# Polarization¶

Polarization can be treated by GROMACS by attaching shell (Drude) particles to atoms and/or virtual sites. The energy of the shell particle is then minimized at each time step in order to remain on the Born-Oppenheimer surface.

## Simple polarization¶

This is implemented as a harmonic potential with equilibrium distance 0. The input given in the topology file is the polarizability $$\alpha$$ (in GROMACS units) as follows:

[ polarization ]
; Atom i  j  type  alpha
1         2  1     0.001


in this case the polarizability volume is 0.001 nm$$^3$$ (or 1 Å$$^3$$). In order to compute the harmonic force constant $$k_{cs}$$ (where $$cs$$ stands for core-shell), the following is used 45:

(1)$k_{cs} ~=~ \frac{q_s^2}{\alpha}$

where $$q_s$$ is the charge on the shell particle.

## Anharmonic polarization¶

For the development of the Drude force field by Roux and McKerell 93 it was found that some particles can overpolarize and this was fixed by introducing a higher order term in the polarization energy:

(2)\begin{split}\begin{aligned} V_{pol} ~=& \frac{k_{cs}}{2} r_{cs}^2 & r_{cs} \le \delta \\ =& \frac{k_{cs}}{2} r_{cs}^2 + k_{hyp} (r_{cs}-\delta)^4 & r_{cs} > \delta\end{aligned}\end{split}

where $$\delta$$ is a user-defined constant that is set to 0.02 nm for anions in the Drude force field 94. Since this original introduction it has also been used in other atom types 93.

[ polarization ]
;Atom i j    type   alpha (nm^3)    delta  khyp
1       2       2       0.001786     0.02  16.736e8


The above force constant $$k_{hyp}$$ corresponds to 4$$\cdot$$10$$^8$$ kcal/mol/nm$$^4$$, hence the strange number.

## Water polarization¶

A special potential for water that allows anisotropic polarization of a single shell particle 45.

## Thole polarization¶

Based on early work by Thole 95, Roux and coworkers have implemented potentials for molecules like ethanol 9698. Within such molecules, there are intra-molecular interactions between shell particles, however these must be screened because full Coulomb would be too strong. The potential between two shell particles $$i$$ and $$j$$ is:

(3)$V_{thole} ~=~ \frac{q_i q_j}{r_{ij}}\left[1-\left(1+\frac{{\bar{r}_{ij}}}{2}\right){\rm exp}^{-{\bar{r}_{ij}}}\right]$

Note that there is a sign error in Equation 1 of Noskov et al.  98:

(4)${\bar{r}_{ij}}~=~ a\frac{r_{ij}}{(\alpha_i \alpha_j)^{1/6}}$

where $$a$$ is a magic (dimensionless) constant, usually chosen to be 2.6 98; $$\alpha_i$$ and $$\alpha_j$$ are the polarizabilities of the respective shell particles.