Computational implementation of integral equations

Integral equations are solved numerically. One has the Ornstein-Zernike relation, $\gamma (12)$ and a closure relation, $c_2(12)$ (which incorporates the bridge function $B(12)$). The numerical solution is iterative;

1. trial solution for $\gamma (12)$
2. calculate $c_2(12)$
3. use the Ornstein-Zernike relation to generate a new $\gamma (12)$ etc.

Note that the value of $c_2(12)$ is local, i.e. the value of $c_2(12)$ at a given point is given by the value of $\gamma (12)$ 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 $\underset{mns}{\overset{\mu \nu }{\gamma }}(r)\to c_{mns}^{\mu \nu }(r)$

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

Perform the summation

$g(12)=g(r_{12},{\omega }_1,{\omega }_2)={\sum }_{mns\mu \nu }{g}_{mns}^{\mu \nu }(r_{12})\underset{\mu \nu s}{\overset{mn}{\mathrm{\Psi }}}({\omega }_1,{\omega }_2)$

where $r_{12}$ 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

$\underset{\mu \nu s}{\overset{mn}{\mathrm{\Psi }}}({\omega }_1,{\omega }_2)=\sqrt{(2m+1)(2n+1)}{\mathcal{D}}_{s\mu }^m({\omega }_1){\mathcal{D}}_{\overline{s}\nu }^n({\omega }_2)$

with $\overline{s}=-s$.

Define the variables

$x_1=\cos {\theta }_1$

$x_2=\cos {\theta }_2$

$z_1=\cos {\chi }_1$

$z_2=\cos {\chi }_2$

$y=\cos {\varphi }_{12}$

Thus

$\gamma (12)=\gamma (r,x_1{x}_2,y,z_1{z}_2)$.

Evaluate

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 $T_{\nu }(cos\varphi )$ and where $z_{1_k},z_{2_k}$ are the $\nu$ roots of the Chebyshev polynomial $T_{\nu }(cos\chi )$ thus

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

where

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

where $d_{s\mu }^m(\theta )$ is the angular, $\theta$, part of the rotation matrix ${\mathcal{D}}_{s\mu }^m(\omega )$, and

$e_s(y)=\exp (is\varphi )$

$e_{\mu }(z)=\exp (i\mu \chi )$

For the limits in the summations

$L_1=max(s,{\nu }_1)$

$L_2=max(s,{\nu }_2)$

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

$\underset{l_{2}m}{\overset{n_1{n}_2}{\gamma }}(r,x_{1_i})=\sum _{l_1=L_1}^M\underset{l_1{l}_{2}m}{\overset{n_1{n}_2}{\gamma }}(r){\hat{d}}_{m{n}_1}^{l_1}(x_{1_i})$

$\underset{m}{\overset{n_1{n}_2}{\gamma }}(r,x_{1_i},x_{2_i})=\sum _{l_2=L_2}^M\underset{l_{2}m}{\overset{n_1{n}_2}{\gamma }}(r,x_{1_i}){\hat{d}}_{\overline{m}{n}_2}^{l_2}(x_{2_i})$

${\gamma }^{n_1{n}_2}(r,x_{1_i},x_{2_i},j)=\sum _{m=-M}^M\underset{m}{\overset{n_1{n}_2}{\gamma }}(r,x_{1_i},x_{2_i})e_m(j)$

${\gamma }^{n_2}(r,x_{1_i},x_{2_i},z_{1_k})=\sum _{n_1=-M}^M{\gamma }^{n_1{n}_2}(r,x_{1_i},x_{2_i},j)e_{n_1}(z_{1_k})$

$\gamma (r,x_{1_i},x_{2_i},z_{1_k},z_{2_k})=\sum _{n_2=-M}^M{\gamma }^{n_2}(r,x_{1_i},x_{2_i},j,z_{1_k})e_{n_2}(z_{2_k})$

Operations involving the $e_m(y)$ and $e_n(z)$ basis functions are performed in complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression 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.

Integrate over angles $c_2(12)$

Use Gauss-Legendre quadrature for $x_1$ and $x_2$ Use Gauss-Chebyshev quadrature for $y$, $z_1$ and $z_2$ thus

$c_{mns}^{\mu \nu }(r)=w^3\sum _{x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}=1}^{NG}{w}_{i_1}{w}_{i_2}{c}_2(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}){\hat{d}}_{s\mu }^m(x_{1_i}){\hat{d}}_{\overline{s}\nu }^n(x_{2_i})e_{\overline{s}}(j)e_{\overline{\mu }}(z_{1_k})e_{\overline{\nu }}(z_{2_k})$

where the Gauss-Legendre quadrature weights are given by

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

while the Gauss-Chebyshev quadrature has the constant weight

$w=\frac{1}{NG}$

===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

• 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

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)