Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory.

Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory Lan Cheng and Jürgen Gauss Citation: The Journal of Chemical Physics 141, 164107 (2014); doi: 10.1063/1.4897254 View online: http://dx.doi.org/10.1063/1.4897254 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analytic first derivatives for a spin-adapted open-shell coupled cluster theory: Evaluation of first-order electrical properties J. Chem. Phys. 141, 104102 (2014); 10.1063/1.4894773 Perturbational treatment of spin-orbit coupling for generally applicable high-level multi-reference methods J. Chem. Phys. 141, 074105 (2014); 10.1063/1.4892060 Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spin-orbit effects at the Hartree-Fock level J. Chem. Phys. 139, 214114 (2013); 10.1063/1.4832739 Analytic second derivatives in closed-shell coupled-cluster theory with spin-orbit coupling J. Chem. Phys. 131, 164113 (2009); 10.1063/1.3245954 Closed-shell coupled-cluster theory with spin-orbit coupling J. Chem. Phys. 129, 064113 (2008); 10.1063/1.2968136

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.2.10.23 On: Sun, 26 Oct 2014 12:04:57

THE JOURNAL OF CHEMICAL PHYSICS 141, 164107 (2014)

Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory Lan Cheng1,a) and Jürgen Gauss2,b) 1

Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, Texas 78712, USA 2 Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany

(Received 18 May 2014; accepted 24 September 2014; published online 24 October 2014) This work deals with the perturbative treatment of spin-orbit-coupling (SOC) effects within the spinfree exact two-component theory in its one-electron variant (SFX2C-1e). We investigate two schemes for constructing the SFX2C-1e SOC matrix: the SFX2C-1e+SOC [der] scheme defines the SOC matrix elements based on SFX2C-1e analytic-derivative theory, hereby treating the SOC integrals as the perturbation; the SFX2C-1e+SOC [fd] scheme takes the difference between the X2C-1e and SFX2C1e Hamiltonian matrices as the SOC perturbation. Furthermore, a mean-field approach in the SFX2C1e framework is formulated and implemented to efficiently include two-electron SOC effects. Systematic approximations to the two-electron SOC integrals are also proposed and carefully assessed. Based on benchmark calculations of the second-order SOC corrections to the energies and electrical properties for a set of diatomic molecules, we show that the SFX2C-1e+SOC [der] scheme performs very well in the computation of perturbative SOC corrections and that the “2eSL” scheme, which neglects the (SS|SS)-type two-electron SOC integrals, is both efficient and accurate. In contrast, the SFX2C-1e+SOC [fd] scheme turns out to be incompatible with a perturbative treatment of SOC effects. Finally, as a first chemical application, we report high-accuracy calculations of the 201 Hg quadrupole-coupling parameters of the recently characterized ethylmercury hydride (HHgCH2 CH3 ) molecule based on SFX2C-1e coupled-cluster calculations augmented with second-order SOC corrections obtained at the Hartree-Fock level using the SFX2C-1e+SOC [der]/2eSL scheme. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897254] I. INTRODUCTION

In chemistry and molecular physics, relativistic effects incorporate scalar-relativistic (SR) and spin-orbit-coupling (SOC) contributions.1 Both of these contributions can be rigorously accounted for in the four-component framework provided by the Dirac equation. Computationally, this leads to the relativistic Dirac-Coulomb and Dirac-Coulomb-Breit approaches depending on the actual choice for the Hamiltonian.2 However, the four-component approaches are computationally demanding due to the presence of positronic degrees of freedom as well as due to spin-symmetry breaking. As a consequence, these four-component schemes have found only limited use in practical applications to problems in chemistry and molecular physics. An obvious route to more costeffective relativistic quantum-chemical approaches consists in the decoupling of the electronic and positronic degrees of freedom and in focusing on the “electrons-only” formulations that lead to the so-called two-component theories.3–16 Another route for ensuring computational efficiency consists in a separate treatment of SR and SOC effects. Spin separation is possible within both the four- and two-component theories and both options have been proposed and used in the literature.3, 17–19 As the spin-free effects usually dominate the a) Electronic mail: [email protected] b) Electronic mail: [email protected]

0021-9606/2014/141(16)/164107/11/$30.00

relativistic corrections, this leads to computational schemes in which a SR calculation is augmented by a perturbative treatment of SOC effects.3, 18, 20–24 Clearly, the separate treatment of SR and SOC effects is the key to efficient relativistic electron-correlation calculations. A major recent achievement in relativistic quantum chemistry is the development of the exact two-component (X2C) theory.5, 6, 16, 25–37 In X2C theory, the matrix representation of the Dirac equation is block diagonalized by a Foldy-Wouthuysen transformation38 in a single step and the subsequent quantum-chemical calculation is then based on the resulting “electrons-only” Hamiltonian matrix. The X2C “electrons-only” Hamiltonian matrix exactly reproduces the spectrum of the corresponding four-component Hamiltonian matrix for a given one-electron basis set. A further advantage of the X2C scheme is that its simple block-diagonalization technique greatly facilitates the construction of analytic derivatives of the X2C Hamiltonian matrix elements.39–44 As the exact block diagonalization of the four-component Hamiltonian matrix requires information about the four-component wave function, the construction of the X2C Hamiltonian matrix elements is as expensive as the solution of the corresponding four-component equation. Therefore, the practical computational efficiency of the X2C approach originates from using the X2C Hamiltonian matrix constructed at a lower level of theory in higher-level quantum-chemical calculations, since the computational cost is usually dominated

141, 164107-1

© 2014 AIP Publishing LLC


164107-2

L. Cheng and J. Gauss

by the level of theory used for treating the electron-electron interactions. The most popular choice is perhaps the X2C-1e scheme, which performs the block diagonalization of the four-component Hamiltonian matrix at the one-electron level and then employs the resulting X2C one-electron Hamiltonian matrix together with untransformed Coulomb interactions in the many-electron treatment. This X2C-1e scheme has been shown to perform very well for the treatment of SR effects,27 as those are dominated by one-electron contributions and the SR two-electron contributions mostly affect the inner shells.45, 46 In particular, the spin-free version of the X2C-1e scheme (SFX2C-1e) has become a useful protocol for treating SR effects on energies and molecular properties. We note that the computational efficiency of the X2C-1e scheme can be further enhanced using local approximations in the block diagonalization combined with density-fitting in the treatment of the electron-electron interactions.47–50 The full X2C scheme with SOC included has been formulated and implemented at the self-consistent-field level for Hartree-Fock and density-functional theory calculations.35, 36 However, such an approach offers no computational advantages in comparison to the corresponding four-component approach. Again, computational benefits are only realized within a one-electron approximation and in this context X2C-1e studies with both variational34, 48, 51, 52 and perturbational22, 23 treatments of SOC effects have been reported. Various types of decoupling schemes for the SOC operator including the X2C, Douglas-Kroll-Hess (DKH), and Barysz-Sadlej-Snijder (BSS) schemes have been analyzed in Ref. 51 and the third-order DKH-type expansion has been recommended as a simple, though accurate scheme. However, a major computational challenge for treating SOC effects is the inclusion of two-electron SOC contributions, which, in contrast to the two-electron SR contributions, are far from negligible. The most efficient, although approximate, scheme is a screening of the nuclear charge in the one-electron SOC integrals. This simple approach of scaling the nuclear charge has become quite popular53–55 but has also been shown to be unreliable in the treatment of heavier-element compounds.56 Therefore, more elaborate screening schemes have been proposed in the literature, e.g., the screened nuclear spin-orbit approximation originally proposed by Boettger57 and the flexible nuclear screening spin-orbit approximation recently developed by Chalupský and Yanai.58 Based on Boettger’s approach, Filatov et al. have recently reported an efficient implementation of a variational X2C-1e scheme.52 However, an explicit inclusion of the two-electron SOC contributions is needed for more rigorous treatments aiming at high accuracy. In this context, a cost-effective approach to include the two-electron SOC contributions is the mean-field approximation for the two-electron SOC integrals as introduced by Hess et al.,21 which has been inspired by earlier work of Blume et al.59, 60 Efficient implementations of both the atomic mean-field approach61–64 and the molecular meanfield approach65, 66 based on the Breit-Pauli Hamiltonian or the DKH approach have been widely used in actual calculations of SOC splittings and various SOC dependent molecular properties. Recently, Li et al.22 outlined a molecular meanfield approach for constructing X2C SOC matrix elements

J. Chem. Phys. 141, 164107 (2014)

together with a scheme for calculating SOC splittings in the framework of time-dependent density-functional theory. An efficient X2C scheme using mean-field SOC contributions obtained from atomic four-component SCF solutions has also recently been implemented by Knecht and coworkers.67 In the present work we report a thorough investigation of a few aspects concerning the perturbative treatment of SOC effects on top of SFX2C-1e calculations, including the construction of the X2C-1e SOC matrix elements from fourcomponent SOC integrals, the mean-field approximation to include two-electron SOC contributions, as well as a variety of integral approximations. In Sec. II A we discuss two schemes for constructing the X2C-1e SOC matrix elements. The first scheme, hereafter referred to as “SFX2C-1e+SOC [fd]” with “fd” being the abbreviation of “full diagonalization,” is probably the most straightforward and consists in taking the difference between the X2C-1e and SFX2C-1e Hamiltonian matrices as the X2C-1e SOC matrix. However, since we are interested in perturbative treatments of SOC effects, we also present an alternative scheme denoted as “SFX2C1e+SOC [der]” with “der” being the abbreviation of “derivative theory,” which only includes the leading contribution of the SOC integrals based on SFX2C-1e analytic-derivative theory, hereby treating the SOC integrals as a perturbation in the X2C block-diagonalization procedure. In Sec. II B, we present a rigorous formulation of the mean-field approach to account for the two-electron SOC contributions that includes all the four-component two-electron SOC integrals. Thereafter, we introduce a variety of integral approximations defined by neglecting certain classes of two-electron SOC integrals. Computational details of our implementations and the calculations are described in Sec. III. In Sec. IV, we report benchmark calculations for the second-order SOC corrections to energies, dipole moments, and electric-field gradients for a set of closed-shell diatomic molecules. In order to assess the accuracy of the SFX2C-1e SOC matrix elements, the perturbative inclusion of SOC effects on top of SFX2C-1e calculations has been compared with the perturbative treatment of SOC effects in the four-component theory.24 We also report a chemical application of the scheme to the computation of the second-order SOC corrections to the 201 Hg quadrupole-coupling constants of the ethylmercury hydride (HHgCH2 CH3 ) molecule recently characterized using rotational spectroscopy.68 In Sec. V we present a summary and an outlook on future development. II. THEORY A. Spin-orbit-coupling matrix elements in the exact two-component theory

The Dirac equation in its matrix representation in terms of a kinetically balanced basis set69 can be written as CL+ CL− CL+ CL− = ESD , (1) hD CS+ CS− CS+ CS− hD =

V T S 0 , SD = . T W−T 0 T/2c2

(2)


164107-3


J. Chem. Phys. 141, 164107 (2014)

In the present notation, bold letters represent 2N × 2N matrices with N as the number of large-component basis functions. CL+ and CS+ denote the large- and small-component coefficients for positive energy state (PES) solutions, and CL− and CS− denote corresponding coefficients for negative energy state (NES) solutions. The nuclear potential V, the kinetic energy T, and the overlap S matrices can be written as V 0 T 0 S 0 V= I, T = I, S = I, 0 V 0 T 0 S I=

(3)

IN×N

0

0

IN×N

with V , T, and S being the corresponding non-relativistic integral matrices and IN × N being the N × N identity matrix. The matrix W includes the relativistic one-electron integrals and accounts for both SR and SOC contributions W = WSF + WSOC , SF

W

=

W SF

0

0

W SF

WSOC =

W

u=x,y,z

SF Wμν =

I,

SOC,u

0

(4)

0 W SOC,u

(5)

σu ,

1 f |p · V p|f ν , 4c2 μ (6)

SOC,u Wμν

σz =

IN×N

0

0

−IN×N

† hFW + = R L+ R,

(10)

L+ = V + X† T + TX + X† (W − T)X,

(11)

together with untransformed Coulomb interactions. The X matrix in Eq. (11) relates the large- and small-component coefficients for the PES solutions CS+ = XCL+ .

CL+ = RC2c .

,

and {fμ } represents the set of large-component one-electron basis functions. For conciseness, we will use in the following the abbreviation: O 0 O 0 (8) I → OI, σ → Oσu . 0 O 0 O u The X2C-1e scheme is based on a one-step block diagonalization of the one-electron Dirac Hamiltonian matrix T hFW 0 + † V U U= (9) T W−T 0 hFW − and the subsequent many-electron treatment employs the “electrons-only” two-component Hamiltonian matrix hFW +

(13)

32

Its explicit form is given by

R = ( S−1 S)1/2 ,

(14)

1 S = S + 2 X† TX. (15) 2c In the spin-free X2C-1e (SFX2C-1e) scheme, the SFX2C-1e SF is obtained by skipping WSOC in Hamiltonian matrix hFW, + the block-diagonalization procedure. The matrices X, R, and SF are then all spin independent. hFW, + Let us now consider the perturbative inclusion of SOC effects after a SFX2C-1e calculation. A straightforward definition of the SOC perturbation within the X2C-1e scheme FW, SF . is possible based on the difference between hFW + and h+ Since {I, σ x , σ y , σ z } forms a complete basis set for twocomponent Hermitian matrices, hFW + can be decomposed in the following way: FW,sr I + hFW,x σx + h+ hFW + = h+ +

(7)

(12)

R is a renormalization matrix and relates the two-component wave functions for the electronic states and the largecomponent wave function for the PES solutions

FW,y

1 = 2 fμ |i(p × V p) u |fν , 4c

where σ x,y,z are the matrix representation of the Pauli spin matrices: 0 IN×N 0 −iIN×N σx = , σy = , IN×N 0 iIN×N 0

given by

σy + hFW,z σz . +

(16)

The SOC matrix can then be written as FW,y

SF = (hFW,sr − hFW, )I + hFW,x σx + h+ hFW,SOC + + + +

σy + hFW,z σz . + (17)

We will refer to the scheme based on Eq. (17) as “SFX2C1e+SOC [fd].” Since we deal in the present work with the perturbative treatment of SOC effects, we also explore an alternative definition of the X2C-1e SOC matrix that includes only the leading-order SOC contribution in the block-diagonalization procedure. This scheme will be referred to as “SFX2C1e+SOC [der].” Here we attach a set of perturbation parame = (λx , λy , λz ) to WSOC : ters λ WSOC → λu W SOC,u σu . (18) u=x,y,z

The matrix series:

hF+W

hFW + [λ]

is then expanded in the following Taylor

∂hFW + = + λu ∂λ u λ=0 u=x,y,z 1 ∂ 2 hFW + + λu λv + · · · . 2 u,v=x,y,z ∂λu ∂λv λ=0 hFW,SF +


164107-4


J. Chem. Phys. 141, 164107 (2014)

The derivatives of hFW + can be obtained from X2C analyticderivative theory. For the first derivatives of hFW + , one obtains

∂hFW +

∂λu λ=0

† ∂L ∂R SF SF = RSF† + RSF + L R + h.c. , + ∂λu λ=0 ∂λu λ=0 (19)

i.e., an expression that involves the first derivatives of the four-component integrals as well as those of the X and R matrices. We note that the first derivatives of hFW + are purely spin ∂hFW dependent, i.e., ∂λ+ can be written as a matrix times σ u : u

∂hFW + = hFW,u(1) σu , + ∂λu

(20)

SOC and that the leading contribution to the scalar part of hFW, , + sr SF − hFW, )I, stems from the second derivatives of i.e., (hFW, + + hFW + :

sr SF (2) (hFW, − hFW, ) I= + +

1 2

∂ 2 hFW + λ2u . 2 ∂λ u λ=0 u=x,y,z

(21)

In the following we consider both the first derivatives of and the scalar part of its second derivatives, since both of them contribute to the second-order SOC correction. hFW +

V T T W−T V + VSOC, 2e, LL → T + VSOC, 2e, SL

T + VSOC, 2e, LS . W − T + VSOC, 2e, SS

(22)

The four-component mean-field two-electron SOC matrices VSOC, 2e, LL , VSOC, 2e, LS , VSOC, 2e, SL , and VSOC, 2e, SS are obtained by using the spin-free Dirac-Coulomb (SFDC) molecular orbitals together with the corresponding two-electron SOC integrals, i.e., the difference between the Dirac-Coulomb and the SFDC two-electron integrals. After spin averaging, this leads to the following spin-dependent and scalar contributions: VSOC, 2e, XY = V SOC, c, XY,u σu + V SOC, c, XY,sr I, X, Y = L, S. u=x,y,z

(23) The “c” in the superscript in Eq. (23) denotes “Coulomb” and the explicit expressions for the spin-dependent matrix elements are SOC, c, LL, u Vμν = 0,

(24)

SOC, c, LS, u Vμν 1 SL =− 2 Dσρ iuvw (fμ fρ |{pv fσ }{pw fν }), 8c σρ v,w=x,y,z

(25) B. A mean-field approach for the two-electron spin-orbit-coupling contributions

The use of a mean-field approximation is the key to an efficient treatment of the two-electron SOC contributions. The basic idea here is to augment the one-electron Dirac Hamiltonian matrix with the mean-field SOC matrix:

SOC, c, SS, u Vμν =

(26)

1 LL Dρσ iuvw (fρ fσ |{pv fμ }{pw fν }) 4c2 σρ v,w=x,y,z +

1 SS Dρσ iuvw ({pf ρ } · {pf σ }|{pv fμ }{pw fν }) 16c4 σρ v,w=x,y,z

−

1 SS Dρσ iuvw ({pf ρ } · {pf ν }|{pv fμ }{pw fσ }) 32c4 σρ v,w=x,y,z

−

1 SS Dρσ iuvw ({pf μ } · {pf σ }|{pv fρ }{pw fν }), 32c4 σρ v,w=x,y,z

in which is the Lévi-Civitá symbol and the Mulliken notation has been employed for the two-electron integrals. In the equations given above, DXY denote the density matrices from the SFDC Hartree-Fock calculation: XY = ni (C+X )μi (C+Y )νi , X, Y = L, S, (28) Dμν i

SOC, c, SL, u Vμν 1 LS =− 2 Dρσ iuvw (fρ fν |{pv fμ }{pw fσ }), 8c σρ v,w=x,y,z

(27)

with the sum running over occupied orbitals and ni as the orbital occupation number, i.e., ni = 2 for doubly occupied orbitals and ni = 1 for singly occupied orbitals. We mention that the expressions for the mean-field SOC matrix elements before spin averaging have been given in Eqs. (21)–(23) of Ref. 24. From the terms that contain two Pauli spin


164107-5


J. Chem. Phys. 141, 164107 (2014)

matrices, we include the scalar contribution in the small-small block: SOC, c, SS,sr Vμν =

1 SS D uvw umn ρσ 32c4 σρ u,v,w,m,n=x,y,z

C. Second-order SOC corrections to energies and electrical properties

The second-order SOC corrections to the Hartree-Fock energy consist of two contributions E SOC = E SOC,sr + E SOC,sd .

×({pv fμ }{pw fσ }|{pm fρ }{pn fν }). We note that the Breit contribution to the SOC can be approximately taken into account by including the following additional terms in VSOC, 2e, XY with “g” in the superscript being the abbreviation of “Gaunt”:70

The “scalar contribution” ESOC,sr is obtained as a simple expectation value FW,sr SF E SOC,sr = (h+ − hFW, )μν Dμν + μν

+

SOC, g, LL,u Vμν

=−

1 SS Dσρ iuvw [(fμ fρ |{pv fσ }{pw fν }) 8c2 σρ v,w=x,y,z

+ (fσ fν |{pv fμ }{pw fρ })],

(29)

SOC, 2e, SS, sr SS Vμν Dμν

(33)

μν SF with respect to (hFW,sr − hFW, )I and VSOC, 2e, SS, sr . The + + SFX2C-1e Hartree-Fock density matrix is given by 2c 2c ni Cμi Cνi , (34) Dμν = i

SOC, g, SS,u

Vμν

=−

(32)

1 LL Dρσ iuvw [(fρ fν |{pv fμ }{pw fσ }) 8c2 σρ v,w=x,y,z

+ (fμ fσ |{pv fρ }{pw fν })].

(30)

Within the SFX2C-1e framework, further approximations to C+L and C+S are needed to ensure an efficient evaluation of the VSOC, 2e, XY matrices. Here we adopt a scheme advocated by Li et al.22 that constructs CL+ and CS+ from the X and R matrices obtained from the solution of the spinfree one-electron Dirac equation and the C2c matrix from a SFX2C-1e Hartree-Fock calculation, i.e., CL+ = RC2c , CS+ = XCL+ .

(31)

This is a well justified approximation, since the X matrix obtained from the solution of the one-electron problem has been shown to be a good approximation to the full X matrix.35, 36 We note that the definition of the SOC matrix elements in our SFX2C-1e+SOC[der] approach differs from that in Ref. 22 in the transformation from the four-component to two-component picture, i.e., we compute the first derivatives of the Hamiltonian matrix elements with respect to the fourcomponent mean-field SOC matrix, while the approach proposed in Ref. 22 adopts the DKH1-type expansion technique that does not include the response of the X matrix. The two approaches thus differ by terms of the order c−4 . Finally, various approximate schemes can be systematically defined by neglecting certain types of the twoelectron SOC integrals. We explore here three different variants: the “2eSL” scheme neglects the (SS|SS)-type ν }|{pv fμ }{pw fσ }) SOC integrals including the ({pf ρ } · {pf and ({pv fμ }{pw fσ }|{pm fρ }{pn fν }) integrals, the “2eSL(1c)” scheme in addition neglects the multi-center two-electron SOC integrals, and the “1e” scheme finally represents the extreme case in which all two-electron SOC integrals are skipped.

and DSS is evaluated using C2c and the one-electron X and R matrices as described in Subsection II B. The determination of the “spin-dependent” contribution ESOC,sd requires the solution of the coupled-perturbed Hartree-Fock (CPHF) equations71–73 with respect to the perσu in order to obtain the CPHF coefficients turbations hFW,u + USOC,u = USOC,u σ u . For closed-shell molecules, the spinintegrated CPHF equation can be written as SOC,u Ubj ((aj |bi) − (ab|j i)) (a − i )UaiSOC,u + bj

= −(hFW,u(1) )ai +

(35)

and ESOC,sd is then given by SOC,u FW,u(1) Uai (h+ )ai . E SOC,sd = 2

(36)

u=x,y,z ai

Second-order SOC corrections to electrical properties such as dipole moment or electric-field gradient are obtained by means of finite-field calculations in which the nuclear attraction operator is augmented with the property operator multiplied by a corresponding field strength λ, i.e., V in Eqs. (3) and (6) is replaced with V + λVprop . The property operator takes the form μ ˆ = −r

(37)

in case of the dipole moment and qˆuv =

2 3(ru − Ru )(rv − Rv ) − δuv |r − R| , u, v = x, y, z, 5 |r − R| (38)

in case of the electric-field gradient, in which r and R represent the positions of the electron and the target nucleus, respectively.


164107-6


J. Chem. Phys. 141, 164107 (2014) ∂C

III. IMPLEMENTATIONAL AND COMPUTATIONAL DETAILS

A suite of computer programs for the calculation of the second-order SOC corrections at the Hartree-Fock level has been implemented within the SFX2C-1e framework into a local version of the CFOUR program package.74 The meanfield SOC contributions to the four-component Hamiltonian matrix are obtained using the SFX2C-1e Hartree-Fock (HF) density matrix, the X and R matrices obtained from the solution of the spin-free Dirac equation, and the two-electron SOC integrals. In the present implementation we considered the two-electron SOC integrals in the Dirac-Coulomb framework and did not include the Breit contribution in Eqs. (29) and (30). The SFX2C-1e SOC matrix elements are then constructed by transforming the four-component SOC matrix elements to the two-component picture using the X2C blockdiagonalization scheme. Here the two schemes described in Sec. II, i.e., the SFX2C-1e+SOC [der] scheme based on Eq. (17) and the SFX2C-1e+SOC [fd] scheme based on Eqs. (20) and (21), have been implemented. As a final step, the required coupled-perturbed HF equations, Eq. (35), are solved and the second-order SOC corrections to energies are computed according to Eqs. (32), (33), and (36). We refer the readers to Refs. 24 and 75 for detailed descriptions of the numerical differentiation procedure needed to compute the dipole moments and electric-field gradients, including the choice of field strength and the polynomial-fitting procedure for obtaining the expansion coefficients. We note that two types of algorithms for computing the derivatives of the X matrix have been described in the literature.40, 51, 76–78 The present authors developed a scheme based on unperturbed and perturbed CL+ and CS+ .40 The first derivatives of the X matrix with respect to a perturbation parameter λ, for example, is obtained by differentiating Eq. (12) as S ∂C+ ∂CL+ ∂X = −X (CL+ )−1 . (39) ∂λ ∂λ ∂λ We mention that in this subsection all the derivatives are always taken at λ = 0. The derivatives of CL = (CL+ , CL− ) and CS = (CS+ , CS− ) are easily obtained from the solution of the perturbed one-electron Dirac equation. More specifically, the perturbed one-electron Dirac equation is first solved in the representation of the unperturbed orbitals: ∂CP Q ∂λ =

∂(hD )P Q ∂λ

−

∂(SD )P Q ∂λ

(hD )QQ /((hD )QQ − (hD )P P ) (40)

with P, Q running over both PES and NES orbitals, and the perturbed coefficients are then transformed back to the atomic-orbital basis representation ∂CL /∂λ CL ∂C/∂λ. (41) = ∂CS /∂λ CS For orbital pairs PQ degenerate with respect to the orbital energies, i.e., (hD )PP − (hD )QQ = 0, the corresponding perturbed

coefficients ∂λP Q are determined by means of the normalization condition ∂(SD )P Q ∂CP Q = −1/2 . (42) ∂λ ∂λ Nearly degenerate orbital pairs with orbital energy differences below a certain given threshold (typically chosen as 10−4 a.u. in actual calculations) are treated as degenerate in order to maintain numerical stability. Algorithms for computing the derivatives of the X matrix can also be derived by analyzing the perturbed decoupling condition.76–78 A non-iterative algorithm involving unperturbed CL and CS for both PES and NES solutions has been developed by Filatov et al.77 and has been extended to higher-order derivatives by Li et al.51 For first derivatives, the corresponding working equation is77 ∂X/∂λ = (CS− − XCL− )OCL+ S

(43)

with the matrix O, which couples PES and NES solutions, given by Opp =

λ (hλD )pp − (SD )pp (hD )p p

L† ∂V

hλ = C−

(44)

(hD )p p − (hD )pp L† ∂T

CL+ + C−

∂λ ∂λ ∂(W − T) S† CS+ , +C− ∂λ

S† ∂V

CS+ + C−

∂λ

CL+ (45)

1 S† ∂T S C+ . C (46) ∂λ 2c2 − ∂λ In Eq. (44), the indices p and p run over PES and NES orbitals, respectively. We expect that the PES coefficients CL+ and CS+ are usually computed more accurately in terms of the number of significant digits than the corresponding NES coefficients: both CL+ and CS+ are of the order c0 ; CL− and CS− are of the order c−1 and c1 , respectively. Therefore, our previous studies on electrical and geometrical properties40, 41 were based on Eqs. (39)–(42). However, the SOC matrix elements that involve the high-lying orbitals localized around the heavy nuclei may acquire large absolute values. Consequently, the treatment of near degeneracies in the case of these orbitals is numerically more demanding in the computation of SOC matrix elements than in the calculation of other properties. Although this introduces no significant effects in the calculation of the SOC contributions involving the first derivatives of hFW + (Eq. (20)), the calculation of the contributions from the scalar part of hFW + (Eq. (21)) becomes numerically problematic in case of finite-field calculations for obtaining electrical properties. Therefore, we have adopted in the present work the alternative scheme of Filatov et al.,77 Eqs. (43)–(46), for computing the first derivatives of the X matrix. With the first derivatives of X thus obtained, the two types of algorithms for computing the second derivatives with respect to the SOC perturbations are both numerically stable. Benchmark calculations were carried out for a set of diatomic molecules consisting of the hydrogen halides (HX, L† ∂S

S λ = C−

CL+ +


164107-7


J. Chem. Phys. 141, 164107 (2014) TABLE I. Second-order spin-orbit-coupling (SOC) corrections to the HF energy (in Hartree). SFDC+SOC (or SFX2C1e+SOC) denotes perturbative SOC corrections on top of a SFDC (or SFX2C-1e) calculation. SFX2C1e+SOC [der] includes only the leading order SFX2C-1e SOC matrix elements; SFX2C-1e+SOC [fd] takes the difference between the X2C-1e and SFX2C-1e Hamiltonian matrix as the SOC matrix. The percentages in the parentheses refer to the ratios with respect to the SFDC+SOC results. All calculations have been carried out using the unc-ANO-RCC basis. The pointlike nuclear model has been used. SFX2C-1e+SOC [der]

FIG. 1. Molecular structure of HHgCH2 CH3 obtained at the SFX2C-1eCCSD(T)/ANO1 level. For the complete set of parameters see Ref. 88.

X=F-At) and the coinage-metal fluorides (MF, M=Cu, Ag, and Au). The geometries for the hydrogen halides and coinage-metal fluorides were taken from Refs. 79 and 24, respectively. The SOC corrections to energies and electrical properties such as dipole moment and electric-field gradient (efg) were computed using the SFX2C-1e+SOC [der] and SFX2C-1e+SOC [fd] schemes and the results were compared with those obtained from corresponding four-component calculations reported in Ref. 24. The accuracy of the three additional approximations in the treatment of the two-electron SOC integrals, i.e., the “2eSL,” “2eSL(1c),” and “1e” variants, was assessed by comparing the results with those of the full calculations. All benchmark calculations were carried out at the HF level using uncontracted ANO-RCC basis sets.80–82 As a first chemical application, we report the 201 Hg quadrupole-coupling parameters in the ethylmercury hydride (HHgCH2 CH3 ) molecule as obtained in high-accuracy calculations. The molecular structure of HHgCH2 CH3 (see Figure 1) was obtained at the SFX2C-1e coupled-cluster singles and doubles (CCSD)84 level augmented by a perturbative inclusion of triple excitations (CCSD(T)).85, 86 ANO basis sets of triple-zeta quality, i.e., with the contraction patterns of 25s22p16d12f4g/8s7p5d3f1g for Hg, of 14s9p4d3f/4s3p2d1f for C, and of 8s4p3d/3s2p1d for H, were used in the geometry optimization. These sets are in the following referred to as ANO1 and have been obtained by recontracting the ANO-RCC primitive sets for the SFX2C-1e scheme.87 The corresponding rotational constants for this geometry, A = 29 656 MHz, B = 2816 MHz, and C = 2654 MHz, are moderately accurate in comparison to the experimental values of A = 29 303 MHz, B = 2755 MHz, and C = 2597 MHz,68 with relative errors of around 1%–2%. Non-relativistic and SFX2C-1e HF, CCSD, and CCSD(T) calculations for the efg at the Hg nucleus were performed to enable a systematic analysis of both scalar-relativistic and electron-correlation effects. In these calculations, an uncontracted ANO-RCC basis set was used for mercury and the ANO1 sets were used in the case of carbon and hydrogen. The second-order SOC corrections for the Hg efg were obtained at the HF level using the SFX2C-1e+SOC/2eSL model. In these calculations, the SARC basis set for mercury83 and the cc-pVTZ basis sets

SFX2C-1e+SOC [fd]

SFDC+SOC24

HF −8.10 × 10−6 (100.0%) −8.09 × 10−6 (99.8%) −8.11 × 10−6 HCl −6.81 × 10−4 (100.0%) −6.74 × 10−4 (99.0%) −6.81 × 10−4 HBr −0.0831 (100.0%) −0.0798 (96.0%) −0.0831 HI −1.2542 (99.9%) −1.1678 (93.0%) −1.2551 HAt −30.2355 (99.9%) −26.7952 (88.5%) −30.2734 CuF −0.0241 (100.0%) −0.0234 (97.1%) −0.0241 AgF −0.5694 (99.9%) −0.5357 (94.0%) −0.5698 AuF −18.1588 (99.9%) −16.2276 (89.3%) −18.1796

for carbon and hydrogen89 were utilized in their fully decontracted form. The calculated efgs were converted to nuclear quadrupole-coupling parameters using a value of 387 mb for the 201 Hg quadrupole moment.90

IV. RESULTS AND DISCUSSIONS A. Benchmark calculations for hydrogen halides (HX, X=F-At) and coinage-metal fluorides (MF, M=Cu, Ag, and Au)

In our benchmark calculations, we first assess the accuracy of the perturbative treatment of SOC effects within the SFX2C-1e scheme (denoted by “SFX2C-1e+SOC [der]” and “SFX2C-1e+SOC [fd]”) by comparing the results with those obtained in the four-component framework (denoted by “SFDC+SOC”). The perturbative SOC corrections to energies, dipole moments, and electric-field gradients are reported in Tables I–III, respectively. It is seen that the

TABLE II. Second-order spin-orbit-coupling (SOC) corrections to the dipole moment (in a.u.) at the HF level. SFDC+SOC (or SFX2C1e+SOC) denotes perturbative SOC corrections on top of a SFDC (or SFX2C-1e) calculation. SFX2C-1e+SOC [der] includes only the leading order SFX2C-1e SOC matrix elements; SFX2C-1e+SOC [fd] takes the difference between the X2C-1e and SFX2C-1e Hamiltonian matrix as the SOC matrix. The percentages in the parentheses refer to the ratios with respect to the SFDC+SOC results. All calculations have been carried out using the unc-ANO-RCC basis. The point-like nuclear model has been used.

HF HCl HBr HI HAt CuF AgF AuF

SFX2C-1e+SOC [der]

SFX2C-1e+SOC [fd]

SFDC+SOC24

−6.72 × 10−6 (100.0%) −7.92 × 10−5 (100.0%) −1.84 × 10−3 (99.9%) −0.0114 (99.9%) −0.1270 (99.8%) −1.74 × 10−4 (99.8%) −9.12 × 10−4 (99.9%) −0.0179 (99.5%)

−6.71 × 10−6 (99.9%) −7.88 × 10−5 (99.5%) −1.81 × 10−3 (98.5%) −0.0110 (96.8%) −0.1188 (93.3%) −1.71 × 10−4 (98.3%) −8.72 × 10−4 (95.5%) −0.0170 (94.3%)

−6.72 × 10−6 −7.92 × 10−5 −1.84 × 10−3 −0.0114 −0.1273 −1.74 × 10−4 −9.14 × 10−4 −0.0180


164107-8


J. Chem. Phys. 141, 164107 (2014)

TABLE III. Second-order spin-orbit-coupling (SOC) corrections to the electric-field gradients (in a.u.) for the heavier nuclei of the diatomic molecules computed at the Hartree-Fock level. SFDC+SOC (or SFX2C1e+SOC) denotes perturbative SOC corrections on top of a SFDC (or SFX2C-1e) calculation. SFX2C-1e+SOC [der] includes only the leading order SFX2C-1e SOC matrix elements; SFX2C-1e+SOC [fd] takes the difference between the X2C-1e and SFX2C-1e Hamiltonian matrix as the SOC matrix. The percentages in the parentheses refer to the ratios with respect to the SFDC+SOC results. All calculations have been carried out using the unc-ANO-RCC basis. The point-like nuclear model has been used.


SFX2C-1e+SOC [der]

SFX2C-1e+SOC [fd]

SFDC+SOC24

−1.10 × 10−5 (100.0%) −1.18 × 10−4 (99.9%) −3.39 × 10−3 (101.3%) −0.0303 (99.7%) −1.1197 (100.3%) −6.89 × 10−5 (104.7%) −8.42 × 10−3 (99.9%) −0.2866 (100.1%)

−1.10 × 10−5 (100.3%) −1.22 × 10−4 (103.6%) −5.27 × 10−3 (157.5%) −0.0549 (180.7%) −1.7273 (154.8%) 1.17 × 10−5 (−17.7%) −4.27 × 10−3 (50.7%) −0.1002 (35.0%)

−1.10 × 10−5 −1.18 × 10−4 −3.35 × 10−3 −0.0304 −1.1161 −6.58 × 10−5 −8.43 × 10−3 −0.2863

“SFX2C-1e+SOC”[der] scheme provides accurate results for the second-order SOC corrections to both energies and electrical properties, with relative errors within 1% for most cases. The largest relative error is observed for the copper electricfield gradient in CuF and amounts to 4%. However, considering the small absolute magnitude of the SOC correction in this case, the error is still negligible for any practical purpose. In contrast to the excellent performance of the “SFX2C1e+SOC [der]” scheme, the “SFX2C-1e+SOC [fd]” scheme is less accurate in reproducing the “SFDC+SOC” results. As shown in Tables I–III, the “SFX2C-1e+SOC [fd]” scheme is unreliable in the case of compounds containing heavy elements as well as in the case of core properties such as the efgs. The poor performance of “SFX2C-1e+SOC [fd]” is tentatively attributed to an unbalanced mixture of contributions higher than second order introduced via the full blockdiagonalization procedure. Further investigations are required to elucidate the performance of the “SFX2C-1e+SOC [fd]” scheme; it might be of interest to analyze as well the performance of the “SFX2C-1e+SOC [fd]” scheme within a nonperturbative treatment of SOC effects. Since we focus in the present work on a perturbative inclusion of SOC effects, we

will consider in the following only the “SFX2C-1e+SOC [der]” scheme. In a further step, we discuss the benchmark results for the various integral approximations within the “SFX2C-1e+SOC [der]” scheme. From the results presented in Tables IV–VI, we see that the neglect of the (SS|SS)-type two-electron SOC integrals in the “2eSL” variant is a safe approximation. The relative errors introduced by this approximation amount to less than 1% for energies and dipole moments and to less than 2% for efgs with the special case of CuF excluded. The neglect of multi-center two-electron SOC integrals within the “2eSL(1c)” variant also performs well in general. However, such approximation breaks down for the HF molecule, probably because some occupied orbitals are here rather diffuse due to the high electronegativity of the fluorine. The onecenter approximation should therefore be used with care in the mean-field SOC treatment. Finally, as expected, the neglect of all two-electron SOC integrals within the “1e” model leads to a significant overestimation of SOC effects. Based on the benchmark results, we conclude that the “SFX2C-1e+SOC [der]” scheme combined with the “2eSL” integral approximation should be considered as a costeffective approach to account for second-order SOC corrections. In Subsection IV B, we further demonstrate the usefulness of this scheme in calculations for the ethylmercury hydride (HHgCH2 CH3 ) molecule. B. The 201 Hg quadrupole-coupling parameters of HHgCH2 CH3

The rotational spectrum of ethylmercury hydride (HHgCH2 CH3 ) has recently been recorded and analyzed by Goubet et al.68 Density functional theory calculations of the 201 Hg quadrupole-coupling constants using the spinfree Douglas-Kroll-Hess (SFDKH) scheme in order to account for scalar-relativistic effects have been used to facilitate the analysis of the spectrum and have been shown to yield qualitatively correct results. It should be noted that adequate treatments of relativistic and electron-correlation effects are required for accurate calculations of 201 Hg quadrupolecoupling parameters.91 In the following we present results from high-accuracy calculations of these parameters by augmenting SFX2C-1e coupled-cluster calculations with

TABLE IV. Second-order spin-orbit-coupling (SOC) corrections to the HF energy (in Hartree) obtained using various integral approximations. “2eSL” neglects the contributions from the (SS|SS)-type SOC integrals; “2eSL(1c)” in addition invokes the one-center approximation for the (SS|LL)-type SOC integrals. “1e” denotes the use of only the one-electron SOC integrals. The percentages in the parentheses refer to the ratios with respect to the results of the full scheme. All calculations have been carried out using the unc-ANO-RCC basis. The point-like nuclear model has been used.


1e

2eSL(1c)

2eSL

Full

−1.20 × 10−5 (148.1%) −7.89 × 10−4 (115.8%) −0.0884 (106.3%) −1.3021 (103.8%) −30.9218 (102.3%) −0.0262 (108.4%) −0.5947 (104.4%) −18.6061 (102.5%)

−3.50 × 10−6 (43.2%) −5.64 × 10−4 (82.8%) −0.0823 (99.0%) −1.2481 (99.5%) −30.1977 (99.9%) −0.0239 (99.0%) −0.5665 (99.5%) −18.1271 (99.8%)

−7.98 × 10−6 (98.4%) −6.72 × 10−4 (98.7%) −0.0825 (99.3%) −1.2488 (99.6%) −30.1983 (99.9%) −0.0239 (99.2%) −0.5666 (99.5%) −18.1274 (99.8%)

−8.10 × 10−6 −6.81 × 10−4 −0.0831 −1.2542 −30.2355 −0.0241 −0.5694 −18.1588


164107-9


J. Chem. Phys. 141, 164107 (2014)

TABLE V. Second-order spin-orbit-coupling (SOC) corrections to the dipole moment (in a.u.) computed at the HF level using various integral approximations. “2eSL” neglects the contributions from the (SS|SS)-type SOC integrals; “2eSL(1c)” in addition invokes the one-center approximation for the (SS|LL)-type SOC integrals. “1e” denotes the use of only the one-electron SOC integrals. The percentages in the parentheses refer to the ratios with respect to the results of the full scheme. All calculations have been carried out using the unc-ANO-RCC basis. The point-like nuclear model has been used.


1e

2eSL(1c)

2eSL

Full

−1.17 × 10−5 (174.1%) −1.08 × 10−4 (136.6%) −2.15 × 10−3 (117.4%) −0.0126 (111.0%) −0.1355 (106.7%) −3.31 × 10−4 (190.6%) −1.18 × 10−3 (129.8%) −0.0219 (122.1%)

3.82 × 10−6 (−56.8%) −7.17 × 10−5 (90.5%) −1.80 × 10−3 (98.3%) −0.0112 (98.1%) −0.1271 (100.1%) −1.69 × 10−4 (97.2%) −9.02 × 10−4 (98.8%) −0.0181 (100.8%)

−6.74 × 10−6 (100.3%) −7.94 × 10−5 (100.3%) −1.84 × 10−3 (100.3%) −0.0114 (100.3%) −0.1276 (100.5%) −1.73 × 10−4 (99.7%) −9.10 × 10−4 (99.7%) −0.0180 (100.6%)

−6.72 × 10−6 −7.92 × 10−5 −1.84 × 10−3 −0.0114 −0.1270 −1.74 × 10−4 −9.12 × 10−4 −0.0179

second-order SOC corrections obtained at the HF level using the “SFX2C-1e+SOC [der]/2eSL” scheme. As seen in Table VII, both scalar-relativistic and electroncorrelation effects are significant for the Hg quadrupolecoupling parameters of HHgCH2 CH3 and a strong coupling between them is noted. The scalar-relativistic correction to χ aa , for example, amounts to −400 MHz at the HF level, which is to be compared with the value of −280 MHz at the CCSD level. A rigorous consideration of both scalarrelativistic and electron-correlation effects at the SFX2C-1eCCSD(T) level yields significantly more accurate results. The SOC contributions are small in terms of absolute magnitude and thus a perturbative treatment is well justified. As shown in Table VII, the SOC corrections obtained using the “SFX2C1e+SOC [der]/2eSL” scheme, which amount to −20 MHz for χ aa and 12 MHz for χ bb , further improves the computational results. Therefore, we have demonstrated that high-accuracy calculations of Hg quadrupole-coupling parameters are possible, provided that scalar-relativistic, electron-correlation, and SOC effects are accurately taken into account. V. SUMMARY AND OUTLOOK

In the present paper, we present a thorough investigation of some aspects concerning the perturbative treatment of SOC

effects on top of a SFX2C-1e calculation. Concerning the SFX2C-1e SOC matrix, we show that the SFX2C-1e+SOC [der] scheme, which defines the SFX2C-1e SOC matrix as analytic derivatives of the SFX2C-1e Hamiltonian matrix with the SOC integrals as perturbation, is the method of choice for the perturbative computation of SOC corrections. In contrast, the SFX2C-1e+SOC [fd] scheme, which considers the difference between X2C-1e and SFX2C-1e Hamiltonian matrices as the SOC matrix, turns out not to be compatible with a perturbative treatment of SOC corrections. Furthermore, among the various approximate treatment of the two-electron SOC integrals, the “2eSL” scheme that neglects the (SS|SS)-type two-electron SOC integrals is shown to be efficient and reliable, as it has only negligible effects on the calculated SOC corrections. Future work will focus on the application of the SFX2C1e+SOC [der] scheme to the calculation of various spindependent molecular properties, in which SOC effects can be treated by means of perturbation theory. One example is the calculation of SOC splittings for spatially degenerate openshell molecules based on perturbation theory. At the same time, it might be of great interest to investigate the accuracy and efficacy of both the SFX2C-1e+SOC [der] and SFX2C1e+SOC [fd] schemes in non-perturbative treatments of SOC effects.

TABLE VI. Second-order spin-orbit-coupling (SOC) corrections to the electric-field gradients (in a.u.) for the heavier nuclei of the diatomic molecules computed at the HF level using various integral approximations. “2eSL” neglects the contributions from the (SS|SS)-type SOC integrals; “2eSL(1c)” in addition invokes the one-center approximation for the (SS|LL)-type SOC integrals. “1e” denotes the use of only the one-electron SOC integrals. The percentages in the parentheses refer to the ratios with respect to the results of the full scheme. All calculations have been carried out using the unc-ANO-RCC basis. The point-like nuclear model has been used.


1e

2eSL(1c)

2eSL

Full

−1.88 × 10−5 (171.0%) −1.66 × 10−4 (141.2%) −4.28 × 10−3 (128.4%) −0.03608 (119.3%) −1.2102 (108.1%) 1.06 × 10−3 (−1536.0%) −9.50 × 10−3 (112.8%) −0.2992 (104.4%)

5.06 × 10−5 (-461.9%) −1.14 × 10−4 (97.3%) −3.09 × 10−3 (92.7%) −0.0290 (96.0%) −1.1210 (100.1%) −1.16 × 10−4 (168.9%) −8.47 × 10−3 (100.6%) −0.2851 ( 99.5%)

−1.12 × 10−5 (101.9%) −1.20 × 10−4 (101.7%) −3.39 × 10−3 (101.6%) −0.0305 (101.0%) −1.1261 (100.6%) −6.48 × 10−5 (94.0%) −8.41 × 10−3 (99.8%) −0.2840 (99.1%)

−1.10 × 10−5 −1.18 × 10−4 −3.34 × 10−3 −0.0303 −1.1197 −6.89 × 10−5 −8.42 × 10−3 −0.2866


164107-10


J. Chem. Phys. 141, 164107 (2014)

TABLE VII. The 201 Hg quadrupole-coupling parameters (in MHz) of HHgCH2 CH3 . The molecular structure is given in Figure 1. In the nonrelativistic and SFX2C-1e calculations, the uncontracted ANO-RCC basis set has been used for mercury and the ANO1 basis sets have been used for carbon and hydrogen. In the computation of the perturbative SOC correction, uncontracted versions of SARC basis set for mercury and ANO1 basis sets for carbon and hydrogen have been employed. The point-like nuclear model has been used. Method Nonrel-HF Nonrel-CCSD SFX2C-1e-HF SFX2C-1e-CCSD SFX2C-1e-CCSD(T) SFX2C-1e-CCSD(T)+SOC SFDKH-B3LYP68 Expt.68

χ aa

χ bb

−1023 −915 −1421 −1193 −1147 −1167 −1376 −1169.50(67)

412 368 572 479 460 472 546 473.33(61)

ACKNOWLEDGMENTS

L.C. is grateful to John F. Stanton (Austin) for the generous support and for careful reading of the manuscript. The part of the work done in Mainz has been supported by the Deutsche Forschungsgemeinschaft (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 F-1283) and U.S. Department of Energy Office of Science, Office of Basic Energy Sciences under Award No. DE-FG02-07ER15884 granted to John F. Stanton. 1 P.

Pyykkö, Chem. Rev. 88, 563 (1988). G. Dyall and K. Fægri, Jr., Relativistic Quantum Chemistry (Oxford University Press, New York, 2007), Pt. III. 3 B. A. Hess, Phys. Rev. A 33, 3742 (1986). 4 E. van Lenthe, E. J. Baerends, and J. G. Snijders, J. Chem. Phys. 99, 4597 (1993). 5 K. G. Dyall, J. Chem. Phys. 106, 9618 (1997). 6 W. Kutzelnigg, Chem. Phys. 225, 203 (1997). 7 T. Nakajima and K. Hirao, Chem. Phys. Lett. 302, 383 (1999). 8 M. Barysz and A. J. Sadlej, J. Chem. Phys. 116, 2696 (2002). 9 M. Iliaš, H. J. Aa. Jensen, V. Kellö, B. O. Roos, and M. Urban, Chem. Phys. Lett. 408, 210 (2005). 10 W. Liu, Mol. Phys. 108, 1679 (2010). 11 T. Saue, Chem. Phys. Chem. 12, 3077 (2011). 12 W. Kutzelnigg, Chem. Phys. 395, 16 (2012). 13 D. Peng and M. Reiher, Theor. Chem. Acc. 131, 1 (2012). 14 M. Reiher, WIREs Comput. Mol. Sci. 2, 139 (2012). 15 T. Nakajima and K. Hirao, Chem. Rev. 112, 385 (2011). 16 W. Liu, Phys. Rep. 537, 59 (2014). 17 A. J. Sadlej and J. G. Snijders, Chem. Phys. Lett. 229, 435 (1994). 18 K. G. Dyall, J. Chem. Phys. 100, 2118 (1994). 19 L. Visscher and E. van Lenthe, Chem. Phys. Lett. 306, 357 (1999). 20 E. Fromager, L. Visscher, L. Maron, and C. Teichteil, J. Chem. Phys. 123, 164105 (2005). 21 B. A. Hess, C. M. Marian, U. Wahlgren, and O. Gropen, Chem. Phys. Lett. 251, 365 (1996). 22 Z. Li, B. Suo, Y. Zhang, Y. Xiao, and W. Liu, Mol. Phys. 111, 3741 (2013). 23 Z. Li, Y. Xiao, and W. Liu, J. Chem. Phys. 141, 054111 (2014). 24 L. Cheng, S. Stopkowicz, and J. Gauss, J. Chem. Phys. 139, 214114 (2013). 25 K. G. Dyall, J. Chem. Phys. 109, 4201 (1998). 26 K. G. Dyall and T. Enevoldsen, J. Chem. Phys. 111, 10000 (1999). 27 K. G. Dyall, J. Chem. Phys. 115, 9136 (2001). 28 M. Filatov and K. G. Dyall, Theor. Chem. Acc. 117, 333 (2007). 29 W. Kutzelnigg and W. Liu, J. Chem. Phys. 123, 241102 (2005). 30 W. Kutzelnigg and W. Liu, Mol. Phys. 104, 2225 (2006). 31 W. Liu and W. Kutzelnigg, J. Chem. Phys. 126, 114107 (2007). 2 K.

32 W.

Liu and D. Peng, J. Chem. Phys. 131, 031104 (2009). J. Aa. Jensen, in Proceeding of International Conference on Relativistic Effects in Heavy Element Chemistry and Physics, Mülheim, Germany, 2005. 34 M. Iliaš and T. Saue, J. Chem. Phys. 126, 064102 (2007). 35 W. Liu and D. Peng, J. Chem. Phys. 125, 044102 (2006). 36 J. Sikkema, L. Visscher, T. Saue, and M. Iliaš, J. Chem. Phys. 131, 124116 (2009). 37 D. Cremer, E. Kraka, and M. Filatov, Chem. Phys. Chem. 9, 2510 (2008). 38 L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). 39 W. Zou, M. Filatov, and D. Cremer, J. Chem. Phys. 134, 244117 (2011). 40 L. Cheng and J. Gauss, J. Chem. Phys. 135, 084114 (2011). 41 L. Cheng and J. Gauss, J. Chem. Phys. 135, 244104 (2011). 42 W. Zou, M. Filatov, and D. Cremer, J. Chem. Phys. 137, 084108 (2012). 43 M. Filatov, W. Zou, and D. Cremer, Int. J. Quantum Chem. 114, 993 (2014). 44 L. Cheng, S. Stopkowicz, and J. Gauss, Int. J. Quantum Chem. 114, 1108 (2014). 45 R. Samzow, B. A. Hess, and G. Jansen, J. Chem. Phys. 96, 1227 (1992). 46 C. van Wüllen and C. Michauk, J. Chem. Phys. 123, 204113 (2005). 47 D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012). 48 D. Peng, N. Middendorf, F. Weigend, and M. Reiher, J. Chem. Phys. 138, 184105 (2013). 49 J. Seino and H. Nakai, J. Chem. Phys. 136, 244102 (2012). 50 J. Seino and H. Nakai, J. Chem. Phys. 139, 034109 (2013). 51 Z. Li, Y. Xiao, and W. Liu, J. Chem. Phys. 137, 154114 (2012). 52 M. Filatov, W. Zou, and D. Cremer, J. Chem. Phys. 139, 014106 (2013). 53 S. Koseki, M. W. Schmidt, and M. S. Gordon, J. Phys. Chem. 96, 10768 (1992). 54 S. Koseki, M. S. Gordon, M. W. Schmidt, and N. Matsunaga, J. Phys. Chem. 99, 12764 (1995). 55 S. Koseki, M. W. Schmidt, and M. S. Gordon, J. Phys. Chem. A 102, 10430 (1998). 56 B. A. Hess, and C. M. Marian, in Computational Molecular Spectroscopy, edited by P. Jensen and P. R. Bunker (Wiley, New York, 2000), p. 169. 57 J. C. Boettger, Phys. Rev. B 62, 7809 (2000). 58 J. Chalupský and T. Yanai, J. Chem. Phys. 139, 204106 (2013). 59 M. Blume and R. E. Watson, Proc. R. Soc. London, Ser. A 270, 127 (1962). 60 M. Blume and R. E. Watson, Proc. R. Soc. London, Ser. A 271, 565 (1963). 61 C. M. Marian and U. Wahlgren, Chem. Phys. Lett. 251, 357 (1996). 62 O. Launilla, B. Schimmelpfennig, H. Fagerli, O. Gropen, A. G. Taklif, and U. Wahlgren, J. Mol. Spectrosc. 186, 131 (1997). 63 U. Wahlgren, M. Sjøvoll, H. Fagerli, O. Gropen, and B. Schimmelpfennig, Theor. Chem. Acc. 97, 324 (1997). 64 B. Schimmelpfennig, L. Maron, U. Wahlgren, C. Teichteil, H. Fagerli, and O. Gropen, Chem. Phys. Lett. 286, 261 (1998). 65 A. Berning, M. Schweizer, H. J. Werner, P. J. Knowles, and P. Palmieri, Mol. Phys. 98, 1823 (2000). 66 F. Neese, J. Chem. Phys. 122, 034107 (2005). 67 S. Knecht, private communication (2014). 68 M. Goubet, R. A. Motiyenko, L. Margulés, and J. Guillemim, J. Phys. Chem. A 116, 5405 (2012). 69 R. E. Stanton and S. Havriliak, J. Chem. Phys. 81, 1910 (1984). 70 W. Schwalbach, L. Mück, L. Cheng, and J. Gauss, “Perturbative treatment of spin-orbit splittings on top of spin-free Dirac-Coulomb calculations,” J. Chem. Phys. (to be published). 71 J. A. Pople, R. Krishnan, H. B. Schlegel, and J. S. Binkley, Int. J. Quantum Chem., Symp. 13, 225 (1979). 72 R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J. Chem. Phys. 38, 550 (1963). 73 J. Gerratt and I. M. Mills, J. Chem. Phys. 49, 1719 (1968). 74 CFOUR, a quantum chemical program package written by J. F. Stanton, J. Gauss, M. E. Harding, P. G. Szalay with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, L. Cheng, O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K. Klein, W. J. Lauderdale, D. A. Matthews, T. Metzroth, L. A. Mück, D. P. O’Neill, D. R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J. D. Watts and the integral packages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van Wüllen. 75 S. Stopkowicz and J. Gauss, J. Chem. Phys. 134, 204106 (2011). 33 H.


164107-11


76 Q. Sun, W. Liu, Y. Xiao, and L. Cheng, J. Chem. Phys. 131, 081101 (2009). 77 W.

Zou, M. Filatov, and D. Cremer, J. Chem. Theor. Comput. 8, 2617 (2012). 78 Q. Sun, Y. Xiao, and W. Liu, J. Chem. Phys. 137, 174105 (2012). 79 F. Neese, A. Wolf, T. Fleig, M. Reiher, and B. A. Hess, J. Chem. Phys. 122, 204107 (2005). 80 B. O. Roos, V. Veryazov, and P. Widmark, Theor. Chem. Acc. 111, 345 (2004). 81 R. Pou-Amérigo, M. Merchán, I. Nebot-Gil, P. Widmark, and B. O. Roos, Theor. Chim. Acta 92, 149 (1995). 82 K. Fægri, Jr., Theor. Chem. Acc. 105, 252 (2001). 83 D. A. Pantazis, X. Chen, C. R. Landis, and F. Neese, J. Chem. Theor. Comput. 4, 908 (2008). 84 G. D. Purvis III and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982). 85 K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 (1989).

J. Chem. Phys. 141, 164107 (2014) 86 R.

J. Bartlett, J. D. Watts, S. A. Kucharski, and J. Noga, Chem. Phys. Lett. 165, 513 (1990). 87 The contracted coefficients of these ANO basis sets were obtained using atomic SFX2C-1e-CCSD density. They are available upon request. 88 The geometrical parameters for HHgCH CH : r 2 3 Hg-H = 1.624 Å, rHg-C1 = 2.086 Å, rC1-C2 = 1.518 Å, rC2-H6 = 1.088 Å, rC1-H2 = rC1-H3 = 1.086 Å, rC2-H4 = rC2-H5 = 1.086 Å, H1-Hg−C1 = 179.7◦ , Hg-C1−C2 = 113.4◦ , ◦ ◦ Hg-C1−H2 = Hg-C1−H3 = 108.0 , C1-C2−H4 = C1-C2−H5 = 111.8 , ◦, ◦, = 117.8 = = 0 C1-C2−H6 C2-C1−Hg−H1 H6-C2−C1−Hg H3-C1−Hg−H1 = - H2-C1−Hg−H1 = 122.8◦ , H5-C2−C1−Hg = - H4-C2−C1−Hg = 60.0◦ . 89 K. A. Peterson, R. A. Kendall, and T. H. Dunning, Jr., J. Chem. Phys. 99, 1930 (1993). 90 P. Pyykkö, Mol. Phys. 106, 1965 (2008). 91 V. Arcisauskaite, S. Knecht, S. P. A. Sauer, and L. Hemmingsen, Phys. Chem. Chem. Phys. 14, 2651 (2012).


Perturbative theory for Brownian vortexes.

Non-perturbative calculation of molecular magnetic properties within current-density functional theory.

Theory of perturbative pulse train based coherent control.

Perturbative Quantum Gravity and its Relation to Gauge Theory.

The exact wavefunction factorization of a vibronic coupling system.

Glass transitions and scaling laws within an alternative mode-coupling theory.

Edge Mode Coupling within a Plasmonic Nanoparticle.

Flow-Induced New Channels of Energy Exchange in Multi-Scale Plasma Dynamics - Revisiting Perturbative Hybrid Kinetic-MHD Theory.

Perturbative treatment of anharmonic vibrational effects on bond distances: an extended Langevin dynamics method.

Theory of excitation-contraction coupling in cardiac muscle.

Asymptotic theory of strong spin-orbit coupling in optical fiber.

Exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media.

Wright-Fisher exact solver (WFES): scalable analysis of population genetic models without simulation or diffusion theory.

Analysis of thin plates with holes by using exact geometrical representation within XFEM.

An Exact Method for Partitioning Dichotomous Items Within the Framework of the Monotone Homogeneity Model.

Effective potential theory for transport coefficients across coupling regimes.

Time-resolved spectroscopy in time-dependent density functional theory: an exact condition.

Modal Coupling of Single Photon Emitters Within Nanofiber Waveguides.

Fréedericksz transition in the director-density coupling theory.

Non-perturbative effects in spin glasses.

Carcinogenesis explained within the context of a theory of organisms.

Tunable and directional plasmonic coupling within semiconductor nanodisk assemblies.

Exact milestoning.

Towards an exact theory of linear absorbance and circular dichroism of pigment-protein complexes: importance of non-secular contributions.