Computational Chemis..

11.21 Commonly Available Force Fields 217V (q,μ) (r)= q aμ b cosθ br 2 (11.33)V (q,Q) (r)= q (aQ b 3cos 2 θ b − 1 )4r 3 (11.34)V (μ,μ) (r)= μ aμ b (2cosθ a cosθ b − sinθ a sin θ b cos(φ a − φ b ))r 3 (11.35)V (μ,Q) (r)= 3μ aQ b[ (cosθa 3cos 24r 4 θ b − 1 ) − 2sinθ a sinθ b cosθ b cos(φ a − φ b ) ]V (Q,Q) (r)= 3Q [aQ b16r 5 1 − 5cos 2 θ a − 5cos 2 θ b − 15cos 2 θ a cos 2 θ b(11.36)+2[sinθ a sinθ b cos(φ a − φ b ) − 4cosθ a cosθ a ] 2] (11.37)where θ a , θ b , φ a ,andφ b define the various orientation angles between the molecules.11.21 Commonly Available Force FieldsSome of the commonly available force fields are mentioned below.11.21.1 MM2, MM3, and MM4The MM family of force fields (MM2, MM3, and MM4) was introduced by Allingeret al. [2,3]. and are widely used for the computations of small molecules. The forcefield can identify sp, sp 2 and sp 3 hybridized carbon atoms, organic intermediatessuch as free radical and carbocation, the carbonyl functional group, and cyclohydrocarbonssuch as cyclopropane and cyclopropene (Leach, 2001). The MM familywas parameterized to fit values obtained through electron diffraction, which providemean distances between atoms averaged over vibrational motion at room temperature.The bond stretching potential is represented by the classic Hooke’s lawexpansion:V (l) = k [ ]2 (l − l 0) 2 1 − k ′ (l − l 0 ) − k ′′ (l − l 0 ) − k ′′′ (l − l 0 ) − ... (11.38)In MM2, expansion is made up to cubic terms, which may cause the cubic functionto pass through a maximum that is far from the reference value. This has leadto disastrous expansion of bonds in some experiments. This defect is overcome inMM3 by limiting the use of the cubic contribution only when the structure is sufficientlyclose to its equilibrium geometry and is inside the actual potential well.Leach includes a quartic term in MM3, which eliminates the inversion problem and