AD for forces

[epolack]

Precompiling is the same (but slow ~100s, I do not understand why; I had this also in #853, then it reverse to normal timings without any changes from my part…).

On a simple silicon supercell of size 2×2×2 the cold and hot timings are similar:
master: 22s then 6.29s
branch: 24.2s then 6.57s

So if the change seems fine for you, I will test this on more robust test cases (ideas are welcome).

[antoine-levitt]

Not too sure about this. Generally, trying to add more AD introduces more headaches down the line. Esp. for forces it's probably possible to use adjoint tricks, which I don't think we do currently but would be probably easier to do with explicit derivatives. Also, here you have to do a lot more FFTs. Is there a motivation to this?

(as an aside, we have the pattern "build a superposition of radial things" quite a lot, we should factor it out eventually)

[epolack]

Flexibility for phonons, as I need also ∫δρδV and ∫ρδ²V. From a comment you made at some point. I used to compute them by hand and you asked if using AD would not simplify the code.

[epolack]

(Was roughly https://github.com/JuliaMolSim/DFTK.jl/compare/6abafdb486eece16e6843e7ee398420bebdb8a5c#diff-0445ce514822986487aa7496af1e6a4a2fb07de6b97071418981f3ea30360ed9R118 I think.)

[mfherbst]

I'm also not too sure about this as we don't actually loose that much code. Also not sure how well this plays with GPU. I think we should have a proper think about how to do this, if we really want to.