Combining the spin-separated exact two-component relativistic Hamiltonian with the equation-of-motion coupled-cluster method for the treatment of spin–orbit splittings of light and heavy elements

Abstract

The spin-separated exact two-component (X2C) relativistic Hamiltonian [sf-X2C+so-DKHn, J. Chem. Phys., 2012, 137, 154114] is combined with the equation-of-motion coupled-cluster method with singles and doubles (EOM-CCSD) for the treatment of spin-orbit splittings of open-shell molecular systems. Scalar relativistic effects are treated to infinite order from the outset via the spin-free part of the X2C Hamiltonian (sf-X2C), whereas the spin-orbit couplings (SOC) are handled at the CC level via the first-order Douglas-Kroll-Hess (DKH) type of spin-orbit operator (so-DKH1). Since the exponential of single excitations, i.e., exp(T), introduces sufficient spin orbital relaxations, the inclusion of SOC at the CC level is essentially the same in accuracy as the inclusion of SOC from the outset in terms of the two-component spinors determined variationally by the sf-X2C+so-DKH1 Hamiltonian, but is computationally more efficient. Therefore, such an approach (denoted as sf-X2C-EOM-CCSD(SOC)) can achieve uniform accuracy for the spin-orbit splittings of both light and heavy elements. For light elements, the treatment of SOC can even be postponed until the EOM step (denoted as sf-X2C-EOM(SOC)-CCSD), so as to further reduce the computational cost. To reveal the efficacy of sf-X2C-EOM-CCSD(SOC) and sf-X2C-EOM(SOC)-CCSD, the spin-orbit splittings of the Π states of monohydrides up to the sixth row of the periodic table are investigated. The results show that sf-X2C-EOM-CCSD(SOC) predicts very accurate results (within 5%) for elements up to the fifth row, whereas sf-X2C-EOM(SOC)-CCSD is useful only for light elements (up to the third row but with some exceptions). For comparison, the sf-X2C-S-TD-DFT-SOC approach [spin-adapted open-shell time-dependent density functional theory, Mol. Phys., 2013, 111, 3741] is applied to the same systems. The overall accuracy (1-10%) is satisfactory.

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

“…It deserves to be mentioned that H so (5) will reduce [135] to the mean-field Breit-Pauli spin-orbit Hamiltonian or the original mean-field so-DKH1 [141,142] if both the decoupling matrix X 0 (11) and the renormalization matrix R +,0 (9) are set to identity or the free-particle counterparts in h l SO,1e (12) and f l SO,2e (13). It is hence clear that so-DKH1 gains accuracy withiout overhead, as demonstrated before [90,91,134,143].…”

“…This is of course due to the lack of spindependent orbital relaxation, just like other two-step approaches. Nonetheless, iCISO may perform well for heavy elements when SOC is quenched to a large extent by chemical bondings [90]. Another point that deserves to be addressed is the sensitivity of iCISO on the number of scalar states.…”

“…Finally, a brief comparison should be made between SOiCI and other related approaches [90,91]. As can be seen from Table 6 for the halogen atoms, only the SOSHCI (spin-orbit stochastic heat-bath CI) value [91] for the SOS of Br is an outlier to the good agreement in between.…”

“…The interplay between SOC and electron correlation can be accounted for in two ways, one-step or two-step [78,79]. The former type of approaches [80][81][82][83][84][85][86][87][88][89][90][91] aims to treat spin-orbit and electron-electron interactions on an equal footing, whereas the latter type of approaches [85, amounts to treating SOC after correlation, by constructing and diagonalizing an effective spinorbit Hamiltonian matrix over a set of close-lying correlated scalar states.…”

iVI‐TD‐DFT: An iterative vector interaction method for exterior/interior roots of TD‐DFT

“…The iVI‐Buf(0 + 3)‐TD‐DFT variant of iVI (to be denoted simply as iVI‐TD‐DFT from now on) has been implemented into the BDF (Beijing Density Functional) program package, which supports essentially all variants of TD‐DFT, including relativistic TD‐DFT (four‐component, two‐component, and composite two‐component; for a comprehensive review see Ref. []), spin‐adapted open‐shell TD‐DFT, spin‐flip TD‐DFT, as well as nonadiabatic couplings between TD‐DFT excited states .…”