Intermolecular pair potential: Difference between revisions
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The '''intermolecular pair potential''' is a widely used approximation. Real intermolecular interactions consist of two-body interactions, three-body interactions, four-body interactions etc. However, the calculation of even three-body interactions is computationally time consuming, and the calculation of only two-body interactions is frequent. | |||
Such "effective" pair potentials often include the higher order interactions implicitly. | |||
Naturally the interaction potential between atoms or molecules remains unchanged regardless of where one is in the phase diagram, be it the low temperature solid, or a high temperature gas. However, when one only uses two-body interactions such 'transferability' is lost, and one may well need to modify the the potential or the parameters of the potential if one is studying a hot gas or a cooler high density liquid. | |||
==Axially symmetric molecules== | |||
In general, the [[intermolecular pair potential]] for axially symmetric molecules, <math>\Phi_{12} </math>, is a function of five coordinates: | In general, the [[intermolecular pair potential]] for axially symmetric molecules, <math>\Phi_{12} </math>, is a function of five coordinates: | ||
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:<math>\left. \Phi_{12} \right. = 4\pi \sum_{L_1 L_2 m} L_1 L_2 m (r) Y_{L_1}^m (\theta_1, \phi_1) Y_{L_2}^m * (\theta_2, \phi_2)</math>, | :<math>\left. \Phi_{12} \right. = 4\pi \sum_{L_1 L_2 m} L_1 L_2 m (r) Y_{L_1}^m (\theta_1, \phi_1) Y_{L_2}^m * (\theta_2, \phi_2)</math>, | ||
where <math> | where <math>Y_L^m(\theta, \phi)</math> are the [[spherical harmonics]]. | ||
==See also== | |||
*[[Idealised models]] | |||
==References== | ==References== | ||
#[http://dx.doi.org/10.1098/rspa.1954.0044 J. A. Pople "The Statistical Mechanics of Assemblies of Axially Symmetric Molecules. I. General Theory", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''221''' pp. 498-507 (1954)] | #[http://dx.doi.org/10.1098/rspa.1954.0044 J. A. Pople "The Statistical Mechanics of Assemblies of Axially Symmetric Molecules. I. General Theory", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''221''' pp. 498-507 (1954)] | ||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
Latest revision as of 15:55, 10 February 2010
The intermolecular pair potential is a widely used approximation. Real intermolecular interactions consist of two-body interactions, three-body interactions, four-body interactions etc. However, the calculation of even three-body interactions is computationally time consuming, and the calculation of only two-body interactions is frequent. Such "effective" pair potentials often include the higher order interactions implicitly. Naturally the interaction potential between atoms or molecules remains unchanged regardless of where one is in the phase diagram, be it the low temperature solid, or a high temperature gas. However, when one only uses two-body interactions such 'transferability' is lost, and one may well need to modify the the potential or the parameters of the potential if one is studying a hot gas or a cooler high density liquid.
Axially symmetric molecules[edit]
In general, the intermolecular pair potential for axially symmetric molecules, 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} } , is a function of five coordinates:
- 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 \left. \Phi_{12} \right. = \Phi_{12}(r, \theta_1, \phi_1, \theta_2, \phi_2) }
The angles 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 \theta_i} and can be considered to be polar angles, with the intermolecular vector, 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} , as the common polar axis. Since the molecules are axially symmetric, the angles 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 \psi_i} do not influence 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 \Phi_{12} } . A very powerful expansion of this pair potential is due to Pople (Ref. 1 Eq. 2.1):
- 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 \left. \Phi_{12} \right. = 4\pi \sum_{L_1 L_2 m} L_1 L_2 m (r) Y_{L_1}^m (\theta_1, \phi_1) Y_{L_2}^m * (\theta_2, \phi_2)} ,
where are the spherical harmonics.
