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

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

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

$e_{s}(y)=\exp(is\phi )$

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

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:

$\gamma _{l_{2}m}^{n_{1}n_{2}}(r,x_{1_{i}})=\sum _{l_{1}=L_{1}}^{M}\gamma _{l_{1}l_{2}m}^{n_{1}n_{2}}(r){\hat {d}}_{mn_{1}}^{l_{1}}(x_{1_{i}})$

$\gamma _{m}^{n_{1}n_{2}}(r,x_{1_{i}},x_{2_{i}})=\sum _{l_{2}=L_{2}}^{M}\gamma _{l_{2}m}^{n_{1}n_{2}}(r,x_{1_{i}}){\hat {d}}_{{\overline {m}}n_{2}}^{l_{2}}(x_{2_{i}})$

$\gamma ^{n_{1}n_{2}}(r,x_{1_{i}},x_{2_{i}},j)=\sum _{m=-M}^{M}\gamma _{m}^{n_{1}n_{2}}(r,x_{1_{i}},x_{2_{i}})e_{m}(j)$

$\gamma ^{n_{2}}(r,x_{1_{i}},x_{2_{i}},z_{1_{k}})=\sum _{n_{1}=-M}^{M}\gamma ^{n_{1}n_{2}}(r,x_{1_{i}},x_{2_{i}},j)e_{n_{1}}(z_{1_{k}})$

$\gamma (r,x_{1_{i}},x_{2_{i}},z_{1_{k}},z_{2_{k}})=\sum _{n_{2}=-M}^{M}\gamma ^{n_{2}}(r,x_{1_{i}},x_{2_{i}},j,z_{1_{k}})e_{n_{2}}(z_{2_{k}})$

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

$w_{i}={\frac {1}{(1-x_{i}^{2})}}[P_{NG}^{'}(x_{i})]^{2}$

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:

1. Conversion from axial reference frame to spatial reference frame, i.e.

$c_{mns}^{\mu \nu }(r)\rightarrow c_{\mu \nu }^{mnl}(r)$

this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}:

$g_{\mu \nu }^{mnl}(r)=\sum _{s=-\min(m,n)}^{\min(m,n)}\left({\begin{array}{ccc}m&n&l\\s&{\overline {s}}&0\end{array}}\right)g_{mns}^{\mu \nu }(r)$

1. Fourier-Bessel Transforms: $c_{\mu \nu }^{mnl}(r)\rightarrow {\tilde {c}}_{\mu \nu }^{mnl}(k)$

${\tilde {c}}_{\mu \nu }^{mnl}(k;l_{1}l_{2}ln_{1}n_{2})=4\pi i^{l}\int _{0}^{\infty }c_{\mu \nu }^{mnl}(r;l_{1}l_{2}ln_{1}n_{2})J_{l}(kr)~r^{2}{\rm {d}}r$

(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 Hankel transform. It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.

$g(q)=2\pi \int _{0}^{\infty }f(r)J_{0}(2\pi qr)r~{\rm {d}}r$

$f(r)=2\pi \int _{0}^{\infty }g(q)J_{0}(2\pi qr)q~{\rm {d}}q$

1. Conversion from the spatial reference frame back to the axial reference frame

i.e.

${\tilde {c}}_{\mu \nu }^{mnl}(k)\rightarrow {\tilde {c}}_{mns}^{\mu \nu }(k)$ this is done using the Blum transformation

$g_{mns}^{\mu \nu }(r)=\sum _{l=|m-n|}^{m+n}\left({\begin{array}{ccc}m&n&l\\s&{\overline {s}}&0\end{array}}\right)g_{\mu \nu }^{mnl}(r)$

OZ Equation} $ \tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$\\ ~\\ For simple fluids: \begin{equation} \tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)} \end{equation} For molecular fluids (see Eq. 19 of Lado \cite{MP_1982_47_0283}) %(see derivation in the thesis of Juan Antonio Anta pp. 105--107): %\begin{equation} %\tilde{\Gamma}_{\chi}(k) = (-1)^{\chi}\rho \left[{\bf I} - (-1)^{\chi} \rho \tilde{\bf C}_{\chi}(k) \right]^{-1} \tilde{\bf C}_{\chi}(k)\tilde{\bf C}_{\chi}(k) %\end{equation} \begin{equation} \tildeTemplate:\bf S_{m}(k) = (-1)^{m}\rho \left[{\bf I} - (-1)^{m} \rho \tilde{\bf C}_{m}(k) \right]^{-1} \tilde{\bf C}_{m}(k)\tilde{\bf C}_{m}(k) \end{equation} where $\tildeTemplate:\bf S_{m}(k)$ and $\tilde{\bf C}_{m}(k)$ are matrices with elements $\tilde{S}_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m$.\\ For mixtures of simple fluids (see \cite{JCP_1988_88_07715} and the thesis of Juan Antonio Anta pp. 107--109): \begin{equation} \tilde{\Gamma}(k) = {\bf D} \left[{\bf I} - {\bf D} \tilde{\bf C}(k)\right]^{-1} \tilde{\bf C}(k)\tilde{\bf C}(k) \end{equation} ~\\ 4) {\bf Conversion back from Fourier space to Real space}: $ \tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r) $\\ (basically the inverse of step 2).\\ i) axial reference frame to spatial reference frame: $ \tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)$\\ ii) Inverse Fourier-Bessel transform: $ \tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)$\\

`Step-up' operations are given by Eq. 53 of  \cite{MP_1982_47_0283}.\\

The inverse Hankel transform is \begin{equation} \gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k \end{equation} iii) Change from spatial reference frame back to axial reference frame:$ \gamma^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)$.

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)

\section{Angular momentum coupling coefficients} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \cite{CPC_1970_1_0337,CPC_1971_2_0381} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Clebsch-Gordon coefficients and Racah's formula} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The Clebsch-Gordon coefficients are defined by \begin{equation} \Psi_{JM}= \sum_{M=M_1 + M_2} C_{M_1 M_2}^J \Psi_{M_1 M_2}, \end{equation} where $J \equiv J_1 + J_2$ and satisfies \begin{equation} (j_1j_2m_1m_2|j_1j_2m)=0 \end{equation} for $m_1+m_2\neq m$.\\ They are used to integrate products of three spherical harmonics (for example the addition of angular momenta).\\ The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-coefficients (Giulio Racah (1909 - 1965)), \begin{equation} V(j_1j_2j;m_1m_2m) \end{equation} (See also the Racah W-coefficients, sometimes simply called the Racah coefficients). \cite{CPC_1974_8_0095}

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)