Scemama at 12:54, 23 October 2009

2009-10-23T12:54:38Z

Scemama at 12:54, 23 October 2009

2009-10-23T12:54:20Z

New page

The Electron Pair Localization Function is a function defined in the three-dimensional space. It measures the degree of pairing of electrons in a molecule, with an increasing value as the electron pairing increases. Therefore chemical bonds, core domains and lone pairs can be visualized.

The EPLF [1] has been designed to describe local electron pairing in molecular systems.
It is defined as a scalar function defined in the three-dimensional space and taking
its values in the [-1,1] range. It is defined as follows:

<math>
{\rm EPLF}(\vec{r}) =
\frac { d_{\sigma \sigma} (\vec{r})
- d_{\sigma {\bar \sigma}} (\vec{r}) }
{ d_{\sigma \sigma} (\vec{r})
+ d_{\sigma {\bar \sigma}} (\vec{r}) }
</math>

where the quantity <math>d_{\sigma \sigma} (\vec{r}) </math> [resp. <math>
d_{\sigma {\bar \sigma}} (\vec{r}) </math>] denotes the quantum-mechanical average of the distance between an
electron of spin <math> \sigma </math> located at <math>\vec{r}</math> and the closest electron of same spin (resp., of opposite spin <math>\bar{\sigma}</math>).

The mathematical definition of these quantities can be written as

<math>
d_{\sigma \sigma}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r_N})
\left[ \sum_{i=1}^N \delta(\vec{r}-\vec{r}_i)
\min_{j\neq i;\sigma_j=\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots d\vec{r}_N
</math>

<math>
d_{\sigma {\bar \sigma}}(\vec{r}) = \int \Psi^2(\vec{r}_1,\dots,\vec{r}_N)
\left[ \sum_{i=1}^N \delta(\vec{r}-\vec{r}_i) \min_{j\ne i;\sigma_j\neq\sigma_i}|\vec{r}_i - \vec{r}_j| \right] d\vec{r}_1 \dots d\vec{r}_N
</math>

where <math>\sigma</math> is the spin (<math>\alpha</math> or <math>\beta</math>),
<math>\bar{\sigma}</math> is the spin opposite to <math>\sigma</math>,
<math>\Psi(\vec{r}_1,\dots,\vec{r}_N)</math> is the wave function, and <math>N</math>
is the number of electrons.

In a region of space, if the shortest distance separating anti-parallel
electrons is smaller than the shortest distance separating electrons of same
spin, the EPLF takes positive values and indicates pairing of anti-parallel
electrons. In contrast, if the shortest distance separating anti-parallel electrons is
larger than the shortest distance separating electrons of same spin, the EPLF
takes negative values and indicates pairing of parallel electrons (which in practice
never happens). If the shortest distance separating anti-parallel
electrons is equivalent to the shortest distance separating electrons of same
spin, the EPLF takes values close to zero and indicates no electron pairing.

The original formulation of EPLF is extremely easy to compute in the quantum
Monte Carlo framework. However, it is not possible to compute it analytically due to
the presence of the ''min'' function in the definitions of <math> d_{\sigma
\sigma} </math> and <math> d_{\sigma {\bar \sigma}} </math>.

----
[1] [http://dx.doi.org/10.1063/1.1765098 '''Electron pair localization function, a practical tool to visualize electron localization in molecules from quantum Monte Carlo data''']<br> A. Scemama, P. Chaquin, M. Caffarel, J. Chem. Phys., vol 121, pp. 1725-1735 (2004)

← Older revision		Revision as of 12:54, 23 October 2009
Line 51:		Line 51:

	----		----
−	[1] [http://dx.doi.org/10.1063/1.1765098 '''Electron pair localization function, a practical tool to visualize electron localization in molecules from quantum Monte Carlo data''']<br> A. Scemama, P. Chaquin, M. Caffarel, J. Chem. Phys., vol 121, pp. 1725-1735 (2004)	+	[1] [http://dx.doi.org/10.1063/1.1765098 '''Electron pair localization function, a practical tool to visualize electron localization in molecules from quantum Monte Carlo data''']<br> A. Scemama, P. Chaquin, M. Caffarel<br>J. Chem. Phys., vol 121, pp. 1725-1735 (2004)

The Electron Pair Localization Function - Revision history

Scemama at 12:54, 23 October 2009

Scemama at 12:54, 23 October 2009