Simulating Spin–Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

Abstract

We present the first implementation of spin−orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QD-NEVPT2). The SO-QDNEVPT2 approach enables the computations of ground-and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin−orbit coupling. In addition to SO-QDNEVPT2 with the full description of one-and two-body spin−orbit interactions at the level of two-component Breit−Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin−orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin−orbit coupling with a relatively low computational cost.

Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules

“…These active spaces mostly consist of all the bonding 2p MOs on oxygen and different numbers of 5f and 6d orbitals on the actinide ion. Later, Gendron et al , and Majumder et al have also used CAS­(7,10) and CAS­(8,10) for [NpO 2 ] 2+ and [PuO 2 ] 2+ , respectively, to obtain vertical excitation energies. For [AmO 2 ] 2+ , SO-CASPT2 calculations have been performed to obtain the excited states using an active space of CAS­(15,13) by Notter et al We have used two different active spaces: (i) CAS­( n , 16) (referred to as AS-I) and (ii) CAS­( m , 10) (referred to as AS-II) throughout our calculations.…”

Consistent Second-Order Treatment of Spin–Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory

“…The two-electron term of F pq BP,ξ in eq 15 also contains contributions from the spin−other orbit operator, which matrix elements can be fully expressed in terms of g pqrs ξ . 71 The g pqrs ξ integrals can be written more compactly in the standard Physicists' notation as The BP Hamiltonian is widely used to incorporate spin− orbit coupling effects in perturbative two-component electronic structure methods. However, it is considered to be a low-Z approximation that is valid when Z 2 α 2 ≪ 1, showing increasingly large errors for elements beyond the third row of periodic table.…”

“…The one- and two-electron integrals calculated in the spatial molecular orbital basis (ϕ p ) represent the one-electron spin–orbit ĥ ξ ( i ) and the two-electron spin–same orbit ĝ ξ,sso ( i , j ) operators where Z A is the charge of nucleus A , r ij and r iA are the coordinates of electron i relative to electron j and nucleus A , respectively, and p̂ ( i ) is the momentum operator for electron i . The two-electron term of F pq BP,ξ in eq also contains contributions from the spin–other orbit operator, which matrix elements can be fully expressed in terms of g pqrs ξ . The g pqrs ξ integrals can be written more compactly in the standard Physicists’ notation as where with respect to o , π ∈ ( x , y , z ) and ϵ oπ ξ is the Levi-Civita symbol.…”

“…The corresponding effective Hamiltonian has the form: where |Ψ I (1) ⟩ is the conventional QDNEVPT2 first-order wave function with real-valued amplitudes determined by solving eq . We note that the BP1-QDNEVPT2 method has been studied in detail in ref , while the DKH1-QDNEVPT2 implementation is reported for the first time. A summary of methods implemented in this work is provided in Table .…”

“…Our approach is based on the second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2), which describes static and dynamic correlation in many electronic states simultaneously, free of intruder-state problems . Previous two-component implementations of QDNEVPT2 have been limited to the first-order treatment of spin–orbit coupling utilizing the BP relativistic Hamiltonian. − Here, we employ the second-order Douglas–Kroll–Hess Hamiltonian (DKH2) in the exact two-component formulation and incorporate all contributions from spin–orbit coupling and dynamic correlation effects up to the second order in perturbation expansion. We demonstrate that this new approach performs consistently well for atoms and molecules across the entire periodic table and is significantly more accurate than the QDNEVPT2 methods with the first-order treatment of spin–orbit coupling.…”