Fully anisotropic rigid molecules - Revision history

Carl McBride: Added a recent publication

2012-03-16T11:02:07Z

Added a recent publication

← Older revision		Revision as of 13:02, 16 March 2012
Line 1:		Line 1:
	The fivefold dependence of the pair functions, <math>\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)</math>, for liquids of rigid, fully anisotropic molecules makes these equations excessively complex for numerical work ~~(see Ref~~. 1).		The fivefold dependence of the pair functions, <math>\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)</math>, for liquids of rigid, fully anisotropic molecules makes these equations excessively complex for numerical work <ref>[http://dx.doi.org/10.1063/1.469615 F. Lado, E. Lomba and M. Lombardero "Integral equation algorithm for fluids of fully anisotropic molecules", Journal of Chemical Physics '''103''' pp. 481-484 (1995)]</ref>.
	The first and essential ingredient for their reduction is a spherical harmonic		The first and essential ingredient for their reduction is a spherical harmonic
	expansion of the correlation functions,		expansion of the correlation functions,
Line 17:		Line 17:
	<math>\Phi_{l_1 l_2 m}^{00}(r_{12})</math> for linear molecules.		<math>\Phi_{l_1 l_2 m}^{00}(r_{12})</math> for linear molecules.
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.~~469615 F. Lado, E~~. ~~Lomba~~ and M. ~~Lombardero~~ "~~Integral equation~~ algorithm for ~~fluids of fully anisotropic molecules~~", Journal of Chemical Physics '''~~103~~''' ~~pp. 481-484~~ (~~1995~~)]		<references/>
			;Related reading
			*[http://dx.doi.org/10.1063/1.3693623 R. Ishizuka and N. Yoshida "Application of efficient algorithm for solving six-dimensional molecular Ornstein-Zernike equation", Journal of Chemical Physics '''136''' 114106 (2012)]
	[[category: integral equations]]		[[category: integral equations]]

Nice and Tidy at 15:12, 10 July 2007

2007-07-10T15:12:13Z

← Older revision		Revision as of 17:12, 10 July 2007
Line 6:		Line 6:

	where the orientations <math>\omega=(\phi,\theta,\chi)</math>, the [[Euler angles]] with respect		where the orientations <math>\omega=(\phi,\theta,\chi)</math>, the [[Euler angles]] with respect
	to the axial line <math>r_{12}</math> between molecular centers, <math>Y_{mn}^l (\omega)</math>		to the axial line <math>{\mathbf r}_{12}</math> between molecular centers, <math>Y_{mn}^l (\omega)</math>
	is a [[Spherical harmonics \| generalized spherical harmonic]] and <math>\overline{m}=-m</math>.		is a [[Spherical harmonics \| generalized spherical harmonic]] and <math>\overline{m}=-m</math>.
	Inversion of this expression provides the coefficients		Inversion of this expression provides the coefficients

Carl McBride: New page: The fivefold dependence of the pair functions, $\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)$ , for liquids of rigid, fully anisotropic molecules makes the...

2007-05-31T11:09:43Z

New page: The fivefold dependence of the pair functions, <math>\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)</math>, for liquids of rigid, fully anisotropic molecules makes the...

New page

The fivefold dependence of the pair functions, <math>\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)</math>, for liquids of rigid, fully anisotropic molecules makes these equations excessively complex for numerical work (see Ref. 1).
The first and essential ingredient for their reduction is a spherical harmonic
expansion of the correlation functions,

:<math>\Phi(12)=\sum_{l_1 l_2 m n_1 n_2} [(2l_1 +1)(2l_2 +1)]^{1/2} \Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12}) Y_{mn_1}^{l_1}(\omega_1) * Y_{\overline{m}n_2}^{l_2}(\omega_2) *</math>

where the orientations <math>\omega=(\phi,\theta,\chi)</math>, the [[Euler angles]] with respect
to the axial line <math>r_{12}</math> between molecular centers, <math>Y_{mn}^l (\omega)</math>
is a [[Spherical harmonics | generalized spherical harmonic]] and <math>\overline{m}=-m</math>.
Inversion of this expression provides the coefficients

:<math>\Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12})= \frac{[(2l_1 +1)(2l_2 +1)]^{1/2}}{64 \pi^4}
\int \Phi(12) Y_{mn_1}^{l_1}(\omega_1) Y_{\overline{m}n_2}^{l_2}(\omega_2)
~{\rm d}\omega_1 {\rm d} \omega_2</math>

Note that by setting <math>n_1 = n_2= 0</math>, one has the coefficients
<math>\Phi_{l_1 l_2 m}^{00}(r_{12})</math> for linear molecules.
==References==
#[http://dx.doi.org/10.1063/1.469615 F. Lado, E. Lomba and M. Lombardero "Integral equation algorithm for fluids of fully anisotropic molecules", Journal of Chemical Physics '''103''' pp. 481-484 (1995)]
[[category: integral equations]]

Fully anisotropic rigid molecules - Revision history

Carl McBride: Added a recent publication

Nice and Tidy at 15:12, 10 July 2007

Carl McBride: New page: The fivefold dependence of the pair functions, \Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2), for liquids of rigid, fully anisotropic molecules makes the...

Carl McBride: New page: The fivefold dependence of the pair functions, $\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)$ , for liquids of rigid, fully anisotropic molecules makes the...