Density Functional Theory (DFT)¶
A functional is a function of a function. This may sound like a scary word, but you’ve definitely worked with them before! (e.g. an integral is a functional \(\int f(x) dx\)). The notation we’re going to use for functionals is square brackets, e.g. F[x] means F is a function of x which is in turn a function of another variable, so F is a functional.
DFT is used to calculate the ground state of a system, and so is independant of time. The most important function in DFT is the electron density, \(n(\textbf{x})\). Once you’ve got the density, you are able to calculate anything you want about the system (more on this later). This vastly simplifies the problem, since the density is only a function of 3 varibles, opposed to the 3N varibles of the full many body wavefunction. The simplest formulations of DFT treat the electron-electron interaction through a mean-field only approach, that is reducing the many-body, interacting electron problem into many non-interacting electrons moving in an effective potential.
Aside - Mean Field Theories¶
If you are familiar with the concept of a mean field theories, you might like to think of DFT in this way but it isn’t necessary to understand the following. We’ll include a quick reminder here of mean field theory here in case you’d like to think of DFT in this way. One can think of this as replacing operators by their expectation values, (or more rigorously, writing an operator as its mean plus fluctuations about the mean and then taking the 0th order term, \(\hat{O} = \langle \hat{O} \rangle + \hat{\delta O} \approx \langle \hat{O} \rangle\).) Clearly this means that the mean field has no fluctuations. This greatly simplifies the problem since you don’t need to keep track of each of the individual particles and instead just one particle moving in the mean field. (Add things about SCF)
Hohenberg-Kohn Theorems¶
DFT is built on two main theorems - the Hohenberg-Kohn theorems [Hohenberg1964] - which we will state without proof here in the interests of brevity, but we very much encourage you to look into their proofs - they’re really not so bad.
From this point on, we are going to specialise into 1 dimension to be faithful to the iDEA code, but these ideas are easily generalised to 3D.
Theorem 1¶
The external potential is a unique functional of the electron density only (up to an additive constant). So the Hamiltonian, and therefore all ground state properties, are determined solely by the electron density.
This is a very far reaching statement! Once we’ve got the electron density, we have got everything we could want to know about the system. It is important to recall how the density is related to the many-body wave function:
where the prefactor of N is included to account for the arbitrary assignment of which of the electrons hasn’t had its coordinate integrated over.
Theorem 2¶
The ground state energy may be obtained variationally, and the density which minimises this energy is the exact ground state energy.
NB There is a nuance here, but a very important one! During the proofs of these theorems, one assumes that the electrons are in their ground states and so DFT is only valid for ground state systems.
Now, taking these two theorems together prove that a universal functional must exist, but sadly don’t even give us a hint as to what it should look like, (or even how to calculate the ground state energy). Indeed, there are no known exact functionals for systems of more than one electron! Also, you shouldn’t get too excited by the electron density being the central parameter, as ever things are more complicated than they seem. Although in principle it is possible to determine all properties of the the system from \(n(\textbf{x})\), in practice it isn’t that easy. The is reason is we often don’t know how to go from \(n(\textbf{x})\) to the quantity we’re interested in finding and so have to revert back to the set of \(N\) wavefunctions.
At this point we pick up the Kohn-Sham formulation of DFT [Kohn1965], which is what opened the door for so much progress in this field.
Kohn-Sham DFT¶
In the Kohn-Sham (KS) formulation of DFT, instead of considering the full system of N interacting electrons, we instead look at a fictitious system of N non-interacting electrons moving in an effective KS potential, \(V_{KS}\). The single-particle KS orbitals are constrained to give the same electron density as that of the real system, so we can then, in theory at least, use theorem 1 to find out anything we want about the system. This (KS density yielding the exact density) is actually an assumption of the KS construction of the fictitious system as no rigorous proofs for realistic systems. other properties of the KS system are not the same as the real system (e.g. the kinetic energy of the auxiliary system won’t, in general, be the same as that of the real system).
Recall that we are treating the electrons as spinless, so we constrain the system such that there is only one electron per KS orbital which gets around any possible problems arising from the Pauli exclusion principal.
We’re going to brush over a few of the formalities here, since, for example, knowledge of how to use Lagrange multipliers to ensure particle conservation doesn’t give greater insight into the physics.
The goal from here is to solve the Schrödinger-like equation
where \(V_{KS}\) is the Kohn-Sham potential that we’ll discuss shortly, and \(\varepsilon_i\) are the single-partle eigenvalues. It is worth emphasising that this equation is, in principal, exact.
The KS potential is given by
where \(V_{ext}\) is the external potential arising from the electron’s interaction with the nuclei, \(V_H\) is the Hartree potenial, \(V_x\) is the exchange potential and \(V_c\) the correlation potential. The last two terms are often lumped together into one exchange-correlation potential, \(V_{xc}\), which we shall use fron now on.
Hartree potential¶
The easiest way to understand the origin of the Hartree potential is to imagine freezing all the electrons in space, and then seeing what the electrostatic potential is due to these electrons.
where \(u\) is the softened coulumb interaction implemented in iDEA. If you examine definition of the Hartree potential you’ll notice that it includes self-intreaction, that is electron’s interacting with other parts of their own charge densities - don’t worry, this gets accounted for later. It’s worth taking a moment here to reflect on where we are so far. On the face of things, it might seem like we have everything we need to solve this problem exactly. We’ve entirely accounted for the Coulomb interaction between the electrons and the nuclei and between the electrons themselves. So why do we need to bother including \(V_{xc}\)? The reason is that in defining \(V_H\), we froze the electrons in place and got an electrostatic potential, but of course the electrons in a real system will be moving, and it is this movement that gives rise to the exchange-correlation potential.
Exchange-Correlation potential¶
The origin of the exchange part of the potential is due to the exchange symmetry of the wavefunction of the system of identical particles (we’ll restrict our treatment to fermions here). When fermions get close to each other they experience “Pauli repulsion”, which causes the expectation values between them to be larger. So when the electrons are moving in the sample, they stay further away from each other than one would naively expect. The correlation of the system is a bit harder to put on explicit physical basis but it is a measure of how much the motion of one electron affects that of another. \(V_{xc}\) also corrects for the self interaction in the Hartree potential.
The problem is that we don’t know what the exchange-correlation functional looks like for any system more complicated than the homogenous electron gas (HEG), which is where KS DFT goes from being an exact theory to an approximate one. We’ll discuss one of these approximations later.
DFT’s strength lies in the fact that \(V_{xc}\) is a relatively small contribution to \(V_{KS}\) so this term only being approximately correct doesn’t change the form of the KS potential too drastically, which gives accruate KS oribitals and hence the electron density given by
The alert reader may notice a problem here. We need the KS oribitals to get the density by the above equation. To get the orbitals we need to solve the Schrödinger-like equation, however, that requires knowledge of the KS potential, which in turn depends on the electron density of the system. So to solve this we put in a guess of the electron density (often the density obtained from the non-interacting electron approximation), then plug this into the Schrödinger-like equation for the orbitals and then get the density from those. You then compare this new density with the old one. If there has been a change, we plug this new density in and try again. We keep this iteration going until we reach a self consistent solution, or in practice that the change from the old density to the new one is very small.
Of course this all assumes we know the form of the KS potential, but as we mentioned earlier, no one knows the form of the exchange-correlation functional which stops us doing this calculations exactly. One of the most common approximations is to use the local density approximation (LDA).
Local Density Approximation (LDA)¶
In the LDA, the functional only depends on the place where we are evaluating the density (hense the ‘local’ part of its name). The energy functional is given by
where \(\varepsilon_{xc}^{HEG}(n)\) is the exchange-correlation energy per particle for the homogenous electron gas. Armed with this functional, we can get \(V_{xc}^{LDA}\) by using a functional derivative, which is written as
Once, we have \(V_{xc}^{LDA}\), we can get the KS potential and go through the process of finding a self consistent solution.
References¶
| [Hohenberg1964] | “Inhomogeneous Electron Gas” P. Hohenberg and W. Kohn (1964) Phys. Rev. 136, B864 |
| [Kohn1965] | “Self-Consistent Equations Including Exchange and Correlation Effects” W. Kohn and L. J. Sham (1965) Phys. Rev. 140, A1133 |
