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 $p_{s}$:

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

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

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

The electrostatic energy and kinetic energy expressions are the same as before:

$$E_{ke} = \sum_{i} \frac{3}{2} \ \frac{1}{s_{i}^{2}}$$

$$E_{nuc \cdot nuc} = \sum_{i<j} \frac{Z_{i} Z_{j}}{R_{ij}}$$

$$E_{nuc \cdot elec} = -\sum_{i,j} \frac{Z_{i}}{R_{ij}}\ \mathrm{Erf}\left(\frac{\sqrt{2} \, R_{ij}}{s_{i}}\right)$$

$$E_{elec \cdot elec} = \sum_{i<j} \frac{1}{x_{ij}}\ \mathrm{Erf}\left( \frac{\sqrt{2} \, x_{ij}}{\sqrt{s_{i}^{2} + s_{j}^{2}}}\right).$$

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:

$$E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2}\ \frac{a_{exch} S_{ij}^{2}}{1 - S_{ij}^{2}} \cdot (t_{11} + t_{22})$$

$$E_{corr} = \sum_{\sigma_{i} \neq \sigma_{j}} \frac{-a_{corr}}{1 + b_{corr} s_{avg}} \cdot S_{ij}^{1/2}$$

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

$$t_{11} + t_{22} = \frac{3}{2}\left(\frac{1}{s_{1}^{2}}+\frac{1}{s_{2}^{2}}\right)$$

$$s_{avg} = \sqrt{(s_{1}^{2} + s_{2}^{2})/2}.$$

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

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

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.