Finding lattice constants using EOS and the stress tensor

HCP

Let’s try to find the \(a\) and \(c\) lattice constants for HCP nickel using the EMT potential.

First, we make a good initial guess for \(a\) and \(c\) using the FCC nearest neighbor distance and the ideal \(c/a\) ratio:

import numpy as np

from ase.build import bulk
from ase.calculators.emt import EMT
from ase.io import Trajectory, read

a0 = 3.52 / np.sqrt(2)
c0 = np.sqrt(8 / 3.0) * a0

and create a trajectory for the results:

traj = Trajectory('Ni.traj', 'w')

Finally, we do the 9 calculations (three values for \(a\) and three for \(c\)):

eps = 0.01
for a in a0 * np.linspace(1 - eps, 1 + eps, 3):
    for c in c0 * np.linspace(1 - eps, 1 + eps, 3):
        ni = bulk('Ni', 'hcp', a=a, c=c)
        ni.calc = EMT()
        ni.get_potential_energy()
        traj.write(ni)

Analysis

Now, we need to extract the data from the trajectory. Try this:

>>> from ase.build import bulk
>>> ni = bulk('Ni', 'hcp', a=2.5, c=4.0)
>>> ni.cell
array([[ 2.5  ,  0.   ,  0.   ],
       [-1.25 ,  2.165,  0.   ],
       [ 0.   ,  0.   ,  4.   ]])

So, we can get \(a\) and \(c\) from ni.cell[0, 0] and ni.cell[2, 2]:

configs = read('Ni.traj@:')
energies = [config.get_potential_energy() for config in configs]
a = np.array([config.cell[0, 0] for config in configs])
c = np.array([config.cell[2, 2] for config in configs])

We fit the energy to this expression:

\[p_0 + p_1 a + p_2 c + p_3 a^2 + p_4 ac + p_5 c^2\]

The best fit is found like this:

functions = np.array([a**0, a, c, a**2, a * c, c**2])
p = np.linalg.lstsq(functions.T, energies, rcond=-1)[0]

and we can find the minimum like this:

p0 = p[0]
p1 = p[1:3]
p2 = np.array([(2 * p[3], p[4]), (p[4], 2 * p[5])])
a0, c0 = np.linalg.solve(p2.T, -p1)

with open('lattice_constant.csv', 'w') as fd:
    fd.write(f'{a0:.3f}, {c0:.3f}\n')

Results:

a	c
2.466	4.023

Using the stress tensor

One can also use the stress tensor to optimize the unit cell. For this we cannot use the EMT calculator.:

from ase.optimize import BFGS
from ase.constraints import StrainFilter
from gpaw import GPAW, PW
ni = bulk('Ni', 'hcp', a=a0,c=c0)
calc = GPAW(mode=PW(200),xc='LDA',txt='Ni.out')
ni.calc = calc
sf = StrainFilter(ni)
opt = BFGS(sf)
opt.run(0.005)

If you want the optimization path in a trajectory, add these lines before calling the run() method:

traj = Trajectory('path.traj', 'w', ni)
opt.attach(traj)