# Charge equilibration for molecular dynamics simulations

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Charge equilibration (QEq) for molecular dynamics simulations [1] [2] is a technique for calculating the distribution of charges within a (large) molecule. This distribution can change with time to match changes in the local environment.

## Electronegativity and electronic hardness

The atomic electronegativity is given by [3]

${\displaystyle \chi ={\frac {\mathrm {IP+EA} }{2}}\approx {\frac {\partial E}{\partial Q}}}$

where IP is the ionisation potential, and EA is the electron affinity. The electronic hardness is given by [4]

${\displaystyle \eta =\mathrm {IP-EA} \approx {\frac {\partial ^{2}E}{\partial Q^{2}}}}$

## Charge equilibration energy

Using the above expressions one has the following second order approximation for the total electrostatic energy ([2] Eq. 6)

${\displaystyle E=\sum _{i}\left(q_{i}\chi _{i}+{\frac {q_{i}^{2}}{2}}\eta _{i}\right)+\sum _{i\neq j}q_{i}q_{j}J_{ij}}$

The last term is a "shielded" Coulombic interaction, where

${\displaystyle J_{ij}({\mathbf {r} }_{ij})=\left\langle \phi _{i}\phi _{j}\left\vert {\frac {1}{|{\mathbf {r} }_{i}-{\mathbf {r} }_{j}|}}\right\vert \phi _{i}\phi _{j}\right\rangle }$

where ${\displaystyle \phi }$ represents a normalised ns Slater-type orbital.