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Approximate Methods: the Hartree and Hartree-Fock Methods

The simplest approximation is the Hartree approximation [42]. The initial ansatz is that we may write the many-body wavefunction as

\Psi(\mathbf{r}_1,\mathbf{r}_2, \ldots, \mathbf{r}_N) = \psi...
...thbf{r}_1)\psi_{2}(\mathbf{r}_2) \ldots \psi_{N}(\mathbf{r}_N)
\end{displaymath} (2.3)

from which it follows that the electrons are independent, and interact only via the mean-field Coulomb potential. This yields one-electron Schrödinger equations of the form

-\frac{\hbar^{2}}{2m} \nabla^{2}\psi_{i}(\mathbf{r}) + V(\mathbf{r})\psi_{i}(\mathbf{r}) = \epsilon_{i}\psi_{i}(\mathbf{r})
\end{displaymath} (2.4)

where $V(\mathbf{r})$ is the potential in which the electron moves; this includes both the nuclear-electron interaction

V_{nucleus}(\mathbf{r}) = -Ze^{2}\sum_{R} \frac{1}{\left\vert\mathbf{r} - \mathbf{R}\right\vert}
\end{displaymath} (2.5)

and the mean field arising from the $N-1$ other electrons. We smear the other electrons out into a smooth negative charge density $\rho(\mathbf{r}^{\prime})$ leading to a potential of the form

V_{electron}(\mathbf{r}) = -e\int d\mathbf{r}^{\prime} \rho(...{1}{\left\vert \mathbf{r} - \mathbf{r}^{\prime} \right\vert}
\end{displaymath} (2.6)

where $\rho(\mathbf{r}) = \sum_{i}\vert\psi(\mathbf{r})\vert^{2}$.

Although these Hartree equations are numerically tractable via the self-consistent field method, it is unsurprising that such a crude approximation fails to capture elements of the essential physics. The Pauli exclusion principle demands that the many-body wavefunction be antisymmetric with respect to interchange of any two electron coordinates, e.g.

\Psi(\mathbf{r}_{1},\mathbf{r}_{2}, \ldots, \mathbf{r}_{N}) = - \Psi(\mathbf{r}_{2},\mathbf{r}_{1}, \ldots, \mathbf{r}_{N})
\end{displaymath} (2.7)

which clearly cannot be satisfied by a non-trivial wavefunction of the form 2.3. This exchange condition can be satisfied by forming a Slater determinant [43] of single-particle orbitals

\Psi(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots, \mathbf{r}_{N})...
...{1})\psi(\mathbf{r}_{2})\ldots\psi(\mathbf{r}_{N}) \right\vert
\end{displaymath} (2.8)

where $\mathcal{A}$ is an anti-symmetrising operator; i.e. it ensures that all possible anti-symmetric combinations of orbitals are taken. Again, this decouples the electrons, leading to the single-particle Hartree-Fock equations [44] of the form

$\displaystyle -\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{i}(\mathbf{r}) + V_{nucleus}(\mathbf{r})\psi_{i}(\mathbf{r}) + V_{electron}(\mathbf{r})\psi_{i}(\mathbf{r})$      
$\displaystyle - \sum_{j} \int d\mathbf{r}^{\prime} \frac{\psi^{\star}_{j}(\math...
...athbf{r} - \mathbf{r}^{\prime} \right\vert} = \epsilon_{i}\psi_{i}(\mathbf{r}).$     (2.9)

The last term on the left-hand side is the exchange term; this looks similar to the direct Coulomb term, but for the exchanged indices. It is a manifestation of the Pauli exclusion principle, and acts so as to separate electrons of the same spin; the consequent depletion of the charge density in the immediate vicinity of a given electron due to this effect is called the exchange hole. The exchange term adds considerably to the complexity of these equations.

The Hartree-Fock equations deal with exchange exactly; however, the equations neglect more detailed correlations due to many-body interactions. The effects of electronic correlations are not negligible; indeed the failure of Hartree-Fock theory to successfully incorporate correlation leads to one of its most celebrated failures: its prediction that jellium is an insulating rather than a metallic system. The requirement for a computationally practicable scheme that successfully incorporates the effects of both exchange and correlation leads us to consider the conceptually disarmingly simple and elegant density functional theory.

next up previous contents
Next: Density Functional Theory Up: The Many-Body Problem Previous: The Many-Body Problem   Contents
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