Conformational energy of the 5-membered ring. Implication of geometric and energetic properties in the conformational characteristics

In a previous article (8) a geometrical study of the five-membered ring showed that: a) for the case of the 20 symmetrical C₂ and C_s conformations, the pseudorotation formulae for the torsion angles are a geometrical property of the ring; b) geometrical considerations alone are unable to define the puckering amplitude, the bond angle values, and the pathway between two symmetrical conformations. Here we examine how the energy equations enable us to define the deformation amplitude _m, establish the bond angles expressions and check the energy invariability along the pseudorotation circuit. The problem is next developed fully in the case where the bond and torsional energy only are considered: the literal expression¹ of _m is then given as a function of the bond angle which cancels out the bond angle energy. A numerical application is carried out on cyclopentane and the values of the parameters K^t, K¹ and used in the Conformational energy calculations are considered.Notations used 1 _i bond lengths 1 in the case of the regular ring - _i torsional angles - _i bond angles - 3/5 = 108 - 4/5 = 144 - ,_{i i} – = complement to the 108 bond angle _i - _T - E Conformational energy of the 5-membered ring - E Conformational energy difference between planar and deformed ring - A_n Coefficients of the energy development in terms of - E_i^l Bond energy relative to atom i (associated with angle _i) - K_i^l Bond constant relative to atom i (associated with angle _i) - E_i^l Torsional energy relative to the i^th bond (associated with angle _i) - k_i^l Torsional constant relative to the i^th bond (associated with angle _i) - _i Angle _i value corresponding to zero bond energy E_i^l (when the 5 atoms of the ring are identical, _i ) - r_ij Distance between atoms i and j - q_i Charge carried by atom i - e Constant of proportionality including the effective dielectric constant - A_ij, B_ij, d_ij Coefficients dependent on the nature of the atoms i and j and accounted for in the Van der Waals energy and hydrogen bond expressions - S (r_ij) Electrostatic contribution to the hydrogen bond energy - P Pseudorotation phase angle - _m Maximum torsional angle value characterising the deformation amplitudeM