Density functional theory for beginners
The quantum mechanical wavefunction contains, in principle, all the information about a given system. For the case of a simple 2-D square potential or even a hydrogen atom we can solve the Schrödinger equation exactly in order to get the wavefunction of the system. We can then determine the allowed energy states of the system.
Unfortunately it is impossible to solve the Schrödinger equation for a N-body system. Evidently, we must involve some approximations to render the problem soluble albeit tricky. Here we have our simplest definition of density functional theory (DFT): A method of obtaining an approximate solution to the Schrödinger equation of a many-body system.
DFT computational codes are used in practise to investigate the structural, magnetic and electronic properties of molecules, materials and defects.
Terminology
Here are some of the ideas used in this field and introduce them to some more. Take, as an example of a many-body problem, the case of a regular crystal. The electrons are not only affected by the nuclei in their lattice sites, but also by the other electrons.
Coulomb potential
This is just the classical potential that arises from any charged object.
Jellium
Jellium is the name given to an homogeneous electron gas. The motion of the electrons in the gas is not only affected by the coulomb interaction between them. There is also a purely quantum mechanical process at work. The motion of the electrons is correlated by the Pauli Exclusion principle. The electrons with parallel spins, must maintain a certain separation. The anti-parallel spin electrons keep apart to lower their mutual coulomb repulsion. So we can gain energy in two ways. Moving parallel spin electron apart will lower the exchange energy, moving anti-parallel spins apart will lower the correlation energy.
For the case of a homogeneous electron gas we can work out the average exchange-correlation energy per electron. In fact, values for the exchange-correlation energy for any reasonable electron density can be taken from parameterisations of calculated data.
The exchange-correlation hole
The exchange-correlation effect gives rise to an exchange-correlation hole around each electron in the gas. The hole is just the exclusion radius caused by the Pauli exclusion principle. The mutual exclusion zone or exchange-correlation hole around an electron in jellium. The radius of the hole corresponds to the exclusion of a single electron thereby giving the hole a single positive charge. The overall charge of the quasiparticle (the electron and its exchange-correlation hole together) is zero. Of course in reality, there is no marked boundary for the exclusion zone as drawn, the boundary is diffuse.
Hartree potential
The interaction between the electrons in the system is approximated by the Coulomb potential arising from a system of fixed electrons. Alternatively we say that each individual electron moves independently of each other, only feeling the average electrostatic field due to all the other electrons plus the field due to the atoms. In other words, this is the potential due to the electron density distribution and ionic lattice but neglects exchange and correlation effects.
Pauli Exclusion Principle
This is the rule that disallows two identical particles to lie in the same quantum state.
Exchange interaction
This is due to the Pauli Exchange Principle. In this case it states that if two electrons have parallel spins then they will not be allowed to sit at the same place at the same time. This phenomenon gives rise to an effective repulsion between electrons with parallel spins. Not only, then do we have two electrons interacting via their electronic charge, but also by their spins.
Correlation interaction
The correlation interaction is also a result of the Pauli Exchange interaction. In this case there is a correlated motion between electrons of anti-parallel spins which arises because of their mutual coulombic repulsion.
Hartree-Fock
This method employs the Hartree potential but also forces the exchange interactions by forcing the antisymmetricity of the electronic wavefunction. This acts to lower the total binding energy of atoms by ensuring that parallel spin electrons stay apart. The downside to the theory is that it neglects correlations in the motion between two electrons with anti parallel spins.
The Principles of Density Functional Theory
Born–Oppenheimer approximation
As a first step we reduce the number of degrees of freedom of the system. The Born–Oppenheimer approximation separates electronic motion from nuclear motion, treating nuclei as fixed relative to the faster electrons.
Since the forces on both the electrons and ions are of the same order of magnitude, their momenta are also comparable. However since the ions are so massive in comparison to the electrons, the kinetic energy of the ions is much smaller than that of the electrons. This idea forms the basis of the Born-Oppenheimer approximation. The electrons are assumed to respond instantaneously to the motion of the ions. For any ionic configuration, we assume that the electrons are in the instantaneous ground-state and calculate the total energy of the system. Varying the ionic positions defines a multi-dimensional ground-state potential energy surface, and the motion of the ions can then be treated as classical particles moving in this potential.
Functionals and the role of electron density
A functional is a function of a function. In DFT the central functional is the electron density, which depends on position (and sometimes time). Unlike Hartree–Fock theory, which works directly with the many-body wavefunction, DFT uses the electron density as the fundamental property.
Using electron density significantly speeds up calculations: while the many-body wavefunction depends on 3N variables (for N atoms), the electron density depends only on x, y, z.
Hohenberg and Kohn proved that the electron density is extremely useful. Their theorem states that the density of any system determines all ground-state properties. Thus, the total ground-state energy of a many-electron system is a functional of the density. If we know the electron density functional, we know the total energy of the system.
Energy decomposition
The total energy can be expressed in terms of several contributions, all of which are functionals of the charge density:
- Ion–electron potential energy
- Ion–ion potential energy
- Electron–electron energy
- Kinetic energy
- Exchange-correlation energy
The two most difficult contributions to calculate accurately are the kinetic energy and the exchange-correlation energy.
The kinetic energy term and Kohn-Sham orbitals
The kinetic energy term could easily be calculated by using the same principle as LDA. That is, we could use the result derived from a homogeneous electron gas. This is the approach followed by Thomas Fermi theory. Unfortunately, the accuracy needed to describe the small energy changes that characterise chemical bonding is not sufficient with this approach. We need another way of getting the kinetic energy.
Two clever chaps, Kohn and Sham introduced a set of orbitals from which the electron density can be calculated. These Kohn-Sham orbitals do not, in general, correspond to the actual electron orbitals. Likewise, the Kohn-Sham eigenvalues are not in general the same as the real energy levels. The only connection the Kohn-Sham orbitals necessarily have to the real electronic wavefunctions is that they both give rise to the same charge density. The Kohn-Sham orbitals are used to calculate the kinetic energy. The property of the orbitals that makes them useful in the derivation is their orthonormality. The tricky problem of a system of interacting electrons has now been mapped onto that of a system of non-interacting electrons moving in an effective potential.
The exchange-correlation term
An approximation to the exchange-correlation term is used. It is called the Local Density Approximation (LDA). For any small region, the exchange-correlation energy is the approximated by that for jellium of the same electron density. In other words, the exchange-correlation hole that is modelled is not the exact one - it is replaced by the hole taken from an electron gas whose density is the same as the local density around the electron.
The interesting point about this approximation is that although the exchange-correlation hole may not be represented well in terms of its shape, the overall effective charge is modelled exactly. This means that the attractive potential which the electron feels at its centre is well described.
Not only does the LDA approximation work for materials with slowly varying or homogeneous electron densities but in practise demonstrates surprisingly accurate results for a wide range of ionic, covalent and metallic materials.
An alternative, slightly more sophisticated approximation is the Generalised Gradient Approximation (GGA) which estimate the contribution of each volume element to the exchange-correlation based upon the magnitude and gradient of the electron density within that element.
