Editing Computational implementation of integral equations
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
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; | ||
Line 46: | Line 46: | ||
====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 | ||
Line 139: | Line 139: | ||
(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. | ||
Line 149: | Line 149: | ||
:<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==== | ||
Line 171: | Line 172: | ||
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 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> | |||
===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> | |||
(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> | :<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)] | ||
Line 197: | Line 197: | ||
*[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)] |