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An ab initio MC-SCF study of the solvent effects in polar and non-polar [2 + 2] cycloadditions
THEO CHEM
F
Journal of Molecular Structure (Theochem) 357 (1995) 33-36
An ab initio MC-SCF study of the solvent effects in polar and non-polar [2 + 21 cycloadditions Fernando
BernardiaT*, Rafael R. Pappalardob, Alessandro Venturini”
Michael A. Robbb,
aDipartimento di Chimica “G. Ciamician”, Universita’ di Bologna, Via Selmi, 2, 40126, Bologna, Italy b Department of Chemistry, King’s College London, Strand, London, WCZR 2LS, UK ‘Istituto dei Composti de1 Carbonio Contenenti Eteroatomi e lore Applicazioni. CNR. via Gobetti 101. 40100 Bologna, Italy
Received 6 March 1995; accepted 10 April 1995
Abstract Solvent effects in non-polar and polar [2 + 21 cycloadditions are examined with MC-SCF methods, using the dimerization of ethylene as a prototype of non-polar cycloaddition and the reaction between dicyanoethylene and hydroxyethylene as a prototype of polar cycloaddition. The effect of the solvent has been included via the continuum model of Rivail. Solvent effects stabilize the gauche transition state relative to the anti transition state in the two-step mechanism. The concerted transition state remains very high in energy, even when solvent effects are considered.
1. Introduction Experimentally, [2 + 21 cycloadditions are an important class of reactions since they represent an effective synthetic approach to the formation of four-membered rings. Thermal [2 + 21 cycloadditions reactions are usually divided into two classes, non-polar and polar cycloadditions [ 11. Non-polar cycloadditions take place at high temperatures (400-700°C) because of the large activation energies involved and all stereochemical and kinetic data can be explained via a two-step biradical mechanism. Polar cycloadditions occur under very mild conditions (25°C in solution), suggesting a low activation barrier. For such reactions, Huisgen and co-workers have suggested a two-step mechanism involving a zwitterionic intermediate * Corresponding author.
SSDI
0166-1280(95)04276-8
[l]. Although the mechanism of the polar cycloadditions remains to be clarified in detail, the experimental results do suggest that there is a mechanistic dichotomy between polar and nonpolar cycloadditions [2]. Recently, we have reported a complete MC-SCF study [3] of the mechanism of polar and non-polar [2 + 21 cycloadditions, where the ethylene dimerization has been taken to represent the prototype of a non-polar [2 + 21 cycloaddition, and the cycloaddition between dicyanoethylene and hydroxyethylene the prototype of a polar [2 + 21 cycloaddition. The polar effect in the latter is emulated by the donor (OH) and acceptor (CN) substituents. This work suggests that the surface topology describing the two-step mechanism turns out to be basically independent of the type of substituent. Two-step paths for gauche and anti attack, involving biradical intermediates, were fully
34
F. Bernardi et al./Journal of Molecular Structure (Theochem)
357 (1995) 33-36
Table 1 Terms of the multipolar expansion (I) for the polar systems: (A) expansion to the sixth order, (B) expansion to the seventh order
(4
I=
Gauche TS Anti TS Gauche M Anti M Concerted TS
la
I=2
I=3
I=4
I = (5-6)
Total
-7.03 -1.91 -7.36 -1.68 -1.94
-4.26 -4.48 -5.01 -5.29 -8.56
-7.21 -9.69 -7.80 -11.63 -10.29
-4.31 -1.94 -4.46 -2.29 -3.06
-3.67 -2.07 -4.20 -2.67 -1.61
-26.41 -20.09 -28.83 -23.56 -31.46
(B)
I=1
1=2
I=3
I=4
I= (5-7)
Total
Gauche TS Anti TS Gauche M Anti M Concerted TS
-7.22
-4.42
-7.51
-4.51
-5.57
-29.24
-1.98 -7.60 -1.77 -8.07
-4.62 -5.22 -5.53 -8.69
-9.95 -8.16 -12.04 -10.45
-2.03 -4.71 -2.44 -3.15
-3.43 -6.42 -4.49 -2.54
-22.02 -32.10 -26.27 -32.91
a I= 1 dipole, I= 2 quadrupole, etc.
characterized. In contrast, the topology of the potential surface describing the concerted mechanism is sensitive to substituents. For the non-polar reaction (i.e. dimerization of ethylene) a concerted reaction path does not exist and the stationary point located along the concerted path corresponds to a local maximum (with two imaginary frequencies). In contrast, for the polar cycloaddition, a true concerted transition state exists, at high energy relative to the biradical region. One expects that solvent effects may play a significant role in the case of polar cycloadditions. The purpose of this work is to evaluate the effect of solvent stabilization of the transition states using a continum model.
2. Computational
polar cycloadditions. In general, the development of the multipole expansion has been done to the sixth order, with some systems also computed to the seventh order to test the convergence. The solvent effect has been computed keeping the geometries of the various transition states frozen at the values optimized in the absence of solvent effects. It can be expected that the molecules studied in this paper, because of the different structures, cannot be fitted by an ellipsoidal cavity at the same level of accuracy. To assess this point, we have computed the contributions of the multipole expansion to the sixth and seventh order (see Table 1). These results show that, even if in both cases the convergence was not as good as we would have desired, the trend of the various contributions remains the same in the two multipole expansions.
details
All computations presented in this paper have been obtained at the MC-SCF/4-31G level using a four electrons/four orbitals active space. The solvent effect has been computed with Rivail’s continuum model [4] using an ellipsoidal cavity. This continuum model has been implemented within the MC-SCF algorithm in the gaussian package [5] using the programs found in SCRFPAC [6]. In the present paper, we have considered a medium with dielectric permittivity 6 corresponding to a polar solvent like acetonitrile (E = 35.9) which has been used as a solvent in experimental studies of [2 + 21
3. Results and discussion
The quantity used here to assess the magnitude of the solvent effect for a given critical point is the difference between the energies of the critical point computed with and without solvent effects. This quantity is denoted as AEs (see Table 2). The dimerization of ethylene, is used as a typical nonpolar [2 + 21 cycloaddition, and the cycloaddition of dicyanoethylene and hydroxyethylene as an example of a polar [2 + 21 cycloaddition. The geometries of the minima and transition states com-
F. Bernardi et al./Journal of Molecular Structure (Theochem) Table 2 Total energy values (au.) computed without (Es) and with (E,) solvent effects at the MC-SCF 4-31G level. Energy barriers (AE,, kcalmol-‘) and solvation index (A&, kcalmol-t) for the dimerization of ethylene (A) and the cycloaddition between the dicyanoethylene with hydroxyethylene (B) Es
E,
AEsa
(A) Reactants Gauche TS Anti TS Gauche M Anti M Concerted SOSP
-155.90608 -155.82029 -155.82395 -155.82135 -155.82439 -155.78299
-155.90838 -155.82105 -155.82476 -155.82218 -155.82531 -155.78353
0. 53.8 51.5 53.2 51.3 77.2
(B) Reactants Gauche TS Anti TS Gauche M Anti M Concerted TS
-412.82434 -413.75479 -413.75647 -413.75580 -413.75707 -413.67612
-413.84559 -413.78753 -413.78224 -413.79049 -413.78619 -413.71731
0. 43.6 42.6 43.0 42.2 93.0
AEsb
-1.44 -0.48 -0.51 -0.52 -0.58 -0.35 -13.1 -20.5 -16.2 -21.8 -18.3 -25.8
a AEs = Eg - Eg (reactants). bAEs=Es-E,.
puted without solvent effects [3,7] are summarized in Figs. 1 and 2. For the dimerization of ethylene (see Fig. 1) there are only two true reaction paths, which correspond to the two-step path involving a biradical transition state (gauche TS and anti TS) and a biradical intermediate (gauche M and anti M). The length of the forming CC bond is very similar in the two transition structures and in the two minima (x1.76A and ~1.64A respectively) [7b]. A dihedral angle C(CC)C (defined by rotation about the direction of the new forming CC bond) of 81.6” and 75.4” characterizes the gauche transition structure and the gauche minimum respectively. The concerted path contains only local maxima, one of DZh symmetry and the other of C,, symmetry, which are both second-order saddle points. Remarkably, the inclusion of donor/acceptor substitutents does not modify the topology of the biradical region for the cycloaddition between dicyanoethylene with hydroxyethylene [3], although the dihedral angle C(CC)C of the ethylenic skeleton is smaller for the gauche TS (see Fig. 2). In contrast, the inclusion of donor/ acceptor substitutents significantly modifies the topology of the concerted region, and an asynchronous
357 (1995) 33-36
35
transition state on the concerted reaction path can be characterized (see Fig. 2). This structure is significantly distorted, the length of the two new forming bonds being 1.931 A and 2.283 A respectively [3]. We now turn to the effect of solvent stabilization. The values AEs for the transition states and intermediates are collected in Table 2. The data for the DZh supra-supra second-order saddle point in the dimerization of ethylene are included for comparison with the concerted transition structure found in the polar cycloaddition. The solvent effects are stabilizing in all cases. As expected, AEs is small (x0.5 kcalmol-‘) in the case of the non-polar cycloaddition, but much larger (Z 20 kcalmol-i) for polar cycloaddition. The gauche transition state and intermediate and the concerted transition state structure are significantly stabilized relative to the corresponding anti critical points. Thus, the solvent ( Anti TS 1
I.001
,80.0
I
Fig. 1. Relevant geometrical parameters (forming CC bond distance (A) and C(CC)C dihedral angle (deg) optimized at the MC-SCF/4-3 lg level for the ethylene dimerization.
F. Bernardi et al/Journal
36
of Molecular Structure (Theochem)
3S7 (1995) 33-36
cycloadditions have been studied using the continuum model of Rivail within MC-SCF theory, with an ellipsoidal cavity corresponding to a dielectric permittivity corresponding to a polar solvent-like acetonitrile. The concerted transition state remains very high in energy and the preferred path turns out to be the gauche approach, which therefore should correspond to the two-step mechanism suggested by Huisgen and co-workers, with the zwitterionic intermediate corresponding to the gauche minimum.
i
, 1.142
Acknowledgement
All calculations were run on an IBM RISC 6000, partly at 1.Co.C.E.A and partly at Kings’ College.
References
111R. Bruckner, R. Huisgen and J. Schmid, Tetrahedron Lett.,
Fig. 2. Relevant geometrical parameters (forming CC bond distance (A) and C(CC)C dihedral angle (deg)) optimized at the MC-SCF/4-3 lg level for the cycloaddition of dicyanoethylene to hydroxyethylene.
effects stabilize the gauche structures relative to the anti structures and the concerted transition state with respect to the two-step transition states. The activation barrier of the concerted approach is very high in energy (x 50 kcal mol-’ above the gauche TS) and the stabilization of x5 kcalmol-’ due to the solvent effect is not significant. These results lead to the suggestion that the gauche approach corresponds to the two-step mechanism suggested by Huisgen and coworkers, and that the gauche intermediate represents the preferred zwitterionic intermediate.
4. Conclusions Solvent effects in non-polar
and polar [2 + 21
31 (1990) 7129; R. Huisgen, Act. Chem. Res., 10 (1977) 117; R. Huisgen, Act. Chem. Res., 10 (1977) 199; R. Huisgen, Pure Appl. Chem., 32 (1980) 2283. 121R.B. Woodward and R. Hoffman, Angew. Chem. Inter. Ed. Engl., 8 (1969) 781; N.D. Epiotis and S. Shaik, J. Am. Chem. Sot., 100 (1978) 9; N.D. Epiotis, R.L. Yates, D. Calberg and F. Bemardi, J. Am. Chem. Sot., 100 (1976) 453; N.D. Epiotis, Theory in Organic Reactions, Springer-Verlag, Berlin, 1979. [31 F. Bernardi, A. Bottoni, M. Olivncci, A. Venturini and M.A. Robb, J. Chem. Sot., Faraday Trans., 90 (1994) 1617. [41 D. Rinaldi and J.L. Rivail, Theoret. Chim. Acta, 32 (1973) 57; D. Rinaldi and J.L. Rivail, Chem. Phys., 18 (1976) 233; J.L. Rivail in J. Bertrain and LG. Csizmadia (Eds.), New Theoretical Concepts for Understanding Organic Reactions, Kluwer, Dordrecht, 1989,pp. 219-231; D. Rinaldi, Comput. Chem., 6 (1982) 155; M.F. Ruiz-Lopez and D. Rinaldi, J. Mol. Struct. (Theochem) 93 (1983) 277; D. Rinaldi, M.F. Ruiz-Lopez and J.L. Rivail, J. Chem. Phys., 78 (1983) 834. PI M.J. Frisch, M. Head-Gordon, J.B. Foresman, G.W. Trucks, K. Raghavachari, H.B. Schlegel, M.A. Robb, J.S. Binkley, C. Gonzalez, D.J. DeFrees, D.J. Fox, R.A. Whiteside, R. Seeger, C.F. Melius, J. Baker, L.R. Kahn, J.J.P. Stewart, E.M. Fluder, S. Topiol and J.A. Pople, GAUSSIAN 92, Gaussian Inc., Pittsburgh, PA, 1992. 161D. Rinaldi and R.R. Pappalardo, SCRFPAC, QCPE progr. No. 622, QCPE Bull., 12 (1992) 69. W F. Bemardi, A. Bottoni, M.A. Robb, H.B. Schlegel and G. Tonachini, Chem. Phys. Lett., 108 (1984) 599. P’bl F. Bernardi, A. Bottoni, M.A. Robb, H.B. Schlegel and G. Tonachini, J. Am. Chem. Sot., 107 (1985) 2260.
F
Journal of Molecular Structure (Theochem) 357 (1995) 33-36
An ab initio MC-SCF study of the solvent effects in polar and non-polar [2 + 21 cycloadditions Fernando
BernardiaT*, Rafael R. Pappalardob, Alessandro Venturini”
Michael A. Robbb,
aDipartimento di Chimica “G. Ciamician”, Universita’ di Bologna, Via Selmi, 2, 40126, Bologna, Italy b Department of Chemistry, King’s College London, Strand, London, WCZR 2LS, UK ‘Istituto dei Composti de1 Carbonio Contenenti Eteroatomi e lore Applicazioni. CNR. via Gobetti 101. 40100 Bologna, Italy
Received 6 March 1995; accepted 10 April 1995
Abstract Solvent effects in non-polar and polar [2 + 21 cycloadditions are examined with MC-SCF methods, using the dimerization of ethylene as a prototype of non-polar cycloaddition and the reaction between dicyanoethylene and hydroxyethylene as a prototype of polar cycloaddition. The effect of the solvent has been included via the continuum model of Rivail. Solvent effects stabilize the gauche transition state relative to the anti transition state in the two-step mechanism. The concerted transition state remains very high in energy, even when solvent effects are considered.
1. Introduction Experimentally, [2 + 21 cycloadditions are an important class of reactions since they represent an effective synthetic approach to the formation of four-membered rings. Thermal [2 + 21 cycloadditions reactions are usually divided into two classes, non-polar and polar cycloadditions [ 11. Non-polar cycloadditions take place at high temperatures (400-700°C) because of the large activation energies involved and all stereochemical and kinetic data can be explained via a two-step biradical mechanism. Polar cycloadditions occur under very mild conditions (25°C in solution), suggesting a low activation barrier. For such reactions, Huisgen and co-workers have suggested a two-step mechanism involving a zwitterionic intermediate * Corresponding author.
SSDI
0166-1280(95)04276-8
[l]. Although the mechanism of the polar cycloadditions remains to be clarified in detail, the experimental results do suggest that there is a mechanistic dichotomy between polar and nonpolar cycloadditions [2]. Recently, we have reported a complete MC-SCF study [3] of the mechanism of polar and non-polar [2 + 21 cycloadditions, where the ethylene dimerization has been taken to represent the prototype of a non-polar [2 + 21 cycloaddition, and the cycloaddition between dicyanoethylene and hydroxyethylene the prototype of a polar [2 + 21 cycloaddition. The polar effect in the latter is emulated by the donor (OH) and acceptor (CN) substituents. This work suggests that the surface topology describing the two-step mechanism turns out to be basically independent of the type of substituent. Two-step paths for gauche and anti attack, involving biradical intermediates, were fully
34
F. Bernardi et al./Journal of Molecular Structure (Theochem)
357 (1995) 33-36
Table 1 Terms of the multipolar expansion (I) for the polar systems: (A) expansion to the sixth order, (B) expansion to the seventh order
(4
I=
Gauche TS Anti TS Gauche M Anti M Concerted TS
la
I=2
I=3
I=4
I = (5-6)
Total
-7.03 -1.91 -7.36 -1.68 -1.94
-4.26 -4.48 -5.01 -5.29 -8.56
-7.21 -9.69 -7.80 -11.63 -10.29
-4.31 -1.94 -4.46 -2.29 -3.06
-3.67 -2.07 -4.20 -2.67 -1.61
-26.41 -20.09 -28.83 -23.56 -31.46
(B)
I=1
1=2
I=3
I=4
I= (5-7)
Total
Gauche TS Anti TS Gauche M Anti M Concerted TS
-7.22
-4.42
-7.51
-4.51
-5.57
-29.24
-1.98 -7.60 -1.77 -8.07
-4.62 -5.22 -5.53 -8.69
-9.95 -8.16 -12.04 -10.45
-2.03 -4.71 -2.44 -3.15
-3.43 -6.42 -4.49 -2.54
-22.02 -32.10 -26.27 -32.91
a I= 1 dipole, I= 2 quadrupole, etc.
characterized. In contrast, the topology of the potential surface describing the concerted mechanism is sensitive to substituents. For the non-polar reaction (i.e. dimerization of ethylene) a concerted reaction path does not exist and the stationary point located along the concerted path corresponds to a local maximum (with two imaginary frequencies). In contrast, for the polar cycloaddition, a true concerted transition state exists, at high energy relative to the biradical region. One expects that solvent effects may play a significant role in the case of polar cycloadditions. The purpose of this work is to evaluate the effect of solvent stabilization of the transition states using a continum model.
2. Computational
polar cycloadditions. In general, the development of the multipole expansion has been done to the sixth order, with some systems also computed to the seventh order to test the convergence. The solvent effect has been computed keeping the geometries of the various transition states frozen at the values optimized in the absence of solvent effects. It can be expected that the molecules studied in this paper, because of the different structures, cannot be fitted by an ellipsoidal cavity at the same level of accuracy. To assess this point, we have computed the contributions of the multipole expansion to the sixth and seventh order (see Table 1). These results show that, even if in both cases the convergence was not as good as we would have desired, the trend of the various contributions remains the same in the two multipole expansions.
details
All computations presented in this paper have been obtained at the MC-SCF/4-31G level using a four electrons/four orbitals active space. The solvent effect has been computed with Rivail’s continuum model [4] using an ellipsoidal cavity. This continuum model has been implemented within the MC-SCF algorithm in the gaussian package [5] using the programs found in SCRFPAC [6]. In the present paper, we have considered a medium with dielectric permittivity 6 corresponding to a polar solvent like acetonitrile (E = 35.9) which has been used as a solvent in experimental studies of [2 + 21
3. Results and discussion
The quantity used here to assess the magnitude of the solvent effect for a given critical point is the difference between the energies of the critical point computed with and without solvent effects. This quantity is denoted as AEs (see Table 2). The dimerization of ethylene, is used as a typical nonpolar [2 + 21 cycloaddition, and the cycloaddition of dicyanoethylene and hydroxyethylene as an example of a polar [2 + 21 cycloaddition. The geometries of the minima and transition states com-
F. Bernardi et al./Journal of Molecular Structure (Theochem) Table 2 Total energy values (au.) computed without (Es) and with (E,) solvent effects at the MC-SCF 4-31G level. Energy barriers (AE,, kcalmol-‘) and solvation index (A&, kcalmol-t) for the dimerization of ethylene (A) and the cycloaddition between the dicyanoethylene with hydroxyethylene (B) Es
E,
AEsa
(A) Reactants Gauche TS Anti TS Gauche M Anti M Concerted SOSP
-155.90608 -155.82029 -155.82395 -155.82135 -155.82439 -155.78299
-155.90838 -155.82105 -155.82476 -155.82218 -155.82531 -155.78353
0. 53.8 51.5 53.2 51.3 77.2
(B) Reactants Gauche TS Anti TS Gauche M Anti M Concerted TS
-412.82434 -413.75479 -413.75647 -413.75580 -413.75707 -413.67612
-413.84559 -413.78753 -413.78224 -413.79049 -413.78619 -413.71731
0. 43.6 42.6 43.0 42.2 93.0
AEsb
-1.44 -0.48 -0.51 -0.52 -0.58 -0.35 -13.1 -20.5 -16.2 -21.8 -18.3 -25.8
a AEs = Eg - Eg (reactants). bAEs=Es-E,.
puted without solvent effects [3,7] are summarized in Figs. 1 and 2. For the dimerization of ethylene (see Fig. 1) there are only two true reaction paths, which correspond to the two-step path involving a biradical transition state (gauche TS and anti TS) and a biradical intermediate (gauche M and anti M). The length of the forming CC bond is very similar in the two transition structures and in the two minima (x1.76A and ~1.64A respectively) [7b]. A dihedral angle C(CC)C (defined by rotation about the direction of the new forming CC bond) of 81.6” and 75.4” characterizes the gauche transition structure and the gauche minimum respectively. The concerted path contains only local maxima, one of DZh symmetry and the other of C,, symmetry, which are both second-order saddle points. Remarkably, the inclusion of donor/acceptor substitutents does not modify the topology of the biradical region for the cycloaddition between dicyanoethylene with hydroxyethylene [3], although the dihedral angle C(CC)C of the ethylenic skeleton is smaller for the gauche TS (see Fig. 2). In contrast, the inclusion of donor/ acceptor substitutents significantly modifies the topology of the concerted region, and an asynchronous
357 (1995) 33-36
35
transition state on the concerted reaction path can be characterized (see Fig. 2). This structure is significantly distorted, the length of the two new forming bonds being 1.931 A and 2.283 A respectively [3]. We now turn to the effect of solvent stabilization. The values AEs for the transition states and intermediates are collected in Table 2. The data for the DZh supra-supra second-order saddle point in the dimerization of ethylene are included for comparison with the concerted transition structure found in the polar cycloaddition. The solvent effects are stabilizing in all cases. As expected, AEs is small (x0.5 kcalmol-‘) in the case of the non-polar cycloaddition, but much larger (Z 20 kcalmol-i) for polar cycloaddition. The gauche transition state and intermediate and the concerted transition state structure are significantly stabilized relative to the corresponding anti critical points. Thus, the solvent ( Anti TS 1
I.001
,80.0
I
Fig. 1. Relevant geometrical parameters (forming CC bond distance (A) and C(CC)C dihedral angle (deg) optimized at the MC-SCF/4-3 lg level for the ethylene dimerization.
F. Bernardi et al/Journal
36
of Molecular Structure (Theochem)
3S7 (1995) 33-36
cycloadditions have been studied using the continuum model of Rivail within MC-SCF theory, with an ellipsoidal cavity corresponding to a dielectric permittivity corresponding to a polar solvent-like acetonitrile. The concerted transition state remains very high in energy and the preferred path turns out to be the gauche approach, which therefore should correspond to the two-step mechanism suggested by Huisgen and co-workers, with the zwitterionic intermediate corresponding to the gauche minimum.
i
, 1.142
Acknowledgement
All calculations were run on an IBM RISC 6000, partly at 1.Co.C.E.A and partly at Kings’ College.
References
111R. Bruckner, R. Huisgen and J. Schmid, Tetrahedron Lett.,
Fig. 2. Relevant geometrical parameters (forming CC bond distance (A) and C(CC)C dihedral angle (deg)) optimized at the MC-SCF/4-3 lg level for the cycloaddition of dicyanoethylene to hydroxyethylene.
effects stabilize the gauche structures relative to the anti structures and the concerted transition state with respect to the two-step transition states. The activation barrier of the concerted approach is very high in energy (x 50 kcal mol-’ above the gauche TS) and the stabilization of x5 kcalmol-’ due to the solvent effect is not significant. These results lead to the suggestion that the gauche approach corresponds to the two-step mechanism suggested by Huisgen and coworkers, and that the gauche intermediate represents the preferred zwitterionic intermediate.
4. Conclusions Solvent effects in non-polar
and polar [2 + 21
31 (1990) 7129; R. Huisgen, Act. Chem. Res., 10 (1977) 117; R. Huisgen, Act. Chem. Res., 10 (1977) 199; R. Huisgen, Pure Appl. Chem., 32 (1980) 2283. 121R.B. Woodward and R. Hoffman, Angew. Chem. Inter. Ed. Engl., 8 (1969) 781; N.D. Epiotis and S. Shaik, J. Am. Chem. Sot., 100 (1978) 9; N.D. Epiotis, R.L. Yates, D. Calberg and F. Bemardi, J. Am. Chem. Sot., 100 (1976) 453; N.D. Epiotis, Theory in Organic Reactions, Springer-Verlag, Berlin, 1979. [31 F. Bernardi, A. Bottoni, M. Olivncci, A. Venturini and M.A. Robb, J. Chem. Sot., Faraday Trans., 90 (1994) 1617. [41 D. Rinaldi and J.L. Rivail, Theoret. Chim. Acta, 32 (1973) 57; D. Rinaldi and J.L. Rivail, Chem. Phys., 18 (1976) 233; J.L. Rivail in J. Bertrain and LG. Csizmadia (Eds.), New Theoretical Concepts for Understanding Organic Reactions, Kluwer, Dordrecht, 1989,pp. 219-231; D. Rinaldi, Comput. Chem., 6 (1982) 155; M.F. Ruiz-Lopez and D. Rinaldi, J. Mol. Struct. (Theochem) 93 (1983) 277; D. Rinaldi, M.F. Ruiz-Lopez and J.L. Rivail, J. Chem. Phys., 78 (1983) 834. PI M.J. Frisch, M. Head-Gordon, J.B. Foresman, G.W. Trucks, K. Raghavachari, H.B. Schlegel, M.A. Robb, J.S. Binkley, C. Gonzalez, D.J. DeFrees, D.J. Fox, R.A. Whiteside, R. Seeger, C.F. Melius, J. Baker, L.R. Kahn, J.J.P. Stewart, E.M. Fluder, S. Topiol and J.A. Pople, GAUSSIAN 92, Gaussian Inc., Pittsburgh, PA, 1992. 161D. Rinaldi and R.R. Pappalardo, SCRFPAC, QCPE progr. No. 622, QCPE Bull., 12 (1992) 69. W F. Bemardi, A. Bottoni, M.A. Robb, H.B. Schlegel and G. Tonachini, Chem. Phys. Lett., 108 (1984) 599. P’bl F. Bernardi, A. Bottoni, M.A. Robb, H.B. Schlegel and G. Tonachini, J. Am. Chem. Sot., 107 (1985) 2260.