Professor Meunier, forgive me if this is a little too in the mathematical weeds, but the definition you're giving in 16:44 is (give or take) the Gateaux Derivative. _In practice_ does the distinction between Gateaux and Frechet derivatives matter in most DFT work? Will you tend to encounter functionals that are Gateaux differentiable but aren't Frechet differentiable? Or where treating them as if they're Frechet differentiable isn't just mathematically imprecise but gives you decent heuristics but will _actively_ return bad, mistaken results?
That's an excellent point! That's also why I wrote (and said!) that the treatment here is "mathematically not 100% rigorous"., in the sense that it is a basic introduction for people with little prior knowledge. However, in practice (in the context of DFT) you concern does not lead to errors or mistakes. Though I can imagine that higher level treatments may call for a clearer description of range of applicability.
Nice introduction to this topic. Can you recommend a text on functionals and their derivatives? (both from the mathematical-theoretical and the physics-application point of view) Thanks
Professor Meunier, forgive me if this is a little too in the mathematical weeds, but the definition you're giving in 16:44 is (give or take) the Gateaux Derivative. _In practice_ does the distinction between Gateaux and Frechet derivatives matter in most DFT work? Will you tend to encounter functionals that are Gateaux differentiable but aren't Frechet differentiable? Or where treating them as if they're Frechet differentiable isn't just mathematically imprecise but gives you decent heuristics but will _actively_ return bad, mistaken results?
That's an excellent point! That's also why I wrote (and said!) that the treatment here is "mathematically not 100% rigorous"., in the sense that it is a basic introduction for people with little prior knowledge. However, in practice (in the context of DFT) you concern does not lead to errors or mistakes. Though I can imagine that higher level treatments may call for a clearer description of range of applicability.
Nice introduction to this topic.
Can you recommend a text on functionals and their derivatives? (both from the mathematical-theoretical and the physics-application point of view)
Thanks
Yes, the appendix of one of the references I gave is quite good: Appendix A, page 404 of "DFT: And Advanced Course" by Engel and Dreizler