# 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.

What about the advantages (and differences) of BDFT?

• The information in this post is useful, but it should be split up into a question and answer. Even though you are self answering, it fits better with the general Q&A format of the site to do it this way.
– Tyberius
May 3, 2020 at 12:58
• Or maybe she is looking for answers from others, in which case, perhaps she could repeat the question's title (with question mark) at the end of the question body? May 3, 2020 at 14:55
• To clary, I am still new in this topic and would like to know other's opinion and/or answer to deepen my understanding. Not sure if this one was not the correct format for asking question (?). May 3, 2020 at 18:14
• @MeilaniWibowo I have edited this to look more like a question, which meant removing "I would like to draw attention" and replacing it with "I would like to ask". Also you summarized at the end, advantages of CDFT but not BDFT, so that's why I put the last line there. You are free to edit further. May 3, 2020 at 20:59
• @MeilaniWibowo is it possible that CDFT can be confused with "constrained DFT" instead of "current DFT" ? May 17, 2020 at 0:03

## 1 Answer

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.