Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 52 (2019)

Improvement of functionals in density functional theory using inverse
Kohn-Sham method and density functional perturbation theory†
D. Ohashi,∗1,∗2 T. Naito,∗1,∗2 and H. Z. Liang∗2,∗10The density functional theory (DFT) is one of themost successful approaches to calculate the groundstate properties. The ground-state density ρgs andenergy Egs are obtained by solving the Kohn-Sham(KS) equation.1,2) The∫ ground-state energy is givenby Egs = T0 [ρgs ] + vext (r) ρgs (r) dr + EHxc [ρgs ],where T0 is the KS kinetic energy, vext is the externalﬁeld, and EHxc [ρ] is the Hartree-exchange-correlationenergy density functional (EDF). However, the accuracy of DFT calculations depends on the determinationof EHxc [ρ] as it is unknown. The inverse Kohn-Sham(IKS) method was proposed toward the improvementof EDFs. Nevertheless, the actual method to improve
EDFs has not been explicitly determined.
In this work, for the ﬁrst time, a new way to improve EDFs by the combination of the IKS methodand density functional perturbation theory (DFPT) isproposed. In this method, a conventional ẼHxc [ρ] isassumed to be close enough to EHxc [ρ] and the diﬀerence is considered in the ﬁrst-order DFPT( )with small(1)
λ as EHxc [ρ] = ẼHxc [ρ] + λEHxc [ρ] + O λ2 . In addition, ρgs (r) and Egs are expanded perturbatively. The
perturbation is assumed to not aﬀect the external ﬁeld,and ρgs (r) is assumed to be given.
Under the assumptions, we calculate Egs based on
the ﬁrst-order DFPT and on the IKS method. By com(1)paring them, the following equation for EHxc [ρ] is obtained:∫(1)
δEHxc [ρgs ](1)(1)
λEHxc [ρ̃gs ] − λEHxc [ρgs ] + λρgs (r) drδρ (r)∫N∑δ ẼHxc [ρgs ]=ρgs (r) dr− Ẽgs ,εi + ẼHxc [ρgs ]−δρ (r)
i=1

∗1
∗2

−30
−40
−50
0.01

target−20

LDA Exchange
1.4
1.3
1.2
1.1
1.0
0.9
0.01 0.1

ε1stx /εxε1stx (a.u.)−100.1111010100100

rs (a.u.)
Fig. 1. Exchange energy density ε1stx (rs ) calculated intarget
the first iteration and ratios ε1st(rs ). Thex (rs ) /εx
Hartree atomic unit is used here.

tionally, they are easy to handle owing to their spherical symmetry.
It is found that A1 and α1 are obtained within 7.2%
and 1.0% errors in He-Ne, and within 2.3% and 0.2%
errors in Xe-Rn, respectively, from the LDA results.
The calculated exchange energy density at the ﬁrst
iteration, ε1stx (rs ), and its ratio to the LDA are shownin Fig. 1 for the pairs of He-Ne and Xe-Rn withdashed and dot lines, respectively, while the LDA
exchange functional is represented with a solid line.
The energy density εx (ρ) and the∫ Wigner-Seitz radius rs are deﬁned as Ex [ρ] = εx (ρ) ρ (r) dr and
1/3
rs = [3/ (4πρ)] , respectively. The Xe-Rn pair reproduces the LDA functional within a few percents.
Moreover, it is found that the ground-state energy
becomes closer to the LDA as the iteration proceeds.
The ground-state energies of He, Ne, Xe, and Rn are
ﬁnally reproduced within 0.4%, 0.003%, 0.002%, and
0.0003% errors, respectively, comparing with 28%, 8%,2%, and 2% errors at the zeroth step. It is also foundthat the ground-state density is improved by the iterations.
To summarize, our method accurately reproduces
the LDA functionals. The accuracy of ground-stateenergies and densities are improved by two to three orders and one to two orders of magnitude, respectively.
This method can be eﬀective for nuclear DFT as well.

where εi is the single-particle energy obtained fromBecause it is diﬃρgs using the IKS method.
(1)
culttosolve
thisequation,
weassume EHxc [ρ] =∫∑αk[ρ (r)] dr and iteratively determine itsk Akvalue. Finally, ẼHxc [ρ] is improved to be ẼHxc [ρ] +(1)λEHxc [ρ].
As benchmark calculations, ρgs (r) is calculated
target[ρ] and we attempt to reprofrom the known EHxc
duce it from a less accurate ẼHxc . Here, we use thetarget
andHartree and LDA exchange functional as EHxc
the Hartree one as ẼHxc . The pairs of noble-gas atoms
He-Ne and Xe-Rn are used because two unknowns Aiand αi should be determined at each iteration. Addi†

Targetfrom He and Ne
from Xe and Rn

References
1) P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964).
2) W. Kohn, L. J. Sham, Phys. Rev. 140, A1133 (1965).
3) Y. Wang, R. G. Parr, Phys. Rev. A 47, R1591 (1993).
4) Q. Zhao, R. G. Parr, J. Chem. Phys. 98, 543 (1993).

Condensed from arXiv:1812.09285 [physics.chem-ph]
Department of Physics, The University of Tokyo
RIKEN Nishina Center- 67 -