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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 \bgroup\color{col1}$ i$\egroup at position \bgroup\color{col1}$ {\bf r}_i$\egroup experiences.

In the Hartree approach one assumes that particle \bgroup\color{col1}$ j$\egroup is described by a wave function (spin orbital) \bgroup\color{col1}$ \psi_{\nu_j}(\xi_j)$\egroup with orbital part \bgroup\color{col1}$ \psi_{\nu_j}({\bf r}_j)$\egroup, and the statistics (anti-symmetrization of all the total \bgroup\color{col1}$ N$\egroup-particle wave function for Fermions, symmetrizatin for Bosons) is neglected. In the following, we discuss electrons.

For electrons interacting via the Coulomb interaction \bgroup\color{col1}$ U(r)=e^2/4\pi\varepsilon_0r$\egroup, the potential seen by an electron \bgroup\color{col1}$ i$\egroup at position \bgroup\color{col1}$ {\bf r}_i$\egroup is given by

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

This is the sum over the potentials generated by all other electrons \bgroup\color{col1}$ j\ne i$\egroup which have a charge density \bgroup\color{col1}$ -e\vert\psi_j({\bf r}_j)\vert^2$\egroup. The corresponding potential energy for electron \bgroup\color{col1}$ i$\egroup is \bgroup\color{col1}$ -eV_{\rm H}({\bf r}_i) $\egroup, and therefore one describes electron \bgroup\color{col1}$ i$\egroup by an effective single particle Hamiltonian,
$\displaystyle H_{\rm Hartree}^{(i)}$ $\displaystyle =$ $\displaystyle H_0^{(i)}+ V_{\rm Hartree}({\bf r}_i)$  
  $\displaystyle =$ $\displaystyle -\frac{\hbar^2}{2m}\Delta_i+ V({\bf r}_i)+\frac{e^2}{4\pi\varepsi...
...r}_j
\frac{\vert\psi_{\nu_j}({\bf r}_j)\vert^2}{\vert{\bf r}_j-{\bf r}_i\vert},$ (1.2)

where \bgroup\color{col1}$ V({\bf r}_i)$\egroup is the usual potential energy due to the interaction with the nucleus. The corresponding Schrödinger equations for the orbital wave functions \bgroup\color{col1}$ \psi_{\nu_i}$\egroup for electron \bgroup\color{col1}$ i$\egroup are
$\displaystyle \left[ -\frac{\hbar^2}{2m}\Delta_i+ V({\bf r}_i)+\frac{e^2}{4\pi\...
..._i\vert}\right] \psi_{\nu_i}({\bf r}_i)=
\varepsilon_i \psi_{\nu_i}({\bf r}_i).$     (1.3)

The total wave function in this Hartree approximation is the simple product
$\displaystyle \Psi_{\rm Hartree} ({\bf r}_1,\sigma_1;...;{\bf r}_N,\sigma_N)
= \psi_{\nu_1}({\bf r}_1,\sigma_1)... \psi_{\nu_N}({\bf r}_N,\sigma_N).$     (1.4)

Remarks:


next up previous contents index
Next: Angular Average, Shells, and Up: The Hartree Equations, Atoms, Previous: The Hartree Equations, Atoms,   Contents   Index
Tobias Brandes 2005-04-26