Partitioning of methyl internal rotational barrier energy of thioacetaldehyde

Transcription

1 Proc. Indian Acad. Sci. (Chem. Sci.), Vol. 114, No. 1, Feruary 2002, pp Indian Academy of Sciences Partitioning of methyl internal rotational arrier energy of thioacetaldehyde 1. Introduction NIKHIL GUCHHAIT Department of Chemistry, Indian Institute of Technology, North Guwahati, Guwahati , India MS received 30 January 2001; revised 2 July 2001 Astract. The nature of methyl internal rotational arrier in thioacetaldehyde has een investigated y relaxation effect, natural ond orital (NBO) analysis and Pauling exchange interactions. The true experimental arrier can e otained y considering fully relaxed rotation. Nuclear-electron attraction term is a arrier forming term in the fully relaxed rotation, ut it appears as an antiarrier for rigid rotation. It is seen that during methyl rotation, the torsional mode is coupled with the aldehydic hydrogen out-of-plane wagging motion. Natural ond orital analysis shows that the principal arrier forming term originates from the C C ond. The lengthening of the C C ond is explained y considering charge transfer interaction etween several onding and antionding oritals in the C C ond region, which leads to higher onding overlap for the eclipsed conformer compared to the staggered conformer. S C(σ)/C me H ip and C H ald /C me H op interactions appear to e the main arrier-forming Pauling exchange terms ut have less contriution to make to the arrier compared to the C C ond interaction. Keywords. Methyl rotation; natural ond oritals; Pauling exchange interaction; thioacetaldehyde. Study of internal rotational arriers of small groups has drawn much attention from oth theoretical and experimental points of view for the last few decades. Experimental information on the arrier and the dynamics on methyl internal rotation is ecoming increasingly availale 1,2. Laser spectroscopic techniques have een coupled with improved system preparation techniques to understand more aout the internal rotation in the ground and higher excited states of molecules. Internal rotation has een studied from the theoretical point of view and the potential energy surfaces governing large amplitude motions have een calculated directly y a initio molecular orital procedures 3 6. Recently, the role of lone-pair electrons on methyl internal rotational arrier of dimethyl ether and its homologues has een analysed theoretically. A initio calculations have een carried out and various contriutions to the arrier (e.g. Pauling exchange steric repulsion, σ-lone-pair reorganization and π-hyperconjugation) have een dissected 7,8. The comined results of natural ond orital, symmetry partitioning and relaxation analyses demonstrate that the σ-lone pair plays an important role in controlling the rotational arrier. The methyl rotational arriers for acetaldehyde and acetone were chosen for in-depth analysis ecause these molecules are simple and exhiit highly 61

2 62 Nikhil Guchhait resolved microwave, Raman and Ryderg spectra 9,10. For example, the methyl rotational arrier for acetaldehyde estimated from the well-resolved microwave and IR spectra was found to e 400 cm 1 (Hougen et al) 11. Simulated a initio calculations revealed the importance of coupling etween methyl torsion and acetaldehyde-hydrogen wagging as a determinant for the arrier shape 12. Well-estalishd π-fragment model y Hehre, Pople and Devaquet 13 proposed that methyl rotational arrier in acetaldehyde is ased on π- interaction. Another suggestion regarding methyl rotational arrier in acetaldehyde included Pauling repulsion forces etween the electron clouds in C H onds involving the C me H CH ald in the metastale state, reaking of the weak covalent ond etween caronyl oxygen lone-pair and the eclipsed methyl hydrogen, and dipole polarization effect on the methyl C H onds driven y the large C=O dipole 14. Goodman et al 15 carried out a initio calculations at several levels and decomposed the arrier energy for methyl internal rotation in acetaldehyde into σ and π symmetry components. These led them to conclude that the σ component of the V ne is the principal arrier forming term. This is, however, contrary to the results from the π-fragment model 13. Very recent NBO analysis 16 showed that the large arrier-forming energy terms arise from the weakening of C C and methyl C H out-of-plane onds. It was also shown that the weakening of the C C ond in acetaldehyde originated from the antiond character introduced into the C C ond region y charge transfer etween the C me H ip onding and C H ald antionding oritals. Thioacetaldehyde is the sulphur analogue of acetaldehyde where oxygen atom of the acetaldehyde is replaced y sulphur atom. Spectroscopic studies on acetaldehyde and acetone were competently carried out in order to know the frequency of torsional and other modes for oth ground and excited states. Interest on thiocaronyl molecules has grown rapidly due to the fact that little is known aout them as compared to the corresponding caronyls. Parallel development in the study of thiocaronyl compounds was hampered y factors, such as, the instaility leading to polymerization and onoxious smell. However, Kroto et al determined the value of the torsional arrier as cm 1 y analysing the rotational line splitting from microwave spectrum 17. Moule et al did visile and UV spectral studies for unstale species like thioacetaldehyde and thioacetone using pyrolysis jet spectroscopic technique Their analysis of the thioacetaldehyde spectrum revealed that the torsional motion in the upper state is coupled with the large amplitude wagging motion of aldehyde hydrogen. The oserved phase shift in the torsional angle on electronic excitation was suggested to e due to the effects of hyperconjugation etween methyl hydrogen and the C S π oritals. They estimated the rotational energy arrier to e cm 1 from the and interval in the visile spectrum. A initio calculation provided the rotational arrier as cm 1 where the wagging mode of aldehydic hydrogen was coupled to the methyl torsion 22. The discrepancy in rotaional arrier energy etween the experimental and the calculated values, especially the lower value otained y the theoretical computation, might owe its origin to the use 21 of a relatively less quantitative model. What was done previously was that the two-dimensional Schroedinger equations were framed for the torsional and wagging motions and susequently solved to otain the arrier energy and shape of the potential energy surfaces. Moreover, the origin of the rotational arrier was, to the est of our knowledge, not addressed earlier. The knowledge of energy involved in the rotational arrier is elieved to e very important in the context of reaction chemistry. In view of the aove considerations, it was felt important to gain in-depth understanding aout the internal rotational arrier during the rotational process y a rather powerful and

3 Partitioning of methyl internal rotational arrier energy 63 relatively more quantitative approach. In the present investigation, the origin of the methyl internal rotational arrier in thioacetaldehyde has een evaluated y first dividing the arrier energy into relaxation effect, NBO analysis and Pauling exchange interactions. The results of our endeavour are presented in this paper. 2. Computational methods HF/6-311g(d) asis set has een employed in geometry optimization and NBO analysis. The virial theorem is satisfied with a discrepency of aout 6% of the arrier, i.e. T = E is 30 cm 1. The calculations have een done y Gaussian 94 program 23 at Tohoku University, Quantum Chemistry La Computer Facility. Geometry optimization has een carried out with HF and MP2 levels with the aove asis set with a very tight option of the Gaussian 94 software. The relaxation calculation has een done at the same level with the same asis set. The natural ond orital (NBO) 24 calculations have een carried out using the NBO program 25 of the Gaussian 94. It may e mentioned that Foster and Wienhold have shown the utility of natural hyrid oritals in descriing the umrella inversion of ammonia 26. Steric calculations have een done y the Badenhoop and Weinhold procedure using NBOs to otain the Pauling exchange terms Results and discussion 3.1 Geometry The MP2/6-311g(d) optimized geometries for oth eclipsed (E) and staggered (S) conformers are shown in figure 1. The eclipsed ground state geometry at the ottom of the arrier, the staggered conformer at the top-of-the arrier and different conformer geometries otained y CH 3 rotation are otained from the a-initio calculation with HF/6-311g(d) asis set. The results are summarized in tale 1. The difference etween staggered and eclipsed conformers is attriuted mainly to the lengthening of the C C ond y Å, and the expansion of the CCH ald ond angle y 1 2º. It is relevant to note that the parallel structural changes induced y methyl rotation in acetaldehyde are the lengthening of the C C ond y Å and the expansion of the CCH ald ond angle y These results are in order ecause the replacement of O y S in going from acetaldehyde to thioacetaldehyde is expected to show a more pronounced effect, as is oserved. Moreover, during the course of methyl rotation in thioacetaldehyde, the out-ofplane hydrogen ond length decreases, whereas the in-plane methyl hydrogen ond increases. Also C S ond length is increased in the staggered conformer y Å. The nonequivalence in the CCH methyl angles in oth eclipsed and staggered conformers is caused y 2 3 and 1.6 tilt angles in the eclipsed and staggered conformers respectively. 3.2 Barrier energy Overall partitioning of total electronic energy change during methyl rotation from the ottom-of-the-well (cf eclipsed) to top-of-the-arrier (cf staggered) conformation is given y the equation, E = T + V ee + V nn + V ne,

4 64 Nikhil Guchhait a Figure 1. MP2/6-311g(d) optimized geometries for thioacetaldehyde. (a) Equilirium eclipsed (E) and () top-of-arrier staggered (S) conformers. The angle of rotation of methyl group along C C ond is expressed y τ. where E is the a initio calculated electronic energy difference etween the thioacetaldehyde staggered (S) and eclipsed (E) conformers, and T, V ee, V nn and V ne are the differences in kinetic, electron-electron repulsion, nuclear-nuclear repulsion and the nuclear-electron attraction energy terms respectively. The attractive and repulsive contriutions to the total energy change are V = V ne and T + V r, respectively, where V r = V ee + V nn. Here the calculations are done for rigid, partially relaxed and fully relaxed methyl group rotations. The quantity of interest is the difference etween the individual energy terms for the staggered and the corresponding eclipsed conformers. The virial theorem, E = T, is satisfied reasonaly well with the HF/6-311g(d) asis set. The torsional angle (τ) dependence of the nuclear virial for the rigid and the fully relaxed forms of the molecule are shown in figure 2. For the rigid rotation, like in the case of acetaldehyde, it increases monotonically as the rotation proceeds. But for the fully relaxed rotation, the change in the nuclear virial with the rotation of the methyl group is very small. Figure 3 shows the τ dependence of the aldehyde hydrogen wagging angle, calculated in a manner similar to that done for acetaldehyde. It predicts a maximum wagging angle of 2 8 for a τ value of 30. The inference drawn from this is that during methyl rotation, the torsional mode is strongly coupled with the out-of-plane wagging motion of the aldehydic hydrogen. Incidentally, a similar oservation was made y Moule et al in their studies involving a two-dimensional Schrodinger equation 22.

5 Partitioning of methyl internal rotational arrier energy 65 Tale 1. Thioacetaldehyde HF/6-311g(d) optimized geometrical parameters for different internal rotations of methyl group conformers a. Torsional angle τ Geometrical parameter Bond length C 2 S C 2 H ald C 2 C C 4 -H ip C 4 H op Bond angle H al C 2 S H 4 C 2 S C 4 C 2 H ald C 2 C 4 H ip C 2 C 4 H op Dihedral angle C 4 C 2 S 1 H ald H op C 4 C 2 H ip Tilt angle a Bond lengths and angles are in Å and degrees respectively; atom numering is shown in figure 1; torsional angle τ refers to the rotation of methyl group along the C C ond axis; tilt angle is the difference in angle etween C me CH ip and C me CH op angles. Figure 2. The dependence of nuclear virial of thioacetaldehyde for rigid ( ) and fully relaxed ( ) rotation on the methyl torsional angle (τ).

6 66 Nikhil Guchhait Figure 3. Dependence of aldehyde hydrogen out-of-plane wagging angle on the methyl torsional angle (τ). Tale 2. Thioacetaldehyde internal rotation arrier partitioned into symmetry terms cm 1 a. Fully Rigid C C C S Methyl relaxed rotation relaxed c relaxed c relaxed c Barrier Kinetic energy ( T) Virial theorem discrepancy Nuclear nuclear repulsion ( V nn ) Electron repulsion ( V ee ) Nuclear electron attraction ( V ne ) a Difference in energy terms (HF/6-311g(d)) etween staggered (180 ) and equilirium eclipsed (0 ) conformers (figure 1). Values are nearest whole numer; methyl group rotated y 180 with all ond lengths and angles are frozen at eclipsed conformer geometry; c rigid rotation followed y given ond length or ond angle relaxed to its fully relaxed value The energy terms are all presented in tale 2. Notale in the context is that Moule et al 22 found and cm 1 as the methyl rotational arrier energy from the results of visile jet spectrum analysis and a initio calculation respectively. Significantly, our calculated values for the fully relaxed rotational arrier is 548 cm 1, showing far etter agreement with the experimental arrier of cm 1 as measured y microwave spectroscopy 17. Partitioning of the arrier energy into kinetic and potential energy terms shows that V ne appears to e the only arrier-forming term in the fully relaxed rotation. For rigid and other relaxed rotations (except C C relaxed), V ne appears as an antiarrier term, while T, V ee and V nn appear as arrier-forming terms. It is seen that the inclusion of C C relaxation to any other relaxation shows V ne to e the arrier-forming

7 Partitioning of methyl internal rotational arrier energy 67 term whereas the rest of the energy terms are antiarrier in nature 28. Thus, the present results state that C C ond relaxation appears to e the major contriuting factor for the arrier in thioacetaldehyde. 3.3 Natural ond orital analysis The natural ond orital (NBO) analysis transforms molecular orital wave functions into one-centre (lone pair) and two-centre (ond) representations 29. This kind of categorization represents a chemically appealing point of view, since it highlights the individual onds and lone pairs that play a role in the chemical process 30. The diagonal elements of the Fock matrix in an NBO representation represent the energies of localized onds, antionds and lone pair, while the off-diagonal elements represent ond antiond, lone-pair antiond and normally small antiond antiond interactions. This scheme is used to differentiate the arrier energy into ond and lone pair energies, ond antiond and lone pair antiond interactions, and steric repulsions. The interactions involving the Ryderg oritals are not considered to e important since these interactions have weak orientational dependence. 3.4 Bond energies The change in ond energy ( ω) during methyl internal rotation for thioacetaldehyde is shown in tale 3 7,8,16, and has een otained from the relation, ω = ε S ρ S ε E ρ E, where ε S and ε E are the NBO energies for the staggered and eclipsed conformers respectively, and ρ S and ρ E are the corresponding NBO electron occupancies 7,16. Since our interest lies in the principal arrier-forming terms, only the important interactions are listed in tale 3. The largest arrier forming term, y far, is the change in ond energy Tale 3. Principal arrier-forming ond and lone-pair energy terms for thioacetaldehyde at rigid, partially and fully relaxed methyl rotational conformers (cm 1 ) a. S C(σ) C C me C me H ip C me H op LP (1) LP (2) Fully relaxed Rigid rotation c C C me relaxed d C S relaxed d C me H ip and C S relaxed d Methyl relaxed d C me H ip and methyl relaxed d C S and CCS angle relaxed d a ω (calculated from equation in the text), the population-weighted natural ond orital energy difference for the staggered and eclipsed conformers, values are nearest whole numers; fully relaxed rotation; c methyl group rotated y 180 with all ond lengths and angles are frozen at eclipsed conformer; d rigid rotation followed y given ond length and angle extended to its fully relaxed staggered conformer Arev. H op : out-of-plane methyl hydrogen atom, H ip : in-plane methyl hydrogen atom, C me : methyl caron atom, LP (1) and LP (2): lone pair.

8 68 Nikhil Guchhait involving the C C ond (2522 cm 1 ) for a fully relaxed rotation. This C C ond energy, however, ecomes anti-arrier for a rigid rotation. Interestingly the investigation reveals that the C C relaxation (row 3, tale 3) alone provides 2622 cm 1 arrier, a value which is quite close to the fully relaxed value. This therefore suggests that C C relaxation is the controlling factor for the arrier. The second highest contriution to the arrier is the C S σ ond energy (800 cm 1 ) which is aout one third that of the C C ond arrier contriution. From tale 3 (row 8) it is clear that the arrier-forming ond energy (800 cm 1 ) originates from oth the C S ond relaxation and the CCS angle relaxation. Unlike in acetaldehyde, C H op ond in thioacetaldehyde appears to e anti-arrier. This can e assumed to arise from the shortening of the C H op ond at the top of the arrier. The C H ip ond appears as a small arrier-forming term of aout 332 cm 1 for the fully relaxed rotation and can easily e gained from the methyl-relaxed rotation. Therefore, it is evident that in addition to the main arrier-forming C C ond energy, the C S ond and the methyl C H ip hydrogen contriute to the methyl rotational arrier to some extent. 3.5 Lone-pair energy There are two localized lone-pairs on the sulphur atom in thioacetaldehyde, similar to that of the oxygen atom in acetaldehyde. One of the lone-pairs is nearly of p character while the other one lies along the C S ond. The sulphur lone-pair energy change in going from one conformer to the other is shown in tale 3. Both are aout 350 cm 1 (arrier-forming) and are not affected apprecialy y relaxation. 3.6 Bond antiond interaction The ond antiond and lone-pair antiond interactions can e calculated y two different procedures following the NBO method 5. The first one involves an indirect procedure initiated y Weinhold, which involves comparison of arrier energies calculated with and without the Fock matrix element, F ij *, etween the onding (or lone pair) NBO and a virtually unoccupied antionding orital, deleted 31. The second one involves the calculation of F ij * 2 /(ε i ε j ) using second-order perturation theory. The results of our calculation using the second-order perturation method are shown in tale 4. There are a numer of arrier-forming ond antiond interaction terms. The most important ones are C H op /S C(σ)*, C H ip /C H al *, C H ald /C H ip *, C H op /S C(π)* and S C(π)/C H op *. The orital contour diagrams show that all these interactions involve charge transfer of electrons from a onding orital to an antionding orital. The highest arrier-forming ond antiond interaction, 1077 cm 1, arises from C H op onding and S C(σ)* antionding oritals. Figure 4a depicts such an interaction for the eclipsed conformer showing significant overlap of the oritals (the Fock matrix element is a.u.). In contrast, a similar orital contour plotted in figure 4 for the staggered conformer shows a decrease in the overlap (Fock matrix is a.u.). A markedly decreased overlap explains why these types of interaction involving the charge transfer of electrons from one onding orital to the other antionding orital are causing the antiarrier. The second highest term is the interaction etween the C H ip ond and the C H ald antiond (1032 cm 1 ) which is incidently similar to that of acetaldehyde (1169 cm 1 ) 16. Figures 5a and show the contour diagrams of these interactions for oth eclipsed and

9 Partitioning of methyl internal rotational arrier energy 69 a Figure 4. Orital contour diagrams for thioacetaldehyde C me H op onding and C S(σ)* antionding pre-nbo (not orthogonal) in (a) eclipsed and () staggered conformations. Tale 4. Principal arrier-forming ond antiond and lone-pair antiond interaction terms from secondorder perturation calculation for fully relaxed rotation of methyl group in thioacetaldehyde (cm 1 ) a. Donor/acceptor Barrier contriution S C(σ)/C H op *,c 129 S C(π)/C H op *,c 283 C H ald /C H ip * 647 C H ip /C H ald * 1032 C H op /S C(σ)*,c 1077 C H op /S C(π)*,c 311 LP (1)/C C* 84 LP (1)/C H ip * 94 LP (2)/C H ald * 133 a See footnote (a) tale 2; where there are multiple identical interactions, the contriution of only one is given; c there are two such interactions staggered conformers. Again, these interactions show good overlap of the orital (Fock matrix element, equal to a.u.) in the eclipsed conformer and a relatively weak overlap (Fock matrix element equal to a.u.) in the staggered conformer. Thus, here too there is charge transfer from the C H ip ond to the C H ald antionding orital. The interaction energy etween C H ald onding and C H ip antionding is 647 cm 1 and it appears as the arrier-forming one. The contour diagram shown in figures 6a and, illustrate similar decreased overlap in going from the eclipsed to the staggered conformer.

10 70 Nikhil Guchhait a Figure 5. Orital contour diagrams showing overlap of thioacetaldehyde C me H ip onding and C H ald * antionding pre-nbos. a Figure 6. Orital contour diagrams showing overlap of thioacetaldehyde C H ald onding and C me H ip * antionding pre-nbos. Three other terms which appear as arrier-forming ond antiond interactions are S C(σ)/C H op *, C S(π)/C H op * and C H op /C S(π)*. Charge transfer interaction from the C S(π) onding orital to the C H op * antiond is important. However, the arrierforming interaction seems to e less important than the σ interaction, and the sum of the arrier-forming π ond antiond interactions is far less than the sum of the σ interactions. Lone-pair antiond interactions which contriute to the arrier are very small. Barrierforming lone-pair antiond interactions are LP(1)/C C*, LP(1)/C H ip * and LP(2)/ C H ald * (tale 4). The contour diagrams for these interactions are shown in figures 7 9.

11 Partitioning of methyl internal rotational arrier energy 71 a Figure 7. Orital contour diagrams showing overlap of thioacetaldehyde LP(1) onding and C C* antionding pre-nbos. a Figure 8. Orital contour diagrams showing overlap of thioacetaldehyde LP (1) onding and C me H ip * antionding pre-nbos. 3.7 Weakening of onds Weakening of the C C ond (0 008 Å ) can only e explained y considering the overlap of the unorthogonalized NBO involved in the arrier-forming ond antiond interaction terms shown in figures 4 6. For all cases the large onding overlap in the eclipsed conformer ecomes the antiarrier in the C C ond region of the staggered conformer

12 72 Nikhil Guchhait a Figure 9. Orital contour diagrams showing overlap of thioacetaldehyde LP(2) onding and C H ald * antionding pre-nbos. resulting in weakening of the C C ond. Weakening of the C C ond in acetaldehyde also arises from a large overlap of the interacting oritals in the equilirium conformer compared to those of the staggered one 16. The contour diagrams (figures 4 6) show that the onding overlap in the equilirium conformation is strongly reduced on going to the top-of-the-arrier. Thus, the decrease in overlap involving the ack and forth charge transfer etween the interacting oritals results in the C C ond lengthening. The cause of C S ond weakening is, however, not so clear. The electron transfer from the C me H op onding orital to the C S(σ)* antionding orital and from the C S(σ) onding orital to the C me H op * antionding orital may e the cause of C S ond lengthening. Addition of electron to antionding orital or removal of electron from the onding orital could also weaken the ond to some degree. 3.8 Steric repulsion The Pauling exchange interaction has een calculated using Bakenhoop and Wienhold formulation. The contriutions of Pauling exchange repulsions to internal rotational arrier are shown in tale 5. The main arrier forming Pauling exchange terms are etween S C(σ) ond/c me H ip ond and C H ald ond/c me H op ond (tale 5). 4. Conclusions Owing to the use of a relatively less powerful model y Moule et al 20 the agreement etween the theoretical evaluated rotational arrier of methyl group in thioacetaldehyde and the experimental found value was rather poor. Our reinvestigation of the prolem in terms of NBO analysis, relaxation effect and Pauling exchange interaction calculations provided a significantly etter concordance enaling us to conclude the following. (1)

13 Partitioning of methyl internal rotational arrier energy 73 Tale 5. Principal Pauling exchange repulsion changes accompanying fully relaxed methyl internal rotations in thioacetaldehyde (cm 1 ) a. Interaction Barrier contriution S C(σ)/C me H ip 899 S C(π)/C me H op 98 C H ald /C me H op 458 C H ald /LP (2) 105 a Values are nearest whole numers; there are two such repulsions The methyl rotational arrier for the fully relaxed rotation originates from the nuclear electron attraction term as it happens in the case of acetaldehyde. This nuclear electron attraction term appears as an anti arrier in the case of rigid rotation. The C C relaxation alone provides the nuclear electron attraction term as methyl rotational arrier. (2) The results of natural ond orital analysis shows that the principal arrier forming term originates from the C C ond. Whereas for acetaldehyde the C C and C Hop ond energies appear to e the arrier forming terms, for thioacetaldehyde C Hop ond appears as an antiarrier. The C S ond contriution to the arrier is one-third that of the C C ond and the ond length increases in the top-of-the-arrier. (3) The lengthening of C C ond in the staggered conformer can, however, e explained y considering several (figures 4 7) ond antiond interaction terms which provide a more favourale onding overlap in the C C ond region for the eclipsed conformer compared to that of the staggered one. The ack and forth charge transfer within these ond antiond oritals make the C C region for the eclipsed conformer to e more onding in character while a similar region for the staggered conformer is antionding in nature. Acknowledgements The author acknowledges his indetedness to Prof Lionel Goodman of Rutgers University, USA, who has initiated him in this type of calculation. He also thanks Prof Naohiko Mikami and Prof T Eata of Tohoku University, Japan for allowing him to carry out this work at their computer facility. Thanks are due to Prof Mihir Chowdhury of the Indian Association for the Cultivation of Science, Calcutta, for helpful discussions. References 1. Groner P, Sullivan J F and Durig J R 1981 Virational spectra and structure (Amsterdam: Elsevier) vol 9, pp Goodman L and Pophristic V 1998 The encyclopedia of computational chemistry (John Wiley & Sons: Chichester) vol. 4, pp ; Goodman L, Pophristic V and Weinhold F Acc. Chem. Res. (accepted) 3. Lister D G, MacDonald J N and Owen N L 1978 Internal rotation and inversion: An introduction to large amplitude motions in molecules (New York: Academic Press) 4. Veillard A 1974 In Internal rotation in molecules (New York: John Wiley and Sons) pp Goodman L and Gu H 1998 J. Chem. Phys Goodman L, Gu H and Pophristic V 1999 J. Chem. Phys

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