Computational implementation of integral equations: Difference between revisions

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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===
===1. Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>===
<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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:<math>\left. y\right.= \cos \phi_{12}</math>
:<math>\left. y\right.= \cos \phi_{12}</math>


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


====Evaluate====
====Evaluate====
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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>e_s(y)=\exp(is\phi)</math>




<math>e_{\mu}(z)= \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>L_1= \max (s,\nu_1)</math>


<math>L_2= \max (s,\nu_2)</math>
:<math>L_2= \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
this expression it is broken down into five one-dimensional transforms:
this expression it is broken down into five one-dimensional transforms:


<math>\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})</math>
:<math>\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})</math>


<math>\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})</math>
:<math>\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})</math>


<math>\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)</math>
:<math>\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)</math>


<math>\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})</math>
:<math>\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})</math>


<math>\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})</math>
:<math>\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})</math>


Operations involving the <math>e_m(y)</math> and <math>e_n(z)</math> basis functions are performed in  
Operations involving the <math>e_m(y)</math> and <math>e_n(z)</math> basis functions are performed in  
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====Integrate over angles <math>c_2(12)</math>====
====Integrate over angles <math>c_2(12)</math>====
~\\
 
Use Gauss-Legendre quadrature for $x_1$ and $x_2$\\
Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math>
Use Gauss-Chebyshev  quadrature for $y$, $z_1$ and $z_2$\\
Use [[Gauss-Chebyshev  quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math>
thus
thus
\begin{equation}
 
c_{mns}^{\mu \nu} (r) = w^3  
:<math>c_{mns}^{\mu \nu} (r) = w^3  
\sum_{x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}=1}^{NG}
\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})
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})
\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})
e_{\overline{s}}(j) e_{\overline{\mu}} (z_{1_k}) e_{\overline{\nu}} (z_{2_k})</math>
 
where the Gauss-Legendre quadrature weights are given by
 
:<math>w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2</math>
 
while the  Gauss-Chebyshev  quadrature has the constant weight
 
:<math>w=\frac{1}{NG}</math>
 
===Perform FFT from Real to Fourier space=== $ c_{mns}^{\mu \nu} (r) \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k)$ \\
~\\
This is non-trivial and is undertaken in three steps:\\
i) Conversion from axial reference frame to spatial reference frame, {\it i.e.}
\begin{equation} 
c_{mns}^{\mu \nu} (r)  \rightarrow  c_{\mu \nu}^{mnl} (r)
\end{equation}
this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}:
\begin{equation}
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)
\end{equation}
{\bf ii}) {\bf Fourier-Bessel Transforms}: $  c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)
$\\
\begin{equation}
\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
\end{equation}
(see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}),
where $J_l(x)$ is a Bessel function of order $l$.\\
`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}.\\
The  Fourier-Bessel transform is also known as a {\bf Hankel transform}.
It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.
\begin{equation}
g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r
\end{equation}
\begin{equation}
f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q
\end{equation}
\end{equation}
where the Gauss-Legendre quadrature weights are given by
%The step-down operator
iii) Conversion from the spatial reference frame back to the  axial reference frame, {\it i.e.}
\begin{equation}
\begin{equation}
w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2
\tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow  \tilde{c}_{mns}^{\mu \nu} (k)  
\end{equation}
\end{equation}
while the Gauss-Chebyshev  quadrature has the constant weight
this is done using the Blum transformation
\begin{equation}
\begin{equation}
w=\frac{1}{NG}
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)
\end{equation}
\end{equation}
===Perform FFT from Real to Fourier space===
 
==Ng acceleration==
==Ng acceleration==


Revision as of 12:10, 30 May 2007

Integral equations are solved numerically. One has the 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 \gamma (12)} and a 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 c_2 (12)} (which incorporates the bridge function 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 B(12)} ). The numerical solution is iterative;

  1. trial solution 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 \gamma (12)}
  2. calculate 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)}
  3. use the Ornstein-Zernike relation to generate a new 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)} 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 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

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:

1. 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)}

(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

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

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

Evaluate

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 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 , 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_s(y)=\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 e_{\mu}(z)= \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 L_1= \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 L_2= \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_{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 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". 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} 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 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)}

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} , 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle w_{i}={\frac {1}{(1-x_{i}^{2})}}[P_{NG}^{'}(x_{i})]^{2}}

while the Gauss-Chebyshev quadrature has the constant weight

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=\frac{1}{NG}}

===Perform FFT from Real to Fourier space=== $ c_{mns}^{\mu \nu} (r) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)$ \\ ~\\ This is non-trivial and is undertaken in three steps:\\ i) Conversion from axial reference frame to spatial reference frame, {\it i.e.} \begin{equation} c_{mns}^{\mu \nu} (r) \rightarrow c_{\mu \nu}^{mnl} (r) \end{equation} this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}: \begin{equation} 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) \end{equation} {\bf ii}) {\bf Fourier-Bessel Transforms}: $ c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k) $\\ \begin{equation} 

\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

\end{equation} (see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}), where $J_l(x)$ is a Bessel function of order $l$.\\ `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}.\\ The Fourier-Bessel transform is also known as a {\bf Hankel transform}. It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel. \begin{equation} g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r \end{equation} \begin{equation} f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q \end{equation} %The step-down operator iii) Conversion from the spatial reference frame back to the axial reference frame, {\it i.e.} \begin{equation} \tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k) \end{equation} this is done using the Blum transformation \begin{equation} 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) \end{equation}

Ng acceleration

References

  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)
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