Intermolecular pair potential - Revision history

Carl McBride: added a little more.

2010-02-10T14:55:17Z

added a little more.

← Older revision		Revision as of 16:55, 10 February 2010
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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.		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.		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==		==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:

Carl McBride: Started introduction

2010-02-10T14:45:58Z

Started introduction

← Older revision		Revision as of 16:45, 10 February 2010
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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.
	==Axially symmetric molecules==		==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:

Carl McBride at 14:48, 14 August 2007

2007-08-14T14:48:22Z

← Older revision		Revision as of 16:48, 14 August 2007
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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>~~Y_{L~~ m}(\theta, \phi)</math> are the [[spherical harmonics]].		where <math>Y_L^m(\theta, \phi)</math> are the [[spherical harmonics]].

	==See also==		==See also==

Carl McBride at 09:48, 30 July 2007

2007-07-30T09:48:36Z

← Older revision		Revision as of 11:48, 30 July 2007
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	where <math>Y_{L m}(\theta, \phi)</math> are the [[spherical harmonics]].		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]]

Carl McBride at 11:30, 7 June 2007

2007-06-07T11:30:50Z

← Older revision		Revision as of 13:30, 7 June 2007
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			==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:

Carl McBride: Axially symmetric molecules moved to Intermolecular pair potential

2007-06-07T11:29:56Z

Axially symmetric molecules moved to Intermolecular pair potential

← Older revision	Revision as of 13:29, 7 June 2007
(No difference)

Carl McBride: New page: In general, the intermolecular pair potential for axially symmetric molecules, $\Phi_{12}$ , is a function of five coordinates: : $\left. \Phi_{12} \right. = \Phi_{12}(...$

2007-06-07T11:29:23Z

New page: In general, the intermolecular pair potential for axially symmetric molecules, <math>\Phi_{12} </math>, is a function of five coordinates: :<math>\left. \Phi_{12} \right. = \Phi_{12}(...

New page

In general, the [[intermolecular pair potential]] for axially symmetric molecules, <math>\Phi_{12} </math>, is a function of five coordinates:

:<math>\left. \Phi_{12} \right. = \Phi_{12}(r, \theta_1, \phi_1, \theta_2, \phi_2) </math>

The angles <math>\theta_i</math> and <math>\phi_i</math> can be considered to be polar angles, with the
intermolecular vector, <math>r</math>, as the common polar axis. Since the molecules
are axially symmetric, the angles <math>\psi_i</math> do not influence the value of <math>\Phi_{12} </math>.
A very powerful expansion of this pair potential is due to Pople (Ref. 1 Eq. 2.1):

:<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>Y_{L m}(\theta, \phi)</math> are the [[spherical harmonics]].
==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)]
[[category: statistical mechanics]]