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Energy expression

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}$:

$\displaystyle \Psi \propto \prod_{j} \,\exp\left[-\left(\frac{1}{s^{2}}-\frac{2...
...(\mathbf{r}-\mathbf{x})^{2}\right]\cdot\exp[i \mathbf{p_{x}} \cdot \mathbf{x}].$ (5.1)

The overall energy is a sum of the Hartree product kinetic energy, Hartree product electrostatic energy, and antisymmetrization (Pauli) energy:

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

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

There are two major changes from the old eFF. First, we divide electrons into core and valence electrons, and assume they do not switch categories over the course of the simulation. We assume an electron is a core electron if $ s_{i} < 1.5 \cdot s_{core}$ and $ \vert x_{i} - \mathbf{R_{nuc}}\vert < 0.5 \cdot s_{core}$, where

$\displaystyle s_{core} = \frac{3 \sqrt{\pi}}{2 \sqrt{2} Z_{nuc} - 1}
$

is the size of a helium-like ion optimized with eFF.

Second, we make the kinetic and Pauli energies depend on the hybridization of an electron, which can vary during the course of the simulation, and depends on the proximity of the electron to the nuclei present in the system. We have

    $\displaystyle E_{ke} = \sum_{i} \frac{3}{2} \ \frac{1}{s_{i}^{2}} \cdot f_{ke}$  
    $\displaystyle E_{Pauli} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2} \frac{S_{ij}}{1-S_{ij}^{2}} \Delta T_{ij} \cdot (f_{repel} + f_{switch})$  

where, as before, $ \Delta T_{ij}$ is a measure of the kinetic energy change upon antisymmetrization, and $ S_{ij}$ is the overlap between two wave packets:
    $\displaystyle \Delta T_{ij} = \frac{3}{2}\left(\frac{1}{s_{1}^{2}}+\frac{1}{s_{...
...t) - \frac{2 (3 (s_{1}^{2}+s_{2}^{2})-2 x_{12}^{2})}{(s_{1}^{2}+s_{2}^{2})^{2}}$  
    $\displaystyle S_{ij} = \left(\frac{2}{s_{i}/s_{j}+s_{j}/s_{i}}\right)^{3/2} \exp(-x_{ij}^{2}/(s_{i}^{2}+s_{j}^{2}))$  

and the factors $ f_{KE}$, $ f_{repel}$, and $ f_{switch}$ are defined as follows:
    $\displaystyle f_{ke} = c_{s-ke} \chi_{ke} + c_{p-ke} (1 - \chi_{ke})$  
    $\displaystyle f_{repel} = \left[c_{repel} + c_{size} \left( \frac{s_{2}}{s_{1}} + \frac{s_{1}}{s_{2}} - 2 \right)\right]\cdot \frac{S_{ij}}{1-S_{ij}^{2}}$  
    $\displaystyle f_{switch} =\left[c_{s-Pauli} \chi_{Pauli} + c_{p-Pauli} (1 - \chi_{Pauli})\right] \cdot (1 - S_{ij})$  

which depend on the hybridization variables $ \chi_{ke}$ and $ \chi_{Pauli}$, which equal one for an electron with $ s$ character only, and zero for an electron with $ p$ character only. Hybridization is a function of an electron's position relative to the protons and cores (nuclei with core electrons on top of them) in the system. We assume that all nuclei with $ Z>2$ have $ 1s^{2}$ core electrons on top of them. The expression for $ \chi_{ke}$ is
$\displaystyle \chi_{ke}$ $\displaystyle =$ $\displaystyle \prod_{protons} \zeta_{proton}\left(\frac{\vert x_{i} - R_{proton...
...ores} \zeta_{core}\left(\frac{\vert x_{i} - R_{core}\vert}{s_{i}}\right)\right]$  
    $\displaystyle \mathrm{if}\ i,j\ \in \mathrm{valence\ electrons,\ 1\ otherwise}$  

while the expression for $ \chi_{Pauli}$ is
$\displaystyle \chi_{Pauli}$ $\displaystyle =$ $\displaystyle \prod_{protons} \zeta_{proton}\left(\frac{\vert x_{i} - R_{proton...
...t) \cdot
\zeta_{proton}\left(\frac{\vert x_{j} - R_{proton}\vert}{s_{j}}\right)$  
    $\displaystyle \cdot \left[1 - \prod_{cores} \zeta_{core}\left(\frac{\vert x_{i}...
...ac{\vert x_{j} - R_{core}\vert}{s_{j}}\right) \sin^{2}\theta_{i-core-j}
\right]$  
    $\displaystyle \mathrm{if}\ i,j\ \in \mathrm{valence\ electrons,\ 1\ otherwise}.$  

The parameters $ c_{repel}$, $ c_{size}$, $ c_{s-Pauli}$, $ c_{p-Pauli}$, $ c_{s-ke}$, $ c_{p-ke}$ are shown in Table 5.1.


Table 5.1: Parameters in the new eFF, in addition to splines in Table 5.2.
Parameter Value Purpose
$ c_{repel}$ 0.5 Prevents electron coalescence
$ c_{size}$ 3 Nearby electrons tend to match size
$ c_{s-Pauli}$ 1 Pauli is repulsive for s-like electrons
$ c_{p-Pauli}$ -1 Pauli is attractive for p-like electrons
$ c_{s-ke}$ 1 No change in kinetic energy for s-like electrons
$ c_{p-ke}$ 1.2 Slightly larger kinetic energy for p-like electrons


The functions $ \zeta_{proton}$ and $ \zeta_{core}$ determine how an electron's $ p$ versus $ s$ character varies with their distance from the protons and cores in the system. They are defined as piecewise quintic splines specified so that the function's value, first, and second derivatives match at the points given:

$\displaystyle \{x_{1}, f(x_{1}), f'(x_{1}), f''(x_{1})\} \ldots \{x_{N}, f(x_{N}), f'(x_{N}), f''(x_{N})\}.
$

Written in terms of the boundary conditions and matching points, we have
    $\displaystyle \zeta_{proton}(r) = \left\{\begin{array}{ll}\mathrm{spline}\left(...
... \right) & \mbox{if $r < 2.5$} \\ 1 & \mbox{if $r \geq 2.5$} \end{array}\right.$  
    $\displaystyle \zeta_{core}(r) = \left\{\begin{array}{ll}\mathrm{spline}\left(\b...
...right) & \mbox{if $r < 1.5$} \\ 0 & \mbox{if $r \geq 1.5$} \end{array}\right. .$  

Explicit polynomial expressions for $ \zeta_{proton}$ and $ \zeta_{core}$ are given in Table 5.2.


Table 5.2: Polynomial coefficients for quintic splines, $ \zeta = \sum_{i} c_{i} r^{i}$.
  $ \zeta_{proton}$   $ \zeta_{core}$
  $ r<2$ $ r>2$   $ r<0.6$ $ 0.6<r<1.5$ $ r>1.5$
$ c_{0}$ 0.500000 1.000000   0.000000 -8.487654 0.000000
$ c_{1}$ 0.550000 0.000000   0.000000 49.897119 0.000000
$ c_{2}$ 0.000000 0.000000   6.944444 -100.222908 0.000000
$ c_{3}$ -0.208000 0.000000   0.000000 97.165066 0.000000
$ c_{4}$ 0.089600 0.000000   -19.290123 46.677336 0.000000
$ c_{5}$ -0.011520 0.000000   12.860082 8.890921 0.000000



next up previous contents
Next: Explanation of the energy Up: Methods Previous: Methods   Contents
Julius 2008-04-29