A closed-shell coupled-cluster treatment of the Breit–Pauli first-order relativistic energy correction

ARTICLES

A closed-shell coupled-cluster treatment of the Breit–Pauli first-order relativistic energy correction

Sonia Coriani

Dipartimento di Scienze Chimiche, Universita` degli Studi di Trieste, Via Licio Giorgieri 1, I-34127 Trieste, Italy

Trygve Helgaker

Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway Poul Jørgensen

Department of Chemistry, University of Aarhus, DK-8000 A˚ rhus C, Denmark Wim Klopper

Lehrstuhl fu¨r Theoretische Chemie, Institut fu¨r Physikalische Chemie, Universita¨t Karlsruhe (TH), D-76128 Karlsruhe, Germany

共Received 17 June 2004; accepted 12 July 2004兲

First-order relativistic corrections to the energy of closed-shell molecular systems are calculated, using all terms in the two-component Breit–Pauli Hamiltonian. In particular, we present the first implementation of the two-electron Breit orbit–orbit integrals, thus completing the first-order relativistic corrections within the two-component Pauli approximation. Calculations of these corrections are presented for a series of small and light molecules, at the Hartree–Fock and coupled-cluster levels of theory. Comparisons with four-component Dirac–Coulomb–Breit calculations demonstrate that the full Breit–Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The Breit interaction is dominated by the spin–spin interaction, the orbit–orbit interaction contributing only about 10% to the total two-electron relativistic correction in molecules consisting of light atoms. However, the relative importance of the orbit–orbit interaction increases with increasing nuclear charge, contributing more than 20% in H₂S. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1788634兴

I. INTRODUCTION

A well-established method for the calculation of relativ- istic corrections to molecular electronic energies is first-order perturbation theory, using the one-electron Pauli Hamiltonian—in particular, the mass-velocity and one- electron Darwin operators. This approach, which is often re- ferred to as the Cowan–Griffin method,¹has been successful for light systems, recovering most of the relativistic correc- tion at little or no cost共even though its use may be problem- atic due to a divergent behavior arising from the mass- velocity term^2,3兲. Indeed, applied at the Hartree–Fock level to molecules with light atoms, the Cowan–Griffin method is usually not in error by more than 10% with respect to the full relativistic correction, as calculated by the full configuration- interaction 共FCI兲 method in the same basis, using all first- order relativistic terms in the Breit–Pauli approximation—

that is, all terms of order O(c^⫺2). This is illustrated in Table I, where various first-order relativistic corrections are dis- played for the H₂S molecule, both at the Hartree–Fock level and at the highly correlated coupled-cluster singles-and- doubles共triples兲CCSD共T兲共full兲level共to mimic FCI兲. To go beyond the Cowan–Griffin Hartree–Fock method, we should take into account both the effect of electron correlation on the relativistic corrections and the effect of the terms in the

Breit–Pauli Hamiltonian that are not included in the Cowan–

Griffin method. In general, each effect is equally important and neither can be neglected if we want to improve on the standard Cowan–Griffin Hartree–Fock approach.⁴ In the present paper, we compute all first-order corrections of the Breit–Pauli Hamiltonian at the coupled-cluster level, for closed-shell molecules containing light atoms 共not heavier than argon兲.

Beyond the Cowan–Griffin model, we only need to con- sider the two-electron Darwin term to arrive at the full Pauli relativistic correction of a closed-shell system. In recent years, such calculations have, for example, been carried out by Tennyson and co-workers in their accurate studies of the rovibrational spectra of triatomics.^5,6However, the Pauli ap- proximation does not exhaust all relativistic terms of order c^⫺2. In the Breit Hamiltonian, there are three additional terms: the two-electron spin–spin and orbit–orbit operators, which contribute to the first-order energy of a closed-shell system, and the spin–orbit operator, which makes no such contribution. The nonzero spin–spin contribution is trivially related to the two-electron Darwin term, being twice as large but of the opposite sign.^7–9The remaining orbit–orbit con- tribution is more complicated, however, requiring the coding of new integrals whose permutational symmetry is different

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from that of the standard two-electron repulsion integrals. Up to now, no implementation of the orbit–orbit contribution has therefore been reported in the literature.

In this paper, we present an implementation of the Breit–Pauli orbit–orbit contribution to the first-order relativ- istic energy of closed-shell systems, enabling us to carry out the first calculations that include all relativistic corrections proportional to c^⫺2. Such calculations are carried out for a selected set of molecules, for which comparisons can be made with four-component Dirac–Coulomb–Breit results.

We note that most four-component calculations do not use the full two-electron Breit Hamiltonian共in standard notation and atomic units兲,

HˆB2⫽1

2

兺

_i⫽j ^{ri j⫺1⫺1}4

兺

_i⫽j ^{共␣i•ri j⫺1␣j⫹␣i"ri jri j⫺3ri j"␣j兲, 共1兲}

but either only the Coulomb two-electron operator or the Coulomb operator augmented with the Gaunt term, which corresponds to twice the first term in the second summation:

Hˆ^Gaunt⫽⫺1/2兺i⫽j␣i•r_{i j}^⫺1␣j. In the reduced two-component form of the Pauli approximation, we here use the full Breit operator, noting that the separate evaluation of the reduced Gaunt term is more difficult than evaluation of the full re- duced Breit operator.

The remainder of this paper consists of three sections: In Sec. II, we discuss the Breit–Pauli Hamiltonian, presenting its constituent operators in second-quantized form and de- scribing the evaluation of the corresponding integrals in the atomic-orbital共AO兲basis. Next, in Sec. III, we present cal- culations on small and light molecular systems at the Hartree–Fock and CCSD levels of theory, comparing with the Dirac–Coulomb–Breit calculations of Ref. 4. Section IV concludes the paper.

II. THEORY

A. The Breit–Pauli Hamiltonian

The Breit–Pauli and Breit Hamiltonian operators are given by^10–13

Hˆ^BP⫽Hˆ^NR⫹Hˆ^MV⫹Hˆ^D1⫹Hˆ^D2⫹Hˆ^SO1⫹Hˆ^SO2⫹Hˆ^Breit, 共2兲 Hˆ^Breit⫽Hˆ^SoO⫹Hˆ^OO⫹Hˆ^SS, 共3兲 where, in atomic units, we have introduced the field-free nonrelativistic Hamiltonian,

Hˆ^NR⫽1

2

兺

_i ^pi2⫺

兺

_i,K r^Z_iK^K⫹1

2

兺

_i⫽j r¹_{i j}⫹1

2_K

兺

_⫽L ^ZR^K_KL^ZL, 共4兲 the one-electron mass-velocity operator,

Hˆ^MV⫽⫺ 1

8c²

兺

_i ^{pi4, 共5兲}

the one- and two-electron Darwin operators, respectively, Hˆ^D1⫽ ␲

2c²

兺

i,K

Z_K␦共r_iK兲, 共6兲

Hˆ^D2⫽⫺ ␲

2c²

兺

_i⫽j ^{␦共ri j兲, 共7兲}

the one- and two-electron spin–orbit operators, respectively, Hˆ^SO1⫽ 1

4c²

兺

_i,K ^ZK␴i•共_r^riK⫻pi兲

iK

3 , 共8兲

Hˆ^SO2⫽⫺ 1

4c²

兺

_i⫽j ^␴i•共r_r^{i j⫻pi兲}

i j

3 , 共9兲

and the Breit spin-other-orbit, orbit–orbit, and spin–spin op- erators, respectively,

Hˆ^SoO⫽ 1 2c²

兺

i⫽j

␴i•共r_{i j}⫻p_j兲

r_{i j}³ , 共10兲

HˆOO⫽⫺ 1

4c²

兺

_i⫽j ^{共pi•ri j⫺1pj⫹pi"ri jri j⫺3ri j"pj兲, 共11兲}

Hˆ^SS⫽ 1

8c²

兺

_i⫽j

冋

^{␴i•ri j⫺3␴j⫺3␴i"ri jri j⫺5ri j"␴j}

⫺8␲

3 ␦共r_{i j}兲␴i"␴j

册

^{. 共12兲}

In these expressions, subscripts i and j are used for the elec- trons, whereas K and L are used for the nuclei. The nuclear charges are denoted by Z_K, the Pauli spin matrices by␴i, the conjugate momentum by p_i, and the velocity of light by c⬇137 (a.u.). We also note that, when the two-electron Gaunt rather than Breit operator is used in Eq.共1兲, then the orbit–orbit operator takes the form^12,13

Hˆ

Gaunt

OO ⫽⫺ 1

2c²

兺

i⫽j 关p_i•r_{i j}^⫺1p_j⫹␲␦共r_{i j}兲兴. 共13兲 No other reduced two-component operators are affected by the use of the Gaunt rather than the full Breit operator.

For the evaluation of molecular integrals, the following expressions in terms of differential operators are more useful than those in Eqs. 共8兲–共12兲:

Hˆ^SO1⫽ i

4c²

兺

_i,K ^{ZK␴i•共“iriK⫺1兲⫻“i, 共14兲}

TABLE I. First-order Breit–Pauli corrections共in Eh) for H2S. At the equi- librium geometry and in the CVDZ basis of Ref. 4.

Correction Hartree–Fock CCSD共T兲共full兲

Mass-velocity ⫺4.549 ⫺4.551

One-electron Darwin 3.474 3.474

Two-electron Darwin ⫺0.034 ⫺0.033

Spin–spin 0.068 0.066

Orbit–orbit 0.019 0.018

Total ⫺1.022 ⫺1.026

Hˆ^SO2⫽⫺ i

4c²

兺

i⫽j ␴i•共“ⁱr_{i j}^⫺1兲⫻“ⁱ, 共15兲 Hˆ^SoO⫽ i

2c²

兺

_i⫽j ^{␴i•共“iri j⫺1兲⫻“j, 共16兲}

Hˆ^OO⫽ 1

4c²

兺

_i⫽j ^{ⵜiT关共I3⌬i⫺“i“iT兲ri j兴“j, 共17兲}

Hˆ^SS⫽ 1 8c²

兺

i⫽j ␴i

T关共I₃⌬i⫺“i“i

T兲r_{i j}^⫺1兴␴j, 共18兲

where we have made use of the relations 共I₃⌬⫺““^T兲r⫽I₃r²⫹rr^T

r³ , 共19兲

共I₃⌬⫺““^T兲r^⫺1⫽I₃r²⫺3rr^T r⁵ ⫺8␲

3 ␦共r兲. 共20兲 The corresponding form of the Gaunt orbit–orbit operator, Eq. 共13兲, is

Hˆ

Gaunt OO ⫽ 1

2c²

兺

_i⫽j ^{关“iTri j⫺1“j⫺␲␦共ri j兲兴. 共21兲}

The Breit spin–spin operator, Eq.共12兲, is often decomposed into the Fermi-contact共FC兲and dipolar共DP兲operators:

Hˆ^SS⫽Hˆ^FC⫹Hˆ^DP, 共22兲 where

HˆFC⫽⫺ ␲

3c²

兺

_i⫽j ^{␦共ri j兲␴i"␴j⫽}_12c^{1 2}

兺

_i⫽j ^{共⌬iri j⫺1兲␴i"␴j,}

共23兲

Hˆ^DP⫽ 1

8c²

兺

_i⫽j ^{共␴i•ri j⫺3␴j⫺3␴i"ri jri j⫺5ri j"␴j兲}

⫽ 1

8c²

兺

_{i j} ^␴iT

冋冉

^{13I3⌬i⫺“i“iT}

冊

^{ri j⫺1}

册

^{␴j. 共24兲}

Whereas the Fermi-contact operator is isotropic, the dipolar spin–spin operator is traceless.

B. Second-quantization representation of the operators

In terms of the singlet excitation operators

E_pq⫽a_p^†_␣a_q␣⫹a_p^†_␤a_q␤, 共25兲 the nonrelativistic electronic Hamiltonian in Eq.共4兲takes the following standard second-quantized form:

Hˆ^NR⫽⫺1

2

兺

_pq ^{具p兩⌬兩q典Epq⫺}

兺

_pq

兺

_K ^{ZK具p兩rK⫺1兩q典Epq}

⫹1

2pqrs

兺

具^pr兩r₁₂^⫺1兩qs典共E_pqE_rs⫺␦qrE_ps兲

⫹1

2_K

兺

_⫽L ^ZR^K_KL^ZL, 共26兲 where we use the following notation for the one- and two- electron integrals:

具^p兩hˆ兩q典⫽

冕

^{␾*p共r兲hˆ␾q共r兲dr, 共27兲}

具^pr兩r12⫺1兩qs典

⫽

冕 冕

^{␾p*共r1兲␾r*共r2兲r12⫺1␾q共r1兲␾s共r2兲dr1dr2. 共28兲}

In the same notation, the singlet mass-velocity and Darwin operators Eqs.共5兲–共7兲become

Hˆ^MV⫽⫺ 1

8c²

兺

_pq ^{具p兩⌬2兩q典Epq, 共29兲}

Hˆ^D1⫽ ␲

2c²

兺

_K ^ZK

兺

_pq ^{具p兩␦共rK兲兩q典Epq, 共30兲}

Hˆ^D2⫽⫺ ␲

2c²pqrs

兺

具^pr兩␦共r₁₂兲兩qs典共E_pqE_rs⫺␦qrE_ps兲, 共31兲 whereas the triplet spin–orbit operators Eqs.共14兲–共16兲take the form

Hˆ^SO1⫽ i

4c²

兺

_K ^ZK

兺

_pq ^{具p兩共“rK⫺1兲⫻“兩q典•Tpq, 共32兲}

Hˆ^SO2⫽⫺ i

4c²_pqrs

兺

^{具pr兩共“1r12⫺1兲⫻“1兩qs典}

•共T_pqE_rs⫺␦qrT_ps兲, 共33兲

Hˆ^SoO⫽ i

2c²_pqrs

兺

^{具pr兩共“1r12⫺1兲⫻“2兩qs典}

•共T_pqE_rs⫺␦qrT_ps兲, 共34兲 where we have introduced vectors containing the Cartesian triplet excitation operators:

T^x_pq⫽¹2共a_p^†_␣a_q␤⫹a_p^†_␤a_q␣兲, 共35兲 T_pq^y ⫽1

2i共a_p^†_␣a_q␤⫺a_p^†_␤a_q␣兲, 共36兲 T^z_pq⫽¹2共a_p␣^† a_q␣⫺a_p␤^† a_q␤兲. 共37兲 In second quantization, the orbit–orbit operator, Eq. 共17兲, and the spin–spin operator, Eq.共18兲, are given by

Hˆ^OO⫽ 1

4c²pqrs

兺

具^pr兩“1

T关共I₃⌬1⫺“¹“1

T兲r₁₂兴“²兩qs典

⫻共E_pqE_rs⫺␦qrE_ps兲, 共38兲 Hˆ^SS⫽ 1

8c²_pqrs

兺 兺

_␮␯ ^{具pr兩关共I3⌬1⫺“1“1T兲r12⫺1兴␮␯兩qs典}

⫻共4T_pq^␮T_rs^␯⫺␦␮␯␦qrE_ps兲. 共39兲 We note that the corresponding Gaunt version of orbit–orbit operator, Eq. 共21兲, is given by

Hˆ

Gaunt OO ⫽ 1

2c²_pqrs

兺

^{具pr兩“1Tr12⫺1“2⫺␲␦共r12兲兩qs典}

⫻共E_pqE_rs⫺␦qrE_ps兲, 共40兲 while the Fermi-contact and dipolar contributions, Eqs. 共23兲 and 共24兲, to the spin–spin operator may be written in the form

Hˆ^FC⫽⫺ ␲

3c²_pqrs

兺

^{具pr兩␦共r12兲兩qs典共4Tpq"Trs⫺3␦qrEps兲,}

共41兲 Hˆ^DP⫽ 1

2c²_pqrs

兺

^{TpqT 具pr兩}

冋冉

^{13I3⌬1⫺“1“1T}

冊

^r12⫺1

册

^兩qs典Trs.

共42兲 There is no contribution from the singlet excitation operators to the traceless dipolar spin–spin operator.

In passing, we note that, for a general triplet two- electron operator of the form

Hˆ

Q⫽ 1

8c²

兺

_i⫽j ^{␴iTQi j␴j, 共43兲}

the corresponding second-quantized operator is given by Hˆ

Q⫽ 1

8c²_pqrs

兺 兺

_␮␯ ^{具pr兩Q␮␯兩qs典}

冉

^{4Tpq␮Trs␯⫺␦qr␦␮␯Eps}

⫺2i␦qr

兺

_␭ ^{⑀␮␯␭Tps␭}

冊

^{, 共44兲}

where ⑀_␮␯␭ is the unit antisymmetric tensor. If Eq. 共43兲 is spatially symmetric Q⫽Q^T, then the terms linear in the trip- let excitation operators vanish; conversely, if it is antisym- metric Q⫽⫺Q^T, then only the linear triplet terms contribute.

C. Expectation values for closed-shell states

Because of the presence of the triplet excitation opera- tors in the Breit–Pauli Hamiltonian Eq.共2兲, many terms van- ish as we take the expectation value of this Hamiltonian for a closed-shell electronic state. To identify the zero contribu- tions and symmetries among the nonzero ones, it is expedi- ent to consider the triplet excitation operators in the spherical-harmonic spin-tensor form:

T^1,1_pq⫽⫺a^†_p␣a_q␤, 共45兲

T^1,0_pq⫽ 1

冑

^{2共ap†␣aq␣⫺a†p␤aq␤兲, 共46兲}

T^1,_pq^⫺1⫽a_p^†_␤a_q␣, 共47兲 to which the Cartesian triplet excitation operators, Eqs.

共35兲–共37兲, multiplied by & are related by a unitary trans- formation:

冑

²

冉

^TTTpqxpqypqz

冊

^⫽

冉

^⫺

^冑

¹⁰

^冑

²¹²

^{冑 冑}

^{11022 001}

冊 ^冉

^{TTT1,pqpq1,1pq1,0⫺1}

^冊

^{. 共48兲}

From the Wigner–Eckart theorem, we note that, for a closed- shell singlet electronic state 兩cs典, the spin-tensor excitation operators satisfy the relations

具^cs兩T_pq^1,0T_rs^1,0兩cs典^⫽⫺具^cs兩T^1,1_pqT_rs^1,⫺1兩cs典

⫽⫺具^cs兩T^1,_pq^⫺1T_rs^1,1兩cs典^{⬅0, 共49兲} all other products of two triplet spin-tensor operators and all single triplet operators having zero expectation values.

Therefore, for some general operator T^T_pqR_pqrsT_rs, where T_pqand T_rscontain the Cartesian triplet excitation operators, only its isotropic component R_pqrs⫽1/3Tr R_pqrs contributes to the expectation value of a singlet state

pqrs

兺

具^cs兩T_pq^T R_pqrsT_rs兩cs典⫽3 2pqrs

兺

R_pqrs具^cs兩T_pq^1,0T_rs^1,0兩cs典^. 共50兲 Taking the expectation value of HˆSS, we then find that only the isotropic Fermi-contact term contributes for singlet states

具^{cs兩HˆSS兩cs}典^⫽⫺␲

c²_pqrs

兺

^{具pr兩␦共r12兲兩qs典}

⫻具^cs兩2Tpq

1,0T_rs^1,0⫺␦qrE_ps兩cs典^{. 共51兲} Next, taking advantage of the special permutational symme- tries among the indices of the integral 具^pr兩␦^(r12)兩qs典

⫽兰␾p*(r)␾q(r)␾r*(r)␾s(r)dr, we find that, for singlet states, the expectation value of the two-electron spin–spin operator may be expressed entirely in terms of singlet exci- tation operators

具^{cs兩HˆSS兩cs}典^{⫽ ␲}

c²_pqrs

兺

^{具pr兩␦共r12兲兩qs典具cs兩EpqErs⫺␦qrEps兩cs典.}

共52兲 Finally, comparing this expression with the singlet two- electron Darwin operator in Eq. 共31兲, we conclude that, for singlet states, the expectation values of the two-electron spin–spin and Darwin operators are related in the following simple manner:

具^cs兩HˆSS兩cs典^⫽⫺2具^cs兩HˆD2兩cs典^{, 共53兲} giving the following expression for the expectation value of the Breit operator for closed-shell electronic state:

具^cs兩HˆBreit兩cs典^⫽具^cs兩HˆOO⫺2HˆD2兩cs典^{, 共54兲} with no contributions from the spin-other-orbit and dipolar spin–spin operators. Consequently, the expectation value of the total Breit–Pauli operator may be calculated as

具^cs兩Hˆ^BP兩cs典⫽具^cs兩Hˆ^NR兩cs典

⫹具^{cs兩HˆMV⫹HˆD1⫺HˆD2⫹HˆOO兩cs}典^{, 共55兲} with a negative sign on the two-electron Darwin term.

Finally, when the Gaunt operator rather than the full Breit operator is used, we need only replace Hˆ^OOby Hˆ

Gaunt OO in Eq.

共55兲. Since the contact term in Hˆ

Gaunt

OO , see Eq. 共40兲, is ex- actly equal to the two-electron Darwin operator Hˆ^D2, Eq.

共31兲, we find that there is no contact contribution to the expectation value of the Hˆ^BPoperator for closed-shell states, Eq. 共55兲, if the Breit operator is replaced by the Gaunt operator.

D. Atomic-orbital integral evaluation

The mass-velocity and Darwin operators, Eqs. 共29兲– 共31兲, present no problems but are given here for complete- ness. Expanding the molecular integrals in AOs, we obtain 共assuming real orbitals兲

Hˆ^MV⫽

兺

_pq

兺

_ab ^{CpaCqbhabMVEpq, 共56兲}

Hˆ^D1⫽

兺

_pq

兺

_ab ^{CpaCqbhabD1Epq, 共57兲}

Hˆ^D2⫽1

2_pqrs

兺

_abcd

兺

^{CpaCqbCrcCsdhabcdD2 共EpqErs⫺␦qrEps兲,}

共58兲 where the C_paare the real molecular-orbital共MO兲expansion coefficients and where the one- and two-electron 共AO兲inte- grals are given by

h_ab^MV⫽⫺ 1

8c²⌬A⌬B具^a兩b典^, 共59兲

h_ab^D1⫽ ␲

2c²

兺

_K ^{ZK具a兩␦共rK兲兩b典, 共60兲}

h_abcd^D2 ⫽⫺␲

c²具^ac兩␦共r₁₂兲兩bd典^. 共61兲 These integrals, which are easily evaluated by standard tech- niques such as the McMurchie–Davidson scheme,¹⁴are sym- metric with permutations to any two of the indices a, b, c, and d.

Next, considering the Breit–Pauli spin–orbit operators, Eqs. 共32兲–共34兲

Hˆ^SO1⫽

兺

_pq

兺

_ab ^{CpaCqbhabSO1"Tpq, 共62兲}

Hˆ^SO2⫽1

2_pqrs

兺

_abcd

兺

^{CpaCqbCrcCsdhabcdSO2}

•共T_pqE_rs⫺␦qrT_ps兲, 共63兲 Hˆ^SoO⫽1

2_pqrs

兺

_abcd

兺

^{CpaCqbCrcCsdhabcdSoO}

•共T_pqE_rs⫺␦qrT_ps兲, 共64兲 we find that the AO integrals may be obtained as derivatives of the Coulomb integrals with respect to the AO centers A, B, C, and D as follows:

h_ab^SO1⫽⫺ i

4c²

兺

_K ^{ZK“A⫻“B具a兩rK⫺1兩b典, 共65兲}

h_abcd^SO2⫽ i

2c²“A⫻“B具^ac兩r₁₂^⫺1兩bd典^, 共66兲 h_abcd^SoO⫽ i

c²“C⫻“D具^ac兩r₁₂^⫺1兩bd典^. 共67兲 In deriving these expressions, we have made extensive use of the relations

具^ac兩共“¹r₁₂^⫺1兲兩bd典^⫽“^P具^ac兩r12⫺1兩bd典^⫽⫺“^Q具^ac兩r12⫺1兩bd典^, 共68兲

“P⫽“A⫹“B, “Q⫽“C⫹“D, 共69兲 where P and Q are given by the expressions

P⫽aA⫹bB

a⫹b , Q⫽cC⫹dD

c⫹d , 共70兲

in terms of the positions A, B, C, and D and the exponents a, b, c, and d of the four AOs in the two-electron integrals 具^ac兩r₁₂^⫺1兩bd典. Note that, whereas the two-electron spin-own- orbit integrals, Eq. 共66兲, are antisymmetric with respect to permutations of the indices ab of the first electron 关like the one-electron spin–orbit integrals Eq. 共65兲兴 but symmetric with respect to permutations of the indices cd of the second electron, the two-electron spin-other-orbit integrals, Eq.共66兲, are symmetric with respect to permutations of ab but anti- symmetric with respect to permutations of cd.

Using the same technique, we find that the AO integrals of the orbit–orbit and spin–spin Breit operators, Eqs. 共38兲 and共39兲,

Hˆ^OO⫽1

2_pqrs

兺

_abcd

兺

^{CpaCqbCrcCsdhabcdOO}

⫻共E_pqE_rs⫺␦qrE_ps兲, 共71兲 Hˆ^SS⫽1

2pqrs

兺

abcd

兺

C_paC_qbC_rcC_sd关h_abcd^SS 兴␮␯

⫻共4T_pq^␮ T_rs^␯⫺␦␮␯␦qrE_ps兲, 共72兲 may be computed as derivatives of the two-electron integrals 具^ac兩r₁₂兩bd典 ^and具^ac兩r₁₂^⫺1兩bd典^:

h_abcd^OO ⫽ 1 2c²ⵜA

T共I₃⌬P⫺“^P“P

T兲“^C具^ac兩r₁₂兩bd典^, 共73兲

h_abcd^SS ⫽ 1

4c²共I₃⌬P⫺“P“P

T兲具^ac兩r₁₂^⫺1兩bd典^. 共74兲 In the Gaunt approximation Eq. 共40兲, the orbit–orbit inte- grals are given by

关h_Gaunt^OO 兴abcd⫽1 c²“A

T“D具^ac兩r₁₂^⫺1兩bd典⫺ ␲

c²具^ac兩␦共r₁₂兲兩bd典^, 共75兲 rather than by Eq.共73兲and all other integrals are unaffected.

Whereas the spin–spin integrals are symmetric in a and b and in c and d, the Breit orbit–orbit integrals are antisym- metric in the same pairs of indices, as follows for a and b from the identities:

共“^A⫹“^B兲^T共I₃⌬P⫺“^P“P T兲

⫽“P

T共I₃⌬P⫺“P“P T兲⫽“P

T⌬P⫺⌬P“P

T⫽0^T. 共76兲 To establish the equivalent antisymmetry of c and d, we pro- ceed in the same manner, using the translational invariance (ⵜP⫹ⵜQ)具^ac兩r₁₂兩bd典^⫽0. By contrast, in the Gaunt orbit–

orbit integrals, there is no permutational symmetry between a and b nor between c and d. To summarize, we have estab- lished the following permutational symmetries of the AO spin–orbit, spin–spin, and共Breit兲orbit–orbit integrals

h_ab^SO1⫽⫺h_ba^SO1,

h_abcd^SO2⫽⫺h_bacd^SO2⫽h_abdc^SO2⫽⫺h_badc^SO2,

h_abcd^SoO⫽h_bacd^SoO⫽⫺h_abdc^SoO⫽⫺h_badc^SoO , 共77兲 h_abcd^SS ⫽h_bacd^SS ⫽h_abdc^SS ⫽h_badc^SS ,

h_abcd^OO ⫽⫺h_bacd^OO ⫽⫺h_abdc^OO ⫽h_badc^OO .

In addition, the spin–spin and orbit–orbit integrals are sym- metric with respect to permutations of ab and cd. Using these permutational symmetries as well as the relations Eq.

共69兲, we find that the spin–orbit, spin–spin, and orbit–orbit integrals may be evaluated in a number of equivalent ways, as different derivatives with respect to the atomic centers A, B, C, and D or the centers P and Q.

E. Coupled-cluster expectation values

We now discuss the evaluation of the first-order Breit–

Pauli energy correction for a closed-shell singlet state using the coupled-cluster singles共CCS兲model, CCSD model, and the CCSD perturbative triples 关CCSD共T兲兴model. For these wave-function models, the evaluation of the molecular gra- dient for a closed-shell singlet case is described by Hald et al. in Ref. 15. As discussed below, the CCSD共T兲 Breit–

Pauli energy correction of Eq.共55兲may be calculated using a simplified version of the molecular-gradient expression given in Eq.共26兲of Ref. 15. The CCSD energy correction is then obtained from the CCSD共T兲expression by omitting all terms

that involve triples amplitudes; likewise, the CCS gradient is obtained by omitting all terms that involve doubles and triples amplitudes.

The main difference between a molecular-gradient cal- culation and the evaluation of the Breit–Pauli energy correc- tion is that perturbation-dependent AOs are used for the mo- lecular gradient, whereas fixed AOs are used for the energy correction. The contributions to the molecular gradient that involve overlap derivatives therefore do not contribute to the Breit-Pauli energy correction. Moreover, the geometrical de- rivatives of the molecular integrals in gradient calculations represent the perturbed operator and must be replaced by the Breit-Pauli integrals of Sec. II D to obtain the first-order en- ergy correction. With these modifications, the first-order Breit-Pauli energy may be straightforwardly calculated with a gradient code.

The only problem that arises with the adaptation of the gradient code of Ref. 15 to the calculation of first-order rela- tivistic corrections is related to the permutational symmetries of the orbit–orbit integrals, which are different from those assumed in the CCSD共T兲 gradient code. Following the dis- cussion of Sec. III C in Ref. 15, the contraction of the triples density matrix elements d_abic^(T) and d_abci^(T) with two-electron integrals g_abic that are either symmetric or antisymmetric with respect to permutations of the last indices i and c may be carried out as

abic

兺

共d_abic^共T兲 g_abic⫹d_abci^共T兲 g_abci兲⫽_abic

兺

^{Sdabic共T兲 gabic,}

共78兲

Sd_abic^共T兲 ⫽d_abic^共T兲 ⫾d_abci^共T兲 ,

where a, b, and c denote here virtual MOs and i occupied MOs. For the efficient calculation of properties, the symme- trized density elements^Sd_abic^(T) are added to the effective den- sity elements d_abic^eff . The plus sign in Eq.共78兲is used when- ever the contraction is made with integrals symmetric in the last two indices 共as for molecular gradients in Ref. 15兲, whereas the minus sign is used for the antisymmetric Breit orbit–orbit integrals. A similar modification must be made for the triples density matrix elements d_{i jka}^(T) and d_{i jak}^(T) where i, j, and k denote occupied and a unoccupied MOs.

III. RESULTS AND DISCUSSION

All calculations presented here have been carried out using a development version of the quantum-chemistry code

DALTON,¹⁶ and a modification of the coupled-cluster molecular-gradient implementation of Ref. 15 as discussed in Sec. II E.

The results presented in Table I were discussed already in Sec. I. We note here that the first-order corrections are dominated by the one-electron mass-velocity and Darwin terms 共the Cowan–Griffin model兲. The two-electron terms are individually much smaller 共about 1%兲 but cancel to a smaller extent and contribute as much as 5% to the total first-order relativistic correction. Of all contributions, the orbit-orbit contribution is the smallest one, constituting less than 1% the total correction.

In Table II, we have listed the first-order Breit correc- tions to the total energy for the same molecules that were studied in Ref. 4, using the same basis sets and geometries.

At the Hartree–Fock level of theory, it is possible to compare with Dirac–Hartree–Fock results. The agreement is in all cases excellent. Moreover, the small difference between the two methods is very systematic, depending in a simple man- ner on the types of atoms in the molecules, for example, each carbon, nitrogen, and oxygen atom contributes about 1, 3, and 7␮^Eh, respectively, to the overall difference; for the heavier Si atom, the contribution is larger 共about 0.15mE_h per atom兲. We also note that, in all cases, the first-order Breit-Pauli treatment gives a too small relativistic correction to the energy. Comparing the Hartree–Fock and all-electron CCSD results, we find that correlation reduces the Breit cor- rection, by 5%–10%共less for the heavier systems兲.

From Table II, we see that the first-order two-electron relativistic correction is dominated by the spin–spin interac- tion, the orbit–orbit interaction contributing only about 10%

to the total Breit correction in molecules consisting of light atoms. However, the orbit–orbit term becomes relatively more important with increasing nuclear charges: in H₂S, for example, it constitutes more than 20% of the total Breit cor-

rection. Both the spin–spin and orbit–orbit contributions are in all cases positive, increasing the total electronic energy.

In Table III, we compare the first-order Breit corrections obtained at the CCSD level of theory by correlating all elec- trons with those obtained by correlating only the valence electrons. Not surprisingly, most of the correlation correction to the Breit term occurs in the core.

IV. CONCLUSIONS

We have presented the first implementation of all terms that contribute to the first-order relativistic correction to the total electronic energy in molecular systems, including the two-electron orbit–orbit Breit interaction. Calculations in- cluding all first-order terms have been presented for a series of small and light molecules, at the Hartree–Fock and coupled-cluster levels of theory. By comparing with four- component Dirac–Coulomb–Breit calculations, we have demonstrated that the full Breit–Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The two-electron interactions are dominated by the spin-spin interaction, the orbit–orbit inter- action contributing only about 10% to the Breit correction in

TABLE II. First-order Breit corrections共in mEh) for selected molecules.

Molecule^a Basis^b

Hartree–Fock CCSD共full兲

E^SS E^OO E^Breit E^SS E^OO E^Breit

C2H6 u-cc-pCVDZ This work 5.312 0.357 5.670 4.855 0.150 5.005

Reference 4 5.312 5.672^c 4.85^d

NH3 u-cc-pCVTZ This work 4.397 0.392 4.789 4.002 0.209 4.211

Reference 4 4.396 4.792^c 4.00^d

H2O u-cc-pCVTZ This work 6.809 0.766 7.575 6.283 0.511 6.793

Reference 4 6.810 7.582^c 6.28^d

HCN u-cc-pCVTZ This work 7.051 0.529 7.580 6.388 0.232 6.620

Reference 4 7.052 7.584^c 6.38^d

HNCC u-cc-pCVDZ This work 13.831 1.292 15.123 12.881 0.854 13.735

Reference 4 13.832 15.134^c 12.88^d

HCOOH u-cc-pCVDZ This work 16.247 1.661 17.909 15.188 1.164 16.352 Reference 4 16.248 17.924^c 15.18^d

SiH3⫺ CVDZ This work 43.771 10.941 54.711 42.078 10.081 52.159

Reference 4 43.770 54.859^c 42.07^d

SiC2 CVDZ This work 49.084 11.193 60.277 46.953 10.138 57.091

Reference 4 49.084 60.427^c 46.95^d

H2S CVDZ This work 68.230 19.100 87.330 65.958 17.843 83.801

Reference 4 68.230 87.639^c 65.95^d

aCalculated equilibrium geometries taken from Ref. 4.

bSee Ref. 4.

cDirac–Hartree–Fock–Coulomb–Breit results obtained with theBerthaprogram.

dFirst-order CCSD共T兲共full兲value.

TABLE III. First-order Breit corrections共in mEh) for the Ne atom in the cc-pCVQZ basis.

Method

Frozen core Correlated core

E^SS E^OO E^Breit E^SS E^OO E^Breit

Hartree–Fock 14.234 2.384 16.618 14.234 2.384 16.618

MP2 14.075 2.282 16.357 13.197 1.865 15.062

CCSD 14.084 2.295 16.379 13.208 1.881 15.089

CCSD共T兲 14.071 2.275 16.347 13.183 1.837 15.021