Two- and four-component relativistic generalized-active-space coupled cluster method: Implementation and application to BiH

Cholesky Decomposition in Spin-Free Dirac-Coulomb Coupled-Cluster Calculations

We present an implementation for the use of Cholesky decomposition (CD) of two-electron integrals within the spin-free Dirac-Coulomb (SFDC) scheme that enables to perform high-accuracy coupled-cluster (CC) calculations at costs almost comparable to those of their nonrelativistic counterparts. While for nonrelativistic CC calculations, atomic-orbital (AO)-based algorithms, due to their significantly reduced disk-space requirements, are the key to efficient large-scale computations, such algorithms are less advantageous in the SFDC case due to their increased computational cost in that case. Here, molecular-orbital (MO)-based algorithms exploiting the CD of the two-electron integrals allow us to reduce disk-space requirements and lead to computational cost in the CC step that is more or less the same as in the nonrelativistic case. The only remaining overhead in a CD-SFDC-CC calculation is due to the need to compute additional two-electron integrals, the somewhat higher cost of the Hartree-Fock calculation in the SFDC case, and additional cost in the transformation of the Cholesky vectors from the AO to the MO representation. However, these additional costs typically amount to less than 5-15% of the total wall time and are thus acceptable. We illustrate the efficiency of our CD scheme for SFDC-CC calculations on a series of illustrative calculations for the X(CO)4 molecules with X = Ni, Pd, Pt.

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.

Theoretical investigation on the low-lying electronic states of diatomic magnesium bismuthide including the spin-orbit coupling effect

High-level ab initio computations have been performed to investigate molecular structures, potential energy curves, vibrational energy levels and spectroscopic constants for twelve Λ-S states of the first four dissociation limits of MgBi. Characterizations of seven Ω states, corresponding to the first and the second Λ-S dissociation limits, have been explored for the first time. The spin-orbit coupling effect is revealed to have introduced a significant impact on the pattern of these electronic states and interactions among them. Our predictions for molecular structures and spectroscopic constants of MgBi are compared with available data of other magnesium-group 18 family species. Regular tendencies of these parameters are clearly exhibited when the group 18 atom is replaced by another one in the group. Information associated with transition dipole moments, Franck-Condon factors, vibrational branching ratios and radiative lifetimes between the Ω states are obtained and their transitional properties are analyzed and discussed. The results and data determined in this work are expected to guide and assist laboratorial detections of MgBi and to extend our understanding for the magnesium-group 18 species.

Correlated ab initio calculations of one-bond 31 P77 Se and 31 P125 Te spin-spin coupling constants in a series of PSe and PTe systems accounting for relativistic effects (part 2)

Synthetic chalcogen-phosphorus chemistry permanently makes new challenges to computational Nuclear Magnetic Resonance (NMR) spectroscopy, which has proven to be a powerful tool of structural analysis of chalcogen-phosphorus compounds. This paper reports on the calculations of one-bond ³¹ P⁷⁷ Se and ³¹ P¹²⁵ Te NMR spin-spin coupling constants (SSCCs) in the series of phosphine selenides and tellurides. The applicability of the combined computational approach to the one-bond ³¹ P⁷⁷ Se and ³¹ P¹²⁵ Te SSCCs, incorporating the composite nonrelativistic scheme, built of high-accuracy correlated SOPPA (CC2) and Coupled Cluster Single and Double (CCSD) methods and the Density Functional Theory (DFT) relativistic corrections (four-component level), was examined against the experiment and another scheme based on the four-component relativistic DFT method. A special J-oriented basis set (acv3z-J) for selenium and tellurium atoms, developed previously by the authors, was used throughout the NMR calculations in this work at the first time. The proposed computational methodologies (combined and 'pure') provided a reasonable accuracy for ³¹ P⁷⁷ Se and ³¹ P¹²⁵ Te SSCCs against experimental data, characterizing by the mean absolute percentage errors of about 4% and 1%, and 12% and 8% for selenium and tellurium species, respectively. The present study reports typical relativistic corrections to ⁷⁷ Se³¹ P and ¹²⁵ Te³¹ P SSCCs, calculated within the four-component DFT formalism for a broad series of tertiary phosphine selenides and tellurides with different substituents at phosphorus.

Laser-cooling with an intermediate electronic state: Theoretical prediction on bismuth hydride

The possibility of laser cooling of bismuth hydride (BiH) molecules has been investigated based on high-level ab initio calculations by considering the core-valence and the spin-orbit coupling (SOC) effects. The potential energy curves of the 12 Λ-S states as well as the 25 Ω states that split from them via SOC are obtained by multireference configuration interaction plus the Davidson correction. The properties of b-X transition are investigated. Based on our calculations, we show that the transition between Ω states b0⁺-X₁0⁺ of BiH is a possible candidate for laser cooling, with consideration of the intermediate Ω state X₂1. An optical cycling scheme is proposed by utilizing four lasers at wavelengths around 471 and 601 nm with 5400 cycles for photon absorption/emission and a sub-microkelvin temperature. Our study should shed some light on searching for possible molecular candidates for laser cooling with the existence of an intermediate electronic state.

Novel strategy to implement active-space coupled-cluster methods

A new approach is presented for the efficient implementation of coupled-cluster (CC) methods including higher excitations based on a molecular orbital space partitioned into active and inactive orbitals. In the new framework, the string representation of amplitudes and intermediates is used as long as it is beneficial, but the contractions are evaluated as matrix products. Using a new diagrammatic technique, the CC equations are represented in a compact form due to the string notations we introduced. As an application of these ideas, a new automated implementation of the single-reference-based multi-reference CC equations is presented for arbitrary excitation levels. The new program can be considered as an improvement over the previous implementations in many respects; e.g., diagram contributions are evaluated by efficient vectorized subroutines. Timings for test calculations for various complete active-space problems are presented. As an application of the new code, the weak interactions in the Be dimer were studied.

Ab initio calculation of energy levels of trivalent lanthanide ions

A fully ab initio computational scheme employing CASSCF/XMCQDPT2/SO-CASSCF for the absorption and emission spectra of trivalent lanthanide complexes is presented.

Equation-of-Motion Coupled-Cluster Theory for Excitation Energies of Closed-Shell Systems with Spin-Orbit Coupling

Excitation energies of closed-shell systems based on the equation-of-motion (EOM) coupled-cluster theory at the singles and doubles (CCSD) level with spin-orbit coupling (SOC) included in the post-Hartree-Fock treatment are implemented in the present work. SOC can be included in both the CC and EOM steps (EOM-SOC-CCSD) or only in the EOM part (SOC-EOM-CCSD). The latter approach is an economical way to account for SOC effects, but excitation energies with this approach are not size-intensive. When the unlinked term in the latter approach is neglected (cSOC-EOM-CCSD), size-intensive excitation energies can be obtained. Time-reversal symmetry and spatial symmetry are exploited to reduce the computational effort. Imposing time-reversal symmetry results in a real matrix representation for the similarity-transformed Hamiltonian, which facilitates the requirement of time-reversal symmetry for new trial vectors in Davidson's algorithm. Results on some closed-shell atoms and molecules containing heavy elements show that EOM-SOC-CCSD can provide excitation energies and spin-orbit splittings with reasonable accuracy. On the other hand, the SOC-EOM-CCSD approach is able to afford accurate estimates of SOC effects for valence electrons of systems containing elements up to the fifth row, while cSOC-EOM-CCSD is less accurate for spin-orbit splittings of transitions involving p1/2 spinors, even for Kr.