High order relativistic corrections on the electric field gradient within the LRESC formalism

Abstract

In this work we present relativistic corrections to the electric field gradient (EFG) given by the Linear Response Elimination of the Small Component (LRESC) scheme at 1/c^2 order and including for the first time spin-dependent (SD) corrections at 1/c^4 order. We show that these new terms improve the performance of LRESC as results with this methodology are very close to those calculated at four-component Dirac-Hartree-Fock (4c-DHF) level. We assess the new corrections in Br Y and At Y di-halogen ( Y = F, Cl, Br, I, At) and XZY bi-lineal molecules ( Z = Zn, Cd, Hg; X, Y = F, Cl, Br, I, At). At 4c-DHF level, we analyse the contributions coming from the large and small components of the relativistic 4c wave function to the electronic part of EFG and compare them with the LRESC corrections to their electronic origin. For Hg X2 ( X = Cl, Br, I) subset, when the SD correcting terms are included, LRESC calculations match very well with 4c-DHF ones and those from the literature, with differences less than 1% for molecules containing up to three heavy atoms. We show that LRESC gives accurate values of EFG allowing the analysis of the electronic origin of relativistic effects in terms of well known nonrelativistic operators.

Electric Field Gradient in Chiral and Tetrahedral Molecules within High-Order LRESC Formalism

“…In this regard, relativistic effects are mainly of SR type for the Br atom since the SD type represents only 7.3% of the total corrections; nevertheless, for the I atom, SD corrections increase to 13.3%, and for the At atom, they reach 22.2%, showing (as in the set of CHFClX chiral molecules) a linear dependence of the percentage of SD correcting terms with the weight of the substituent halogen atom. The inclusion of the SD terms allows for a proper description of the trend of relativist effects, as was demonstrated in ref .…”

“…The nuclear contribution is independent of the electronic wave function, and it depends only on the position of the other nuclei in the molecule; in contrast, the electronic contribution does depend on the ground state of the molecule (|Ψ 0 ⟩). ⟨ V z z false( boldR N false) ⟩ = ⟨ Ψ 0 false| q z z ( R N ) false| Ψ 0 ⟩ + normalΩ nuc ( R N ) The development of the LRESC methodology describing EFG is given elsewhere. , The starting point is the mean value of the electronic part of the EFG operator over the 4c N -particle states within the N -particle manifold of Dirac–Fock space solutions, followed by the application of the kinetic balance prescription, the normalization of the large component of the 4c wave function, and the description of the normalized eigenstates of Pauli Hamiltonian as nonrelativistic states (|ϕ i Sch ⟩) plus corrections given by the perturbative Hamiltonians mass velocity ( H Mv ), Darwin ( H Dw ), and spin–orbit ( H SO ). Keeping terms up to the 1/ c 4 order and grouping them according to their response order, the electronic contribution to the EFG of eq given by the LRESC scheme, in atomic units ( e = m = ℏ = 1), is false⟨ q z z ( R N ) false⟩ LRESC = false⟨ q z z ( R N ) false⟩ nr …”

“…Δ SO–S is a linear response function among the derivative of the q zz operator and the H SO Hamiltonian, while Δ SO–L is a quadratic response function among the q zz operator and the H SO Hamiltonians. It was shown previously that the spin-dependent electronic mechanisms improve the EFG results, and in this work, correlation effects to those terms are included for the first time. The relativistic corrections of the LRESC method will be distinguished by their spin character as the leading scalar relativistic (SR) terms at the 1/ c 2 order and the leading spin-dependent (SD) terms at the 1/ c 4 order .…”

“…LRESC was first presented to describe the NMR nuclear magnetic shielding tensor as corrections of the relativistic molecular ground state with a nonconserving total number of particles. In the past decade, LRESC was developed to calculate several properties, and more recently, it was applied for EFG calculation. − …”

Calculation of electric field gradients with the exact two-component (X2C) quasi-relativistic method and its local approximations

“…35 Yet, four-component calculations are very computationally demanding even for small molecules. Therefore, a number of two-component or scalar quasirelativistic approaches for EFG calculation have been developed in the most recent two decades, such as the zeroth-order regular approximation (ZORA), 40,50 linear response elimination of the small component (LRESC) formalism, 51 second-order Douglas-Kroll-Hess transformation (DKH2), 52 high-order DKH, 41,53 direct perturbation theory (DPT), 54 and exact twocomponent (X2C), 41,55,56 as well as the closely related infiniteorder two-component (IOTC) and Barysz-Sadlej-Snijders (BSS) transformation-based methods. 41 Among these quasi-relativistic methods, X2C is gaining more traction due to its advantages of ''simplicity, accuracy, and efficiency'' 57 and the availability of analytic calculation of the one-electron properties.…”