next up previous contents
Next: The Exchange-Correlation Term Up: The Many Body Problem Previous: Approximate Methods: the Hartree   Contents

Density Functional Theory

The density functional theory (DFT) treats the electron density as the central variable rather than the many-body wavefunction. This conceptual difference leads to a remarkable reduction in difficulty: the density is a function of three variables, i.e. the three Cartesian directions, rather than $3N$ variables as the full many-body wavefunction is. An early density functional theory was proposed by Thomas and Fermi [45]. This took the kinetic energy to be a functional of the electron density, but in common with the Hartree and Hartree-Fock methods, only incorporated electron-electron interactions via a mean field potential: as such it neglected both exchange and correlation; a subsequent proposal by Dirac [46], formulating an expression for the exchange energy in terms of the electron density failed to significantly improve the method. Here we consider the Hohenberg-Kohn-Sham formulation of DFT; this technique is one of the choice state-of-the-art methods routinely applied in electronic structure theory, and has enjoyed success in fields ranging from quantum chemistry and condensed matter physics to geophysics. It is based upon the following remarkable and deceptively simple theorems:

The many-body Hamiltonian H fixes the groundstate of the system under consideration, i.e. it determines the groundstate many-body wavefunction $\Psi$, and thus the above theorem ensures that this itself is also a unique functional of the groundstate density. Consequently, the kinetic and electron-electron interaction energies will also be functionals of $n(\mathbf{r})$. One may therefore define the functional $F[n(\mathbf{r})]$


\begin{displaymath}
F[n(\mathbf{r})] = \langle\Psi\vert(T + V_{ee})\vert\Psi\rangle
\end{displaymath} (2.10)

where $T$ is the kinetic energy operator, and $V_{ee}$ is the electron-electron interaction operator. This functional $F$ is a universal functional in the sense that it has the same dependence on the electron density for any system, independent of the external potential concerned. The exact density dependence of this functional is, however, unknown.

Using this functional, one may then define, for a given external potential $v(\mathbf{r})$, the energy functional


\begin{displaymath}
E\left[n(\mathbf{r})\right] = \int d\mathbf{r} n(\mathbf{r})v(\mathbf{r}) + F\left[n(\mathbf{r})\right]
\end{displaymath} (2.11)

where $F\left[n(\mathbf{r})\right]$ is a universal but unknown functional of the electron density. We can write the system energy (for a non-degenerate groundstate) in terms of the groundstate many-body wavefunction $\Psi$ as


\begin{displaymath}
E\left[n(\mathbf{r})\right] = \langle \Psi \left\vert H \right\vert \Psi \rangle
\end{displaymath} (2.12)

with the Hamiltonian given by


\begin{displaymath}
H = F + V
\end{displaymath} (2.13)

where $V$ is the operator corresponding to the external potential, and $F$ is the electronic Hamiltonian


\begin{displaymath}
F = T + V_{ee}.
\end{displaymath} (2.14)

Although these two theorems prove the existence of a universal functional, they do not give any idea as to the nature of the functional, or how to actually calculate the groundstate density. In order to do so, we must discuss the Kohn-Sham formulation [38]. This is based upon a sleight of hand whereby we map the fully interacting system of N-electrons onto a fictitious auxiliary system of N non-interacting electrons moving within an effective Kohn-Sham potential, $v_{KS}(\mathbf{r})$, thereby coupling the electrons. The single-particle Kohn-Sham orbitals are constrained to yield the same groundstate density as that of the fully-interacting system, so the Hohenberg-Kohn-Sham theorems are still valid.

Variation of the total energy functional in 2.11 with respect to the electron density, subject to the constraint of fixed particle number, i.e.


\begin{displaymath}
\int \delta n(\mathbf{r}) d\mathbf{r} = 0
\end{displaymath} (2.20)

yields


\begin{displaymath}
\delta \bigg[ F\left[n(\mathbf{r})\right] + \int v_{ext}(\ma...
...mu \bigg( \int n(\mathbf{r}) d\mathbf{r} - N \bigg) \bigg] = 0
\end{displaymath} (2.21)

where $\mu$ is a Lagrange multiplier associated with our constraint condition 2.20. The Euler-Lagrange equation associated with minimisation of this functional is then


\begin{displaymath}
\mu = \frac{\delta F\left[n(\mathbf{r})\right]}{\delta n(\mathbf{r})} + v_{ext}(\mathbf{r}).
\end{displaymath} (2.22)

The Kohn-Sham formulation allows us to write the universal functional $F\left[n(\mathbf{r})\right]$ as


\begin{displaymath}
F\left[n(\mathbf{r})\right] = T_{s}\left[n(\mathbf{r})\right...
...}\left[n(\mathbf{r})\right] + E_{xc}\left[n(\mathbf{r})\right]
\end{displaymath} (2.23)

where the last term is the exchange-correlation energy, to which we will return presently, and $T_{s}\left[n(\mathbf{r})\right]$ is the kinetic energy, which may be written in terms of the non-interacting single-particle orbitals as


\begin{displaymath}
T_s\left[n(\mathbf{r})\right] = -\frac{1}{2} \sum_{i=1}^{N} ...
...^{\star}(\mathbf{r})\nabla^{2}\psi_{i}(\mathbf{r})d\mathbf{r}.
\end{displaymath} (2.24)

It is important to note that this is the kinetic energy of the auxiliary non-interacting system, not the kinetic energy of the actual physical system under consideration. $E_{H}\left[n(\mathbf{r})\right]$ is the classical Hartree energy of the electrons


\begin{displaymath}
E_{H}\left[n(\mathbf{r})\right] = \frac{1}{2} \int \int \fra...
...thbf{r}^{\prime} \right\vert} d\mathbf{r} d\mathbf{r}^{\prime}
\end{displaymath} (2.25)

which includes a self-interaction term.

Thus the Euler-Lagrange equation 2.21 becomes


\begin{displaymath}
\mu = \frac{\delta T_{s}\left[n(\mathbf{r})\right]}{\delta n(\mathbf{r})} + v_{KS}(\mathbf{r})
\end{displaymath} (2.26)

where $v_{KS}(\mathbf{r})$ is the effective Kohn-Sham potential


\begin{displaymath}
v_{KS}(\mathbf{r}) = v_{ext}(\mathbf{r}) + v_H(\mathbf{r}) + v_{xc}(\mathbf{r}).
\end{displaymath} (2.27)

The Hartree potential $v_H(\mathbf{r})$ is given by


\begin{displaymath}
v_H(\mathbf{r}) = \frac{\delta E_{H}\left[n(\mathbf{r})\righ...
...hbf{r} - \mathbf{r}^{\prime} \right\vert} d\mathbf{r}^{\prime}
\end{displaymath} (2.28)

with the exchange-correlation potential $v_{xc}(\mathbf{r})$


\begin{displaymath}
v_{xc}(\mathbf{r}) = \frac{\delta E_{xc}\left[ n(\mathbf{r}) \right]}{\delta n(\mathbf{r})}.
\end{displaymath} (2.29)

The Euler-Lagrange equation is now of exactly the same form as that which leads to the Hartree equations 2.4. Therefore we are required to solve the Schrödinger-type equations


\begin{displaymath}
\bigg( -\frac{1}{2} \nabla^{2} + v_{KS}(\mathbf{r}) \bigg)\psi_{i}(\mathbf{r}) = \epsilon_{i}\psi_{i}(\mathbf{r})
\end{displaymath} (2.30)

where the $\epsilon_i$ correspond to the eigenvalues of the single-particle states and the charge density $n(\mathbf{r})$ is constructed from the Kohn-Sham orbitals as


\begin{displaymath}
n(\mathbf{r}) = \sum_{i=1}^{N} \psi^{\star}_{i}(\mathbf{r})\psi_{i}(\mathbf{r}).
\end{displaymath} (2.31)

Similarly, the many-electron wavefunction of the system may be constructed as a Slater determinant of the Kohn-Sham orbitals.

The Kohn-Sham formulation thus succeeds in transforming the $N$-body problem into $N$ single-body problems, each coupled via the Kohn-Sham effective potential. It is worth noting that formally there is no physical interpretation of these single-particle Kohn-Sham eigenvalues and orbitals: they are merely mathematical artefacts that facilitate the determination of the true groundstate density. The exception is the highest occupied state, for which it can be shown that [47] the eigenvalue corresponding to the highest occupied state yields the ionisation energy of the system.


next up previous contents
Next: The Exchange-Correlation Term Up: The Many Body Problem Previous: Approximate Methods: the Hartree   Contents
Web Page Administrator 2004-12-16