Alternative expressions for energies and forces due to angle bending and torsional energy

We have derived alternative expressions for computing the energies and forces associated with angle bending and torsional energy terms commonly used in molecular mechanics and molecular dynamics computer programs. Our expressions address the problems of singularities that are intrinsic in popular angle energy functions and that occur from other chain rule derivations of force expressions. Most chain rule derivations of expressions for Cartesian forces due to angle energies make use of relations such as (Formula Presented.) where ϕ is a bond or torsion angle, E(ϕ) is energy, and ∂/∂x represents a derivative with respect to some Cartesian coordinate. This expression leads to singularities from the middle term, −1/sin ϕ, when ϕ is 0 or π. This is a problem that prevents the use of torsional energy expressions that have phase angles, ϕ°, other than 0 or π, such as in E(ϕ) = κ[1 + cos(nϕ − phsi;°)]. Our derivations make use of a different, but equivalent, form of the chain rule: (Formula Presented.) This form still possesses singularities for the bond angle forces since the last factor is undefined when ϕ is 0 or π. However, the alternate form may be used to great advantage for the torsional angle forces where no such problem arises. The new expressions are necessary if one desires the use of torsional energy expressions with general phase angles. Even for energy expressions in common use, i.e., with phase angles of 0 or π, our force expressions are as computationally efficient as the standard ones. The new expressions are applicable to all molecular simulations that employ restrained, or phase‐shifted, torsional angle energy expressions.