Catalyst Force Field and Potential Energy Function

Introduction

This document provides an overview of the Catalyst implementation of potential energy functions for describing physical properties of molecular systems. Empirically, the potential energy of a molecular system can be described in terms of a sum of the energies due to different interactions within the molecular system. The Catalyst implementation contains energy contributions from the following types of interactions:

: (a) bond stretching
: (b) angle bending
: (c) linear deformation
: (d) out-of-plane deformation
: (e) dihedral torsion
: (f) van der Waals interactions

The functional forms (and hence the parameters) traditionally used to describe these interactions differ. Catalyst uses a version derived from common functional forms (CHARMm, AMBER, MM3, . . .), as described in Section 3. References for the force field and energy relationships are listed in Section 4.

The Catalyst Force Field Parameter Set

The Catalyst parameter set is derived from the CHARMm values, with redundancies removed and several errors corrected. These parameters truncate all force field components after the quadratic term, and generally express the dihedral expansion as a single term. Van der Waals interactions use the Lennard-Jones form discussed in Section 3(f.1).

This parameter set performs well in reproducing known experimental conformational trends. For those who are familiar with the format of the CHARMm parameters, the entire exhaustive set of parameter values are listed in the Catalyst force field file, $CATALYST_CONF/forceField.data.

Energy Function

This section describes the general form of the terms used in the Catalyst energy function. The comments indicate typical approximations that are made, while achieving sufficient accuracy for "well-behaved" molecules.

(a) Bond Stretching (Taylor Series Expansion to Morse Potential):

where Kn are the force constants for the nth term in the expansion, deltar = (r - r0), r is the current actual bond length, and r0 is the equilibrium bond length (in picometers). The termination of the series at the fourth power is common, and usually justified as being the highest degree to which experimental data can be rationally fit.

(b) Angle Bending (Taylor Series Expansion):

where (in radians), alpha is the current actual angle, alpha0 is the equilibrium angle, and Hn is the force constant for the nth power in the expansion. Usually only terms up to the fourth (quartic) power are implemented, and the choice of the cubic and quartic terms are, in a majority of cases, extensions from hydrocarbons.

(c) Linear Deformation:

where Ln are the force constants for the nth term of the expansion (in joules per mole per square radian), is the actual deformation (in radians). A peculiarity of linear bends is that they are defined in terms of a "phantom" vector originating at the central atom and perpendicular to the ideal linear axis. For completeness, two linear bends for each triplet of atoms are required.

(d) Out-Of-Plane Deformation:

where Wn are the force constants for the nth term of the expansion (in joules per mole per square radian), is the equilibrium out-of-plane angle (in radians), and omega is the current actual out-of-plane angle (in radians).

(e) Dihedral Torsion (Fourier Expansion):

where tau is the dihedral angle, and Yn are the force constants for the nth term of the expansion of the dihedral potential curve. The dihedral potential curve is usually determined experimentally by using a full quantum mechanics study of the system. The fourth and sixth power terms are rare in hydrocarbons and simple organic molecules, but common with sulfur, phosphorus, and metal compounds.

(f.1) van der Waals Interactions (Lennard-Jones):

where eAB is the energy of the minimum (deepest) point on the Van der Waals curve for the atom pair AB (in joules per mole, is the separation distance between the atom pair AB at the energy minima (in picometers, sigmaAB = 0.5(sigmaA + sigmaB)20.166667), rAB is the current actual distance between atom A and atom B (in picometers), and sigmaA and sigma are the van der Waals cross section dimensions for atoms A and B. These terms are not evaluated for atoms that are "1-3" (connected to a common atom) or atoms that are hydrogen bonded.

(f.2) Van der Waals Interactions (Buckingham):

where A, B, and C are constants, and rAB is the interatomic distance between the two atoms being evaluated.

Force Field References

S. Lifson and A. Warshel, "Consistent Force Field for Calculations of Conformations, Vibrational Spectra, and Enthalpies of Cycloalkane and n-Alkane Molecules." J. Chem. Phys., Vol. 49, 5116-5129 (1968).

A. Warshel, M. Levitt, and S. Lifson, "Consistent Force Field for Calculation of Vibrational Spectra and Conformations of Some Amides and Lactam Rings." J. Molecular Spectrosc., Vol. 33, 84-99 (1970).

N. Allinger, J. Am. Chem. Soc., Vol. 99, 8127 (1977).

N. Allinger, Y.H. Yuh, and J-H. Lii, "Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 1." J. Am. Chem. Soc., Vol. 111, 8551-8566 (1989).

J-H. Lii and N.L. Allinger, "Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 2. Vibrational Frequencies and Thermodynamics." J. Am. Chem. Soc., Vol. 111, 8566-8575 (1989).

J-H. Lii and N.L. Allinger, "Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 3. The van der Waals' Potentials and Crystal Data for Aliphatic and Aromatic Hydrocarbons." J. Am. Chem. Soc., Vol. 111, 8576-8582 (1989).

B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, and M. Karplus, "CHARMm: A Program for Macromolecular Energy, Minimization, and Dynamics Calculations." J. Comp. Chem., Vol. 4, 187-217 (1983).

S.J. Weiner, P.A. Kollman, D.A. Case, U.C. Singh, C. Ghio, G. Alagona, S. Profeta Jr., and P. Weiner, "A New Force Field for Molecular Mechanical Simulation of Nucleic Acids and Proteins." J. Am. Chem. Soc., Vol. 106, 765-784 (1984).

S.J. Weiner, P.A. Kollman, D.T. Nguyen, and D.A. Case, "An All Atom Force Field for Simulations of Proteins and Nucleic Acids." J. Comp. Chem., Vol. 7, 230-252 (1986).

L. Nilsson and M. Karplus, "Empirical Energy Functions for Energy Minimization and Dynamics of Nucleic Acids." J. Comp. Chem., Vol. 7, 591-616 (1986).

J.C. Smith and M. Karplus, "Empirical Force Field Study of Geometries and Conformational Transitions of Some Organic Molecules." J. Am. Chem. Soc., Vol. 114, 801-812 (1992).

T.A. Halgren, "Maximally Diagonal Force Constants in Dependent Angle-Bending Coordinates. 2. Implications for the Design of Empirical Force Fields." J. Am. Chem. Soc., Vol. 112, 4710-4723 (1990).

Last updated April 18, 1996 at 11:55am PDT.

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