Hartree Self-Consistent Field Method

The Hartree Self-Consistent Field (SCF) Method is a variational approach for computing the Fock product,

$$\Phi= g_1(r_1, \theta_1, \phi_1)g_2(r_2, \theta_2, \phi_2) ... g_n(r_n, \theta_n, \phi_n),$$

that minimizes the variational integral,

$$I=\frac{<\Phi\vert\hat{H}\vert\Phi>}{<\Phi\vert\Phi>}.$$

Functions $g_i(r_i, \theta_i, \phi_i)$ are one electron functions characterized by a set of variational parameters (e.g., the effective nuclear charge, when such functions are defined as hydrogenlike orbitals). The initial guess of the n-electron product function,

$$\Phi = S_1(r_1, \theta_1, \phi_1)S_2(r_2, \theta_2, \phi_2)...S_n(r_n, \theta_n, \phi_n),$$   (Fock Product)$$,$$

is used to compute the potential energy,

$$V_1(r_1, \theta_1, \phi_1) = \sum_{j=2}^n Q_1 \int \frac{\rho_j}{r_{1j}}d\sigma_j -\frac{ze^2}{r_1},$$

where $Q_1=-e$ and $\rho_j=-e\vert S_j\vert^2$. Then, it is assumed that the effective potential acting on an electron can be adequately described by the average of the potential $V_1(r_1, \theta_1, \phi_1)$ over angles $\theta$ and $\phi$,

$$V_1(r_1)= \frac{1}{4 \pi}\int d\theta_1 \int d \phi_1$$   sin$$\theta_1 V_1(r_1, \theta_1, \phi_1).$$

Such potential function is used to solve the one-electron Schrödinger equation,

$$\left [ -\frac{\hbar^2}{2m}\nabla_i^2 +V_1(r_1) \right ] t_1(1) = \epsilon_1 t_1(1),$$

according to the variational approach. The eigenfunctions $t_1(1)$ are improved version of the initially guessed functions $S_1$. The procedure is then repeated, after replacing the initial trial function $\phi$ by the improved trial wavefunction $\tilde{\phi} = t_1 S_2 ... S_n$, and $t_2$ is obtained as an improved version of $S_2$. The procedure is repeated to obtain $t_3$, etc., until $S_n$ is replaced by $t_n$. The whole procedure is iterated (i.e., starting with $t_1$ ..., etc.) until there is no further change from one iteration to the next one. The converged wave function gives the Hartree SCF solution of the eigenvalue problem with energy,

$$E= \sum_i \epsilon_i -\sum_i \sum_{j>i} J_{ij}.$$

The last term in this equation involves the Coulombic integrals $J_{ij}$ and discounts all of the interactions that have been counted twice.