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Ab initio molecular orbital studies on a singlet-triplet splitting of C3H6 and C4H8 molecules
Journal of Molecub Structure (Theo&em), 251 (1991) 141-151 Elsevier Science Publishers B.V., Amsterdam
141
Ab initio molecular orbital studies on a singlet-triplet splitting of C,H, and C,H, molecules A.A. Ovchinnikov,
I.L. Shamovsky and K.V. Bozhenko
Institute of Chemical Physics of the USSR Academy of Sciences, Kosygin str., 4, Moscow 117334 (USSR) (Received 22 March 1991)
Abstract The vertical singlet-triplet (ST) splitting of C3H6 and C4H8 molecules containing two sp’ carbon atoms separated by one or two methylene groups was investigated by means of ab initio calculations. The molecular geometry was either taken as that corresponding to the UHF/6-31G* energy minimum of the triplet electron configuration, or extracted from the structure of the ferrocarbon crystal, which has been determined previously by the molecular mechanics method. Different basis sets (6-31G* and 6-31 lG**) and various techniques of taking into account electron correlation (multiconfiguration self-consistent field (MC-SCF) and fourth-order Msller-Plesset (MP4) ) were applied. The MP4 method provides more reliable ST splitting values than does the MC-SCF method which involves only pair excitations, but MP4 requires the use of an extremely extended basis set if quantitative estimates are to be obtained. The ST splitting appears to be determined not only by the distance between sp’ carbon centres but also to a great extent by the mutual orientation of these centres. The presence of three-dimensional ferromagnetic ordering in the previously suggested intermediate graphite-diamond structure is proved.
INTRODUCTION
It has been suggested [ 1,2 ] that there is a structure of the crystalline carbon phase which should exhibit ferromagnetic properties. This phase, which has a structure intermediate between that of diamagnetic graphite and diamond, possesses the maximum possible concentration of unpaired electron carriers, and its theoretically expected magnetization exceeds the maximum magnetization of a-Fe. By using the molecular mechanics technique it has been established [ 31 that the minimum potential energy of a crystal having such a structure corresponds to the orthorhombic symmetry (ar = p= y= 90 o ). In addition, the coordinates of the atoms in the unit cell and the parameters of the unit cell itself (a, b and c) have been determined. Each sp2 carbon atom was found to be separated by two covalent bonds from six analogous sp’ atoms and by three bonds from eight analogous atoms; it should be noted that there are two and
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142 four physically non-equivalent l-3 and l-4 exchange interactions between radicals localized on the sp2 carbon atoms, respectively. To prove the ferromagnetic ordering in the structure under investigation, it is necessary and sufficient to prove the presence of ferromagnetic exchange between two radicals in the elementary fragments [ 21. The established ferromagnetic properties of such a material suggest that most fragments containing two adjacent sp2 atoms possess a triplet ground state. In the previous study ab initio molecular orbital (MO) calculations were performed to determine the vertical singlet-triplet (ST) splitting in C&H6and C&H*.These two molecules were used to simulate l-3 and l-4 exchange interactions between radicals in the ferrocarbon. The molecular geometry used corresponded either to the total energy minimum of the triplet state at the UHF/ 6-31G* level, or to the configuration of the ferrocarbon crystal fragments [ 3 1.
METHODS Ab initio calculations of the lowest singlet (S) and triplet (T) states of molecules were performed using the programs MONSTERGAUSS 88 and GAUSSIAN 82 and the 6-31G* and 6-311G** polarized split-valence basis sets. Ab initio unrestricted Hartree-Fock gradient techniques, as incorporated in GAUSSIAN 82, using a 6-31G* basis set were used to optimize the geometry of the T state. The electron correlation energy was estimated by means of multiconfiguration self-consistent field (MC-SCF) and fourth-order Marller-Plesset perturbation theory (MP4 ) calculations. The MC-SCF method took into account only electron pair excitations, i.e. excitations of electron pairs from doubly occupied MOs to vacant MOs. The number of doubly occupied MOs in the S state exceeds by one the number in the T state. The same is true for vacant MOs. The active space (n x n) included the n highest occupied MOs (HOMOs) and n lowest unoccupied MOs (LUMOs). The most adequate comparison of the total energies of the S and T states corresponds to an equal number of “frozen” MOs, i.e. those not included in the active space [ 41. Thus the active space of the S state includes one more doubly occupied MO and one more vacant MO than the active space of the T state.
RESULTS The ST splitting values for C3H6and C.&H8 were determined for their various conformations using different basis sets and techniques of calculating the electron correlation energy. Energy minimization of C&H6was carried out within
143
TABLE 1
UHF andMP4 energies of CsHs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal
No.
M”
S/T
Symmetry
Energy (a.u.)/ST
& (deg)
MP4SDQ
HF 6-31G*
UHF/6-31G* minimum-energy geometry of 1 0 s C 2” 113.6 T Splitting 2 0 s C, 113.2 T Splitting 3 0 s C, 113.2 T Splitting
splitting (eV)
triplet - 116.847438 - 116.999692 4.143 - 116.855759 - 117.000422 3.936 - 116.840547 - 117.000259 4.346
Geometry extracted from ferrocarbon crystal 4 2 s C. 120.0 T Splitting 5 4 s CI 105.6 T Splitting
-116.847603 - 116.989979 3.874 - 116.869369 - 116.990002 3.283
6-311G**
6-31G*
6-311G**
- 116.886209 -117.031371 3.950 - 116.894126 - 117.031968 3.751 - 116.879271 - 117.031865 4.152
- 117.354156 - 117.382797 0.779 -117.353491 - 117.383132 0.807 - 117.347750 - 117.383176 0.964
- 117.440848 - 117.473882 0.899 - 117.441405 - 117.474334 0.896 - 117.434650 -117.474404 1.082
-116.886782 - 117.022078 3.682 - 116.906221 - 117.021756 3.144
-117.350862 -117.373320 0.611 - 117.344444 - 117.374166 0.809
-117.438650 - 117.464955 0.716 -117.434963 -117.465705 0.837
“The number of interactions of this kind for each sps carbon atom in the ferrocarbon crystal.
the framework of the C,,, C, and C, point symmetry groups. The molecular geometry of CIHS was defined with a fixed value of the C-C-C-C dihedral angle: @= 0’ and @= 60’ corresponding to the cis and gauche conformations (C, and C, symmetry point groups) respectively. The results of the ab initio calculations performed on the molecules under investigation using the GAUSSIAN 82 program are listed in Tables 1 and 2. Electron correlation was taken into account by means of MP4 perturbation theory calculations. The results obtained with the MONSTERGAUSS 88 program are listed in Tables 3 and 4. Electron correlation was taken into account by means of the MC-SCF method. All the physically non-equivalent geometries of C&H, and CIH8 characteristic of the ferrocarbon crystal were also investigated, the results are summarized in Tables l-4. The C3H6 molecule which was used to model the l-3 exchange interaction within the same quasi-graphite plane of the crystal was considered in two configurations. The C!,H, molecule which was used to model
Ma
S/T
Conformation
‘See footnote to Table 1.
Geometry extracted from the ferrocarbon crystal 104.0 3 1 s Cis T Splitting 118.6 4 1 s Cis T Splitting 104.0 5 2 s Eclipsed T Splitting 104.0 6 4 s Eclipsed T Splitting 0.0
0.0
126.4
116.8
118.6
104.0
118.6
0.0
117.3
104.0
60.0
114.2
- 155.941809 - 155.972566 0.837 - 155.842460 - 156.011329 4.595 - 155.862421 - 156.004377 3.863 - 155.851693 - 156.008443 4.265
- 155.865822 - 156.033158 4.553 - 155.884141 - 156.024362 3.816
6-31G*
- 155.985924 - 156.013697 0.756 - 155.888523 - 156.052649 4.466 - 155.909659 - 156.045712 3.702 - 155.898892 - 156.049661 4.108
- 155.914191 - 156.074112 4.352 - 155.932755 - 156.064999 3.598
6-311G**
HF
@
0,
6s
Energy (a.u.)/ST splitting (eV)
Angle (deg)
UHFf6-31G* minimum-energy geometry of triplet 114.2 1 0 s Gauche T Splitting 117.3 2 0 s Cis T Splitting
No.
- 156.537615 - 156.504126 -0.911 - 156.498045 - 156.538419 1.099 - 156.513850 - 156.533224 0.527 - 156.509536 - 156.536387 0.731
- 156.540385 - 156.559051 0.508 - 156.528809 - 156.549908 0.574
6-31G*
MP4SDQ
- 156.660626 - 156.611203 - 1.345 - 156.615719 - 156.645682 0.815 - 156.631605 - 156.640712 0.248 - 156.626798 - 156.643835 0.464
- 156.666747 0.348 - 156.646092 - 156.656711 0.289
- 156.653942
6-311G**
UHF and MP4 energies of CIHs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal
TABLE 2
g
145 TABLE 3 Energies of C3Hs in the singlet (S) and triplet (T) states at the UHF/6-31G* tries of the triplet Symmetry
C2”
0, (deg)
T
S
ST splitting (ev)
Active space
MC-SCF energy (a.u.)
116.98872 116.99094 116.99770 117.00204 117.00381 117.00786 117.01382 117.01266 117.01945
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
116.99969 116.99441 117.00741 117.01543 117.01866 117.00963 117.03599 117.03527 117.04540
0.298
-
116.99023 116.99393 116.99942 117.00396 117.00572 117.00963 117.01570 117.01761 117.02174
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
117.00042 116.99860 117.00874 117.01698 117.02007 117.02675 117.03692 117.04008 117.04672
0.277 0.127 0.253 0.354 0.390 0.466 0.578 0.611 0.680
-
116.98894 116.99090 116.99725 117.00454 117.00636 117.01008 117.01817 117.01994 117.02364
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
117.00026 116.99532 117.00840 117.02249 117.02595 117.03276 117.04747 117.05055 117.05678
0.308 0.120 0.303 0.489 0.533 0.617 0.797 0.833 0.902
Active space
MC-SCF energy (a.u.)
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
-
113.2
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
113.2
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
113.6
optimized geome-
0.094
0.264 0.359 0.404 0.487 0.603 0.615 0.706
the l-4 exchange interaction between radicals of adjacent quasi-graphite planes was considered in four configurations. The number A4 of such interactions for each sp2 radical centre of the crystal is listed in Tables 1 and 2. In these cases the C-H bond orientations were established in accordance with the “cut-off’ C-C bond orientations of the crystal structure.
146 TABLE 4 Energies of C,Hs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal Conformation
Angle (deg) 6,
6,
S @
Active space
T MC-SCF energy (a.u.)
UHF/6-31G* minimum-energy geometry of triplet Gauche 114.2 114.2 60.0 1x1 - 156.02421 2x2 - 156.03175 3x3 - 156.03901 4x4 - 156.04646 5x5 - 156.05461 7x7 -156.07104 8x8 - 156.07914 9x9 - 156.08856 117.3
MC-SCF energy (a.u.)
UHF 1x1 2x2 3x 3 4x4 6x6 7x7 8X8
-
156.03316 156.03946 156.05342 156.06716 156.08163 156.10916 156.12296 156.13656
0.244 0.210 0.392 0.563 0.735 1.037 1.176 1.306
156.01763 156.02626 156.03441 156.04271 156.05210
UHF 1x1 2x2 3x3 4x4
-
156.02436 156.03116 156.04502 156.05861 156.07355
0.183 0.133 0.289 0.433 0.584
Geometry extracted from the ferrocarbon crystal Cis 104.0 104.0 0.0 1x1 2x2 3x3 4x4 5x5 8X8 -
156.00634 156.01900 156.03091 156.04385 156.06835 156.09262
UHF 1x1 2x2 3x 3 4x4 7x7
-
155.97256 155.97763 155.99185 155.00661 155.03574 155.06325
-0.919 - 1.126 - 1.063 - 1.013 - 0.887 - 0.799
- 155.99775 - 156.00605 - 156.01443 -156.02298 - 156.03232 - 156.04183 - 156.06983
UHF 1x1 2x2 3x3 4x4 5x5 8X8
-
156.00438 156.00994 156.02418 156.03821 156.05301 156.06851 156.10816
0.180 0.106 0.265 0.414 0.563 0.741 1.043
Eclipsed
104.0
104.0
0.0 1x1 2x2 3x3 4x4 5x5
Active space
-
CiS
117.3
ST splitting (ev)
126.4 lx 1 2x2 3x3 4x4 5x5 6X6 9x9
DISCUSSION
It can be seen from Tables l-4 that the ST splitting value for all the calculated C,H, and C4H8 configurations studied cannot be determined even qual-
147
itatively at the Hartree-Fock level using a one-determinant wavefunction for the S state. As electron correlation is taken into account its value diminishes by 3.0-3.5 eV which is easily explained in the analysis of the complete wavefunction obtained within the framework of the MC-SCF approach for each molecule. Thus, if the active space for the S state is expanded from 1 x 1 to 9 x 9, the wavefunction of this state appears to be actually determined for each C3Hs and C4Hs conformation by two reference determinants entering into the function with weightings of 0.66-0.74. The T state is defined by a practically one-determinant wavefunction: if the active space is expanded from 1 x 1 to 8 x 8 for the T state of both molecules the weighting of the determinant is 0.950.99, with negligible ( 10-4-10-2) weightings for the other determinants. The distance between the sp2 carbon atoms varies in the molecules considered from 2.29 to 3.55 A, which greatly exceeds the usual length of a single CC bond (1.54 A). In this case the HOMO and LUMO appear to be approximately degenerate and their occupation by an electron pair is almost equally probable. Hence an adequate description of the S state requires the use of a two-determinant wavefunction, whilst one determinant is sufficient to describe the T state. The energy difference between the singlet (two-determinant wavefunction) and the triplet in the UHF approximation is the value of ST splitting at the Hartree-Fock level with wavefunctions of correct symmetry (see also ref. 4). It can be seen from Tables 3 and 4 that the value of the ST splitting does not reach saturation even upon maximum expansion of the active space for all C&H6and C4H8 configurations. For C&H6the ST splitting varies from 0.3 to 0.7-0.9 eV, and for C4Hs from 0.2 to l-O-1.3 eV, with the exception of the cis configuration (Table 2, no. 3) where the S is the ground state. The value of the ST splitting lacks saturation because the MC-SCF method takes into account a small amount of electron correlation energy due to the fact that the exclusive consideration of pair excitations is obviously insufficient. Nevertheless, the MC-SCF method provides results qualitatively similar to those provided by MP4 perturbation theory. First, both approaches yield the same sign for the ST splitting for all the configurations considered. For one configuration (Table 2, no. 3) both approaches predict an S ground state. This configuration is a fragment of a six-membered ring in the boat form, the sp2 carbon atoms being displaced to the vertices of the boat. Unlike all other C,H, configurations, the bond order between these atoms is non-zero and is approximately 0.46. Thus, owing to the small distance between atoms 1 and 4 in the given configuration (R = 2.29 A), a chemical bond leading to the formation of cyclobutane “starts” to form between them, leading to a sharp decrease in the S energy. All the other C,H, configurations and all the C&H6configurations investigated possess a T ground state, irrespective of the method used to account for electron correlation. Secondly, in all the configurations considered
148
the ST splitting obtained with the MC-SCF method at a moderate expansion of the active space approximates the results obtained with the MP4 approach. The results obtained testify to the fact that the value of the ST splitting is rather sensitive to the method used to account for electron correlation, which is consistent with the data obtained in ab initio investigations of CH, [5-71 and HCO,+ [ 41. The value of the ST splitting is known to converge very slowly to the experimentally observed value by increasing the size of the basis set and the accuracy of the electron correlation energy calculation. In order to reproduce precisely the experimentally observed value of the ST splitting of CH, [ 7,8] extremely large basis sets (including g functions), complete geometry optimization and second-order configuration interaction (CI ) were used. Compared with such comprehensive investigations, the basis sets and methods of calculating the electron correlation energy used in the present work seem quite poor. Nevertheless, let us determine the boundaries of the ideal values of the vertical ST splitting for C4Hs and C3H6.To this end, analogous calculations were undertaken for O2 and CHz for which the ST splitting has been determined experimentally [ 8,9]. The results of the ab initio investigations of the ST splitting for O2 and CH2 are listed in Tables 5 and 6. The molecular geometry corresponded to the energy minimum at UHF/6-31G* level. The results given in Tables 5 and 6 can be used to compare how efficiently methods such as MC-SCF, MP4 and CISD take into account electron correlation with regard to reproducing the experimentally determined ST splitting value. The expansion of the active space in the MC-SCF method leads to a good approximation of the ideal value for CH2, but not for 0,. As CHz contains only three doubly occupied MOs it is impossible to identify the “subsequent” TABLE 5 Comparison of various ab initio techniques for calculating the ST splitting of CH, Level
S (a.u.)
T (a.u.)
Splitting (eV)
HF/6-31G* MP2/6-31G* MP3/6-31G* MP4/6-31G* CISD/G-31G* CISD/G-311G** MP4/6-311G** MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G*
- 38.872370 - 38.969886 - 38.987720 - 38.991986 - 38.993677 - 39.042067 -39.028103 - 38.893435 - 38.909401 - 38.925016 -38.926355
- 38.921497
1.337 0.910 0.803 0.760 0.775 0.674 0.634 0.764 0.568 0.494 0.496 0.390
EXP.
181
(Sl x 1) @2x2) @3x3) @4x4)
-39.003312 -39.017221 -39.019911 - 39.022169 - 39.066848 -39.051393 -38.921497 - 38.930257 - 38.943153 - 38.944591
149 TABLE 6 Comparison of the various ab initio techniques for calculating the ST splitting of O1 Level
S (a.u.)
T (a.u.)
Splitting (eV)
HF/6-31G* MP2/6-31G* MP3/6-31G* MP4/6-31G* CISD/G-31G* CISD/G-311G* MP4/6-311G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G*
- 149.532997
-
2.311 1.491 1.578 1.529 1.652 1.634 1.481 1.806 1.489 0.155 0.127 0.060 0.024 0.048 0.054 0.974
-
(Sl x 1) (52 x 2) (S3x3) (S4x4) (S5 x 5) @6x6) (57 x 7) @8x8)
149.886537 149.881832 149.891582 149.865828 149.971115 149.969999 149.550709 149.566883 149.618549 149.621327 149.625698 149.628668 149.629753 149.631889
EXP. [91
149.617908 149.941328 149.939833 149.947771 149.926540 150.031158 150.024424 149.617082 149.621619 149.624232 149.626029 149.627893 149.629557 149.631533 149.633869
TABLE 7 Dependence of the ST splitting at the MP4SDQ/6-311G** hybridized carbon atoms No.
Interatomic distance (A)
1 2 3 4 5 6 7 8 9 10 11
2.288 2.398 2.504 2.507 2.509 2.608 2.937 3.001 3.088 3.469 3.552
level on the separation between sp’
ST splitting (eV) 1.345 0.837 1.082 0.896 0.899 0.716 0.289 0.815 0.348 0.248 0.464
behaviour of the ST splitting. However, the MC-SCF method can be said to give results that are generally in the right direction. The other two approaches give similar results, but in both cases MP4 appears to be more efficient than
150
CISD. The expansion of the basis set in both cases improves the results obtained with the MP4 and CISD methods. Now let us estimatethe bounds of the ideal valuesof the ST splittingfor the molecules studied, assuming that the above noted features of the MP4 and MC-SCF methods are applicable here also. The data listed in Table 1 testify to the fact that expansionof the basis set leads to increasedST splittingvalues (on averageby 12% ) obtained with the MP4 method for all C3Hsconfigurations. Conversely,the ST splittingvaluesfor all C4Hsconfigurationsdecrease (on averageby 41% ) upon the expansion of the basis set (Table 2). In all cases the sign of the splittingdoes not change. The ST splitting values for C,H, provided by the MC-SCF and MP4 techniqueson expansion of the basis set exhibit the same direction of the trend. In fact, splittingsin this case may be even largerthan those presentedin Table 1. The directionof the trendsfor all C4Hsconfigurationsarethe opposite to those for C3H6:expansion of the active space in the MC-SCF approach leads to an increasein splitting,while expansion of the basis set leads to a decreasein the splitting values at the MP4 level. We consider the results obtained with the MP4 method to be the most reliableones. The ideal ST splittingfor the C4H8 configurationsis apparentlybetween0.15 and 0.40 eV, excludingconfiguration 3 (Table 2) where the singlet has a lower energy than the triplet by approximately 1.5 eV. More precise resultscan be obtained only after the application of still largerbasis sets and more accuratemethods of calculatingthe electron correlationenergy (CAS-SCF, MRD-CI or full CI). It is well known that the ST splitting is equal to the exchange integralbetween two radical centres. As these centres are infinitely separated the ST splittingshould vanish. It is naturalto assumethat this tendency will also be seen at finite distances. The distance between sp2 hybridized atoms and the correspondingvalues of the ST splittingat the MP4/6-311G** level are given in Table 7 for all the molecular systems investigated.The expected tendency towardsdecreasedsplittingwith increasingdistancebetweensp2carbon atoms is not observed. Consequently,a more important factor for determining ST splitting at such distances is the mutual orientation of the singly occupied p orbitalsof the sp2atoms. The tendencytowardsdecreasedsplittingcan be seen by comparingmoleculesthat have the same mutualorientationof these MOs. Indeed,in two configurationsof C3H6(Table 1, nos. 1 and 5) these orbitals are paralleland as the C-C-C angle increasesfrom 113.6” to 120.0” the ST splitting decreasesfrom 0.899 to 0.716 eV. It has been found that two cis configurations (Table 2, nos. 2 and 4) of C,Hs have the largest and smallest (negative) splittingsamong the configurations considered. The main factor determiningthe ST splitting for these contigurations is the C-C-C bond angle. When this angle decreasesto 90” the singly occupied p orbitals tend to arrangein one straight line and an ordinary twoelectron a-bond is formed.
151
In summary, we carried out ab initio investigations of the exchange interactions between pairs of radicals located at sp’ carbon atoms and separated by one and two sp2 carbon atoms in the intermediate graphite-diamond structure which has been suggested previously to be a ferromagnetic three-dimensional system. All six l-3 interactions of each radical centre in the structure exhibit ferromagnetic exchange. Of the eight l-4 interactions seven exhibit ferromagnetic and one exhibits antiferromagnetic exchange. These results prove the existence of three-dimensional ferromagnetic ordering in the given carbon phase. In future work we will consider this three-dimensional ferromagnet in the mean-field approximation. ACKNOWLEDGEMENTS
Helpful discussions with Professor A.I. Boldyrev are much appreciated. The help of Dr. V.G. Zakzhevsky in installing the FORTRAN version of GAUSSIAN 82 is acknowledged.
REFERENCES 1 A.A. Ovchinnikov and V.N. Spector, in T. Saegusa, T. Higashimura and A. Abe (Eds.), Frontiers of Macromolecular Science, IUPAC, Blackwell Scientific Publications, 1989, p. 455. 2 A.A. Ovchinnikov and V.N. Spector, Synth. Metals, 27 (1988) B615. 3 A.A. Ovchinnikov and I.L. Shamovsky, J. Mol. Struct. (Theochem), 251 (1991) 133. 4 M. Scarlett and P.R. Taylor, Chem. Phys., 101 (1986) 17. 5 E.A. Carter and W.A. Goddard III, J. Chem. Phys., 86 (1987) 862. 6 E.A. Carter and W.A. Goddard III, J. Chem. Phys., 88 (1988) 1752. 7 C.W. Bauschlicher, S.R. Langhoff and P.R. Taylor, J. Chem. Phys., 87 (1987) 387. 8 P.R. Bunker, P. Jensen, W.P. Kracmer and R. Beardsworth, J. Chem. Phys., 85 (1986) 3724. 9 F.L. Gilmore, Quantum Spectrosc. Radiat. Trans., 5 (1965) 369.
141
Ab initio molecular orbital studies on a singlet-triplet splitting of C,H, and C,H, molecules A.A. Ovchinnikov,
I.L. Shamovsky and K.V. Bozhenko
Institute of Chemical Physics of the USSR Academy of Sciences, Kosygin str., 4, Moscow 117334 (USSR) (Received 22 March 1991)
Abstract The vertical singlet-triplet (ST) splitting of C3H6 and C4H8 molecules containing two sp’ carbon atoms separated by one or two methylene groups was investigated by means of ab initio calculations. The molecular geometry was either taken as that corresponding to the UHF/6-31G* energy minimum of the triplet electron configuration, or extracted from the structure of the ferrocarbon crystal, which has been determined previously by the molecular mechanics method. Different basis sets (6-31G* and 6-31 lG**) and various techniques of taking into account electron correlation (multiconfiguration self-consistent field (MC-SCF) and fourth-order Msller-Plesset (MP4) ) were applied. The MP4 method provides more reliable ST splitting values than does the MC-SCF method which involves only pair excitations, but MP4 requires the use of an extremely extended basis set if quantitative estimates are to be obtained. The ST splitting appears to be determined not only by the distance between sp’ carbon centres but also to a great extent by the mutual orientation of these centres. The presence of three-dimensional ferromagnetic ordering in the previously suggested intermediate graphite-diamond structure is proved.
INTRODUCTION
It has been suggested [ 1,2 ] that there is a structure of the crystalline carbon phase which should exhibit ferromagnetic properties. This phase, which has a structure intermediate between that of diamagnetic graphite and diamond, possesses the maximum possible concentration of unpaired electron carriers, and its theoretically expected magnetization exceeds the maximum magnetization of a-Fe. By using the molecular mechanics technique it has been established [ 31 that the minimum potential energy of a crystal having such a structure corresponds to the orthorhombic symmetry (ar = p= y= 90 o ). In addition, the coordinates of the atoms in the unit cell and the parameters of the unit cell itself (a, b and c) have been determined. Each sp2 carbon atom was found to be separated by two covalent bonds from six analogous sp’ atoms and by three bonds from eight analogous atoms; it should be noted that there are two and
0166-1280/91/$03.50
0 1991 Elsevier Science Publishers
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142 four physically non-equivalent l-3 and l-4 exchange interactions between radicals localized on the sp2 carbon atoms, respectively. To prove the ferromagnetic ordering in the structure under investigation, it is necessary and sufficient to prove the presence of ferromagnetic exchange between two radicals in the elementary fragments [ 21. The established ferromagnetic properties of such a material suggest that most fragments containing two adjacent sp2 atoms possess a triplet ground state. In the previous study ab initio molecular orbital (MO) calculations were performed to determine the vertical singlet-triplet (ST) splitting in C&H6and C&H*.These two molecules were used to simulate l-3 and l-4 exchange interactions between radicals in the ferrocarbon. The molecular geometry used corresponded either to the total energy minimum of the triplet state at the UHF/ 6-31G* level, or to the configuration of the ferrocarbon crystal fragments [ 3 1.
METHODS Ab initio calculations of the lowest singlet (S) and triplet (T) states of molecules were performed using the programs MONSTERGAUSS 88 and GAUSSIAN 82 and the 6-31G* and 6-311G** polarized split-valence basis sets. Ab initio unrestricted Hartree-Fock gradient techniques, as incorporated in GAUSSIAN 82, using a 6-31G* basis set were used to optimize the geometry of the T state. The electron correlation energy was estimated by means of multiconfiguration self-consistent field (MC-SCF) and fourth-order Marller-Plesset perturbation theory (MP4 ) calculations. The MC-SCF method took into account only electron pair excitations, i.e. excitations of electron pairs from doubly occupied MOs to vacant MOs. The number of doubly occupied MOs in the S state exceeds by one the number in the T state. The same is true for vacant MOs. The active space (n x n) included the n highest occupied MOs (HOMOs) and n lowest unoccupied MOs (LUMOs). The most adequate comparison of the total energies of the S and T states corresponds to an equal number of “frozen” MOs, i.e. those not included in the active space [ 41. Thus the active space of the S state includes one more doubly occupied MO and one more vacant MO than the active space of the T state.
RESULTS The ST splitting values for C3H6and C.&H8 were determined for their various conformations using different basis sets and techniques of calculating the electron correlation energy. Energy minimization of C&H6was carried out within
143
TABLE 1
UHF andMP4 energies of CsHs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal
No.
M”
S/T
Symmetry
Energy (a.u.)/ST
& (deg)
MP4SDQ
HF 6-31G*
UHF/6-31G* minimum-energy geometry of 1 0 s C 2” 113.6 T Splitting 2 0 s C, 113.2 T Splitting 3 0 s C, 113.2 T Splitting
splitting (eV)
triplet - 116.847438 - 116.999692 4.143 - 116.855759 - 117.000422 3.936 - 116.840547 - 117.000259 4.346
Geometry extracted from ferrocarbon crystal 4 2 s C. 120.0 T Splitting 5 4 s CI 105.6 T Splitting
-116.847603 - 116.989979 3.874 - 116.869369 - 116.990002 3.283
6-311G**
6-31G*
6-311G**
- 116.886209 -117.031371 3.950 - 116.894126 - 117.031968 3.751 - 116.879271 - 117.031865 4.152
- 117.354156 - 117.382797 0.779 -117.353491 - 117.383132 0.807 - 117.347750 - 117.383176 0.964
- 117.440848 - 117.473882 0.899 - 117.441405 - 117.474334 0.896 - 117.434650 -117.474404 1.082
-116.886782 - 117.022078 3.682 - 116.906221 - 117.021756 3.144
-117.350862 -117.373320 0.611 - 117.344444 - 117.374166 0.809
-117.438650 - 117.464955 0.716 -117.434963 -117.465705 0.837
“The number of interactions of this kind for each sps carbon atom in the ferrocarbon crystal.
the framework of the C,,, C, and C, point symmetry groups. The molecular geometry of CIHS was defined with a fixed value of the C-C-C-C dihedral angle: @= 0’ and @= 60’ corresponding to the cis and gauche conformations (C, and C, symmetry point groups) respectively. The results of the ab initio calculations performed on the molecules under investigation using the GAUSSIAN 82 program are listed in Tables 1 and 2. Electron correlation was taken into account by means of MP4 perturbation theory calculations. The results obtained with the MONSTERGAUSS 88 program are listed in Tables 3 and 4. Electron correlation was taken into account by means of the MC-SCF method. All the physically non-equivalent geometries of C&H, and CIH8 characteristic of the ferrocarbon crystal were also investigated, the results are summarized in Tables l-4. The C3H6 molecule which was used to model the l-3 exchange interaction within the same quasi-graphite plane of the crystal was considered in two configurations. The C!,H, molecule which was used to model
Ma
S/T
Conformation
‘See footnote to Table 1.
Geometry extracted from the ferrocarbon crystal 104.0 3 1 s Cis T Splitting 118.6 4 1 s Cis T Splitting 104.0 5 2 s Eclipsed T Splitting 104.0 6 4 s Eclipsed T Splitting 0.0
0.0
126.4
116.8
118.6
104.0
118.6
0.0
117.3
104.0
60.0
114.2
- 155.941809 - 155.972566 0.837 - 155.842460 - 156.011329 4.595 - 155.862421 - 156.004377 3.863 - 155.851693 - 156.008443 4.265
- 155.865822 - 156.033158 4.553 - 155.884141 - 156.024362 3.816
6-31G*
- 155.985924 - 156.013697 0.756 - 155.888523 - 156.052649 4.466 - 155.909659 - 156.045712 3.702 - 155.898892 - 156.049661 4.108
- 155.914191 - 156.074112 4.352 - 155.932755 - 156.064999 3.598
6-311G**
HF
@
0,
6s
Energy (a.u.)/ST splitting (eV)
Angle (deg)
UHFf6-31G* minimum-energy geometry of triplet 114.2 1 0 s Gauche T Splitting 117.3 2 0 s Cis T Splitting
No.
- 156.537615 - 156.504126 -0.911 - 156.498045 - 156.538419 1.099 - 156.513850 - 156.533224 0.527 - 156.509536 - 156.536387 0.731
- 156.540385 - 156.559051 0.508 - 156.528809 - 156.549908 0.574
6-31G*
MP4SDQ
- 156.660626 - 156.611203 - 1.345 - 156.615719 - 156.645682 0.815 - 156.631605 - 156.640712 0.248 - 156.626798 - 156.643835 0.464
- 156.666747 0.348 - 156.646092 - 156.656711 0.289
- 156.653942
6-311G**
UHF and MP4 energies of CIHs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal
TABLE 2
g
145 TABLE 3 Energies of C3Hs in the singlet (S) and triplet (T) states at the UHF/6-31G* tries of the triplet Symmetry
C2”
0, (deg)
T
S
ST splitting (ev)
Active space
MC-SCF energy (a.u.)
116.98872 116.99094 116.99770 117.00204 117.00381 117.00786 117.01382 117.01266 117.01945
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
116.99969 116.99441 117.00741 117.01543 117.01866 117.00963 117.03599 117.03527 117.04540
0.298
-
116.99023 116.99393 116.99942 117.00396 117.00572 117.00963 117.01570 117.01761 117.02174
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
117.00042 116.99860 117.00874 117.01698 117.02007 117.02675 117.03692 117.04008 117.04672
0.277 0.127 0.253 0.354 0.390 0.466 0.578 0.611 0.680
-
116.98894 116.99090 116.99725 117.00454 117.00636 117.01008 117.01817 117.01994 117.02364
UHF 1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8
-
117.00026 116.99532 117.00840 117.02249 117.02595 117.03276 117.04747 117.05055 117.05678
0.308 0.120 0.303 0.489 0.533 0.617 0.797 0.833 0.902
Active space
MC-SCF energy (a.u.)
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
-
113.2
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
113.2
1x1 2x2 3x3 4x4 5x5 6X6 7x7 8X8 9x9
113.6
optimized geome-
0.094
0.264 0.359 0.404 0.487 0.603 0.615 0.706
the l-4 exchange interaction between radicals of adjacent quasi-graphite planes was considered in four configurations. The number A4 of such interactions for each sp2 radical centre of the crystal is listed in Tables 1 and 2. In these cases the C-H bond orientations were established in accordance with the “cut-off’ C-C bond orientations of the crystal structure.
146 TABLE 4 Energies of C,Hs in the singlet (S) and triplet (T) states at the UHF/6-31G* optimized geometry of the triplet and at the geometry of the extracted fragment of the ferrocarbon crystal Conformation
Angle (deg) 6,
6,
S @
Active space
T MC-SCF energy (a.u.)
UHF/6-31G* minimum-energy geometry of triplet Gauche 114.2 114.2 60.0 1x1 - 156.02421 2x2 - 156.03175 3x3 - 156.03901 4x4 - 156.04646 5x5 - 156.05461 7x7 -156.07104 8x8 - 156.07914 9x9 - 156.08856 117.3
MC-SCF energy (a.u.)
UHF 1x1 2x2 3x 3 4x4 6x6 7x7 8X8
-
156.03316 156.03946 156.05342 156.06716 156.08163 156.10916 156.12296 156.13656
0.244 0.210 0.392 0.563 0.735 1.037 1.176 1.306
156.01763 156.02626 156.03441 156.04271 156.05210
UHF 1x1 2x2 3x3 4x4
-
156.02436 156.03116 156.04502 156.05861 156.07355
0.183 0.133 0.289 0.433 0.584
Geometry extracted from the ferrocarbon crystal Cis 104.0 104.0 0.0 1x1 2x2 3x3 4x4 5x5 8X8 -
156.00634 156.01900 156.03091 156.04385 156.06835 156.09262
UHF 1x1 2x2 3x 3 4x4 7x7
-
155.97256 155.97763 155.99185 155.00661 155.03574 155.06325
-0.919 - 1.126 - 1.063 - 1.013 - 0.887 - 0.799
- 155.99775 - 156.00605 - 156.01443 -156.02298 - 156.03232 - 156.04183 - 156.06983
UHF 1x1 2x2 3x3 4x4 5x5 8X8
-
156.00438 156.00994 156.02418 156.03821 156.05301 156.06851 156.10816
0.180 0.106 0.265 0.414 0.563 0.741 1.043
Eclipsed
104.0
104.0
0.0 1x1 2x2 3x3 4x4 5x5
Active space
-
CiS
117.3
ST splitting (ev)
126.4 lx 1 2x2 3x3 4x4 5x5 6X6 9x9
DISCUSSION
It can be seen from Tables l-4 that the ST splitting value for all the calculated C,H, and C4H8 configurations studied cannot be determined even qual-
147
itatively at the Hartree-Fock level using a one-determinant wavefunction for the S state. As electron correlation is taken into account its value diminishes by 3.0-3.5 eV which is easily explained in the analysis of the complete wavefunction obtained within the framework of the MC-SCF approach for each molecule. Thus, if the active space for the S state is expanded from 1 x 1 to 9 x 9, the wavefunction of this state appears to be actually determined for each C3Hs and C4Hs conformation by two reference determinants entering into the function with weightings of 0.66-0.74. The T state is defined by a practically one-determinant wavefunction: if the active space is expanded from 1 x 1 to 8 x 8 for the T state of both molecules the weighting of the determinant is 0.950.99, with negligible ( 10-4-10-2) weightings for the other determinants. The distance between the sp2 carbon atoms varies in the molecules considered from 2.29 to 3.55 A, which greatly exceeds the usual length of a single CC bond (1.54 A). In this case the HOMO and LUMO appear to be approximately degenerate and their occupation by an electron pair is almost equally probable. Hence an adequate description of the S state requires the use of a two-determinant wavefunction, whilst one determinant is sufficient to describe the T state. The energy difference between the singlet (two-determinant wavefunction) and the triplet in the UHF approximation is the value of ST splitting at the Hartree-Fock level with wavefunctions of correct symmetry (see also ref. 4). It can be seen from Tables 3 and 4 that the value of the ST splitting does not reach saturation even upon maximum expansion of the active space for all C&H6and C4H8 configurations. For C&H6the ST splitting varies from 0.3 to 0.7-0.9 eV, and for C4Hs from 0.2 to l-O-1.3 eV, with the exception of the cis configuration (Table 2, no. 3) where the S is the ground state. The value of the ST splitting lacks saturation because the MC-SCF method takes into account a small amount of electron correlation energy due to the fact that the exclusive consideration of pair excitations is obviously insufficient. Nevertheless, the MC-SCF method provides results qualitatively similar to those provided by MP4 perturbation theory. First, both approaches yield the same sign for the ST splitting for all the configurations considered. For one configuration (Table 2, no. 3) both approaches predict an S ground state. This configuration is a fragment of a six-membered ring in the boat form, the sp2 carbon atoms being displaced to the vertices of the boat. Unlike all other C,H, configurations, the bond order between these atoms is non-zero and is approximately 0.46. Thus, owing to the small distance between atoms 1 and 4 in the given configuration (R = 2.29 A), a chemical bond leading to the formation of cyclobutane “starts” to form between them, leading to a sharp decrease in the S energy. All the other C,H, configurations and all the C&H6configurations investigated possess a T ground state, irrespective of the method used to account for electron correlation. Secondly, in all the configurations considered
148
the ST splitting obtained with the MC-SCF method at a moderate expansion of the active space approximates the results obtained with the MP4 approach. The results obtained testify to the fact that the value of the ST splitting is rather sensitive to the method used to account for electron correlation, which is consistent with the data obtained in ab initio investigations of CH, [5-71 and HCO,+ [ 41. The value of the ST splitting is known to converge very slowly to the experimentally observed value by increasing the size of the basis set and the accuracy of the electron correlation energy calculation. In order to reproduce precisely the experimentally observed value of the ST splitting of CH, [ 7,8] extremely large basis sets (including g functions), complete geometry optimization and second-order configuration interaction (CI ) were used. Compared with such comprehensive investigations, the basis sets and methods of calculating the electron correlation energy used in the present work seem quite poor. Nevertheless, let us determine the boundaries of the ideal values of the vertical ST splitting for C4Hs and C3H6.To this end, analogous calculations were undertaken for O2 and CHz for which the ST splitting has been determined experimentally [ 8,9]. The results of the ab initio investigations of the ST splitting for O2 and CH2 are listed in Tables 5 and 6. The molecular geometry corresponded to the energy minimum at UHF/6-31G* level. The results given in Tables 5 and 6 can be used to compare how efficiently methods such as MC-SCF, MP4 and CISD take into account electron correlation with regard to reproducing the experimentally determined ST splitting value. The expansion of the active space in the MC-SCF method leads to a good approximation of the ideal value for CH2, but not for 0,. As CHz contains only three doubly occupied MOs it is impossible to identify the “subsequent” TABLE 5 Comparison of various ab initio techniques for calculating the ST splitting of CH, Level
S (a.u.)
T (a.u.)
Splitting (eV)
HF/6-31G* MP2/6-31G* MP3/6-31G* MP4/6-31G* CISD/G-31G* CISD/G-311G** MP4/6-311G** MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G*
- 38.872370 - 38.969886 - 38.987720 - 38.991986 - 38.993677 - 39.042067 -39.028103 - 38.893435 - 38.909401 - 38.925016 -38.926355
- 38.921497
1.337 0.910 0.803 0.760 0.775 0.674 0.634 0.764 0.568 0.494 0.496 0.390
EXP.
181
(Sl x 1) @2x2) @3x3) @4x4)
-39.003312 -39.017221 -39.019911 - 39.022169 - 39.066848 -39.051393 -38.921497 - 38.930257 - 38.943153 - 38.944591
149 TABLE 6 Comparison of the various ab initio techniques for calculating the ST splitting of O1 Level
S (a.u.)
T (a.u.)
Splitting (eV)
HF/6-31G* MP2/6-31G* MP3/6-31G* MP4/6-31G* CISD/G-31G* CISD/G-311G* MP4/6-311G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G* MC-SCF/G-31G*
- 149.532997
-
2.311 1.491 1.578 1.529 1.652 1.634 1.481 1.806 1.489 0.155 0.127 0.060 0.024 0.048 0.054 0.974
-
(Sl x 1) (52 x 2) (S3x3) (S4x4) (S5 x 5) @6x6) (57 x 7) @8x8)
149.886537 149.881832 149.891582 149.865828 149.971115 149.969999 149.550709 149.566883 149.618549 149.621327 149.625698 149.628668 149.629753 149.631889
EXP. [91
149.617908 149.941328 149.939833 149.947771 149.926540 150.031158 150.024424 149.617082 149.621619 149.624232 149.626029 149.627893 149.629557 149.631533 149.633869
TABLE 7 Dependence of the ST splitting at the MP4SDQ/6-311G** hybridized carbon atoms No.
Interatomic distance (A)
1 2 3 4 5 6 7 8 9 10 11
2.288 2.398 2.504 2.507 2.509 2.608 2.937 3.001 3.088 3.469 3.552
level on the separation between sp’
ST splitting (eV) 1.345 0.837 1.082 0.896 0.899 0.716 0.289 0.815 0.348 0.248 0.464
behaviour of the ST splitting. However, the MC-SCF method can be said to give results that are generally in the right direction. The other two approaches give similar results, but in both cases MP4 appears to be more efficient than
150
CISD. The expansion of the basis set in both cases improves the results obtained with the MP4 and CISD methods. Now let us estimatethe bounds of the ideal valuesof the ST splittingfor the molecules studied, assuming that the above noted features of the MP4 and MC-SCF methods are applicable here also. The data listed in Table 1 testify to the fact that expansionof the basis set leads to increasedST splittingvalues (on averageby 12% ) obtained with the MP4 method for all C3Hsconfigurations. Conversely,the ST splittingvaluesfor all C4Hsconfigurationsdecrease (on averageby 41% ) upon the expansion of the basis set (Table 2). In all cases the sign of the splittingdoes not change. The ST splitting values for C,H, provided by the MC-SCF and MP4 techniqueson expansion of the basis set exhibit the same direction of the trend. In fact, splittingsin this case may be even largerthan those presentedin Table 1. The directionof the trendsfor all C4Hsconfigurationsarethe opposite to those for C3H6:expansion of the active space in the MC-SCF approach leads to an increasein splitting,while expansion of the basis set leads to a decreasein the splitting values at the MP4 level. We consider the results obtained with the MP4 method to be the most reliableones. The ideal ST splittingfor the C4H8 configurationsis apparentlybetween0.15 and 0.40 eV, excludingconfiguration 3 (Table 2) where the singlet has a lower energy than the triplet by approximately 1.5 eV. More precise resultscan be obtained only after the application of still largerbasis sets and more accuratemethods of calculatingthe electron correlationenergy (CAS-SCF, MRD-CI or full CI). It is well known that the ST splitting is equal to the exchange integralbetween two radical centres. As these centres are infinitely separated the ST splittingshould vanish. It is naturalto assumethat this tendency will also be seen at finite distances. The distance between sp2 hybridized atoms and the correspondingvalues of the ST splittingat the MP4/6-311G** level are given in Table 7 for all the molecular systems investigated.The expected tendency towardsdecreasedsplittingwith increasingdistancebetweensp2carbon atoms is not observed. Consequently,a more important factor for determining ST splitting at such distances is the mutual orientation of the singly occupied p orbitalsof the sp2atoms. The tendencytowardsdecreasedsplittingcan be seen by comparingmoleculesthat have the same mutualorientationof these MOs. Indeed,in two configurationsof C3H6(Table 1, nos. 1 and 5) these orbitals are paralleland as the C-C-C angle increasesfrom 113.6” to 120.0” the ST splitting decreasesfrom 0.899 to 0.716 eV. It has been found that two cis configurations (Table 2, nos. 2 and 4) of C,Hs have the largest and smallest (negative) splittingsamong the configurations considered. The main factor determiningthe ST splitting for these contigurations is the C-C-C bond angle. When this angle decreasesto 90” the singly occupied p orbitals tend to arrangein one straight line and an ordinary twoelectron a-bond is formed.
151
In summary, we carried out ab initio investigations of the exchange interactions between pairs of radicals located at sp’ carbon atoms and separated by one and two sp2 carbon atoms in the intermediate graphite-diamond structure which has been suggested previously to be a ferromagnetic three-dimensional system. All six l-3 interactions of each radical centre in the structure exhibit ferromagnetic exchange. Of the eight l-4 interactions seven exhibit ferromagnetic and one exhibits antiferromagnetic exchange. These results prove the existence of three-dimensional ferromagnetic ordering in the given carbon phase. In future work we will consider this three-dimensional ferromagnet in the mean-field approximation. ACKNOWLEDGEMENTS
Helpful discussions with Professor A.I. Boldyrev are much appreciated. The help of Dr. V.G. Zakzhevsky in installing the FORTRAN version of GAUSSIAN 82 is acknowledged.
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