Relativistic correction of the Coulomb interaction in the local density approximation for energies and radii in doubly-magic nuclei

Effects of the relativistic correction of the Coulomb interaction on doubly-magic nuclei are discussed with Skyrme Hartree--Fock calculations. The relativistic correction is treated by using the local density approximation. It is found that the correction to the total energy is about $ 2.4 \, \mathrm{MeV} $ for $ {}^{208} \mathrm{Pb} $, while proton and neutron radii do not change significantly. This difference is larger than the difference of the Coulomb exchange (Fock) energy calculated with the local density (Hartree--Fock--Slater) approximation and that with the exact treatment and the neutron finite-size effect. Effects of the correction are also compared to the correction due to the vacuum polarization. It is shown that the two contributions to the total energy are comparable in light nuclei, but the latter dominates in heavy nuclei, while the contribution of the relativistic correction to the total energy is non-negligible compared to the target accuracy of the DFT calculation.

GGA Coulomb energy. The Breit correction had been taken into account for an exchange [59] or exchange-correlation [60] EDF in the LDA level, and was already taken into account fully for atomic DFT [51]. It was found in Ref. [51] that the Breit correction contributes to the total energies of atoms in almost the same magnitude as but the opposite to the density gradient effect. In the context of nuclear physics, it was shown in Refs. [39,61,62] that difference between the Coulomb exchange energy calculated by the exact-Fock term or GGA and that in LDA is non-negligible in nuclear binding energies. Thus, it should be tested to what extent the Breit correction contributes to nuclear binding energies.
The covariant density functional theory, the nuclear DFT in the relativistic scheme, is based on the covariant Lagrangian with nucleon, meson, and photon fields. Hence, effects beyond the Coulomb interaction, such as magnetic interaction and retardation, have often been considered [42,43,63,64]. Therefore, it is interesting to take into account the relativistic correction of the Coulomb interaction in non-relativistic SHF calculations.
The relativistic correction of the Coulomb interaction was already discussed in Refs. [65,66]. As discussed later, the correction used in these works is the same as the non-relativistic reduced form of the Breit correction for electronic systems [67,68], while anomalous magnetic moments of nucleons are, of course, considered. The aim of this paper is modern calculation of Refs. [65,66]: Deriving the EDF for such corrections and performing the self-consistent calculation. I shall discuss contributions to proton and neutron radii, total energies, and single-particle energies. The LDA is used to consider the relativistic correction; some terms of the correction vanish in the approximation, as will be discussed. Note that it seems that "the Breit correction" refers to the whole terms of the non-relativistic reduced form of the correction [Eq. (5)] in the context of atomic physics, while it refers to only a term of the correction in the context of nuclear physics. In order to avoid any confusion, hereinafter, I will use a term "the relativistic correction", instead of "the Breit correction". This paper is organized as follows: In Sec. 2, the theoretical framework for the relativistic correction will be shown. In Sec. 3, the relativistic correction to the total energy will be estimated analytically, and in Sec. 4, numerical results will be shown. Finally, in Sec. 5, conclusion and perspectives will be given.

Theoretical Framework
In this section, I show the theoretical framework for the relativistic correction and the DFT. Throughout this paper, an unit with 4πε 0 = 1 is used and c denotes the speed of light.
Quantum many-body systems can be described by using the Hamiltonian where T , V ext , and V int denote the kinetic energy operator, an external potential, and an interaction, respectively. In the case of electronic systems, V int is identical to the EM interaction V EM , while in the case of atomic nuclei, V int includes a nuclear interaction V nucl and the EM interaction V EM with V ext ≡ 0. In most works, the Coulomb interaction between protons is used for V EM , where r j and r k are the spatial coordinates of the protons j and k, r jk = r j − r k , and r jk = |r jk |. According to the quantum electrodynamics, the leading-order 3/19 instantaneous interaction, in which a photon mediates between two particles with c = ∞, is the Coulomb interaction.

Breit correction for electronic systems
In electronic systems, the next-leading-order interaction (O 1/c 2 ) in the Coulomb gauge is called the Breit correction, whose form is [50,[55][56][57]69] where α is 4 × 4 matrix defined by with the Pauli matrices σ x , σ y , and σ z and the 2 × 2 zero matrix O 2 . In isolated atoms, the vacuum polarization of the Coulomb field appears in higher order than the Breit correction [58]. The Breit correction includes the magnetic (current-current) interaction and the retardation in the Coulomb gauge [69]. In the Lorenz (covariant) gauge, the next-leadingorder interaction corresponds to the magnetic interaction and the retardation [69][70][71].
which will be referred to as the Coulomb-Breit interaction.

Beyond Coulomb interaction for nuclear systems
Corrections beyond the Coulomb interaction for nuclear systems were discussed several decades ago [65,66]. The correction was where M ≈ 938.919 MeV/c 2 is the nucleon mass [76], δ pτ is the Kronecker delta, which is equal to 1 for τ = p and 0 for τ = n, e τ and µ τ are the charge and magnetic moment of a nucleon τ , andμ τ is the magnetic moment of the unit of the nuclear magneton, i.e., [76]  If one assumes that these two particles are electrons instead of nucleons, and accordingly, m e , −e, and µ e , are used instead of M , e τ , and µ τ , one can find that Eq. (7) is identical to Eq. (5). Here, the relation between the g-factor and the magnetic moment µ e µ e = e 4m e g e (9) and g e = 2 is used, apart from the second term of Eq. (7b), in which the g-factor for the orbital motion of electrons g e = 1 is used. In more detail, the correspondence between Eqs. (5) and (7)

Density functional theory for relativistic correction
In nuclear DFT, the ground-state energy is obtained by where T 0 is the Kohn-Sham kinetic energy, E nucl and E EM are EDFs of nuclear and EM interactions, respectively, and ρ p gs and ρ n gs are the ground-state densities of protons and neutrons. If the finite charge distributions of nucleons are considered, E EM should depend on the ground-state charge distribution, instead of ρ gs p [40]. If one only considers the Coulomb interaction to V EM , E EM in LDA reads [77] where the density ρ is the electron density ρ e for electronic systems and the proton density ρ p for nuclear systems.
Recently, an EDF for the Breit correction [Eq. (5)] for electronic systems in the LDA was developed in Refs. [51,60]. In this paper, I consider the relativistic correction [Eq. (7)] in the nuclear DFT using the idea of such an EDF and the correspondence shown in Table 1. This newly developed EDF enables one to perform the self-consistent calculation. To isolate the relativistic correction, I shall not consider the nucleon finite-size effects 1 .
The Hartree and LDA exchange EDFs for the Breit correction in the electronic systems [Eq. (5)] were proposed in Ref. [51] and Ref. [59], respectively. Their forms are respectively, and accordingly, the total correction is The idea of the LDA is that the energy density is approximated to that of homogeneous systems; hence, contributions of Eqs. (5c), (5d), and (5e) vanish. Consequently, Eqs. (12a) and (12b) contain only effects originating from Eqs. (5a) and (5b) [60]. In addition, due to the symmetry, Eq. (12a) does not contain the effect of the retardation [Eq. (5b)]. I shall construct an EDF for Eq. (7) in the LDA, using the knowledge of Eqs. (12a) and (12b). Because of the idea of the LDA, the EDF contains only the effect of Eqs. (7a) and (7g), as in the electronic systems. Note that the spin-orbit interaction originating from the magnetic form factors of nucleons [Eqs. (7b), (7e), and (7f)] have been discussed in Refs. [7,40,78]; the spin-magnetic interaction [Eqs. (7c) and (7d)] has not been considered yet for the self-consistent calculation and it cannot be considered in the LDA, which is left for the future study.
Then, I will construct the LDA EDF for Eq. (7). Referring the result on Eq. (12), one can find that Eq. (7g) contributes both the Hartree and the exchange terms, whereas Eq. (7a) contributes the exchange term only.
The Hartree and the exchange EDFs of Eq. (7g), respectively, read where and accordingly The exchange EDF for Eq. (7a) is constructed from Eq. (12b). Equation (12b) does not include only a contribution from Eq. (5b) but also that from Eq. (5a). Hence, the exchange EDF for Eq. (5b) reads since the Darwin EDF for electronic systems is To apply the nuclear systems, with replacing the electron mass m e to the nucleon mass M , one obtains Combining Eqs. (13) and (18), the total EDF for the correction reads

Analytical Estimation
Before performing numerical many-body calculations with the relativistic correction, I estimate contribution of the correction to the total energy with a simple model and discuss the systematic behavior. As in Ref. [40], I assume hard-sphere distributions for both proton and neutron distribution ρ p and ρ n : where R τ is the proton (τ = p) or neutron (τ = n) radius defined by with N p = Z and N n = N . For simplicity, the half of the saturation density of atomic nuclei ρ p 0 = ρ n 0 ≈ 0.08 fm −3 is assumed. With this density, one can estimate the Coulomb Hartree and exchange energies as [40] E H Coul = 3e 2 5 respectively. Since relations hold with R < = R p for R p < R n and R < = R n for R p > R n , one, accordingly, obtains In the case of R p < R n , Eqs. (25a) and (25d) read in the case of R p > R n , they read Comparing Eqs. (23b) and (26b), one can find that the relativistic correction to the total energy E tot Rel is 15 times smaller than the Coulomb exchange energy |E x Coul |. This means that although E tot Rel is small compared to the other contributions, one can estimate that E tot Rel for 208 Pb may be around 2 MeV, which is a non-negligible contribution compared to the desired accuracy of E nucl (O (100) keV).
In atomic physics, the vacuum polarization to the Coulomb potential formed by an atomic nucleus is the next-leading order to the Breit correction, i.e., the relativistic correction of electron-electron interaction [58]. On the other hand, in nuclear structure, both the vacuum polarization and the relativistic correction are corrections of the proton-proton Coulomb interaction. Thus, it is worthwhile to compare them. In Ref. [40], the vacuum polarization was taken into account in a nuclear structure calculation using the Uehling potential [79,80]. The contribution of the vacuum polarization was estimated as [40] E VP ≃ 0.0047Z 5/3 MeV.
Hence, the relativistic correction and the vacuum polarization provide comparable contributions to the total energy in the light nuclei (e.g., for Z = 20, E tot Rel ≈ 0.64 MeV and E VP ≈ 0.69 MeV) but the latter dominates in heavy nuclei (e.g., for Z = 100, E tot Rel ≈ 3.2 MeV and E VP ≈ 10 MeV). In the next section, I shall confirm this by performing numerical calculations.

Skyrme Hartree-Fock Calculation
To calculate the ground-state density distribution and energy numerically, one needs to use a many-body calculation technique. In this paper, the Skyrme Hartree-Fock method is used.
The EDF for the correction [Eq. (19)] is implemented to the self-consistent Skyrme Hartree-Fock calculation code skyrme_rpa [81]. In this work, doubly-magic nuclei are focused on; thus, the pairing interaction is not considered, and the spherical symmetry is assumed. The calculations are performed with a box of 15 fm with 0.1 fm mesh. The SAMi EDF [82] is adopted for the nuclear EDF.
It is worthwhile to mention here that the non-relativistic Coulomb GGA exchange EDF, more precisely, the modified Coulomb GGA-Perdew-Burke-Ernzerhof (PBE) EDF with λ = 1.25 [39], reproduces the exact Coulomb Fock energy less than 100 keV error, which is much less than the relativistic correction, which I will discuss; thus, in this paper, the exact Coulomb Fock energy is approximated to the Coulomb GGA exchange energy, Coul . First of all, I shall mention effects to nuclear radii. Neither the proton nor the neutron radii are found to be changed more than 0.002 fm due to the relativistic correction and the vacuum polarization. Therefore, the effect of the correction on nuclear radii can be safely neglected.
Next, the ground-state energies for doubly-magic nuclei are shown in Table 2. The columns labelled "Coulomb (NR-LDA)" and "Coulomb (R-LDA)" show results with the non-relativistic Coulomb LDA [Eq. (11)] and the relativistic Coulomb LDA [Eqs. (11) and (19)], respectively. For comparison, total energies calculated with the vacuum polarization on top of the relativistic Coulomb LDA and with the non-relativistic Coulomb GGA [39] are shown in a column It was shown in Refs. [39,62] that the exact treatment or its simplified treatment called GGA of the Coulomb exchange energy makes 40 Ca bound more by around 0.32 MeV and 208 Pb by around 0.74 MeV, which can be understood as the correction of the density gradient on the LDA. Hence, the relativistic correction of the Coulomb interaction is almost comparable with or even larger than the gradient correction but opposite direction to total energies, as in the case of atoms [51]. Thus, if both contributions are considered simultaneously, even it is not obvious whether the total energy decreases or increases. Indeed, this fact was pointed out in the context of atomic physics in Ref. [51], while no first-principles relativistic Coulomb GGA EDF is available, which remains an open question.
I also compare the relativistic correction with the vacuum polarization. The vacuum polarization also makes the atomic nuclei less bound, and contribution to the total energy is approximately 0.4 MeV in 40 Ca and 3.7 MeV in 208 Pb. Thus, as discussed in Sec. 3, the vacuum polarization gives a comparable contribution to the total energy to the relativistic correction in light nuclei, but it contributes more in heavy nuclei.
The systematic behavior of the relativistic correction and the vacuum polarization shall be discussed next. The contributions of the relativistic correction and the vacuum polarization can be seen as the difference of the second and the third columns and the third and the fourth 11/19 Table 2 Ground-state energies calculated with the SAMi EDF [82] shown in MeV. The columns labelled "Coulomb (NR-LDA)" and "Coulomb (R-LDA)" show results calculated with the non-relativistic Coulomb LDA [Eq. (11)] and the relativistic Coulomb LDA [Eqs. (11) and (19) Table 2, respectively. The proton number Z dependence of these contributions are shown in Fig. 1. These energies are fitted with E = aZ b , where a and b are adjustable parameters. The results are which are consistent with Eqs. (26b) and (28). The breakdown of the relativistic correction [Eq. (19)] is shown in hence, [2.292ρ p (r) − 1.913ρ n (r)] determines the sign of E H Darwin . In N ≤ Z nuclei, ρ p (r) ρ n (r) holds and, accordingly, it is positive (i.e., E H Darwin < 0), whereas, in N > Z nuclei, ρ p (r) ρ n (r) holds and accordingly, it can be negative (i.e., E H Darwin > 0) in contrast to Eq. (25a). The Darwin and retardation exchange energies are also fitted with E = aZ b , where the results are which are consistent to Eqs. (25b) and (25c).  The exchange contribution to the EM energy shall be further discussed. The ratio of the Coulomb or relativistic exchange energy to the non-relativistic Coulomb GGA exchange energy is shown in Fig. 2 (18)], respectively. It is seen that the absolute value of the relative change |∆E x | for the difference between the exact and LDA non-relativistic Coulomb exchange energy gets smaller as A increases in light nuclei and reaches almost constant ∆E x ≈ −3 % in heavy nuclei (A 100). The absolute value of the relative change |∆E x | for the difference between the non-relativistic and relativistic LDA Coulomb exchange energy behaves similarly, while its value is larger and reaches almost constant ∆E x ≈ −9 % in heavy nuclei. The Darwin contribution and the retardation one are almost the same size.
This difference has been studied in the relativistic (covariant) DFT [63,64]. The exact Coulomb exchange energy in this paper corresponds to Eq. (4) in Ref. [64], which does not seem to include the relativistic correction. They also show that the difference between the non-relativistic and relativistic Coulomb LDA exchange energies is approximately 1 MeV. With considering such difference, E x Coul + E x Rel calculated in this work is almost consistent qualitatively with, but almost twice of Figs. 1 and 4  Lastly, a change of the single-particle energies is discussed, for which 208 Pb is taken as an example. The single-particle energies calculated with the non-relativistic and relativistic Coulomb LDA are shown in Table 4. For comparison, those calculated with the vacuum polarization on top of the relativistic LDA are shown. It is seen that the proton single-particle energies change by about 0.04 MeV in inner shells and 0.01 MeV in outer shells, due to the relativistic correction. In contrast, the change due to the vacuum polarization does not depend on orbitals strongly, and its value is about 0.1 MeV. This difference is due to the nature of interaction; the relativistic correction is the point-coupling interaction, while the vacuum polarization is finite range. Since the Darwin Hartree term contributes to the neutron meanfield potential [see Eq. (20)], the neutron single-particle energies change by about 0.04 MeV in inner shells and 0.01 MeV in outer shells, due to the relativistic correction. In contrast, the vacuum polarization changes neutron single-particle energies only by less than 0.01 MeV.

Conclusion and Perspectives
In this paper, the relativistic correction of the Coulomb interaction was introduced in Skyrme Hartree-Fock calculations using the local density approximation (LDA). The correction contains the Darwin term, the retardation, the spin-orbit interaction, and the spin-magnetic interaction, while only the first and second terms were considered in this paper because the remaining terms vanish in the LDA.
It was found that the relativistic correction makes 40 Ca less bound by 0.4 MeV and 208 Pb by 2.4 MeV, which are comparable with or even larger than the difference between the exact Coulomb exchange (exact-Fock), or its simplified treatment called the generalized gradient approximation (GGA), and the LDA (Hartree-Fock-Slater approximation) Coulomb exchange energies [38,39,61,62], but with the opposite sign. Thus, once both the exact treatment and the relativistic correction are considered at the same time, even the sign of the total correction is not obvious, as was pointed out also in the context of atomic physics [51]. In addition, since the desired accuracy of the nuclear EDF is 100 keV order, the relativistic correction is often non-negligible, as well as the correction due to the density gradient.
The correction was also compared with the vacuum polarization. Since the proton-number dependence of the vacuum polarization is stronger than that of the relativistic correction, these two contributions to the total energy are comparable in light nuclei, but the former dominates in heavy nuclei. This behavior is in contrast to atoms, in which the vacuum polarization is sub-dominant to the Breit correction [58]. Although the contribution of vacuum polarization to the total energy is usually tiny, it is non-negligible to discuss mirror nuclei mass difference or the isobaric analog energy [7,11,40], whereas the relativistic correction does not contribute to such properties.
In this calculation, the spin-orbit and spin-magnetic interactions were not able to be considered since they vanish in the LDA. The former has been discussed in Refs. [7,40,78], while the latter has not been considered in the self-consistent calculation. Consideration of the latter is left for a future task. The relativistic Coulomb GGA functional is indispensable to achieve more accurate calculation in both atomic and nuclear structure calculations, which is also left for a future perspective.
The Coulomb correlation energy, which originates from many-body correlations [83][84][85][86][87], is also an important and interesting topic. In electronic systems, the Coulomb correlation 15/19 energy can be defined without any ambiguity as the difference between the "exact" energy and the Hartree-Fock energy, which can be calculated by using the Monte Carlo calculation and some other many-body techniques [15][16][17][88][89][90][91][92]. Reference [60] had derived the relativistic Coulomb exchange-correlation EDF in the LDA level, while Ref. [51] discussed that the relativistic correction of the Coulomb correlation energy is almost negligible. Meanwhile, in nuclear systems, there exists both the nuclear and Coulomb interaction; hence, there is still ambiguity in defining the Coulomb correlation energy, while there have been discussed in several works [38,[93][94][95]. Thus, it should be discussed whether the relativistic correction of the Coulomb correlation energy in nuclear systems is negligible, while it is left for a future perspective.
Once all the relevant corrections to the Coulomb energy are taken into account for the nuclear DFT calculation on top of the non-relativistic Coulomb LDA exchange EDF, the correction may be more than 1 MeV for heavy nuclei, such as 208 Pb. For instance, the nucleon finite-size effects and the vacuum polarization contribute to the binding energy of 208 Pb, in total, in about 4 MeV [40], the relativistic correction discussed in this paper is about 2 MeV, and the Coulomb correlation energy may reach an order of 1 MeV [94]. The current accuracy of nuclear EDFs is about 1-5 MeV in binding energies [31,96], which is the same order as or even smaller than the Coulomb corrections. Therefore, in order to achieve accurate calculation of binding energies using nuclear EDF quantitatively, the nuclear EDF should be, in principle, refitted by considering all the relevant Coulomb corrections.