Treating spin-orbit coupling at different levels in equation-of-motion coupled-cluster calculations

Error of relativistic effective core potentials for closed-shell diatomic molecules of p-block heavy and superheavy elements in DFT and TDDFT calculations

Pseudopotentials (PP) are extensively used in electronic structure calculations, particularly for molecules containing heavy elements. Parameters in PPs are mainly determined from ab initio results, and errors of such PPs in density functional theory (DFT) calculations have been studied previously. However, PP errors on results with spin-orbit coupling and those in time-dependent DFT (TDDFT) calculations have not been reported previously. In this work, we investigate the error of the small-core energy-consistent Stuttgart/Koln pseudopotentials in DFT and TDDFT calculations with and without spin-orbit coupling. Ground state bond lengths, harmonic frequencies, dissociation energies, and vertical excitation energies for a series of closed-shell diatomic heavy and superheavy p-block molecules are calculated using several popular exchange-correlation functionals. PP errors are estimated by comparing with results using the all-electron Dirac-Coulomb (-Gaunt) Hamiltonian. Our results show that the difference between ground state properties and most excitation energies in scalar-relativistic calculations with the PP and those of all-electron calculations is quite small. This difference becomes somewhat larger when spin-orbit coupling (SOC) is present, especially for properties that are affected by SOC to some extent. In addition, the errors of the PPs are insensitive to the employed exchange-correlation functionals in most cases. Our results indicate that reasonable DFT and TDDFT results can be obtained using the small-core energy-consistent Stuttgart/Koln pseudopotentials for heavy and super-heavy p-block molecules.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Intermediate Hamiltonian Fock-space coupled-cluster theory for excitation energies, double ionization potentials, and double electron attachments with spin–orbit coupling

The intermediate Hamiltonian Fock-space coupled-cluster methods at the singles and doubles level (IHFSCCSD) for excitation energies in the (1p, 1h) sector, double ionization potentials in the (0p, 2h) sector, and double electron attachments in the (2p, 0h) sector of the Fock space are implemented based on the CCSD method with spin–orbit coupling (SOC) included in the post-Hartree–Fock treatment using a closed-shell reference in this work. The active space is chosen to contain those orbitals that have the largest contribution to principal ionized or electron-attached states obtained from the equation-of-motion coupled-cluster calculations. Both time-reversal symmetry and spatial symmetry are exploited in the implementation. Our results show that the accuracy of IHFSCCSD results is closely related to the active space, and the sufficiency of the active space can be assessed from the percentage of transitions within the active space. In addition, unreasonable results may be encountered when the ionized or electron-attached states with a somewhat larger contribution from double excitations are included to determine the active space and cluster operators in the (0p, 1h) or (1p, 0h) sector of the Fock space. A larger active space may be required to describe SO splitting reliably than that in the scalar-relativistic calculations in some cases. The IHFSCCSD method with SOC developed in this work can provide reliable results for heavy-element systems when a sufficient active space built upon the principal ionization potential/electron affinity states is adopted.

Equation-of-motion coupled-cluster theory for double electron attachment with spin-orbit coupling

We report implementation of the equation-of-motion coupled-cluster (EOM-CC) method for double electron-attachment (DEA) with spin-orbit coupling (SOC) at the CC singles and doubles (CCSD) level using a closed-shell reference in this work. The DEA operator employed in this work contains two-particle and three-particle one-hole excitations, and SOC is included in post-Hartree-Fock treatment. Time-reversal symmetry and spatial symmetry are exploited to reduce computational cost. The EOM-DEA-CCSD method with SOC allows us to investigate SOC effects of systems with two-unpaired electrons. According to our results on atoms, double ionization potentials (DIPs), excitation energies (EEs), and SO splittings of low-lying states are calculated reliably using the EOM-DEA-CCSD method with SOC. Its accuracy is usually higher than that of EOM-CCSD for EEs or DIPs if the same target can be reached from single excitations by choosing a proper closed-shell reference. However, performance of the EOM-DEA-CCSD method with SOC on molecules is not as good as that for atoms. Bond lengths for the ground and the several lowest excited states of GaH, InH, and TlH are underestimated pronouncedly, although reasonable EEs are obtained, and splittings of the ³Σ^- state from the π² configuration are calculated to be too small with EOM-DEA-CCSD.