Wenjian Liu

Quasirelativistic theory equivalent to fully relativistic theory

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to a quasirelativistic Hamiltonian matrix, that has the same electronic eigenstates as the original Dirac matrix. This transformation involves a matrix X, for which an exact identity is derived, and which can be constructed either in a noniterative way or by various iteration schemes, without requiring an expansion parameter. The convergence behavior of five different iteration schemes is studied numerically, with very promising results.

Ideas of relativistic quantum chemistry

The basic ideas of relativistic quantum chemistry are highlighted, with the most important ingredients summarised as follows. (1) The restricted kinetic balance (RKB) condition, being both necessary and sufficient, serves as the cornerstone for the matrix representation of the Dirac-based Hamiltonian. (2) The concept of matrix transformation plays the key role in formulating two-component relativistic theories. Some popular ones, albeit presented as operators, are de facto matrix formulations in terms implicitly of the RKB condition. They merely make simple things complicated as a one-step block-diagonalization of the matrix Dirac equation can do the whole job. (3) The computational efficiency for both four- and two-component relativistic theories can be gained by means of the simple chemical idea of ‘from atoms to molecule’ without recourse to mathematical tricks. The two branches of relativistic theories have thus been made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. It is concluded that the best relativistic electrons-only Hamiltonian has been found, which can be combined with any know-how correlation methods for electronic structure calculations of all the atoms in the Periodic Table. Most amazingly, the new quantum mechanical equation serves as a seamless bridge between the Schrödinger and Dirac equations. In short, the ‘relativity problem’ in chemistry has been solved.

Exact two-component Hamiltonians revisited

Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy-Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schrodinger picture. The two formulations become equivalent after the mistake is corrected.

Quasirelativistic theory. II. Theory at matrix level

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to the matrix representation of a quasirelativistic Hamiltonian that has the same electronic eigenstates as the original Dirac matrix (but no positronic eigenstates). This transformation involves a matrix X, for which an exact identity is derived and which can be constructed either in a noniterative way or by various iteration schemes, not requiring an expansion parameter. Both linearly convergent and quadratically convergent iteration schemes are discussed and compared numerically. The authors present three rather different schemes, for each of which even in unfavorable cases convergence is reached within three or four iterations, for all electronic eigenstates of the Dirac operator. The authors present the theory both in terms of a non-Hermitian and a Hermitian quasirelativistic Hamiltonian. Quasirelativistic approaches at the matrix level known from the literature are critically analyzed in the frame of the general theory.

Quasirelativistic theory I. Theory in terms of a quasi-relativistic operator

In this first part of a general analysis of a quasi-relativistic theory, i.e. a relativistic theory for electrons only, the transformation of the Dirac operator to an operator with two-component spinor solutions is studied, while a forthcoming second part will be devoted to a transformation at matrix level. We start with a simple derivation of the key relation between the upper (ϕ) and lower (χ) components of the Dirac bispinor (ψ) for both electrons and positrons. The three possible choices of a non-hermitian quasi-relativistic Hamiltonian L, a hermitian one with non-unit metric, and a hermitian one L + with unit metric are compared. The eigenfunctions of the first two are the upper components ϕ of ψ, while those of L + are the Foldy–Wouthuysen-type spinorsφ. Some general properties of quasi-relativistic Hamiltonians and their eigenfunctions are discussed, especially the behaviour near the position of a nucleus. Exact solutions, and even variational ones are only obtained if the orbital basis describes the weak singularities at the positions of the nuclei correctly. A new derivation of the quasi-relativistic version of direct perturbation theory (DPT) is given, followed by the theory of quasi-relativistic effective Hamiltonians, both non-hermitian and hermitian ones. The classical Foldy–Wouthuysen transformation is then presented as the singular limit of quasi-relativistic effective Hamiltonians with the model space extended to the entire space of positive-energy states. Finally, the problems that arise for a Douglas–Kroll transformation or the regular approximation at operator level are studied in detail. In part 2, it will be shown that everything becomes much simpler if one performs the transformation from relativistic to quasi-relativistic theory at matrix level. ¶This paper is dedicated to Andrzej Sadlej on the occasion of his 65th birthday.

Time-dependent four-component relativistic density functional theory for excitation energies

Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.

The Beijing Density Functional (BDF) Program Package: Methodologies and Applications

The Beijing Density Functional (BDF) 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 this account, the methodologies adopted in the code will be discussed in great detail and some applications of the code will be briefly presented.

Time-dependent four-component relativistic density-functional theory for excitation energies. II. The exchange-correlation kernel.

We extend our previous formulation of time-dependent four-component relativistic density-functional theory [J. Gao, W. Liu, B. Song, and C. Liu, J. Chem. Phys. 121, 6658 (2004)] by using a noncollinear form for the exchange-correlation kernel. The new formalism can deal with excited states involving moment (spin)-flipped configurations which are otherwise not accessible with ordinary exchange-correlation functionals. As a first application, the global potential-energy curves of 16 low-lying omega omega-coupled electronic states of the AuH molecule have been investigated. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are grossly in good agreement with those of ab initio multireference second-order perturbation theory and the available experimental data.

Four-component relativistic theory for nuclear magnetic shielding: Magnetically balanced gauge-including atomic orbitals

It is recognized only recently that the incorporation of the magnetic balance condition is absolutely essential for four-component relativistic theories of magnetic properties. Another important issue to be handled is the so-called gauge problem in calculations of, e.g., molecular magnetic shielding tensors with finite bases. It is shown here that the magnetic balance can be adapted to distributed gauge origins, leading to, e.g., magnetically balanced gauge-including atomic orbitals (MB-GIAOs) in which each magnetically balanced atomic orbital has its own local gauge origin placed on its center. Such a MB-GIAO scheme can be combined with any level of theory for electron correlation. The first implementation is done here at the coupled-perturbed Dirac-Kohn-Sham level. The calculated molecular magnetic shielding tensors are not only independent of the choice of gauge origin but also converge rapidly to the basis set limit. Close inspections reveal that (zeroth order) negative energy states are only important for the expansion of first order electronic core orbitals. Their contributions to the paramagnetism are therefore transferable from atoms to molecule and are essentially canceled out for chemical shifts. This allows for simplifications of the coupled-perturbed equations.

Four-component relativistic theory for nuclear magnetic shielding constants: the orbital decomposition approach.

The authors present a scheme to simplify four-component relativistic calculations of nuclear magnetic shielding constants. The central idea is to decompose each first order orbital into two terms, one is magnetically balanced and directly leads to the diamagnetic term, and the other is, to leading order of relativity, kinetically balanced and can therefore simply be represented in the basis of unperturbed positive energy states. As a matrix formulation, the present approach is far simpler than other operator theories. Combined with the Dirac-Kohn-Sham ansatz, the nuclear magnetic shielding constants for the Kr, Xe, and Rn atoms as well as the HBr and HI molecules are calculated, and the results compare favorably with those of other schemes.