Editing Computational implementation of integral equations
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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 | and a 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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Note: convergence is poor for liquid densities. (See Ref.s 1 to 6). | Note: convergence is poor for liquid densities. (See Ref.s 1 to 6). | ||
==Picard iteration== | ==Picard iteration== | ||
===Closure relation=== | |||
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:\\ | ||
~\\ | |||
(Note: for linear fluids | 1) {\bf Closure relation}: $\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)$\\ | ||
(Note: for linear fluids $\mu = \nu =0$)\\ | |||
~\\ | |||
i) Perform the summation | |||
\begin{equation} | |||
g(12)=g({\bf r}_{12},\omega_1,\omega_2)=\sum_{mns\mu \nu} g_{mns}^{\mu \nu}({\bf r}_{12}) \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2) | |||
where | \end{equation} | ||
where ${\bf r}_{12}$ is the separation between molecular centers and | |||
$\omega_1,\omega_2$ the sets of Euler angles needed to specify the orientations of the two molecules, with | |||
\begin{equation} | |||
\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 | \end{equation} | ||
with $\overline{s} = -s$.\\ | |||
~\\ | |||
ii) Define the variables | |||
\begin{equation} | |||
x_1= \cos \theta_1 | |||
\end{equation} | |||
\begin{equation} | |||
x_2= \cos \theta_2 | |||
\end{equation} | |||
Thus | \begin{equation} | ||
z_1 = \cos \chi_1 | |||
\end{equation} | |||
\begin{equation} | |||
Evaluations of | z_2 = \cos \chi_2 | ||
where the | \end{equation} | ||
where | \begin{equation} | ||
and where | y= \cos \phi_{12} | ||
\end{equation} | |||
Thus $\gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)$.\\ | |||
~\\ | |||
iii) Evaluations of $\gamma (12)$ are performed at the discrete points $x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}$\\ | |||
where the $x_i$ are the $\nu$ roots of the Legendre polynomial $P_\nu(cos \theta)$ | |||
~\\ | |||
where $y_j$ are the $\nu$ roots of the Chebyshev polynomial $T_{\nu}(\ cos \phi)$\\ | |||
and where $z_{1_k},z_{2_k}$ are the $\nu$ roots of the Chebyshev polynomial $T_{\nu}(\ cos \chi)$\\ | |||
~\\ | |||
thus | thus | ||
\begin{equation} | |||
\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 | \sum_{\nu , \mu , s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M | ||
\gamma_{mns}^{\mu \nu} (r) | \gamma_{mns}^{\mu \nu} (r) | ||
\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_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k}) | e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k}) | ||
\end{equation} | |||
where | where | ||
\begin{equation} | |||
\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta) | |||
\end{equation} | |||
where $d_{s \mu}^m(\theta)$ is the angular, $\theta$, part of the | |||
rotation matrix $\mathcal{D}_{s \mu}^m (\omega)$,\\ | |||
where | |||
rotation matrix | |||
and | and | ||
\begin{equation} | |||
e_s(y)=\exp(is\phi) | |||
\end{equation} | |||
\begin{equation} | |||
e_{\mu}(z)= \exp(i\mu \chi) | |||
\end{equation} | |||
For the limits in the summations | For the limits in the summations | ||
\begin{equation} | |||
\begin{equation} | |||
L_1= \max (s,\nu_1) | |||
\end{equation} | |||
\begin{equation} | |||
L_2= \max (s,\nu_2) | |||
\end{equation} | |||
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: | ||
\begin{equation} | |||
\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}) | |||
\end{equation} | |||
\begin{equation} | |||
\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}) | |||
\end{equation} | |||
\begin{equation} | |||
\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) | |||
\end{equation} | |||
\begin{equation} | |||
\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 | \end{equation} | ||
\begin{equation} | |||
\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}) | |||
\end{equation} | |||
Operations involving the $e_m(y)$ and $e_n(z)$ basis functions are performed in | |||
complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression | complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression | ||
and thus constitutes a ``fast separable transform". | and thus constitutes a ``fast separable transform". | ||
$NG$ and $M$ are parameters; $NG$ is the number of nodes in the Gauss integration, and $M$ the the max index in the truncated rotational invariants expansion.\\ | |||
~\\ | |||
iv) Integrate over angles $c_2(12)$:\\ | |||
~\\ | |||
Use | Use Gauss-Legendre quadrature for $x_1$ and $x_2$\\ | ||
Use | Use Gauss-Chebyshev quadrature for $y$, $z_1$ and $z_2$\\ | ||
thus | |||
\begin{equation} | |||
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}) | ||
\end{equation} | |||
where the Gauss-Legendre quadrature weights are given by | where the Gauss-Legendre quadrature weights are given by | ||
\begin{equation} | |||
w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2 | |||
\end{equation} | |||
while the Gauss-Chebyshev quadrature has the constant weight | while the Gauss-Chebyshev quadrature has the constant weight | ||
\begin{equation} | |||
w=\frac{1}{NG} | |||
\end{equation} | |||
===Perform FFT from Real to Fourier space=== | |||
==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)] | ||
==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)] | ||