Effective Average Potential

The basic idea here is to replace the complicated interactions among the electrons by an effective, average potential energy that each electron $i$ at position ${\bf r}_i$ experiences.

In the Hartree approach one assumes that particle $j$ is described by a wave function (spin orbital) $\psi_{\nu_j}(\xi_j)$ with orbital part $\psi_{\nu_j}({\bf r}_j)$, and the statistics (anti-symmetrization of all the total $N$-particle wave function for Fermions, symmetrizatin for Bosons) is neglected. In the following, we discuss electrons.

For electrons interacting via the Coulomb interaction $U(r)=e^2/4\pi\varepsilon_0r$, the potential seen by an electron $i$ at position ${\bf r}_i$ is given by

$$V_{\rm H}({\bf r}_i) = \frac{-e}{4\pi\varepsilon_0}\sum_{j=1(\ne ... ...r}_j \frac{\vert\psi_{\nu_j}({\bf r}_j)\vert^2}{\vert{\bf r}_j-{\bf r}_i\vert}.$$ (1.1)

This is the sum over the potentials generated by all other electrons $j\ne i$ which have a charge density $-e\vert\psi_j({\bf r}_j)\vert^2$. The corresponding potential energy for electron $i$ is $-eV_{\rm H}({\bf r}_i)$, and therefore one describes electron $i$ by an effective single particle Hamiltonian,

$$H_{\rm Hartree}^{(i)} = H_0^{(i)}+ V_{\rm Hartree}({\bf r}_i)$$

$$= -\frac{\hbar^2}{2m}\Delta_i+ V({\bf r}_i)+\frac{e^2}{4\pi\varepsi... ...r}_j \frac{\vert\psi_{\nu_j}({\bf r}_j)\vert^2}{\vert{\bf r}_j-{\bf r}_i\vert},$$ (1.2)

where $V({\bf r}_i)$ is the usual potential energy due to the interaction with the nucleus. The corresponding Schrödinger equations for the orbital wave functions $\psi_{\nu_i}$ for electron $i$ are

$$\left[ -\frac{\hbar^2}{2m}\Delta_i+ V({\bf r}_i)+\frac{e^2}{4\pi\... ..._i\vert}\right] \psi_{\nu_i}({\bf r}_i)= \varepsilon_i \psi_{\nu_i}({\bf r}_i).$$ (1.3)

The total wave function in this Hartree approximation is the simple product

$$\Psi_{\rm Hartree} ({\bf r}_1,\sigma_1;...;{\bf r}_N,\sigma_N) = \psi_{\nu_1}({\bf r}_1,\sigma_1)... \psi_{\nu_N}({\bf r}_N,\sigma_N).$$ (1.4)

Remarks:

• The Hartree equation Eq. (IV.1.3) is a set of $i=1,...,N$ non-linear coupled integro-differential equations.
• As the solutions $\psi_{\nu_i}$ of the equations appear again as terms (the Hartree potential) in the equations, these are called self-consistent equations. One way to solve them is by iteration: neglect the Hartree term first, find the solutions $\psi_{\nu_i}^{(0)}$, insert them in the Hartree potential, solve the new equations for $\psi_{\nu_i}^{(1)}$, insert these again, and so on until convergence is reached.
• The Pauli principle is not properly accounted for in this approach, as we do not have a Slater determinant but only a product wave function. This can be improved by the Hartree-Fock equations which we derive in the next section.