Electric field gradient

Definition

A given charge distribution of electrons and nuclei, ρ(r), generates an

${\displaystyle V_{ij}={\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}.}$

For each nucleus, the components V_ij are combined as a symmetric 3 × 3 matrix. Under the assumption that the charge distribution generating the electrostatic potential is external to the nucleus, the matrix is

${\displaystyle \varphi _{ij}=V_{ij}-{\frac {1}{3}}\delta _{ij}\nabla ^{2}V,}$

where ∇²V(r) is evaluated at a given nucleus.

As V (and φ) is symmetric it can be

${\displaystyle \eta ={\frac {V_{xx}-V_{yy}}{V_{zz}}}.}$

with ${\displaystyle \vert V_{zz}\vert \geq \vert V_{yy}\vert \geq \vert V_{xx}\vert }$ and ${\displaystyle V_{zz}+V_{yy}+V_{xx}=0}$, thus ${\displaystyle 0\leq \eta \leq 1}$.