Exchange functional by a range-separated exchange hole

An approximation to the exchange-hole density is proposed for the evaluation of the exact exchange energy in electronic structure calculations within the density-functional theory and the Kohn-Sham scheme. Based on the localized nature of density matrix, the exchange hole is divided into the short-range (SR) and long-range (LR) parts by using an adequate filter function, where the LR part is deduced by matching of moments with the exactly calculated SR counterpart, ensuring the correct asymptotic $-1/r$ behavior of the exchange potential. With this division, the time-consuming integration is truncated at a certain interaction range, largely reducing the computation cost. The total energies, exchange energies, exchange potentials, and eigenvalues of the highest-occupied orbitals are calculated for the noble-gas atoms. The close agreement of the results with the exact values suggests the validity of the approximation.

Figure 1

Exchange hole of a neon atom at $r=0.31$ a.u. The exact hole function (solid line), the hole in the present scheme (dashed line) with the parameter $\mu =0.1\hspace{0.5em}\mathrm{a}.{\mathrm{u}}_{.}^{-1}$, and the SR part of the hole in the present scheme (dotted line) are plotted after multiplied by $s^2$.

Figure 2

Exchange hole of an argon atom at $r=2.48$ a.u. The exact hole function (solid line), the hole in the present scheme (dashed line) with the parameter $\mu =0.1\hspace{0.5em}\mathrm{a}.{\mathrm{u}}_{.}^{-1}$, and the SR part of the hole in the present scheme (dotted line) are plotted after multiplied by $s^2$.

Figure 3

Exchange potential of a helium atom, calculated by the exact (solid line), LDA (dashed line), GGA [5] (dotted line), and the present (thick dashed line) method.