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| Integral equations are solved numerically.
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| One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
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| and a [[closure relations | closure relation]], <math>c_2 (12)</math> (which
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| incorporates the [[bridge function]] <math>B(12)</math>).
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| The numerical solution is iterative;
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|
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| # trial solution for <math>\gamma (12)</math>
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| # calculate <math>c_2 (12)</math>
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| # use the [[Ornstein-Zernike relation]] to generate a new <math>\gamma (12)</math> ''etc.''
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|
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| Note that the value of <math>c_2 (12)</math> is '''local''', ''i.e.''
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| the value of <math>c_2 (12)</math> at a given point is given by
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| the value of <math>\gamma (12)</math> at this point. However, the [[Ornstein-Zernike relation]] is '''non-local'''.
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| The way to convert the [[Ornstein-Zernike relation]] into a local equation
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| is to perform a [[Fast Fourier transform |(fast) Fourier transform]] (FFT).
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| Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).
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| ==Picard iteration==
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|
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| Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration.
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| Here are the four steps used to solve integral equations:
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| ===Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>===
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| (Note: for linear fluids <math>\mu = \nu =0</math>)
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|
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| ====Perform the summation====
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|
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| :<math>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)</math>
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|
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| where <math>r_{12}</math> is the separation between molecular centers and
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| <math>\omega_1,\omega_2</math> the sets of [[Euler angles]] needed to specify the orientations of the two molecules, with
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|
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| :<math>\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)</math>
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|
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| with <math>\overline{s} = -s</math>.
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|
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| ====Define the variables====
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|
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| :<math>\left. x_1 \right.= \cos \theta_1</math>
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| :<math>\left. x_2\right.= \cos \theta_2</math>
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| :<math>\left. z_1 \right.= \cos \chi_1</math>
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| :<math>\left. z_2 \right.= \cos \chi_2</math>
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| :<math>\left. y\right.= \cos \phi_{12}</math>
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|
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| Thus
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| :<math>\left. \gamma(12) \right. =\gamma (r,x_1x_2,y,z_1z_2)</math>.
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|
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| ====Evaluate====
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| Evaluations of <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>
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| where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials |Legendre polynomial]] <math>P_\nu(cos \theta)</math>
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| where <math>y_j</math> are the <math>\nu</math> roots of the [[Chebyshev polynomials |Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
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| and where <math>z_{1_k},z_{2_k}</math> are the <math>\nu</math> roots of the Chebyshev polynomial
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| <math>T_{\nu}(\ cos \chi)</math>
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| thus
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|
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| :<math>\gamma(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})=
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| \sum_{\nu , \mu , s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M
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| \gamma_{mns}^{\mu \nu} (r)
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| \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i})
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| e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k})</math>
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|
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| where
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|
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|
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| :<math>\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)</math>
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|
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|
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| where <math>d_{s \mu}^m(\theta)</math> is the angular, <math>\theta</math>, part of the
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| rotation matrix <math>\mathcal{D}_{s \mu}^m (\omega)</math>,
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| and
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|
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| :<math>\left. e_s(y) \right.=\exp(is\phi)</math>
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|
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| :<math>\left. e_{\mu}(z) \right.= \exp(i\mu \chi)</math>
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|
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| For the limits in the summations
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|
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| :<math>\left. L_1 \right.= \max (s,\nu_1)</math>
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|
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| :<math>\left. L_2 \right.= \max (s,\nu_2)</math>
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|
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| The above equation constitutes a separable five-dimensional transform. To rapidly evaluate
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| this expression it is broken down into five one-dimensional transforms:
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|
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| :<math>\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})</math>
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|
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| :<math>\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})</math>
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|
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| :<math>\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)</math>
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|
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| :<math>\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})</math>
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|
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| :<math>\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})</math>
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|
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| Operations involving the <math>e_m(y)</math> and <math>e_n(z)</math> basis functions are performed in
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| complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression
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| and thus constitutes a ``fast separable transform".
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| <math>NG</math> and <math>M</math> are parameters; <math>NG</math> is the number of nodes in the Gauss integration, and <math>M</math> the the max index in the truncated rotational invariants expansion.
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|
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| ====Integrate over angles <math>c_2(12)</math>====
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|
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| Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math>
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| Use [[Gauss-Chebyshev quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math>.
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| Thus
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|
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| :<math>c_{mns}^{\mu \nu} (r) = w^3
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| \sum_{x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}=1}^{NG}
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| w_{i_1}w_{i_2}c_2(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})
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| \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i})
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| e_{\overline{s}}(j) e_{\overline{\mu}} (z_{1_k}) e_{\overline{\nu}} (z_{2_k})</math>
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|
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| where the Gauss-Legendre quadrature weights are given by
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|
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| :<math>w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2</math>
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|
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| while the Gauss-Chebyshev quadrature has the constant weight
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|
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| :<math>w=\frac{1}{NG}</math>
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|
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| ===Perform FFT from Real to Fourier space <math>c_{mns}^{\mu \nu} (r) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)</math>===
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|
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| This is non-trivial and is undertaken in three steps:
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|
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| ====Conversion from axial reference frame to spatial reference frame====
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|
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| :<math>c_{mns}^{\mu \nu} (r) \rightarrow c_{\mu \nu}^{mnl} (r)</math>
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|
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| this is done using the Blum transformation (Refs 7, 8 and 9):
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|
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| :<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left(
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| \begin{array}{ccc}
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| m&n&l\\
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| s&\overline{s}&0
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| \end{array}
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| \right)g_{mns}^{\mu \nu} (r)</math>
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|
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| ====Fourier-Bessel Transforms====
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| :<math>c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)</math>
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|
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| :<math>\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</math>
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|
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| (see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),
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| where <math>J_l(x)</math> is a [[Bessel functions |Bessel function]] of order <math>l</math>.
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| `step-down' operations can be performed by way of sin and cos operations
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| of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3.
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| The Fourier-Bessel transform is also known as a '''Hankel transform'''.
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| It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.
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|
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| :<math>g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r</math>
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|
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|
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| :<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>
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|
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| ====Conversion from the spatial reference frame back to the axial reference frame====
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| :<math>\tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k) </math>
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| this is done using the Blum transformation
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|
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| :<math>g_{mns}^{\mu \nu} (r)
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| = \sum_{l=|m-n|}^{m+n} \left(
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| \begin{array}{ccc}
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| m&n&l\\
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| s&\overline{s}&0
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| \end{array}
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| \right)
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| g_{\mu \nu}^{mnl}(r)</math>
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|
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| ===Ornstein-Zernike relation <math>\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)</math>===
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|
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| For simple fluids:
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|
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| :<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)}</math>
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|
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| For molecular fluids (see Eq. 19 of Lado Ref. 3)
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|
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| :<math>\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)</math>
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|
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| where <math>\tilde{{\mathbf S}}_{m}(k)</math> and <math>\tilde{\mathbf C}_{m}(k)</math> are matrices
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| with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.
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|
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| For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):
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|
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| :<math>\tilde{\Gamma}(k) = {\mathbf D} \left[{\mathbf I} - {\mathbf D} \tilde{\mathbf C}(k)\right]^{-1} \tilde{\mathbf C}(k)\tilde{\mathbf C}(k)</math>
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|
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| ===Conversion back from Fourier space to Real space===
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| :<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>
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| (basically the inverse of step 2).
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| ====Axial reference frame to spatial reference frame====
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| :<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
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| ====Inverse Fourier-Bessel transform====
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| :<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)</math>
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| 'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is
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| :<math>\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</math>
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| ====Change from spatial reference frame back to axial reference frame====
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| :<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>.
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|
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| ==Ng acceleration==
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| *[http://dx.doi.org/10.1063/1.1682399 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)]
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| ==Angular momentum coupling coefficients==
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| *[http://dx.doi.org/10.1016/0010-4655(70)90034-2 Taro Tamura "Angular momentum coupling coefficients", Computer Physics Communications '''1''' pp. 337-342 (1970)]
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| *[http://dx.doi.org/10.1016/0010-4655(71)90030-0 J. G. Wills "On the evaluation of angular momentum coupling coefficients", omputer Physics Communications '''2''' pp. 381-382 (1971)]
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| ==References== | | ==References== |
| #[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)] | | #[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)] |
| #[http://dx.doi.org/10.1080/00268978500102651 Stanislav Labík, Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation", Molecular Physics '''56''' pp. 709-715 (1985)] | | #[http://dx.doi.org/10.1080/00268978500102651 Stanislav Labík, Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation" Molecular Physics '''56''' pp. 709-715 (1985)] |
| #[http://dx.doi.org/10.1080/00268978200100202 F. Lado "Integral equations for fluids of linear molecules I. General formulation", Molecular Physics '''47''' pp. 283-298 (1982)]
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| #[http://dx.doi.org/10.1080/00268978200100212 F. Lado "Integral equations for fluids of linear molecules II. Hard dumbell solutions", Molecular Physics '''47''' pp. 299-311 (1982)]
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| #[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)]
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| #[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)] | | #[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)] |
| #[http://dx.doi.org/10.1063/1.1676864 L. Blum and A. J. Torruella "Invariant Expansion for Two-Body Correlations: Thermodynamic Functions, Scattering, and the Ornstein—Zernike Equation", Journal of Chemical Physics '''56''' pp. pp. 303-310 (1972)]
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| #[http://dx.doi.org/10.1063/1.1678503 L. Blum "Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model", Journal of Chemical Physics '''57''' pp. 1862-1869 (1972)]
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| #[http://dx.doi.org/10.1063/1.1679655 L. Blum "Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions", Journal of Chemical Physics '''58''' pp. 3295-3303 (1973)]
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| #[http://dx.doi.org/10.1063/1.454286 P. G. Kusalik and G. N. Patey " On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration", Journal of Chemical Physics '''88''' pp. 7715-7738 (1988)]
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| [[category: integral equations]]
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