Efficient computation of potential energy first and second derivatives for molecular dynamics, normal coordinate analysis, and molecular mechanics calculations

Abstract

SUMMARY:By using two-and three-body internal coordinates and their derivatives as intermediates, it is possible to enormously simplify formulas for three-and four-body internal coordinates and their derivatives. Using this approach, simple formulas are presented for stretch (two-body), two types of bend (three-body), and wag and two types of torsion (four-body) internal coordinates and their first and second derivatives. The formulas are eminently suitable for economizing molecular dynamics and molecular mechanics calculations and normal coordinate analysis of complicated potential energy surfaces. Efficient methods for computing derivatives of entire potential energy terms, and in particular cross terms or terms with switching functions, are presented.

Classical and Quantum Molecular Simulations in Nanotechnology Applications

“…Recognizing this, many research groups have put considerable effort into speeding up the calculation of internal coordinates and derivatives for stretch and other chemically bonded interactions (see, for example, [14-Our group has found it most advantageous computationally to take advantage of the highly connected nature of the bond networks for the systems we simulate [8][9][10][11]. For example, in a diamondoid bond network one bond stretch can be part of as many as 152 …”

“…The use of importance sampling [19] introduces a level of control that allowed larger systems to be treated. The essential idea of importance sampling is to introduce a guiding function (also called a trial function) and to define by = Introducing this trial function and defining = it/h yields (9) The first and last terms on the right hand side are diffusion and first order terms. The middle term corresponds to what is called drift or quantum force.…”

“…Our group has optimized molecular simulation methods particularly suitable for systems with highly interconnected bond networks. One area of improvement is in the calculation of forces and second derivatives common to many types of simulation [8][9][10][11]. Another is a method for automatically keeping track of bend, torsion, and other interactions in any bond network [12].…”

Computation of internal coordinates, derivatives, and gradient expressions: torsion and improper torsion

“…Swope and Ferguson 3 give a particularly complete discussion of mathematical issues for torsion interactions (and also for three-atom bend angles); Bekker et al 5 and Wilson, Decius, and Cross 12 also discuss these issues. Different computational schemes have been implemented by, among others, Miller et al, 4 Bekker et al, 5 Jung, 6 Blondel and Karplus, 9 and Tuzun et al 8 The recently developed internal coordinate quantum Monte Carlo (ICQMC) method 13 requires, in addition to first derivatives, Laplacians and gradient products of internal coordinates. Because the Laplacians often have particularly simple formulas, they can also be used to economize calculation of second derivatives required in classical simulation methods.…”

“…3 -9 Our group has evolved an approach (named GSF-geometric statement function method) in which three-and four-atom internal coordinates and derivatives by using are computed using those for two-and three-body interactions contained therein as intermediates. 7,8,10,11 Our recently developed general bond network method automates the bookkeeping required for this part of the calculation while making optimal use of our formulas. 12 Because of the large computational effort, mathematical singularities, and the well-known 1/sin τ problem, the torsion angle (τ , cos τ , sin τ ) has attracted much attention.…”