## MOLECULAR MECHANICS AND DYNAMICS

04/13/16

# E. Non-Bonded Interaction Energy

The non-bonded energy is calculated for all
possible pairs of nonbonded atoms, i and j:

**Enonbonding**
= Σi Σj [ -Bij/rij^{6} +A ij/rij^{12}
] + Σi Σj (qi qj) / rij

The first term represents van der Waals interactions while
the second terms represents Coloumbic electrostatic interactions.

You should remember that van der
Waals interactions are short range and occur among all atoms. The 6-12
energy equation based on the Lennard-Jones' potential, shows a negative
(attractive) term proportional to -1/r^{6} and a repulsive term proportional to
+1/r^{12} :

I

The A and B parameters control the depth and position (interatomic
distance) of the potential energy well for a given pair of non-bonded
interacting atoms (e.g. C:C, O:C, etc.). In effect, A determined the degree
of stickiness of the van der Waals attraction, and B determines the degree
of hardness of the atoms (e.g. marshmallow-like, billiard ball-like, etc.).

The B parameter is related to the "stickiness" of the interactions and is
related to the polarization of the atoms. B can be obtained from
atomic polarizability measurements, or it can be calculated quantum
mechanically. The A parameter is empirically derived to fit nonbonded
contacts between atoms in crystal structures.

Wolfram
Mathematica CDF Player - Interactive Graph of 6-12 Lennard Jones Potential
(free plugin required)

Interactive SageMath
Graph: Interactive Graph of 6-12 Lennard Jones Potential

Lennard-Jones
Interactive Applet

Couloumb's Law is used to calculate the electrostatic interactions based
on appropriate dielectric constants (for buried or water accessible ion-ion
pairs).

Enonbonding (electrostatic) =
Σi Σj
(qi qj) / rij

A higher dielectric constant reflect more shielding by polar solvents of
charge pairs. Partial charges on atoms are calculated using ab initio or
semiempirical quantum mechanics. (usually MOPAC or AMPAC).
The equation for the electrostatic potential is:

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