Appendix F - Time-Dependent Current DFT

Summary

In this appendix I outline the main tenets of time-dependent current DFT (TD-CDFT). A pedagogical and extended account of this approach can be found in the book by Giuliani and Vignale (2005).

The current density as the main variable

In Appendix E I have hinted at the problem of obtaining a local representation of the exchange-correlation scalar potential V_xc (the one appearing in the Kohn–Sham equations E.4) in terms of the density. Without going into details, I just mention that the physical reason why such a representation does not exist for an inhomogeneous electron liquid is related to the strong non-local (in space) functional dependence of this potential on the density.

A way out of this problem is to realize that due to gauge invariance a time-dependent scalar potential V(r,t) can always be represented by a longitudinal vector potential A(r,t).1 We also know that the density is related to the current density via the continuity equation 1.11. Putting these two facts together, we therefore see that the response of the density operator to an external scalar potential can be expressed as the response of the current density operator to an external vector potential, i.e., we can reformulate time-dependent density-functional theory in terms of the current density instead of the density. It turns out that a local representation of the exchange-correlation vector potential in terms of the current density can be derived.