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The Hartree-Fock equations in the position representation are

$\displaystyle \left[ \hat{H_0} + {\sum_i \int d{\bf r'} \vert\psi_{\nu_{i}}({\bf r'})\vert^2 U(\vert{r-r'}\vert)} \right]
\psi_{\nu_{j}}({\bf r})$      
$\displaystyle - {\sum_i \int d{\bf r'} \psi_{\nu_{i}}^*({\bf r'}) U(\vert{r-r'}...
...t) \psi_{\nu_{j}}({\bf r'}) \psi_{\nu_{i}}({\bf r}) \delta_{\sigma_i
\sigma_j}}$ $\displaystyle =$ $\displaystyle \varepsilon_j \psi_{\nu_{j}}({\bf r}) .$ (4.2)

In this problem, we consider the case where the interaction is constant, \bgroup\color{col1}$ U$\egroup=const.

a) Simplify the Hartree-Fock equations by using the orthonormality of the \bgroup\color{col1}$ \psi_{\nu_{i}}$\egroup.

b) Thus solve the Hartree-Fock equations for the ground state wave function \bgroup\color{col1}$ \vert\Psi\rangle_{\rm HF}$\egroup explicitly.

c) Using b), calculate the Hartree-Fock ground state energy

$\displaystyle E_{\Psi}=\frac{1}{2}\sum_{i=1}^{N} \left[ \varepsilon_i+ \langle\nu_{i}\vert\hat{H}_0 \vert \nu_{i}\rangle\right].$     (4.3)

d) Compare the Hartree-Fock ground state energy \bgroup\color{col1}$ E_{\Psi}$\egroup with the exact ground state energy \bgroup\color{col1}$ E_0$\egroup from the exact solution of the previous problem and briefly discuss your result.



Tobias Brandes 2005-04-26