Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spinorbit effects at the Hartree-Fock level Lan Cheng, Stella Stopkowicz, and Jürgen Gauss Citation: The Journal of Chemical Physics 139, 214114 (2013); doi: 10.1063/1.4832739 View online: http://dx.doi.org/10.1063/1.4832739 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/21?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS 139, 214114 (2013)

Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spin-orbit effects at the Hartree-Fock level Lan Cheng,1,a) Stella Stopkowicz,2,b) and Jürgen Gauss3,c) 1

Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, Texas 78712, USA 2 Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, N-0315 Oslo, Norway 3 Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany

(Received 13 September 2013; accepted 8 November 2013; published online 5 December 2013) A perturbative approach to compute second-order spin-orbit (SO) corrections to a spin-free DiracCoulomb Hartree-Fock (SFDC-HF) calculation is suggested. The proposed scheme treats the difference between the DC and SFDC Hamiltonian as perturbation and exploits analytic second-derivative techniques. In addition, a cost-effective scheme for incorporating relativistic effects in high-accuracy calculations is suggested consisting of a SFDC coupled-cluster treatment augmented by perturbative SO corrections obtained at the HF level. Benchmark calculations for the hydrogen halides HX, X = F-At as well as the coinage-metal fluorides CuF, AgF, and AuF demonstrate the accuracy of the proposed perturbative treatment of SO effects on energies and electrical properties in comparison with the more rigorous full DC treatment. Furthermore, we present, as an application of our scheme, results for the electrical properties of AuF and XeAuF. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832739] I. INTRODUCTION

Advanced techniques to treat electron-correlation effects, especially those of the coupled-cluster (CC) type,1 are widely used for accurate calculations of atomic and molecular energies and properties for systems comprising first- and secondrow elements.2–7 In such calculations, relativistic effects are often neglected or at most treated with low-level approximations. Extending the applicability of these high-accuracy treatments to systems containing heavier elements, however, necessitates a more sophisticated treatment of relativistic effects.8, 9 Fully relativistic four-component schemes are the most rigorous option for quantum-chemical calculations. They are typically used together with kinetically balanced basis sets10 and the no-pair Dirac-Coulomb(-Breit) (DC-(B)) Hamiltonian11 in the electron-correlation treatment. The implementation of four-component CC methods, in the following referred to as DC-CC, has been reported by Visscher et al.,12, 13 Kállay et al.,14 as well as Fleig et al.15, 16 However, the high computational requirements of DC-CC calculations confine the applications of these approaches to relatively small systems: the DC-CC method in its singles and doubles (DC-CCSD) approximation is around 30 times more costly than its non-relativistic counterpart.12 One should emphasize at this point that the additional computational overhead in the rate-determining step of a DC-CC calculation (the solution of the CC amplitude equations) arises mainly from spinsymmetry breaking due to spin-orbit (SO) coupling. In contrast, the solution of the CC amplitude equations in a scalar-

a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]

0021-9606/2013/139(21)/214114/11/$30.00

relativistic framework is usually not more costly than in the non-relativistic case.17 At the same time, scalar-relativistic and SO effects have distinct physical origins and give rise to different effects in atoms and molecules. Scalar-relativistic effects originate from relativistic kinematics and lead to a stabilization and spatial contraction of the s- and p-type orbitals as well as to a destabilization and spatial extension of the d- and f-type orbitals.8 On the other hand, the SO interaction can be viewed as a consequence of the interactions between the electron spin and the nuclear or electron current.18 They are responsible for level splittings and phenomena related to transitions between SOsplit states. In systems containing 6p (Tl-Rn) or 7p (Eka-Tl to Eka-Rn) elements, SO interactions furthermore make significant contributions to molecular properties.19–22 However, since scalar relativity affects the shape of the wave function more than SO interactions, scalar-relativistic contributions are typically larger than SO effects. Based on the above analysis, a cost-effective scheme for high-level calculations in the four-component framework may be obtained by combining a rigorous treatment of the (large and computationally cheap) scalar-relativistic effects with an approximate treatment of the (smaller and computationally expensive) spin-orbit effects by means of perturbation theory. Dyall proposed a scheme to separate the DC Hamiltonian into a spin-free (SF) and a spin-dependent (SD) part by applying a simple change of metric to the small component wave function.23 Visscher and van Lenthe pointed out that the spin-separation scheme in the four-component framework is not unique since there exist other possibilities for the metric change leading to different formulations.24 However, Dyall’s original scheme is naturally compatible with the use of kinetically balanced basis sets and is also the most

139, 214114-1

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computationally tractable. Saue and Visscher presented a serviceable implementation of the spin-free Dirac-Coulomb (SFDC) approach within the framework of a DC code by zeroing out the imaginary parts of the Fock matrix in the quaternion representation within a DC Hartree-Fock (HF) calculation.25 Note that such an implementation leads to a scheme that is slightly different from the one proposed by Dyall, since the real part of the exchange matrix elements on the small-small block still contains a spin-dependent contribution. Corresponding CC and configuration interaction calculations were reported by Fleig and Visscher26 and Knecht et al.27 More recently, an efficient implementation of the SFDC approach was presented based solely on the calculation and manipulation of the spin-free one- and two-electron integrals.28 In addition, an analytic scheme for SFDC-CC calculations of first-order electrical properties was reported.28 The use of the SFDC approach in several applications furthermore has demonstrated that this scheme can furnish an accurate treatment of relativistic effects on electrical properties for systems containing first- to fourth-row elements.29, 30 Nevertheless, consideration of SO effects is essential when one would like to go further down in the periodic table or when higher accuracy is desired.22, 31 In this paper, we present a scheme for the perturbative treatment of SO effects in order to augment a SFDC calculation. An implementation of such a scheme for computing second-order SO energy corrections at the HF level is reported. In Sec. II, we briefly summarize the spin-separation scheme in the DC framework and derive an expression for the perturbative second-order SO corrections to the SFDCHF energies. Computational details are given in Sec. III, while benchmark results for energies, dipole moments, and electric-field gradients of the hydrogen-halide series (HX, X = F, Cl, Br, I, and At) and the coinage-metal fluoride series (CuF, AgF, and AuF) are presented and discussed in Sec. IV. The applicability of our perturbative approach to SO corrections in chemical problems is demonstrated in Sec. IV, in which SFDC-CC results for the electrical properties of AuF and XeAuF are augmented by corresponding perturbative SO corrections computed at the SFDC-HF level. A summary and outlook is given in Sec. V.

J. Chem. Phys. 139, 214114 (2013)

basis set10 ψP =

ψPL



 =

ψPL

 

 =

μ



L CμP fμ



. (1) S σ ·p CμP f 2c μ  S fμ the soNote that we introduce here via φPL = μ CμP called “pseudo-large” component.23 This component has the same symmetry properties as the large component and is related to the small component through a metric change. Spin separation is most easily accomplished when applied to the Hamiltonian matrix elements expressed in terms of the large- and pseudo-large components of the spinors. The matrix elements of the one-electron part of the DiracCoulomb Hamiltonian can then be written as    L   L  L   L  L  + ψP T φQ + φP T ψQ ψP |hD |ψQ  = ψPL V ψQ ψPS

+

σ ·p L φ 2c P

μ

    L  L  1  L σ · p)φ  Q ( σ · p)φ  PL V ( − φP T φQ , 2 4c (2)

while those for the two-electron part, i.e., the instantaneous Coulomb interactions, are given by (ψP ψQ |ψR ψS )   L L L ψR ψS = ψPL ψQ +

 1  L L L L ψ ( σ · p)φ  ( σ · p)φ  ψ P Q R S 4c2

+

1  L L  L L ( σ · p)φ  ψ ( σ · p)φ  ψ P Q R S 4c2

+

 1  L L L L ( σ · p)φ  ( σ · p)φ  ( σ · p)φ  ( σ · p)φ  . P Q R S 16c4

(3)

Using the Dirac relation  σ · B)  = A · B + i σ · (A × B),  ( σ · A)(

(4)

the Hamiltonian matrix elements in Eqs. (2) and (3) can be split into a SF part and a SD part    L   L  L   L  L  ψP |hD |ψQ SF = ψPL V ψQ + ψP T φQ + φP T ψQ +

II. THEORY

In this section, we derive an expression for the secondorder SO correction in terms of the second derivative of the SFDC-HF energy. The International System of Units (SI) based atomic units are used throughout. The indices {P, Q, . . . } and {i, j, . . . } are used to denote arbitrary and occupied four-component spinors, respectively, while {A, B, . . . } are used to refer to the joint set of positive energy state (PES) virtual and all negative energy state (NES) spinors. In general, the term “spinor” (or “orbital”) denotes a one-electron wave function obtained from calculations with (or without) spin-orbit coupling. However, the acronym “AO” (or “MO”) is also used to refer to both atomic (or molecular) orbitals and atomic (or molecular) spinors. In a DC calculation, the four-component molecular spinors are expanded in terms of a kinetically balanced



ψP |hD |ψQ SD =

  L  1  L   L L  Q pφ  P V · pφ − φP |T |φQ , 4c2 1 4c2



(5)

  L , iuvw pv φPL |V |σu pw φQ

u,v,w=x,y,z

(6) (ψP ψQ |ψR ψS )SF   1  L  L  L L L L L ψR ψS + 2 pφ  P · pφ  Q ψR ψS = ψPL ψQ 4c +

1  L L  L  L  R · pφ ψ ψ pφ  S 4c2 P Q

+

1  L  L  L  L  R · pφ pφ  P · pφ  Q pφ  S , 16c4

(7)

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J. Chem. Phys. 139, 214114 (2013)

(ψP ψQ |ψR ψS )SD

   1 L  L L ψR ψS = 2 iuvw pv φPL σu pw φQ 4c u,v,w=x,y,z   L L  + ψP ψQ  pv φRL σu pw φSL − ×

1 16c4 



second and fourth index, e.g.,  Q }{pψ  R } · {pψ  S }) ({pψ  P } · {pψ = ({pψ  P } · {pψ  Q }|{pψ  R } · {pψ  S }) − ({pψ  P } · {pψ  S }|{pψ  R } · {pψ  Q }).

uvw lmn

u,v,w,l,m,n=x,y,z

   L  pm φRL σl pn φSL , pv φPL σu pw φQ

(8)

where  is the Levi-Civita. In the following, the spinseparation scheme will be applied to the DC-HF method.32–34 The DC-HF equations are given by       S 0 F LL F LS CL CL = E, (9) F SL F SS CS CS 0 2c12 T with S and E being the overlap matrix and the diagonal matrix containing orbital energies, respectively, and the various blocks of the Fock matrix defined as

LL LL = fμ |V |fν  + Dρσ (fμ fν fρ fσ ) Fμν ρσ

1 SS D (fμ fν |( σ · p)f  ρ ( σ · p)f  σ ), + 2 4c ρσ ρσ LS Fμν

(10)

1 SL = fμ |T |fν  − 2 D (fμ fρ |( σ · p)f  σ ( σ · p)f  ν ), 4c ρσ σρ (11)  LS ∗ SL = Fνμ , Fμν

(12)

1 ( σ · p)f  μ |V |( σ · p)f  ν  − fμ |T |fν  4c2 1 LL D (( σ · p)f  μ ( σ · p)f  ν |fρ fσ ) + 2 4c ρσ ρσ

SS Fμν =

The AO density matrices that appear in Eqs. (10)–(13) are defined as

 ∗ XY X Y Dμν Cμi = Cνi , (16) where X and Y can be eitheri L or S. Using the spin-separation scheme discussed for the DC Hamiltonian, the Fock matrix in Eqs. (10)–(13) can be decomposed into its SF and SD terms XY XY F XY = FSF + FSD ,

ρσ

1 SS D (fμ fσ |{pf  ρ } · {pf  σ }), + 2 4c ρσ ρσ  LS FSF μν = fμ |T |fν  −

u,v,w=x,y,z

iuvw pv fμ |V |σu pw fν  +

1 SL D (fμ fρ |{pf  σ } · {pf  ν }), 4c2 ρσ σρ

(19)

 SS 1 FSF μν = 2 pf  μ |V | · pf  ν  − fμ |T |fν  4c 1 LL D ({pf  μ } · {pf  ν }|fρ fσ ) + 2 4c ρσ ρσ 1 SS D ({pf  μ } · {pf  ν }{pf  ρ } · {pf  σ }), 16c4 ρσ ρσ

(14)

and the SD part containing the SD contributions written as  LL 1

D SS iuvw FSD μν = 2 4c ρσ u,v,w=x,y,z ρσ × (fμ fν |{pv fρ }σu {pw fσ }),

(21)

 LS 1

FSD μν = − 2 D SL iuvw 4c ρσ u,v,w=x,y,z σρ

This notation will be used in the following also in a more general sense to denote antisymmetrization with respect to the

(18)

(20)

For the sake of conciseness, we use the notation (PQ||RS) to denote antisymmetrized two-electron integrals

 SS 1 FSD μν = 2 4c

(17)

with the SF part containing the SF Hamiltonian matrix elements given by

 LL LL FSF μν = fμ |V |fν  + Dρσ (fμ fν fρ fσ )

+

1 SS D (( σ ·p)f  μ ( σ ·p)f  ν ( σ ·p)f  ρ ( σ ·p)f  σ ). + 16c4 ρσ ρσ (13)

(P QRS) = (P Q|RS) − (P S|RQ).

(15)

× (fμ fρ |{pv fσ }σu {pw fν }),

(22)

1 LL

D iuvw ({pv fμ }σu {pw fν }|fρ fσ ) 4c2 ρσ ρσ u,v,w=x,y,z

+

1 SS

1 SS

Dρσ iuvw ({pv fμ }σu {pw fν }{pf  ρ }·{pf  σ }) + D iuvw ({pf  μ }·{pf  ν }{pv fρ }σu {pw fσ }) 4 16c ρσ 16c4 ρσ ρσu,v,w=x,y,z u,v,w=x,y,z

−

1 SS D uvw lmn ({pv fμ }σu {pw fν }{pm fρ }σl {pn fσ }). ρσ 16c4 ρσ u,v,w,l,m,n=x,y,z

(23)

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J. Chem. Phys. 139, 214114 (2013)

Our suggestion is to solve first the SFDC-HF equations with the Fock matrix given in Eqs. (18)–(20) and then to account for the SD terms as a perturbation. In the latter, we define the order of perturbation in the SD terms in terms of the number of Pauli matrices that appear. For closed-shell systems, the first-order SO contribution to the energy vanishes. Therefore, we focus on the corresponding second-order correction that can be formulated in terms of a second derivative of the SFDC-HF energy. A general expression for the second derivative of the HF energy can be found in Ref. 35. Here, we rewrite it in the following form for the DC approach:



1

∂ 2E λ,χ λ,χ = (hD )ii + (ψi ψi ψj ψj )λ,χ − i (SD )ii ∂λ∂χ 2 i ij i  

χ

χ λ + UAi FAi + c.c. − (SD )j i Fjλi Ai

−



ji



χ i UP i (SD )λP i

+ c.c. +

iP



 χ i (SD )λii

, (24)

i

in which the  i ’s are the orbital energies of the canonical DC-HF spinors and SD represents the four-component overlap matrix. We adopt the MO representation as the starting point of our derivation due to its compactness in the context of a four-component theory. The corresponding AO representation would involve explicit reference to the large- and smallcomponent basis functions and thus would be rather cumbersome in appearance. In Eq. (24), the partial derivatives of the Hamiltonian, metric, and Fock matrix elements involve derivatives of the AO integrals and unperturbed MO coefficients. For example, the first two terms on the right-hand side of Eq. (24) correspond to the lengthy AO basis expressions 2 2



λ,χ LL ∂ fμ |V |fν  LS ∂ fμ |T |fν  + (hD )ii = Dμν Dμν ∂λ∂χ ∂λ∂χ μν μν i +



SL Dμν

μν

+ −

∂ 2 fμ |T |fν  ∂λ∂χ

SS Dμν

μν

∂ 2 fμ |T |fν  , ∂λ∂χ

(25)

1

(ψi ψi ψj ψj )λ,χ 2 ij =

1 LL LL ∂ 2 (fμ fν fσ fρ ) D D 2 μν,σρ μν σρ ∂λ∂χ

+

σ ·p)f  σ ( σ ·p)f  ρ) 1 LL SS ∂ 2 (fμ fν |( Dμν Dσρ 2 4c μν,σρ ∂λ∂χ

σ ·p)f  σ ( σ ·p)f  ν) 1 LS SL ∂ 2 (fμ fρ |( D D − 2 4c μν,σρ μν σρ ∂λ∂χ +

(27) It should be noted here that the set of MOs ψ Q ’s consisting of both PES and NES spinors span the same one-particle space as the joint set of the large- and small-component AOs. Simplifications can be made in Eq. (24) for the calculation of SO corrections. Since the SO perturbation does not affect the metric, all terms involving the partial derivatives of the overlap matrix elements vanish. Furthermore, as can be seen from Eqs. (25), (26), and (4), the one-electron Dirac Hamiltonian matrix is linear in the SO perturbation and second-order terms only appear in the two-electron part. The contribution of the second-order terms to the second-order energy correction is thus 1

(ψi ψi ψj ψj )SO(2) 2 ij

σ ·p)f  μ ( σ ·p)f  ν ( σ ·p)f  σ ( σ ·p)f  ρ) 1 SS SS ∂ 2 (( . Dμν Dσρ 2 32c μν,σρ ∂λ∂χ (26)

1 SS SS Dρσ Dμν uvw lmn 4 32c ρσ u,v,w,l,m,n=x,y,z

=

× ({pv fμ }σu {pw fσ }|{pm fρ }σl {pn fν }).

(28)

The first-order terms in the Hamiltonian contribute via the partial derivative of the Fock matrix, ∂FAi /∂λ, which is identical to FSD (see Eqs. (21)–(23)) in the representation of unperturbed MOs. Therefore, gathering all contributions, one obtains the following expression for the second-order SO contribution to the SFDC-HF energy: E SO =

σ · p)V  ( σ · p)|f  ν 1 SS ∂ 2 fμ |( Dμν 2 4c μν ∂λ∂χ

The contribution from the response of the MO coefficients with respect to the external perturbation are incorporated through the coupled perturbed SFDC-HF (CPSFDC-HF) coχ efficients UAi ’s defined as     S L

∂CμP

χ

∂CμP 0 fμ UQP ψQ . = + σ  · p  ∂χ ∂χ 0 f μ μ Q 2c μ

1 SS SS D D uvw lmn ({pv fμ }σu ρσ μν 32c4 ρσ u,v,w,l,m,n=x,y,z

× {pw fσ }|{pm fρ }σl {pn fν })  

SO + UAi (FSD )Ai + c.c. .

(29)

Ai SO The coefficients UAi that account for the first-order change of the MO coefficients with respect to the SO perturbation are obtained by solving the corresponding CPSFDC-HF equations35–37

SO SO∗ (A − i )UAi + UBj (Aj Bi)

+



Bj SO UBj (ABj i) = −(FSD )Ai .

(30)

Bj

Note that the indices A and B in the CPSFDC-HF equations run over the virtual and NES spinors. The NES spinors need to be considered, since they are part of the complete one-particle space in the SFDC treatment and make non-negligible contributions to the perturbed spinors.38

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214114-5

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J. Chem. Phys. 139, 214114 (2013)

III. IMPLEMENTATIONAL AND COMPUTATIONAL DETAILS

TABLE I. Equilibrium bond distances (in Å) obtained at SFX2C1e/CCSD(T)/unc-ANO-RCC level.

A suite of programs to compute second-order SO corrections to the SFDC-HF energies based on Eq. (29) have been incorporated in the CFOUR program package.39 The SO inte μ } · {pf  ν }|{pv fρ }σu {pw fσ }) were grals of the type iuvw ({pf implemented as part of the present work, while all other SO integrals were already available due to the previous work on fourth-order direct perturbation theory (DPT4).31, 40, 41 The CPSFDC-HF equations were solved in the same way as the non-relativistic CPHF equations35–37 except that the index range of the virtual orbitals was extended to cover both virtual and NES orbitals. The required integral transformation for the two-electron integrals involving NES orbitals was already implemented in the previous work on SFDC electrical properties.28 To assess the accuracy of the perturbative scheme for treating SO effects, benchmark calculations were carried out for the hydrogen-halide series (HX, X = F, Cl, Br, I, and At) and the coinage-metal halides CuF, AgF, and AuF using uncontracted ANO-RCC (unc-ANO-RCC) basis sets.42–44 Both point-like and Gaussian nuclear models45 were employed in our calculations. The H–X bond distances for the hydrogen halides were taken from Ref. 46, while the metal-fluorine distances for the coinage-metal fluorides, on the other hand, were obtained at the CCSD level47 augmented with a perturbative treatment of triple excitation (CCSD(T))48, 49 using the uncANO-RCC basis. Scalar-relativistic effects were accounted for by using the spin-free version of the exact two-component theory in its one-electron variant (SFX2C-1e).50, 51 The geometry optimization employed the available development of analytic gradients for the SFX2C-1e scheme52, 53 and the CC approaches.54–58 The equilibrium bond distances are summarized in Table I. The perturbative SO corrections are compared with the difference between the DC-HF and SFDC-HF results. In the finite-field calculations of dipole moments and electric-field gradients, the nuclear-attraction potential V was augmented with the corresponding property operators Vprop multiplied by a set of field strengths {nλ, n = ±1, ±2, . . . }, i.e., V → V + nλVprop ,

(31)

 pV  · p → p(V  + nλVprop ) · p,

(32)

 i σ · (pV  × p)  → i σ · (p(V  + nλVprop ) × p).

(33)

We adopted the values of λ recommended in Ref. 31, i.e., λ = 7.5 × 10−5 for the calculation of dipole moments and λ = 1 × 10−5 for the calculation of electric-field gradients. However, the SFDC wave function turns out to be more sensitive to the electric-field gradient perturbation than the non-relativistic wave function. Therefore, for calculating the heavy-element electric-field gradients in HAt and AuF, we employed a smaller λ value of 1 × 10−7 . A two-point formula with n = ±1 is sufficient to obtain accurate values for the dipole moments. However, it is necessary to use a six-point or even an eight-point formula for computing the electric-field

XeAuF

Molecule Bond

CuF Cu–F

AgF Ag–F

AuF Au–F

Xe–Au

Au–F

Cal. Exp.

1.747 1.745

1.968 1.983

1.925 1.918

2.543 2.548a

1.928 1.918b

a b

r0 value. Fixed to the distance in AuF.

gradients. We refer the readers to Ref. 31 for details concerning the implementation of the property integrals as well as the numerical differentiation schemes applied to obtain accurate values for the electric-field gradients. All SFDC and perturbative SO calculations were performed using the CFOUR program package,39 while the DC calculations were done using the DIRAC program package.59 Apart from the benchmark studies, we also carried out a first application concerning the SO effects on the dipole moments and the nuclear quadrupole-coupling constants of the molecules AuF and XeAuF. High accuracy is obtained here by augmenting results from SFDC-CC calculations with a perturbative estimate for SO effects computed at the SFDCHF level. The geometry of XeAuF was obtained at the SFX2C-1e/CCSD(T)/unc-ANO-RCC level. While the uncANO-RCC basis sets have been shown to provide accurate results for dipole moments and electric-field gradients of maingroup elements,29 additional steep functions are required for the accurate determination of metal electric-field gradients.30 To obtain a suitable basis set for the calculation of Au electricfield gradients, we performed SFDC-CCSD calculations on the AuF molecule with systematically enlarged basis sets for Au and the unc-ANO-RCC basis set for F. We started from the unc-ANO-RCC basis set of Au, which corresponds to a 24s21p15d11f4g2h primitive set. BAS0 was then obtained by replacing the three rather loosely spaced p functions (those with the exponents 2 743 047.34, 620 746.976, and 163 862.987) by four p functions (those with the exponents 5 111 450, 1 597 328, 499 165, and 155 989). In a second step, we added up to nine sets of steep p-, d-, f-type functions; the corresponding sets were named BAS1, BAS2, etc. In a further step, BAS10 to BAS13 were obtained by adding up to four additional sets of steep g- and h-type functions to BAS8. With the further addition of two i-type functions with exponents 4.3515 and 1.4505 to BAS12, the BAS14 set is defined. Finally, two further i-type functions with exponents 13.0545 and 0.5802 were added to BAS14 to obtain BAS15, while one additional i-type functions with the exponent 39.1635 was added to BAS15 to generate our largest basis set named BAS16. In the procedure described above, the exponents of the additional steep functions were obtained by multiplying the steepest functions in the previous set by a factor of 3. Since BAS14 (24s30p23d19f7g5h2i) was shown to provide converged SFDC-CCSD results for the Au electricfield gradient in AuF, we carried out SFDC-CCSD(T) calculations for both AuF and XeAuF using the BAS14 set for Au together with unc-ANO-RCC basis sets for Xe and F in order to obtain our best scalar-relativistic results. All electrons were

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214114-6

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J. Chem. Phys. 139, 214114 (2013)

TABLE II. Spin-orbit contributions to the HF energy (in Hartree). “SO(2) ” denotes the perturbative spin-orbit correction to the SFDC approach, while “DC-SFDC” denotes the difference between the DC and SFDC results. All calculations were carried out using the unc-ANO-RCC basis. Point-like nuclear model

HF HCl HBr HI HAt CuF AgF AuF

Gaussian nuclear model

SO(2)

DC-SFDC

SO (2) DC − SF DC

SO(2)

DC-SFDC

SO (2) DC − SF DC

−8.105 × 10−6 −6.810 × 10−4 −0.0831 −1.2551 −30.2734 −0.0241 −0.5697 −18.1613

−8.111 × 10−6 −6.828 × 10−4 −0.0841 −1.2914 −33.2121 −0.0243 −0.5824 −19.6041

0.999 0.997 0.988 0.972 0.912 0.992 0.978 0.926

−8.105 × 10−6 −6.810 × 10−4 −0.0831 −1.2549 −30.2306 −0.0241 −0.5698 −18.1613

−8.111 × 10−6 −6.828 × 10−4 −0.0841 −1.2911 −33.1371 −0.0243 −0.5823 −19.5754

0.999 0.997 0.988 0.972 0.912 0.992 0.978 0.926

correlated in the calculations for AuF, while the frozen-core approximation was used in the calculations for XeAuF with a small core consisting of Au 1s, 2s, 2p, 3s, 3p, 3d orbitals and the Xe 1s, 2s, 2p, 3s, 3p orbitals. The perturbative SO corrections to the properties were computed using unc-ANO-RCC basis sets. The nuclear quadrupole-coupling constants of the nucleus K was obtained from the computed electric-field gradients using the following relation: χij (K) = −eQqij (K)/κ,

(34)

κ = 1/0.2349647,

(35)

where qij (K) is the electric-field gradient component at the position of the target nucleus K. We used the standard values of the 197 Au and 131 Xe nuclear quadrupole moments, i.e., Q(197 Au) = −521.5(5.0)mb60 and Q(131 Xe) = −114(1)mb.61 IV. RESULTS AND DISCUSSIONS A. Benchmark calculations for the HX series (X = F, Cl, Br, I, and At) and CuF, AgF, and AuF

In Table II, we list the SO corrections to the SFDC-HF energies of the molecules considered in the present benchmark study. The results obtained from the perturbative treatment are also compared there with the corresponding non-

perturbative values obtained as the differences between the DC-HF and SFDC-HF results. It is seen that the perturbative treatment tends to underestimate the SO corrections to the SFDC-HF energies. As expected, the perturbative scheme performs extremely well for systems comprising first and second-row elements, while the accuracy gradually deteriorates when the elements become heavier. However, even for systems containing fifth-row elements such as HAt and AuF, the perturbational calculations recover more than 90% of the total SO energy correction. The SO corrections to absolute energies include significant contributions from the core orbitals. However, many properties of chemical interest are mostly determined by the valence-shell contributions. In Tables III and IV, we therefore demonstrate the efficacy of the present perturbative approach to compute SO corrections to electrical properties such as dipole moments and electric-field gradients. From the results in Tables III and IV, it is obvious that SO effects on properties are more pronounced for the p-block elements than for coinage metals. For example, the SO correction to the dipole moment of HAt is one order of magnitude larger than that of AuF, because the SO effects on the 6p orbitals of astatine are much more pronounced than those on the 6s or 5d orbitals of gold. It is interesting to note that the SO corrections to the total energies of HAt and AuF are of similar magnitude. This is probably due to the fact that the SO energy corrections are dominated by the inner-shell contributions. As expected,

TABLE III. Spin-orbit contributions to the dipole moment (in a.u.) at the HF level. “SO(2) ” denotes the perturbative spin-orbit correction to the SFDC approach, while “DC-SFDC” denotes the difference between the DC and SFDC results. All calculations were carried out using the unc-ANO-RCC basis. Point-like nuclear model SO(2) HF HCl HBr HI HAt CuF AgF AuF

10−6

−6.72 × −7.92 × 10−5 −0.00184 −0.0114 −0.1273 −1.74 × 10−4 −9.14 × 10−4 −0.0179

DC-SFDC 10−6

−6.71 × −7.90 × 10−5 −0.00181 −0.0110 −0.1092 −1.72 × 10−4 −9.18 × 10−4 −0.0176

Gaussian nuclear model SO (2) DC − SF DC 1.001 1.002 1.013 1.035 1.165 1.011 0.995 1.017

SO(2)

DC-SFDC

SO (2) DC − SF DC

−6.72 × 10−6 −7.92 × 10−5 −0.00184 −0.0114 −0.1272 −1.74 × 10−4 −9.13 × 10−4 −0.0180

−6.71 × 10−6 −7.90 × 10−5 −0.00181 −0.0110 −0.1091 −1.72 × 10−4 −9.18 × 10−4 −0.0177

1.001 1.003 1.013 1.035 1.166 1.012 0.995 1.017

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214114-7

Cheng, Stopkowicz, and Gauss

J. Chem. Phys. 139, 214114 (2013)

TABLE IV. Spin-orbit contributions to the electric-field gradients (in a.u.) at the HF level. “SO(2) ” denotes the perturbative spin-orbit correction to the SFDC approach, while “DC-SFDC” denotes the difference between the DC and SFDC results. All calculations were carried out using the unc-ANO-RCC basis. Point-like nuclear model

HF HCl HBr HI HAt CuF AgF AuF

Gaussian nuclear model

SO(2)

DC-SFDC

SO (2) DC − SF DC

SO(2)

DC-SFDC

SO (2) DC − SF DC

−1.097 × 10−5 −1.176 × 10−4 −0.00335 −0.03036 −1.1161 −6.58 × 10−5 −0.0084 −0.2863

−1.095 × 10−5 −1.173 × 10−4 −0.00319 −0.02574 −0.6958 −7.64 × 10−5 −0.0088 −0.3180

1.002 1.003 1.047 1.180 1.604 0.861 0.959 0.900

−1.097 × 10−5 −1.176 × 10−4 −0.00335 −0.0303 −1.0968 −6.68 × 10−5 −0.0084 −0.2812

−1.095 × 10−5 −1.175 × 10−4 −0.00319 −0.0257 −0.6638 −7.60 × 10−5 −0.0088 −0.3171

1.002 1.001 1.048 1.180 1.652 0.879 0.959 0.887

the perturbative treatment of SO effects is more satisfactory for dipole moments, a valence property, than for electric-field gradients which can be considered as a core property. For instance, the relative error of the perturbative SO correction in comparison to the non-perturbative value is around 20% for the dipole moment of HAt, but it amounts to 60% for the astatine electric-field gradient of HAt. Therefore, the perturbative approach to SO effects can yield accurate results for valence properties even for compounds containing 6p-block elements, while for core properties the perturbative scheme works well only for compounds containing elements up to the 5d block. Although the results in Tables III and IV demonstrate that the perturbation scheme proposed in this work generally provides a good description of SO effects on molecular properties, a few results were at first sight surprising. The perturbative scheme tends to overshoot the SO corrections to dipole moments and to the halogen electric-field gradients in the hydrogen halides, while it underestimates the SO contributions to the metal electric-field gradients. Moreover, the perturbative treatment recovers only 86% of the total SO contribution to the Cu electric-field gradient in CuF, while it works much better for AgF and even for AuF. To understand these observations, we analyze the SO corrections to the electrical properties by decomposing them into their two individual contributions: (a) the first is due to the non-relativistic and scalar-relativistic property integrals given in Eqs. (31) and (32) and is denoted in Tables V and VI as “V+PVP,”

(b) the second arises from the SO property integrals given in Eq. (33) and is denoted as “PXVP.” It can be seen from the data given in Table V that the SO corrections to the dipole moments are mainly of the “P+PVP” type. The perturbative treatment tends to overestimate this contribution except in the case of the AgF molecule. On the other hand, as is seen from Table VI, the total SO corrections to the electric-field gradients involve a partial cancellation between the two individual contributions. The total values of the SO corrections to the halogen electric-field gradients in the hydrogen halides have the same sign as the corresponding “V+PVP” contributions, while the opposite is the case for the metal electric-field gradients in CuF, AgF, and AuF. Therefore, the overestimation of the halogen electric-field gradients and underestimation of the metal electric-field gradients both indicate that the perturbative SO treatment on top of SFDC calculations tends to overestimate the “V+PVP” contribution more than the “PXVP” contribution. This is also consistent with the observation for the dipole moments mentioned above. In the case of CuF, a major cancellation between the two individual contributions leads to a total value that amounts to only a fraction of the individual ones. Therefore, the perturbative treatment appears to be inaccurate for the SO correction to the Cu electric-field gradient in CuF, although the individual terms are reproduced reasonably well. We note that in most cases DPT4 tends to underestimate both individual contributions and as a consequence also the total value. The only exception is that the

TABLE V. Individual contributions to the spin-orbit effects on the dipole moments (in a.u.) at the HF level. “V+PVP” denotes the contribution from the non-relativistic and scalar-relativistic property integrals. “PXVP” represents the contribution from spin-orbit property integrals. All calculations were carried out using the point-like nuclear model and the unc-ANO-RCC basis. SO(2) V+PVP HF HCl HBr HI HAt CuF AgF AuF

−6.72 × 10−6 −7.92 × 10−5 −0.00184 −0.0114 −0.1273 −1.74 × 10−4 −9.14 × 10−4 −0.0179

PXVP −0.00 × 10−6 0.00 × 10−5 0.00000 0.0000 0.0000 0.00 × 10−4 0.00 × 10−4 0.0000

DPT4-SO Total

V+PVP

−6.72 × 10−6 −7.92 × 10−5 −0.00184 −0.0114 −0.1273 −1.74 × 10−4 −9.14 × 10−4 −0.0179

−6.68 × 10−6 −7.79 × 10−5 −0.00168 −0.0092 −0.0678 −1.51 × 10−4 −8.43 × 10−4 −0.0076

PXVP −0.00 × 10−6 0.00 × 10−5 0.00000 0.0000 0.0000 0.00 × 10−4 0.00 × 10−4 0.0000

Total

DC-SFDC

−6.68 × 10−6 −7.79 × 10−5 −0.00168 −0.0092 −0.0678 −1.51 × 10−4 −8.43 × 10−4 −0.0076

−6.71 × 10−6 −7.90 × 10−5 −0.00181 −0.0110 −0.1092 −1.72 × 10−4 −9.18 × 10−4 −0.0176

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214114-8

Cheng, Stopkowicz, and Gauss

J. Chem. Phys. 139, 214114 (2013)

TABLE VI. Individual contributions to the spin-orbit effects on the electric-field gradients (in a.u.) at the HF level. “V+PVP” denotes the contribution from the non-relativistic and scalar-relativistic property integrals. “PXVP” represents the contribution from spin-orbit property integrals. All calculations were carried out using the point-like nuclear model and the unc-ANO-RCC basis. SO(2) V+PVP HF HCl HBr HI HAt CuF AgF AuF

−1.316 × 10−5 −1.718 × 10−4 −0.00986 −0.10543 −2.9724 0.001403 0.0044 0.3402

PXVP 0.219 × 10−5 0.542 × 10−4 0.00652 0.07507 1.8564 −0.001469 −0.0128 −0.6265

DPT4-SO Total

V+PVP

−1.097 × 10−5 −1.176 × 10−4 −0.00335 −0.03036 −1.1161 −6.6 × 10−5 −0.0084 −0.2863

−1.304 × 10−5 −1.668 × 10−4 −0.00851 −0.07324 −0.9924 0.00096 0.0033 0.1030

Cu electric-field gradient of CuF is grossly overestimated, as DPT4 does not describe the individual terms as accurately. The perturbative inclusion of SO effects to augment SFDC calculations accounts for the coupling between scalarrelativistic and SO effects and thus provides SO contributions that are more accurate than those obtained at the DPT4 level. We should emphasize that, although the coupling between scalar-relativistic and SO effects is of third order in terms of c−2 and appears only at the DPT6 level, its contribution to the SO corrections turns out to be important due to the significant influence of scalar-relativistic effects on the spatial distribution of the orbitals.

B. Dipole moments and nuclear quadrupole-coupling constants of AuF and XeAuF

The molecule XeAuF is of considerable chemical interest, since it involves a covalent bond between two assumed “inert” elements, Xe and Au. Based on quantum-chemical calculations, a covalent bond between Xe and Au was first predicted for the closely related XeAu+ ion by Pyykkö62 and later for XeAuF by Lovallo and Klobukowski.63 The existence of the covalent Xe–Au bond in XeAuF was confirmed by the rather short experimental bond distance found in the analysis of the corresponding rotational spectrum by Cooke and Gerry.64 Furthermore, the surprisingly large difference between the Au quadrupole-coupling constants in XeAuF and AuF suggested a major reorganization of the charge distribution in the formation of the Xe–Au bond.64, 65 Accurate calculations of molecular properties for gold-containing compounds, especially Au quadrupole-coupling constants, remain challenging due to the importance of both relativistic and electron-correlation effects.66–72 The most accurate calculations for XeAuF reported so far were carried out using DCCC methods by Belpassi et al.66, 67 However, the high computational cost and the use of finite-difference techniques to obtain the electron-correlation contributions to the Au electricfield gradient confined the treatment of electron correlation to a space spanned by a rather small number of occupied and virtual spinors and also hampered the use of steep functions in the CC calculations. In the present study, we demonstrate that our SFDC-CC approach augmented by a perturbative treatment of SO effects

PXVP 0.218 × 10−5 0.533 × 10−4 0.00590 0.05800 0.8343 −0.00118 −0.0098 −0.2218

Total

DC-SFDC

−1.086 × 10−5 −1.135 × 10−4 −0.00264 −0.01524 −0.1580 −0.00022 −0.0065 −0.1190

−1.095 × 10−5 −1.173 × 10−4 −0.00319 −0.02574 −0.6958 −7.6 × 10−5 −0.0088 −0.3180

at the SFDC-HF level is capable of providing accurate values for the properties of AuF and in particular XeAuF. We exploit thereby the efficiency of the SFDC-CC scheme which allows the use of large basis sets and thus to investigate the effects of core correlation and the use of steep basis functions. In addition, the availability of an analytic scheme for computing first-order properties at the SFDC-CC level significantly facilitates these calculations. The results shown in Table VII indicate that the Au electric-field gradient in AuF is more sensitive to core correlation than the dipole moment. Correlating 24 valence and semi-core electrons including the 5s, 5p, 5d, and 6s orbitals of Au and the 2s and 2p orbitals of F yields an accurate value for the dipole moment of AuF, essentially indistinguishable from the value obtained in the corresponding all-electron treatment. However, the same frozen-core treatment turns out to be insufficient for the Au electric-field gradient. The calculation yields a value of 0.019 a.u., which should be compared to the value of 0.241 a.u. from the all-electron treatment. Freezing only the innermost 1s, 2s, 2p, 3s, 3p, 3d orbitals of the gold atom reduces the error significantly down to 0.024 a.u.. Errors of this magnitude are acceptable in the case of XeAuF, but they are still non-negligible for AuF due to its rather small total value of the Au electric field gradient. Therefore, in our calculations we correlated all electrons in the case of AuF, while we used the frozen-core approximation in the case of XeAuF with a small core consisting of the 1s, 2s, 2p, 3s, 3p, 3d orbitals of Au and the 1s, 2s, 2p, 3s, 3p orbitals of Xe. To analyze the basis-set effects in the calculation of Au electric-field gradients in some detail, we plotted in Figure 1 the convergence pattern of such calculations for AuF when going from BAS0 to BAS16. It is immediately obvious that it is necessary to add steep functions to the unc-ANO-RCC

TABLE VII. Dipole moments μ (in Debye) and electric-field gradient q at the Au nucleus (in a.u.) of the AuF molecule obtained at SFDC/CCSD(T)/unc-ANO-RCC level. All calculations were carried out using the Gaussian nuclear model. No. of correlated electrons μ q

88

60

40

24

16

4.27 0.241

4.27 0.217

4.29 0.109

4.27 0.019

4.39 − 2.057

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214114-9

Cheng, Stopkowicz, and Gauss

J. Chem. Phys. 139, 214114 (2013)

FIG. 1. Basis-set convergence of the Au electric-field gradient in AuF calculated at the SFDC-CCSD level. The definition of the Bas-n sets (n = 0-16) are given in Sec. III.

set for reaching convergence. We also note that BAS3, a rather small basis set, gives a result very close to the basisset limit, which also explains the good results obtained in Ref. 66, as Dyall’s valence quadruple-zeta basis used there covers a similar range of exponents as BAS3. Since BAS14 provides essentially converged results for the Au electric-field gradient in the case of AuF, we carried out SFDC-CCSD(T) calculations for both AuF and XeAuF using this basis for Au and the unc-ANO-RCC sets for F and Xe. The results are given in Table VIII. Concerning the dipole moments, we note that the electron-correlation contributions amount to 30% in the case of AuF and to 8% in the case of XeAuF. On the other hand, SO effects on the dipole moments are rather small, e.g., 1% in the case of AuF and even negligible as for XeAuF. Our results agree well with both the DCCCSD(T) results67 and the latest experimental data for AuF.73 The effective-core-potential (ECP) based CCSD(T) calculations reported in Ref. 68 also provided reasonable values for the dipole moments of AuF and XeAuF. The Au quadrupole-coupling constants are more sensitive to electron-correlation effects than the dipole moments. As is seen in Table VI, the SFDC-HF results are unreliable: the electron-correlation contributions to the Au quadrupolecoupling constants amount to −560 MHz in the case of AuF and to −460 MHz in the case of XeAuF. The SO contributions are also significant, i.e., they amount to around −35 MHz

(two third of the total value) for AuF and to −37 MHz (6% of the total value) for XeAuF. Therefore, it turns out essential to consider SO corrections in computational studies aiming at an accurate theoretical prediction of Au quadrupolecoupling constants. The relative errors of our best results in comparison to the experimental data are 11% in the case of AuF and less than 3% for XeAuF. The remaining discrepancy might be attributed to the neglect of the Breit interaction in

TABLE VIII. Dipole moments μ (in Debye) and nuclear quadrupolecoupling constants −eQq (in MHz). BAS14 for Au and unc-ANO-RCC basis sets for Xe and F were used in the SFDC calculations. The perturbative SO corrections were obtained at the HF level using unc-ANO-RCC basis sets. All calculations were carried out using the Gaussian nuclear model. −eQq(Au)

μ

SFDC-HF SFDC-CCSD(T) +SO(2) ECP-CCSD(T)68 NR-CCSD(T)+Rel.70 NESC-CCSD69 DC-CCSD(T)66, 67 Exp.64, 65, 73

AuF

XeAuF

AuF

XeAuF

–eQq(Xe) XeAuF

5.46 4.27 4.23 4.44 ... ... 4.29 4.13

7.22 6.70 6.69 6.64 ... ... 6.76 ...

− 587 − 12 − 47 ... ... − 54 − 46 − 53.2

− 943 − 480 − 517 ... − 823 − 549 − 539 − 527.6

− 110 − 135 − 136 ... − 122 ... ... − 134.5

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214114-10

Cheng, Stopkowicz, and Gauss

the present treatment, which was shown to make a contribution of around −7 MHz in the case of AuF and −5 MHz in the case of XeAuF.66 Our result for XeAuF (−517 MHz) differs from the DC-CCSD(T) result reported in Ref. 66 (−539 MHz) by around 22 MHz. This is mainly due to the use of a large core in the frozen-core approximation in Ref. 66: when we used the same frozen-core approximation as Ref. 66,74 the SFDC-CCSD(T) result decreased by 22 MHz in comparison to the value given in Table VI. It should be furthermore noted that a non-relativistic CCSD(T) calculation augmented with a relativistic correction obtained at the HF level yielded a rather inaccurate value of −823 MHz.70 This underlines the importance of the coupling between scalarrelativistic and electron-correlation effects due to the significant influence of scalar-relativistic effects on the wave function. Note that in the present scheme we neglect only the coupling between SO and electron-correlation effects. This is a rather safe approximation, since SO effects on the wave function are relatively small. We should finally mention that the good agreement between the NESC-CCSD results in Ref. 69 and the experimental data arises from a major cancellation of the contributions from triple excitations, SO effects, and basis-set effects and thus is fortuitous. By comparing our SFDC-CCSD and SFDC-CCSD(T) result, we can see that the contributions from triple excitations amount to around 100 MHz in the case of AuF and to around 80 MHz in the case of XeAuF. This also corroborates the findings in Ref. 66. Finally, for the Xe quadrupole-coupling constants in AuXeF, SFDC-HF calculations already provide a qualitatively correct result and SFDC-CCSD(T) calculations yield a value (−134 MHz) that compares favorably with the experimental data (−134.5 MHz). The SO correction for the Xe quadrupole-coupling constant (−1 MHz) is less than 1% of the total value and thus negligible. V. SUMMARY AND OUTLOOK

In the present paper, a perturbative scheme for calculating second-order SO corrections to SFDC-HF energies has been formulated and implemented. The scheme takes the difference between the DC and SFDC Hamiltonians as perturbation and is formulated in terms of energy derivatives, i.e., the secondorder SO corrections are obtained as second derivatives of the SFDC-HF energy. For high-accuracy treatments including relativistic effects, we suggest to augment electron-correlated SFDC calculations, e.g., of the coupled-cluster type, by perturbative SO corrections computed at the SFDC-HF level. Benchmark calculations for the hydrogen-halide series as well as for the coinage-metal fluorides indicate the good performance of our proposed scheme. For energies and dipole moments, the perturbative scheme provides accurate SO corrections even for compounds containing fifth-row elements. As for electric-field gradients, an example for core properties, the perturbative treatment performs well for systems not involving 6p elements. In a first application to AuF and XeAuF, we demonstrate the feasibility of our approach for high-accuracy calculations, as the obtained dipole moments and gold electric-field gradients turn out to be in good agreement with the experimental results available in the literature,

J. Chem. Phys. 139, 214114 (2013)

provided sufficiently large basis sets are used and SO effects are incorporated. A limitation of the present scheme is that SO effects are not treated at an electron-correlated level. Our scheme thus works well when the coupling of SO interactions and electron correlation is small, which is probably often the case. Nevertheless, we plan to extend the current perturbative scheme to electron-correlation treatments at the coupled-cluster level. A further issue is the neglect of the Breit term and consequently of the spin-other-orbit interaction due to the current restriction to the DC Hamiltonian. Therefore, another future extension is to incorporate the Breit term for the proposed approach. Additional efforts here include the formulation of a spin separation for the Breit term as well as from the computational side the implementation of additional relativistic two-electron integrals.

ACKNOWLEDGMENTS

L.C. is grateful to John F. Stanton (Austin) for the support and for careful reading of the paper. The part of the work done in Mainz has been supported by the Deutsche Forschungsgemeinschaft (DFG) (DFG, GA 370/5-1) and the Fonds der Chemischen Industrie. The part of the work done in Austin has been supported by Robert A. Welch Foundation (Grant No. F-1283) and (U.S.) Department of Energy (DOE), Office of Basic Energy Sciences (Contract No. DEFG02-07ER15884) granted to John F. Stanton. 1 I.

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214114-11 24 L.

Cheng, Stopkowicz, and Gauss

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