GROMACS can simulate polarizability using the shell model of Dick and

Overhauser 43. In such models a shell particle

representing the electronic degrees of freedom is attached to a nucleus

by a spring. The potential energy is minimized with respect to the shell

position at every step of the simulation (see below). Successful

applications of shell models in GROMACS have been published for

(N_2) 44 and water45.

## Optimization of the shell positions

The force (mathbf{F})(_S) on a shell

particle (S) can be decomposed into two components

where (mathbf{F}_{bond}) denotes the

component representing the polarization energy, usually represented by a

harmonic potential and (mathbf{F}_{nb}) is the sum of Coulomb

and van der Waals interactions. If we assume that

(mathbf{F}_{nb}) is almost constant we

can analytically derive the optimal position of the shell, i.e. where

(mathbf{F}_S) = 0. If we have the shell S connected to atom A we have

In an iterative solver, we have positions (mathbf{x}_S(n)) where (n) is

the iteration count. We now have at iteration (n)

and the optimal position for the shells (x_S(n+1)) thus follows from

if we write

we finally obtain

which then yields the algorithm to compute the next trial in the

optimization of shell positions

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