Computational implementation of integral equations

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;

1. 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)}
2. 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)}
3. 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 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 r_{12}} 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, 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 \theta} , 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)}

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_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:

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})}

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},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 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; 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} 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 y} , 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

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=\frac{1}{NG}}

Perform FFT from Real to Fourier spaceFailed 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) \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.

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) \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}:

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_{\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: 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_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)}

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 \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}

(see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}), 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 J_l(x)} is a Bessel function of order 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} . `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.

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(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r}

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 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.

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

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_{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)