Gauge-Invariant Method for Calculating the Electric Current Density Induced in Molecules and Solids by an External Magnetic Field

Abstract

In the approximation in which the nuclei are fixed in space, the electric current density induced in a molecule or solid by a uniform external magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}$ is shown to have the form $\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}=\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ where, to first order in $\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=(\frac{{e}^{2}\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}}{2mc})\ensuremath{\Sigma}{i}^{}{g}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. There is one function ${g}_{i}$ for each electron. These functions are independent of the gauge of the vector potential, and they are determined by the zero-field configuration-space probability density.