# Question on density functional theory

1. May 26, 2013

### pillow47

Dear PF,

I'm reading a book on DFT, and it says that only ground-state wave function is a unique functional of the ground-state density, n(r). However, if in DFT the potential, v(r), is a unique functional of n(r), then shouldn't all wave functions be functionals of n(r), because you can just solve for the excited-state wave functions from the v(r) determined from n(r)?

Thanks

2. May 26, 2013

### daveyrocket

DFT is built on the idea of constructing an auxiliary non-interacting problem that has the same ground-state density as the interacting problem. There is nothing in the theories of DFT that guarantee that the auxiliary problem will have the same excited states as the actual interacting problem that you're interested in studying.

There are extensions to DFT, like TDDFT (time-dependent -) which put the calculation of excited states on firmer theoretical footing.

3. May 27, 2013

### cgk

OP, I would recommend not worrying about these things too much. There is all that talk about DFT being "in principle" exact and deep theorems and so on, but for all practical purposes, real world DFT is a hacked up Hartree-Fock. There is little insight to be gained by studying those theorems.

Regarding in principle exact': Note that the ground state density uniquely determines the positions of the atomic cores (due to their nuclear cusps), and these in turn uniquely determine the external potential. If you have the external potential, you can, theoretically, just calculate the exact many-body wave function from it, including excited states and all properties. And with the same argument as used in Levy's constrained search formalism you can show that classical force fields are in principle exact (if you'd just know the right energy function...).
Now is that a deep theoretical insight? Many people seem to think yes. In my opinion, if Levy's constrained search formalism has shown one thing, then it is that the Hohenberg-Kohn-Theorem is pointless[1].

[1] (but the Kohn-Sham approach is not. In contrary to real DFT, it actually offers a practical way of estimating the kinetic energy... by not actually being a /density functional/ theory, but calculating a Hartree-Fock-like wave function).

4. May 27, 2013

### Useful nucleus

5. May 27, 2013

### cgk

Useful nucleus, I think the original reference is this: http://www.pnas.org/content/76/12/6062.short and this one: http://dx.doi.org/10.1002/qua.22895 gives some historical context. The constrained search formalism is nowadays often employed in generalizations of DFT (in my personal optionion mainly because it's an almost trivial way to prove that almost everything you can come up with is "in principle exact")

6. May 27, 2013

### Useful nucleus

Thank you, cgk! I will study those references.