A concise review of the landscape of Relativistic Quantum Chemistry is given, and related to the papers in the present issue.

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The article discusses a highly efficient and accurate theory for electronic-state calculations of large-scale heavy-element compounds. After overviewing the current relativistic quantum-chemical theories, the authors describe what points should be considered and improved in order to attain the objective. In addition, they have developed an efficient and accurate two-component relativistic scheme, termed local unitary transformation (LUT) scheme based on infinite-order Douglas-Kroll-Hess method. Numerical applications show the effectiveness of the scheme. Furthermore, the extension of the LUT scheme to the divide-and-conquer (DC) method achieves overall linear-scaling computational cost. In particular, the present scheme would realize a paradigm shift from a non-relativistic (NR) framework to a relativistic one in the quantum chemistry field because the scheme gives close results to four-component relativistic ones with NR computational costs.

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The concept of energy density has been formulated in terms of stress tensor in general relativity. The spin vorticity of electron has been hidden in the energy-momentum tensor and plays a significant role in the dynamics of electron. The dynamics of electron spin is driven by the antisymmetric component of the stress tensor of electron through the vorticity. The symmetric component of the stress tensor of electron drives the tensorial energy density of chemical reactivity.

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The four-component relativistic general multiconfigurational quasidegenerate perturbation theory (GMC-QDPT) and its applications to some atomic and molecular systems are reviewed. An efficient and accurate approximation to the relativistic GMC-QDPT is also presented. In the approximation, the terms including core to virtual excitations in the second-order effective Hamiltonian are replaced with those of the conventional quasidegenerate perturbation theory. The approximation form, which we call semi-approximate second-order form, is applied to some molecular systems. The computed excitation energies and potential energy curves were in good agreement with the original GMC-QDPT values as well as available experimental data.

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We have developed an efficient direct spin-orbit configuration interaction (SOCI) program system with one-component L-S coupled configurations. To realize the high performance, we have employed various aspects of group theories. The formulas of coupling coefficients for the U (2n) generators were expressed in terms of segment values of GUGA. With the time reversal symmetry, we can define "real" spin functions for the even number of electrons system. The use of such spin-functions is essential to make the CI matrix real symmetric and also to facilitate the double point group applications. For the odd number of electrons system, we add a virtually non-interacting electron to make the total number of electrons even, enabling us to use the "real" spin functions for the real arithmetic, and keeping the high efficiency of computation.

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The Douglas–Kroll (DK) approach decouples the large and small components of the Dirac spinors in the presence of an external potential by repeating several unitary transformations. The DK transformation is a variant of the Foldy–Wouthuysen (FW) transformation, and adopts the external potential V as an expansion parameter instead of the speed of light c that is used in the FW transformation. In the present review, we will describe both the theoretical and practical aspects of the DK approach.

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The significance of the relativistic density functional theory (RDFT) grows with the trend that the target of the theoretical chemistry gradually moves to metal surfaces and metal complexes. Since electronic configurations containing a lot of electrons should be quantitatively reproduced to approach the reactions and properties of large systems including metals,it is reasonable to consider that the RDFT is one of the best theories for theoretical studies on large metallic systems. The RDFT is a certain theory founded on quantum electrodynamics and composes the major proportion of relativistic calculations in chemistry. In this review,the foundations of the RDFT are briefly explained and then a specific way to state-of-the-art two-component RDFTs is revealed. This intends to show a way that the RDFT would track in the near future.

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We have developed quantum multi-component ab initio methods, such as multi-component molecular orbital and multi-component quantum Monte Calro methods, for theoretical calculation of positronic compounds. We have carried out the accurate calculation of positron affinity (PA) and pair annihilation rate for positronic compounds by using these multi-component methods, and found that these values are in reasonable agreement with the corresponding experimental ones. We found that (i) the positronic orbital is much more delocalized than the electronic highest occupied molecular orbital, and (ii) there exist the strong correlation between PA and dipole moment.

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Isotope separation is important in industry, such as uranium enrichment. It is also very important in geochemistry and cosmo-chemistry because heavy and light isotopes are gradually fractionated in global equilibrium processes. For isotopes of heavy elements such as uranium and gadolinium, this fractionation occurs mainly owing to the difference of nuclear charge radii between the isotopes. This is because electronic states of isotopologues are different because of the difference of nuclear charge radii. Nuclear volume terms (lnKnv) in the equilibrium constants of isotope fractionations can be calculated from the electronic energy difference of isotopologues, obtained by relativistic quantum chemical calculations with finite-size nuclear models. In this review, we summarize our theoretical results in uranium isotope separations, using U (III)-U (IV) and U (IV)-U (VI) redox reaction systems. Calculated nuclear volume term (lnKnv) is reasonably close to the ones estimated from experimental analysis of temperature dependence in equilibrium constants. We used the four-component Dirac-Coulomb Hartree-Fock method with the Fermi and Gaussian-type nuclear models in our calculations. We also briefly mention how the nuclear volume effect is important in geochemistry or cosmo-chemistry referring to recent relevant works.

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The thermodynamic properties and electronic structure of hydrated Ra²⁺ have been investigated using ab initio quantum chemical calculations that apply the relativistic model core potential method and compared with those of the other hydrated divalent alkaline earth metal ions (Mg²⁺, Ca²⁺, Sr²⁺, and Ba²⁺). The solvation free energies calculated for [Ra (H₂O)_n]²⁺ (n = 1–9) in a continuum dielectric media (semi-continuum model) showed that the hydration number of Ra²⁺ is in the range of 7–9. Natural population analysis (NPA), natural bond orbital (NBO) analysis and localized molecular orbital energy decomposition analysis (LMO-EDA) showed that the dominant interaction between hydrated Ra²⁺ ions and solvent water molecules is electrostatic interaction to form coordination bonds which have a strong ionic bond character. On the other hand, not only electrostatic interaction but also covalent interaction accompanying charge-transfer from solvent water molecules to the central ion are important in the interaction between hydrated Mg²⁺ or Ca²⁺ (lighter divalent alkaline earth metal ions) and solvent water molecules.

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