Editing Computational implementation of integral equations
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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 [[ | and a [[Closures | 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 [[ | where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math> | ||
where <math>y_j</math> are the <math>\nu</math> roots of the [[ | where <math>y_j</math> are the <math>\nu</math> roots of the [[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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(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3), | (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 [[ | where <math>J_l(x)</math> is a [[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 Ref. 3. | of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3. | ||
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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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For molecular fluids (see Eq. 19 of Lado Ref. 3) | For molecular fluids (see Eq. 19 of Lado Ref. 3) | ||
:<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) | |||
</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 Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109): | For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109): | ||
:<math>\tilde | :<math>\tilde \Gamma (k) = D \left[ I - D \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math> | ||
===Conversion back from Fourier space to Real space=== | ===Conversion back from Fourier space to Real space=== | ||
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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)] |