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Energy expressions

As before, the system is composed of point nuclei with coordinates $ \mathbf{R}$ and momenta $ \mathbf{P}$, and of electrons defined by spherical Gaussian wave packets with positions $ \mathbf{x}$, translational momenta $ \mathbf{p_{x}}$, sizes $ s$, and radial momenta $ p_{s}$:

$\displaystyle \Psi \propto \prod_{j} \,\exp\left[-\left(\frac{1}{s^{2}}-\frac{2...
...(\mathbf{r}-\mathbf{x})^{2}\right]\cdot\exp[i \mathbf{p_{x}} \cdot \mathbf{x}].$ (6.2)

Then the overall energy is a sum of the Hartree product kinetic energy, Hartree product electrostatic energy, and exchange and correlation energies:

$\displaystyle E = E_{ke} + E_{nuc \cdot nuc} + E_{nuc \cdot elec} + E_{elec \cdot elec} + E_{exch} + E_{corr}.
$

The electrostatic energy and kinetic energy expressions are the same as before:
$\displaystyle E_{ke}$ $\displaystyle =$ $\displaystyle \sum_{i} \frac{3}{2} \ \frac{1}{s_{i}^{2}}$  
$\displaystyle E_{nuc \cdot nuc}$ $\displaystyle =$ $\displaystyle \sum_{i<j} \frac{Z_{i} Z_{j}}{R_{ij}}$  
$\displaystyle E_{nuc \cdot elec}$ $\displaystyle =$ $\displaystyle -\sum_{i,j} \frac{Z_{i}}{R_{ij}}\ \mathrm{Erf}\left(\frac{\sqrt{2} \, R_{ij}}{s_{i}}\right)$  
$\displaystyle E_{elec \cdot elec}$ $\displaystyle =$ $\displaystyle \sum_{i<j} \frac{1}{x_{ij}}\ \mathrm{Erf}\left( \frac{\sqrt{2} \, x_{ij}}{\sqrt{s_{i}^{2} + s_{j}^{2}}}\right).$  

We define an exchange energy as a pairwise sum over same-spin electrons, and a correlation energy as a pairwise sum over opposite-spin electrons. For the uniform electron gas, we use exchange and correlation functions defined as:
    $\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2}\ \frac{a_{exch} S_{ij}^{2}}{1 - S_{ij}^{2}} \cdot (t_{11} + t_{22})$  
    $\displaystyle E_{corr} = \sum_{\sigma_{i} \neq \sigma_{j}} \frac{-a_{corr}}{1 + b_{corr} s_{avg}} \cdot S_{ij}^{1/2}$  

where the parameters are $ a_{exch} = 1/2$, $ a_{corr} = 0.111283$ hartrees, $ b_{corr} = 0.110253 \mathrm{bohr^{-1}}$; and the kinetic energy sum $ t_{11} + t_{22}$ and average electron size $ s_{avg}$ are defined as
    $\displaystyle t_{11} + t_{22} = \frac{3}{2}\left(\frac{1}{s_{1}^{2}}+\frac{1}{s_{2}^{2}}\right)$  
    $\displaystyle s_{avg} = \sqrt{(s_{1}^{2} + s_{2}^{2})/2}.$  

For systems with nuclei, we use a modified exchange function:

$\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2}\ \frac{S_{i...
...c_{exch} \right) (t_{11} + t_{22}) - c_{exch} \ \frac{2 t_{12}}{S_{ij}}\right)
$

where $ f_{size} = s_{1}/s_{2} + s_{2}/s_{1} - 2$, and we set the parameters to be $ a_{exch} = 0.4$, $ b_{exch} = 0.15$, and $ c_{exch} = 1$. We use as the correlation function the uniform electron gas correlation function multiplied by three.


next up previous contents
Next: On the exchange and Up: Development of an electron Previous: Introduction   Contents
Julius 2008-04-29