Valence Angle Bending

Vibrational spectroscopy reveals that, for small displacements from equilibrium, energy variations associated with bond angle deformation are as well modeled by polynomial expansions as are variations associated with bond stretching. Thus, the typical force field function for angle strain energy is 

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As the most notable contribution of ab initio studies, it was revealed that the different modes of molecular deformation (i.e. bond stretching, valence angle bending and internal rotation) are excited simultaneously and not sequentially at different levels of stress. Intuitive arguments, implied by molecular mechanics and other semi-empirical procedures, lead to the erroneous assumption that the relative extent of deformation under stress of covalent bonds, valence angles and internal rotation angles (Ar A0 AO) should be inversely proportional to the relative stiffness of the deformation modes which, for a typical polyolefin, are 100 10 1 [15]. A completly different picture emerged from the Hartree-Fock calculations where the determined values of Ar A0 AO actually vary in the ratio of 1 2.4 9 [91]. 

The units for the force constants are usually given in mdyn A-1 (bond length), mdyn A rad"2 (valence angle bending and out-of-plane distortion) and mdyn A (torsion angle deformation). 

EAS (Engler, Andose, Schleyer) [184] is quite an old force field designed to model alkanes exclusively. The harmonic potential is used for the bond stretching and cubic anharmonic for the valence angle bending. No out of plane, electrostatic or cross terms are included. The nonbonded interactions are represented by the Buchingham potential. 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

MM2 [189] uses cubic anharmonic potential to represent the bond stretching, up to sixth power expansion for the valence angle bending, and harmonic field for the out-of-plane deformations. The stretch-bending cross term is included. 

Now we consider the valence angle bending force field as it appears from the DMM picture. For this end the geometry variation given by the vectors eq. (3.139) must be inserted in eq. (3.72) and the required elasticity constant can be obtained by extracting the second order contribution in vectors 6[Pg.260]

In this equation, the first sum accounts for bond stretching, the second sum for valence angle bending, the third (double) sum for torsions, and the fourth, where the sum goes only over nonbonded and nongeminal atoms, for van der Waals interactions and nonbonded Coulomb forces. 

The energy of valence interactions accounts for bond stretching (bond), valence angle bending (angle), dihedral angle torsion (torsion), and inversion, also called out of plane interactions (oop) ... 

The harmonic term for (valence) angle bending can be written as... 

Valence Angle Bending. Another example is the increase in the angle between two valence bonds attached to the same atom 3. Let St be this... 

In order to systematize the tabulation of formulas, a system of classification and of notation for the different matrix elements must be devised. First, the matrix elements are classified in terms of the general types of coordinates involved. In what follows, only two types of coordinates will appear bond stretching and valence angle bending. These types have been described in Sec. 4-2. 

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