There are no wavefunctions in DFT

Question

I have seen this phrase several times across DFT textbooks. However, I am not sure if it still holds. Was there a change or a theorem that proved it otherwise? Several programs display wavefunctions for DFT calculations. For example, Quantum Espresso has the option to collect wave functions. Some of the people that I have to talked to mention the Kohn Sham orbitals as if they were a real physical thing. Shouldn't it not be possible to obtain the wavefunction for a certain state? For example, I thought it was not possible to tell that a Kohn Sham orbital belongs to a wavefunction near the surface of a material.

I know that the Kohn Sham orbitals lack a physical meaning. They are simply wavefunctions that are used to obtain the correct density. They do not have an analogue of Koopmans' Theorem. In addition, "the exact wave function of the target system is not available in density functional theory" (Koch, 2001).

So why do people need/use wavefunctions (after all most DFT programs can group such wavefunctions)? I thought that they were Kohn Sham orbitals. Therefore, there should be seldom any use for them.

Answer 1

There are no wavefunctions in DFT.

I don't like that "wave function" is used here in the plural form and I feel like it reflects OP's misconception. The right way to say this is

There is no wave function in DFT.

See, there exist only one wave function, the function that describes the state of a system in question and can be used to calculate its properties. When it comes to a many-electron system, the wave function is a function that describes the state of all the electrons present in the system interacting with each other. This is the wave function people have in their minds when they say the it is absent in DFT. And that is true: the whole point of DFT is to get rid of this function as a description of state since it is too complex to work with and difficult to visualise.

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But that is not the whole story. In practise the Kohn-Sham DFT is used which introduces a fictious system of non-interacting electrons. Each electron in the fictious system is described by a Kohn-Sham orbital, a single-particle wave function, while the whole fictious system is described by the Kohn–Sham wave function, a Slater determinant constructed from a set of Kohn-Sham orbitals. Still there is no wave function in KS-DFT in the above mentioned sense:

• KS orbitals don't qualify as the wave function just because they do not describe the state of a whole system;
• the Kohn–Sham wave function doesn't qualify because it describes a different system (fictious non-interacting rather than real interacting).

So, the KS wave function is not the wave function, since it doesn't describe the (real) system. You can't use it to calculate the properties of the (real) system, you have to use the electron density to do so. In this sence, the statement that there is no wave function in DFT holds true for KS-DFT as well.

Can the KS wave functions (both single- and many-particle one) be somehow useful in any other way is a different story. See, for instance, the discussion of the usefulness of KS-orbitals here.

Answer 2

To address the question whether "there is a a wave funtion in Kohn-Sham theory", one has to decide first whether there is a wave function or not in Hartree Fock theory. I argue below that the answer will be the same in both cases. In HF theory, we seek the noninteracting wave function (Slater determinant) for which the expectation value of the interacting Hamiltonian H is minimum. Obviously this wave function is only an approximation and it is very different from the true ground state interacting wavefunction Ψ. In KS theory, there is a similar point of view using just wave functions and the Rayleigh Ritz variational principle and never mentioning the density. Let us consider the class of effective Hamiltonians, Hv=T+V, with kinetic energy T and an effective local potential V but with no interaction. Let us denote by Ev the ground state energy of Hv. There is an infinity of effective Hamiltonians Hv (differing by V) and we can ask which one of these effective Hamiltonians Hv adopts the true interacting ground state Ψ optimally as its approximate ground state. Mathematically, we can use the variational principle: <Ψ|Hv|Ψ>-Ev > 0. On the l.h.s. the wave function Ψ is given, so it makes sense to minimise the l.h.s. over the effective potential V to find that which makes the l.h.s. minimum. It turns out that the minimizing effective potential is the Kohn-Sham potential. So, from this point of view the ground state of the minimizing Hv (i.e. the exact KS wave funtion) is a meaningful approximate wave function for Ψ, in the same way that the HF wave function is a meaningful approximate wave function for Ψ. See the reference for more details: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.83.040502