Functional form

Functional form

The following functional abstraction, termed an interatomic potential function or force field in chemistry, calculates the molecular system’s potential energy (E) in a given conformation as a sum of individual energy terms.

 

{\displaystyle \ E=E_{\text{covalent}}+E_{\text{noncovalent}}\,}

where the components of the covalent and noncovalent contributions are given by the following summations:

 

{\displaystyle \ E_{\text{covalent}}=E_{\text{bond}}+E_{\text{angle}}+E_{\text{dihedral}}}

 

{\displaystyle \ E_{\text{noncovalent}}=E_{\text{electrostatic}}+E_{\text{van der Waals}}}

The exact functional form of the potential function, or force field, depends on the particular simulation program being used. Generally the bond and angle terms are modeled as harmonic potentials centered around equilibrium bond-length values derived from experiment or theoretical calculations of electronic structure performed with software which does ab-initio type calculations such as Gaussian. For accurate reproduction of vibrational spectra, the Morse potential can be used instead, at computational cost. The dihedral or torsional terms typically have multiple minima and thus cannot be modeled as harmonic oscillators, though their specific functional form varies with the implementation. This class of terms may include improper dihedral terms, which function as correction factors for out-of-plane deviations (for example, they can be used to keep benzene rings planar, or correct geometry and chirality of tetrahedral atoms in a united-atom representation).

The non-bonded terms are much more computationally costly to calculate in full, since a typical atom is bonded to only a few of its neighbors, but interacts with every other atom in the molecule. Fortunately the van der Waals term falls off rapidly. It is typically modeled using a 6–12 Lennard-Jones potential, which means that attractive forces fall off with distance as r−6 and repulsive forces as r−12, where r represents the distance between two atoms. The repulsive part r−12 is however unphysical, because repulsion increases exponentially. Description of van der Waals forces by the Lennard-Jones 6–12 potential introduces inaccuracies, which become significant at short distances. Generally a cutoff radius is used to speed up the calculation so that atom pairs which distances are greater than the cutoff have a van der Waals interaction energy of zero.

The electrostatic terms are notoriously difficult to calculate well because they do not fall off rapidly with distance, and long-range electrostatic interactions are often important features of the system under study (especially for proteins). The basic functional form is the Coulomb potential, which only falls off as r−1. A variety of methods are used to address this problem, the simplest being a cutoff radius similar to that used for the van der Waals terms. However, this introduces a sharp discontinuity between atoms inside and atoms outside the radius. Switching or scaling functions that modulate the apparent electrostatic energy are somewhat more accurate methods that multiply the calculated energy by a smoothly varying scaling factor from 0 to 1 at the outer and inner cutoff radii. Other more sophisticated but computationally intensive methods are particle mesh Ewald (PME) and the multipole algorithm.

In addition to the functional form of each energy term, a useful energy function must be assigned parameters for force constants, van der Waals multipliers, and other constant terms. These terms, together with the equilibrium bond, angle, and dihedral values, partial charge values, atomic masses and radii, and energy function definitions, are collectively termed a force field. Parameterization is typically done through agreement with experimental values and theoretical calculations results. Norman L. Allinger’s force field in the last MM4 version calculate for hydrocarbons heats of formation with a rms error of 0.35 kcal/mol, vibrational spectra with a rms error of 24 cm−1, rotational barriers with a rms error of 2.2°, C-C bond lengths within 0.004 Å and C-C-C angles within 1°.

Later MM4 versions cover also compounds with heteroatoms such as aliphatic amines.

Each force field is parameterized to be internally consistent, but the parameters are generally not transferable from one force field to another.

Functional form

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