Fully anisotropic rigid molecules: Difference between revisions
Carl McBride (talk | contribs) (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...) |
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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> | 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 | ||
Revision as of 17:12, 10 July 2007
The fivefold dependence of the pair functions, 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)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)} , 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,
- 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)=\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) *}
where the orientations 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 \omega=(\phi,\theta,\chi)} , the Euler angles with respect to the axial line between molecular centers, 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 Y_{mn}^l (\omega)} is a generalized spherical harmonic and 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 \overline{m}=-m} . Inversion of this expression provides the coefficients
- 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_{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}
Note that by setting 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 n_1 = n_2= 0} , one has the coefficients for linear molecules.
