CASTEP

What is Density Functional Theory?

Density Functional Theory (DFT) was developed in the 1960s by Walter Kohn, amongst various others. The chief discovery was that the total energy of a system of electrons in an external potential is a unique functional of the groundstate density; if you find the density, then you can calculate the energy. Unfortunately this functional is not known, but it turns out that even relatively crude approximations can give excellent results. The particular "external potential" we are interested in is that generated by nuclei, but this is not necessary for density functional theory itself.

A Many-Body Wavefunction

In order to study our system of interacting electrons and nuclei, we need to find the density. This requires us to construct a suitable many-body wavefunction for the system. In principle this wavefunction is a function of time and all nuclear and electronic coordinates, but the first thing we do is to make an adiabatic separation.

The Born-Oppenheimer Approximation

Because the nuclei are typically more than 2000 times more massive than the electrons, we can decouple the electronic and nuclear motion and say that the electrons respond instantaneously to any change in nuclear coordinates. This approximation allows us to rewrite the full many-body wavefunction as the product of a nuclear and an electronic wavefunction. Since the electronic wavefunction only depends on the instantaneous nuclear configuration, and not on time, we can describe its behaviour with the time-independent Schrodinger equation.

The nuclei are massive enough to be treated as classical particles, responding to the electronic forces according to Newton's laws. This semi-classical approximation, coupled with adiabatic separation of variables, is called the Born-Oppenheimer approximation.

N.B. under some circumstances this approximation can be a little too crude for accurate calculations. The zero point motion of light nuclei such as hydrogen and lithium can have substantial effects on the groundstate of some materials, effects which we have neglected.

Bloch's Theorem

If we confine our studies to periodic systems, then we can make use of a very useful theorem to simplify our problem. Bloch's theorem says that any wavefunction of a periodic system must be the product of a cell-periodic part and a phase factor, in order to preserve the translational symmetry of the density. The phase factor takes the form of a plane wave, whose wavevector is a linear combination of reciprocal lattice vectors.

Plane Waves

Following on from Bloch's theorem it is natural to use a plane-wave basis to express the wavefunction as a whole (both the phase factor and the cell-periodic part). This has the additional advantage that the basis functions are orthogonal, and that operations on them are computationally efficient. In particular it is straightforward to Fourier transform our wavefunction from real to reciprocal space or vice versa.

Supercells

Many of the systems we wish to study are not periodic, and so we construct what is known as a supercell. This is essentially just a large unit cell, repeated periodically in space, but containing a "spacer" region to separate the region of interest from its periodic images. Typically the "spacer" is just vacuum.

By using supercells to represent our system, we can take advantage of Bloch's theorem even for non-periodic systems. Clearly we must ensure that our supercell is a good approximation of the original, aperiodic system, and this is done by checking that our results are invariant with respect to small changes in the supercell size.

Pseudopotentials

The fermionic nature of electrons means that every electronic state of our system must be orthogonal to every other state. As we fill higher and higher states, this orthogonality condition forces the wavefunctions of these states to have increasing numbers of nodes. This dramatically increases the width of the Fourier spectrum, and hence the number of plane waves we need to adequately represent our wavefunction.

Many of the nodes of the atomic states lie in a "core region" close to the nucleus. The wavefunctions in this core region are relatively unchanging, regardless of the chemical environment of the atom, and many of the lower energy states are localised in this region.

In the pseudopotential approximation, the electrons whose wavefunctions are localised in the core region are removed ("pseudised"), and replaced with an effective potential which, when combined with the nuclear coulomb potential, is called the pseudopotential. This means that the remaining electrons' wavefunctions only have to be orthogonal to each other, so the number of nodes is reduced. The pseudopotential is constructed such that the wavefunctions outside the core region are unchanged.

The use of pseudopotentials dramatically reduces the number of plane waves required to represent the wavefunctions, whilst still giving excellent results.