Bond Angle Bending Functions

The simplest functional form of the angle-bending potential energy function includes only a quadratic term (the harmonic approximation) as in Eq. [6]  

The TRIPOS, Chem-X, CHARMm, COSMIC, and CVFF force fields all use only the harmonic approximation, whereas in the DREIDING force field, angle bending is described by a harmonic cosine function where 0 and 0o in Eq. [6] are replaced by cos 0 and cos 0q, respectively, to avoid a zero slope as 0 approaches 180°. In the UFF force field, angle bending is described with a small Fourier expansion in 0 (Eq. [7]). This functional form was selected to better describe large-amplitude motions. 

To better reproduce experimental data for bond angles with large deviations from their reference values, higher order terms may be added to Eq. [6]. In addition to the quadratic term, MM2 (and the MM2-based programs) use a quadratic and a sixth-power term (Eq. [8]), 

The MMFF93 force field uses a quadratic and a cubic term for angle bending, whereas the CFF91 force field includes quadratic, cubic, and fourth-power terms. 

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Figure 5 CHARMM and MM3 bond angle bend functions = 39 kcal-mol -rad and 9o = 109.5°).

According to the namre of the empirical potential energy function, described in Chapter 2, different motions can take place on different time scales, e.g., bond stretching and bond angle bending vs. dihedral angle librations and non-bond interactions. Multiple time step (MTS) methods [38-40,42] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast movements most effectively. 

For example, the r0 value for the Co-N bond in cobalt(III)-amine complexes is smaller in parameterization schemes where 1,3-nonbonded interactions between the ligating atoms are included, than in force fields where only L-M-L angle bending functions are used. This is because the 1,3-nonbonded interactions in such complexes are highly repulsive, promoting an extension of the Co-N bonds. Thus, a smaller value for the ideal Co-N bond is required in order to reproduce the experimentally ob-... 

Some formulations of the potential energy function (e.g., References 27 and 79, and AMBER, CHARMM, and DISCOVER, as well as most force fields used in small-molecule studies) include terms that allow for bond stretching and bond angle bending, that is, for flexible geometry. Hence, the terms in Eq. [1] are augmented by the expression27... 

Although the basic methodology for calculating structures of oligopeptides is now in place, there is still room for improvement of potential functions, not only in the parameters used but also in the forms of the functions themselves. In the near future, we can expect to see the introduction of anhar-monicity in bond angle bending, a treatment involving many-body interactions (rather than pairwise interactions), polarization (and possibly distributed multipoles) to treat electrostatics, and improved treatment of hydration. In addition, we will undoubtedly see considerably more use of parallelism in computer hardware and software. 

Figure 8 MM3 and UFF bond angle bend potential energy functions = 39 kcal-mol ...

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