Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian

Abstract

We report the implementation of analytic energy gradients for the evaluation of first-order electrical properties and nuclear forces within the framework of the spin-free (SF) exact two-component (X2c) theory. In the scheme presented here, referred to in the following as SFX2c-1e, the decoupling of electronic and positronic solutions is performed for the one-electron Dirac Hamiltonian in its matrix representation using a single unitary transformation. The resulting two-component one-electron matrix Hamiltonian is combined with untransformed two-electron interactions for subsequent self-consistent-field and electron-correlated calculations. The "picture-change" effect in the calculation of properties is taken into account by considering the full derivative of the two-component Hamiltonian matrix with respect to the external perturbation. The applicability of the analytic-gradient scheme presented here is demonstrated in benchmark calculations. SFX2c-1e results for the dipole moments and electric-field gradients of the hydrogen halides are compared with those obtained from nonrelativistic, SF high-order Douglas-Kroll-Hess, and SF Dirac-Coulomb calculations. It is shown that the use of untransformed two-electron interactions introduces rather small errors for these properties. As a first application of the analytic geometrical gradient, we report the equilibrium geometry of methylcopper (CuCH(3)) determined at various levels of theory.

Perspective: Relativistic effects

“…[84][85][86] For spin-free (scalar relativistic) X2C and NESC, analytic methods for calculations of nuclear gradients and second derivatives have been reported. [87][88][89] Regarding magnetic properties such as NMR shielding, the concept of "magnetic balance" 51 becomes important: The lower-component basis set should not only contain {σ · pχ } functions, but also {σ · Aχ } (see Eq. (6) (Ref.…”

Relativistic calculations of magnetic resonance parameters: background and some recent developments

“…Cheng & Gauss [82] implemented a scalar X2C approach and techniques for first-and secondorder derivative properties [82,157]. Representative EFG data for the HX series are listed in table 1 along with SF 4c data, EFGs that are correct to order c −2 , and NR results that were reported in the same paper.…”

“…For scalar (SF) X2C and NESC techniques, calculations of analytic nuclear gradients and second derivatives, along with other derivative properties, have been implemented [82,[157][158][159][160]. These methods therefore give access to the traditional exploration of potential energy surfaces in studies of reaction mechanisms that make up a vast portion of applications of computational chemistry methods.…”

“…Contraction of A with the twocomponent density matrix in the AO basis gives the PC-transformed property, whereas contraction of A UU with the density matrix gives the property without PC correction. Cheng & Gauss [82] pointed out that in calculations on many-electron systems the PC-corrected property operator should also contain contributions from the decoupling transformation to first order in the field. For EFGs, the contributions were calculated to be small (scalar X2C) when compared with the magnitude of relativistic and PC effects themselves.…”

“…Point nuclei except where indicated. a NR data, and relativistic SF data for direct relativistic perturbation theory of order c −2 (DPT), X2C with (*) and without electric field contributions to the decoupling transformation of the EFG operator, SF DC (4c) [82]. b Seventh-order SF DKH with PC corrections (DKH (7,7)) and DC (4c) with SO coupling and finite-nucleus model [81].…”