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.