Effective Hamiltonian for near‐degenerate states in direct relativistic perturbation theory. I. Formalism

Perturbation Theory Treatment of Spin-Orbit Coupling, Part I: Double Perturbation Theory Based on a Single-Reference Initial Approximation

We develop a perturbation theory for solving the many-body Dirac equation within a given relativistic effective-core potential approximation. Starting from a scalar-relativistic unrestricted Hartree-Fock (SR UHF) solution, we carry out a double perturbation expansion in terms of spin-orbit coupling (SOC) and the electron fluctuation potential. Computationally convenient energy expressions are derived through fourth order in SOC, second order in the electron fluctuation potential, and a total of third order in the coupling between the two. Illustrative calculations on the halogen series of neutral and singly positive diatomic molecules show that the perturbation expansion is well-converged by taking into account only the leading (nonvanishing) term at each order of the electron fluctuation potential. Our perturbation theory approach provides a computationally attractive alternative to a two-component self-consistent field treatment of SOC. In addition, it includes coupling with the fluctuation potential through third order and can be extended (in principle) to multireference calculations, when necessary for both closed- and open-shell cases, using quasi-degenerate perturbation theory.

Perturbation Theory Treatment of Spin-Orbit Coupling II: A Coupled Perturbed Kohn-Sham Method

A noncanonical coupled perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment of spin-orbit coupling (SOC) is provided. We take the scalar-relativistic KS-DFT/HF solution, obtained with a relativistic effective core potential, as the zeroth-order approximation. Explicit expressions are given for the total energy through the 4th order, which satisfy the 2n + 1 rule. Second-order expressions are provided for orbital energies and density variables of spin-current DFT. Test calculations are carried out on the halogen homonuclear diatomic and hydride molecules, including 6p and 7p elements, as well as open-shell negative ions. The computed properties through second or third order match well with those from reference two-component self-consistent field calculations for total and orbital energies as well as spin-current densities. In only one case (At₂^-) did a significant deviation occur for the remaining density variables. Our coupled perturbation theory approach provides an efficient way of adding the effect of SOC to a scalar-relativistic single-reference KS-DFT/HF treatment, in particular because it does not require diagonalization in the two-component spinor basis, leading to saving factors on the number of required floating-point operations that may exceed one order of magnitude.

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.

Comprehensive relativistic ab initio and density functional theory studies on PtH, PtF, PtCl, and Pt(NH(3))(2)Cl(2)

Platinum monohydride is taken as an example to compare the performance of various relativistic and correlation approaches, such as all-electron DPT (direct perturbation theory), ECP (effective core potential); RSPT2, RSPT3 (second- and third-order multireference Rayleigh-Schrödinger perturbation theory), CCSD(T) (coupled-cluster with singles, doubles, and perturbative triples), as well as the four-component relativistic density functional theory. It is shown that first-order DPT performs significantly better than the (first-order) Breit-Pauli Hamiltonian. The performance of different approaches for the excitation energies of the platinum diatomics is discussed critically. The molecular spectroscopic constants for PtF and PtCl are predicted for the first time. The geometric data for several isomers of cis- and trans-Pt(NH(3))(2)Cl(2) are reported. The corresponding energetic data are calculated at relativistic all-electron and ECP-CCSD(T) as well as four-component relativistic density functional levels of theory. Contrary to previous results, it is found that the two C(2v) isomers of cis-Pt(NH(3))(2)Cl(2) are marginally separated in energy, which could be ascribed to Cl-H interactions.