Code has been adapted from the original 4G implementation and
this: https://github.com/Jonas-Finkler/ewald-summation
For stress see https://wanglab.hosted.uark.edu/DLPOLY2/node65.html.
The original code was created by Jonas A. Finkler on 04.02.20.
IMPORTANT: if the first element of sigma is set to a value <= 0,
point charges are used. In this case self interaction of the charges
is omitted.
Lattice convention: lat(:,i) is the i-th lattice vector
author: Jonas A. Finkler
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functionget_optimal_eta_matrix( & lat, & sigma & )
Compute the optimal value of eta for the calculation of A or its derivatives.
When the matrix is built, each reciprocal term takes
\f$ \mathcal{O}(nat^2) \f$ computations.
Therefore reciprocal is not cheaper than real anymore and the
optimal choice of eta is different (it is independant of nat).
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functionget_optimal_eta_gauss_charges( & eta, & sigma & )
Compute the optimal value of eta for gaussian charges
Adjust the optimal eta to take into account the smearing of the charges.
Using maxval(sigma) (most diffuse Gaussian) → smallest correction,
slight oversmoothing for tighter Gaussians.
Using minval(sigma) (tightest Gaussian) → largest correction,
could be closer to “truly optimal” for the worst-case species, with more change in η.
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Compute Ewald energy, forces and stress
Gaussian charges (and also the compensating Ewald Gaussians) are of the form
\f$ \frac{1}{(2 \PI \sigma^2)^{3/2}} \exp\left(\frac{-r^2}{(2 \sigma^2)}\right) \f$
-> \f$ \sigma = \sqrt{2} \alpha_\text{cent} \f$
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subroutineewald_a( & nat, & pos, & lat, & sigma, & cutoff_real, & eta, & cutoff_recip, & a_matrix & )
Calculates the Coulomb matrix A, such that E = 1/2 * q^t * A * q
the periodic A matrix is not unique (depends on eta)
however, because sum(q)==0, A*q is independent of eta
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subroutineewald_a_recip( & nat, & pos, & lat, & eta, & cutoff, & a_matrix & )
Reciprocal space part of the Coulomb matrix calculation
Please not that this routine has a different scaling that
than the one for the energy force and stress calculation,
as each reciprocal lattice point requires O(nat**2) instead of O(nat) computations
hence the different optimal ewald eta for the matrix
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Compute the matrix sum_j dA_ij/dlat_jk * q_j
There is a dependence of A on eta. The partial derivative of this
is not considered here since it cancels out when sum(q_i) == 0.
These routines are not optimized, but they work and are tested vs finite
differences.
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subroutineewald_da_dlat_times_q_real( & nat, & pos, & lat, & charges, & sigma, & cutoff, & eta, & da_dlat_times_q & )
Real space part of the dA/dlat derivatives
This does not take the dependence on eta into account. However it should
cancel out with the reciprocal part.
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