Recent Advances in Relativistic Molecular Theory

The following sections are included:

• PREFACE

• AUTHOR LIST

• CONTENTS

The energy-consistent pseudopotential variant of the relativistic effective core potential method is briefly reviewed. The corresponding available valence model Hamiltonian parametrizations for lanthanide and actinide elements as well as related valence basis sets are discussed. Calibration studies for atoms and diatomic molecules assessing the reliability of the outlined approaches are summarized and selected examples for recent applications to problems of chemical interest are provided. Finally, future directions for the development of energy-consistent pseudopotentials for f-elements are mentioned.

We present in this paper some recent developments of relativistic model core potential (MCP) method. We briefly outline the determination procedure of the MCP parameters and valence basis sets. Some characteristics of MCPs and valence basis sets for the lanthanide elements recently developed are discussed and the applications to the ground and low-lying excited states of diatomic molecules containing lanthanide elements are reported. The calculated results show that the MCP method can predict quantitatively not only the ground state properties but also the term values of low-lying excited states. We also outline the development of generally contracted Gaussian-type function sets for relativistic correlating functions of Ga-Kr, In-Xe, and Tl-Rn atoms. The correlating functions are constructed using atomic natural orbital approach based on scalar relativistic CI calculations with MCPs and are applied to several atoms and the TlH molecule. The spectroscopic constants of the TlH molecule yielded by relativistic MCP calculations including spin-orbit effects are in excellent agreement with the experimental results. It is demonstrated that the MCP method can properly evaluate the valence correlation owing to the nodal structures of the associated valence orbitals, and can appropriately include the relativistic effects.

Two computational methods are described for carrying out configuration interaction (SO–CI) calculations in which spin-orbit and other relativistic effects are included in the Hamiltonian by means of effective core potentials. A many-electron basis of eigenfunctions is employed in each case, and advantage is taken of the Wigner–Eckart theorem and the spin ladder operator formalism to minimize the computational effort required to obtain matrix elements of the spin-orbit interaction for different M_S combinations. Special attention is given to procedures which have been implemented to make efficient use of double group symmetry relationships in the calculations. One SO–CI method employs a contracted basis of Λ–S eigenfunctions obtained from a conventional MRD–CI treatment, and thus requires the solution of only relatively small secular equations. The MR–SO–CI method by contrast is carried out in a much larger basis of simple linear combinations of Slater determinants, and thus in principle is capable of a more balanced treatment of Coulomb and spin-orbit effects. Examples of both types of calculations are considered for systems containing heavy atoms such as lead and bismuth, and the results demonstrate that each method is capable of making reliable predictions of the energy and radiative lifetimes of such systems over a broad range of electronic states and nuclear conformations.

Various one and two electron spin-orbit coupling Hamiltonians are discussed, along with symmetry properties of their matrix elements. Alternatives to full all-electron treatments are given in some detail, most importantly, the effective charge approach, an approach based on model core potentials and the partial two-electron method. Assessment of the applicability of these methods is based on numerous tests on atoms and molecules. Spin-orbit multi-configuration quasi-degenerate theory (SO-MCQDPT) is briefly introduced and several applications illustrating its promising performance are given. The vibrational dependence of spin-orbit coupling constants is investigated in some diatomics.

The fundamental theory underlying relativistic quantum chemistry is quantum electrodynamics (QED). For almost all chemical situations, however, electronic interactions with the radiation field or with the positronic degrees of freedom, which are inherently included in QED, are of negligible importance. Relativistic effects are hence very satisfactorily described by the traditional 4-component Dirac equation, which is currently computationally still too expensive for most systems of chemical interest, i.e., for molecules with more than two heavy nuclei. It is thus highly desirable to further reduce the computational requirements. This may most conveniently be achieved by a transition to two-component formulations, either by elimination techniques for the small component or by suitably chosen unitary transformations which decouple the Hamiltonian. Only these further simplifications of the theoretical framework enable a transgression of traditional theory boundaries, opening relativistic quantum chemistry to a far larger class of systems by application of 2-component and 1-component electron-only theories.

After a brief discussion of the most important reduction techniques for the Dirac equation, we focus on the generalized Douglas–Kroll (DK) transformation. Its central idea is the expansion of the Hamiltonian in even terms of definite order in the external potential, achieved by the application of a sequence of unitary transformations, which eliminate the odd terms of the Hamiltonian step by step. For this purpose, the most general parametrization of the unitary matrices is used, and subsequently applied in order to derive the fifth-order approximation. While DKH2 – DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy–Wouthuysen transformation. The freedom in the choice of this expansion coefficient may be employed to seek for an optimum unitary transformation.

Various DK approximations, DKH1 to DKH5, are applied to both one-electron hydrogen-like ions and many-electron atomic systems in order to investigate their numerical performance. Whilst the higher-order approximations yield remarkable accuracy for the one-electron systems as compared to the exact Dirac ground state energy, the deviations of the energy from Dirac–Fock results is systematically increasing for many-electron calculations. This behavior is due to the complete neglect of spin-dependent terms and of the transformation of the two-electron terms in the standard protocol of the DK transformation.

The generalized-UHF (GUHF) theory is reviewed in the framework of the two-component quasi-relativistic molecular orbital theory. The GUHF orbitals in which alpha and beta spin functions have independent complex spatial functions can describe the magnetic interactions of spinning electrons, which are important for magnetic chemistry. Spin-orbit (SO) and Zeeman interactions are examples. This study presents the SO-UHF and SO-GUHF theories in which the SO interaction is considered and the quasi-relativistic (QR)-GUHF theory in which the DKH (Douglas-Kroll-Hess) quasi-relativistic Hamiltonian is used. We review the theories on the magnetic shielding constants based on the SO-UHF, SO-GUHF and QR-GUHF methods and explain the applications to the ¹H shielding constants of hydrogen halides, the ¹³C chemical shifts of methyl halides, ¹¹⁹Sn chemical shifts of tin halides, the shielding constants of noble gas atoms, and the ¹⁹⁹Hg shielding constants of mercury halides.

The formulation and computational aspects of relativistic electronic structure theories that are efficient enough to make calculations on heavy-atom polyatomic systems tractable are discussed. This chapter contains surveys of both fully relativistic four-component and quasirelativistic two-component methods. First the four-component approach for the relativistic ab initio molecular orbital (MO) methods is reviewed. Emphasis is placed on efficient computational schemes that take advantage of generally contracted spherical harmonic Gaussian-type spinors (GTSs) and use sophisticated integral evaluation methods. Illustrative calculations, performed with a new four-component relativistic ab initio MO program package, REL4D, demonstrate the efficiency attainable in the Dirac-Hartree-Fock (DHF) and Dirac-Hartree-Fock (DKS) methods. Next, in the two-component quasirelativistic framework, two relativistic Hamiltonians, RESC and higher-order Douglas-Kroll (DK), are introduced and illustrative calculations are shown. Numerical results for several systems show that good accuracy can be obtained with our third-order DK (DK3) Hamiltonian.

A program suite, PROPHET4R, consists of program codes for four-component relativistic atomic and molecular calculations and associated nonrelativistic calculations. Outlines of the codes are described along with the calculated results using them.

The BDF (Beijing Density Functional) program package is such a code that can perform nonrelativistic, one-, two-, and four-component relativistic density functional calculations on medium-sized molecular systems with various functionals in most compact and yet sufficient basis set expansions. The mergence of different approaches in a single code facilitates direct and systematic comparisons between different Hamiltonians, since they share all the same numerical and technical issues. In the present review, the salient features and perspectives of the code will be discussed.

High-accuracy results for energy levels of heavy and superheavy atoms are presented. The Dirac-Coulomb-Breit Hamiltonian serves as the starting point, and electron correlation is treated by the Fock-space coupled cluster method. The recent intermediate Hamiltonian approach opens the way to larger P (model) spaces, which may be varied to convergence, thus enhancing accuracy and allowing applications to states not accessible before. Very large basis sets, going up to l = 8, are used. High-l orbitals are particularly important for transitions involving f electrons. The outer 20–40 electrons are correlated. The Breit term is significant for fine-structure splittings and for f transitions. Representative applications are described, including electron affinities of alkali atoms, obtained within 5 meV of known experimental values; the gold atom, with relativistic effects of 3–4 eV on transition energies; and Pr³⁺, where the many ƒ² levels are reproduced with great precision. The most exciting aspect of the high accuracy provided by the method is the possibility of reliable predictions for superheavy elements, where level ordering (and therefore chemistry) may differ from that of the lighter homologues. Thus, the ground state of eka-gold (E111) is 6d⁹7s², rather than the (n - 1)d¹⁰ns exhibited by other group-11 elements; in Rf (E104), opposite effects of relativity and correlation lead finally to a 7s²6d² ground state, ~0.3 eV below the 7s²6d7p predicted by MCDF; eka-lead (E114), a potential member of the "island of stability" forecast by nuclear physics, is expected to have ionization potentials higher than all other group-14 atoms except carbon, showing therefore more inert and less metallic character than lead; and eka-radon (E118) has a unique property for a rare gas, binding an electron with an affinity of 0.064(2) eV. QED corrections reduce this value by 0.0059(5) eV or 9%, the largest relative QED effect reported for a neutral or weakly ionized species. Finally, some molecular applications are described.