Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 52 (2019)

Coulomb exchange functional with generalized gradientapproximation for self-consistent Skyrme Hartree-Fock calculations†T. Naito,∗1,∗2 X. Roca-Maza,∗3,∗4 G. Colò,∗3,∗4 and H. Z. Liang∗2,∗110LDA (%)∆ECx

Recently, it was shown that the Coulomb energy
density functionals of the generalized gradient approximation (GGA) give almost the same accuracy as thatof the exact-Fock energy1) by using the experimental charge density distribution as inputs of the functional. As a step further, we carry out the corresponding self-consistent Skyrme Hartree-Fock calculation byusing the Perdew-Burke-Ernzerhof GGA (PBE-GGA)
Coulomb exchange functional2) instead of the exactFock. The GGA Coulomb exchange functionals havebeen proposed as|∇ρch |,
2kF ρch()1/3kF = 3π 2 ρch.

(2)

κ,
1 + µs2 /κ

∗1
∗2
∗3
∗4exact (%)∆ECxCaSnOCaSn

20

40

0.5
0.0
−0.5
−1.0
−1.5
−2.006080

100 120 140 160 180

Mass Number AFig. 1. Deviation between the PBE-GGA and the LDA,LDA∆ECx, and the deviation between the PBE-GGA andexact
the exact-Fock ∆ECx
, defined as Eq. (4).
exact
ECx − ECx,ECx(4)are shown as functions of mass number A in Figs. 1(a)and (b), respectively.
In conclusion, λ does not have an obvious isospin
dependence, and λ = 1.25 reproduces the exact-Fockcalculation well in general. Here, note that the PBEGGA Coulomb potential is the local potential andhence the(numericalcost of the self-consistent calcula)tion is O N 3 , while the exact-Fock Coulomb potential is the(non-local
potential and hence the numerical)
cost is O N 4 . This scheme helps to achieve a better
description and understanding of the observables inwhich the Coulomb interaction plays important roles,
such as the mass diﬀerence of the mirror nuclei and theenergy of the isobaric analog state.
LDA=
∆ECx

(3)

in order to satisfy some physical conditions.3) Accordingly, the parameter κ = 0.804 is determined for any
value of µ by the Hölder inequality. In contrast, two
diﬀerent values of µ have been widely used in the studies of atoms2) and solids.4) For the PBE-GGA functional, µ = 0.21951 is determined by the random phaseapproximation of the homogeneous electron gas. Since
this µ can be a diﬀerent value for nuclear systems, thefree parameter of the PBE-GGA Coulomb exchange
functional, µ, is multiplied by a factor λ. For the nuclear part, the SAMi functional5) is used in the selfconsistent calculation. For comparison, the exact-Fockenergies are also calculated, where the exact-Fock calculation is carried out using the ﬁrst-order perturbation theory.6)
The deviation of the Coulomb exchange energy ECx
LDAin the PBE-GGA from that in the LDA, ∆ECx, andexact,
the deviation from that in the exact-Fock ∆ECx
†O(b)

1.0

In particular, the enhancement factor F in the PBEGGA Coulomb exchange functional is assumed to be2)
F (s) = 1 + κ −40
1.5( )1/3∫3e34/3[ρch (r)] F (s (r)) dr,ECx [ρch ] = −16πε0 π(1)where F is the enhancement factor due to the density
gradient, F ≡ 1 is the hold for the local density approximation (LDA), and ρch is the charge distribution.
Here, s denotes the dimensionless density gradient

exact-Fock
GGA (λ = 1.00)
GGA (λ = 1.25)
GGA (λ = 1.50)

SAMi

622

s=(a)

8LDA
ECx − ECx,
ECx

exact=∆ECx

References
1) T. Naito, R. Akashi, H. Liang,Phys. Rev. C
97, 044319 (2018).
2) J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett.
77, 3865 (1996).
3) J. P. Perdew, K. Burke, Y. Wang,Phys. Rev.
B 54, 16533 (1996).
4) J. P. Perdew, et al ., Phys. Rev. Lett. 100, 136406(2008).
5) X. Roca-Maza, G. Colò, H. Sagawa, Phys. Rev. C 86,031306 (2012).
6) X. Roca-Maza, L.-G. Cao, G. Colò, H. Sagawa, Phys.
Rev. C 94, 044313 (2016).

Condensed from the article in Phys. Rev. C 99, 024309(2019)
Department of Physics, The University of Tokyo
RIKEN Nishina Center
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano- 68 -