Computational implementation of integral equations

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Integral equations are solved numerically. One has the Ornstein-Zernike relation, $γ(12)$ and a closure relation, $c 2 (12)$ (which incorporates the bridge function $B (12)$ ). The numerical solution is iterative;

trial solution for $γ(12)$
calculate $c 2 (12)$
use the Ornstein-Zernike relation to generate a new $γ(12)$ 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 $γ(12)$ 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).

1 Picard iteration
2 Ng acceleration
3 Angular momentum coupling coefficients
4 References

[edit] 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:

[edit] Closure relation $\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)$

(Note: for linear fluids $μ = ν = 0$ )

[edit] 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 $ω 1,ω 2$ 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 $\overline{s} = -s$ .

[edit] 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

$\left. \gamma(12) \right. =\gamma (r,x_1x_2,y,z_1z_2)$ .

[edit] Evaluate

Evaluations of $γ(12)$ 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 $ν$ roots of the Legendre polynomial $P ν (c o s θ)$ where $y j$ are the $ν$ roots of the Chebyshev polynomial $T_{\nu}(\ cos \phi)$ and where $z_{1_k},z_{2_k}$ are the $ν$ 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, $θ$ , part of the rotation matrix $\mathcal{D}_{s \mu}^m (\omega)$ , and

$\left. e_s(y) \right.=\exp(is\phi)$

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

For the limits in the summations

$\left. L_1 \right.= \max (s,\nu_1)$

$\left. L_2 \right.= \max (s,\nu_2)$

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

$\gamma_{l_2m}^{n_1n_2}(r,x_{1_i})=\sum_{l_1=L_1}^M \gamma_{l_1 l_2 m}^{n_1 n_2}(r) \hat{d}_{m n_1}^{l_1} (x_{1_i})$

$\gamma_{m}^{n_1n_2}(r,x_{1_i},x_{2_i})=\sum_{l_2=L_2}^M \gamma_{l_2 m}^{n_1 n_2}(r,x_{1_i}) \hat{d}_{\overline{m} n_2}^{l_2} (x_{2_i})$

$\gamma^{n_1n_2}(r,x_{1_i},x_{2_i},j)=\sum_{m=-M}^M \gamma_{m}^{n_1 n_2}(r,x_{1_i},x_{2_i}) e_m(j)$

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

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

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". $N G$ and $M$ are parameters; $N G$ is the number of nodes in the Gauss integration, and $M$ the the max index in the truncated rotational invariants expansion.

[edit] 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

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

[edit] 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:

[edit] Conversion from axial reference frame to spatial reference frame

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

this is done using the Blum transformation (Refs 7, 8 and 9):

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

[edit] Fourier-Bessel Transforms

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

$\tilde{c}_{\mu \nu}^{mnl} (k; l_1 l_2 l n_1 n_2) = 4\pi i^l \int_0^{\infty} c_{\mu \nu}^{mnl} (r; l_1 l_2 l n_1 n_2) J_l (kr) ~r^2 {\rm d}r$

(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3), 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 Ref. 3. 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.

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

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

[edit] Conversion from the spatial reference frame back to the axial reference frame

$\tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)$

this is done using the Blum transformation

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

[edit] Ornstein-Zernike relation $\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$

For simple fluids:

$\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)}$

For molecular fluids (see Eq. 19 of Lado Ref. 3)

$\tilde{{\mathbf S}}_{m}(k) = (-1)^{m}\rho \left[{\mathbf I} - (-1)^{m} \rho \tilde{\mathbf C}_{m}(k) \right]^{-1} \tilde{\mathbf C}_{m}(k)\tilde{\mathbf C}_{m}(k)$

where $\tilde{{\mathbf S}}_{m}(k)$ and $\tilde{\mathbf C}_{m}(k)$ are matrices with elements $\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m$ .