Analytic energy derivatives in relativistic quantum chemistry



In this review, we discuss the current status of analytic derivative theory in relativistic quantum chemistry. A brief overview of the basic theory for the available relativistic quantum chemical methods as well as the state-of-the-art development of their analytic energy derivatives is given. Among the various relativistic quantum chemical methods, cost-effective approaches based on spin separation and/or on the matrix representation of two-component theory have been proven particularly promising for the accurate and efficient treatment of relativistic effects in electron correlation calculations. We highlight analytic derivative techniques for these cost-effective relativistic quantum chemical approaches including direct perturbation theory, the spin-free Dirac-Coulomb approach, and the exact two-component theory in its one-electron variant. An outlook is given on future developments of analytic energy derivative techniques for relativistic quantum chemical methods, again with an emphasis on cost-effective schemes. © 2014 Wiley Periodicals, Inc.


Analytic schemes for evaluating energy derivatives[1-6] play an essential role in quantum chemical calculations of molecular properties. Apart from being obviously more reliable in accuracy than corresponding numerical schemes, analytic schemes are advantageous in terms of computational efficiency. A further advantage of analytic schemes lies in the calculation of energy derivatives with respect to perturbations with imaginary matrix elements in the nonrelativistic (or scalar relativistic) framework. For example, in the computation of nuclear magnetic resonance (NMR) chemical shieldings as analytic second derivatives of the energy, the magnetic perturbations involved can be efficiently treated within real algebra.[7-9]

Since the ground-breaking contribution by Pulay in 1969[10] analytic derivative theory has been extensively developed in combination with nonrelativistic quantum chemical methods. Analytic first and second derivatives of the energy have been well established and are widely used in computations of molecular geometries, vibrational frequencies, as well as electrical and magnetic properties. In particular, the efficient implementation of analytic first and second derivatives for electron correlation methods,[11-22] especially coupled cluster (CC) theory,[23] based on a density matrix formulation[24] has enabled high-accuracy calculations of various molecular properties.

Although nonrelativistic quantum chemical methods have proven very successful in chemical applications for systems comprising light elements (first and second rows of the periodic table), it is generally accepted that relativistic effects should be taken into account for reliable calculations of heavy-element compounds,[25-29] as special relativity has a pronounced influence on the properties of these systems. Scalar [spin-free (SF)] relativity causes radial contraction of the s- and p-type orbitals and subsequent extension of the d- and f-type orbitals[27] and, thus, can significantly affect the chemical and physical properties of heavy-element species. In addition, spin–orbit (SO) coupling effects are responsible for energy-level splittings and also for so-called “spin-forbidden” chemical processes.[30, 31]

To accurately account for relativistic effects in chemical applications, a plethora of relativistic quantum chemical methods have been developed in the past several decades.[32, 33] The most rigorous quantum chemical treatment of relativistic effects is perhaps offered by the four-component Dirac-Coulomb (DC) approach based on the one-electron Dirac Hamiltonian augmented by instantaneous Coulomb interactions between electrons.[32-34] Implementations and applications of the DC approach have been reported at the Hartree–Fock (HF) and density functional theory (DFT) level[35-44] and also in combination with various electron correlation treatments.[45-64] However, as the DC approach is computationally demanding due to spin-symmetry breaking as well as the necessity of dealing with small-component integrals (see Tables 1 and 2), its chemical applications have been limited to relatively small molecules. Thus, more cost-effective schemes are highly desired for practical calculations. A natural route to a more efficient treatment of relativistic effects is to compute relativistic corrections to the nonrelativistic energy by means of perturbation theory.[65] Such perturbative schemes enable the exploitation of spin symmetry and also allow to deal with small-component integrals in an efficient manner. The most promising scheme is probably the direct perturbation theory (DPT) developed by Rutkowski[66-69]and Kutzelnigg,[65, 70-73] which is formulated directly in the four-component framework and can also be extended in a straightforward manner to higher orders. Nonperturbative cost-effective relativistic quantum chemical approaches also have been extensively studied, which separately treat the SF and spin-dependent terms in the relativistic Hamiltonian and/or decouple the large- and small-component degrees of freedom. The most rigorous scheme to separate SF and spin-dependent terms can be achieved in the DC framework, as originally proposed by Dyall.[74] The resulting SF Dirac-Coulomb (SFDC) approach is particularly suitable for efficient scalar relativistic electron correlation treatments due to the preservation of spin symmetry and its rigorousness in treating scalar relativistic effects, although it is still expensive at the HF and second-order Møller-Plesset perturbation theory (MP2) levels due to the presence of the small component. Thus, the SFDC approach has been mostly used in CC and configuration interaction (CI) calculations since the presence of the small component here does not lead to an increase in the computational cost in the rate-determining steps.[75, 76]

The small-component degrees of freedom can be eliminated by a block diagonalization of the four-component Hamiltonian, leading to a variety of two-component methods.[77-96] In particular, SF two-component methods, for example, the SF second-order Douglas–Kroll–Hess (SF-DKH2) method[80] and the zeroth-order regular approximation (SF-ZORA) approach[97] have become very popular in relativistic quantum chemical applications. Note that all the SF approaches can be effectively rewritten in a one-component form.

Table 1. Features of various relativistic quantum chemical methods.
Spin separationYesYesYesYesYesNo
Elimination of small componentYesYesYesNoNoNo
Matrix representationNoYesYes
One-step block diagonalizationNoNoYes
Table 2. Computational costs of relativistic quantum chemical methods relative to the nonrelativistic case.
MethodFormal Rate-Determining StepNonrelSFX2C-1eSFDCDPT2DC[44, 51]
HFIntegral evaluation11∼10∼3∼60
MP2Integral transformation11∼6∼1∼16
CCSDCC equations11∼1∼2∼32

In view of the essential role of analytic derivative techniques in the calculation of molecular properties[1-6] and the recent advances in relativistic quantum chemistry, it can be expected that analytic derivative techniques in combination with relativistic quantum chemical methods will be very useful for the efficient and accurate calculation of molecular properties for heavy-element compounds. However, within the relativistic framework analytic derivative theory has been explored significantly less than in the nonrelativistic case, due to the following reasons:

  1. Application of the expensive fully relativistic four-component approach has been so far limited mainly to small molecules, where the efficiency of analytic schemes is less manifested.
  2. For perturbative schemes such as DPT, property calculations require higher derivatives of the energy due to the fact that the relativistic energy corrections are themselves already given in terms of derivatives of the nonrelativistic energy.[98-100]
  3. Analytic energy derivatives for traditional two-component schemes are plagued by the “picture change” issue,[101] that is, the transformation from the four- to the two-component picture has to be also performed for the property operators. As traditional two-component methods use multiple-step tranformation schemes and/or operator expansions for the transformation operator, the corresponding transformation of the property operator often leads to complicated formulations.
  4. The need for additional integrals that involve the respective property operator enclosed by the Pauli spin matrices and momentum operators complicates the implementation of relativistic corrections to properties.

These difficulties have been significantly alleviated by recent theoretical and algorithmic innovations. In the two-component framework, the newly developed exact two-component (X2C) theory[95, 102-120] based on a one-step block diagonalization of the matrix representation of the Dirac equation has established a simple procedure for the transformation from the four- to the two-component picture within a matrix representation. In this way, the “picture change” for the property matrix elements is considerably simplified in comparison to that in traditional two-component theories. For DPT, its recent formulation in terms of energy derivatives[99, 121] has made higher-order energy corrections as well as corresponding corrections to properties more easily accessible. Finally, the SFDC approach has for the first time been efficiently implemented by exclusively calculating and manipulating SF relativistic integrals, thus, enabling a full exploitation of the computational efficiency of this scheme.[122] Apart from the development of the cost-effective relativistic quantum chemical schemes mentioned above, advances have been reported for the most rigorous DC approach, for example, the recent implementations using density-fitting techniques and parallelization of the computer code by Kelley et al.[44], Belpassi et al.[123], and Repisky et al.[124] have the potential for significantly extending the applicability of four-component DC-HF and DFT calculations.

Based on those recent advancements, the past several years have witnessed significant progress in the development of analytic derivative techniques for relativistic quantum chemical methods. We expect a growing interest and increasing activity in this field. This review intends to provide a summary of the state of the art and an outlook on future development for this topic. In the following section, we present a brief summary of available relativistic quantum chemical methods, in which we first discuss the basic theory for relativistic quantum chemistry and then introduce the rigorous DC approach as well as various more efficient but approximate approaches. Thereafter, we provide a brief overview of the development of analytic energy derivatives for various relativistic quantum chemical methods and highlight recent work on analytic derivative techniques for cost-effective relativistic quantum chemical approaches. This is followed by a discussion of benchmark results and of some example applications, before we present an outlook on future developments.

Overview of Relativistic Quantum Chemical Methods

The relativistic wave equation for electrons, that is, the Dirac equation, serves as the basis for relativistic quantum chemical methods.[32] For the present purpose, it is sufficient to use the time-independent Dirac equation:

display math(1)

with the Dirac Hamiltonian math formula and the one-electron wave function ψ given by

display math(2)

In Eq. (2), math formula accounts for the nuclear–electron interactions, c is the speed of light, math formula denotes the momentum operator, and math formula represents the vector of the Pauli matrices given by

display math(3)

Note that math formula is a math formula operator matrix and ψ is a “four-component” wave function, that is, math formula and math formula are two-component spinors

display math(4)

Unless otherwise specified, we use atomic units throughout, in which the elementary charge e, the mass of an electron me, the reduced Plank constant ℏ, and the Coulomb's constant math formula are equal to unity. The Dirac equation naturally includes SO coupling and simultaneously provides positive energy state (PES) and negative energy state (NES) solutions that correspond to electronic and positronic states, respectively.

As the large- and small-component wave function are related by math formula weighted by an energy-dependent operator denominator

display math(5)

it is necessary to use a “kinetically balanced” basis set[125] to guarantee variational stability and to obtain the correct nonrelativistic limit within a finite basis-set representation:

display math(6)
display math(7)

The large-component atomic-orbital (AO) basis functions are denoted here by math formula, while math formula and math formula are the expansion coefficients for large and small components, respectively.

The matrix representation of the Dirac equation is then written as:

display math(8)

where W represents the relativistic integral matrix with the individual elements given by

display math(9)

and T, V, and S denote the usual kinetic energy, potential energy, and overlap matrix, respectively.

For treating many-electron systems, the one-electron Dirac Hamiltonian can be augmented by the instantaneous Coulomb interactions between the electrons, which gives rise to the DC Hamiltonian:

display math(10)

Further inclusion of the leading-order relativistic correction to the electron–electron interactions, that is, of the Breit term,[126] leads to the DC-Breit (DCB) Hamiltonian. In the following, we will only explicitly refer to the DC Hamiltonian, though most of the discussions about the DC Hamiltonian also apply to the DCB Hamiltonian.

It has been shown that the DC Hamiltonian should be projected onto the PES spinors for electron correlation treatments.[127] The resulting “no-pair” DC Hamiltonian can be expressed in occupation number representation as:

display math(11)

where math formula and math formula are matrix elements of the Dirac operator and the two-electron Coulomb interactions, respectively,

display math(12)

and the labels math formula run over all PES spinors. math formula can be rewritten in normal order with respect to a single Slater determinant, usually the HF determinant:

display math(13)

where math formula denotes the Fock matrix elements given by

display math(14)

and math formula is the DC-HF energy. Thus, math formula takes a form similar to the nonrelativistic Hamiltonian in occupation number representation. However, it should be noted that the one- and two-electron matrix elements in math formula involve small-component contributions and also include SO coupling.

The “no-pair” DC Hamiltonian forms the basis for the DC approach,[32] which is perhaps the most rigorous relativistic quantum chemical method and will be discussed briefly in the first of the following subsections. However, the DC approach is computationally costly; its computing time is typically an order of magnitude larger than for the corresponding nonrelativistic methods.[128] Therefore, efficient but approximate relativistic quantum chemical methods are very useful (and essential) for practical chemical applications and have been extensively studied. Thus, we will continue after the discussion of the DC approach with a brief summary of cost-effective relativistic quantum chemical approaches including DPT, the SFDC approach, traditional two-component theories, and the exact two-component theory (see Table 1 for a summary of their underlying features). At this point, it is also appropriate to mention that it is an interesting topic to formulate a more rigorous relativistic Hamiltonian than the “no-pair” DC(-Breit) Hamiltonian by resorting to quantum electrodynamics. This area of research has recently attracted a lot of attention.[120, 129-132] This topic, however, is beyond the scope of this review and, thus, will not be further discussed.

The Dirac-Coulomb approach

The DC approach directly based on the “no-pair” DC Hamiltonian perhaps offers the most rigorous treatment of relativistic effects in quantum chemistry. As the “no-pair” DC Hamiltonian in occupation number representation shares a generic form with the nonrelativistic Hamiltonian, the available nonrelativistic quantum chemical formulations to treat electron correlation can in principle be used as well in the DC framework, provided that complex algebra and double-group symmetry are used.[133] The DC approach has been formulated and implemented at the HF self-consistent field (HF-SCF),[36-38, 41, 44] DFT,[39, 40, 42, 43] MP2,[45, 46] multiconfigurational SCF,[58, 60-62] CI,[59-62] and CC levels of theory.[51, 52, 54, 55] However, the DC approach is computationally demanding in comparison with the nonrelativistic case due to spin-symmetry breaking and the presence of the positronic degrees of freedom; the computing time of a DC calculation is at least one order of magnitude larger than that of a corresponding nonrelativistic calculation (see Table 2). Consequently, the application of the DC approach has been so far limited to relatively small molecules.

Direct perturbation theory

An obvious choice for a cost-effective treatment is offered by perturbation theory. Such an approach is in particular advantageous when relativistic effects are not too pronounced and is a well justified choice for treating relativistic effects in compounds containing only lighter elements. So far, most calculations that include a perturbative treatment of relativistic effects are based on the use of the mass–velocity and Darwin terms as perturbations.[134] However, a more rigorous treatment is offered by the so-called DPT introduced and formulated by Rutkowski[66-69] and Kutzelnigg.[65, 70-73]

In DPT, relativistic effects are treated by expanding the Dirac equation, Eq. (1) with respect to 1/c as perturbation parameter starting from the nonrelativistic limit. As it is easily seen, this limit ( math formula) is not well defined for the Dirac equation. However, it should be noted that for electronic solutions the magnitude of the large component is similar to that of the small component multiplied by the speed of light, that is,

display math(15)

To obtain a well-defined perturbation expansion, a metric change is introduced,[135]

display math(16)

that leads to the following modified form of the Dirac equation

display math(17)
display math(18)

With the perturbation parameter math formula, the above equation can be decomposed

display math(19)


display math(20)

and provides a suitable starting point for a perturbative expansion in orders of math formula. The nonrelativistic limit, given in form of the Lévy–Leblond equation,[136] is defined via the terms with index 0, while the relativistic corrections are introduced by terms with index 2. Using a Taylor expansion, the DPT energy corrections may be obtained via energy derivatives, that is,

display math(21)
display math(22)

Along this route, it is more convenient to differentiate a suitable Lagrangian L instead of the energy itself.[99] Restricting the discussion to the one-electron level, L is obtained by augmenting the energy with the orthonormality condition premultiplied by a Lagrange multiplier ε

display math(23)

Differentiating this expression with respect to math formula leads to the various orders of the DPT corrections.

It is in principle possible to keep the small component in the equations as long as the kinetic-balance condition is accounted for.[99] However, for the implementation of DPT within a nonrelativistic quantum chemical framework it is advantageous to eliminate the small component, preferably straight from the beginning, to exploit the nonrelativistic analytic derivative machinery. With the help of an math formula operator relating math formula and math formula

display math(24)

and taking into account the equivalence of the nonrelativistic wave function φ and the large component math formula in the nonrelativistic limit, the derivation of DPT may be carried out starting from a nonrelativistic Lagrangian[99]

display math(25)

Relativistic effects are then folded in by replacing the nonrelatistic operators math formula and math formula

display math(26)
display math(27)


display math(28)
display math(29)

Expansion of the operators math formula, and math formula in terms of math formula then renders it possible to obtain the DPT energy corrections as derivatives of the Lagrangian in Eq. (25)

display math(30)
display math(31)

For the many-electron treatment, the procedure is analogous.[99] Note that in this case, the two-electron interaction also depends on math formula and needs to be expanded accordingly.

Furthermore, we note that using the Dirac identity[32]

display math(32)

DPT leads in a natural manner to spin separation. For closed-shell and open-shell systems with a spatially nondegenerate electronic state, the leading-order correction as given in Eq. (30) and usually referred to as DPT2, accounts only for scalar relativistic effects. At second order, that is, DPT4, scalar relativistic and SO contributions appear in a completely decoupled fashion.[99] This suggests that the scalar relativistic and SO effects can be treated in a good approximation independently as the couplings between those two contributions are third-order and higher order effects. Concerning the DPT treatment of degenerate electronic states, we refer the reader to the literature.[137-139]

The SFDC approach

The computational overhead of the DC approach in electron correlation calculations, for example, when using CC theory, is mainly due to spin-symmetry breaking, which effectively increases the dimension of the spinors to be correlated and generally necessitates the use of complex algebra. A cost-effective strategy for relativistic correlated calculations, therefore, may be based on a separation of the SF and SO terms in the Hamiltonian and consists of a rigorous treatment of the SF part augmented by a perturbative consideration of the SO effects. The spin-separation scheme for the DC Hamiltonian originally proposed by Dyall[74] is most convenient for this purpose and is best illustrated within the matrix representation of the Dirac equation, Eq. (8), in which the relativistic integral W is the only spin-dependent quantity. Using the Dirac identity, Eq. (32), W is readily decomposed into a SF and a SO contribution

display math(33)


display math(34)
display math(35)

leading to a natural separation of the SF and SO terms in the one-electron matrix elements math formula. A similar separation can be carried out for the two-electron matrix elements math formula. Thus, one obtains a separation of the no-pair DC Hamiltonian into a SF and a SO part:

display math(36)

The SFDC approach is then obtained by retaining only math formula.

While the presence of the small component in the SFDC formulation still introduces a considerable computational overhead in the SCF and the integral-transformation steps, the computational cost of a SFDC-CC calculation is identical to that of a corresponding nonrelativistic one, provided that only molecular orbital (MO)-based algorithms are used. Thus, the SFDC approach offers an economical and rigorous treatment of scalar relativistic effects within the CC framework.[122]

SO contributions can be incorporated on top of a SFDC calculation by means of perturbation theory, treating thereby math formula as a perturbation.[140] For spatially nondegenerate electronic states, the leading SO correction is then obtained via second-order perturbation theory. Computationally, this correction is best implemented using analytic derivative techniques. At the HF level, for example, this requires the solution of the coupled-perturbed SFDC-HF equations. Note that unlike in the DPT4 treatment of SO effects, the treatment on top of a SFDC calculation incorporates couplings between scalar relativistic and SO effects and, therefore, is more reliable than DPT4 for systems containing heavier elements. The perturbative treatment of SO effects after a SFDC calculation[140] has great potential for applications in heavy-element chemistry. In particular, this scheme renders it possible to combine high-level treatments of the SF relativistic effects with a perturbative SO treatment at a more cost-effective, that is, lower level of theory.[140]

Traditional two-component theories

An important and often used route to a more efficient, though approximate treatment of relativistic effects is to focus solely on the electronic solutions by decoupling the electronic and positronic degrees of freedom in the one-electron Dirac equation, or in other words, by eliminating the small-component degrees of freedom. Formally, the decoupling of the electronic and positronic blocks in the one-electron Dirac equation can be achieved by a Foldy–Wouthuysen (FW) unitary transformation[77]:

display math(37)

where math formula and math formula denote the electronic and positronic two-component blocks of the Hamiltonian, respectively. In subsequent applications to many-electron systems, math formula is then used together with untransformed two-electron interactions:

display math(38)

The two-component approaches achieve their computational efficiency in comparison with the four-component schemes by the fact that the small-component integrals are no longer required. We should mention that the FW transformation in principle is also applicable to the two-electron interactions.[141-145] However, this leads to complicated two-component formulations that are even more involved than the parent four-component theory.

Spin separation can be performed for math formula in a straightforward manner. The resulting SF two-component methods provide the possibility for accurate treatments of scalar relativistic effects, as the scalar two-electron picture-change effects have been shown to be usually small.[143] The efficiency of the SF two-component methods is obvious; in comparison with the nonrelativistic case, only the nonrelativistic one-electron Hamiltonian integrals need to be replaced with the matrix elements of math formula.

The unitary transformation math formula in Eq. (37) is in principle determined by the math formula operator as defined in Eq. (5), which relates the large and small component of the one-electron wave function

display math(39)

However, a closed analytic form for math formula is only available for the free-particle (FP) case with the corresponding math formula given by:

display math(40)

Various approximate two-component theories have thus been proposed and tested for practical calculations.

In the Douglas–Kroll–Hess (DKH)[78-80, 84-89] and the Barysz–Sadlej–Snijders (BSS) methods,[91-93] a multiple-step procedure is used, that is, the FP–FW transformation is first applied to the one-electron Dirac Hamiltonian and a second transformation is then performed to block diagonalize the FP–FW transformed Hamiltonian:

display math(41)

In the DKH approach, math formula is expanded order by order in terms of the nuclear-attraction potential:

display math(42)

For example, the second-order DKH approach (DKH2) uses math formula and the decoupling is correct to second-order in terms of the nuclear-attraction potential. We should mention that the implementation of the DKH sequential transformations starting from math formula can be conveniently done in the “momentum space,” that is, practically by using eigenfunctions of a matrix representation of the kinetic-energy operator as basis functions.[79]

In the BSS method, the FP–FW-transformed Hamiltonian is cast into a matrix representation. In the original work,[91] the remaining FW transformation for block diagonalizing this matrix Hamiltonian is expanded in terms of math formula, whereas in later developments,[92, 93] it is obtained by solving a system of nonlinear matrix equations.

Another popular two-component method is the regular-approximation approach[81-83] based on the following expansion of the math formula operator:

display math(43)

The leading-order approximation, that is, the zeroth-order regular approximation (ZORA) method is obtained by skipping the higher-order terms with respect to math formula in math formula:

display math(44)

The rate of convergence for the regular approximation is quite appealing and the ZORA method has been successfully used in many applications.[146-148] However, due to the presence of math formula in the denominator, it is necessary to use numerical-quadrature techniques for evaluating the ZORA Hamiltonian matrix elements. Consequently, the application of the ZORA method has typically been limited to DFT calculations. Furthermore, the appearance of math formula in the denominator breaks the gauge invariance in the ZORA method. This problem can be dealt with by using a scaled-ZORA variant[149] or by replacing V with a model potential[148] or a truncated nuclear potential.[150] Higher-order regular approximations including the first-order (FORA)[83] and infinite-order (IORA) variant[151] have also been formulated. It should be noted that the regular-approximation approach as described so far is the only pure operator-based two-component scheme in the sense that it yields a two-component Hamiltonian operator that is then discretized using a finite basis set. In contrast, all other two-component methods resort to matrix representations during the block-diagonalization procedure. In this context, we also mention the matrix formulation of ZORA and IORA as suggested by Filatov[152] as well as Filatov and Cremer.[153] However, as already pointed out in Ref. [96] these methods sacrifice the operator character of the ZORA approach and thus, from a conceptional perspective, should be considered as different theories. They can be considered as approximate versions of the exact two-component theory described in the next section.

At this point, one should also mention the relativistic elimination of small component (RESC) approach suggested by Nakajima and Hirao.[94] In this method, the energy dependence of the denominator in Eq. (5) is eliminated using a simple classical correspondence

display math(45)

Despite its rather good performance in example molecular calculations,[154] the RESC method has been shown to suffer from variational instability for heavier systems when steep functions are included in the basis set.[155]

Exact two-component theory

During the development of the above mentioned two-component methods, it has been realized that the decoupling of the electronic and positronic degrees of freedom in the one-electron Dirac equation can be carried out at the matrix level in one step, thereby exploiting information provided by the straightforward solution of the one-electron Dirac equation in its matrix representation. This finding is the basis for the so-called “exact two-component” (X2C) theory.[102, 108, 112, 113]

In X2C theory, the matrix representation of the one-electron Dirac Hamiltonian is block-diagonalized via a unitary matrix transformation

display math(46)

The X matrix relates the large- and small-component MO coefficients of the PES orbitals and can be determined via

display math(47)

The renormalization matrix R for the electronic block relating the two-component wave function and the large component

display math(48)

is given by[111]

display math(49)

and ensures that the relativistic treatment preserves the nonrelativistic metric. The “electrons-only” two-component matrix Hamiltonian math formula can then be written as

display math(50)


display math(51)

The SF version of X2C theory (SFX2C) is finally obtained by excluding the spin-dependent part in W during the solution of the four-component matrix equation as well as in Eq. (51).

The construction of the X2C matrix Hamiltonian only requires the determination of the X matrix as well as a sequence of manipulations of four-component one-electron Hamiltonian integrals. The former can be achieved using iterative schemes[102, 106, 108] or by adopting approximations based on the matrix representation of the regular approximations.[153, 156] However, in practice, the preferred way is the direct solution of the Dirac equation in its matrix representation, Eq. (8), via diagonalization and the subsequent construction of X directly from math formula and math formula. This approach is both simple and robust and the associated computational cost is essentially negligible in many-electron treatments. The X2C approach thus is theoretically simpler and computationally more efficient than the traditional higher-order two-component schemes mentioned in the previous subsection based on multiple-step decoupling schemes and/or operator expansions. Furthermore, X2C achieves an exact block diagonalization of the one-electron Dirac Hamiltonian in its matrix representation and thus is more accurate than approximate two-component schemes.

For many-electron treatments, the SFX2C one-electron Hamiltonian matrix can be augmented by untransformed two-electron interactions, which gives rise to the SFX2C-1e scheme. At this point, we should mention that the word “exact” in the X2C-1e scheme indicates the exact decoupling of the electronic and positronic degrees of freedom at the one-electron level, while the computational efficiency of SFX2C-1e originates from the use of untransformed electron–electron interactions, which introduces the underlying approximation of neglecting the scalar two-electron picture-change effects. Thus, the SFX2C-1e scheme provides an accurate treatment of scalar relativistic effects, as scalar two-electron “picture-change” errors are typically rather small.[143] Furthermore, as the implementation of SFX2C-1e only requires a replacement of the nonrelativistic one-electron Hamiltonian integrals by the matrix elements of math formula, the SFX2C-1e scheme can use the available nonrelativistic quantum chemical machinery in a straightforward manner at the same computational cost. Because of its accuracy and computational efficiency, the SFX2C-1e scheme has emerged as a very promising candidate for practical chemical applications.

State-of-the-Art Analytic Derivative Theory in Relativistic Quantum Chemistry

Basic aspects of analytic derivative theory

Analytic energy derivative techniques are important prerequisites for the routine chemical applications of quantum chemical methods, as analytic schemes for calculating molecular properties as energy derivatives are numerically more accurate and computationally more efficient than corresponding numerical schemes. The computational efficiency of analytic schemes stems from Wigner's 2n+1 rule,[157] that is, the fact that knowledge of up to the nth order derivatives of the wave function parameters is sufficient for the analytic evaluation of the 2n+1th derivatives of the energy. A similar 2n+2 rule holds for Lagrange multipliers in case of nonvariational methods.[158]

To illustrate the efficiency of analytic energy derivatives and also to set up the notation for the subsequent sections, we briefly discuss analytic first derivatives of the energy[11, 13, 16, 17] for the nonrelativistic CC approach.[23] The CC wave function is parameterized via an exponential ansatz:

display math(52)

where math formula is the HF reference determinant and math formula represents the cluster operator, for example, in the singles and doubles (CCSD) approximation[159]

display math(53)

We use math formula to denote occupied spin orbitals, math formula to represent virtual spin orbitals, and math formula for arbitrary spin orbitals. The CC amplitudes and energy are determined by amplitude and energy equations, respectively, given by

display math(54)
display math(55)

where math formula denotes excited determinants

display math(56)

First derivatives of the CC energy can conveniently be obtained by differentiating the following Lagrangian[19, 160-162]

display math(57)

Λ is here a set of de-excitation operators, for example, in the CCSD approximation

display math(58)

in which the λ amplitudes serve as the Lagrange multipliers for the CC equations and can be determined by the following stationary condition

display math(59)

that leads to the so-called “lambda equations”[11, 163]

display math(60)

Zai and Ipq are the Lagrange multipliers for the Brillouin condition and the orthonormality condition for the MOs, respectively, and can be determined via the stationary condition of the Lagrangian with respect to the MO coefficients. The first derivatives of CC energy are then obtained by partial differentiation of the Lagrangian with respect to the perturbation parameter χ:

display math(61)

where D and Γ are CC effective one- and two-particle density matrices

display math(62)
display math(63)

whereas math formula, and math formula involve one- and two-electron AO derivative integrals and the unperturbed MO coefficients:

display math(64)
display math(65)
display math(66)


display math(67)

as the HF density matrix.

As the analytic evaluation of first derivatives of the energy only requires the solution of the unperturbed amplitude and lambda equations, the computational cost of a CC analytic gradient calculation is approximately twice that of a single-point energy calculation independent of the number of perturbations considered. This is particularly important for the calculation of nuclear forces, as here the corresponding schemes based on numerical differences of energies require two or more single-point energy calculations for each geometrical degree of freedom. The advantage of the analytic scheme is therefore obvious, in particular with increasing system size. The computational efficiency of analytic schemes is even more prominent for computing higher order derivatives;[164] the scaling behaviors of analytic schemes for evaluating nth order geometrical derivatives of the energy are math formula and math formula for odd and even derivatives, respectively, where math formula denotes the number of perturbation parameters. This should be compared with the math formula scaling of the corresponding numerical schemes.

The concepts of analytic derivative theory are general and can be applied to relativistic quantum chemistry in a straightforward manner. In the following, we first present an overview of the current status of analytic energy derivatives for available relativistic quantum chemical methods. Then we focus on recent progress of analytic derivative theory for cost-effective relativistic approaches including DPT as well as the SFDC and the SFX2C-1e approaches.

Overview of analytic energy derivatives for relativistic quantum chemical methods

Analytic energy derivatives for the DC approach have been widely used in the computation of electrical and magnetic properties at the SCF (HF and/or DFT) level.[165-170] In particular we mention here the recent development of a general framework for calculating DC-SCF energy derivatives of arbitrary order.[171] An analytic scheme for calculating first-order properties at DC-MP2 level has been reported by van Stralen et al.[172] Nuclear forces for DC-DFT calculations have been implemented by Fricke and coworkers[173] for the special case of numerical basis functions. Analytic nuclear gradients for the recent resolution-of-identity implementation of the DC-HF method have been reported by Shiozaki,[174] thereby aiming at geometry optimizations for large molecules (100 atoms and more) containing heavy atoms.

For DPT, analytic schemes for the calculation of nuclear forces[175-177] as well as first-order electrical properties[178] have been implemented at the DPT2 level in the framework of HF, DFT, MP2, and CC calculations. These corrections are obtained as first derivatives of the DPT2 energy. As the latter is already itself a first derivative of the energy, the analytic evaluation of first-order properties and nuclear forces requires the use of analytic second-derivative techniques.[178]

Analytic first derivatives for the SFDC-CC energy have recently been formulated and implemented for electrical perturbations.[122] These developments have enabled efficient scalar relativistic CC calculations of first-order properties for heavy-element compounds. In addition, perturbative SO corrections at HF level have been formulated and implemented as analytic second derivatives of the SFDC-HF energy, taking the difference between DC and SFDC Hamiltonians as the perturbation.[140] A more detailed discussion will be given in the following subsection.

Although the DKH method has been often used in chemical applications, analytic energy derivatives for the DKH approach are not well established. For the calculation of electrical properties, different formulations have been proposed depending on how the derivatives of the unitary transformation are included.[179-181] Analytic schemes for calculating nuclear magnetic shielding constants have been reported[182, 183] using a special scheme based on independent block diagonalizations for the nuclear attraction potential and the magnetic vector potential, respectively[184] Analytic nuclear gradients and Hessians for the SF version of the second-order DKH (SF-DKH2) method were originally reported by Rösch and coworkers,[185, 186] whereas the later work on analytic SF-DKH2 gradients by de Jong et al.[187] resorted to a mixed analytic and numerical scheme due to the complexity of the DKH transformation. Recently, an analytic implementation for the computation of nuclear gradients within the infinite-order DKH approach has been reported by Nakajima et al.[188] based on the use of a local approximation for the DKH transformation. The analytic evaluation of electrical[181] and magnetic properties[182, 189] using the BSS method has been reported in the literature. However, it should be noted that the BSS formulation for the nuclear magnetic shieldings is not unique, as it depends on whether the nuclear attraction operator and the external magnetic field are treated separately[182] or simultaneously[189] in the decoupling procedure. Analytic ZORA gradients and Hessians have been well established[148, 190-192] and are nowadays widely used in DFT calculations. In addition, analytic schemes for calculating magnetic properties have been developed for the ZORA method.[193-195] For the matrix formulations of the ZORA and IORA approaches, analytic schemes for nuclear gradients, electrical and magnetic properties were implemented and reported by Filatov and Cremer.[196-198] Finally, we mention that analytic nuclear gradients for the RESC method have been implemented by Fedorov et al.[199]

As is discussed in one of the following subsections in some more detail, the formulation of analytic X2C energy derivatives is more straightforward than for the traditional two-component methods, as the one-step block diagonalization of the one-electron Dirac Hamiltonian at the matrix level in the X2C scheme is much simpler than the multiple-step procedures and/or the operator expansion techniques in traditional two-component methods. Analytic energy gradients for the SFX2C-1e scheme have recently been formulated and implemented for the calculation of nuclear forces[200, 201] and first-order properties.[201-203] Analytic second derivatives have also been explored in the computation of nuclear Hessians[204, 205] as well as NMR chemical shieldings.[206] Initial applications of SFX2C-1e energy derivatives in the calculation of the properties of heavy-element compounds have turned out promising.[207-215] A more detailed discussion of analytic SFX2C-1e energy derivatives is presented in the following. We also refer the reader to a recent review on this topic by Filatov et al.[216] Finally, we mention that the full X2C formulation, which includes SO interactions, has been used for DFT calculation of electrical properties[181] and NMR chemical shieldings.[217]

Analytic energy derivatives for cost-effective relativistic quantum chemical approaches

Direct perturbation theory

For the calculation of first-order electrical properties at the DPT level, the nuclear-electron potential needs to be augmented in the following way

display math(68)

with math formula as the corresponding perturbation operator and ε as the associated field strength. In the case of the dipole moment, for example, the additional term comprises the dipole operator math formula and the electric field math formula and has a negative sign.

With math formula as potential, two new terms appear in the one-electron Hamiltonian, Eq. (28), used for the DPT expansion,[178]

display math(69)

This means that besides the usual perturbation term math formula there appears also a relativistic correction, in which the property operator math formula is bracketed by two math formula operators. To obtain the DPT correction to the first-order property, the modified one-electron Hamiltonian is inserted into the Lagrangian and the latter is differentiated with respect to ε and as often as required by the DPT expansion with respect to math formula. For DPT2, the differentiation is best carried out first with respect to ε, as the second differentiation step is the one that introduces the perturbed wave function parameters. In this way, one deals only with the perturbed density matrix concerning the scalar relativistic perturbation instead of several perturbed density matrices, one for each property perturbation.

In an AO formulation, the DPT2 correction for a property O is then given as[178]

display math(70)

with math formula as the elements of the one-electron density matrix for the quantum chemical method used and the basis functions math formula. At the HF level, the perturbed density matrix is determined by solving the coupled-perturbed HF equations[218, 219] for the relativistic perturbation math formula, whereas in the case of a CC calculation the perturbed amplitude and lambda equations need to be solved as well.[178]

For the analytic computation of nuclear forces at the DPT2 level, the formula[177] includes additional terms that take into account the change of the metric with the geometrical perturbation as well as the corresponding two-electron contributions. Implementations have been reported for DFT[176] as well as HF, MP2, and CC calculations,[177] thereby exploiting existing analytic second-derivative techniques.

The SFDC approach

Analytic schemes for computing first-order electrical properties within the SFDC approach have been reported for HF, MP2, as well as various CC levels of theory.[122] In the following, we restrict the discussion to analytic SFDC-CC energy derivatives, as this is the most important SFDC scheme.

Analytic expressions for the first derivatives of the SFDC-CC energy can be obtained using a Lagrangian similar to the one used in nonrelativistic CC theory,

display math(71)

However, there are some differences to be noted and the most important one concerns the summation range of the involved indices. Within the framework of the “no-pair” Hamiltonian, the electron correlation treatment only deals with the PES orbitals. Their space has the same dimension as the one spanned by the nonrelativistic spin orbitals and they are labeled by small letters, that is, math formula denote occupied PES orbitals, math formula virtual PES orbitals, and math formula is used when the occupation of the PES orbital is undetermined. However, unlike in the nonrelativistic case, the full orbital space also comprises the NES orbitals and they are needed for the expansion of the perturbed PES orbitals in terms of the unperturbed one-particle functions. Whenever the index runs over both PES and NES orbitals, capital letters are used. The convention is that math formula consists of both virtual orbitals and NES orbitals, whereas math formula comprises arbitrary PES and NES orbitals. Concerning the Lagrangian in Eq. (71), we note that the Brillouin condition with the SFDC Fock matrix FPQ need to be enforced for all A,i pairs, whereas the orthonormality constraint should hold for all P,Q pairs.

For the actual computation of SFDC-CC gradients, we note that not only the amplitude equations take the same form as in the nonrelativistic case but also the lambda equations and the expressions for the one- and two-particle density matrix. As a consequence, these steps of a gradient calculation are easily performed using the available nonrelativistic analytic derivative techniques, provided MO-based algorithms are used. This also means that the cost of a SFDC-CC gradient calculation is, considering just its rate-determining steps, similar to that in the nonrelativistic case.

Additional work is required for the treatment of orbital-relaxation effects, as the NES orbitals need to be included here. These effects are handled in a Z-vector type manner[220] starting from the coupled-perturbed SFDC-HF equations. The inclusion of the NES orbitals in this step requires the transformation of additional two-electron integrals into the MO basis involving NES orbitals, evaluation of the contributions of these integrals to the intermediate quantities, as well as the extension of the solver for the Z-vector equations to include NES orbitals in the virtual space.

For a first-order electrical property O, the final analytic expression at the SFDC-CC level then takes the form

display math(72)

In this expression, the matrix elements math formula consist of contributions from the usual nonrelativistic property integrals as well as those from relativistic property integrals in which the property operator is bracketed by the momentum operators:

display math(73)

Besides the computation of first-order properties, analytic SFDC derivative techniques have also been used for the perturbative inclusion of SO effects on top of a SFDC calculation. In such a treatment, the difference between the DC and the SFDC Hamiltonian, that is, math formula in Eq. (36), is taken as the perturbation. The SO corrections to the energy can then be formulated and implemented via a second derivative of the SFDC energy. For details of a recent implementation at the HF level, we refer the readers to Ref. [140].

The SF exact two-component theory in its one-electron variant

The formulation of analytic energy derivatives for the SFX2C-1e scheme only requires one major modification in comparison with the nonrelativistic case. In the same way as for energy calculations the nonrelativistic one-electron Hamiltonian matrix h is replaced by the SFX2C-1e Hamiltonian math formula, the derivatives of h need to be substituted by those of math formula in order to obtain the desired analytic derivative expressions. Therefore, the analytic SFX2C-1e gradient expression can be written in the same way as in Eq. (61)

display math(74)

except that math formula now contains derivatives of math formula instead of those of h:

display math(75)

The required expression for the first derivative of math formula with respect to a perturbation χ can be obtained by straightforward differentiation of the expression given in Eq. (50),[201]

display math(76)


display math(77)

and thus involves the derivatives of the four-component integrals T, V, and W, that is, math formula, and math formula, as well as the derivatives of the X and R matrices that define the unitary transformation to decouple electronic and positronic states. The derivative of X is easily obtained via[201]

display math(78)

with the derivatives of the four-component MO coefficients determined from the solution of the perturbed one-electron Dirac equation in its matrix representation. The derivative of the R matrix is given by

display math(79)

that is, obtained by straightforward differentiation of Eq. (49), and involves the derivatives of S and math formula as intermediate quantities. It should be mentioned that in principle a Z-vector type formulation[220] can be used to avoid solving the perturbed one-electron Dirac equation, as noted and implemented by Filatov et al.[221] At least for electron correlation calculations, there is no need for such a formulation, as the cost for solving the perturbed one-electron equations is negligible in comparison to those of the many-electron treatment.

The discussion so far is easily applied to electrical properties. For a first-order electrical property O, the expression is simplified to:

display math(80)

In addition to analytic first derivatives, analytic SFX2C-1e second derivatives for the calculation of the Hessian[204, 205] or magnetic properties such as nuclear magnetic shielding tensors[206] have been reported in the literature. Again, the major modification in comparison to a nonrelativistic treatment consists in the replacement of the derivatives of the nonrelativistic one-electron Hamiltonian by the SFX2C-1e counterparts. The required expressions can be derived in a straightforward manner by differentiating the expressions given in Eqs. (76)(79) with respect to a second perturbation parameter.

Benchmark Results and Example Applications of Cost-Effective Relativistic Quantum Chemical Methods

In this section, we present results that illustrate the importance of including relativistic effects in quantum chemical computations of energies and properties as well as the applicability, accuracy, and limitations of the cost-effective methods discussed in this review. We start with benchmark results for the hydrogen halides HX (X = F, Cl, Br, I, and At) as well as the coinage-metal fluorides MF (M = Cu, Ag, and Au), before we discuss a few examples where relativistic quantum chemical calculations have been used in actual chemical applications.

Benchmark results

Benchmark results for energies, dipole moments, as well as the electric field gradients of the hydrogen halides HX (X = F, Cl, Br, I, and At) and the coinage metal fluorides MF (M = Cu, Ag, and Au) are given in Tables 35. The calculations have been carried out at the HF level with the geometries given in Ref. [179] in the case of HX and in Ref. [140] in the case of MF using uncontracted versions of the ANO-RCC basis sets.[222-224].

Table 3. Energies (in Hartree) computed at the HF level using uncontracted ANO-RCC basis sets.
 Scalar Relativistic EffectsSO Effects
  1. The percentages of scalar relativistic effects recovered in comparison to SFDC results are given in parentheses in the SFX2C-1e and DPT2 columns. The percentages of SO effects obtained by the perturbative treatments in comparison to the DC results are given in parentheses in the DPT4-SO and SFDC-SO(2) columns.

HF−100.070−0.087 (95%)−0.092(100%)−0.092−8.09 × 10−6 (100%)−8.11 × 10−6 (100%)−8.11 × 10−6
HCl−460.111−1.413 (97%)−1.445 (99%)−1.455−6.74 × 10−4 (99%)−6.81 × 10−4 (100%)−6.83 × 10−4
HBr−2573.050−32.073 (99%)−31.575 (97%)−32.515−0.079 (94%)−0.083 (99%)−0.084
HI−6918.569−194.985 (99%)−183.845 (93%)−196.737−1.104 (86%)−1.255 (97%)−1.291
HAt−21,267.456−1607.641 (99%)−1338.174 (83%)−1616.867−21.126 (64%)−30.273 (91%)−33.212
CuF−1738.464−14.320 (98%)−14.274 (98%)−14.564−0.023 (97%)−0.024 (99%)−0.024
AgF−5297.180−115.362 (99%)−110.524 (95%)−116.539−0.516 (89%)−0.570 (98%)−0.582
AuF−17,964.857−1147.511 (99%)−983.817 (85%)−1154.517−13.395 (68%)−18.161 (93%)−19.604

Starting with the energies for the hydrogen halides reported in Table 3, we observe that, as expected, relativistic effects drastically increase when going down in the periodic table. While for HF the relativistic correction is only of the order of 0.1% of the total energy, the importance of the relativistic contribution increases to 0.3% for HCl, 1.3% for HBr, 2.8% for HI, and up to more than 7% in the case of HAt. We furthermore note that the scalar relativistic effects are the dominating contribution. The SO interactions contribute less than 0.01% to the relativistic corrections in the case of HF but even for HAt the relative contribution of the SO interations only amounts to 2%. A similar trend as for the hydrogen halides is also seen for the coinage-metal fluorides, with the only exception that SO effects are even less important.

The trend seen for the energies is also observed for the first-order electrical properties reported in Tables 4 and 5. The properties chosen are the dipole moment, a valence property, and the electric-field gradient, a so-called core property, which is mostly determined by the tails of the valence orbitals in the core region close to the considered nucleus. In comparison to the energies in Table 3, we note that relativistic effects are here more pronounced. In the case of the dipole moment, they amount to 0.2% for HF, 1% for HCl, 6% for HBr, and 28% for HI. For HAt, the relativistic effects even lead to a change of sign for the dipole moment. The SO contributions are here also more important, that is, they amount to 4% of the relativistic change in the case of HF and 43% in the case of HAt. The results for the coinage-metal fluoride confirm these findings, again with the only additional remark that SO effects are for these compounds somewhat less relevant. For the halogen electric field gradients (see Table 5), the corresponding relative values for the relativistic contribution are 0.3% for HF, 1% for HCl, 7% for HBr, 17% for HI, and 41% for HAt. SO effects are again less negligible when going down in the periodic table. They contribute 0.1% to the relativistic effects in the case of HF but 7% in the case of HAt. The findings for the hydrogen halides are again supported by the results for the MF compounds. There is only one peculiarity to be noted, namely that the relativistic corrections to the metal electric field gradient changes sign when going from CuF to AuF.

Table 4. Dipole moments (in a.u.) computed at HF level using uncontracted ANO-RCC basis sets.
 Scalar Relativistic EffectsSO Effects
  1. The percentages of scalar relativistic effects recovered in comparison to SFDC results are in parentheses in the SFX2C-1e and DPT2 columns. The percentages of SO effects obtained in the perturbative treatments in comparison to the DC results are given in parentheses in the DPT4-SO and SFDC-SO(2) columns.

HF0.7562−0.0014 (97%)−0.0014 (100%)−0.0014−6.68 × 10−6 (100%)−6.72 × 10−6 (100%)−6.71 × 10−6
HCl0.4648−0.0048 (98%)−0.0048 (99%)−0.0049−7.79 × 10−5 (99%)−7.92 × 10−5 (100%)−7.90 × 10−5
HBr0.3733−0.0185 (99%)−0.0176 (95%)−0.0186−0.0017 (93%)−0.0018 (101%)−0.0018
HI0.2555−0.0450 (99%)−0.0398 (88%)−0.0453−0.0092 (84%)−0.0114 (104%)−0.0110
HAt0.1871−0.1427 (99%)−0.1003 (70%)−0.1437−0.0678 (62%)−0.1273 (117%)−0.1092
CuF2.4364−0.0332 (99%)−0.0325 (97%)−0.0336−1.51 × 10−4 (88%)−1.74 × 10−4 (101%)−1.72 × 10−4
AgF2.7317−0.0912 (99%)−0.0834 (91%)−0.0919−8.43 × 10−4 (92%)−9.14 × 10−4 (100%)−9.18 × 10−4
AuF2.4974−0.3470 (99%)−0.2389 (68%)−0.3494−0.0076 (43%)−0.0179 (101%)−0.0176
Table 5. Halogen electric-field gradients (in a.u.) in hydrogen halides and metal electric-field gradients (in a.u.) in coinage-metal fluorides computed at HF level using uncontracted ANO-RCC basis sets.
 Scalar Relativistic EffectsSO Effects
  1. The percentages of scalar relativistic effects recovered in comparison to SFDC results are given in parentheses in the SFX2C-1e and DPT2 columns. The percentages of SO effects obtained in the perturbative treatments in comparison to the DC results are given in parentheses in the DPT4-SO and SFDC-SO(2) columns.

HF2.85500.0093 (97%)0.0097 (100%)0.0097−1.086 × 10−5 (99%)−1.097 × 10−5 (100%)−1.095 × 10−5
HCl3.58920.0469 (98%)0.0474 (99%)0.0478−1.135 × 10−4 (97%)−1.176 × 10−4 (100%)−1.173 × 10−4
HBr7.06950.5287 (99%)0.5047 (95%)0.5338−0.0026 (83%)−0.0034 (105%)−0.0032
HI9.72351.9601 (99%)1.7128 (87%)1.9725−0.0152 (59%)−0.0304 (118%)−0.0257
HAt15.402211.3138 (99%)7.4383 (65%)11.3657−0.1580 (23%)−1.1161 (160%)−0.6958
CuF−1.29840.0568 (99%)0.0451 (78%)0.0576−0.0002 (290%)−6.58 × 10−5 (86%)−7.64 × 10−5
AgF−2.4959−0.0881 (99%)−0.1103 (124%)−0.0887−0.0065 (74%)−0.0084 (96%)−0.0088
AuF−5.24920.9284 (99%)−0.4769 (−51%)0.9424−0.1190 (37%)−0.2863 (90%)−0.3180

Moving to the perturbative treatment of scalar relativistic effects, the DPT2 results in Tables 35 show that such an approach works well for compounds containing only rather light elements. In the case of the hydrogen halides, DPT2 recovers more than 95% of the relativistic contributions to energies and first-order properties up to HBr, but the results start to deteriorate for HI and in particular for HAt. For these cases, a second-order treatment clearly is no longer sufficient and higher-order contributions need to be taken into account. Corresponding results have been reported in the literature[99, 121, 225] and provide evidence for the convergence of the DPT expansion at the HF level. However, there are also indications that the DPT series faces some convergence problems in electron correlation treatments.[226, 227] The results for the coinage metal compounds support our findings for DPT2 in the case of energies and dipole moments. However, the DPT2 results for the Ag and Au electric field gradients in the corresponding fluorides turn out unreliable and substantial in error.

The benchmark results for the SFX2C-1e scheme demonstrates that this approach provides a robust and accurate treatment of scalar relativistic effects across the entire periodic table. The scalar two-electron picture-change error, monitored by the difference between the SFDC and SFX2C-1e results, is in all cases small and close to be negligible. Nevertheless, it is interesting to note that for the compounds containing only lighter elements such as HF and HCl this error amounts to about a few percent and that the DPT2 treatment is in these cases more accurate. However, one should also note that the magnitude of the relativistic corrections is for light-element compounds rather small, so that the errors of the SFX2C-1e treatment in these cases are rarely of practical relevance. More important is that the relative magnitude of the scalar relativistic two-electron picture-change effects decreases for heavier element compounds. The error is consistently less than 1% starting from HBr and SFX2C-1e clearly outperforms DPT2 for heavier elements. The results for the coinage metal compounds support these conclusions for both energies and first-order properties and SFX2C-1e indeed offers for those compounds an accurate and cost-effective way for including relativistic effects.

As it has been already noted, scalar relativistic contributions are in general much larger in absolute magnitude than the SO effects. For this reason, it appears justified to treat the latter in a perturbative manner, preferably on top of a SFDC calculation, as it has been already discussed before. To validate such a scheme, we report in Tables 35 also SO corrections to energies and first-order properties and compare those obtained in perturbative treatments, that is, either via DPT4 or via second-order perturbation theory on top of a SFDC-HF calculation, to rigorous SO results obtained via the difference between a full DC- and a SFDC-HF treatment. We note that in almost all cases the perturbative consideration of SO effects yields more than satisfactory results. Even for the gold electric-field gradient in AuF, the perturbative SO treatment after a SFDC calculation recovers 90% of the total SO contribution. The only exception is the astatine electric-field gradient in HAt, for which the perturbative treatment on top of a SFDC calculation overestimate the SO corrections by around 60%. We furthermore note that the treatment of SO interactions by means of DPT4 level[99] also works well in general, but due to the missing coupling between scalar relativistic and SO effects, the results deteriorate in the case of the heavier elements.

Example applications

In the following, we present a few examples in which cost-effective relativistic quantum chemical calculations have been performed in the context of chemical applications. The purpose here is to demonstrate that relativistic calculations, in particular of molecular properties, are feasible in computational studies and that the cost-effective schemes discussed in this review are suitable for routine usage. The examples to be discussed are from our own work and the first two of them are part of joint experimental and theoretical investigations in the area of rotational spectroscopy. The focus is in particular on the nuclear quadrupole coupling, which causes in rotational spectra additional splittings[228, 229] and, which due to its close relationship to the electric-field gradients at the corresponding nucleus, is rather sensitive to relativistic effects.

Bromine quadrupole coupling constants and DPT2 calculations

To investigate the fine structure due to bromine quadrupole couplings, Cazzoli et al.[230, 231] recently measured and analyzed the rotational spectra of CH2FBr and CHF2Br. The quantum chemical calculations were essential for the recording, analysis, and interpretation of the rather complicated spectra (for details, see Refs. [230, 231]). Considering that relativistic effects amount for the bromine quadrupole coupling to about 7% (see Table 5 and Ref. [178]), inclusion of scalar relativistic effects is essential in these calculations when aiming at high accuracy and reliable results. As bromine is still a rather light element (Z = 35), a perturbative treatment of relativistic contributions using lowest-order DPT is justified and this view is well supported by our benchmark results for HBr. Table 6 summarizes the results and compares the computational results with those derived from experiment. For completeness and as the experimental bromine quadrupole-coupling tensor has been reported in the literature,[232] in Table 6 we consider also CH3Br. The computations were carried out at the CCSD(T) level using the uncontracted ANO-RCC basis sets; vibrational contributions were estimated at the nonrelativistic level using second-order vibrational perturbation theory.[229]

Table 6. Comparison of computed and experimental 79Br quadrupole-coupling tensors (χ, in MHz) for CH3Br, CH2FBr, and CHF2FBr.
 NonrelativisticRelativistic CorrectionsVibr. CorrectionsaTheory (best estimate)Exp.b
  1. Calculations were performed at the CCSD(T) level using the uncontracted ANO-RCC basis with a DPT2 treatment of relativistic effects.

  2. a

    Computed by means of second-order vibrational perturbation theory. at the nonrelativistic MP2/cc-pCVTZ level.

  3. b

    Experimental values for CH3Br were taken from Ref. [232], for CH2FBr from Ref. [230], and for CHF2Br from Ref. [231].

math formula534.536.13.9574.5577.1343(31)
math formula412.327. 50.4440.2443.431(8)
math formula141.89. 42.5153.6153.566(26)
math formula−258.117.4−2.2277.7±278.56(19)
math formula487.632.4−0.3519.7521.257(10)
math formula−34.7−2.4−0.5−37.5−38.350(40)
math formula−170.2−11.3−1.6−183.1±185.87(28)

The results in Table 6 demonstrate in a convincing manner the high accuracy that can be reached in computations of bromine quadrupole-coupling constants. Clearly, the use of an electron correlation method such as CCSD(T) together with a sufficiently large basis set is an important prerequisite for this level of accuracy, but the results also document the importance of including relativistic corrections. They amount up to more than 30 MHz and contribute about 6% to the total value. Vibrational corrections, however, are much less important; the largest contribution is here seen in the case of CH3Br and amounts to 3.9 MHz.

It should be mentioned that, motivated by our investigations of CH2FBr, we recently revised the bromine quadrupole moment value[207] based on high-level calculations and the available experimental quadrupole coupling constants for HBr and the bromine atom. The revised value of 308.7(20) mb for 79Br is slightly lower than the previously accepted value of 313(3) mb by Bieroń et al.[233] and has been already used for the determination of the bromine quadrupole couplings reported in Table 6.

Iodine quadrupole coupling constants and SFDC calculations

As relativistic effects are significantly larger for iodine than for bromine, DPT2 calculations are less reliable for the computational investigation of iodine compounds. For this reason, the joint theoretical and experimental investigation of CH2FI was based on SFDC-CC calculations.[208, 234] In Table 7, our computational results are summarized and compared with the values derived from the experimental measurements. In addition, we report calculated and experimental values for CH3I with the latter taken from Ref. [235]. As expected from our benchmark results already discussed, relativistic effects are substantial for the iodine quadrupole coupling and are of the order of 17%. The agreement of the SFDC results with experiment is more than satisfactory with remaining deviations of the order of less than 0.5%. Concerning DPT2, we note that the perturbative treatment underestimates relativistic effects and leads in the case of CH2FI to remaining discrepancies to experiment of the order of 2%.

Table 7. Comparison of computed and experimental 127I quadrupole coupling tensors (χ, in MHz) for CH3I and CH2FI.
  1. Calculations were performed at the CCSD(T) level using the uncontracted ANO-RCC basis sets.

display math
display math
display math

At this point, it is also appropriate to remark that SO effects are of minor importance for the reliable prediction of iodine quadrupole couplings.[236] This is well documented by our benchmark results for HI; the SO corrections amount here to less than 2% of the total relativistic correction. This small magnitude of the SO interactions is the justification for using the cost-effective SFDC approach in the present example.

Equilibrium geometry, harmonic vibrational frequencies, and copper quadrupole coupling constant of CuOH and SFX2C-1e calculations

Relativistic effects are in particular pronounced for the coinage metals and their compounds.[237, 238] We demonstrate here the necessity of a relativistic quantum chemical treatment for a copper compound, that is, CuOH, by comparing results from nonrelativistic and SFX2C-1e calculations for the structure, harmonic frequencies, and the quadrupole coupling constant with experiment.[239, 240] The calculations were performed at HF, MP2, and CCSD(T) level to show the additional impact of electron correlation and were carried out with large uncontracted ANO-RCC basis sets.

As it is seen from Table 8, scalar relativistic effects shorten the Cu[BOND]O distance by about 0.02 Å. However, electron correlation effects are also significant, leading to a further shortening of about 0.06 Å. Unlike for the Cu[BOND]O distance, relativistic effects are more or less negligible for the O[BOND]H distance. The CuOH angle is slightly enlarged by inclusion of relativistic effects. However, electron correlation leads here to a reduction by about 1°. Our results are in good agreement with those obtained in corresponding DKH calculations reported in Ref. [241]. The changes in the geometry due to relativistic effects are also reflected in the computed harmonic frequencies. In line with the shortening of the Cu[BOND]O distance, the corresponding stretching frequency is increased by 20 cm−1. The bending frequency also increases due to relativistic effects, whereas the O[BOND]H stretching frequency is relatively insensitive to the inclusion of relativistic effects in the calculation. The relativistic contribution to the Cu quadrupole coupling constant is about 4 MHz at the HF level and more than 10 MHz at electron correlation levels. As the value for math formula is only about 10 MHz, the relativistic contribution is essential and no longer a small correction. The comparison of the results obtained at HF, MP2, and CCSD(T) level, furthermore, documents the well-known nonadditivity of electron correlation and relativistic effects.[47, 80, 242, 243] This means that one has to deal with both the effects on an equal footing in a quantum chemical calculation. Besides the SFDC treatment at the CC level, the corresponding SFX2C-1e scheme also offers a reliable and robust treatment at a computational cost comparable to those of a nonrelativistic treatment.

Table 8. Equilibrium geometry (distances in Å, angles in degrees), harmonic frequencies (in cm−1), and 63Cu quadrupole coupling constants (MHz) of CuOH.
 HFMP2CCSD(T)Exp.[241, 242]
  1. Calculations were performed at HF, MP2, and CCSD(T) level using the uncontracted ANO-RCC basis sets. Relativistic effects were included via the SFX2C-1e scheme.

math formula(Cu[BOND]O[BOND]H)120.3118.8112.0110.7111.3110.2110.1
math formula62.458.014.82.720.29.610.572(49)
math formula−39.4−37.6−17.4−11.8−20.1−15.2
math formula0.−0.1−0.9
math formula420341993859384338433831
math formula648669765807735770
math formula553584639671607629

Dipole moment curve of HI and perturbative treatment of SO corrections

As an example for the importance of SO effects on molecular properties, we computed the dipole moment curve for HI, see Figure 1. Van Stralen et al.[244] have shown for this example, based on full DC calculations, that the geometrical dependence of the HI dipole moment is to a large extent due to SO effects. Our calculations confirm their finding that both nonrelativistic and SFDC-CCSD(T) treatments are unable to yield the proper geometrical dependence even on a qualitative level. However, as the absolute magnitude of the SO contribution is in this case still rather small, a perturbartive treatment of SO seems to be justified. As demonstrated in Figure 1, the perturbative inclusion of SO effects rectifies the problems of the SFDC-CCSD(T) calculations and recovers the correct trend. The remaining nonparallelity error is probably due to the neglect of the coupling between electron correlation and SO effects and thus should not be attributed to the second-order treatment of SO effects. We expect that a perturbative consideration of SO effects at electron correlation levels will improve the accuracy.

Figure 1.

Dipole moment curve of the HI molecule. The experimental data were taken from Ref. [244]. The SFDC results were obtained at CCSD(T) level, whereas the perturbative SO corrections were calculated at HF level. Uncontracted ANO-RCC basis sets were used in all calculations.

Dipole moment and Au quadrupole coupling constants in gold compounds and SO contributions

Gold is well known to exhibit extraordinarily strong relativistic effects[245-251] and consequently accurate quantum chemical computations of the properties of gold-containing compounds have been considered as a suitable testing ground for theoretical methods to treat relativistic as well as electron correlation effects.[215, 252-257] Therefore, to fully demonstrate the applicability of the cost-effective relativistic quantum chemical schemes advocated in this article, we present as a last example CCSD(T) calculations for the dipole moments and Au quadrupole coupling constants of two gold-containing compounds, namely AuF and XeAuF. XeAuF is chosen here as it involves a somewhat unusual covalent bond between Xe and Au and thus is of significant chemical interest.[253, 258]

Nonrelativistic, SFX2C-1e, and SFDC calculations were carried out using large uncontracted basis sets (BAS14 for Au as described in Ref. [140] and uncontracted ANO-RCC basis sets for Xe and F) to be compared with DC calculations as well as the experimental data available in the literature. The results are summarized in Table 9 and reveal the importance of scalar relativistic effects, electron correlation, as well as SO corrections. Concerning the scalar relativistic effects, the SFDC calculations clearly reveal the unreliability of corresponding nonrelativistic computations. However, the SFX2C-1e scheme provides here an efficient and robust treatment of these effects. The remaining two-electron picture-change errors (measured by the difference between SFDC and SFX2C-1e results, for the used definition of the two-electron picture change error see Ref. [115]) amounts to only 1–2 MHz in case of the Au quadrupole coupling and is essentially negligible for the dipole moments. The importance of electron correlation is evident when one compares the HF and CCSD(T) results. The difference amounts to as much as 570 MHz for the quadrupole coupling constant of AuF and 460 MHz in the case of XeAuF. A rigorous treatment of both scalar relativity and electron correlation on an equal footing thus turns out to be mandatory for the present example. The SFDC- and SFX2C-1e-CCSD(T) calculations are here adequate choices, as can be seen from the computed values in Table 9. However, the numbers reported in this table also indicate the significance of SO corrections. While the corresponding effects are rather small in case of the dipole moments, they amount to about −35 MHz for the Au quadrupole coupling constant of AuF and to about −37 MHz for that of XeAuF. Their inclusion based on perturbative calculations at the HF level clearly improves the agreement with experiment.[258-260] With respect to the DC-CCSD(T) calculations reported in the literature for AuF and XeAuF,[252] it is noted that the present SFDC-CCSD(T)+SO(2) treatment is by far more economical and also more accurate, despite the perturbative treatment of the SO effects, as the reduced computational costs allow the use of larger basis sets and to correlate more electrons. In our opinion, this is more important than a rigorous treatment of SO effects at the DC level.

Table 9. Dipole moment μ (in Debye) and gold quadrupole coupling constant −eQq (in MHz) of AuF and XeAuF.
  1. Calculations were performed with the BAS14 set for Au as described in Ref. [140] and the uncontracted ANO-RCC basis sets for Xe and F. The perturbative SO corrections were obtained at the HF level using the uncontracted ANO-RCC basis sets. All calculations were carried out using the Gaussian nuclear model.[261]

DC-CCSD(T)[252, 253]4.296.76−46−539

Concluding Remarks and Outlook

The last years have witnessed a rapid development of cost-effective quantum chemical methods for the treatment of relativistic effects. The key concepts behind these developments are the separate treatment of scalar relativistic and SO effects via spin separation and the formulation of two-component schemes via a one-step block diagonalization of the one-electron Dirac equation in its matrix form. A further important aspect is the recent revival of perturbative treatments of relativistic effects.

Along with these developments, there has been an increased interest in the efficient computation of properties within relativistic methods using analytic derivative techniques. Such techniques are well established for nonrelativistic methods but have so far been explored to a much lesser extent in relativistic quantum chemistry. This review discusses the recent developments concerning analytic energy derivatives in relativistic electronic structure theory. A special focus has been, thereby, on cost-effective schemes for treating relativistic effects.

While the recent developements (i.e., gradients for DPT2, analytic first-order properties for SFDC-CC, analytic first and second derivatives for SFX2C-1e, etc.) are important for establishing the underlying methods as standard tools in computational chemistry, more work is needed to reach a similar level of applicability and efficiency as in nonrelativistic quantum chemistry. In particular, there are a number of challenges, which require further theoretical as well as implementational efforts. Those are the implementation of geometrical first and second derivatives for the SFDC-CC approach, in this way rendering this most rigorous scheme for treating scalar relativistic effects more applicable, the inclusion of SO corrections in analytic schemes for computing properties, and the formulation of generally applicable schemes for computing magnetic properties with an adequate consideration of SO effects, which are here often more important than for energies and electrical properties. It should be noted that the consideration of two-electron SO contributions to molecular properties is particularly challenging, as it involves the evaluation and processing of additional relativistic two-electron integrals. The significance of SO two-electron picture-change effects furthermore requires the use of transformed two-electron SO interactions in two-component calculations.[262-264] A further issue is the extension of the applicability of the cost-effective schemes presented here to open-shell systems that necessitate a multideterminantal treatment, as the present developments are mostly concerned with closed-shell or high-spin open-shell systems. Complications are here in particular expected as soon as SO splittings occur. For high-accuracy calculations of molecular properties, it is also necessary to consider Breit interactions, which requires the evaluation and processing of additional relativistic two-electron integrals[265] and, thus, will be computationally challenging. Finally, the consideration of QED corrections in energy and property calculations might be an interesting topic, though this is probably not warranted for most chemical applications.

We conclude that the recent developments in relativistic quantum chemistry made it clear that there is a bright future for relativistic methods in computational chemistry. Nevertheless, further work on implementing analytic energy derivatives (as well as response-theory techniques) is essential to secure this applicability of relativistic quantum chemical methods and to develop the corresponding schemes as general tools for solving problems in but not only in heavy-element chemistry.