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

$\gamma _{mns}^{\mu \nu }(r)\rightarrow c_{mns}^{\mu \nu }(r)$ (Note: for linear fluids $\mu =\nu =0$)

Perform the summation

$g(12)=g(r_{12},\omega _{1},\omega _{2})=\sum _{mns\mu \nu }g_{mns}^{\mu \nu }(r_{12})\Psi _{\mu \nu s}^{mn}(\omega _{1},\omega _{2})$

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

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

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

Define the variables

$\left.x_{1}\right.=\cos \theta _{1}$

$\left.x_{2}\right.=\cos \theta _{2}$

$\left.z_{1}\right.=\cos \chi _{1}$

$\left.z_{2}\right.=\cos \chi _{2}$

$\left.y\right.=\cos \phi _{12}$

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

$\gamma (r,x_{1_{i}},x_{2_{i}},j,z_{1_{k}},z_{2_{k}})=\sum _{\nu ,\mu ,s=-M}^{M}\sum _{m=L_{2}}^{M}\sum _{n=L_{1}}^{M}\gamma _{mns}^{\mu \nu }(r){\hat {d}}_{s\mu }^{m}(x_{1_{i}}){\hat {d}}_{{\overline {s}}\nu }^{n}(x_{2_{i}})e_{s}(j)e_{\mu }(z_{1_{k}})e_{\nu }(z_{2_{k}})$

where \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 $\mathcal{D}_{s \mu}^m (\omega)$,\\ 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 \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 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}) \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 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 Gauss-Legendre quadrature for $x_1$ and $x_2$\\ 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} 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}) \end{equation} 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 \begin{equation} w=\frac{1}{NG} \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)