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 | Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math> | ||
Use Gauss-Chebyshev quadrature for | Use [[Gauss-Chebyshev quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math> | ||
thus | thus | ||
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} | ||
%The step-down operator | |||
iii) Conversion from the spatial reference frame back to the axial reference frame, {\it i.e.} | |||
\begin{equation} | \begin{equation} | ||
\tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k) | |||
\end{equation} | \end{equation} | ||
this is done using the Blum transformation | |||
\begin{equation} | \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} | \end{equation} | ||
==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;
- 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)}
- 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)}
- 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 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
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 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 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, , 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 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)}
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})}
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". 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; 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)}
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 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=== $ 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
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