Computational implementation of integral equations: Difference between revisions

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Integral equations are solved numerically.
Integral equations are solved numerically.
One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
and a closure relation, <math>c_2 (12)</math> (which
and a [[closure relations | closure relation]], <math>c_2 (12)</math> (which
incorporates the [[bridge function]] <math>B(12)</math>).
incorporates the [[bridge function]] <math>B(12)</math>).
The numerical solution is iterative;  
The numerical solution is iterative;  
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Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration.
Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration.
Here are the four steps used to solve integral equations:
Here are the four steps used to solve integral equations:
===1. Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>===
===Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>===
(Note: for linear fluids <math>\mu = \nu =0</math>)
(Note: for linear fluids <math>\mu = \nu =0</math>)


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Thus  
Thus  
:<math>\gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)</math>.
:<math>\left. \gamma(12) \right. =\gamma (r,x_1x_2,y,z_1z_2)</math>.


====Evaluate====
====Evaluate====
Evaluations of  <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>
Evaluations of  <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>
where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math>
where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials |Legendre polynomial]] <math>P_\nu(cos \theta)</math>
where <math>y_j</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
where <math>y_j</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomials |Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
and where <math>z_{1_k},z_{2_k}</math>  are the  <math>\nu</math> roots of the [[Chebyshev polynomial]]
and where <math>z_{1_k},z_{2_k}</math>  are the  <math>\nu</math> roots of the Chebyshev polynomial
<math>T_{\nu}(\ cos \chi)</math>
<math>T_{\nu}(\ cos \chi)</math>
thus
thus
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where
where


:<math>\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)</math>
:<math>\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)</math>


where <math>d_{s \mu}^m(\theta)</math> is the angular, <math>\theta</math>, part of the
where <math>d_{s \mu}^m(\theta)</math> is the angular, <math>\theta</math>, part of the
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and
and


:<math>e_s(y)=\exp(is\phi)</math>
:<math>\left. e_s(y) \right.=\exp(is\phi)</math>


 
:<math>\left. e_{\mu}(z) \right.= \exp(i\mu \chi)</math>
:<math>e_{\mu}(z)= \exp(i\mu \chi)</math>


For the limits in the summations
For the limits in the summations


:<math>L_1= \max (s,\nu_1)</math>
:<math>\left. L_1 \right.= \max (s,\nu_1)</math>


:<math>L_2= \max (s,\nu_2)</math>
:<math>\left. L_2 \right.= \max (s,\nu_2)</math>


The above equation constitutes a separable five-dimensional transform. To rapidly evaluate
The above equation constitutes a separable five-dimensional transform. To rapidly evaluate
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Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math>
Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math>
Use [[Gauss-Chebyshev  quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math>
Use [[Gauss-Chebyshev  quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math>.
thus
Thus


:<math>c_{mns}^{\mu \nu} (r) = w^3  
:<math>c_{mns}^{\mu \nu} (r) = w^3  
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:<math>w=\frac{1}{NG}</math>
:<math>w=\frac{1}{NG}</math>


===Perform FFT from Real to Fourier space<math>c_{mns}^{\mu \nu} (r) \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k)</math>====
===Perform FFT from Real to Fourier space <math>c_{mns}^{\mu \nu} (r) \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k)</math>===


This is non-trivial and is undertaken in three steps:
This is non-trivial and is undertaken in three steps:
#Conversion from axial reference frame to spatial reference frame, ''i.e.''
 
====Conversion from axial reference frame to spatial reference frame====


:<math>c_{mns}^{\mu \nu} (r)  \rightarrow  c_{\mu \nu}^{mnl} (r)</math>
:<math>c_{mns}^{\mu \nu} (r)  \rightarrow  c_{\mu \nu}^{mnl} (r)</math>


this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}:
this is done using the Blum transformation (Refs 7, 8 and 9):


:<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left(  
:<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left(  
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\right)g_{mns}^{\mu \nu} (r)</math>
\right)g_{mns}^{\mu \nu} (r)</math>


#'''Fourier-Bessel Transforms''': <math>c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)</math>
====Fourier-Bessel Transforms====
:<math>c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)</math>


:<math>\tilde{c}_{\mu \nu}^{mnl} (k; l_1 l_2 l n_1 n_2) = 4\pi i^l \int_0^{\infty}  c_{\mu \nu}^{mnl} (r; l_1 l_2 l n_1 n_2) J_l (kr) ~r^2 {\rm d}r</math>
:<math>\tilde{c}_{\mu \nu}^{mnl} (k; l_1 l_2 l n_1 n_2) = 4\pi i^l \int_0^{\infty}  c_{\mu \nu}^{mnl} (r; l_1 l_2 l n_1 n_2) J_l (kr) ~r^2 {\rm d}r</math>


(see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}),
(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),
where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.
where <math>J_l(x)</math> is a [[Bessel functions |Bessel function]] of order <math>l</math>.
`step-down' operations can be performed by way of sin and cos operations
`step-down' operations can be performed by way of sin and cos operations
of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado  \cite{MP_1982_47_0283}.
of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado  Ref. 3.
The  Fourier-Bessel transform is also known as a '''Hankel transform'''.
The  Fourier-Bessel transform is also known as a '''Hankel transform'''.
It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.
It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.


<math>g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r</math>
:<math>g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r</math>




<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>
:<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>


 
====Conversion from the spatial reference frame back to the  axial reference frame====
#Conversion from the spatial reference frame back to the  axial reference frame
:<math>\tilde{c}_{\mu \nu}^{mnl} (k)  \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k) </math>
''i.e.''
 
<math>\tilde{c}_{\mu \nu}^{mnl} (k)  \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k)  
</math>
this is done using the Blum transformation
this is done using the Blum transformation


<math>g_{mns}^{\mu \nu} (r)
:<math>g_{mns}^{\mu \nu} (r)
= \sum_{l=|m-n|}^{m+n} \left(  
= \sum_{l=|m-n|}^{m+n} \left(  
\begin{array}{ccc}
\begin{array}{ccc}
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g_{\mu \nu}^{mnl}(r)</math>  
g_{\mu \nu}^{mnl}(r)</math>  


===OZ Equation=== <math>\tilde{c}_{mns}^{\mu \nu} (k)  \rightarrow  \tilde{\gamma}_{mns}^{\mu \nu} (k)</math>===
===Ornstein-Zernike relation <math>\tilde{c}_{mns}^{\mu \nu} (k)  \rightarrow  \tilde{\gamma}_{mns}^{\mu \nu} (k)</math>===


For simple fluids:  
For simple fluids:  


<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho  \tilde{c}_2 (k)}</math>
:<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho  \tilde{c}_2 (k)}</math>


For molecular fluids (see Eq. 19 of Lado \cite{MP_1982_47_0283})
For molecular fluids (see Eq. 19 of Lado Ref. 3)


:<math>\tilde{{\mathbf S}}_{m}(k) = (-1)^{m}\rho \left[{\mathbf I} - (-1)^{m} \rho \tilde{\mathbf C}_{m}(k) \right]^{-1} \tilde{\mathbf C}_{m}(k)\tilde{\mathbf C}_{m}(k)</math>


:<math>\tilde{S}_{m}(k) = (-1)^{m}\rho \left[I - (-1)^{m} \rho \tilde C_{m}(k) \right]^{-1} \tilde C_{m}(k)\tilde C_{m}(k)
where <math>\tilde{{\mathbf S}}_{m}(k)</math> and <math>\tilde{\mathbf C}_{m}(k)</math> are matrices
</math>
where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices
with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.
with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.
For mixtures of simple fluids  (see \cite{JCP_1988_88_07715} and the thesis of Juan Antonio Anta pp. 107--109):


<math>\tilde \Gamma (k) =  D  \left[ I - D  \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>
For mixtures of simple fluids  (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):


:<math>\tilde{\Gamma}(k) =  {\mathbf D}  \left[{\mathbf I} -  {\mathbf D}  \tilde{\mathbf C}(k)\right]^{-1} \tilde{\mathbf C}(k)\tilde{\mathbf C}(k)</math>


#Conversion back from Fourier space to Real space
===Conversion back from Fourier space to Real space===
<math>\tilde{\gamma}_{mns}^{\mu \nu} (k)  \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>  
:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k)  \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>  
(basically the inverse of step 2).
(basically the inverse of step 2).
i) axial reference frame to spatial reference frame: <math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow  \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
====Axial reference frame to spatial reference frame====
ii) Inverse Fourier-Bessel transform: <math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow  \gamma^{mnl}_{\mu \nu} (r)</math>
:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow  \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
`Step-up' operations are given by Eq. 53 of \cite{MP_1982_47_0283}.\\
====Inverse Fourier-Bessel transform====
The inverse Hankel transform is
:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow  \gamma^{mnl}_{\mu \nu} (r)</math>  
 
'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is
<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty  \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>
:<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty  \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>
 
====Change from  spatial reference frame back to  axial reference frame====
iii) Change from  spatial reference frame back to  axial reference frame: <math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow  \gamma_{mns}^{\mu \nu} (r)</math>.
:<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow  \gamma_{mns}^{\mu \nu} (r)</math>.


==Ng acceleration==
==Ng acceleration==
*[http://dx.doi.org/10.1063/1.1682399  Kin-Chue Ng "Hypernetted chain solutions for the classical one-component plasma up to Gamma=7000", Journal of Chemical Physics '''61''' pp. 2680-2689  (1974)]
*[http://dx.doi.org/10.1063/1.1682399  Kin-Chue Ng "Hypernetted chain solutions for the classical one-component plasma up to Gamma=7000", Journal of Chemical Physics '''61''' pp. 2680-2689  (1974)]
==Angular momentum coupling coefficients==
==Angular momentum coupling coefficients==
*CPC_1970_1_0337,CPC_1971_2_0381}
*[http://dx.doi.org/10.1016/0010-4655(70)90034-2  Taro Tamura  "Angular momentum coupling coefficients", Computer Physics Communications  '''1''' pp.  337-342 (1970)]
==Clebsch-Gordon coefficients and Racah's formula==
*[http://dx.doi.org/10.1016/0010-4655(71)90030-0 J. G. Wills  "On the evaluation of angular momentum coupling coefficients", omputer Physics Communications  '''2''' pp. 381-382 (1971)]
 
The Clebsch-Gordon coefficients are defined by
 
<math>\Psi_{JM}= \sum_{M=M_1 + M_2} C_{M_1 M_2}^J \Psi_{M_1 M_2},</math>
 
where $J \equiv J_1 + J_2$ and satisfies
 
<math>(j_1j_2m_1m_2|j_1j_2m)=0</math>
 
for $m_1+m_2\neq m$.
They are used to integrate products of three spherical harmonics (for example the addition of
angular momenta).\\
The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-coefficients (Giulio Racah (1909 - 1965)),
 
<math>V(j_1j_2j;m_1m_2m)</math>
 
(See also the [[Racah W-coefficients]], sometimes simply called the Racah coefficients).
\cite{CPC_1974_8_0095}
 
==References==
==References==
#[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)]
#[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)]
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#[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)]
#[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)]
#[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)]
#[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)]
#[http://dx.doi.org/10.1063/1.1676864 L. Blum and A. J. Torruella "Invariant Expansion for Two-Body Correlations: Thermodynamic Functions, Scattering, and the Ornstein—Zernike Equation", Journal of Chemical Physics '''56''' pp. pp. 303-310  (1972)]
#[http://dx.doi.org/10.1063/1.1678503 L. Blum "Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model", Journal of Chemical Physics '''57''' pp. 1862-1869 (1972)]
#[http://dx.doi.org/10.1063/1.1679655 L. Blum "Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions", Journal of Chemical Physics '''58''' pp. 3295-3303 (1973)]
#[http://dx.doi.org/10.1063/1.454286    P. G. Kusalik and G. N. Patey " On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration",  Journal of Chemical Physics '''88''' pp. 7715-7738 (1988)]
[[category: integral equations]]

Latest revision as of 15:57, 31 January 2008

Integral equations are solved numerically. One has the Ornstein-Zernike relation, and a closure relation, (which incorporates the bridge function ). The numerical solution is iterative;

  1. trial solution for
  2. calculate
  3. use the Ornstein-Zernike relation to generate a new etc.

Note that 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 c_2 (12)} is local, i.e. 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 c_2 (12)} at a given point is given by 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 \gamma (12)} at this point. However, the Ornstein-Zernike relation is non-local. The way to convert the Ornstein-Zernike relation into a local equation is to perform a (fast) Fourier transform (FFT). Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).

Picard iteration[edit]

Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration. Here are the four steps used to solve integral equations:

Closure relation 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 \gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)} [edit]

(Note: for linear fluids 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 \mu = \nu =0} )

Perform the summation[edit]

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 g(12)=g(r_{12},\omega_1,\omega_2)=\sum_{mns\mu \nu} g_{mns}^{\mu \nu}(r_{12}) \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2)}

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_{12}} is the separation between molecular centers 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 \omega_1,\omega_2} the sets of Euler angles needed to specify the orientations of the two molecules, with

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_{\mu \nu s}^{mn}(\omega_1,\omega_2) = \sqrt{(2m+1)(2n+1)} \mathcal{D}_{s \mu}^m (\omega_1) \mathcal{D}_{\overline{s} \nu}^n (\omega_2)}

with 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{s} = -s} .

Define the variables[edit]

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. x_1 \right.= \cos \theta_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. x_2\right.= \cos \theta_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 \left. z_1 \right.= \cos \chi_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. z_2 \right.= \cos \chi_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 \left. y\right.= \cos \phi_{12}}

Thus

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. \gamma(12) \right. =\gamma (r,x_1x_2,y,z_1z_2)} .

Evaluate[edit]

Evaluations 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 \gamma (12)} are performed at the discrete points 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 x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}} where the 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 x_i} are the 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 \nu} roots of the Legendre polynomial 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 P_\nu(cos \theta)} 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 y_j} are the 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 \nu} roots of the Chebyshev polynomial 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 T_{\nu}(\ cos \phi)} and 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 z_{1_k},z_{2_k}} are the 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 \nu} roots of the Chebyshev polynomial 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 T_{\nu}(\ cos \chi)} thus

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 \gamma(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})= \sum_{\nu , \mu , s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M \gamma_{mns}^{\mu \nu} (r) \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i}) e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k})}

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 \hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)}


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 d_{s \mu}^m(\theta)} is the angular, 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} , part of the rotation matrix 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 \mathcal{D}_{s \mu}^m (\omega)} , 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 \left. e_s(y) \right.=\exp(is\phi)}
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. e_{\mu}(z) \right.= \exp(i\mu \chi)}

For the limits in the summations

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. L_1 \right.= \max (s,\nu_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. L_2 \right.= \max (s,\nu_2)}

The above equation constitutes a separable five-dimensional transform. To rapidly evaluate this expression it is broken down into five one-dimensional transforms:

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 \gamma_{l_2m}^{n_1n_2}(r,x_{1_i})=\sum_{l_1=L_1}^M \gamma_{l_1 l_2 m}^{n_1 n_2}(r) \hat{d}_{m n_1}^{l_1} (x_{1_i})}
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 \gamma_{m}^{n_1n_2}(r,x_{1_i},x_{2_i})=\sum_{l_2=L_2}^M \gamma_{l_2 m}^{n_1 n_2}(r,x_{1_i}) \hat{d}_{\overline{m} n_2}^{l_2} (x_{2_i})}
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 \gamma^{n_1n_2}(r,x_{1_i},x_{2_i},j)=\sum_{m=-M}^M \gamma_{m}^{n_1 n_2}(r,x_{1_i},x_{2_i}) e_m(j)}
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 \gamma^{n_2}(r,x_{1_i},x_{2_i},z_{1_k})=\sum_{n_1=-M}^M \gamma^{n_1 n_2}(r,x_{1_i},x_{2_i},j) e_{n_1}(z_{1_k})}
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 \gamma(r,x_{1_i},x_{2_i},z_{1_k},z_{2_k})=\sum_{n_2=-M}^M \gamma^{n_2}(r,x_{1_i},x_{2_i},j,z_{1_k}) e_{n_2}(z_{2_k})}

Operations involving the 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 e_m(y)} 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 e_n(z)} basis functions are performed in complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression and thus constitutes a ``fast separable transform". 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 M} are parameters; 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 NG} is the number of nodes in the Gauss integration, 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 M} the the max index in the truncated rotational invariants expansion.

Integrate over 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 c_2(12)} [edit]

Use Gauss-Legendre quadrature for 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 x_1} 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 x_2} Use Gauss-Chebyshev quadrature for 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} , 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 z_1} 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 z_2} . Thus

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 c_{mns}^{\mu \nu} (r) = w^3 \sum_{x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}=1}^{NG} w_{i_1}w_{i_2}c_2(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}) \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i}) e_{\overline{s}}(j) e_{\overline{\mu}} (z_{1_k}) e_{\overline{\nu}} (z_{2_k})}

where the Gauss-Legendre quadrature weights are given by

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 w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2}

while the Gauss-Chebyshev quadrature has the constant weight

Perform FFT from Real to Fourier space 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 c_{mns}^{\mu \nu} (r) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)} [edit]

This is non-trivial and is undertaken in three steps:

Conversion from axial reference frame to spatial reference frame[edit]

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 c_{mns}^{\mu \nu} (r) \rightarrow c_{\mu \nu}^{mnl} (r)}

this is done using the Blum transformation (Refs 7, 8 and 9):

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 g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left( \begin{array}{ccc} m&n&l\\ s&\overline{s}&0 \end{array} \right)g_{mns}^{\mu \nu} (r)}

Fourier-Bessel Transforms[edit]

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 c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)}

(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3), 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 J_l(x)} is a Bessel function of order 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 l} . `step-down' operations can be performed by way of sin and cos operations of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3. The Fourier-Bessel transform is also known as a Hankel transform. It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.

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 g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r}


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 f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q}

Conversion from the spatial reference frame back to the axial reference frame[edit]

this is done using the Blum transformation

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 g_{mns}^{\mu \nu} (r) = \sum_{l=|m-n|}^{m+n} \left( \begin{array}{ccc} m&n&l\\ s&\overline{s}&0 \end{array} \right) g_{\mu \nu}^{mnl}(r)}

Ornstein-Zernike relation 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 \tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)} [edit]

For simple fluids:

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 \tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)}}

For molecular fluids (see Eq. 19 of Lado Ref. 3)

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 \tilde{{\mathbf S}}_{m}(k) = (-1)^{m}\rho \left[{\mathbf I} - (-1)^{m} \rho \tilde{\mathbf C}_{m}(k) \right]^{-1} \tilde{\mathbf C}_{m}(k)\tilde{\mathbf C}_{m}(k)}

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 \tilde{{\mathbf S}}_{m}(k)} 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 \tilde{\mathbf C}_{m}(k)} are matrices with elements 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 \tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m} .

For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):

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 \tilde{\Gamma}(k) = {\mathbf D} \left[{\mathbf I} - {\mathbf D} \tilde{\mathbf C}(k)\right]^{-1} \tilde{\mathbf C}(k)\tilde{\mathbf C}(k)}

Conversion back from Fourier space to Real space[edit]

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 \tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r)}

(basically the inverse of step 2).

Axial reference frame to spatial reference frame[edit]

Inverse Fourier-Bessel transform[edit]

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 \tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)}

'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is

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 \gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k}

Change from spatial reference frame back to axial reference frame[edit]

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 \gamma^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)} .

Ng acceleration[edit]

Angular momentum coupling coefficients[edit]

References[edit]

  1. M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics 38 pp. 1781-1794 (1979)
  2. Stanislav Labík, Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation", Molecular Physics 56 pp. 709-715 (1985)
  3. F. Lado "Integral equations for fluids of linear molecules I. General formulation", Molecular Physics 47 pp. 283-298 (1982)
  4. F. Lado "Integral equations for fluids of linear molecules II. Hard dumbell solutions", Molecular Physics 47 pp. 299-311 (1982)
  5. F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics 47 pp. 313-317 (1982)
  6. Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics 68 pp. 87-95 (1989)
  7. L. Blum and A. J. Torruella "Invariant Expansion for Two-Body Correlations: Thermodynamic Functions, Scattering, and the Ornstein—Zernike Equation", Journal of Chemical Physics 56 pp. pp. 303-310 (1972)
  8. L. Blum "Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model", Journal of Chemical Physics 57 pp. 1862-1869 (1972)
  9. L. Blum "Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions", Journal of Chemical Physics 58 pp. 3295-3303 (1973)
  10. P. G. Kusalik and G. N. Patey " On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration", Journal of Chemical Physics 88 pp. 7715-7738 (1988)
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