# What is the current state of research into $v$-representability?

In their proof, Hohenberg and Kohn (Phys Rev 136 (1964) B864) established that the ground state density, $\rho_\text{gs}$, uniquely determines the Hamiltonian. This had the effect of establishing an implicit relationship between $\rho_\text{gs}$ and the external potential (e.g. external magnetic field, crystal field, etc.), $V$, as the form of the kinetic energy and particle-particle interaction energy functionals are universal since they are only functions of the density. This implicit relationship defines a set of densities which are called $v$-representable. What is surprising is that there are "a number of 'reasonable' looking densities that have been shown to be impossible to be the ground state density for any $V$." (Martin, p. 130) On the surface, this restriction looks like it would reduce the usefulness of density functional theory, but, in practice, that is not the case. (See the proof by Levy - PNAS 76 (1979) 6062, in particular.) However, research continues into the properties of the $v$-representable densities, and I was wondering if someone could provide a summary of that work.

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Requested by author chat.stackexchange.com/transcript/message/4529045#4529045 to be migrated to chem.SE. I, personally think there's no need to do so, but we do have activity out there (public beta now) at the moment, so it may well get an answer. What say? –  Manishearth May 9 '12 at 3:02
We would do this if the question is off topic here. But as far as I can tell, it's not, so there's no cause for migration. If anyone knows someone on chem.SE who can address this, the best course of action is to tip them off that there is a question waiting for their expertise here ;-) Alternatively, rcollyer, you could consider re-asking on chem.SE (but make sure to link to this one when you do). –  David Z May 9 '12 at 6:02
Additionally, if you do cross post, please tailor it to be slightly more chem in tone (it that's possible) –  Manishearth May 9 '12 at 7:30
@DavidZaslavsky crossposting is actually what I intended, initially. The suggestion at migration was a side effect. –  rcollyer May 9 '12 at 10:45
Just to avoid confusion, an external magnetic field can not be derived from just the density. One needs both the spin-up and spin-down densities, even if it was just to fix the sign of the magnetic field. –  Toon Verstraelen Sep 7 '12 at 20:52

There are three places to address the $v$-representability:

1) The Hohenberg-Kohn theorem for the one-to-one mapping from ground-state electronic density to potential $v$

2) The Hohenberg-Kohn variational theorem

3) The Kohn-Sham scheme

As far as I know,

1) The $v$-representability is automatically guaranteed. Since the HK theorem starts from the exact ground-state density, which comes from the eigenfunction with $v$

2) The domain of $v$-representable density is still unknown. It can be avoided by Levy's constrained search (as you quoted paper). In the constrained search, one only needs $N$-representable density, the domain is known.

3) The problem is non-interacting $v$-representability (associated with the potential of non-interacting system). If one restricts to integer occupation of the orbitals/single Slater determinant, the domain is unknown. If one relaxes the integer occupation, the domain is known by Lieb's work.

One recent summary can be found from Density Functional Theory: An Advanced Course (Springer, 2011 edition) by Eberhard Engel and Reiner M. Dreizler. The book includes references to Lieb's paper.

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