Current density functional framework for spin–orbit coupling

02 September 2022, Version 2
This content is a preprint and has not undergone peer review at the time of posting.


Relativistic two-component and four-component density functional calculations are carried out in a non-collinear formalism to describe spin-orbit interaction. The exchange-correlation functional itself is constructed as a straightforward generalization of the non-relativistic density functional approximation. However, spin-orbit coupling is a form of magnetic induction and generally leads to a non-vanishing paramagnetic current density. This means that functionals depending on the kinetic energy density such as meta-generalized gradient approximations and local hybrid functionals should be constructed in the framework of current density functional theory (CDFT), which is (almost) exclusively used in the regime of strong magnetic fields. Herein, we present a CDFT approach for spin-orbit coupling, which leads to current-dependent terms in the potential and its derivatives. The ansatz is implemented for the exchange-correlation potential and the exchange-correlation kernel, making it applicable to a wide range of molecular properties. We assess the importance of the current density terms for ground-state energies, excitation energies, nuclear magnetic resonance (NMR) shielding and spin-spin coupling constants, as well as hyperfine coupling constants and ∆g-shifts in electron paramagnetic resonance (EPR) spectroscopy. The most notable changes are found for EPR properties. The impact of the current-dependent terms rises with the number of unpaired electrons and consequently EPR properties are more sensitive towards CDFT. Considerable changes in the results are observed for the strongly constrained and appropriately normed (SCAN) functionals, as well as the B97M family and TASK. The current density terms are less important when exact exchange is incorporated. Consequently, the results with (local) hybrid functionals are not substantially affected. At the same time, the current density does not notably increase the computational costs for hybrid functionals, but the current-dependent kernel ensures the stability of response calculations in all cases. We therefore strongly recommend to use the framework of CDFT for self-consistent spin-orbit calculations.


Density Functional Theory
Current Density
Relativistic Effects
Local Hybrid Functionals

Supplementary materials

Supporting Information
Theory for NMR shieldings with X2C. Further data and data for the statistical evaluation presented in the main text.


Comments are not moderated before they are posted, but they can be removed by the site moderators if they are found to be in contravention of our Commenting Policy [opens in a new tab] - please read this policy before you post. Comments should be used for scholarly discussion of the content in question. You can find more information about how to use the commenting feature here [opens in a new tab] .
This site is protected by reCAPTCHA and the Google Privacy Policy [opens in a new tab] and Terms of Service [opens in a new tab] apply.