Development, Implementation, and Application of an Analytic Second Derivative Formalism for the Normalized Elimination of the Small Component Method

Abstract

Analytical second derivatives for the normalized elimination of the small component (NESC) method are derived for the first time and implemented for the routine calculation of NESC vibrational frequencies and other second order molecular properties using the scalar relativistic form of NESC. Using response theory, the second derivatives of the transformation matrix U connecting the large and the pseudolarge components of the relativistic wave function are correctly derived. The 24 derivative terms involving the NESC Hamiltonian and the NESC renormalization matrix are individually tested, and their contributions to the energy Hessian are calculated. The influence of a finite nucleus model and that of the picture change is determined. Different ways of speeding up the calculation of the NESC second derivatives are tested. It is shown that second order properties can routinely be calculated in combination with Hartree-Fock, density functional theory, Moller-Plesset perturbation theory, and any other electron correlation corrected quantum chemical method provided analytic second derivatives are available in the nonrelativistic case. The general applicability of the analytic NESC Hessian is demonstrated by benchmark calculations for NESC/DFT calculations involving up to 1500 basis functions.

Analytical energy gradient for the two-component normalized elimination of the small component method

“…In Ref. 9, it was proved that the renormalization matrix G used in the spin-free NESC method is positive definite (p.d.). This is also true for a complex matrix G of 2c-NESC.…”

“…5 Our previous method development work sets the basis for the routine scalar relativistic calculation of molecular geometries, 2 electric dipole moments, 3 EPR hyperfine structure constants, 6 contact densities for the calculation of Mössbauer isomer shifts, 7 or electric field gradients for nuclear quadrupole coupling constants. 8 As for the calculation of second order response properties, we developed the methodology for the analytic calculation of vibrational frequencies, 3,9 static electric polarizabilities, or infrared intensities 3 utilizing the 1c-NESC method. We applied these methods to predict molecular properties of mercury, 10 gold, 11 uranium containing molecules, 3 or other relativistic molecules.…”

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

“…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.…”

Communication: On the isotope anomaly of nuclear quadrupole coupling in molecules

“…22,23 The EFG is calculated as a derivative of the total electronic energy with respect to the NQM, where the NQI Hamiltonian is explicitly included when calculating the energy. 12 When calculating the electronic energy and the NQI operator, the finite size nuclear model with the charge distribution described by a Gaussian function is employed.…”