On the origin and contribution of the diamagnetic term in four-component relativistic calculations of magnetic properties

Abstract

The relativistic Dirac Hamiltonian that describes the motion of electrons in a magnetic field contains only paramagnetic terms ͑i.e., terms linear in the vector potential A͒ while the corresponding nonrelativistic Schrödinger Hamiltonian also contains diamagnetic terms ͑i.e., those from an A 2 operator͒. We demonstrate that all diamagnetic terms relativistically arise from second-order perturbation theory and that they correspond to a ''redressing'' of the electrons by the magnetic field. If the nonrelativistic limit is taken with a fixed no-pair Hamiltonian ͑no redressing͒, the diamagnetic term is missing. The Schrödinger equation is normally obtained by taking the nonrelativistic limit of the Dirac one-electron equation, we show why nonrelativistic use of the A 2 operator is also correct in the many-electron case. In nonrelativistic approaches, diamagnetic terms are usually considered in first-order perturbation theory because they can be evaluated as an expectation value over the ground state wave function. The possibility of also using an expectation value expression, instead of a second-order expression, in the relativistic case is investigated. We also introduce and discuss the concept of ''magnetically balanced'' basis sets in relativistic calculations.

Formal relations connecting different approaches to calculate relativistic effects on molecular magnetic properties

“…1,2 The first ones 1 were obtained within the Sternheim approximation. 3 It has been shown formally 4,5 that the diamagnetic component of magnetic properties is only recovered in the nonrelativistic limit when electron-positron rotations are taken into account. Therefore in the four-component linear-response theory the paramagnetic term is identified with the linear-response related to electron-electron rotations, while the diamagnetic one is identified with the contribution associated to electronpositron rotations.…”

Breit interaction effects in relativistic theory of the nuclear spin-rotation tensor

“…The calculation of the propagator involves excitations to virtual electronic states (e-e excitations) and virtual positronic states (e-p excitations). Both kinds of terms can be obtained separately with adequate decomposition of the corresponding blocks of the propagator, 19,20 as implemented in the DIRAC 10 code. For the equilibrium molecular geometry, the first order electronic contribution and the nuclear contribution h (2)nuc ω,μ N can be gathered together in the simple (nuclear) term…”

Relativistic effects on the nuclear magnetic shielding tensor

“…In the last few years an ever increasing number of new formalisms and calculations for the evaluation of relativistic effects on molecular properties from four-, two-, or onecomponent response schemes or perturbation theory approaches have been published. [1][2][3][4][5][6][7][8][9][10][11] It was shown that the inclusion of such effects in the calculation of some molecular properties is mandatory when one wants to reproduce experimental trends. 4,10 Relativistic spin-orbit ͑SO͒ effects on magnetic molecular properties were thought to be the most important ones until recent calculations of Visscher et al 4 Numerical results for nuclear magnetic shieldings obtained by four-component calculations and their counterpart from RayleighSchrödinger perturbation theory ͑RSPT͒ only match each other for the shielding of heavy atoms X in HX compounds when a new term different from SO is included.…”