One-step treatment of spin-orbit coupling and electron correlation in large activespaces

Bastien Mussard1, ∗ and Sandeep Sharma1, †

1Department of Chemistry and Biochemistry, University of Colorado Boulder, Boulder, CO 80302, USA

In this work we demonstrate that the heat bath configuration interaction (HCI) and its semis-tochastic extension can be used to treat relativistic effects and electron correlation on an equalfooting in large active spaces to calculate the low energy spectrum of several systems includinghalogens group atoms (F, Cl, Br, I), coinage atoms (Cu, Au) and the Neptunyl(VI) dioxide radical.This work demonstrates that despite a significant increase in the size of the Hilbert space due tospin symmetry breaking by the spin-orbit coupling terms, HCI retains the ability to discard largeparts of the low importance Hilbert space to deliver converged absolute and relative energies. Forinstance, by using just over 107 determinants we get converged excitation energies for Au atom inan active space containing (150o,25e) which has over 1030 determinants. We also investigate theaccuracy of five different two-component relativistic Hamiltonians in which different levels of ap-proximations are made in deriving the one-electron and two-electrons Hamiltonians, ranging fromBreit-Pauli (BP) to various flavors of exact two-component (X2C) theory. The relative accuracyof the different Hamiltonians are compared on systems that range in atomic number from first rowatoms to actinides.

I. INTRODUCTION

Relativistic effects play an important role in avariety of photochemical processes, various mag-netic spectroscopies and are responsible for un-usual phase behaviors in transition metal oxides. Inpractice, the most accurate procedure for treatingrelativistic effects in molecular systems is throughthe solution of the Dirac-Coulomb-Breit (DCB)equation116,138,168,176,189,190,200. Although recent devel-opments have made it possible to perform self-consistentfield158,188,216,223, density functional theory120,182,201,222,coupled cluster185,211, explicitly correlated125,171,215 andmultireference114,119,139,141,163,208 calculations on theDCB Hamiltonian, these methods remain problematicdue to their high computational cost.

An alternative is to use two-component Hamiltoni-ans which are capable of delivering quantitative accu-racy for relativistic problems. They are derived by ei-ther eliminating the small component (ESC) or by usinga unitary transformation called the Foldy-Wouthuysen(FW) transformation142 to decouple the large and thesmall components in the one-body Dirac or Dirac-FockHamiltonian. These two are of course related and FWtransformation can be viewed as a normalized elimina-tion of small component (NESC). This transformationis also applied to the two-electron integrals to generatea relativistically correct electron interaction. The re-sulting two-component Hamiltonian can be partitionedinto the Schrodinger equation and additional relativis-tic spin-free and spin-dependent terms. The Schrodingerequation together with the spin-free terms are referredto as the scalar relativistic Hamiltonian and calculatingthe ground state energy of this Hamiltonian is usually nomore expensive than that of the Schrodinger equation.The remaining spin-dependent terms, known as the spin-orbit coupling (SOC), are usually treated using pertur-bation theory. However, for heavier elements these terms

can become large and it is more appropriate to treat themon an equal footing with the spin-free terms and electroncorrelation. In this work we follow the latter approach.

The three most commonly used two-componentHamiltonians are the Douglass-Kroll-Hess130,150,184,192,the Barysz-Sadlej-Snijders117,118,155 and the exact two-component (X2C) Hamiltonians132,156,169,177. BothDouglass-Kroll-Hess and Barysz-Sadlej-Snijders Hamil-tonians carry out the transformation analytically. Thetransformation operator contains a complicated functionof the momentum operator and its integrals cannot becalculated analytically, instead the matrix representationof the momentum operator in a finite basis is calculated.In X2C, one forgoes the analytic transformation and theentire process is carried out algebraically using the ma-trix representation of operators, starting from the solu-tion of the Dirac equation. The commonly used approx-imation, known as X2C-1e, consists of solving the spin-free non-interacting Dirac equation in one step (withoutelectron-electron interaction a self-consistent procedureis not needed) and the transformation matrix is derivedfrom this solution. The motivation for doing so comesfrom the fact that only a small fraction of the total costof performing a correlated calculation is used in the so-lution of the one-body problem and thus solving it toexactly decouple the large and small component is rela-tively cheap.

To get quantitative accuracy one also needs to includethe relativistically correct two-body terms. Such termsare derived from the transformation of the Coulomb andGaunt operators, which gives rise to several spin-free andspin-dependent terms. In this work we will ignore the rel-ativistic correction to the spin-free two-body terms andonly some of the spin-dependent terms are included thatappear at order α2. The effect of those two-body SOCterms is often treated approximately using the spin-orbitmean field (SOMF) approximation151,181, which replacesthe two-body terms by an effective fock-like one-body

arX

iv:1

710.

0025

9v1

[ph

ysic

s.ch

em-p

h] 3

0 Se

p 20

17

2

term. Although the approximation sounds drastic, inpractice it is known to be extremely accurate and is al-most universally used. The SOMF approximation sim-plifies the correlated calculation considerably by reducingthe memory and CPU requirement of storing and trans-forming the different sets of relativistic two-electron in-tegrals.

Several algorithms and programs are now available forperforming relativistic calculations with two-componentHamiltonians115,122,123,146,162,179,187,198,218,220,224. Someof these methods include the spin-orbit coupling inthe self-consistent field (SCF) calculation, due towhich the orbital relaxation effects are fully in-cluded, but the resulting orbitals are complex-valuedspinors133,140,159,160,183. In the intermediate approachthe spin-orbit coupling terms are only introduced dur-ing the correlated calculation135,144,162,209,221, and aretreated on an equal footing with the dynamical electroncorrelation. A further approximation is possible which isusually called the quasi-degenerate perturbation theoryor the state interaction approach, whereby one first cal-culates several electronic state wavefunctions of the spin-free Hamiltonian in different spin sectors. The matrixrepresentation in the basis of these states of the com-plete two-component Hamiltonian including spin-orbitcoupling is then diagonalized to obtain the spin-orbit cou-pled results143,164,194,196,202. The accuracy of the methodis limited by the number of states included in the stateinteraction approach, which is usually on the order of afew tens of states and in rare cases of heavy elements afew hundred175. A related approach is the EOM(SOC)-CC128,161 where the spin-orbit coupling terms are in-cluded only during the equation of motion part of the cal-culation and thus are effectively treated perturbatively.

In this work we perform multireference calculations inlarge active spaces where the SOC is treated on an equalfooting as the electron correlation. Heat-bath configu-ration interaction (HCI) algorithm152,205 is the multiref-erence method which is used to calculate the zero-fieldsplittings using real-valued orbitals obtained from a non-relativistic SCF calculation. HCI is an efficient variantof the general class of methods in which a selected con-figuration interaction is performed which is followed byperturbation theory. As we will show in the results sec-tion, HCI is able to systematically discard large parts ofthe Hilbert space to deliver accurate excitation energiesat a much reduced cost relative to a full configuration in-teraction (FCI) calculation. To the best of our knowledgecurrent calculations and the density matrix renormaliza-tion group (DMRG) theory calculations performed bySayfutyarova et al.202 and Knecht et al.165 are the onlyones in literature where relativistic calculations are per-formed on an active space larger than 16 electrons in 16orbitals. However, unlike the DMRG calculations of Say-futyarova, here we treat SOC non-perturbatively on anequal footing with the electron correlation.

The rest of the article is organized as follows. In Sec-tion II, we present the derivation and the working equa-

tions of the Breit-Pauli and X2C Hamiltonians used inthis work. In Section III, we describe the Heat-bath Con-figuration Interaction algorithm and the extensions to itthat allow us to treat the SOC terms. In Section IV andV, we give the computational details and results respec-tively.

II. THEORY

In this section, we present the working equations ofthe two-component Hamiltonians used in this work: theBreit-Pauli (BP) and the exact two-component (X2C)Hamiltonians. To give some context, we start with thematrix formulation of the one-electron four-componentDirac Hamiltonian(

Vne c σσσ · pc σσσ · p Vne − 2mc2

)Ψ = EΨ, (1)

where Vne is the electron-nucleus potential, p is the mo-mentum and σσσ = {σσσx,σσσy,σσσz} is the set of Pauli matri-ces. The eigenvectors Ψ of the Dirac Hamiltonian arefour-component bispinors which contain the “large” ΨL

and “small” ΨS components that are themselves two-component wavefunctions:

Ψ =

(ΨL

ΨS

). (2)

Although the Dirac Hamiltonian is fully Lorentz invari-ant, introducing the Coulomb interaction between theelectrons breaks this invariance and one would need togo to quantum electrodynamics (QED) to obtain a fullyLorentz invariant theory of interacting electrons, but it isnot obvious that a fully Lorentz invariant many-electronHamiltonian can be derived from QED. In practice, theDirac-Coulomb-Breit Hamiltonian with the non-retardedelectron-electron interaction

Vee(i, j) =

(1

rij

)−(αααi ·αααjrij

)+

(αααi ·αααj

2rij− (αααi · rij)(αααj · rij)

2r3ij

)(3)

is used (in brackets are the Coulomb, Gaunt and Gaugeterms, while αααi are the Dirac matrices) and is expectedto be sufficiently accurate for chemical applications.

In principle, it is possible to obtain exact two-component Hamiltonians by solving the DCB Hamilto-nian and using the many-body solutions to decouple thesmall and the large components exactly. This is of courseimpractical because the full solution to the DCB Hamil-tonian would be required. Instead, one tries to decouplethe large and small components at the one-body level,which can be done approximately analytically (as is doneto derive the Breit-Pauli Hamiltonian) or exactly alge-braically (as is done to derive the exact two-component

3

Hamiltonian). This class of procedures is tractable be-cause one needs to solve at most the one-body problem.The transformation obtained this way is then applied tothe two-body Coulomb, Gaunt and Breit terms to derivetwo-component two-body SOC Hamiltonians.

A. Breit-Pauli Hamiltonian

The Breit-Pauli (BP) Hamiltonian used in this workis a two-component Hamiltonian obtained through ananalytic FW transformation performed on the four-component one-body Dirac Hamiltonian of Eq. (1) withtwo-body terms obtained by contributions from thetransformed Coulomb and Gaunt terms (first and secondterms of Eq. (3)). The final expression of the Breit-PauliHamiltonian131,193 is

HBP = HBPSF + HBP

SO(1) + HBP

MF(1), (4)

where the three terms are defined in Eqs. (5), (6) and(15).

1. One-body part

The one-body part of the BP Hamiltonian, knownas the Pauli Hamiltonian, contains the kinetic energy,electron-nuclear interaction, one-body spin-orbit cou-pling, mass-velocity and Darwin terms. In this work wewill ignore the divergent mass-velocity and Darwin termsand are thus left with HPauli = HBP

SF +HBPSO

(1), where thespin-free part

HBPSF = T + Vne (5)

contains the kinetic energy T and the electron nuclearattraction Vne. The one-body spin-orbit coupling termis

HBPSO

(1) = −iα2

4

∑iA

(piZAriA× pi

)· σσσi, (6)

where, α is the fine structure constant, pi is the momen-tum of electron i, Za is the atomic number of nucleus A,and riA the distance between electron i and nucleus A.In second quantization the one-body SOC term can bewritten as

HBPSO

(1) =− iα2

4

∑iApqµνλ

ελµν〈pµ|ZAriA|qν〉σλpq (7)

=− iα2

4

∑pqλ

BPSλpqσλpq, (8)

where λ, µ, ν can be (x, y, z), p and q are orbitals, pµ =∂p∂µ are nuclear derivatives of the orbitals, and ελµν is the

Levi-Civita symbol. Note that the three σ operators are

σxpq = a†pαaqβ + a†pβ aqα

σypq = −i(a†pαaqβ − a†pβ aqα) (9)

σzpq = a†pαaqα − a†pβ aqβ .

We can infer from the definition of BPSλpq that the one-body spin-orbit coupling integrals are anti-hermitian.

2. Two-body part

The application of the FW transformation to theCoulomb and Gaunt operators spawns several spin-freeand spin-dependent terms, out of which in this work wewill only include the spin-same-orbit and spin-other-orbitinteractions, i.e. respectively the first and second termin

HBPSO

(2) =iα2

4

∑i 6=j

(pi

1

rij× pi

)· (σσσi + 2σσσj). (10)

In second quantization, this takes the form

HBPSO

(2) =iα2

4

∑pqrsλ

(2Jλrspq + Jλpqrs

)Dλpqrs, (11)

where the two-electron integrals are

Jλpqrs =∑µν

ελµν〈pµr|qνs〉, (12)

and the operator is

Dλpqrs = σλpqErs − δqrσλps. (13)

For real orbitals one can utilize the 4-fold symmetry

Jλpqrs = −Jλqprs = −Jλqpsr = Jλpqsr (14)

to reduce the memory cost of storing the two electronintegrals.

Note that the two-body spin-orbit coupling term is ofopposite sign to the one-body term of Eq. (8) and actsas a screening potential. Several other terms includingthe spin-spin dipole interactions and Fermi contact in-teraction are ignored. For most applications such sim-plifications are justified, however some noteworthy ex-ceptions exist such as the oxygen molecule170,217, wherethe spin-spin dipole interactions are of the same order ofmagnitude as the spin-orbit coupling terms.

Although the two-body terms can be easily incorpo-rated in a calculation, the cost of storing and trans-forming integrals for large molecules become too expen-sive. This motivates the use of the spin-orbit mean fieldapproximation151,174,181, where one writes an effectiveone-body operator by integrating out the spins of twoout of the four orbitals in the two-electron integrals of

4

Eq. (12). Such an integration leads to the effective one-body term151 written in second quantization as

HBPMF

(1) = iα2

4

∑pq

BPSλpqσλps, (15)

where

BPSλpq =∑rs

γrs

(Jλpqrs −

3

2Jλprsq −

3

2Jλsqpr

)(16)

and γ is the one-body density matrix.Although it is not used here, the cost of calculating the

two-body terms can be further substantially reduced byusing the one-center approximation which makes use ofthe local nature of the spin-orbit interaction and intro-duces only a small error121.

B. X2C Hamiltonian

The derivation of the X2C Hamiltonian begins by usingthe restricted kinetic balance212 that replaces the smallcomponent (ψS) with the pseudo-large component (φL)

ψS =σσσ · p

2cφL, (17)

which gives rise to the modified four-component one-electron Dirac equation(

Vne T

T W −T

)(ψL

φL

)=

(S 0

0 α2

4 T

)(ψL

φL

)E, (18)

where W = α2

4 (σσσ·p)Vne(σσσ·p) and S is the non-relativisticmetric. The restricted kinetic balance transformation en-sures that the large and the pseudo-large componentshave the same symmetry properties and can be expandedwith a common basis set. Also, the two components be-come equal to each other in the non-relativistic limit.

The only spin-dependent term in the modified Diracequation of Eq. (18) is found in W, which can be exactlypartitioned into a spin-free and spin-dependent term

W =α2

4p · Vnep + i

α2

4σσσ · (pVne × p)

=WSF + WSD. (19)

This allows us to separate the spin-free (H4cSF) and spin-

dependent (H4cSD) terms in the four-component modified

Dirac equation

H4cSF =

(Vne T

T WSF −T

)(20)

H4cSD =

(0 0

0 WSD

). (21)

An FW transformation can be derived by solving onlythe spin-free part of the one-body Hamiltonian (Eq. 20)to obtain the X2C-1 one-body SOC terms172–174. An-other possibility is to obtain a complex-valued FW trans-formation by solving the total one-body Hamiltonian(containing the total W of Eq. 19) to obtain the X2C-None-body SOC terms129. In both cases the two-body partis obtained by applying the X2C-1 FW transformation tothe Coulomb and Gaunt operators. The final expressionof the resulting exact two-component Hamiltonians usedin this work is thus:

HX2C-x = HX2CSF + HX2C-x

SO(1) + HX2C

MF(1), (22)

where “x” distingishes between the X2C-1 and X2C-N one-body terms. The HX2C

SF is defined in Eq. (28),

HX2C-1SO

(1) is defined in Eq. (29), HX2C-NSO

(1) is defined in

Eq. (31) and the HX2CMF

(1) is defined in Eq. (32).

1. One-body part: X2C-1 scheme

As stated above, in the X2C-1 scheme one uses onlythe spin-free four-component Hamiltonian H4c

SF to solvethe modified Dirac equation of Eq. (18) in a single step.The data of the obtained four-component positive en-ergy solutions Ψ+ is used to apply an FW transformationto block-diagonalize H4c

SF and retain its large componentblock. The same FW transformation is subsequently usedon H4c

SD and on the Coulomb and Gaunt terms to obtainthe complete X2C-1 Hamiltonian.

Without going into the detailed derivation which canbe found for example in Ref. 178, it can be shown that thefollowing FW transformation matrix block-diagonalizesthe spin-free modified Dirac equation

U =

(1 X

X 1

)(R 0

0 R

), (23)

where the first matrix does the decoupling and the secondrenormalizes the states to bring them to the Schrodingermetric. Only X and R are required to determine thelarge component of the block-diagonalized modified DiracHamiltonian. The matrix X is defined through the rela-tion

φL+ = XψL+ (24)

between the large component ψL+ and the pseudo large

component φL+ of the positive energy bispinors

Ψ+ =

(ψL+

φL+

)(25)

and the matrix R is defined as

R = (S−1S)−1/2 (26)

S = S +α2

2X†TX. (27)

5

Using this unitary transformation, one can show that thespin-free two-component X2C Hamiltonian HX2C

SF is givenas

HX2CSF = R†(Vne + TX + X†T + X†(WSF −T)X)R.

(28)

The same FW transformation is subsequently applied tothe spin-dependent four-component Hamiltonian H4c

SD toobtain the one-body spin-orbit coupling operator

HX2C-1SO

(1) = −iα2

4

∑Ai

(XR)†(piZairAi× pi

)· σσσi(XR),

(29)

which involves integrals

X2CSλpq =∑rs

(XR)†prBPSλpq(XR)sq, (30)

that are equal to the integrals found in the Breit-Paulicase transformed by the regularizing matrices X and R.

2. One-body part: X2C-N scheme

An alternative scheme is to solve the complete mod-ified Dirac equation found in Eq. (18), including boththe spin-free and spin-dependent parts, to calculate thecorresponding complex-valued FW transformation. Theexpressions for calculating the matrices X and R areformally equivalent to Eqs. (24) and (26) respectively,with the difference that these are now 2n × 2n dimen-sional matrices. The X and R matrices can be used inEqs. (28) and (29) to obtain the complete two-componentone-body hamiltonian containing both the spin-free andspin-dependent terms

HX2C-N = HX2C-NSF + HX2C-N

SO(1). (31)

It is worth pointing out that the HX2C-NSF contains con-

tributions from high powers of WSD and is not the sameas the genuine spin-free Hamiltonian HX2C

SF of Eq. (28).However, as expression in Eq. (22) indicates, we never-theless use HX2C

SF in the X2C-N scheme.

3. Two-body part

The two-body operator and its mean field approxima-tion for the X2C theory are derived in great detail inRef. 173 and 174 and we merely give a brief outline ofthe derivation.

The X2C two-body operators are obtained by per-forming the same two sequential transformations (therestricted kinetic balance transformation and the spin-free unitary transformation) described in the “One-bodypart: X2C-1 scheme” section. As a result, one obtainsseveral terms, the first of which is the modified spin-free

Coulomb term. This term is customarily replaced by thebare Coulomb term and the difference is ignored. Out ofthe remaining spin-dependent terms only the spin-same-orbital and spin-other-orbital are retained in our work.These are similar to their Breit-Pauli counterparts mul-tiplied by the regularizing matrices X and R.

The mean field approximation to the two-body partis derived by performing the X2C transformation onthe spin-orbit contribution of the two-electron integralsin the modified Dirac-Fock-Coulomb-Breit operator174.The resulting X2C mean field operator is

HX2CMF

(1) = iα2

4

∑pq

X2CSλpqσλps, (32)

where the one-body integrals are

X2CSλpq = Rpr(gLL,λ + gLS,λ + gSL,λ + gSS,λ

)rsRsq.

(33)

The matrices g are

gLL,λmn =−∑pq

2Kλpmqnγ

SSpq

gLS,λmn =−∑pqr

(Kλmpqr +Kλ

pmqr

)γLSpq Xrn = gSL,λnm (34)

gSS,λmn =−∑pqrs

2(Kλrspq +Kλ

rsqp −Kλrpsq

)γLLpq X

†mpXqn,

the matrices X and R are formally defined in Eq. (24)and (26) and the matrices γγγ are

γγγLL = R†γγγSAR (35)

γγγLS = γγγLLX = γγγSL† (36)

γγγSS = X†γγγX. (37)

where γγγSA = 12 (γγγα + γγγβ) is the spin-averaged density

matrix. Finally, the two-electron integrals appearing inEq. (34) are defined as

Kλpqrs =

∑µν

ελµν〈pµrν |qs〉. (38)

The mean field X2C Hamiltonian exactly reduces to themean field BP Hamiltonian when R = X = I.

III. HEAT-BATH CONFIGURATIONINTERACTION

Heat-bath Configuration Interaction (HCI) is arecently-developed152 variant of the general class ofmethods that perform a selected configuration interac-tion calculation followed by perturbation theory (SCI-PT)113,124,126,127,134,145,149,154,157,166,175,186,203,213,219.HCI, similar to other SCI-PT methods, consists ofa variational step and a perturbative step. In the

6

variational step, a set of important determinants isiteratively identified and the Hamiltonian is diagonalizedin the space of these determinants. In the perturbativestep, the energy obtained during the variational stepis corrected using Epstein-Nesbet perturbation theory.For details of the algorithm we refer the reader to ourrecent publications152,153,205,210, here we briefly describethe key aspects of the method relative to other SCI-PTapproaches.

A. The heat-bath criterion

At each iteration of the variational stage, the multiref-erence ground state wavefunction

|Ψ〉 =∑Di∈V

ci|Di〉 (39)

with energy E0 is calculated by diagonalizing the Hamil-tonian in the current space V of important determinants|Di〉 to obtain their coefficient ci. The space V is aug-mented with a set of new determinants |Da〉 that satisfythe HCI criterion

maxDi∈V

|Haici| > ε1, (40)

where Hai = 〈Da|H|Di〉 and ε1 is a user-defined param-eter. The HCI criterion is different from the one usedin CIPSI134,154, which is based on the contribution ofa determinant Da to the perturbative correction to thewavefunction ∣∣∣∣∣

∑|Di〉∈V Haici

E0 − Ea

∣∣∣∣∣ > ε1. (41)

Although the HCI criterion is in principle suboptimal atpicking out the most important determinants, in practicethe difference is minimal. Moreover, this slight differenceis more than made up for by the significant advantage ofthe HCI criterion, which is that one generates only thedeterminants |Da〉 that satisfy the criterion in Eq. (40),and no resources are spent on generating determinantsthat would have been discarded. The same criterion canalso be used to speed up perturbation theory step.

B. Stochastic perturbation theory

The Epstein-Nesbet perturbative correction is evalu-ated as

E2(ε2) =∑|Da〉

1

E0 − Ea

(ε2)∑|Di〉∈V

Haici

2

, (42)

where the symbol∑(ε2) designates a “screened sum” in

which terms smaller in magnitude than a user-defined

parameter ε2 are discarded and where the first sum isover the determinants |Da〉 that meet this criterion. Theexact perturbation correction is recovered in the limitof ε2 → 0, and an ε2 much smaller than ε1 is needed inorder to obtain a good approximation to the perturbativecorrection.

Although the screened sum allows one to discard alarge fraction of determinants, in most cases this is notsufficient to eliminate the memory bottleneck of havingto store all the determinants |Da〉 and their perturba-tive contributions in memory. To overcome this memorybottleneck, we use semistochastic perturbation theory205

to estimate the perturbative correction. In the semis-tochastic perturbation theory, an initial deterministiccalculation is performed with a relatively loose param-eter εd2 to obtain an approximate perturbative correctionED

2 (εd2). The error in this calculated energy is then cor-rected stochastically by performing several iterations inwhich a stochastic perturbative correction is evaluatedusing the loose parameter εd2 and a much tighter param-eter ε2. The near-exact perturbative correction with thetight parameter ε2 can then be calculated as

E2(ε2) = ED2 (εd2) +

[ES

2 (ε2)− ES2 (εd2)

]. (43)

The key to the success of the semistochastic perturba-tion theory is that both the loose and tight stochasticperturbative corrections, ES

2 (εd2) and ES2 (ε2), are calcu-

lated with the exact same set of sampled determinants.This correlated sampling significantly reduces the vari-ance in the estimated perturbative corrections, therebyreducing the stochastic error.

C. Spin-Orbit coupling and excited states

With the introduction of the spin-orbit coupling, theHamiltonian does not commute with the Sz operator andconsequently 〈Sz〉 is no longer a good quantum number.The SOC terms introduce non-zero matrix elements be-tween determinants that contain different numbers of αand β electrons, and thus break the degeneracy betweenthe 2S + 1 states of a spin multiplet. The energy split-ting can be measured experimentally to determine thezero-field splitting (ZFS), and these energy differencesare usually a small fraction of the absolute energies of themolecule. Thus two complications need to be addressedwith the HCI algorithm: forming the wavefunction con-sisting of determinants with different numbers of α andβ electrons, and the special care needed to calculate thesmall energy differences with sufficient accuracy.

To address the first complication, let us recall that thevariational step of the HCI algorithm includes three op-erations, identifying the important determinants to addto the space V, finding the non-zero Hamiltonian matrixelements between all the determinants of V and diago-nalizing the Hamiltonian. Each of these three operationsare modified due to the addition of the SOC terms. In

7

identifying the important determinants, one now also hasto include those that can be generated with an α → βor a β → α excitation. On the other hand, the advan-tageous search protocol for doubly excited determinantsthat meet the HCI criterion does not change because theSOC terms only contain one-body operators. The Hamil-tonian is stored in the sparse storage format and mostof the terms in it are constructed using the same algo-rithm as before205, which avoids having to perform theexpensive double loop over all determinants in the spaceV. Additionally, matrix elements that break Sz symme-try are identified for each determinant by looping overall possible α-occupied β-unoccupied and β-occupied α-unoccupied orbital pairs to generate new determinantsand doing a binary search over V to check if the de-terminant is present. Finally, the generalization of theDavidson algorithm to diagonalize the complex-valuedHamiltonian poses no serious problem.

Calculating the small energy splitting accurately re-quires that the energy of each state is calculated with asimilar accuracy. The standard procedure for doing so isthe state-average algorithm and was recently introducedin the context of HCI by modifying the HCI criterionof Eq. (40)153. In this work we introduce yet anotherdifferent criterion that is more suitable for targeting de-generate states. The updated HCI criterion is

maxDi∈V |Haici| > ε1

ci =√∑

n |cni |2

(44)

where ci is the square root of the sum of the squares of thecoefficients cni of determinant |Di〉 in the nth state. Thisensures that unitary transformations between degeneratestates do not change the determinants that are added toV.

IV. COMPUTATIONAL DETAILS

Several different combinations of one-body scalar rela-tivistic, two-body scalar relativistic, one-body SOC andtwo-body mean field SOC terms can be used. Here weperform calculations with 5 different Hamiltonians wherethe two-body scalar relativistic term is the Coulomb in-teraction while the other three terms are specified below:

• “bp-bp” uses the scalar relativistic, one-body andmean field two-body parts of the Breit-Pauli Hamil-tonian. This involves the integrals in Eqs. (8) and(16).

• “x2cn-bp” uses the X2C-1 scalar relativistic Hamil-tonian (Eq. (28)), the one-body part of the X2CHamiltonian derived with the X2C-N scheme andthe two-body part of the Breit-Pauli Hamiltonian.This involves terms in Eqs. (31) and (16).

• “x2cn-x2c” uses the X2C-1 scalar relativisticHamiltonian (Eq. (28)), the one-body part of the

X2C Hamiltonian derived with the X2C-N schemeand the two-body part of the X2C Hamiltonian.This involves the terms in Eqs. (31) and (33).

• “x2c1-bp” uses the X2C-1 scalar relativistic Hamil-tonian (Eq. (28)), the one-body part of the X2CHamiltonian derived with the X2C-1 scheme andthe two-body part of the Breit-Pauli Hamiltonian.This involves the integrals in Eqs. (30), (16).

• “x2c1-x2c” uses the X2C-1 scalar relativisticHamiltonian (Eq. (28)), the one- an two-body partsof the X2C Hamiltonian derived with the X2C-1scheme. This involves the integrals in Eqs. (30),and (33).

In all calculations we begin by performing state-average HCISCF210 (which is a CASSCF-like methodwhere FCI is replaced by HCI) calculations targeting therelevant states with an equal weight. These HCISCF cal-culations are performed with a scalar relativistic Hamil-tonian and SOC terms are not included at this stage. Theoptimized active space and orbitals are then used to per-form a one-step SHCI calculation where both the scalarrelativistic and SOC terms are included. Thus the spin-orbit coupling and electron correlation are treated on anequal footing during the correlated calculation. In somecalculations the initial HCISCF calculation and the sub-sequent one-step SHCI calculation are performed with adifferent active space. This is reasonable because the aimof the HCISCF calculations is to generate orbitals thatare equally weighted towards all the targeted states; therelevant SOC calculation is the one-step SHCI calcula-tion.

All calculations were performed with a combination ofthe PySCF191 and Dice codes. The BP and X2C integralsand the routines for performing HCISCF calculations areimplemented in PySCF and the SHCI calculations areperformed using the Dice code. The SOC integrals areimplemented using the automated code generator of thelibcint library214, which provides an efficient implemen-tation of the standard integrals and their nuclear deriva-tives.

In the following it is important to note that SHCI hasstochastic error associated with it, which is sometimeshown in parentheses, but is in any case much smallerthan the targeted zero-field splitting.

V. RESULTS

We present results of calculations done on the halo-gen group atoms (F, Cl, Br, I), the coinage metals groupatoms (Cu, Au) as well as on the Neptunyl(VI) diox-ide radical, NpO2+

2 , using the different Hamiltonians de-scribed in Section IV.

8

TABLE I. Zero-field splitting results (cm−1) for the atomsof the halogen group using SHCI with the ANO basis andusing several active spaces and spin-orbit coupling methods.The stochastic errors on the zero-field splitting are systemat-ically lower than 1 cm−1 and the numbers in brackets are ourestimates of the error in ZFS relative to the fully convergedresults calculated using the extrapolation scheme (see text).The results are compared to previously reported data fromRef. 196, 128 and 174 and from experimental data from Ref.167.

Method Active Spaces Ref.

F (4o,7e) (87o,7e)

bp-bp 405 399 403a

x2c1-bp 404 397 398b

x2cn-bp 404 398 421c

x2c1-x2c 405 399 404d

x2cn-x2c 405 399

Cl (4o,7e) (95o,7e) (100o,17e)

bp-bp 834 805 873 823a

x2c1-bp 825 804 867 876b

x2cn-bp 822 802 858 907c

x2c1-x2c 827 806 866(1) 882d

x2cn-x2c 825 804 865

Br (4o,7e) (95o,7e) (100o,17e)

bp-bp 3680 3655 3797 3404a

x2c1-bp 3407 3382 3432 3649b

x2cn-bp 3373 3346 3396 3723c

x2c1-x2c 3428 3403 3454(93) 3685d

x2cn-x2c 3394 3366 3415

I (4o,7e) (116o,7e) (121o,17e)

bp-bp 8816 9346 9752 6961a

x2c1-bp 6951 7199 7453 7755b

x2cn-bp 6816 7049 7246 7752c

x2c1-x2c 7021 7277 7487(62) 7603d

x2cn-x2c 6886 7127 7379a Ref. 196b EOM-CCSD(SOC) from Ref. 128c X2Cmmf-FSCCSD from Ref. 174d Exp. from Ref. 167

A. Halogens

We begin by performing a state-average HCISCF cal-culation where we target the three doubly-degenerateground states corresponding to the singly occupied px, pyor pz orbitals using a valence active space of 4 electrons in7 orbitals with the ANO basis set195. We then perform aone-step SHCI calculation including SOC terms to calcu-late the six lowest energy states. We find a 4-fold degen-erate 2P3/2 ground state and a 2-fold degenerate 2P1/2

excited state, and their energy difference corresponds to

FIG. 1. Convergence of the total energies (Ha+2604) of the2P1/2 and 2P3/2 states (blue and red dots in the upper part

of the graph) and the zero-field splitting (cm−1+3685, blackdots in the lower part of the graph) with the SHCI perturba-tive correction. The dotted lines are linear fits through thesepoints which are used to extrapolate the energies to vanishingperturbation energies. There is a small noise (less than 1%)associated with the ZFS which can be traced to a combinationof stochastic noise and uncertainty of the SHCI energies.

-0.845

-0.840

-0.835

-0.830

-0.825

-0.820

-25 -20 -15 -10 -5 0

-300

-200

-100

To

tal e

ne

rgy (

Ha

+2

60

4)

ZF

S (

cm

-1+

36

85

)

PT correction (mHa)

the zero-field splitting measured experimentally and tab-ulated in the NIST reference table of Ref. 167 (displayedin the last column of Table I).

It is important to note that the Hilbert space corre-sponding to the larger active spaces in Table I can beenormous, for example the number of determinants thatcontribute to the ground and excited state of the Iodineatom with an active space of (121o,17e) is greater than1024 (this number is 242C17, which assumes that no sym-metry other than the particle number symmetry is used).With this kind of enormous active space, the individualenergies of the targeted states are not fully converged,however, the zero-field splitting is converged to betterthan 1% of its FCI value except in the case of Br whereit is accurate to 3%. The estimated error in the energiesis calculated using the extrapolation technique describedin Ref. 153, which uses the fact that near convergence theSHCI energies are nearly perfectly linear with respect tothe SHCI perturbative correction. Such an extrapolationis shown in Figure 1, where this trend seems to hold truein presence of the SOC terms as well. Such an extrapo-lation is used to estimate the error in the ZFS energiescalculated using SHCI and is given in the bracket.

From Table I it is possible to compare the perfor-mance of the various SOC Hamiltonians. We find thatthe “x2c1-x2c” scheme delivers the most accurate ZFSenergies for most of the atoms. Interestingly, the sim-ple Breit-Pauli results are extremely accurate for loweratomic weight species including Florine and Chlorine,however the results progressively start to deteriorate, sig-nificantly over-estimating the ZFS with the increase inthe molecular weight of the atom. This is to be expected

9

TABLE II. Zero-field splitting results (eV) for the atoms of the coinage metal group using SHCI with the ANO basis and the“x2c1-x2c” spin-orbit coupling method. We report also some previous theoretical work, from Ref. 180 and Ref. 202.

States Active Spaces Ref.199

Cu CASSCF-SO180 CASPT2-SO180 DMRG-SISO202 SHCI

(11o,11e) (11o,11e) (45o,19e) (172o,19e)2D5/2 1.49 1.43 1.31 1.39 1.392D3/2 1.75 1.69 1.57 1.67 1.642D5/2 − 2D3/2 0.26 0.26 0.26 0.28 0.25

Au CASSCF-SO180 CASPT2-SO180 DMRG-SISO202 SHCI

(11o,11e) (11o,11e) (57o,43e) (150o,25e)2D5/2 1.71 0.97 1.02 1.15 1.142D3/2 3.22 2.49 2.55 2.64 2.662D5/2 − 2D3/2 1.51 1.52 1.53 1.49 1.52

because the Breit-Pauli SOC Hamiltonian is unboundedand thus over-estimates the energy splittings of a heavyatom in a variational calculation132.

The disagreement between the experimental and“x2c1-x2c” ZFS energies can be attributed to threecauses, basis set incompleteness error, neglect of core-correlation and error in the relativistic Hamiltonian. Forthe Fluorine atom the core is fully correlated and it issurprising to note that the agreement with experimentbecomes worse as we increase the active space to includethe core electrons. This has to be attributed to basis-setincompleteness and deficiencies of the relativistic treat-ment, which include neglect of various two-body scalarrelativistic and spin-orbit coupling terms and the use ofthe SOMF approximation. At the moment it is not pos-sible to disentangle the two because the current imple-mentation of SHCI in Dice only works with the one-bodySOC terms, however, in future publications we intend toaddress this point in more detail. Similar errors can beseen with other Halogens as well. In the cases of Bromineand Iodine, the improvement in agreement with the ex-periment when one increases the active space electronsfrom 7 to 17 suggests that including inner core electronsin the active space might further improve the agreementwith experiments.

B. Coinage metals

We have performed SHCI calculations on the morechallenging coinage elements, Cu and Au. In all thesecalculations we again first perform a scalar relativisticstate-average HCISCF calculation using Roos’s ANO ba-sis set197 with the 11 valence electrons in 11 orbitals (in-cluding the valence s and d orbitals along with the vir-tual d orbitals to account for the double-d shell effect)to simultaneously optimize with equal weight the 6 low-est lying states, 5 of which are degenerate, where thefive 3d and the 4s orbital are singly occupied. The ob-

tained optimized orbitals are then used to perform a one-step SHCI calculation with the “x2c1-x2c” SOC Hamil-tonian to obtain a 2-fold degenerate 2S1/2 ground state,

a 6-fold degenerate 2D5/2 state and a 4-fold degenerate2D3/2 state. Two such calculations were performed, onein which the same (11o,11e) active space was used andanother in which all the virtual orbitals were included inthe active space in addition to semi-core electrons. Wereport the result of these calculations in Table II whichalso contains the reference data from Ref. 199 and resultsof previous theoretical calculations.

It is interesting to note (see Figure 2) that although theZFS splitting of the 2D state into the 2D5/2 and 2D3/2

states, calculated by SHCI and other methods, are quiteaccurate, the excitation energies of these states relativeto the 2S ground state are very different. The agree-ment between the SHCI calculations and experiments isquite good with the maximum error being of just 0.03 eV,as opposed to other methods including the large activespace DMRG-SISO202 calculations where the error canbe as large as 0.12 eV. This difference in accuracy can beattributed to the use of the ANO-TZ basis set and theuse of perturbative treatment of the SOC terms in theDMRG-SISO calculations. We note that the accuracyof CASPT2-SO180 is significantly better than CASSCF-SO180, which points to the fact that dynamical and corecorrelations are important.

C. Neptunyl(VI)

The ability to offer insight on magnetic and chemicalproperties of lanthanide and actinide containing complexfound in single molecular magnets is important becauseexperiments on those compounds prove difficult due totheir high toxicity. Here, we perform relativistic calcula-tions using the SHCI algorithm to calculate the groundand low lying excited states of NpO2+

2 , the Neptunyl(VI)

10

FIG. 2. Errors (eV) in the individual 2D5/2 and 2D3/2 statesof the coinage atoms (bars), as well as of the ZFS betweenthe two (black line) of different methods previously reportedand of SHCI calculations from this work. Although the ZFSis well reproduced by all methods, SHCI performs well in itsdescription of the individual states.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Err

or

(eV

)

CASSCF-SO

(11o,11e)

CASPT2-SO

(11o,11e)

DMRG-SISO

(45o,19e)

HCI

(172o,19e)

CASSCF-SO

(11o,11e)

CASPT2-SO

(11o,11e)

DMRG-SISO

(57o,43e)

HCI

(150o,25e)

Cu Au

2D5/22D3/2

ZFS

TABLE III. Relative energies of the electronic states ofNpO2+

2 calculated with “x2c1-x2c” SOC method for two dif-ferent active spaces. We have also shown the results obtainedpreviously using the atomic mean field spin-orbit Hamilto-nian as implemented in Molcas and the SOC terms treatedperturbatively.

States Energies Ref.147

(4o,1e) (143o,17e)2Φ5/2 0 0 02∆3/2 4527 3857 30112Φ7/2 8568 8675 80922∆5/2 10659 10077 9192

dioxide radical147,148,164 using the ANO basis set trun-cated to the triple zeta level. The molecule has C∞vpoint group symmetry and the Np-O bond is of length1.70 A, taken from the work of Gendron et al.147. Webegin by performing a state-average HCISCF calculationwith a (4o,1e) active space in which the energies of thedoubly degenerate 2Φ and 2∆ states are optimized usingthe X2C scalar relativistic Hamiltonian and we find anenergy splitting equal to 2108 cm−1. The energy differ-ence between these two states is indicative of the strengthof the crystal field splitting since in the absence of theaxial oxygen atoms and relativistic effects, these stateswould be exactly degenerate. Next we perform the one-step SHCI calculations with two different active spaces,(4o,1e) and (143o,17e), to obtain the 4 doubly degeneratestates which are shown in Table III. It is interesting tonote that with the use of the same active space and geom-etry, the ZFS calculated here using the X2C Hamiltonianis significantly different from the one calculated by Gen-dron et al.147, where the DKH Hamiltonian along with

atomic mean field spin-orbit coupling as implemented inMolcas was used. The ZFS splitting changes very littlerelative to this result even when slightly different geom-etry was used by Knecht et al.164. The difference be-tween our results and those of Gendron and Knecht canbe attributed to the difference in the Hamiltonian usedto treat the relativistic effect. For the Np atom, it willcertainly be interesting to perform a full four-componentcalculation with the DCB Hamiltonian to calculate theZFS and compare the accuracy of the various approxi-mate two-component Hamiltonians. Such an MRCI cal-culation was performed by Knecht, however only the cal-culated g-tensors was reported, and was in good agree-ment with the DKH results calculated there. It shouldbe pointed out that even though the energy differencesare quite different, we find that the ground state energycalculated with the (4o,1e) active space consists of 89%2Φ state and 11% 2∆ state, in agreement with the re-sults of Knecht et al.. This indicates that the g-tensorscalculated using SHCI would most likely have agreed wellwith those of DKH and four-component MRCI calcula-tions performed by Knecht et al..

FIG. 3. Energies of the electronic states of NpO2+2 , relative

to the SOC ground-state energy. On the left are spin-freeand spin-orbit coupling states calculated using SHCI and the“x2c1-x2c” method, and an active space of (4o,1e) (black) and(143o,17e) (red), and on the right are reference data from Ref.147.

−2000

0

2000

4000

6000

8000

10000

12000

Rela

tive

ener

gy (c

m−1

) 2 ∆

2 ∆3/2

2 Φ7/2

2 ∆5/2

2 Φ

2 Φ5/2

2 Φ5/2

2 ∆3/2

2 Φ7/2

2 ∆5/2

2 Φ

2 ∆

spin-free SOC SOC spin-freeHCI Ref.

VI. CONCLUSION

In this work we have extended the SHCI algorithm totreat the relativistic Hamiltonians containing spin-orbitcoupling terms, by allowing one-body integrals and con-figuration interaction coefficients to be complex-valuednumbers. The agreement with experimental results forseveral atomic species has been shown to be quite goodand superior to previously calculated results. This is be-cause not only are we treating large active spaces but we

11

are also treating the electron correlation and relativisticeffects on an equal footing. The main shortcoming of thecurrent methodology is that we cannot effectively includecore correlation and treat larger systems because we donot have the ability to add dynamical correlation. Workin this direction is under way and we are looking for waysto combine SHCI with the recently-developed multiref-erence linearized coupled cluster theory204,206,207, whichis formulated as a perturbation theory and uses Fink’spartitioning136,137 of the Hamiltonian. We are also look-ing to extend the methodology so that the relativisticeffects can already be included at the self-consistent field

cycle to obtain complex-valued spinors. This will reducethe burden on the correlated calculations because a largepart of the relativistic effects will be treated at the SCFlevel.

ACKNOWLEDGMENTS

The research was supported through the startup pack-age from the University of Colorado, Boulder. We wouldalso like to thank several useful discussions and insightfulcomments from Zhendong Li, Qiming Sun and JuergenGauss.

∗ bastien.mussard@colorado.edu† sanshar@gmail.com1 M. L. Abrams and C. D. Sherrill. Important configu-

rations in configuration interaction and coupled-clusterwave functions. Chemical Physics Letters, 412(1-3):121–124, 2005.

2 A. Almoukhalalati, S. Knecht, H. Jørgen, A. Jensen, K. G.Dyall, T. Saue, A. Almoukhalalati, S. Knecht, H. Jørgen,A. Jensen, and K. G. Dyall. Electron correlation withinthe relativistic no-pair approximation Electron correlationwithin the relativistic no-pair approximation. The Journalof Chemical Physics, 145:074104, 2016.

3 F. Aquilante, J. Autschbach, R. K. Carlson, L. F. Chibo-taru, M. G. Delcey, L. De Vico, I. Fdez. Galvan, N. Ferre,L. M. Frutos, L. Gagliardi, M. Garavelli, A. Giussani,C. E. Hoyer, G. Li Manni, H. Lischka, D. Ma, P. A.Malmqvist, T. Muller, A. Nenov, M. Olivucci, T. B.Pedersen, D. Peng, F. Plasser, B. Pritchard, M. Rei-her, I. Rivalta, I. Schapiro, J. Segarra-Martı, M. Sten-rup, D. G. Truhlar, L. Ungur, A. Valentini, S. Vancoil-lie, V. Veryazov, V. P. Vysotskiy, O. Weingart, F. Zap-ata, and R. Lindh. Molcas 8: New capabilities for mul-ticonfigurational quantum chemical calculations acrossthe periodic table. Journal of Computational Chemistry,37(5):506–541, feb 2016.

4 M. Barysz. Relativistic Methods for Chemists. InM. Barysz and Y. Ishikawa, editors, Relativistic Methodsfor Chemists, chapter Two-Compon. Springer, 2010.

5 M. Barysz, L. Mentel, and J. Leszczynski. Recoveringfour-component solutions by the inverse transformationof the infinite-order two-component wave functions. TheJournal of Chemical Physics, 130(16):164114, apr 2009.

6 M. Barysz and A. J. Sadlej. Infinite-order two-componenttheory for relativistic quantum chemistry. The Journal ofChemical Physics, 116(7):2696–2704, feb 2002.

7 J. E. Bates and T. Shiozaki. Fully relativistic completeactive space self-consistent field for large molecules : Fullyrelativistic complete active space self-consistent field forlarge molecules : Quasi-second-order minimax optimiza-tion. The Journal of Chemical Physics, 142:044112, 2015.

8 L. Belpassi, L. Storchi, M. Quiney, and F. Tarantelli. Re-cent advances and perspectives in four-component DiracKohn Sham calculations. Physical Chemistry ChemicalPhysics, 13:12368, 2011.

9 U. o. S. Bernd Schimmelpfennig. AMFI, an atomic mean-field spin-orbit integral program, 1996.

10 A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles,and P. Palmieri. Spin-orbit matrix elements for inter-nally contracted multireference configuration interactionwavefunctions. Molecular Physics, 98(21):1823–1833, nov2000.

11 R. J. Buenker, A. B. Alekseyev, H.-P. Liebermann, R. Lin-gott, and G. Hirsch. Comparison of spin-orbit configura-tion interaction methods employing relativistic effectivecore potentials for the calculation of zero-field splittingsof heavy atoms with a 2Po ground state. The Journal ofChemical Physics, 108(9):3400–3408, mar 1998.

12 R. J. Buenker and S. D. Peyerimhoff. Individualized con-figuration selection in CI calculations with subsequent en-ergy extrapolation. Theoretica Chimica Acta, 35(1):33–58, 1974.

13 M. Bylicki, G. Pestka, and J. Karwowski. Relativis-tic Hylleraas configuration-interaction method projectedinto positive-energy space. Physical Review A, 77(Febru-ary):044501, 2008.

14 L. Bytautas and K. Ruedenberg. A priori identificationof configurational deadwood. Chemical Physics, 356(1-3):64–75, 2009.

15 M. Caffarel, T. Applencourt, E. Giner, and A. Scemama.Using CIPSI Nodes in Diffusion Monte Carlo. ACS Sym-posium Series, 1234(Dmc):15–46, 2016.

16 Z. Cao, Z. Li, F. Wang, and W. Liu. Combining thespin-separated exact two-component relativistic Hamilto-nian with the equation-of-motion coupled-cluster methodfor the treatment of spin-orbit splittings of light andheavy elements. Physical Chemistry Chemical Physics,19(5):3713–3721, 2017.

17 L. Cheng and J. Gauss. Perturbative treatment of spin-orbit coupling within spin-free exact two-component the-ory. The Journal of Chemical Physics, 141(16):164107,2014.

18 M. Douglas and N. M. Kroll. Quantum electrodynami-cal corrections to the fine structure of helium. Annals ofPhysics, 82(1):89–155, 1974.

19 K. Dyall and K. J. Faegri. Introduction to RelativisticQuantum. Oxford University Press, 2007.

20 K. G. Dyall. Interfacing relativistic and nonrelativisticmethods. I. Normalized elimination of the small com-ponent in the modified Dirac equation. The Journal of

12

Chemical Physics, 106(23):9618–9626, jun 1997.21 M. Esser, W. Butscher, and W. H. E. Schwarz. Complex

two- and four-index transformation over two-componentkramers-degenerate spinors. Chemical Physics Letters,77(2):359–364, 1981.

22 S. Evangelisti, J. P. Daudey, and J. P. Malrieu. Conver-gence of an improved CIPSI algorithm. Chemical Physics,75:91–102, 1983.

23 D. G. Fedorov and M. S. Gordon. A study of the rel-ative importance of one and two-electron contributionsto spinorbit coupling. The Journal of Chemical Physics,112(13):5611, 2000.

24 R. F. Fink. Two new unitary-invariant and size-consistentperturbation theoretical approaches to the electron corre-lation energy. Chemical Physics Letters, 428(4-6):461–466, 2006.

25 R. F. Fink. The multi-reference retaining the excitationdegree perturbation theory: A size-consistent, unitary in-variant, and rapidly convergent wavefunction based abinitio approach. Chemical Physics, 356(1-3):39–46, 2009.

26 T. Fleig. Invited review: Relativistic wave-function basedelectron correlation methods. Chemical Physics, 395,2012.

27 T. Fleig, H. J. A. Jensen, J. Olsen, and L. Visscher. Thegeneralized active space concept for the relativistic treat-ment of electron correlation . III . Large-scale configura-tion interaction and multiconfiguration self- consistent-field four-component methods with application to Thegeneralized active space. The Journal of ChemicalPhysics, 124:104106, 2006.

28 T. Fleig, J. Olsen, and C. M. Marian. The generalized ac-tive space concept for the relativistic treatment of electroncorrelation . I . Kramers-restricted two-component con-figuration interaction. The Journal of Chemical Physics,114:4775, 2001.

29 T. Fleig, J. Olsen, and L. Visscher. The generalized ac-tive space concept for the relativistic treatment of electroncorrelation . II . Large-scale configuration interaction im-plementation based on relativistic 2- and 4-spinors and itsapplication The generalized active space concept for the.The Journal of Chemical Physics, 119:2963, 2003.

30 L. L. Foldy and S. A. Wouthuysen. On the Dirac Theory ofSpin 1/2 Particles and Its Non-Relativistic Limit. PhysicalReview, 78(1):29–36, apr 1950.

31 D. Ganyushin and F. Neese. First-principles calculationsof zero-field splitting parameters. The Journal of Chemi-cal Physics, 125(2):24103, 2006.

32 D. Ganyushin and F. Neese. A fully variational spin-orbit coupled complete active space self-consistent fieldapproach: application to electron paramagnetic reso-nance g-tensors. The Journal of Chemical Physics,138(10):104113, 2013.

33 Y. Garniron, A. Scemama, P.-F. Loos, and M. Caf-farel. Hybrid stochastic-deterministic calculation of thesecond-order perturbative contribution of multireferenceperturbation theory. The Journal of Chemical Physics,147(3):034101, 2017.

34 J. Gauss, M. Kallay, and F. Neese. Calculation of Elec-tronic g -Tensors using Coupled Cluster Theory . TheJournal of Physical Chemistry A, 113(43):11541–11549,oct 2009.

35 F. Gendron, D. Paez-Hernandez, F. P. Notter,B. Pritchard, H. Bolvin, and J. Autschbach. Magneticproperties and electronic structure of neptunyl(VI) com-

plexes: Wavefunctions, orbitals, and crystal-field models.Chemistry - A European Journal, 20(26):7994–8011, 2014.

36 F. Gendron, B. Pritchard, H. Bolvin, and J. Autschbach.Magnetic Resonance Properties of Actinyl CarbonateComplexes and Plutonyl(VI)-tris-nitrate. InorganicChemistry, 53(16):8577–8592, aug 2014.

37 R. J. Harrison. Approximating full configuration interac-tion with selected configuration interaction and perturba-tion theory. The Journal of Chemical Physics, 94(7):5021–5031, 1991.

38 B. A. Hess. Relativistic electronic-structure calcula-tions employing a two-component no-pair formalism withexternal-field projection operators. Physical Review A,33(6):3742–3748, jun 1986.

39 B. A. Heß, C. M. Marian, U. Wahlgren, and O. Gropen.A mean-field spin-orbit method applicable to correlatedwavefunctions. Chemical Physics Letters, 251(5):365–371,mar 1996.

40 A. A. Holmes, N. M. Tubman, and C. J. Umrigar. Heat-bath configuration interaction: An efficient selected ci al-gorithm inspired by heat-bath sampling. The Journal ofChemical Theory and Computation, 12:3674, 2016.

41 A. A. Holmes, C. J. Umrigar, and S. Sharma. Excitedstates using semistochastic heat-bath configuration inter-action. arXiv preprint arXiv:1708.03456, 2017.

42 B. Huron. Iterative perturbation calculations of groundand excited state energies from multiconfigurationalzeroth-order wavefunctions. The Journal of ChemicalPhysics, 58(12):5745, 1973.

43 M. Ilias, A. J. H. Jensen, V. Kelo, B. O. Roos, and M. Ur-ban. Theoretical study of PbO and the PbO anion. Chem-ical Physics Letters, 408(4-6):210–215, jun 2005.

44 M. Ilias and T. Saue. An infinite-order two-componentrelativistic Hamiltonian by a simple one-step transforma-tion. The Journal of Chemical Physics, 126(6):64102, feb2007.

45 J. Ivanic and K. Ruedenberg. Identification of dead-wood in configuration spaces through general direct con-figuration interaction. Theoretical Chemistry Accounts,106(5):339–351, 2001.

46 M. S. Kelley, T. Shiozaki, M. S. Kelley, and T. Shiozaki.Large-scale Dirac Fock Breit method using density fittingand 2-spinor basis functions Large-scale Dirac Fock Breitmethod using density fitting and 2-spinor basis functions.The Journal of Chemical Physics, 138:204113, 2013.

47 I. Kim and Y. S. Lee. Two-component multi-configurational second-order perturbation theory withkramers restricted complete active space self-consistentfield reference function and spin-orbit relativistic effec-tive core potential. The Journal of Chemical Physics,141(16):164104, 2014.

48 I. Kim, Y. C. Park, H. Kim, and Y. S. Lee. Spin-orbitcoupling and electron correlation in relativistic configura-tion interaction and coupled-cluster methods. ChemicalPhysics, 395(1):115–121, 2012.

49 K. Klein and J. Gauss. Perturbative calculation of spin-orbit splittings using the equation-of-motion ionization-potential coupled-cluster ansatz. The Journal of ChemicalPhysics, 129(19):194106, nov 2008.

50 M. Kleinschmidt, J. Tatchen, and C. M. Marian.SPOCK.CI: A multireference spin-orbit configuration in-teraction method for large molecules. The Journal ofChemical Physics, 124(12):124101, mar 2006.

13

51 S. Knecht, A. J. H. Jensen, and A. Jensen. group gen-eral active space implementation with application to BiHLarge-scale parallel configuration interaction . II . Two-and four-component double-group general active spaceimplementation with application to BiH. The Journal ofChemical Physics, 132:014108, 2010.

52 S. Knecht, S. Keller, J. Autschbach, and M. Reiher.A Nonorthogonal State-Interaction Approach for MatrixProduct State Wave Functions. Journal of Chemical The-ory and Computation, 12(12):5881–5894, 2016.

53 S. Knecht, O. Legeza, and M. Reiher. Communication:Four-component density matrix renormalization group.The Journal of Chemical Physics, 140(4):41101, jan 2014.

54 P. J. Knowles. Compressive sampling in configurationinteraction wavefunctions. Molecular Physics, 113(13-14):1655–1660, 2015.

55 A. Kramida, Y. Ralchenko, J. Reader, and N. A. S. D.Team. NIST Atomic Spectra Database (version 5.4).http://physics.nist.gov/asd.

56 W. Kutzelnigg. Solved and unsolved problems in rela-tivistic quantum chemistry. Chemical Physics, 395:16–34,2012.

57 W. Kutzelnigg and W. Liu. Quasirelativistic theory equiv-alent to fully relativistic theory. The Journal of ChemicalPhysics, 123(24):241102, dec 2005.

58 S. R. Langhoff. Ab initio evaluation of the fine structureof the oxygen molecule. The Journal of Chemical Physics,61(5):1708–1716, sep 1974.

59 Z. Li, S. Shao, and W. Liu. Relativistic explicit corre-lation : Coalescence conditions and practical suggestionsRelativistic explicit correlation : Coalescence conditions.The Journal of Chemical Physics, 136:144117, 2012.

60 Z. Li, B. Suo, Y. Zhang, Y. Xiao, and W. Liu. Combiningspin-adapted open-shell TD-DFT with spinorbit coupling.Molecular Physics, 111(24):3741–3755, dec 2013.

61 Z. Li, Y. Xiao, and W. Liu. On the spin separation ofalgebraic two-component relativistic Hamiltonians. TheJournal of Chemical Physics, 137(5):154114, oct 2012.

62 Z. Li, Y. Xiao, and W. Liu. On the spin separa-tion of algebraic two-component relativistic Hamiltonians:Molecular properties. The Journal of Chemical Physics,141(5):54111, 2014.

63 J. Liu, Y.-C. Chen, J.-L. Liu, V. Vieru, L. Ungur, J.-H.Jia, L. F. Chibotaru, Y. Lan, W. Wernsdorfer, S. Gao, X.-M. Chen, and M.-L. Tong. A Stable Pentagonal Bipyra-midal Dy(III) Single-Ion Magnet with a Record Magneti-zation Reversal Barrier over 1000 K. Journal of the Amer-ican Chemical Society, 138(16):5441–5450, apr 2016.

64 W. Liu. Ideas of relativistic quantum chemistry. Molecu-lar Physics, 108(13):1679–1706, 2010.

65 W. Liu and W. Kutzelnigg. Quasirelativistic theory. II.Theory at matrix level. The Journal of Chemical Physics,126(11):114107, mar 2007.

66 W. Liu and D. Peng. Exact two-component Hamiltoniansrevisited. The Journal of Chemical Physics, 131(3):31104,jul 2009.

67 S. Mai, T. Muller, F. Plasser, P. Marquetand, H. Lis-chka, and L. Gonzalez. Perturbational treatment of spin-orbit coupling for generally applicable high-level multi-reference methods. The Journal of Chemical Physics,141(7):74105, aug 2014.

68 P. A. Malmqvist, B. O. Roos, and B. Schimmelpfennig.The restricted active space (RAS) state interaction ap-

proach with spinorbit coupling. Chemical Physics Letters,357(3):230–240, 2002.

69 C. M. Marian and U. Wahlgren. A new mean-field andECP-based spin-orbit method. Applications to Pt andPtH. Chemical Physics Letters, 251(5-6):357–364, mar1996.

70 W. Mizukami, T. Nakajima, K. Hirao, and T. Yanai.A dual-level approach to four-component relativisticdensity-functional theory. Chemical Physics Letters,508(1-3):177–181, may 2011.

71 Myeong Cheol Kim, Sang Yeon Lee, and Yoon Sup Lee.Spin-orbit effects calculated by a configuration interac-tion method using determinants of two-component molec-ular spinors: test calculations on Rn and T1H. ChemicalPhysics Letters, 253(3):216–222, 1996.

72 T. Nakajima and K. Hirao. The DouglasKrollHess Ap-proach. Chemical Reviews, 112(1):385–402, jan 2012.

73 H. S. Nataraj, M. Kallay, L. Visscher, H. S. Nataraj,M. Kallay, and L. Visscher. General implementation of therelativistic coupled-cluster method General implementa-tion of the relativistic coupled-cluster method. The Jour-nal of Chemical Physics, 133:234109, 2010.

74 F. Neese. A spectroscopy oriented configuration inter-action procedure. The Journal of Chemical Physics,119(18):9428, 2003.

75 F. Neese. The ORCA program system. Wiley Inter-disciplinary Reviews: Computational Molecular Science,2(1):73–78, 2012.

76 M. Pernpointner, L. Visscher, W. A. de Jong, andR. Broer. Parallelization of four-component calculations.I. Integral generation, SCF, and four-index transforma-tion in the DiracFock package MOLFDIR. Journal ofComputational Chemistry, 21(13):1176–1186, 2000.

77 P. Pyykko. Relativistic Effects in Chemistry: More Com-mon Than You Thought. Annual Review of PhysicalChemistry, 63(1):45–64, 2012.

78 P. Pyykko. The Physics behind Chemistry and the Peri-odic Table. Chemical Reviews, 112(1):371–384, 2012.

79 S. Qiming, T. C. Berkelbach, N. S. Blunt, G. H. Booth,S. Guo, Z. Li, J. Liu, J. McClain, E. R. Sayfutyarova,S. Sharma, S. Wouters, and G. K.-L. Chan. The Python-based Simulations of Chemistry Framework (PySCF).arXiv preprint arXiv:1701.08223, jan 2017.

80 M. Reiher. DouglasKrollHess Theory: a relativisticelectrons-only theory for chemistry. Theoretical Chem-istry Accounts, 116(1):241–252, 2006.

81 M. Reiher and A. Wolf. Relativistic Quantum Chemistry,Second Edition. Wiley Online Library, 2014.

82 M. Roemelt. Spin orbit coupling for molecular ab initiodensity matrix renormalization group calculations: Ap-plication to g-tensors. The Journal of Chemical Physics,143(4):044112, jul 2015.

83 B. O. Roos, R. Lindh, P.-A. Malmqvist, V. Veryazov, andP.-O. Widmark. Main Group Atoms and Dimers Studiedwith a New Relativistic ANO Basis Set. The Journal ofPhysical Chemistry A, 108(15):2851–2858, apr 2004.

84 B. O. Roos and P.-A. Malmqvist. Relativistic quantumchemistry: the multiconfigurational approach. PhysicalChemistry Chemical Physics, 6(11):2919–2927, 2004.

85 B. O. Roos, Roland Lindh, Per-Ake Malmqvist, V. Verya-zov, and P.-O. Widmark. New Relativistic ANO BasisSets for Transition Metal Atoms. Journal of PhysicalChemistry A, 109:6575, 2005.

14

86 M. S. Gordon and M. W. Schmidt. Advances in Elec-tronic Structure Theory: GAMESS a Decade Later. InC. E. Dykstra, G. Frenking, K. S. Kim, and G. E. Scuse-ria, editors, Theory and Applications of ComputationalChemistry, pages 1167–1189. Elsevier, Amsterdam, 2005,2005.

87 J. E. Sansonetti and W. C. Martin. Hand-book of basic atomic spectroscopic data.http://www.nist.gov/pml/data/handbook/index.cfm.

88 T. Saue. Relativistic Hamiltonians for Chemistry: APrimer. Chem. Phys. Chem., 12(17):3077–3094, 2011.

89 T. Saue and T. Helgaker. Four-component relativisticKohnSham theory. Journal of Computational Chemistry,23(8):814–823, 2002.

90 E. R. Sayfutyarova and G. K.-L. Chan. A state inter-action spin-orbit coupling density matrix renormaliza-tion group method. The Journal of Chemical Physics,144(23):234301, jun 2016.

91 J. B. Schriber and F. A. Evangelista. Communication: Anadaptive configuration interaction approach for stronglycorrelated electrons with tunable accuracy. The Journalof Chemical Physics, 144(16):161106, 2016.

92 S. Sharma and A. Alavi. Multireference linearized cou-pled cluster theory for strongly correlated systems usingmatrix product states. The Journal of Chemical Physics,143(10):102815, 2015.

93 S. Sharma, A. A. Holmes, G. Jeanmairet, A. Alavi, andC. J. Umrigar. Semistochastic heat-bath configurationinteraction method: selected configuration interactionwith semistochastic perturbation theory. The Journalof Chemical Theory and Computation, 13(4):1595–1604,2017.

94 S. Sharma, G. Jeanmairet, and A. Alavi. Quasi-degenerate perturbation theory using matrix productstates. The Journal of Chemical Physics, 144(3):034103,2016.

95 S. Sharma, G. Knizia, S. Guo, and A. Alavi. Combininginternally contracted states and matrix product states toperform multireference perturbation theory. Journal ofChemical Theory and Computation, 13(2):488–498, 2017.PMID: 28060507.

96 T. Shiozaki and W. Mizukami. Relativistic InternallyContracted Multireference Electron Correlation Meth-ods. Journal of Chemical Theory and Computation,11(10):4733–4739, oct 2015.

97 M. Sjøvoll, O. Gropen, and J. Olsen. A determinantal ap-proach to spin-orbit configuration interaction. TheoreticalChemistry Accounts, 97(1):301–312, 1997.

98 J. E. T. Smith, A. A. Holmes, B. Mussard, and S. Sharma.Cheap and near exact CASSCF with large active spaces.arXiv preprint arXiv:1708.07544v2, 2017.

99 L. K. Sørensen, J. Olsen, and T. Fleig. Two- andfour-component relativistic generalized-active-space cou-pled cluster method : Implementation and applicationto BiH Two- and four-component relativistic generalized-active-space coupled cluster method : Implementationand application to BiH. The Journal of Chemical Physics,134:214102, 2011.

100 R. E. Stanton and S. Havriliak. Kinetic balance: A partialsolution to the problem of variational safety in Dirac cal-culations. The Journal of Chemical Physics, 81(4):1910–1918, 1984.

101 M. M. Steiner, W. Wenzel, K. G. Wilson, and J. W.Wilkins. The efficient treatment of higher excitations in

CI calculations. A comparison of exact and approximateresults. Chemical Physics Letters, 231(2-3):263–268, 1994.

102 Q. Sun. Libcint: An efficient general integral libraryfor Gaussian basis functions. Journal of ComputationalChemistry, 36:1664–1671, 2015.

103 S. Ten-no and D. Yamaki. Communication : Explic-itly correlated four-component relativistic second-orderMøller- Plesset perturbation theory Communication :Explicitly correlated four-component relativistic second-order Møller-Plesset perturbation theory. The Journal ofChemical Physics, 137:131101, 2012.

104 J. Thyssen, T. Fleig, and A. J. H. Jensen. A di-rect relativistic four-component multiconfiguration self-consistent-field method for molecules. The Journal ofChemical Physics, 129:034109, 2008.

105 O. Vahtras, O. Loboda, B. Minaev, H. Agren, andK. Ruud. Ab initio calculations of zero-field splitting pa-rameters. Chemical Physics, 279(2-3):133–142, 2002.

106 V. Vallet, L. Maron, C. Teichteil, and J.-P. Fla-ment. A two-step uncontracted determinantal effectiveHamiltonian-based SOCI method. The Journal of Chem-ical Physics, 113(4):1391–1402, jul 2000.

107 W. Wenzel, M. M. Steiner, and K. G. Wilson. Multirefer-ence Basis-Set Reduction. International Journal of Quan-tum Chemistry, 30:1325–1330, 1996.

108 H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby,M. Schutz, and Others. MOLPRO, version 2015.1, a pack-age of ab initio programs, 2015.

109 S. Yabushita, Z. Zhang, and R. M. Pitzer. SpinOrbit Con-figuration Interaction Using the Graphical Unitary GroupApproach and Relativistic Core Potential and SpinOr-bit Operators. The Journal of Physical Chemistry A,103(29):5791–5800, jul 1999.

110 T. Yanai, H. Iikura, T. Nakajima, Y. Ishikawa, and K. Hi-rao. A new implementation of four-component relativis-tic density functional method for heavy-atom polyatomicsystems. The Journal of Chemical Physics, 115(18):8267–8273, oct 2001.

111 T. Yanai, T. Nakajima, Y. Ishikawa, and K. Hirao. A newcomputational scheme for the DiracHartreeFock methodemploying an efficient integral algorithm. The Journal ofChemical Physics, 114(15):6526–6538, apr 2001.

112 T. Yanai, H. Nakano, T. Nakajima, T. Tsuneda, S. Hi-rata, Y. Kawashima, Y. Nakao, M. Kamiya, H. Sekino,and K. Hirao. UTChem A Program for ab initio Quan-tum Chemistry. In P. M. A. Sloot, D. Abramson, A. V.Bogdanov, Y. E. Gorbachev, J. J. Dongarra, and A. Y.Zomaya, editors, Computational Science, pages 84–95.Springer Berlin Heidelberg, Berlin, Heidelberg, 2003.

113 M. L. Abrams and C. D. Sherrill. Important configu-rations in configuration interaction and coupled-clusterwave functions. Chemical Physics Letters, 412(1-3):121–124, 2005.

114 A. Almoukhalalati, S. Knecht, H. Jørgen, A. Jensen, K. G.Dyall, T. Saue, A. Almoukhalalati, S. Knecht, H. Jørgen,A. Jensen, and K. G. Dyall. Electron correlation withinthe relativistic no-pair approximation Electron correlationwithin the relativistic no-pair approximation. The Journalof Chemical Physics, 145:074104, 2016.

115 F. Aquilante, J. Autschbach, R. K. Carlson, L. F. Chibo-taru, M. G. Delcey, L. De Vico, I. Fdez. Galvan, N. Ferre,L. M. Frutos, L. Gagliardi, M. Garavelli, A. Giussani,C. E. Hoyer, G. Li Manni, H. Lischka, D. Ma, P. A.Malmqvist, T. Muller, A. Nenov, M. Olivucci, T. B.

15

Pedersen, D. Peng, F. Plasser, B. Pritchard, M. Rei-her, I. Rivalta, I. Schapiro, J. Segarra-Martı, M. Sten-rup, D. G. Truhlar, L. Ungur, A. Valentini, S. Vancoil-lie, V. Veryazov, V. P. Vysotskiy, O. Weingart, F. Zap-ata, and R. Lindh. Molcas 8: New capabilities for mul-ticonfigurational quantum chemical calculations acrossthe periodic table. Journal of Computational Chemistry,37(5):506–541, feb 2016.

116 M. Barysz. Relativistic Methods for Chemists. InM. Barysz and Y. Ishikawa, editors, Relativistic Methodsfor Chemists, chapter Two-Compon. Springer, 2010.

117 M. Barysz, L. Mentel, and J. Leszczynski. Recoveringfour-component solutions by the inverse transformationof the infinite-order two-component wave functions. TheJournal of Chemical Physics, 130(16):164114, apr 2009.

118 M. Barysz and A. J. Sadlej. Infinite-order two-componenttheory for relativistic quantum chemistry. The Journal ofChemical Physics, 116(7):2696–2704, feb 2002.

119 J. E. Bates and T. Shiozaki. Fully relativistic completeactive space self-consistent field for large molecules : Fullyrelativistic complete active space self-consistent field forlarge molecules : Quasi-second-order minimax optimiza-tion. The Journal of Chemical Physics, 142:044112, 2015.

120 L. Belpassi, L. Storchi, M. Quiney, and F. Tarantelli. Re-cent advances and perspectives in four-component DiracKohn Sham calculations. Physical Chemistry ChemicalPhysics, 13:12368, 2011.

121 U. o. S. Bernd Schimmelpfennig. AMFI, an atomic mean-field spin-orbit integral program, 1996.

122 A. Berning, M. Schweizer, H.-J. Werner, P. J. Knowles,and P. Palmieri. Spin-orbit matrix elements for inter-nally contracted multireference configuration interactionwavefunctions. Molecular Physics, 98(21):1823–1833, nov2000.

123 R. J. Buenker, A. B. Alekseyev, H.-P. Liebermann, R. Lin-gott, and G. Hirsch. Comparison of spin-orbit configura-tion interaction methods employing relativistic effectivecore potentials for the calculation of zero-field splittingsof heavy atoms with a 2Po ground state. The Journal ofChemical Physics, 108(9):3400–3408, mar 1998.

124 R. J. Buenker and S. D. Peyerimhoff. Individualized con-figuration selection in CI calculations with subsequent en-ergy extrapolation. Theoretica Chimica Acta, 35(1):33–58, 1974.

125 M. Bylicki, G. Pestka, and J. Karwowski. Relativis-tic Hylleraas configuration-interaction method projectedinto positive-energy space. Physical Review A, 77(Febru-ary):044501, 2008.

126 L. Bytautas and K. Ruedenberg. A priori identificationof configurational deadwood. Chemical Physics, 356(1-3):64–75, 2009.

127 M. Caffarel, T. Applencourt, E. Giner, and A. Scemama.Using CIPSI Nodes in Diffusion Monte Carlo. ACS Sym-posium Series, 1234(Dmc):15–46, 2016.

128 Z. Cao, Z. Li, F. Wang, and W. Liu. Combining thespin-separated exact two-component relativistic Hamilto-nian with the equation-of-motion coupled-cluster methodfor the treatment of spin-orbit splittings of light andheavy elements. Physical Chemistry Chemical Physics,19(5):3713–3721, 2017.

129 L. Cheng and J. Gauss. Perturbative treatment of spin-orbit coupling within spin-free exact two-component the-ory. The Journal of Chemical Physics, 141(16):164107,2014.

130 M. Douglas and N. M. Kroll. Quantum electrodynami-cal corrections to the fine structure of helium. Annals ofPhysics, 82(1):89–155, 1974.

131 K. Dyall and K. J. Faegri. Introduction to RelativisticQuantum. Oxford University Press, 2007.

132 K. G. Dyall. Interfacing relativistic and nonrelativisticmethods. I. Normalized elimination of the small com-ponent in the modified Dirac equation. The Journal ofChemical Physics, 106(23):9618–9626, jun 1997.

133 M. Esser, W. Butscher, and W. H. E. Schwarz. Complextwo- and four-index transformation over two-componentkramers-degenerate spinors. Chemical Physics Letters,77(2):359–364, 1981.

134 S. Evangelisti, J. P. Daudey, and J. P. Malrieu. Conver-gence of an improved CIPSI algorithm. Chemical Physics,75:91–102, 1983.

135 D. G. Fedorov and M. S. Gordon. A study of the rel-ative importance of one and two-electron contributionsto spinorbit coupling. The Journal of Chemical Physics,112(13):5611, 2000.

136 R. F. Fink. Two new unitary-invariant and size-consistentperturbation theoretical approaches to the electron corre-lation energy. Chemical Physics Letters, 428(4-6):461–466, 2006.

137 R. F. Fink. The multi-reference retaining the excitationdegree perturbation theory: A size-consistent, unitary in-variant, and rapidly convergent wavefunction based abinitio approach. Chemical Physics, 356(1-3):39–46, 2009.

138 T. Fleig. Invited review: Relativistic wave-function basedelectron correlation methods. Chemical Physics, 395,2012.

139 T. Fleig, H. J. A. Jensen, J. Olsen, and L. Visscher. Thegeneralized active space concept for the relativistic treat-ment of electron correlation . III . Large-scale configura-tion interaction and multiconfiguration self- consistent-field four-component methods with application to Thegeneralized active space. The Journal of ChemicalPhysics, 124:104106, 2006.

140 T. Fleig, J. Olsen, and C. M. Marian. The generalized ac-tive space concept for the relativistic treatment of electroncorrelation . I . Kramers-restricted two-component con-figuration interaction. The Journal of Chemical Physics,114:4775, 2001.

141 T. Fleig, J. Olsen, and L. Visscher. The generalized ac-tive space concept for the relativistic treatment of electroncorrelation . II . Large-scale configuration interaction im-plementation based on relativistic 2- and 4-spinors and itsapplication The generalized active space concept for the.The Journal of Chemical Physics, 119:2963, 2003.

142 L. L. Foldy and S. A. Wouthuysen. On the Dirac Theory ofSpin 1/2 Particles and Its Non-Relativistic Limit. PhysicalReview, 78(1):29–36, apr 1950.

143 D. Ganyushin and F. Neese. First-principles calculationsof zero-field splitting parameters. The Journal of Chemi-cal Physics, 125(2):24103, 2006.

144 D. Ganyushin and F. Neese. A fully variational spin-orbit coupled complete active space self-consistent fieldapproach: application to electron paramagnetic reso-nance g-tensors. The Journal of Chemical Physics,138(10):104113, 2013.

145 Y. Garniron, A. Scemama, P.-F. Loos, and M. Caf-farel. Hybrid stochastic-deterministic calculation of thesecond-order perturbative contribution of multireferenceperturbation theory. The Journal of Chemical Physics,

16

147(3):034101, 2017.146 J. Gauss, M. Kallay, and F. Neese. Calculation of Elec-

tronic g -Tensors using Coupled Cluster Theory . TheJournal of Physical Chemistry A, 113(43):11541–11549,oct 2009.

147 F. Gendron, D. Paez-Hernandez, F. P. Notter,B. Pritchard, H. Bolvin, and J. Autschbach. Magneticproperties and electronic structure of neptunyl(VI) com-plexes: Wavefunctions, orbitals, and crystal-field models.Chemistry - A European Journal, 20(26):7994–8011, 2014.

148 F. Gendron, B. Pritchard, H. Bolvin, and J. Autschbach.Magnetic Resonance Properties of Actinyl CarbonateComplexes and Plutonyl(VI)-tris-nitrate. InorganicChemistry, 53(16):8577–8592, aug 2014.

149 R. J. Harrison. Approximating full configuration interac-tion with selected configuration interaction and perturba-tion theory. The Journal of Chemical Physics, 94(7):5021–5031, 1991.

150 B. A. Hess. Relativistic electronic-structure calcula-tions employing a two-component no-pair formalism withexternal-field projection operators. Physical Review A,33(6):3742–3748, jun 1986.

151 B. A. Heß, C. M. Marian, U. Wahlgren, and O. Gropen.A mean-field spin-orbit method applicable to correlatedwavefunctions. Chemical Physics Letters, 251(5):365–371,mar 1996.

152 A. A. Holmes, N. M. Tubman, and C. J. Umrigar. Heat-bath configuration interaction: An efficient selected ci al-gorithm inspired by heat-bath sampling. The Journal ofChemical Theory and Computation, 12:3674, 2016.

153 A. A. Holmes, C. J. Umrigar, and S. Sharma. Excitedstates using semistochastic heat-bath configuration inter-action. arXiv preprint arXiv:1708.03456, 2017.

154 B. Huron. Iterative perturbation calculations of groundand excited state energies from multiconfigurationalzeroth-order wavefunctions. The Journal of ChemicalPhysics, 58(12):5745, 1973.

155 M. Ilias, A. J. H. Jensen, V. Kelo, B. O. Roos, and M. Ur-ban. Theoretical study of PbO and the PbO anion. Chem-ical Physics Letters, 408(4-6):210–215, jun 2005.

156 M. Ilias and T. Saue. An infinite-order two-componentrelativistic Hamiltonian by a simple one-step transforma-tion. The Journal of Chemical Physics, 126(6):64102, feb2007.

157 J. Ivanic and K. Ruedenberg. Identification of dead-wood in configuration spaces through general direct con-figuration interaction. Theoretical Chemistry Accounts,106(5):339–351, 2001.

158 M. S. Kelley, T. Shiozaki, M. S. Kelley, and T. Shiozaki.Large-scale Dirac Fock Breit method using density fittingand 2-spinor basis functions Large-scale Dirac Fock Breitmethod using density fitting and 2-spinor basis functions.The Journal of Chemical Physics, 138:204113, 2013.

159 I. Kim and Y. S. Lee. Two-component multi-configurational second-order perturbation theory withkramers restricted complete active space self-consistentfield reference function and spin-orbit relativistic effec-tive core potential. The Journal of Chemical Physics,141(16):164104, 2014.

160 I. Kim, Y. C. Park, H. Kim, and Y. S. Lee. Spin-orbitcoupling and electron correlation in relativistic configura-tion interaction and coupled-cluster methods. ChemicalPhysics, 395(1):115–121, 2012.

161 K. Klein and J. Gauss. Perturbative calculation of spin-orbit splittings using the equation-of-motion ionization-potential coupled-cluster ansatz. The Journal of ChemicalPhysics, 129(19):194106, nov 2008.

162 M. Kleinschmidt, J. Tatchen, and C. M. Marian.SPOCK.CI: A multireference spin-orbit configuration in-teraction method for large molecules. The Journal ofChemical Physics, 124(12):124101, mar 2006.

163 S. Knecht, A. J. H. Jensen, and A. Jensen. group gen-eral active space implementation with application to BiHLarge-scale parallel configuration interaction . II . Two-and four-component double-group general active spaceimplementation with application to BiH. The Journal ofChemical Physics, 132:014108, 2010.

164 S. Knecht, S. Keller, J. Autschbach, and M. Reiher.A Nonorthogonal State-Interaction Approach for MatrixProduct State Wave Functions. Journal of Chemical The-ory and Computation, 12(12):5881–5894, 2016.

165 S. Knecht, O. Legeza, and M. Reiher. Communication:Four-component density matrix renormalization group.The Journal of Chemical Physics, 140(4):41101, jan 2014.

166 P. J. Knowles. Compressive sampling in configurationinteraction wavefunctions. Molecular Physics, 113(13-14):1655–1660, 2015.

167 A. Kramida, Y. Ralchenko, J. Reader, and N. A. S. D.Team. NIST Atomic Spectra Database (version 5.4).http://physics.nist.gov/asd.

168 W. Kutzelnigg. Solved and unsolved problems in rela-tivistic quantum chemistry. Chemical Physics, 395:16–34,2012.

169 W. Kutzelnigg and W. Liu. Quasirelativistic theory equiv-alent to fully relativistic theory. The Journal of ChemicalPhysics, 123(24):241102, dec 2005.

170 S. R. Langhoff. Ab initio evaluation of the fine structureof the oxygen molecule. The Journal of Chemical Physics,61(5):1708–1716, sep 1974.

171 Z. Li, S. Shao, and W. Liu. Relativistic explicit corre-lation : Coalescence conditions and practical suggestionsRelativistic explicit correlation : Coalescence conditions.The Journal of Chemical Physics, 136:144117, 2012.

172 Z. Li, B. Suo, Y. Zhang, Y. Xiao, and W. Liu. Combiningspin-adapted open-shell TD-DFT with spinorbit coupling.Molecular Physics, 111(24):3741–3755, dec 2013.

173 Z. Li, Y. Xiao, and W. Liu. On the spin separation ofalgebraic two-component relativistic Hamiltonians. TheJournal of Chemical Physics, 137(5):154114, oct 2012.

174 Z. Li, Y. Xiao, and W. Liu. On the spin separa-tion of algebraic two-component relativistic Hamiltonians:Molecular properties. The Journal of Chemical Physics,141(5):54111, 2014.

175 J. Liu, Y.-C. Chen, J.-L. Liu, V. Vieru, L. Ungur, J.-H.Jia, L. F. Chibotaru, Y. Lan, W. Wernsdorfer, S. Gao, X.-M. Chen, and M.-L. Tong. A Stable Pentagonal Bipyra-midal Dy(III) Single-Ion Magnet with a Record Magneti-zation Reversal Barrier over 1000 K. Journal of the Amer-ican Chemical Society, 138(16):5441–5450, apr 2016.

176 W. Liu. Ideas of relativistic quantum chemistry. Molecu-lar Physics, 108(13):1679–1706, 2010.

177 W. Liu and W. Kutzelnigg. Quasirelativistic theory. II.Theory at matrix level. The Journal of Chemical Physics,126(11):114107, mar 2007.

178 W. Liu and D. Peng. Exact two-component Hamiltoniansrevisited. The Journal of Chemical Physics, 131(3):31104,

17

jul 2009.179 S. Mai, T. Muller, F. Plasser, P. Marquetand, H. Lis-

chka, and L. Gonzalez. Perturbational treatment of spin-orbit coupling for generally applicable high-level multi-reference methods. The Journal of Chemical Physics,141(7):74105, aug 2014.

180 P. A. Malmqvist, B. O. Roos, and B. Schimmelpfennig.The restricted active space (RAS) state interaction ap-proach with spinorbit coupling. Chemical Physics Letters,357(3):230–240, 2002.

181 C. M. Marian and U. Wahlgren. A new mean-field andECP-based spin-orbit method. Applications to Pt andPtH. Chemical Physics Letters, 251(5-6):357–364, mar1996.

182 W. Mizukami, T. Nakajima, K. Hirao, and T. Yanai.A dual-level approach to four-component relativisticdensity-functional theory. Chemical Physics Letters,508(1-3):177–181, may 2011.

183 Myeong Cheol Kim, Sang Yeon Lee, and Yoon Sup Lee.Spin-orbit effects calculated by a configuration interac-tion method using determinants of two-component molec-ular spinors: test calculations on Rn and T1H. ChemicalPhysics Letters, 253(3):216–222, 1996.

184 T. Nakajima and K. Hirao. The DouglasKrollHess Ap-proach. Chemical Reviews, 112(1):385–402, jan 2012.

185 H. S. Nataraj, M. Kallay, L. Visscher, H. S. Nataraj,M. Kallay, and L. Visscher. General implementation of therelativistic coupled-cluster method General implementa-tion of the relativistic coupled-cluster method. The Jour-nal of Chemical Physics, 133:234109, 2010.

186 F. Neese. A spectroscopy oriented configuration inter-action procedure. The Journal of Chemical Physics,119(18):9428, 2003.

187 F. Neese. The ORCA program system. Wiley Inter-disciplinary Reviews: Computational Molecular Science,2(1):73–78, 2012.

188 M. Pernpointner, L. Visscher, W. A. de Jong, andR. Broer. Parallelization of four-component calculations.I. Integral generation, SCF, and four-index transforma-tion in the DiracFock package MOLFDIR. Journal ofComputational Chemistry, 21(13):1176–1186, 2000.

189 P. Pyykko. Relativistic Effects in Chemistry: More Com-mon Than You Thought. Annual Review of PhysicalChemistry, 63(1):45–64, 2012.

190 P. Pyykko. The Physics behind Chemistry and the Peri-odic Table. Chemical Reviews, 112(1):371–384, 2012.

191 S. Qiming, T. C. Berkelbach, N. S. Blunt, G. H. Booth,S. Guo, Z. Li, J. Liu, J. McClain, E. R. Sayfutyarova,S. Sharma, S. Wouters, and G. K.-L. Chan. The Python-based Simulations of Chemistry Framework (PySCF).arXiv preprint arXiv:1701.08223, jan 2017.

192 M. Reiher. DouglasKrollHess Theory: a relativisticelectrons-only theory for chemistry. Theoretical Chem-istry Accounts, 116(1):241–252, 2006.

193 M. Reiher and A. Wolf. Relativistic Quantum Chemistry,Second Edition. Wiley Online Library, 2014.

194 M. Roemelt. Spin orbit coupling for molecular ab initiodensity matrix renormalization group calculations: Ap-plication to g-tensors. The Journal of Chemical Physics,143(4):044112, jul 2015.

195 B. O. Roos, R. Lindh, P.-A. Malmqvist, V. Veryazov, andP.-O. Widmark. Main Group Atoms and Dimers Studiedwith a New Relativistic ANO Basis Set. The Journal ofPhysical Chemistry A, 108(15):2851–2858, apr 2004.

196 B. O. Roos and P.-A. Malmqvist. Relativistic quantumchemistry: the multiconfigurational approach. PhysicalChemistry Chemical Physics, 6(11):2919–2927, 2004.

197 B. O. Roos, Roland Lindh, Per-Ake Malmqvist, V. Verya-zov, and P.-O. Widmark. New Relativistic ANO BasisSets for Transition Metal Atoms. Journal of PhysicalChemistry A, 109:6575, 2005.

198 M. S. Gordon and M. W. Schmidt. Advances in Elec-tronic Structure Theory: GAMESS a Decade Later. InC. E. Dykstra, G. Frenking, K. S. Kim, and G. E. Scuse-ria, editors, Theory and Applications of ComputationalChemistry, pages 1167–1189. Elsevier, Amsterdam, 2005,2005.

199 J. E. Sansonetti and W. C. Martin. Hand-book of basic atomic spectroscopic data.http://www.nist.gov/pml/data/handbook/index.cfm.

200 T. Saue. Relativistic Hamiltonians for Chemistry: APrimer. Chem. Phys. Chem., 12(17):3077–3094, 2011.

201 T. Saue and T. Helgaker. Four-component relativisticKohnSham theory. Journal of Computational Chemistry,23(8):814–823, 2002.

202 E. R. Sayfutyarova and G. K.-L. Chan. A state inter-action spin-orbit coupling density matrix renormaliza-tion group method. The Journal of Chemical Physics,144(23):234301, jun 2016.

203 J. B. Schriber and F. A. Evangelista. Communication: Anadaptive configuration interaction approach for stronglycorrelated electrons with tunable accuracy. The Journalof Chemical Physics, 144(16):161106, 2016.

204 S. Sharma and A. Alavi. Multireference linearized cou-pled cluster theory for strongly correlated systems usingmatrix product states. The Journal of Chemical Physics,143(10):102815, 2015.

205 S. Sharma, A. A. Holmes, G. Jeanmairet, A. Alavi, andC. J. Umrigar. Semistochastic heat-bath configurationinteraction method: selected configuration interactionwith semistochastic perturbation theory. The Journalof Chemical Theory and Computation, 13(4):1595–1604,2017.

206 S. Sharma, G. Jeanmairet, and A. Alavi. Quasi-degenerate perturbation theory using matrix productstates. The Journal of Chemical Physics, 144(3):034103,2016.

207 S. Sharma, G. Knizia, S. Guo, and A. Alavi. Combininginternally contracted states and matrix product states toperform multireference perturbation theory. Journal ofChemical Theory and Computation, 13(2):488–498, 2017.PMID: 28060507.

208 T. Shiozaki and W. Mizukami. Relativistic InternallyContracted Multireference Electron Correlation Meth-ods. Journal of Chemical Theory and Computation,11(10):4733–4739, oct 2015.

209 M. Sjøvoll, O. Gropen, and J. Olsen. A determinantal ap-proach to spin-orbit configuration interaction. TheoreticalChemistry Accounts, 97(1):301–312, 1997.

210 J. E. T. Smith, A. A. Holmes, B. Mussard, and S. Sharma.Cheap and near exact CASSCF with large active spaces.arXiv preprint arXiv:1708.07544v2, 2017.

211 L. K. Sørensen, J. Olsen, and T. Fleig. Two- andfour-component relativistic generalized-active-space cou-pled cluster method : Implementation and applicationto BiH Two- and four-component relativistic generalized-active-space coupled cluster method : Implementationand application to BiH. The Journal of Chemical Physics,

18

134:214102, 2011.212 R. E. Stanton and S. Havriliak. Kinetic balance: A partial

solution to the problem of variational safety in Dirac cal-culations. The Journal of Chemical Physics, 81(4):1910–1918, 1984.

213 M. M. Steiner, W. Wenzel, K. G. Wilson, and J. W.Wilkins. The efficient treatment of higher excitations inCI calculations. A comparison of exact and approximateresults. Chemical Physics Letters, 231(2-3):263–268, 1994.

214 Q. Sun. Libcint: An efficient general integral libraryfor Gaussian basis functions. Journal of ComputationalChemistry, 36:1664–1671, 2015.

215 S. Ten-no and D. Yamaki. Communication : Explic-itly correlated four-component relativistic second-orderMøller- Plesset perturbation theory Communication :Explicitly correlated four-component relativistic second-order Møller-Plesset perturbation theory. The Journal ofChemical Physics, 137:131101, 2012.

216 J. Thyssen, T. Fleig, and A. J. H. Jensen. A di-rect relativistic four-component multiconfiguration self-consistent-field method for molecules. The Journal ofChemical Physics, 129:034109, 2008.

217 O. Vahtras, O. Loboda, B. Minaev, H. Agren, andK. Ruud. Ab initio calculations of zero-field splitting pa-rameters. Chemical Physics, 279(2-3):133–142, 2002.

218 V. Vallet, L. Maron, C. Teichteil, and J.-P. Fla-ment. A two-step uncontracted determinantal effectiveHamiltonian-based SOCI method. The Journal of Chem-ical Physics, 113(4):1391–1402, jul 2000.

219 W. Wenzel, M. M. Steiner, and K. G. Wilson. Multirefer-ence Basis-Set Reduction. International Journal of Quan-tum Chemistry, 30:1325–1330, 1996.

220 H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby,M. Schutz, and Others. MOLPRO, version 2015.1, a pack-age of ab initio programs, 2015.

221 S. Yabushita, Z. Zhang, and R. M. Pitzer. SpinOrbit Con-figuration Interaction Using the Graphical Unitary GroupApproach and Relativistic Core Potential and SpinOr-bit Operators. The Journal of Physical Chemistry A,103(29):5791–5800, jul 1999.

222 T. Yanai, H. Iikura, T. Nakajima, Y. Ishikawa, and K. Hi-rao. A new implementation of four-component relativis-tic density functional method for heavy-atom polyatomicsystems. The Journal of Chemical Physics, 115(18):8267–8273, oct 2001.

223 T. Yanai, T. Nakajima, Y. Ishikawa, and K. Hirao. A newcomputational scheme for the DiracHartreeFock methodemploying an efficient integral algorithm. The Journal ofChemical Physics, 114(15):6526–6538, apr 2001.

224 T. Yanai, H. Nakano, T. Nakajima, T. Tsuneda, S. Hi-rata, Y. Kawashima, Y. Nakao, M. Kamiya, H. Sekino,and K. Hirao. UTChem A Program for ab initio Quan-tum Chemistry. In P. M. A. Sloot, D. Abramson, A. V.Bogdanov, Y. E. Gorbachev, J. J. Dongarra, and A. Y.Zomaya, editors, Computational Science, pages 84–95.Springer Berlin Heidelberg, Berlin, Heidelberg, 2003.