## Introduction

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.

SF-ZORA | SF-DKH(BSS,RESC) | SFX2C-1e | SFDC | DPT2 | DC | |
---|---|---|---|---|---|---|

Spin separation | Yes | Yes | Yes | Yes | Yes | No |

Elimination of small component | Yes | Yes | Yes | No | No | No |

Matrix representation | No | Yes | Yes | – | – | – |

One-step block diagonalization | No | No | Yes | – | – | – |

Method | Formal Rate-Determining Step | Nonrel | SFX2C-1e | SFDC | DPT2 | DC[44, 51] |
---|---|---|---|---|---|---|

HF | Integral evaluation | 1 | 1 | ∼10 | ∼3 | ∼60 |

MP2 | Integral transformation | 1 | 1 | ∼6 | ∼1 | ∼16 |

CCSD | CC equations | 1 | 1 | ∼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:

- 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.
- 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]
- 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.
- 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.