Electrical response of molecular chains in density functional theory: Ultranonlocal response from a semilocal functional

An exchange potential functional is constructed from semi-local quantities and is shown to reproduce hydrogen chain polarizabilities with the same accuracy as exact exchange methods. We discuss the exchange potential features that are essential for accurate polarizability calculations, i.e., derivative discontinuities and the potential step structure. The possibility of a future generalization of the methods into a complete semi-local exchange-correlation functional is discussed.

Figure 1

Total and exchange energy of a Mg ion versus electron occupation number, calculated with the fully self-consistent exchange-only LDA. New electron shells are started when $N$ is equal to 2 and 10. The figure shows that even for exchange-only LDA, both $dE∕dN$ and $d{E}_{\mathrm{x}}∕dN$ are discontinuous at these points.

Figure 2

(Color online) The exchange potential calculated with Slater exchange plus the BJ correction term for fully doubly ionized magnesium atom ${\mathrm{Mg}}^{2+}$ (fully drawn black line) and the same system but with small fractions of an electron added, ${10}^{-2}$ (dotted red) and ${10}^{-6}$ (dashed blue). The last two lines are virtually indistinguishable, which means that the BJ correction term makes the graph discontinuously shifted just as the exact exchange does—precisely when a new orbital shell is opened. The detailed shape of the shift may not be absolutely accurate compared to an exact exchange calculation (not shown, compare to Ref. 20, Fig. 10), but the fact that such a shift is at all present makes for an important difference to other semi-local methods. Also note that the step moves to larger values of $r$ with decreasing occupation number, as it should.

Figure 3

(Color online) The difference between the exchange potential for calculations with and without an external electrical field of strength $F=0.005\text{hartree}∕\text{bohr}$ (shown in the figure). The upper figure is for a ${\mathrm{H}}_4$ chain, and the lower for a ${\mathrm{H}}_6$ chain (atom positions are indicated by circles). The figure shows that calculations using the BJ correction term are not very similar to the exact exchange as approximated by KLI nor does it behave properly far outside the system.

Figure 4

The BJ correction term $v_{\mathrm{x}\sigma }^{\mathrm{BJ}}$ of Eq. (8) for a single hydrogen atom in an electrical field $F=0.005\text{hartree}∕\text{bohr}$. The BJ correction term is here calculated from self-consistent KLI KS orbitals (the other figures show Slater $\text{exchange}+\mathrm{BJ}$ correction done fully self-consistently). The figure shows how the BJ term slopes linearly outside a system in an electrical field.

Figure 5

(Color online) As in Fig. 3, the difference between the exchange potential for calculations with and without an external electrical field of strength $F=0.005\text{hartree}∕\text{bohr}$ for a ${\mathrm{H}}_4$ and ${\mathrm{H}}_6$ chain. However, now, the exchange functional used has been further corrected with the additional linear term derived in Eq. (20), and the similarity with exact exchange (as approximated by KLI) is apparent.