Other built-in calculators¶
- rc: float
Cutoff radius for Coulomb part.
- width: float
Width for cutoff function for Coulomb part.
TIP4P potential for water.
Requires an atoms object of OHH,OHH, … sequence Correct TIP4P charges and LJ parameters set automatically.
Virtual interaction sites implemented in the following scheme: Original atoms object has no virtual sites. When energy/forces are requested:
virtual sites added to temporary xatoms object
energy / forces calculated
forces redistributed from virtual sites to actual atoms object
This means you do not get into trouble when propagating your system with MD while having to skip / account for massless virtual sites.
This also means that if using for QM/MM MD with GPAW, the EmbedTIP4P class must be used.
Lennard Jones potential calculator
The fundamental definition of this potential is a pairwise energy:
u_ij = 4 epsilon ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )
For convenience, we’ll use d_ij to refer to “distance vector” and
r_ijto refer to “scalar distance”. So, with position vectors \(r_i\):
r_ij = | r_j - r_i | = | d_ij |
The derivative of u_ij is:
d u_ij / d r_ij = (-24 epsilon / r_ij) ( sigma^12/r_ij^12 - sigma^6/r_ij^6 )
We can define a pairwise force
f_ij = d u_ij / d d_ij = d u_ij / d r_ij * d_ij / r_ij
The terms in front of d_ij are often combined into a “general derivative”
du_ij = (d u_ij / d d_ij) / r_ij
The force on an atom is:
f_i = sum_(j != i) f_ij
There is some freedom of choice in assigning atomic energies, i.e. choosing a way to partition the total energy into atomic contributions.
We choose a symmetric approach:
u_i = 1/2 sum_(j != i) u_ij
The total energy of a system of atoms is then:
u = sum_i u_i = 1/2 sum_(i, j != i) u_ij
The stress can be written as ( \((x)\) denoting outer product):
sigma = 1/2 sum_(i, j != i) f_ij (x) d_ij = sum_i sigma_i ,with atomic contributions
sigma_i = 1/2 sum_(j != i) f_ij (x) d_ij
For computational efficiency, we minimise the number of pairwise evaluations, so we iterate once over all the atoms, and use NeighbourList with bothways=False. In terms of the equations, we therefore effectively restrict the sum over \(i != j\) to \(j > i\), and need to manually re-add the “missing” \(j < i\) contributions.
Another consideration is the cutoff. We have to ensure that the potential goes to zero smoothly as an atom moves across the cutoff threshold, otherwise the potential is not continuous. In cases where the cutoff is so large that u_ij is very small at the cutoff this is automatically ensured, but in general, \(u_ij(rc) != 0\).
In order to catch this case, this implementation shifts the total energy
u'_ij = u_ij - u_ij(rc)
which ensures that it is precisely zero at the cutoff. However, this means that the energy effectively depends on the cutoff, which might lead to unexpected results!
Default values chosen to be similar as Lennard-Jones.