1. Lecture 6: density functional theory
In the previous two lectures we discussed several examples of mean-field theory where the variational class is the class of gaussian states or single-particle product states. It turns out that mean-field theory is extremely useful in practice and can give powerful insights into the physics of many strongly interacting systems. This is because it leads to equations which are essentially tractable analytically. Additionally, the solutions often have a direct physical interpretation. Thirdly, the results obtained can often be very good indeed and often match experimentally obtained values to a high precision. Indeed, I’m sure that mean-field theory will continue to be a very useful tool for many years to come, and it should certainly be the first port of call when trying to understand a new system.
In this lecture we turn our attention to another method known as density functional theory (DFT). This can be viewed as, in part, an application of the variational method, although it is not a pure application in the sense of our previous examples.
The pdf version of the lecture notes can be found here.
2. The Hohenberg-Kohn theorem
Throughout the next two lectures we consider many-fermion systems in the second-quantised formalism. Thus we formulate everything in terms of the tuple of quantum field operators , , obeying the CAR
In the case that one can think of each of as modelling the two spin components of the electron field.
As we’ve seen, a non-relativistic many fermion system is characterised by a second quantised hamiltonian with form
where is some fixed two-particle interaction. In all the cases we’re interested in is simply the Coulomb interaction.
The fundamental building block of DFT is the Hohenberg-Kohn theorem. At an abstract level this result simply trades one set of variational parameters for another, the local density, via a Legendre transform. The impact of this apparently trivial operation cannot be understated, however, as it unlocks a new way to study electronic systems. One motivation for this transformation is that the many-particle state vector is not directly accessible in experiments. Rather, experiments can only access the one-particle density
The first important observation is that essentially the only thing that changes from physical situation to physical situation is the local one-particle potential . Since we are typically interested in the ground-state properties our first step is to construct a special set , the set of all one-particle potentials with the property that the solution of the eigenvalue equation
leads to a non-degenerate ground state for a system of fermions. We have thus defined, via solution of the time-independent Schrödinger equation, a map from to a subset of hilbert space. This map is surjective onto by construction.
The next step is to calculate the ground-state density for each state in , this gives a map to the set of densities via
This map is, again, surjective by construction.
The first statement of the Hohenberg-Kohn theorem is then that the maps and , and hence , are injective, and hence bijective. To prove that is injective one needs to show that for any two potentials and always lead to different ground states whenever they differ by more than a constant, i.e.,
This is because potentials differing by a constant are physically equivalent. The Schrödinger equation then yields
so that the assumption gives.
Since acts via multiplication (in the position basis) we obtain, assuming that the wavefunction is nowhere vanishing, that
contradicting our assumption. (This can be made rigourous, see e.g., Lieb (1982)).
To show that the map is injective one needs to show that for two ground states then . Note that the variational characterisation of the ground eigenvalue implies that
Also, we have that
Similarly, repeating the argument for gives us
Adding these two inequalities and supposing that we conclude that
which is a contradiction.
Thus we’ve learnt that the map is invertible, meaning that any density defines a unique state via
Hence we obtain the first statement of the Hohenberg-Kohn theorem: the ground-state expectation value of any observable is a unique functional of the ground-state energy density:
The full inverse of the map shows us that the ground-state density completely determines the external potential of the system (up to a constant):
Since the kinetic energy and the interparticle interaction is already specified, the inverse map completely specifies the hamiltonian.
The second statement of the Hohenberg-Kohn theorem establishes a variational property of the functional
where is the external potential of interest with ground-state density and ground-state energy . The variational method directly implies that
We learn that the ground-state energy can be found via minimisation of with respect to instead of :
Since the map doesn’t depend on we have that
Remarkably is universal meaning that it doesn’t depend on . It is exactly the same whether we talk about atoms, molecules, solids, or gases since is given in all cases by the Coulomb repulsion for electrons.
3. The Kohn-Sham scheme
The Hohenberg-Kohn theorem allows one to determine the ground-state density of a specific many particle system via a direct variation with respect to densities. In practice there are considerable advantages in replacing the direct variation with respect to the density with an intermediate orbital calculation. This results in a self-consistent scheme that is now the cornerstone of the density functional formalism.
To describe the Kohn-Sham scheme we consider an auxiliary system of non-interacting particles with hamiltonian
According to the Hohenberg-Kohn theorem there is a unique energy functional
such that the variational extremum given by yields the exact ground-state density . Here denotes the universal kinetic-energy functional for non-interacting particles.
The main assertion of the Kohn-Sham scheme is: for any interacting system there is a local single-particle potential such that the exact ground-state density of the interacting system is identical to the ground-state density of the auxiliary problem, i.e.,
If the ground state of is nondegenerate, the ground-state density is uniquely represented by
The degenerate case is more subtle, and we don’t discuss it here (see, e.g., Dreizler (1991) for the complete case). Having assumed the existence of the uniqueness of follows immediately follows from the Hohenberg-Kohn theorem. This implies that the single-particle orbitals are also unique functionals of the density
and hence so is the non-interacting kinetic energy via
Now we are going to show how to obtain the single-particle potential of our auxiliary non-interacting system which generates the ground-state density of a given interacting system with potential .
We start with the single-particle equations
We then rewrite the total energy functional for the interacting system by adding and subtracting and a Hartree term:
where the exchange-correlation functional is defined to be
Via the Hohenberg-Kohn theorem we know that is stationary with respect to small variations :
There is a subtle point to mention here: not all variations are allowed, instead, only those which could have arisen from a local potential. Now our main assumption ensures that this variation can be represented with a unique variation of the local potential of the single-particle problem. This, in turn, leads to a variation of the single-particle orbitals. The variation of may now be expressed in terms of these orbital variations:
which follows from Green’s theorem. Since the orbitals obey the Schrödinger equation we obtain, to first order,
The first term vanishes thanks to the normalisation of the wavefunctions. This, combined with (42), leads to
This is a self-consistent equation which can be solved via iteration.