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Theory: interaction potentials

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This page describes interaction potentials supported by pyOZ, together with the description of their properties and possible pecularities in their handling.

Hard spheres potential (HS)

Hard spheres potentialpotential: U_{ij}^{\mathrm{HS}}(r) = \left\{ \begin{array}{l} \infty, \quad r < \sigma_{ij} \\ 0, \quad r > \sigma_{ij}}\end{array} \right.
derivative: infinite at the discontinuity, zero otherwise; special treatment needed
parameters: hard sphere diameters σ

Very simple potential with complicated treatment due to the discontinuity at the hard sphere diameter. It is worth recalling, that a Fourier series of a function converges at a place of discontinuity to the value half way between the left and right neighborhood ε of the discontinuity σ. Then

\displaystyle\mathrm{FT}(f(\sigma)) = \mathrm{FT}\left(\frac{1}{2}\left(f(\sigma-\varepsilon) + f(\sigma + \varepsilon)\right)\right)

Please note that for even in case of discrete functions, this is not the average of values at previous and next discretization point-only the infinitesimal neighborhood of the discontinuity has to be taken into account!

Coulomb potential (Coul)

Coulomb potentialpotential: \displaystyle U_{ij}^{\mathrm{Coul}}(r) = \frac{e^2}{4 \pi \varepsilon_0 \varepsilon_r}\frac{z_i z_j}{r}

derivative: \displaystyle\frac{d}{dr}U_{ij}^{\mathrm{Coul}}(r) = -\frac{e^2}{4 \pi \varepsilon_0 \varepsilon_r}\frac{z_i z_j}{r^2} = -\frac{U_{ij}^{\mathrm{Coul}}(r)}{r}

parameters: particle charges z_i, z_j, relative permittivity (dielectric constant) of the solvent (material) \varepsilon_r

Interaction between charged particles. Due to its long-ranged character, its treatment is complicated (discrete Fourier transform of a long-ranged function is not an easy task since this function doesn’t vanish at integration limits). See the article related to the Ng-renormalization for some information about how this problem can be solved.

When we convert the potential to the units of kT, we can write

\displaystyle U_{ij}^{\mathrm{Coul}}(r) = \lambda_B \frac{z_i z_j}{r}

where \lambda_B is the so called Bjerrum length-separation where Coulombic interaction is of thermal strength (kT). It follows from the previous text, that it is defined as

\displaystyle \lambda_B = \frac{e^2}{4 \pi \varepsilon_0 \varepsilon_r kT}

Charge-induced dipole interaction (CD)

potential: \displaystyle U_{ij}^{\mathrm{CD}} = -\frac{q_i^2 \alpha_j \lambda_B}{8 \pi \varepsilon_0 r^4} -\frac{q_j^2 \alpha_i \lambda_B}{8 \pi \varepsilon_0 r^4} = -\frac{\lambda_B q_i^2 \alpha_j^*}{2 r^4} -\frac{\lambda_B q_j^2 \alpha_i^*}{2 r^4}
derivative: \displaystyle \frac{d}{dr}U_{ij}^{\mathrm{CD}} = -\frac{4 U_{ij}^{\mathrm{CD}}}{r^5}

This potential corresponds to the interaction of a charge with induced dipole, represented by the value of the polarizabity \alpha. In pyOZ, the so-called excess polarizabilities \alpha^*=\alpha/4\pi\varepsilon_0 are used. As all potentials of the type r^{-n}, the evaluation of the derivative is very easy and straightforward.

Lennard-Jones potential (LJ)

Lennard-Jones potentialpotential: \displaystyle U_{ij}^{\mathrm{LJ}}(r) = 4 \epsilon_{ij} \left( \left(\frac{\sigma}{r}\right)^{12}-\left( \frac{\sigma}{r}\right)^6 \right)

derivative: \displaystyle \frac{d}{dr} U_{ij}^{\mathrm{LJ}}(r) = -24 \epsilon_{ij} \frac{1}{r} \left( 2 \left(\frac{\sigma}{r}\right)^{12}-\left( \frac{\sigma}{r}\right)^6 \right)

parameters: potential well depth \epsilon_{ij}, separation where the potential is zero \sigma_{ij}

This potential is frequently used as an approximation for the description of van der Waals (dispersion) forces. The repulsion part has no theoretical justification (it should be exponential, instead), but the attractive part describes the dispersion relatively well.

Normally, only ϵ and σ values for interaction between the same type of particles are given. The cross-parameters are calculated using the so-called combination rules. In practice (and in pyOZ as well), the arithmetic or geometric averages are used. In the so called Lorentz-Berthelot rules, geometric average is used for ϵ and arithmetic average for σ.

\displaystyle \sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}

\displaystyle \epsilon_{ij} = \sqrt{\epsilon_{ii}\epsilon_{jj}}

LJ potential provides analytical solution of the Ornstein-Zernike equation when Percus-Yevick closure is used.

Potential of mean force (PMF)

Potential read in from an external file, coming e.g., from a MD simulation of a pair of particles (“infinite dilution”) – the force between these particles is measured and converted into a potential). Derivative of this potential has to be calculated numerically.


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