and one f function (exp 0.32) and contracted to [11s, 7p, 2d, 1f]. For carbon, the van Duijneveldt15 (13s, 8p) has been augmented by one diffuse s (exp 0.04) and p (exp 0.028), and the (2d, 1f) basis set of Dunning17 and contracted to [9s, 6p, 2d, 1f]. This basis set included 142 contracted Gaussian functions. The second basis set B has been selected in the following way: for silicon, the atomic orbital basis set of Woon and Dunning16 consisting of the (16s, 11p, 3d, 2f, 1g) set has been contracted to [6s, 5p, 3d, 2f, 1g] and for carbon the (12s, 6p, 3d, 2f, 1g) set of Dunning17 has been contracted to [5s, 4p, 3d, 2f, 1g], respectively. This resulted in 173 generally contracted Gaussian functions. For the electronic ground state, five three-dimensional potential energy functions have been generated. The first potential has been obtained with the basis set A and the multiconfiguration self-consistent field calculations (CASSCF)18,19 followed by large scale internally contracted multireference configuration interaction (MRCI).20,21 The computations in the C2V symmetry subgroup employed the active space of the 6-10a1, 2-3b1, 5-8b2, and 2a2 molecular orbitals, or their equivalents in higher or lower symmetries. The reference wave functions for the MRCI calculation have been selected in different geometry regions of the potential energy functions according to a coefficient threshold of 0.01 in the CASSCF configuration expansion, leading to the equivalent of an expansion with about 8 × 107 uncontracted configurations. In our previous publication similar computations were performed, but for computational reasons the lowest three valence © 1996 American Chemical Society
10056 J. Phys. Chem., Vol. 100, No. 24, 1996
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TABLE 1: Expansion Coefficients of the Three-Dimensional PEF of the Ground Electronic State of Si2C (in au)a C000 C110 C210 C102 C220 C112 C410 C221 C113 C510 C411 C222 C114
-616.0591879 -0.1190490 -0.0972602 -0.0381265 -1.1440244 0.1623227 1.2908063 -2.2179940 -0.1075169 -1.2031400 -0.5164905 2.2894590 0.0399005
-0.0000000 0.0400060 -1.3492648 -0.0074480 -0.0431232 0.0173016 1.8241345 0.0938292 -0.0096615 26.5859348 -1.6007972 -0.0776228 0.0040390
C100 C101 C030 C003 C301 C103 C320 C302 C104 C420 C321 C303 C105
C001 C002 C201 C400 C211 C004 C401 C212 C005 C330 C402 C213 C006
0.0000000 0.0095667 0.0027248 -0.9741418 -0.0891412 0.0009763 -0.3917938 -0.0481981 -0.0009814 -8.3269247 0.3512359 0.0445525 0.0018667
C200 C300 C111 C310 C202 C500 C311 C203 C600 C501 C312 C204
1.8142112 -1.3492648 -0.0379871 0.2135138 -0.0262435 0.2174484 -0.4476587 -0.0129267 0.2144928 -0.4308317 0.0810586 0.0330802
a The calculated total energies for 74 geometries were fitted by a sextic polynomial expansion V(R R ,R) ) ∑ijkCijkQi1Qj2Qk3 in the bond lengths 1 2 R1 and R2 and the included angle R. For Q1 and Q2 the Simon-Parr-Finlan coordinate Qi ) 1 - Rref/Ri and for Q3 the Carter-Handy coordinate ref ref ) 114.992°. C Q3 ) A0Θ + A1Θ2 + A2Θ3, with Θ ) R-Rref, was employed. Rref ijk ) Cjik. A0 ) 1.7, A1 ) -0.511 1 ) R2 ) 3.220 bohr. R rad-1, A2 ) -0.191 rad-2. The above expansion is valid in the range 2.6 bohr e Ri e 4.2 bohr and 75° e R e 180°.
TABLE 2: Spectroscopic Constants of the X1A1 State of Si2C this work CCSD
CCSDb
837 139 1240 63836 4295 4024 1.704
867 156 1256 63332 4387 4103 1.693
876 159 1263 63305 4412 4124 1.690
839 152 1208 61163 4357 4067 1.705
847 155 1215 61115 4383 4089 1.702
830 144 1220 63895 4281 4012 1.706
618 116.6 -83.067 -0.049 -0.013 1.904
876 115.9 -64.582 -0.042 -0.013 1.532
905 115.7 -61.237 -0.042 -0.013 1.482
864 115.1 -60.005 -0.044 -0.013 1.503
859 115.0 -56.335 -0.043 -0.013 1.440
762 117.1
constant /cm-1
ω1 ω2/cm-1 ω3/cm-1 Ae/MHz Be/MHz Ce/MHz Re(SiC)/Å barrier to linearity/cm-1 Re/deg τAAAA/MHz τBBBB/MHz τABAB/MHz τAABB/MHz a
CI
a
a
CCSD(T)a
CCSD(T)b
ref 11
Basis set A. b Basis set B.
molecular orbitals could not be correlated. This resulted in a too large stabilization of the linear structures leading to a too small barrier to linearity. This time, all valence electrons have been correlated. Four additional PEF’s have been obtained with the CCSD and CCSD(T) approaches.22 The second and third PEF’s have been obtained with basis set A, the fourth and fifth PEF’s with the large basis set B. The vertical excitation energies have been obtained with basis set A at the internally contracted MRCI level of theory, the PEF’s of the electronically excited states have been calculated by state-averaged CASSCF method, also with basis set A. The study has been performed with the MOLPRO program suite.23 III. Electronic Ground State The five calculated PEF’s have been fitted to sextic polynomial expansions and used in perturbational calculations of the spectroscopic constants, and three of those PEF’s have been used in variational24 calculations of the rovibrational energy levels. The expansion coefficients are given in Table 1 only for the most accurate PEF obtained with the basis set B and CCSD(T) approach. Depending on the electron correlation approach and the AO basis set used, it has been found for Si2C a bent global minimum,7,8,10 or bent and linear minima separated by a barrier,12 or even only a linear minimum.7,10 Grev et al.7 and Rittby10 have shown that it is necessary to use a large basis set including enough polarization functions. The barrier to linearity has been calculated to lie between 309 cm-1 8 and 753 cm-1.11 Our most accurate barrier height (CCSD(T); basis B) has been calculated to be 859 cm-1 (cf. Table 2), larger than the value deduced from the PEF of Barone et al.11 of 753 cm-1.
This energy difference is very sensitive to the electron correlation effects. For instance, the CCSD and CCSD(T) values for the large basis set B still deviate by 46 cm-1 (cf. Table 2). It is evident that the barrier height and the flat shape of the PEF strongly influence the positions of the higher bending and bending-combination levels. The near-equilibrium properties such as the equilibrium structure and fundamental harmonic wavenumbers are easier to calculate. The calculated equilibrium rotational constants, the harmonic wavenumbers for the fundamentals, and the set of τ constants at equilibrium are given in Table 2. The equilibrium distance and the bond angle are smaller than the best previous theoretical results of Barone et al.11 The harmonic wavenumbers for the fundamentals obtained from the PEF in Table 1 deviate from previous theoretical values by 5-17 cm-1,11 by 10-30 cm-1,10 and by 60-200 cm-1,7 respectively. In Table 3 the anharmonic vibrational levels are given for all levels up to 1500 cm-1. The deviations between the matrix values for the experimentally known three vibrational transitions and our values lie in the range of 3-10 cm-1. The five PEF’s of the present study obtained from highly correlated electronic wave functions and flexible AO basis sets show strong deviations in the calculated positions of the bending modes with increasing bending quantum numbers. Similarly, deviations between the results of the present study and those of Barone et al.11 obtained from a MP2 PEF will rapidly get larger for higher bending levels. Very important for the check of accuracy of the theoretical PEF would be an experimental rotationally resolved emission spectrum, for instance, of the A1Πu-X1A1 transition (cf. next section). As demonstrated by Dixon,13 the Ka energy level structure changes in the energy region close to
Electronic States of Si2C
J. Phys. Chem., Vol. 100, No. 24, 1996 10057
TABLE 3: Vibrational Energy Levels in the X1A1 State of Si2Ce this work ν1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 0
ν2 1 2 3 4 5 6 7 0 8 1 9 2 10 3 0 11 4 5 1 12 6 7 2 13 8 3
ν3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1
CIa 1100.1 128.5 247.4 354.6 446.0 515.8 586.2 676.1 814.9 780.2 923.3 895.4 1015.6 1020.2 1080.8 1216.1 1153.8 1133.6 1208.5 1355.7 1295.3 1298.6 1400.6 1487.0 1444.4 1513.1 1609.2
f
CCSD(T)a
CCSD(T)b
1094.8f
1103.1f
142.8
145.3 281.1 405.8 519.0 621.0 710.3 788.8 829.4 877.1 959.1 982.5 1075.5 1100.4 1176.5 1198.4 1228.4 1262.0 1333.4 1352.1 1365.1 1406.8 1497.7 1498.0 1509.8 1602.9 1634.9
821.2 948.2
1193.2
1346.5
expc,d
ref 11 1085.7f 133.4 258.7 376.2 485.0
839.5
811.1 928.3
1188.4
1195.3
1354.8
1342.4
Figure 2. Contour plot (Q1 ) 1/21/2(R1 + R2), Q2 ) R) of the (3, 11, 0) vibrational level of Si2C at 3487 cm-1.
a Basis set A. b Basis set B. c References 5 and 6. d Experiment in rare-gas matrix. e Calculated variationally. Units ) cm-1. f Zero-point vibrational energies.
Figure 3. CASSCF dissociation paths of the singlet states of Si2C in linear structure correlating with the lowest dissociation asymptotes (R2(CSi) ) 3.4 bohr). Figure 1. One-dimensional cut of the CCSD(T) potential energy function of Si2C along the bending coordinate (RCSi ) 3.216 bohr) with the calculated energy positions of the bending levels for V2 ) 0-14 and Ka ) 0-5.
the electronic barrier such as in Si2C. The ∆Ka energy differences increase slowly below and rapidly above the barrier to linearity. In Figure 1 such changes appear for the bending levels between V2 ) 4 and 6. This energy level diagram has been obtained from variational calculations of the rovibrational states up to J ) 5. As mentioned by Barone et al.,11 the vibrational states of Si2C remain almost unperturbed by anharmonic resonances. For instance, from the contour plot (cf. Figure 2) of the (V1 ) 3, V2 ) 11, V3 ) 0) state calculated at 3487 cm-1, the assignment by the harmonic vibrational quantum numbers is straightforward.
IV. Electronically Excited States In Table 4a the calculated MRCI vertical excitation energies for the singlet states are given for two different RCSi distances (3.2 and 3.4 bohr) and the bond angle of 120°. The results are compared with those of Sabin et al.9 obtained with smaller AO basis set and second-order polarization propagator calculations for RCSi 3.24 bohr and a broad angle of 118°. Both sets of values agree within about 0.2-0.3 eV. The present results obtained from high-quality electronic wave functions are seen to be more accurate. For comparison, also the CASSCF vertical excitation energies are given in Table 4a. The differences between the MRCI and the CASSCF values amount to about 0.2-0.4 eV. In Table 4b the calculated energy differences between the bent electronic ground state and linear centrosymmetric structures, the information about the equilibrium struc-
10058 J. Phys. Chem., Vol. 100, No. 24, 1996
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TABLE 4: (a) Vertical Excitation Energies for the Singlet States of Si2Ca and (b) Energy Differences between the Bent Electronic Ground State and the Linear Centrosymmetric Structures of the Singlet States of Si2Ca (a) Vertical Excitation Energies ∆E (eV) states
CASSCF (3.2, 3.2, 120°)
MRCI (3.2, 3.2, 120°)
MRCI (3.4, 3.4, 120°)
ref 9 (3.24, 3.24, 118°)
1A
0.00 4.51 5.76 4.14 5.05 6.50 3.94 5.44 6.31 3.70 4.48 5.50
0.00 4.22 5.18 3.84 4.57 5.83 3.54 4.71 5.70 3.48 4.19 5.46
0.25 4.09 4.87 3.44 4.48 5.62 3.52 4.33 5.37 3.17 3.69 5.30
0.00 3.99 4.89 3.64 4.25 5.26 3.28 4.44 5.73 3.34 4.12 5.68
X 1 21A1 31A1 11B2 21B2 31B2 11B1 21B1 31B1 11A2 21A2 31A2
(b) Energy Differences ∆E (eV)
a
states
MRCI (3.2, 3.2, 180°)
MRCI (3.4, 3.4, 180°)
bent or linear
X1Σ+ g 11∆u 1 1 Σu 11Πu 11Πg 11Σ+ u 11Σg 11∆g
0.04 3.04 3.03 3.41 5.01 5.17 5.59 5.65
0.31 2.77 2.76 3.51 5.02 4.93 5.18 5.24
B L, L L L, B B, L L L B, B
R < 180° 1A 1
1
1B 2 1A 2
+ 1A2
1
+ 1A2
A1 + 1B1 B2 1B 2
1B 1 1 A1
+ 1B1
MRCI ground state energy ) -616.007 319 15 hartrees.
TABLE 5: (a) Vertical Excitation Energies for the Triplet States of Si2Ca and (b) Energy Differences between the Bent Electronic Ground State and the Linear Centrosymmetric Structures of the Triplet States of Si2Ca (a) Vertical Excitation Energies ∆E (eV) states
CASSCF (3.2, 3.2, 120°)
MRCI (3.2, 3.2, 120°)
MRCI (3.4, 3.4, 120°)
ref 9 (3.24, 3.24, 118°)
13A1 23A1 33A1 13B2 23B2 33B2 13B1 23B1 33B1 13A2 23A2 33A2
3.74 4.26 5.54 3.55 3.88 4.33 3.11 4.84 5.68 3.42 4.36 4.79
3.46 3.85 5.05 3.22 3.64 4.03 2.86 4.29 5.24 3.22 4.12 4.64
3.46 3.54 4.61 2.88 3.46 3.84 2.95 3.90 4.84 3.01 3.64 4.71
2.91 3.38 4.92 2.82 3.02 3.82 2.42 4.04 5.33 2.82 3.93 4.25
(b) Energy Differences ∆E (eV)
a
states
MRCI (3.2, 3.2, 180°)
MRCI (3.4, 3.4, 180°)
bent or linear
13Σ+ u 13∆u 13Σu 13Πu 13Πg 13Σ+ g 13∆g 3 1 Σg
2.65 2.87 3.06 2.75 4.10 4.80 5.14 5.34
2.46 2.65 2.81 2.93 4.36 4.35 4.66 4.81
L L, L L L, L B, L B B, B L
R < 180°
3A 1 3A 1 3A 1 3 B1
3B 2 3B 2 3 A2
+ 3A2
3B 2
+ 3A2
+ 3B1 + 3B1
MRCI ground state energy ) -616.007 319 15 hartrees.
tures, and the Renner-Teller pairs are given. In Table 5a,b similar results are presented for the triplet states. Using the CASSCF approach, one-dimensional cuts of the potential energy functions have been calculated in order to characterize the electronically excited states of the Si2C molecule in more details. As discussed before, the CASSCF energy differences are less accurate than the corresponding MRCI
values, but the mapping of the potential energy functions by the MRCI approach turned out to be computationally too demanding. In Figure 3 the collinear dissociation paths of some low-lying singlet states are shown. All states are bound with respect to predissociation and are calculated to dissociate smoothly to their asymptotes. The regions in which these potentials cross are regions of conical intersections in lower
Electronic States of Si2C
Figure 4. CASSCF dissociation paths of the triplet states of Si2C in linear structure correlating with the lowest dissociation asymptotes (R2(CSi) ) 3.4 bohr).
J. Phys. Chem., Vol. 100, No. 24, 1996 10059
Figure 6. CASSCF PEF’s of the triplet states of Si2C in linear structures along the symmetric displacement coordinate.
Figure 7. CASSCF PEF’s for the 1A1 and 1B2 states of Si2C along the bending displacement coordinate (R1(CSi) ) R2(CSi) ) 3.2 bohr). Figure 5. CASSCF PEF’s of the singlet states of Si2C in linear structures along the symmetric displacement coordinate.
symmetries accompanied by the vibronic coupling effects. In Figure 4 similar results are displayed for the triplet states. Also, along the symmetric stretching coordinate (cf. Figure 5 for the singlet state and Figure 6 for the triplet states) there are no avoided crossing regions for the investigated singlet and triplet states. The allowed 11Πu - X1Σ+ g absorption transition will be 1 influenced by the vibronic coupling with the 11Σu and ∆u states close to the Franck-Condon region (Figures 3 and 5). 1 + Also, the most intense (cf. next paragraph) 11Σ+ u - X Σg transition in Si2C will exhibit vibronic coupling effects due to the conical intersections with several electronically excited states (cf. Figure 5). The shapes of the PEF’s for the electronically excited states become much more complicated along the bending displacement coordinate. For the electronically degenerate states both linear-linear and bent-bent Renner-Teller systems are
found. Moreover, around 110°, i.e., very close to the FranckCondon region of the absorption spectrum, the lowest 1A1 and 1B states form an avoided crossing (cf. Figure 6), which will 2 lead to perturbations particularly in the bending levels of the excited states. The avoided crossings lead to double minimum structures already in the lowest singlet states (Figure 7). The crossings of the PEF’s in C2V symmetries become conical intersections in Cs symmetries leading to vibronic couplings via the antisymmetric stretching mode (Figures 7 and 8). For the 3A and 3B states (Figure 9), the PEFs along the bending 1 2 coordinate have even more complicated shapes than for the singlet states. Also for the lowest triplet states double minimum PEF’s have been calculated. The 3A2 and 3B1 states (Figure 10) will be also coupled by the antisymmetric stretching mode. The complicated changes in the electronic structures accompanying the avoided crossings are responsible for complicated shapes of the electronic transition moment functions. In Figure 9 only the one-dimensional cuts along the bending
10060 J. Phys. Chem., Vol. 100, No. 24, 1996
Figure 8. CASSCF PEF’s for the 1B1 and 1A2 states of Si2C along the bending displacement coordinate (R1(CSi) ) R2(CSi) ) 3.2 bohr).
Figure 9. CASSCF PEF’s for the 3A1 and 2B2 states of Si2C along the bending displacement coordinate (R1(CSi) ) R2(CSi) ) 3.2 bohr).
coordinate for the singlet-singlet transitions are displayed. In agreement with the results of Sabin et al.,9 the largest transition 1 moment (Figure 11) is found for the 31B2(1Σ+ u )-X A1 transition 1 1 1 1 followed by 2 A1-X A1 and 2 B2-X A1.
Spielfiedel et al.
Figure 10. CASSCF PEF’s for the 3B1 and 3A2 states of Si2C along the bending displacement coordinate (R1(CSi) ) R2(CSi) ) 3.2 bohr).
Figure 11. CASSCF electronic transition moment functions for the singlet states of Si2C along the bending displacement coordinate (R1(CSi) ) R2(CSi) ) 3.2 bohr).
tigations of Si2C, which are expected to be involved in the chemistry of the carbon star atmospheres, molecular clouds, and silicon-containing discharges and plasmas.
V. Conclusions New PEF’s for the electronic ground state of Si2C have been generated and used in variational calculations of rovibrational energies by an approach which takes fully into account the anharmonic and rovibrational coupling effects. For the accurate prediction of higher bending and combination levels, a precise value of the barrier to linearity is needed, which can hardly be obtained with sufficient accuracy from ab initio calculations. It has been suggested that the 11Πu - X1Σ+ g emission spectrum could provide the necessary experimental information. It has been shown that the intrepretation of the absorption and emission spectra of the Si2C will require the inclusion of vibronic coupling effects already for the lowest singlet and triplet states. The present theoretical study should aid future spectroscopic inves-
Acknowledgment. The computations were performed on the CRAYs of the computer centers IDRIS and UMLV and on the CONVEX of the computer center of Observatoire de Paris. This work was supported by the EC grant ERBCHBG CT93-0387. References and Notes (1) Thaddeus, P.; Cummins, S. E.; Linke, R. A. Ap. J. 1984, 283, L45. (2) Snyder, L. E.; Henkel, C.; Hollis, J. M.; Lovas, F. G. Ap. J. 1985, 290, L29. (3) Cernicharo, J.; Kahane, C.; Gomez-Gonzalez, J.; Gue´lin, M. Astron. Astrophys. 1986, 167, L9. (4) Cernicharo, J.; Gottlieb, C. A.; Gue´lin, M.; Thaddeus, P.; Vrtilek, J. M. Ap. J. 1989, 341, L25.
Electronic States of Si2C (5) Presilla-Marquez, J. D.; Graham, W. R. M. J. Chem. Phys. 1991, 95, 5612. (6) Kafafi, Z. H.; Hauge, R. H.; Fredin, L.; Margrave, J. J. Phys. Chem. 1983, 87, 797. (7) Grev, R. S.; Schaefer, H. F., III. J. Chem. Phys. 1985, 82, 4126. (8) Diercksen, G. H. F.; Gru¨ner, N. E.; Oddershede, J.; Sabin, J. R. Chem. Phys. Lett. 1985, 117, 29. (9) Sabin, J. R.; Oddershede, J.; Diercksen, G. H. F.; Gru¨ner, N. E. J. Chem. Phys. 1986, 84, 354. (10) Rittby, C. M. L. J. Chem. Phys. 1991, 95, 5609. (11) Barone, V.; Jensen, P.; Minichino, C. J. Mol. Spectrosc. 1992, 154, 252. (12) Gabriel, W.; Chambaud, G.; Rosmus, P.; Spielfiedel, A.; Feautrier, N. Ap. J. 1992, 398, 706. (13) Dixon, R. N. Trans. Faraday Soc. 1964, 60, 1363. (14) Huzinaga, S. Technical Report, University of Alberta, 1971.
J. Phys. Chem., Vol. 100, No. 24, 1996 10061 (15) van Duijneveldt, F. B. IBM Res. Rep. 1971, RJ945. (16) Woon, D. E.; Dunning, T. H. J. Chem. Phys. 1993, 98, 1369. (17) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (18) Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1985, 82, 5053. (19) Knowles, P. J.; Werner, H.-J. Chem. Phys. Lett. 1985, 115, 259. (20) Werner, H.-J.; Knowles, P. J. J. Chem. Phys. 1988, 89, 5803. (21) Knowles, P. J.; Werner, H.-J. Chem. Phys. Lett. 1988, 145, 514. (22) Hampel, C.; Peterson, K.; Werner, H.-J. Chem. Phys. Lett. 1992, 190, 1. (23) MOLPRO is a package of ab initio programs written by H.-J. Werner, P. J. Knowles, with contributions of J. Almlo¨f, R. D. Amos, M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. Peterson, R. Pitzer, A. J. Stone, and P. R. Taylor. (24) Carter, S.; Handy, N. C. Mol. Phys. 1984, 52, 1367.
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