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 :math:`\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\ :math:`^3` (or 1 Å\ :math:`^3`). In order to compute the harmonic force constant :math:`k_{cs}` (where :math:`cs` stands for core-shell), the following is used \ :ref:`45 `: .. math:: k_{cs} ~=~ \frac{q_s^2}{\alpha} :label: eqnsimplepol where :math:`q_s` is the charge on the shell particle. Anharmonic polarization ~~~~~~~~~~~~~~~~~~~~~~~ For the development of the Drude force field by Roux and McKerell \ :ref:`93 ` it was found that some particles can overpolarize and this was fixed by introducing a higher order term in the polarization energy: .. math:: \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} :label: eqnanharmpol where :math:`\delta` is a user-defined constant that is set to 0.02 nm for anions in the Drude force field \ :ref:`94 `. Since this original introduction it has also been used in other atom types \ :ref:`93 `. :: [ polarization ] ;Atom i j type alpha (nm^3) delta khyp 1 2 2 0.001786 0.02 16.736e8 The above force constant :math:`k_{hyp}` corresponds to 4\ :math:`\cdot`\ 10\ :math:`^8` kcal/mol/nm\ :math:`^4`, hence the strange number. Water polarization ~~~~~~~~~~~~~~~~~~ A special potential for water that allows anisotropic polarization of a single shell particle \ :ref:`45 `. Thole polarization ~~~~~~~~~~~~~~~~~~ Based on early work by Thole :ref:`95 `, Roux and coworkers have implemented potentials for molecules like ethanol \ :ref:`96 `\ :ref:`98 `. 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 :math:`i` and :math:`j` is: .. math:: 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] :label: eqntholepol **Note** that there is a sign error in Equation 1 of Noskov *et al.*  :ref:`98 `: .. math:: {\bar{r}_{ij}}~=~ a\frac{r_{ij}}{(\alpha_i \alpha_j)^{1/6}} :label: eqntholsignerror where :math:`a` is a magic (dimensionless) constant, usually chosen to be 2.6 \ :ref:`98 `; :math:`\alpha_i` and :math:`\alpha_j` are the polarizabilities of the respective shell particles.