Computational implementation of integral equations: Difference between revisions

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Integral equations are solved numerically.
Integral equations are solved numerically.
One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
and a [[Closures | closure relation]], <math>c_2 (12)</math> (which
and a [[closure relations | closure relation]], <math>c_2 (12)</math> (which
incorporates the [[bridge function]] <math>B(12)</math>).
incorporates the [[bridge function]] <math>B(12)</math>).
The numerical solution is iterative;  
The numerical solution is iterative;  
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====Evaluate====
====Evaluate====
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>
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>
where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math>
where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials |Legendre polynomial]] <math>P_\nu(cos \theta)</math>
where <math>y_j</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
where <math>y_j</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomials |Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
and where <math>z_{1_k},z_{2_k}</math>  are the  <math>\nu</math> roots of the [[Chebyshev polynomial]]
and where <math>z_{1_k},z_{2_k}</math>  are the  <math>\nu</math> roots of the Chebyshev polynomial
<math>T_{\nu}(\ cos \chi)</math>
<math>T_{\nu}(\ cos \chi)</math>
thus
thus
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:<math>c_{mns}^{\mu \nu} (r)  \rightarrow  c_{\mu \nu}^{mnl} (r)</math>
:<math>c_{mns}^{\mu \nu} (r)  \rightarrow  c_{\mu \nu}^{mnl} (r)</math>


this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}:
this is done using the Blum transformation (Refs 7, 8 and 9):


:<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left(  
:<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left(  
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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>
:<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>


(see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}),
(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),
where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.
where <math>J_l(x)</math> is a [[Bessel functions |Bessel function]] of order <math>l</math>.
`step-down' operations can be performed by way of sin and cos operations
`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}.
of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado  Ref. 3.
The  Fourier-Bessel transform is also known as a '''Hankel transform'''.
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.
It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.
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:<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>
:<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>


====Conversion from the spatial reference frame back to the  axial reference frame====
====Conversion from the spatial reference frame back to the  axial reference frame====
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:<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho  \tilde{c}_2 (k)}</math>
:<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho  \tilde{c}_2 (k)}</math>


For molecular fluids (see Eq. 19 of Lado \cite{MP_1982_47_0283})
For molecular fluids (see Eq. 19 of Lado Ref. 3)


:<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>


:<math>\tilde{S}_{m}(k) = (-1)^{m}\rho \left[I - (-1)^{m} \rho \tilde C_{m}(k) \right]^{-1} \tilde C_{m}(k)\tilde C_{m}(k)
where <math>\tilde{{\mathbf S}}_{m}(k)</math> and <math>\tilde{\mathbf C}_{m}(k)</math> are matrices
</math>
where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices
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>.
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>.
For mixtures of simple fluids  (see \cite{JCP_1988_88_07715} and the thesis of Juan Antonio Anta pp. 107--109):


<math>\tilde \Gamma (k) =  D  \left[ I - D  \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>
For mixtures of simple fluids  (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):


:<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>


===Conversion back from Fourier space to Real space===
===Conversion back from Fourier space to Real space===
<math>\tilde{\gamma}_{mns}^{\mu \nu} (k)  \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>  
:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k)  \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>  
(basically the inverse of step 2).
(basically the inverse of step 2).
====Axial reference frame to spatial reference frame====
====Axial reference frame to spatial reference frame====
:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow  \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow  \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
====Inverse Fourier-Bessel transform====
====Inverse Fourier-Bessel transform====
:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow  \gamma^{mnl}_{\mu \nu} (r)</math> `Step-up' operations are given by Eq. 53 of \cite{MP_1982_47_0283}. The inverse Hankel transform is
:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow  \gamma^{mnl}_{\mu \nu} (r)</math>  
'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is
:<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>
:<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>
====Change from  spatial reference frame back to  axial reference frame====
====Change from  spatial reference frame back to  axial reference frame====
:<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow  \gamma_{mns}^{\mu \nu} (r)</math>.
:<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow  \gamma_{mns}^{\mu \nu} (r)</math>.
==Ng acceleration==
==Ng acceleration==
*[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)]
*[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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*[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)]
*[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)]
*[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)]
*[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)]
==Clebsch-Gordon coefficients and Racah's formula==
The Clebsch-Gordon coefficients are defined by
:<math>\Psi_{JM}= \sum_{M=M_1 + M_2} C_{M_1 M_2}^J \Psi_{M_1 M_2},</math>
where <math>J \equiv J_1 + J_2</math> and satisfies <math>(j_1j_2m_1m_2|j_1j_2m)=0</math>
for <math>m_1+m_2\neq m</math>.
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,
<math>V(j_1j_2j;m_1m_2m)</math>
(See also the [[Racah W-coefficients]], sometimes simply called the Racah coefficients).
*[http://dx.doi.org/10.1016/0010-4655(74)90059-9  Robert E. Beck and Bernard Kolman "Racah's outer multiplicity formula", Computer Physics Communications  '''8''' pp.  95-100 (1974)]
==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)]
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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)]
#[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)]
#[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)]
#[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)]
#[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)]
#[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)]
[[category: integral equations]]
[[category: integral equations]]

Latest revision as of 15:57, 31 January 2008

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[edit]

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:

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)} [edit]

(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[edit]

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 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[edit]

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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 \left. \gamma(12) \right. =\gamma (r,x_1x_2,y,z_1z_2)} .

Evaluate[edit]

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 \left. e_s(y) \right.=\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 \left. e_{\mu}(z) \right.= \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 \left. L_1 \right.= \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 \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:

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 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 [edit]

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 space 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 \tilde{c}_{mns}^{\mu \nu} (k)} [edit]

This is non-trivial and is undertaken in three steps:

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

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 (Refs 7, 8 and 9):

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)}

Fourier-Bessel Transforms[edit]

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 in Ref. 7 or Lado Eq. 39 in Ref. 3), 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 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.

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}

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

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)}

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 \tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)} [edit]

For simple 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 \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)

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{{\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 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{{\mathbf S}}_{m}(k)} 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 \tilde{\mathbf C}_{m}(k)} are matrices with elements 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 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 Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):

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{\Gamma}(k) = {\mathbf D} \left[{\mathbf I} - {\mathbf D} \tilde{\mathbf C}(k)\right]^{-1} \tilde{\mathbf C}(k)\tilde{\mathbf C}(k)}

Conversion back from Fourier space to Real space[edit]

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{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r)}

(basically the inverse of step 2).

Axial reference frame to spatial reference frame[edit]

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

Inverse Fourier-Bessel transform[edit]

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{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)}

'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is

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;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}

Change from spatial reference frame back to axial reference frame[edit]

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^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)} .

Ng acceleration[edit]

Angular momentum coupling coefficients[edit]

References[edit]

  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)
  7. 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)
  8. 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)
  9. 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)
  10. 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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