Nonempirical Construction of Current-Density Functionals from Conventional Density-Functional Approximations

Density-functional approximations for the exchange-correlation energy $E_{\mathrm{xc}}[n]$ of a many-electron ground state are highly developed and widely useful. When a paramagnetic current ${\mathbf{j}}_p(\mathbf{r})$ is present, Vignale and Rasolt have extended the Kohn-Sham theorems and presented an additive correction valid to second order in the gauge-invariant vorticity $\mathbf{\nu }=\nabla \times ({\mathbf{j}}_p/n)$: $E_{\mathrm{xc}}[n,{\mathbf{j}}_p]=E_{\mathrm{xc}}[n,{\mathbf{j}}_p=0]+\Delta E_{\mathrm{xc}}^{\mathrm{VR}}[n,\mathbf{\nu }]$. Apart from spin-polarization effects, their correction is unambiguous for a generalized gradient approximation (GGA). But for a meta-GGA (MGGA), one needs to know how to go back from the orbital kinetic energy density $\tau ([n,{\mathbf{j}}_p];\mathbf{r})$ to $\tau ([n,0];\mathbf{r})$; we show how to do this here. Numerical tests on the degeneracies for open-shell atoms show that current-density corrections reduce the error of GGA from 2 to $1\mathrm{kcal}/\mathrm{mol}$, and of MGGA from 5 to $2\mathrm{kcal}/\mathrm{mol}$.