The Kohn-Sham Formulation

The Kohn-Sham formulation centres on mapping the full interacting system with the real potential, onto a fictitious non-interacting system whereby the electrons move within an effective ``Kohn-Sham'' single-particle potential $v_{\hbox{{KS}}}({\bf r})$. The Kohn-Sham method is still exact since it yields the same groundstate density as the real system, but greatly facilitates the calculation.

First consider the variational problem presented in the second Hohenberg-Kohn theorem - the groundstate energy of a many-electron system can be obtained by minimising the energy functional (1.30), subject to the constraint that the number of electrons $N$ is conserved, which leads to,

$$\delta \left[ F[n({\bf r})] + \int \, v_{\hbox{{ext}}}({\bf r}) n... ... d{\bf r} - \mu \left( \int n({\bf r}) \, d{\bf r} - N \right) \right] = 0 \, ,$$ (1.40)

and the corresponding Euler equation is given by,

$$\mu = \frac{\delta F[n({\bf r})]}{\delta n({\bf r})} + v_{\hbox{{ext}}}({\bf r}) \, ,$$ (1.41)

where $\mu$ is the Lagrange multiplier associated with the constraint of constant $N$. The idea of Kohn and Sham was to set up a system where the kinetic energy could be determined exactly, since this was a major problem in Thomas-Fermi theory. This was achieved by invoking a non-interacting system of electrons. The corresponding groundstate wavefunction $\Psi_{\hbox{{KS}}}$ for this type of system is given exactly by a determinant of single-particle orbitals $\psi_i({\bf r}_i)$,

$$\Psi_{\hbox{{KS}}} = \frac{1}{\sqrt{N!}} \, {\hbox{det}} [\psi_1({\bf r}_1) \psi_2({\bf r}_2) \dots \psi_N({\bf r}_N)] \,.$$ (1.42)

The universal functional $F[n({\bf r})]$ was then partitioned into three terms, the first two of which are known exactly and constitute the majority of the energy, the third being a small unknown quantity,

$$F[n({\bf r})] = T_{\hbox{{s}}}[n({\bf r})] + E_{\hbox{{H}}}[n({\bf r})] + E_{\hbox{{XC}}}[n({\bf r})] \, .$$ (1.43)

$T_{\hbox{{s}}}[n({\bf r})]$ is the kinetic energy of a non-interacting electron gas of density $n({\bf r})$, $E_{\hbox{{H}}}[n({\bf r})]$ is the classical electrostatic (Hartree) energy of the electrons,

$$E_{\hbox{{H}}}[n({\bf r})] = \frac{1}{2} \int \int \frac{n({\bf r}) n({\bf r'})}{\vert \, {\bf r}-{\bf r'} \, \vert} \, d{\bf r} \, d{\bf r'} \, ,$$ (1.44)

and $E_{\hbox{{XC}}}[n({\bf r})]$ is the exchange-correlation energy, which contains the difference between the exact and non-interacting kinetic energies and also the non-classical contribution to the electron-electron interactions, of which the exchange energy is a part. In the Kohn-Sham prescription the Euler equation given in (1.43) now becomes,

$$\mu = \frac{\delta T_{\hbox{{s}}}[n({\bf r})]}{\delta n({\bf r})} + v_{\hbox{{KS}}}({\bf r}) \, ,$$ (1.45)

where the Kohn-Sham potential $v_{\hbox{{KS}}}({\bf r})$ is given by,

$$v_{\hbox{{KS}}}({\bf r}) = v_{\hbox{{ext}}}({\bf r}) + v_{\hbox{{H}}}({\bf r}) + v_{\hbox{{XC}}}({\bf r}) \, ,$$ (1.46)

with the Hartree potential $v_{\hbox{{H}}}({\bf r})$,

$$v_{\hbox{{H}}}({\bf r}) = \frac{\delta E_{\hbox{{H}}}[n({\bf r})]... ...= \int \frac{n({\bf r'})}{\vert \, {\bf r}-{\bf r'} \, \vert} \, d{\bf r'} \, ,$$ (1.47)

and the exchange-correlation potential $v_{\hbox {{XC}}}({\bf r})$,

$$v_{\hbox{{XC}}}({\bf r}) = \frac{\delta E_{\hbox{{XC}}}[n({\bf r})]}{\delta n({\bf r})} \, .$$ (1.48)

The crucial point to understand in Kohn-Sham theory is that (1.47) is just a rearrangement of (1.43), so the density obtained when solving the alternative non-interacting Kohn-Sham system is the same as the exact groundstate density. The groundstate density is obtained in practice by solving the $N$ one-electron Schrödinger equations,

$$\left[ -\frac{1}{2} \nabla^2 + v_{\hbox{{KS}}}({\bf r}) \right] \psi_i({\bf r}) = \varepsilon_i \psi_i({\bf r}) \, ,$$ (1.49)

where $\varepsilon_i$ are Lagrange multipliers corresponding to the orthonormality of the $N$ single-particle states $\psi_i({\bf r})$, and the density is constructed from,

$$n({\bf r}) = \sum_{i=1}^N \, \vert\psi_i({\bf r})\vert^2 \, .$$ (1.50)

The non-interacting kinetic energy $T_{\hbox{{s}}}[n({\bf r})]$ is therefore given by,

$$T_{\hbox{{s}}}[n({\bf r})] = - \frac{1}{2} \sum_{i=1}^N \int \, \psi_i^*({\bf r}) \nabla^2 \psi_i({\bf r}) \, d{\bf r} \, .$$ (1.51)

Since $v_{\hbox{{KS}}}({\bf r})$ depends on the density through the exchange-correlation potential, relations (1.48), (1.51) and (1.52), which are known as the Kohn-Sham equations, must be solved self-consistently as in the Hartree-Fock scheme described in Sec. 1.4.1.

In order to handle the kinetic energy in an exact manner, $N$ equations have to be solved in Kohn-Sham theory to obtain the set of Lagrange multipliers $\{\varepsilon_i\}$, as opposed to one equation that determines $\mu$ when solving for the density directly, as in the Thomas-Fermi approach. However an advantage of the Kohn-Sham method is that as the complexity of a system increases, due to $N$ increasing, the problem becomes no more difficult, only the number of single-particle equations to be solved increases.

Although exact in principle, Kohn-Sham theory is approximate in practice because of the unknown exchange-correlation functional $E_{\hbox{{XC}}}[n({\bf r})]$. An implicit definition of $E_{\hbox{{XC}}}[n({\bf r})]$ can be given through (1.45) as,

$$E_{\hbox{{XC}}}[n({\bf r})] = T[n({\bf r})] - T_{\hbox{{s}}}[n({\bf r})] + E_{\hbox{{ee}}}[n({\bf r})] - E_{\hbox{{H}}}[n({\bf r})]$$ (1.52)

where $T[n({\bf r})]$ and $E_{\hbox{{ee}}}[n({\bf r})]$ are the exact kinetic and electron-electron interaction energies respectively. The intention of Kohn and Sham was to make the unknown contribution to the total energy of the non-interacting system as small as possible, and this is indeed the case with the exchange-correlation energy, however it is still an important contribution since the binding energy of many systems is about the same size as $E_{\hbox{{XC}}}[n({\bf r})]$, so an accurate description of exchange and correlation is crucial for the prediction of binding properties. Present approximations for the exchange-correlation energy are far from satisfactory, consequently the development of improved exchange-correlation functionals is essential. An in-depth discussion of the exact properties of $E_{\hbox{{XC}}}[n({\bf r})]$, and the approximations presently used will be discussed in Chapter $2$. Before this is presented, the remainder of this chapter concentrates on the implementation of Kohn-Sham theory for periodic systems.