# What is the difference between BDFT and CDFT (magnetic-field DFT and current DFT)?

Following the previous discussion on DFT in strong fields, I would like to ask about the difference between those two alternatives (BDFT and CDFT). By CDFT I do not mean (constrained DFT, but current DFT). The mathematical background of these two can be found in:

• Vignale, et al., Density-functional theory in strong magnetic fields, Phys. Rev. Lett. 1987, 59, 2360.
• Grayce, et al., Magnetic-field density functional theory, Phys. Rev. A, 1994, 50, 3089.
• Tellgren, et al., Choice of basic variables in current-density functional theory, Phys. Rev. A 2012, 86, 062506.
• Bates, et al., Harnessing the meta-generalised gradient approximation for time-dependent density functional theory, J. Chem. Phys. 2012, 137, 164105.
• Furness, et al., Current density functional theory using meta-generalised gradient exchange-correlation functionals, J. Comp. Theory Comput. 2015, 11, 4169.
• Reimann, et al., Magnetic-field density functional theory (BDFT): Lessons from the adiabatic connection, J. Comp. Theory Comput. 2017, 13, 4089.
• Reimann, et al., Kohn-Sham energy decomposition for molecules in a magnetic field, Mol. Phys. 2018.

To summarise, Kohn-Sham CDFT has the advantages that it is universal (can be applied for any magnetic field / vector potential A), a non-perturbative implementation can be applied to arbitrary field strengths, and functionals can be generated from existing meta-GGAs. However, functional dependence on the physical current, j_p is not widely studied, hence, new functionals are required.

Magnetic-Field-Density-Functional-Theory (BDFT) is an alternative to CDFT in which the notion of universality of the DFT functional is relaxed (Grayce and Harris, 1994). It can be thought of as regarding the external magnetic field $$\textbf{B}$$ or equivalently the vector potential $$\textbf{A}$$ as fixed. In practice, it means approximations must be developed that change with $$\textbf{B}$$ / $$\textbf{A}$$ directly and are specific to a type of external field. If the dependence on $$\textbf{B}$$ can be modelled, then BDFT may provide a simpler route to a DFT useful for systems in a magnetic field.