Variational treatment of the Dirac–Coulomb–Gaunt or Dirac–Coulomb–Breit two-electron interaction at the Dirac–Hartree–Fock level is the starting point of high-accuracy four-component calculations of atomic and molecular systems. In this work, we introduce, for the first time, the scalar Hamiltonians derived from the Dirac–Coulomb–Gaunt and Dirac–Coulomb–Breit operators based on spin separation in the Pauli quaternion basis. While the widely used spin-free Dirac–Coulomb Hamiltonian includes only the direct Coulomb and exchange terms that resemble nonrelativistic two-electron interactions, the scalar Gaunt operator adds a scalar spin–spin term. The spin separation of the gauge operator gives rise to an additional scalar orbit-orbit interaction in the scalar Breit Hamiltonian. Benchmark calculations of Aun (n = 2–8) show that the scalar Dirac–Coulomb–Breit Hamiltonian can capture 99.99% of the total energy with only 10% of the computational cost when real-valued arithmetic is used, compared to the full Dirac–Coulomb–Breit Hamiltonian. The scalar relativistic formulation developed in this work lays the theoretical foundation for the development of high-accuracy, low-cost correlated variational relativistic many-body theory.

Variationally solving the four-component Dirac–Coulomb–Gaunt (DCG) or Dirac–Coulomb–Breit (DCB) Hartree–Fock equation is the starting point for high-accuracy relativistic calculations.1–9 It is generally well understood that the Breit operator is more accurate than the Gaunt operator.1,2,10–13 However, due to the large computational cost of the Breit term, the application of the Dirac–Coulomb–Breit Hamiltonian to molecules mostly remains in mean-field theory or is used as a perturbative treatment in correlated methods.14–18 

Spin separation of the Dirac–Coulomb–Gaunt and Dirac–Coulomb–Breit Hamiltonians was proposed by Dyall in 1994.19 For example, by applying the kinetic balance condition and the Dirac identity, the Coulomb interaction can be separated into spin-free and spin-own-orbit parts. The spin-free part of the Dirac–Coulomb Hamiltonian captures the majority of the interaction energy, whereas the spin-own-orbit term contributes to the energy level splitting. Due to their low computational cost, Hamiltonians based on the spin-free Dirac–Coulomb approximation have been widely used with perturbative spin–orbit coupling in four-component relativistic electronic structure methods.20–24 In addition, the spin-free exact-two-component Hamiltonian has been successfully applied in multi-reference calculations.25–27 However, to the best of our knowledge, the scalar Gaunt and scalar Breit Hamiltonians have never been developed and implemented for molecular calculations due to the high computational cost and complex mathematical form of the separation of the two-electron Gaunt and Breit operators.

In these Hamiltonians, there are terms that are not spin-free but are nevertheless spin-separated. These terms are products of a scalar spin operator and a scalar spatial operator, both of which transform into the totally symmetric irreducible representation of the point group. The spin–spin interaction, for example, couples states of the same spin, and the associated spatial operator couples states of the same spatial symmetry. Such operators are candidates for inclusion along with the spin-free terms in a Hamiltonian that preserves both spin and spatial symmetry. While the phrase “scalar relativistic” has been used interchangeably with “spin-free relativistic,” the contributions from spin-dependent terms could justifiably be considered “scalar relativistic” as they can be incorporated into the scalar relativistic methods that traditionally have included only the spin-free contributions.

In the mathematical derivation of these terms, the use of the Dirac relation for spin reduction of the one-electron modified Dirac Hamiltonian19 produces a scalar product, which is spin free, and a scalar triple product, which is the spin–orbit term. Likewise, the spin reduction of the electronic Coulomb interaction produces a spin-free term and a spin–orbit term from the (LL|SS) integrals and various more complex terms from the (SS|SS) integrals involving products of spin-free and spin–orbit terms. The spin reduction of the Gaunt and Breit interactions involves multiple applications of the Dirac relation and results in scalar triple and quadruple products as well as regular scalar products. The triple products are all of the spin–orbit type, which is not spin-free and does not preserve spin and spatial symmetry. The scalar quadruple product is a scalar product of vector products, which can be written in terms of the products of two scalar products. Of these, one term separates the spin from the spatial operators completely so that they can be treated separately, and the other includes terms that preserve spatial and spin symmetry, and from these terms arise contributions to the scalar relativistic Hamiltonian. Therefore, we propose that the term “scalar” not be considered synonymous with “spin-free” but include terms that contribute to the diagonal spin and spatial symmetry blocks of the Hamiltonian or Fock matrix. It should be noted that the quaternion-based SCF methods cannot separate out the double spin–orbit terms from the (SS|SS) integrals because these terms are even in the number of spin operators. The same is true for the spin–spin terms derived from the Gaunt interaction. Hence, these methods are not really spin-free but fall under our proposed definition of scalar relativistic Hamiltonians.

As mentioned, the Gaunt and Breit terms,
gG(i,j)=αiαjrij,
(1)
gB(i,j)=12αiαjrij+αirijαjrijrij3,
(2)
can be decomposed into spin-free, spin-other-orbit, spin–spin, and orbit-orbit interactions.3,28,29 Here, {i, j} are electron indices, and the components of the α matrices are defined as
αi,q=02σqσq02,q={x,y,z},
(3)
where the σq are the Pauli matrices,
σx=0110,σy=0ii0,σz=1001,
(4)
and 02 is the 2 × 2 zero matrix.

Recent advances using the Pauli matrix quaternion basis have achieved the minimum floating-point count algorithm for building the full DCG and DCB Hamiltonians in spin-separated forms28,29 and opened a pathway for the development of fully correlated DCG/DCB many-body methods. The current work builds upon the spin- and component-separation using the Pauli matrix quaternion representation and, for the first time, introduces the low-scaling scalar Breit Hamiltonian that allows for practical applications of the fully relativistic electronic structure theory. The mathematical derivations are lengthy and complex, especially for the spin- and component-separation of the Breit operator in the Pauli quaternion representation. We refer readers to Refs. 28 and 29 for detailed derivations. In this work, we present the key discussion and final working expressions (see the  Appendix) of the scalar Breit Hamiltonian with benchmark calculations.

In the Dirac–Coulomb Hamiltonian, the spin-free O(c2) integrals can be easily derived using the Dirac identity [Eq. (A2)] in the restricted kinetic balance condition,30–32 
μν(μν|κλ),
(5)
where μ operates on an atomic basis function μ. Equation (5) is the conventional spin-free term that does not depend on spin densities. The contraction of spin-free integrals with the large-component and small-component scalar densities gives rise to the spin-free Dirac–Coulomb term in the Pauli quaternion representation [Eq. (A7) and Eq. (A12) in the  Appendix]. The O(c4) spin-free contribution can be derived similarly from the (SS|SS) integrals and leads to the integral (μ · ν) (κ · λ) (μν|κλ), which provides contributions like the nonrelativistic direct and exchange terms [Eqs. (A13)(A16) in the  Appendix] in the small-component block of the Fock matrix. It is easy to see that the spin-free Dirac–Coulomb Hamiltonian accounts for the direct O(c2) Coulomb interaction between the large- and small-component electron densities and O(c4) Coulomb and exchange interactions in the small-component space.
For the Gaunt operator, a spin-free term that is similar to the spin-free Dirac–Coulomb term arises after applying the Dirac identity twice. This term is of the form
κν(μν|κλ).
(6)
There are four terms of this kind, corresponding to the possible combinations of indices. Just like the spin-free Dirac–Coulomb case, this term accounts for O(c2) direct and exchange contributions in the scalar Gaunt Hamiltonian. However, the scalar Gaunt term has more contributions from the Gaunt operator than the spin-free interaction. In the derivation of spin separation for the Gaunt term, the reduction of the scalar quadruple product of the spin–spin interaction yields two terms,
(σ(2)×κ)(σ(1)×ν)(μν|κλ)=[(σ(2)σ(1))(κν)(σ(2)ν)(σ(1)κ)](μν|κλ).
(7)
The first term is a product of a spin operator and a spatial operator, each of which preserves symmetry as they are tensors of rank 0. This term resembles in form the contribution from the spin-free integrals [Eq. (6)], except that it is contracted with spin densities instead of scalar densities. The second term couples the spin and spatial operators and contains operators that preserve spin and spatial symmetry (tensors of rank 0) and operators that break spin or spatial symmetry (tensors of rank 1 and 2). Both of these terms contribute to the Fermi contact and the spin–spin dipole terms of the Breit–Pauli Hamiltonian. In this work, only the contributions from tensors of rank 0 are included in the scalar Gaunt Hamiltonian.

In other words, in addition to the direct Coulomb-like interaction, the spin–spin interaction also contributes to the scalar Gaunt term. Compared to the spin-free Dirac–Coulomb term, scalar Gaunt contributions predominantly appear as exchange interactions in the LL, SS, and LS blocks of the Fock matrix. The working expressions of density-integral contractions of Eq. (7) are presented in Eqs. (A18)(A23) of  Appendix A 3.

The derivation of the scalar Breit operator is more complicated, mainly due to the gauge term in Eq. (2), which has the form (αirij)(αjrij)rij3. The all-scalar product of the gauge operator in the restricted kinetic balance condition naturally leads to a scalar gauge contribution that resembles the direct Coulomb interaction,
(μν|νr12r12κ|κλ)3,
(8)
where the symbol ()3 stands for the integral of operator 1r123 for simplicity. However, this all-scalar gauge term is incomplete for the scalar Breit operator. Another scalar term arises from the derivation of spin separation for the gauge operator,
(μν|(r12×ν)(r12×κ)|κλ)3.
(9)
Since r × p is the angular momentum operator, (r12 × ν) · (r12 × κ) is the scalar orbit-orbit interaction. This scalar quadruple product reduces to a product of scalar products, as for the spin–spin term. Like the Gaunt operator, the scalar Breit operator contributes predominantly through exchange interactions in all the component blocks of the Fock matrix.

In summary, the spin-free Dirac–Coulomb operator consists of direct and exchange terms that resemble non-relativistic two-electron interactions, and the scalar Gaunt operator adds scalar spin–spin interactions along with other direct and exchange terms. The scalar Breit operator adds a scalar orbit-orbit interaction. All working expressions in the Pauli quaternion representation are presented in the  Appendix.

Figure 1 presents the computational scaling of Aun chains and the accuracy of spin-free/scalar terms with respect to the full Dirac–Coulomb, Gaunt, and Breit Hamiltonians. All calculations in this section are performed with a development version of the Chronus Quantum software package.33 The speed of light utilized is 137.035 999 679 94 a.u. The uncontracted ANO-RCC basis34–36 set was utilized, resulting in 297 basis functions per Au atom with up to h orbital angular momentum (2376 basis functions for Au8 with 16 h, 32 g, 88 f, 120 d, 168 p, and 192 s functions). The scalar and full relativistic calculations are carried out in complex arithmetic for a direct algorithmic and energy comparison.

FIG. 1.

Cost comparison of full (dark colors) and spin-free/scalar (light colors) Dirac-Coulomb-Breit Hamiltonian components. (a) Dirac–Coulomb (DC) without SSSS; (b) SSSS; (c) Gaunt; (d) Gauge. Cost is measured relative to the LLLL only calculation, TTLLLL, where T and TLLLL are the CPU times to build the corresponding relativistic and the LLLL term. (d) Speedup factor = TfullTSF, where Tfull and TSF are the CPU times to build the full and spin-free/scalar (SF) Hamiltonian term. Schwarz integral screening threshold of 10−14 is used in the calculations.

FIG. 1.

Cost comparison of full (dark colors) and spin-free/scalar (light colors) Dirac-Coulomb-Breit Hamiltonian components. (a) Dirac–Coulomb (DC) without SSSS; (b) SSSS; (c) Gaunt; (d) Gauge. Cost is measured relative to the LLLL only calculation, TTLLLL, where T and TLLLL are the CPU times to build the corresponding relativistic and the LLLL term. (d) Speedup factor = TfullTSF, where Tfull and TSF are the CPU times to build the full and spin-free/scalar (SF) Hamiltonian term. Schwarz integral screening threshold of 10−14 is used in the calculations.

Close modal

It is evident from Fig. 1 that spin-free and scalar Hamiltonians are computationally much cheaper than the full Hamiltonian. For the Dirac–Coulomb case, building the spin-free Fock matrix is only ∼27% of the cost of the full Hamiltonian, including integral formation and integral-density contraction, for Au2. The scalar Gaunt and scalar gauge costs are only ∼16% and ∼24%, respectively, of the full Hamiltonian for Au2.

As the system size increases, the advantage of using the scalar Hamiltonian decreases, although it is still much computationally cheaper than the full Hamiltonian. This is mainly due to the increased effectiveness of integral screening in the full Hamiltonian build. The scalar Breit interaction is relatively more local than non-relativistic operators, dominated by the nearest-neighbor interaction, and as a result, less sensitive to the increased integral screening as the system size increases.

While exhibiting a strong computational advantage, Fig. 2 shows that these scalar Hamiltonians capture ∼99% of the total two-electron interaction energy compared to those computed using the full Hamiltonian. It is worth emphasizing that ignoring the scalar spin–spin interaction from the Gaunt interaction or the scalar orbit-orbit interaction in the gauge term leads to a large discrepancy in total energy, strongly supporting the use of these terms in the scalar Breit Hamiltonian. For Aun tests, the orbit-orbit interaction accounts for ∼73% of the scalar gauge term, and the spin–spin interaction is ∼56% of the scalar Gaunt term.

FIG. 2.

Percent error of scalar Dirac-Coulomb-Breit interaction components relative to corresponding full operators. The percent error is computed as |ΔEfullΔESF||ΔEfull|, where ΔE = EELLLL is the relativistic energy contribution. The energy E is obtained by only adding the corresponding relativistic operator separately to the LLLL term, followed by a full self-consistent-field wave function optimization. Schwarz integral screening threshold of 10−14 is used in the calculations.

FIG. 2.

Percent error of scalar Dirac-Coulomb-Breit interaction components relative to corresponding full operators. The percent error is computed as |ΔEfullΔESF||ΔEfull|, where ΔE = EELLLL is the relativistic energy contribution. The energy E is obtained by only adding the corresponding relativistic operator separately to the LLLL term, followed by a full self-consistent-field wave function optimization. Schwarz integral screening threshold of 10−14 is used in the calculations.

Close modal

In order to study the scalar Hamiltonian’s effect on total calculated energy in molecules, four-component Dirac–Hartree–Fock calculations were performed on three members of the actinyl oxycation series utilizing either the full Dirac–Coulomb–Breit or the scalar Dirac–Coulomb–Breit Hamiltonian. In the scalar case, the Dirac–Coulomb, Gaunt, and gauge terms are all chosen to be scalar only. The results in Table I show that over 99.99% of the total energy is captured by the scalar Breit Hamiltonian. Note that the percent errors shown here are much less than those in Fig. 2 because the errors here are displayed as a percentage of total energy, not as a percentage of the relativistic energy contribution attributable to a single operator.

TABLE I.

Total energy (Hartrees) comparison between scalar Dirac–Coulomb–Breit and full Dirac–Coulomb–Breit calculations with the ANO-RCC basis set across the actinyl oxycation series. A Schwarz integral screening threshold of 10−14 was used in the uncontracted calculations. All molecules are in the linear form with An-O bond lengths of 1.76, 1.75, and 1.74 Å for UO22+, NpO22+, and PuO22+, respectively. For scalar calculations, the spin multiplicities are singlet, doublet, and triplet for UO22+, NpO22+, and PuO22+, respectively.

FormulaFull HamiltonianScalarAbsolute errorPercent error (%)
UO22+ −28 164.71 −28 165.47 0.75 0.0027 
NpO22+ −28 957.31 −28 958.14 0.83 0.0029 
PuO22+ −29 765.31 −29 766.19 0.88 0.0030 
FormulaFull HamiltonianScalarAbsolute errorPercent error (%)
UO22+ −28 164.71 −28 165.47 0.75 0.0027 
NpO22+ −28 957.31 −28 958.14 0.83 0.0029 
PuO22+ −29 765.31 −29 766.19 0.88 0.0030 

The scalar Dirac–Coulomb–Breit Hamiltonian introduced in this work includes direct and exchange spin-free terms from the Coulomb and Gaunt interactions and the scalar spin–spin and scalar orbit-orbit interactions from the Breit interaction, obtained through spin separation in the Pauli quaternion basis. Benchmark calculations show that the scalar Dirac–Coulomb–Breit Hamiltonian captures ∼99% of the total relativistic two-electron interaction energy at only a fraction of the computational cost compared to the full Hamiltonian.

The scalar Dirac–Coulomb–Breit Hamiltonian consists of only real-valued matrix elements. Therefore, the computational cost can be further reduced by using real-valued arithmetic, resulting in another twofold reduction in computational cost in the Pauli quaternion basis relative to the full Dirac–Coulomb–Breit Hamiltonian. The reduction is only twofold because the full Dirac–Coulomb–Breit Hamiltonian in the Pauli quaternion basis already utilizes real-valued integrals in an optimal FLOP (floating point operation) algorithm.

Although important physics (including the spin-own-orbit, spin-other-orbit, and more complex spin-dependent interactions) is separated out, the scalar Dirac–Coulomb–Breit Hamiltonians are ideal for preparing zeroth-order four-component wave functions for the perturbative treatment of the spin physics in the state interaction framework. In addition, we suggest that the scalar Dirac–Coulomb–Breit operator can be used to capture relativistic electron correlation in many-body theory, laying the theoretical foundation for the future development of fully correlated Dirac–Coulomb–Breit ultra-high-accuracy computational methods.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, in the Heavy-Element Chemistry program (Grant No. DE-SC0021100) for the development of variational relativistic multi-reference methods and in the Computational and Theoretical program (Grant No. DE-SC0006863) for the development of the Breit operator for Dirac and quantum field theory.

The authors have no conflicts to disclose.

Shichao Sun: Conceptualization (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Jordan N. Ehrman: Data curation (equal); Validation (equal); Visualization (lead); Writing – review & editing (equal). Tianyuan Zhang: Software (equal); Validation (equal). Qiming Sun: Software (equal); Validation (equal). Kenneth G. Dyall: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Xiaosong Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The development of the scalar Gaunt and Breit Hamiltonians is based on the restricted kinetic balance (RKB) condition for the large component (χL) and small component (χS) basis,30–32, χS=12cσpχL, to ensure the correct nonrelativistic limit of the positive energy states,37,38 where c is the speed of light and p = −i is the linear momentum operator. Mathematical derivations are lengthy and complex for the spin- and component-separation of the Dirac–Coulomb, Gaunt, and Breit operators in the Pauli quaternion representation. We refer to readers Refs. 28 and 29 for detailed derivations. We only present the key equations and the final working expressions here.

1. Pauli quaternion representation

In order to build the scalar Dirac–Coulomb–Breit Hamiltonian on the Pauli quaternion basis, all integrals (Dirac–Coulomb, Gaunt, and Breit) and density matrices need to be expressed in the Pauli matrix representation. The Pauli components of the density matrix can be transformed from the spin-blocked matrices,39 
Ds=Dαα+Dββ,Dz=DααDββ,Dx=Dαβ+Dβα,Dy=i(DαβDβα),
for each component-block (LL, SS, LS, SL). These density matrices are contracted with scalar integrals to build Fock matrices, all in the Pauli quaternion representation. Collecting all Pauli components, the four-component Fock matrix can be constructed,
FXY=12FsXY+FzXYFxXYiFyXYFxXY+iFyXYFsXYFzXY,
(A1)
where X, Y ∈ {L, S}.

2. Pauli quaternion spin-free Dirac–Coulomb

The derivation of the spin-free Dirac–Coulomb is straightforward, utilizing the Dirac identity,
(σμ)(σν)=Iμν+iσμ×ν,
(A2)
in the two-electron Coulomb integrals with the RKB condition. The first term in Eq. (A2) is known as the conventional spin-free contribution. The resulting Pauli quaternion spin-free Dirac–Coulomb expressions, including the O(c4) order contribution from the (SS|SS) term, are
Vμν,sC,LL=Dλκ,sLL[2(μν|κλ)(μλ|κν)],
(A3)
Vμν,zC,LL=Dλκ,zLL(μλ|κν),
(A4)
Vμν,xC,LL=Dλκ,xLL(μλ|κν),
(A5)
Vμν,yC,LL=Dλκ,yLL(μλ|κν),
(A6)
Vμν,sC,LL=2Dλκ,sSSκλ(μν|κλ),
(A7)
Vμν,sC,LS=Dλκ,sLSκν(μλ|κν),
(A8)
Vμν,zC,LS=Dλκ,zLSκν(μλ|κν),
(A9)
Vμν,xC,LS=Dλκ,xLSκν(μλ|κν),
(A10)
Vμν,yC,LS=Dλκ,yLSκν(μλ|κν),
(A11)
Vμν,sC,SS=2Dλκ,sLLμν(μν|κλ),
(A12)
Vμν,sC,SS=2Dλκ,sSS(μν)(κλ)(μν|κλ)(μλ)(κν)(μλ|κν),
(A13)
Vμν,zC,SS=Dλκ,zSS(μλ)(κν)(μλ|κν),
(A14)
Vμν,xC,SS=Dλκ,xSS(μλ)(κν)(μλ|κν),
(A15)
Vμν,yC,SS=Dλκ,ySS(μλ)(κν)(μλ|κν),
(A16)
where κ is the nuclear coordinate derivative of the atomic basis function χκ.

3. Pauli quaternion scalar Gaunt

The mathematical expressions for the scalar Gaunt and Breit Hamiltonians cannot be easily obtained using the Dirac identity [Eq. (A2)]. Unlike the Dirac–Coulomb term, both the Gaunt and Breit operators [Eqs. (1) and (2)] are intrinsically dependent on Pauli matrices. As a result, the following vector identity:
μ×νκ×λ=μκνλμλνκ,
(A17)
must be applied, in addition to the Dirac identity, to achieve fully spin-separated expressions. The working expressions for the scalar Gaunt Hamiltonian in the Pauli quaternion representation are
Vμν,sG,LL=3Dλκ,sSSλκ(μλ|κν),
(A18)
Vμν,sG,SS=3Dλκ,sLLμν(μλ|κν),
(A19)
Vμν,sG,LS=2Dλκ,sLSκν(μν|κλ)+Dλκ,sSLνλ(μν|κλ)+Dλκ,sSLλν(μλ|κν),
(A20)
Vμν,zG,LS=2Dλκ,zSLλν(μλ|κν)+Dλκ,zLSκν(μν|κλ)+Dλκ,zSLνλ(μν|κλ),
(A21)
Vμν,xG,LS=2Dλκ,xSLλν(μλ|κν)+Dλκ,xLSκν(μν|κλ)+Dλκ,xSLνλ(μν|κλ),
(A22)
Vμν,yG,LS=2Dλκ,ySLλν(μλ|κν)+Dλκ,JLSκν(μν|κλ)+Dλκ,ySLνλ(μν|κλ).
(A23)

4. Pauli quaternion scalar gauge

We introduce the following short notations for the gauge integrals that contribute to the scalar term:
(ss)(μν̄|κ̄λ)3=μν|νr12r12κ|r12|3|κλ,
(A24)
(σσ)JK(μν̄|κ̄λ)3=μν|(r12×ν)J(κ×r12)K|r12|3|κλ,J,K{x,y,z},
(A25)
(σσ)(μν̄|κ̄λ)3=[(σσ)xx+(σσ)yy+(σσ)zz](μν̄|κ̄λ)3,
(A26)
where subscript 3 denotes the |r12|−3 operator for the gauge integral.
The scalar gauge Hamiltonian in the Pauli quaternion representation is
Vμν,sg,LL=12Dλκ,sSS(ss)+Dλκ,sSS(σσ)(μλ̄|κ̄ν)3,
(A27)
Vμν,zg,LL=12Dλκ,zSS(ss)(μλ̄|κ̄ν)3,
(A28)
Vμν,xg,LL=12Dλκ,xSS(ss)(μλ̄|κ̄ν)3,
(A29)
Vμν,yg,LL=12Dλκ,ySS(ss)(μλ̄|κ̄ν)3,
(A30)
Vμν,sg,SS=12Dλκ,sLL(ss)+Dλκ,sLL(σσ)(μ̄λ|κν̄)3,
(A31)
Vμν,zg,SS=12Dλκ,zLL(ss)(μ̄λ|κν̄)3,
(A32)
Vμν,xg,SS=12Dλκ,xLL(ss)(μ̄λ|κν̄)3,
(A33)
Vμν,yg,SS=12Dλκ,yLL(ss)(μ̄λ|κν̄)3,
(A34)
Vμν,sg,LS=Dλκ,sLS(ss)+Dλκ,sSL(ss)(μν̄|κλ̄)312Dλκ,sSL(ss)+Dλκ,sSL(σσ)(μλ̄|κν̄)3,
(A35)
Vμν,zg,LS=12Dλκ,zSL(ss)(μλ̄|κν̄)3,
(A36)
Vμν,xg,LS=12Dλκ,xSL(ss)(μλ̄|κν̄)3,
(A37)
Vμν,yg,LS=12Dλκ,ySL(ss)(μλ̄|κν̄)3.
(A38)
For the SL blocks of Dirac–Coulomb, Gaunt, and Breit, we have the symmetry
VSL=VLS.
1.
I. P.
Grant
and
W. G.
Penney
, “
Relativistic self-consistent fields
,”
Proc. R. Soc. London, Ser. A
262
,
555
576
(
1961
).
2.
E.
Lindroth
,
A.-M.
Martensson-Pendrill
,
A.
Ynnerman
, and
P.
Oster
, “
Self-consistent treatment of the Breit interaction, with application to the electric dipole moment in thallium
,”
J. Phys. B: At., Mol. Opt. Phys.
22
,
2447
2464
(
1989
).
3.
K. G.
Dyall
and
K.
Fægri
, Jr.
,
Introduction to Relativistic Quantum Chemistry
(
Oxford University Press
,
2007
).
4.
M.
Reiher
and
A.
Wolf
,
Relativistic Quantum Chemistry
,
2nd ed.
(
Wiley VCH
,
2015
).
5.
I. P.
Grant
,
Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation
(
Springer Science & Business Media
,
2007
), Vol. 40.
6.
I.
Lindgren
,
Relativistic Many-Body Theory: A New Field-Theoretical Approach
(
Springer
,
Switzerland
,
2016
).
7.
W.
Liu
, “
Advances in relativistic molecular quantum mechanics
,”
Phys. Rep.
537
,
59
89
(
2014
).
8.
W.
Liu
, “
Essentials of relativistic quantum Chemistry
,”
J. Chem. Phys.
152
,
180901
(
2020
).
9.
C. E.
Hoyer
,
L.
Lu
,
H.
Hu
,
K. D.
Shumilov
,
S.
Sun
,
S.
Knecht
, and
X.
Li
, “
Correlated Dirac–Coulomb–Breit multiconfigurational self-consistent-field methods
,”
J. Chem. Phys.
158
,
044101
(
2023
).
10.
J. B.
Mann
and
W. R.
Johnson
, “
Breit interaction in multielectron atoms
,”
Phys. Rev. A
4
,
41
51
(
1971
).
11.
K.-N.
Huang
,
M.
Aoyagi
,
M. H.
Chen
,
B.
Crasemann
, and
H.
Mark
, “
Neutral-atom electron binding energies from relaxed-orbital relativistic Hartree-Fock-Slater calculations 2 ≤ Z ≤ 106
,”
At. Data Nucl. Data Tables
18
,
243
291
(
1976
).
12.
C. T.
Chantler
,
T. V. B.
Nguyen
,
J. A.
Lowe
, and
I. P.
Grant
, “
Convergence of the Breit interaction in self-consistent and configuration-interaction approaches
,”
Phys. Rev. A
90
,
062504
(
2014
).
13.
K.
Kozioł
,
C. A.
Giménez
, and
G. A.
Aucar
, “
Breit corrections to individual atomic and molecular orbital energies
,”
J. Chem. Phys.
148
,
044113
(
2018
).
14.
E.
Eliav
,
U.
Kaldor
, and
Y.
Ishikawa
, “
Open-shell relativistic coupled-cluster method with Dirac-Fock-Breit wave functions: Energies of the gold atom and its cation
,”
Phys. Rev. A
49
,
1724
1729
(
1994
).
15.
Y.
Ishikawa
and
K.
Koc
, “
Relativistic many-body perturbation theory based on the No-pair Dirac-Coulomb-Breit Hamiltonian relativistic correlation energies for the Noble-gas sequence through Rn (Z=86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence
,”
Phys. Rev. A
50
,
4733
4742
(
1994
).
16.
M.
Vilkas
,
Y.
Ishikawa
, and
K.
Hirao
, “
Ionization energies and fine structure splittings of highly correlated systems: Zn, zinc-like ions and copper-like ions
,”
Chem. Phys. Lett.
321
,
243
252
(
2000
).
17.
I. P.
Grant
and
H. M.
Quiney
, “
Application of relativistic theories and quantum electrodynamics to chemical problems
,”
Int. J. Quant. Chem.
80
,
283
297
(
2000
).
18.
C.
Zhang
and
L.
Cheng
, “
Atomic mean-field approach within exact two-component theory based on the Dirac–Coulomb–Breit Hamiltonian
,”
J. Phys. Chem. A
126
,
4537
4553
(
2022
).
19.
K. G.
Dyall
, “
An exact separation of the spin-free and spin-dependent terms of the Dirac–Coulomb–Breit Hamiltonian
,”
J. Chem. Phys.
100
,
2118
2127
(
1994
).
20.
L.
Visscher
and
T.
Saue
, “
Approximate relativistic electronic structure methods based on the quaternion modified Dirac equation
,”
J. Chem. Phys.
113
,
3996
4002
(
2000
).
21.
T.
Fleig
and
L.
Visscher
, “
Large-scale electron correlation calculations in the framework of the spin-free Dirac formalism: The Au2 molecule revisited
,”
Chem. Phys.
311
,
113
120
(
2005
).
22.
L.
Cheng
and
J.
Gauss
, “
Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian
,”
J. Chem. Phys.
134
,
244112
(
2011
).
23.
L.
Cheng
,
S.
Stopkowicz
, and
J.
Gauss
, “
Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spin-orbit effects at the Hartree-Fock level
,”
J. Chem. Phys.
139
,
214114
(
2013
).
24.
F.
Lipparini
and
J.
Gauss
, “
Cost-effective treatment of scalar relativistic effects for multireference systems: A CASSCF implementation based on the spin-free Dirac–Coulomb Hamiltonian
,”
J. Chem. Theory Comput.
12
,
4284
4295
(
2016
).
25.
L.
Cheng
and
J.
Gauss
, “
Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian
,”
J. Chem. Phys.
135
,
084114
(
2011
).
26.
Z.
Li
,
Y.
Xiao
, and
W.
Liu
, “
On the spin separation of algebraic two-component relativistic Hamiltonians
,”
J. Chem. Phys.
137
,
154114
(
2012
).
27.
P. K.
Tamukong
,
M. R.
Hoffmann
,
Z.
Li
, and
W.
Liu
, “
Relativistic GVVPT2 multireference perturbation theory description of the electronic states of Y2 and Tc2
,”
J. Phys. Chem. A
118
,
1489
1501
(
2014
).
28.
S.
Sun
,
T. F.
Stetina
,
T.
Zhang
,
H.
Hu
,
E. F.
Valeev
,
Q.
Sun
, and
X.
Li
, “
Efficient four-component Dirac–Coulomb–Gaunt Hartree–Fock in the Pauli spinor representation
,”
J. Chem. Theory Comput.
17
,
3388
3402
(
2021
).
29.
S.
Sun
,
J. N.
Ehrman
,
Q.
Sun
, and
X.
Li
, “
Efficient evaluation of the Breit operator in the Pauli spinor basis
,”
J. Chem. Phys.
157
,
064112
(
2022
).
30.
R. E.
Stanton
and
S.
Havriliak
, “
Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations
,”
J. Chem. Phys.
81
,
1910
1918
(
1984
).
31.
Y.
Ishikawa
,
R.
Binning
, and
K.
Sando
, “
Dirac-Fock discrete-basis calculations on the Beryllium atom
,”
Chem. Phys. Lett.
101
,
111
114
(
1983
).
32.
K. G.
Dyall
and
K.
Fægri
, “
Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set
,”
Chem. Phys. Lett.
174
,
25
32
(
1990
).
33.
D. B.
Williams-Young
,
A.
Petrone
,
S.
Sun
,
T. F.
Stetina
,
P.
Lestrange
,
C. E.
Hoyer
,
D. R.
Nascimento
,
L.
Koulias
,
A.
Wildman
,
J.
Kasper
,
J. J.
Goings
,
F.
Ding
,
A. E.
DePrince
III
,
E. F.
Valeev
, and
X.
Li
, “
The Chronus quantum (ChronusQ) software package
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
10
,
e1436
(
2020
).
34.
B. O.
Roos
,
R.
Lindh
,
P.-Å.
Malmqvist
,
V.
Veryazov
, and
P.-O.
Widmark
, “
Main group atoms and dimers studied with a new relativistic ANO basis set
,”
J. Phys. Chem. A
108
,
2851
2858
(
2004
).
35.
B. O.
Roos
,
R.
Lindh
,
P.-Å.
Malmqvist
,
V.
Veryazov
, and
P.-O.
Widmark
, “
New relativistic ANO basis sets for transition metal atoms
,”
J. Phys. Chem. A
109
,
6575
6579
(
2005
).
36.
B. O.
Roos
,
R.
Lindh
,
P.-Å.
Malmqvist
,
V.
Veryazov
,
P.-O.
Widmark
, and
A. C.
Borin
, “
New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3
,”
J. Phys. Chem. A
112
,
11431
11435
(
2008
).
37.
W.
Liu
, “
Ideas of relativistic quantum Chemistry
,”
Mol. Phys.
108
,
1679
1706
(
2010
).
38.
Q.
Sun
,
W.
Liu
, and
W.
Kutzelnigg
, “
Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations
,”
Theor. Chem. Acc.
129
,
423
436
(
2011
).
39.
A.
Petrone
,
D. B.
Williams-Young
,
S.
Sun
,
T. F.
Stetina
, and
X.
Li
, “
An efficient implementation of two-component relativistic density functional theory with torque-free auxiliary variables
,”
Eur. Phys. J., B
91
,
169
(
2018
).