The Electric Field Gradient Produced by a Gaussian Charge Density Distribution

Abstract

An analytical formula for the distance dependence of the electric field gradient produced by a Gaussian charge density distribution n(r) is derived. This charge density is displaced by z ₀ along the z-axis. The system has cylindrical symmetry; hence it suffices to calculate V _zz(0). It turns out that V _zz(0) is always smaller than the value with the total charge shrunk into a point. For distances larger than about four times the Gaussian width σ the expression approaches the point charge value. For z₀ → 0, i.e., a spherically symmetric charge distribution around the origin, V _zz(0) vanishes quadratically, as required by symmetry. A slab-wise calculation in cylindrical coordinates is presented which shows the contribution to V _zz(0) for infinitesimally thin slabs as a function of distance from the origin. This analytical formula allows for a fast computation of electric field gradients from a given charge density distribution for Gaussian expansions of Slater-type orbitals. An example for a hydrogen atom will be given.