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Theory: Ng renormalization

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Fourier transformation is problematic for long range potentials (for example Coulomb interaction) since they usually do not become zero on the discretization interval. Different approaches have to be used in such cases. Please note, that this text concerns only the Hypernetted Chain closure relation at the moment. Percus-Yevick related discussion will be added later (PY shouldn’t be used with long ranged (Coulomb) potential, since it doesn’t perform well!).

Ng renormalization (J. Chem. Phys. 1974, 61, 2680)

As suggested by Ng, for potentials in the form U(r) = A/r we use modified functions

U^s(r) = U(r)-U^{\mathrm{corr}}(r)

c^s(r) = c(r) + U^{\mathrm{corr}}(r)

\Gamma^s(r) = h(r)-c^s(r) = \Gamma (r)-U^{\mathrm{corr}}(r)

where U^{\mathrm{corr}}(r) is some suitable well-behaved function. The expression for the direct correlation function can be easily proven. c(r) is given within the Hypernetted Chain approximation by

c = \exp (-\beta U + h-c)-h + c -1 = \exp (-\beta U + h-c^s + U^{\mathrm{corr}})-h + c^s-U^{\mathrm{corr}}-1

c + U^{\mathrm{corr}} = c^s = \exp (-\beta U + U^{\mathrm{corr}} + \Gamma^s )-\Gamma^s-1

The second expression is used as a new (short-ranged) version of the HNC closure relation for the Ornstein-Zernike equation.

comparison of long- and short-ranged potentialsRegarding the suitable functions to be used in this procedure, there are some possibilities. We are looking for a function that is capable of counteracting the long range tail of the interaction potential. One of such functions is

\displaystyle U^{\mathrm{corr}} = \frac{A}{r} \mathrm{erf} (\alpha r)

with \alpha being adjustable parameter (pyOZ uses 1.08). This function agrees with the A/r potential overall with the exception of small distances. Therefore, subtracting this function from the interaction potential leads to a short ranged function (see figure for comparison, click for bigger version).

For the purpose of pyOZ, we are interested in the Fourier-Bessel transform of this function, which is performed using the sine transform (see the links if you are not familiar with these transforms!).

\displaystyle\mathrm{FBT}(U^{\mathrm{corr}}(r)) = \^U^{\mathrm{corr}}(k) = \frac{A}{k} 4 \pi \int\limits_0^\infty \frac{1}{r} \mathrm{erf} (\alpha r)\ r \sin (kr) dr

Integration per parts leads to

\displaystyle\^U^{\mathrm{corr}}(k) = \frac{A}{k} 4\pi \left( \left [ \mathrm{erf} (\alpha r) \frac{-\cos (kr)}{k} \right ]_0^\infty-\int\limits_0^\infty\frac{2}{\sqrt{\pi}} \exp (-(\alpha r)^2) \frac{\alpha}{k} (-1) \cos (kr) dr\right)

Since the first term vanishes at integration limits, we are left with

\displaystyle\^U^{\mathrm{corr}}(k) = \frac{A}{k^2} 4\pi \frac{\alpha}{\sqrt{\pi}}\ 2\int\limits_0^\infty \exp (-(\alpha r)^2) \cos (kr) dr

Solution of the integral is known, since it is equivalent to the Fourier transform of a gaussian

\displaystyle\mathrm{FT}\left(\exp \left(-(\alpha r)^2)\right\right) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty \exp (-(\alpha r)^2) e^{ikr} dr = \frac{2}{\sqrt{2\pi}} \int\limits_0^\infty \exp (-(\alpha r)^2) \cos (kr) dr = \frac{1}{\sqrt{2\alpha^2}} \exp \left( -\frac{k^2}{4\alpha^2}\right)

Since gaussian is an even function, the Fourier transform involves only cosine terms and the integral from -\infty to \infty can be replaced by twice the integral from zero to infinity. The used FT was unitary (having the same normalization factor for forward and inverse transform). In order to have non-unitary transform (used i pyOZ), we need to divide by the normalization factor 1/\sqrt{2\pi}, yielding

\displaystyle\mathrm{FT}(\exp (-(\alpha r)^2)) = \frac{\sqrt{2\pi}}{\sqrt{2\alpha^2}} \exp \left(-\frac{k^2}{4\alpha^2}\right) = \frac{\sqrt{\pi}}{\alpha} \exp \left(-\frac{k^2}{4\alpha^2}\right)

Taking this expression and substituting the integral discussed above yields

\displaystyle\^U^{\mathrm{corr}}(k) = \frac{A}{k^2} 4\pi \frac{\alpha}{\sqrt{\pi}}  \frac{\sqrt{\pi}}{\alpha} \exp \left(-\frac{k^2}{4\alpha^2}\right) = \frac{A}{k^2}  4 \pi  \exp \left(-\frac{k^2}{4\alpha^2}\right)

Last expression is the sought Fourier-Bessel transform of the function U^{\mathrm{corr}}.

Modified solution procedure can be described as follows (see Ornstein-Zernike equation theory and the pyOZ algorithm description to get more information). The O-Z equation is written by means of short-ranged functions

\mathbf{H} = \mathbf{C} + \mathbf{C}*\mathbf{H}

\mathbf{H} = \mathbf{C}^s-U^{\mathbf{corr}} + (\mathbf{C}^s-U^{\mathbf{corr}})*\mathbf{H}

Fourier transformation on short-range functions (the indirect correlation function uses modified closure defined previously) is performed discretely without problems. Recall, that Fourier transform of U^{\mathbf{corr}} is known analytically.

\^{\mathbf{H}} = \^{\mathbf{C}}^s-\^U^{\mathbf{corr}} + (\^{\mathbf{C}}^s-\^U^{\mathbf{corr}}) \^{\mathbf{H}}

Then, uncorrected (original) functions in the Fourier space are restored (FT is linear, i.e., FT(f+g) = FT(f)+FT(g))

\^{\mathbf{H}} = \^{\mathbf{C}}  + \^{\mathbf{C}}\^{\mathbf{H}}

This equation is solved for \^{\mathbf{H}} and indirect short-ranged correlation function \Hat{\Gamma}^s is calculated.

\Hat{\Gamma}^s = \^{\mathbf{H}}-\^{\mathbf{C}}^s

After transforming back to real space, short-ranged Gamma is used in the closure relation, new short-ranged direct correlation function is calculated and used for next iteration (until convergence). In reality, pyOZ uses this algorithm for convenience even in case of short-range only potentials. In such a case, the correction (and its Fourier transform as well) are set to zero, restoring thereby the non-modified Ornstein-Zernike equation.

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