The Self-Consistent Field

The Hartree-Fock groundstate energy $E_0^{\hbox{{HF}}}$ is obtained by minimising (1.26) with respect to the variation of the orbitals, subject to the constraint that the orbitals remain orthonormal (1.24). This is another constrained minimisation problem that can be performed using the Euler-Lagrange method. The corresponding stationary condition is given by:

$$\delta \Bigg(E_0^{\hbox{{HF}}} - \sum_i^N \sum_j^N \varepsilon_{ij} (\langle \psi_i \vert \psi_j \rangle - 1) \Bigg) = 0 \, ,$$ (1.25)

where the Lagrange multipliers, $\varepsilon_{ij}$, form a Hermitian matrix which can be diagonalised by a unitary transformation of the orbitals. The so-called Hartree-Fock (HF) equations (in canonical form) are therefore given by,

$${\left(- \frac{1}{2} \nabla^2 + v_{\hbox{{ext}}}({\bf r}) + \sum_... ...r'})\vert^2}{\vert{\bf r}-{\bf r'}\vert} \, d{\bf r'} \right) \psi_i({\bf r}) }$$

$$\hspace{2.5cm} - \sum_{j}^{N} \int \frac{\psi_i({\bf r'}) \psi_j^... ...\vert} \, \delta_{s_i s_j} \, d{\bf r'} = \varepsilon_i \psi_i({\bf r}) \, . \,$$ (1.26)

In general the Hartree-Fock equations cannot be solved analytically. One exception is for the homogeneous electron gas, where the constant external potential leads to plane wave solutions that result in the local exchange energy derived by Dirac (1.16). In other situations, the Hartree-Fock equations are solved using an iterative process known as the self-consistent field procedure. Since the desired orbitals also make up their own one-electron effective potential in (1.28), the set of orbitals $\{\psi_i({\bf r}) \}$ that give rise to the same set after solving (1.28) are known as the self-consistent orbitals, and they are the groundstate orbitals for that system within the Hartree-Fock approximation. The self-consistent procedure starts with an initial guess for the orbitals, and successive iterations are performed with new orbitals until the self-consistent condition is achieved.