sarkas.potentials.egs#

Module for handling Exact Gradient corrected Screened (EGS) Potential.

Potential#

The exact-gradient screened (EGS) potential introduces new parameters that can be easily calculated from initial inputs. Density gradient corrections to the free energy functional lead to the first parameter, \(\nu\),

\[\nu = - \frac{3\lambda}{\pi^{3/2}} \frac{4\pi \bar{e}^2 \beta }{\Lambda_{e}} \frac{d}{d\eta} \mathcal I_{-1/2}(\eta),\]

where \(\lambda\) is a correction factor; \(\lambda = 1/9\) for the true gradient corrected Thomas-Fermi model and \(\lambda = 1\) for the traditional von Weissaecker model, \(\mathcal I_{-1/2}[\eta_0]\) is the Fermi Integral of order \(-1/2\), and \(\Lambda_e\) is the de Broglie wavelength of the electrons.

In the case \(\nu < 1\) the EGS potential takes the form

\[U_{ab}(r) = \frac{Z_a Z_b \bar{e}^2 }{2r}\left [ ( 1+ \alpha ) e^{-r/\lambda_-} + ( 1 - \alpha) e^{-r/\lambda_+} \right ],\]

with

\[\lambda_\pm^2 = \frac{\nu \lambda_{\textrm{TF}}^2}{2b \pm 2b\sqrt{1 - \nu}}, \quad \alpha = \frac{b}{\sqrt{b - \nu}},\]

where the parameter \(b\) arises from exchange-correlation contributions, see below.n On the other hand \(\nu > 1\), the pair potential has the form

\[U_{ab}(r) = \frac{Z_a Z_b \bar{e}^2}{r}\left [ \cos(r/\gamma_-) + \alpha' \sin(r/\gamma_-) \right ] e^{-r/\gamma_+}\]

with

\[\gamma_\pm^2 = \frac{\nu\lambda_{\textrm{TF}}^2}{\sqrt{\nu} \pm b}, \quad \alpha' = \frac{b}{\sqrt{\nu - b}}.\]

Neglect of exchange-correlational effects leads to \(b = 1\) otherwise

\[b = 1 - \frac{2}{8} \frac{1}{k_{\textrm{F}}^2 \lambda_{\textrm{TF}}^2 } \left [ h\left ( \Theta \right ) - 2 \Theta h'(\Theta) \right ]\]

where \(k_{\textrm{F}}\) is the Fermi wavenumber and \(\Theta = (\beta E_{\textrm{F}})^{-1}\) is the electron degeneracy parameter` calculated from the Fermi energy.

\[h \left ( \Theta \right) = \frac{N(\Theta)}{D(\Theta)}\tanh \left( \Theta^{-1} \right ),\]
\[N(\Theta) = 1 + 2.8343\Theta^2 - 0.2151\Theta^3 + 5.2759\Theta^4,\]
\[D \left ( \Theta \right ) = 1 + 3.9431\Theta^2 + 7.9138\Theta^4.\]

Force Error#

The EGS potential is always smaller than pure Yukawa. Therefore the force error is chosen to be the same as Yukawa’s

\[\Delta F = \frac{q^2}{4 \pi \epsilon_0} \sqrt{\frac{2 \pi n}{\lambda_{-}}}e^{-r_c/\lambda_-}\]

This overestimates it, but it doesn’t matter.

Potential Attributes#

The elements of sarkas.potentials.core.Potential.matrix are

if sarkas.core.Parameters.nu less than 1:

matrix[0] = q_iq_j/4pi eps0
matrix[1] = nu
matrix[2] = 1 + alpha
matrix[3] = 1 - alpha
matrix[4] = 1.0 / lambda_minus
matrix[5] = 1.0 / lambda_plus

else

matrix[0] = q_iq_j/4pi eps0
matrix[1] = nu
matrix[2] = 1.0
matrix[3] = alpha prime
matrix[4] = 1.0 / gamma_minus
matrix[5] = 1.0 / gamma_plus

Functions

calc_force_error_quad(a, rc, pot_matrix)

Calculate the force error by integrating the square modulus of the force over the neglected volume.n The force error is calculated from

egs_force(r_in, pot_matrix)

Numba'd function to calculate the potential and force between particles using the EGS Potential.

force_error_integrand(r, pot_matrix)

potential_derivatives(r, pot_matrix)

Calculate the first and second derivatives of the potential.

pretty_print_info(potential)

Print potential specific parameters in a user-friendly way.

update_params(potential, species)

Assign potential dependent simulation's parameters.