9-6 Lennard-Jones potential: Difference between revisions

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(Corrected equation (r_m not sigma))
(Added Eq in terms of sigma)
 
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:<math> \Phi_{12}(r) = \epsilon  \left[ 2\left(\frac{r_m}{r} \right)^{9} -  3\left( \frac{r_m}{r}\right)^6 \right] </math>
:<math> \Phi_{12}(r) = \epsilon  \left[ 2\left(\frac{r_m}{r} \right)^{9} -  3\left( \frac{r_m}{r}\right)^6 \right] </math>
or in terms of <math> \sigma </math> (the value of <math>r</math> at which <math> \Phi_{12}(r)=0</math>) one has:
:<math> \Phi_{12}(r) = 6.75 \epsilon  \left[ \left(\frac{\sigma}{r} \right)^{9} -  \left( \frac{\sigma}{r}\right)^6 \right] </math>


where
where
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles or ''sites''
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles or ''sites''
* <math> r_m </math> is the  distance, <math>r</math>, at which <math> \Phi_{12}(r)</math> is a minimum.
* <math> r_m </math> is the  distance, <math>r</math>, at which <math> \Phi_{12}(r)</math> is a minimum, which corresponds to <math> r_m =    1.5^{1/3} \sigma</math>.
* <math> \epsilon </math> is the well depth (energy)
* <math> \epsilon </math> is the well depth (energy)


Latest revision as of 13:41, 4 February 2014

The 9-6 Lennard-Jones potential (also known as the 6-9 potential) is a variant the more well known Lennard-Jones model. It is used for computing non-bonded interactions. The potential is given by [1] :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) = \epsilon \left[ 2\left(\frac{r_m}{r} \right)^{9} - 3\left( \frac{r_m}{r}\right)^6 \right] }

or in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } (the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} at which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)=0} ) one has:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) = 6.75 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{9} - \left( \frac{\sigma}{r}\right)^6 \right] }

where

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r) } is the intermolecular pair potential between two particles or sites
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_m } is the distance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , at which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)} is a minimum, which corresponds to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_m = 1.5^{1/3} \sigma} .
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } is the well depth (energy)

It is worth noting that the inclusion of an odd power (here the 9) adds an additional computational overhead, and the 8-6 Lennard-Jones potential has been suggested as a viable alternative.

References[edit]

  1. ↑ Arieh Warshel and Shneior Lifson "Consistent Force Field Calculations. II. Crystal Structures, Sublimation Energies, Molecular and Lattice Vibrations, Molecular Conformations, and Enthalpies of Alkanes", Journal of Chemical Physics 53 pp. 582-594 (1970)
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