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

Abstract

The analytical gradient for the two-component Normalized Elimination of the Small Component (2c-NESC) method is presented. The 2c-NESC is a Dirac-exact method that employs the exact two-component one-electron Hamiltonian and thus leads to exact Dirac spin-orbit (SO) splittings for one-electron atoms. For many-electron atoms and molecules, the effect of the two-electron SO interaction is modeled by a screened nucleus potential using effective nuclear charges as proposed by Boettger [Phys. Rev. B 62, 7809 (2000)]. The effect of spin-orbit coupling (SOC) on molecular geometries is analyzed utilizing the properties of the frontier orbitals and calculated SO couplings. It is shown that bond lengths can either be lengthened or shortened under the impact of SOC where in the first case the influence of low lying excited states with occupied antibonding orbitals plays a role and in the second case the jj-coupling between occupied antibonding and unoccupied bonding orbitals dominates. In general, the effect of SOC on bond lengths is relatively small (≤5% of the scalar relativistic changes in the bond length). However, large effects are found for van der Waals complexes Hg2 and Cn2, which are due to the admixture of more bonding character to the highest occupied spinors.

Relevance of effective bond orders in heterodiatomic molecules and role of the spin-orbit coupling in theAtX(X=At–F)series

“…After having settled the good accuracy of the results reported in Table II, it is worth commenting on the role of the scalar-relativistic and of the spin-dependent (mainly the SOC) relativistic effects on the bond distances of interest. It is clear that in most of the systems (X = At -Cl), the scalar-relativistic effects lead to a (moderate) bond shortening, as expected for diatomic molecules made of p elements [62]. The AtF molecule appears as an outlier within the series, displaying an apparent bond lengthening induced by scalar-relativistic effects.…”

“…Actually, this bond distance enlargement is not seen at the Hartree-Fock level, and thus results from the addition and interplay between electron correlation and scalar-relativistic effects. Consequently, it is of different origin than the "classical" scalar-relativistic bond lengthenings that may occur for instance in diatomic molecules with f elements [62]. The differences between the groundstate energies obtained at both the nonrelativistic and scalarrelativistic geometries and at the nonrelativistic and scalarrelativistic Hartree-Fock and CCSD(T) levels being small (the four resulting energy differences being of the order of 1k Jm o l −1 ), it is difficult to draw conclusive insight into the bond lengthening in AtF and we thus do not discuss it here in further detail.…”

Calculations of atomic magnetic nuclear shielding constants based on the two-component normalized elimination of the small component method

“…The modified screenednuclear-spin-orbit (mSNSO) approach was employed to approximate the influence of the two-electron spin-orbit terms. 20,22 The finite (F) model of the nucleus based on a Gaussian charge distribution 105 also performed. A value of 137.035 999 070(98) was used for the velocity of light c. 106 The DIRAC12 program package 107 was employed to obtain 4c-sMB NMR shielding constants.…”

“…The routine calculation of geometries was made possible by deriving the analytical 2c-NESC gradient. 22 In follow-up work, we used the energy gradient algorithm to reliably determine SOcorrected electrical properties such as the molecular electrical moments 7 and to develop the 2c-NESC Hessian with regard to the components of the electric field for the calculation of the static electric dipole polarizability, 7 which required the use of coupled perturbed GHF (CPGHF) 23 and a coupled perturbed general Kohn-Sham (CPGKS) approach. [24][25][26][27] Our work on NESC has to be seen on the background of a multi-pronged approach to obtain exact relativistic descriptions, which has involved several research groups.…”

“…87 In the present work, we extend the spin-free X1C-NMR method of Cheng and co-workers 85 to the 2c-relativistic level by using the 2c-NESC methodology developed in previous work. 11,12,16,22 To reduce the high cost of previous approaches including the relativistic two-electron terms, 27,88 we use the non-relativistic two-electron term of the GHF method. 7,89 As a result, we need the renormalization matrix G 66 to transform the Dirac picture (with the 4c-relativistic metric) to the Schrödinger picture (with the 2c-relativistic metric).…”