Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 51 (2018)

Application of a Coulomb energy density functional for atomic
nuclei: Case studies of local density approximation and generalized
gradient approximation†
T. Naito,∗1,∗2 R. Akashi,∗1 and H. Z. Liang∗2,∗10εxρch (MeV fm−3)

The exchange (x) and correlation (c) energy density functionals Ex [ρ] and Ec [ρ] formulated for electron systems are tested in the context of atomic nuclei,respectively. Both the local density approximation(LDA) and generalized gradient approximation (GGA)
functionals are investigated. For quantitative calculations, we employed the experimental charge-densitydistributions ρch 1) of the selected nuclei as inputs ofground-state density distributions.
When it is assumed that the energy density εi depends only on the density at r locally as∫
Ei [ρ] = εi (ρ (r)) ρ (r) dr(i = x, c),(1)

∗202

4

68

10B88
PW91
PBE
PBEsol∆Ex (%)

1086420050100

150200250AFig. 2. Deviation between the LDA and GGA in Ex definedas Eq. (3) as a function of A.

self-consistent calculations. In contrast, the correlation Coulomb energy density functionals of electron
systems are not applicable for atomic nuclei, because
these functionals are not separable and the nuclear interaction determines the properties of atomic nuclei.

ranges from around 11% in 4 He to around 2.2% in
Pb, by taking the PBE functional as an exampleof the GGA. From light to heavy nuclei, it is seen that
∆Ex shown in Fig. 2 behaves in a very similar way as
the deviation between the Hartree-Fock-Slater approximation and the exact Hartree-Fock given by Le Bloaset al .6) In this sense, the GGA exchange functionalsof electron systems can be applied in a straightforward manner with practical accuracy to atomic nuclei.
( 3)
Furthermore, the numerical cost of GGA is (O N) ,whereas that cost of exact Hartree-Fock is O N 4 for∗1−0.04

12(3)

CLDA
B88
PW91
PBE
PBEsol

−0.03

Fig. 1. GGA exchange energy densities weighted with ρchfor 208 Pb as a function of r.

208

Condensed from the article in Phys. Rev.
044319 (2018)Department of Physics, the University of Tokyo
RIKEN Nishina CenterSurfacer (fm)(2)
The GGA exchange energy density weighted withρch (r) for 208 Pb is shown in Fig. 1. The LDA result is shown with the long-dashed line, and thosegiven by the GGA functionals B88,2) PW91,3) PBE,4)and PBEsol5) are shown with the short-dashed, dotdashed, solid, and dot-dot-dashed lines, respectively.
The surface is defined as the region that has a density
between 90% and 10% of the maximum density.
For the exchange Coulomb energies, it is found thatthe deviation between the LDA and GGA,†−0.02−0.05

this approximation is called the LDA. In the GGA,
the energy density depends not only on the density
distribution ρ but also on its gradient |∇ρ| at r locallyas∫
Ei [ρ] = εi (ρ (r) , |∇ρ (r)|) ρ (r) dr(i = x, c).

E GGA − ExLDA,∆Ex = x
ExGGA

208Pb

−0.01

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