Spin–orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories

Abstract

We revise formal and numerical aspects of collinear and non-collinear density functional theory (DFT) in the context of a two-component self-consistent treatment of spin-orbit coupling (SOC). While the extension of the standard one-component theory to a non-collinear magnetization is formally well-defined within the local density approximation (LDA), and therefore results in a numerically stable theory, this is not the case within the generalized gradient approximation (GGA). Previously reported formulations of non-collinear DFT based on GGA exchange-correlation potentials have several limitations: i) they fail at reducing (either formally or numerically) to the proper collinear limit (i.e. when the magnetization is parallel to the z axis everywhere in space); ii) they fail at ensuring a quantitative rotational invariance of the total energy and even a qualitative rotational invariance of the spatial distribution of the magnetization when a SOC operator is included in the Hamiltonian; iii) they are numerically very unstable in regions of small magnetization. All of the above mentioned problems are here shown (both formally and through test examples) to be solved by using instead a new formulation of non-collinear DFT for GGA functionals, which we call the "signed canonical" theory, as combined with an effective screening algorithm for unstable terms of the exchange-correlation potential in regions of small magnetization. All methods are implemented in the Crystal program and tests are performed on simple molecules to compare the different formulations of non-collinear DFT.

Fundamental Role of Fock Exchange in Relativistic Density Functional Theory

“…Therefore, relativistic DFT calculations must be performed by modifying existing nonrelativistic, one-component functionals, for which several formulations have been suggested. − The most straightforward way of generalizing existing nonrelativistic functionals to the two-component relativistic approach is known as the “collinear formulation” of the DFT, where only the particle-number density and one of the three components of the magnetization are used (namely, n and m z ) . The collinear formulation does not result in a rotationally invariant theory.…”

“…The collinear formulation does not result in a rotationally invariant theory. Otherwise, the one-component, nonrelativistic functionals are typically “hacked” in such a way to plug into them the particle-number density and the three components of the magnetization vector (namely, n and m ) within so-called “non-collinear formulations” of the DFT. − ,− In the four-component approach, the wave function is usually decomposed into n and m z or n and m as if it were a two-component wave function, using a very similar procedure. , At variance with previous attempts, we have recently shown that the procedure from which the full n and m are plugged into the functional can result in a fully rotationally invariant theory using a new noncollinear prescription . However, the problem remains that such noncollinear formulations of the DFT still do not make any explicit use of the orbital current density j in the definition of the exchange–correlation functional.…”

“…We now discuss the effect of Fock exchange within a relativistic DFT approach on the description of the noncollinear magnetization and orbital current density, through test calculations on a representative example: the I 2 + open-shell molecule. All relativistic DFT calculations, with inclusion of SOC, are performed with a developmental version of the public Crystal program, , where we have recently implemented a self-consistent treatment of SOC in a two-component formalism. , We also perform spin–orbit configuration–interaction (SO-CI) calculations with the Epciso and Cipsi programs , to have a reference description from a correlated wave function approach. Computational details on these calculations are provided in the Supporting Information.…”

“…28,45 At variance with previous attempts, we have recently shown that the procedure from which the full n and m are plugged into the functional can result in a fully rotationally invariant theory using a new noncollinear prescription. 30 However, the problem remains that such noncollinear formulations of the DFT still do not make any explicit use of the orbital current density j in the definition of the exchange−correlation functional. The orbital current density j is a current of charges induced by the magnetic field created through the SOC effect.…”

“…All relativistic DFT calculations, with inclusion of SOC, are performed with a developmental version of the public CRYSTAL program, 50,51 where we have recently implemented a self-consistent treatment of SOC in a twocomponent formalism. 25,30 We also perform spin−orbit configuration−interaction (SO-CI) calculations with the EPCISO and CIPSI programs 52,53 to have a reference description from a correlated wave function approach. Computational details on these calculations are provided in the Supporting Information.…”