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where q is x, y, or z. Savings are even more dramatic for torsion and other four-body interaction types.

Translational invariance and symmetry of second derivatives are commonly used to speed up simulation calculations. In addition, simple expressions for gradients of internal coordinates we derived for ICQMC calculations can also be used to speed up MM and other calculations. For example, once the second x and y derivatives of a bond angle are calculated, . we obtain the z derivative by

2 COS #123

One difficulty our group has addressed is the bookkeeping required for keeping track of internal coordinates and their derivatives in different types of bond networks. Previously it was necessary to account for the type of bond network in order to optimize simulation codes. For example, it makes sense to think of carbon nanotubes in terms of rings, polymer chains in terms of monomer units, and so on. However, codes constructed in this manner are not easily portable to different applications. We have found a way to automatically construct tables of stretch, bend, torsion, and other interactions for any bond network beginning with a connection table [12]. The tables are split according to the numerical order of the atom numbers (for example, 123 vs. 213 vs. 312 for bend interactions). Once this is done, our formulas for internal coordinates and derivatives can be implemented with no loss of efficiency. In addition, the code can be used for different applications with little or no modification. Although it is still necessary to account for the functional form of the potential energy surface, the general bond network method can aid in the classification of interaction types (for example, according to atom types in the most general codes).

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