Magnesium

import profess
import ase_tools
import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize
import spglib

Simplfied energy landscape

At ambient pressure, magnesium crystallizes in the hexagonal close-packed (HCP) structure.

We begin by plotting the energy of HCP Mg for a range of volumes and c/a ratios.

volumes = np.linspace(20,28,12)
c_a_ratios = np.linspace(1.4,2,12)
energies = np.empty([volumes.size, c_a_ratios.size])

for i, vol_per_atom in enumerate(volumes):
    for j, c_a_ratio in enumerate(c_a_ratios):
        
        # generate lattice constant, cell vectors, and ion positions
        a = ((2*vol_per_atom) / (np.sqrt(3)/2*c_a_ratio))**(1/3)
        cell_vectors = np.array([[   a,              0,           0],
                                 [-a/2, np.sqrt(3)/2*a,           0],
                                 [   0,              0, c_a_ratio*a]])
        frac_coords = np.array([[  0,   0,   0],
                                [2/3, 1/3, 1/2]])
        xyz_coords = (cell_vectors.T).dot(frac_coords.T).T
        # create a profess system
        energy_cutoff = 600
        system = (
            profess.System.create(cell_vectors, energy_cutoff, ['a','ev'])
            .add_ions('potentials/mg.gga.recpot', xyz_coords, 'a')
            .add_electrons()
            .add_wang_teter_functional()
            .add_hartree_functional()
            .add_perdew_burke_ernzerhof_functional() 
            .add_ion_electron_functional()
            .add_ion_ion_interaction()
        )
        # compute the energy per atom
        energies[i,j] = system.minimize_energy().energy('ev')/2
    
fig = plt.figure(figsize=(10,7))
ax = plt.axes(projection='3d')
x, y = np.meshgrid(volumes, c_a_ratios, sparse=False, indexing='ij')
ax.contour3D(x, y, energies, 80, cmap='gist_earth', zorder=1)
ax.set_xlabel('Volume [$A^3$/atom]', labelpad=10)
ax.set_ylabel('c/a', labelpad=10)
ax.set_zlabel('Energy [eV/atom]', labelpad=15);
ax.view_init(25, 300)

Structure optimization

To find the optimal volume and c/a ratio, we relax the geometry, eliminating forces on ions and/or stress on the unit cell.

ase_tools.minimize_forces_stress(system, 'BFGSLineSearch', 1e-4);

vol_per_atom = system.volume('a3')/2

# standarize the relaxed lattice to extract c/a ratio (uses spglib)
cell_vectors = np.array(system.box_vectors('a'))
xyz_coords = np.array(system.ions_xyz_coords('a'))
frac_coords = np.linalg.inv(cell_vectors.T).dot(xyz_coords.T).T
standardized_cell, _, _ = spglib.standardize_cell((cell_vectors,frac_coords,(1,1)))
c_a_ratio = standardized_cell[2,2] / standardized_cell[0,0]

print('\n')
print('Relaxed volume per atom:  {:5.2f} A3'.format(vol_per_atom))
print('Relaxed c/a:              {:5.2f}'.format(c_a_ratio))

x = vol_per_atom*np.ones(10)
y = c_a_ratio*np.ones(10)
ax.plot(x, y, np.linspace(energies.min(),energies.max(),10), color='red', linewidth=2, label='optimal', zorder=2)
ax.legend(loc='upper left')
fig

                Step[ FC]     Time          Energy          fmax
BFGSLineSearch:    0[  0] 09:55:16       -1.769792        0.0884

BFGSLineSearch:    1[  4] 09:55:18       -1.777926        0.0316

BFGSLineSearch:    2[  6] 09:55:19       -1.778383        0.0420

BFGSLineSearch:    3[  8] 09:55:20       -1.780262        0.0055

BFGSLineSearch:    4[  9] 09:55:21       -1.780307        0.0009

BFGSLineSearch:    5[ 10] 09:55:21       -1.780308        0.0003

BFGSLineSearch:    6[ 12] 09:55:22       -1.780308        0.0000

Relaxed volume per atom:  23.05 A3
Relaxed c/a:               1.63

Alternate approach

To illustrate another strategy, we reproduce the previous result by reformulating the task as an abstract optimization problem, which we then solve with a simplex algorithm from scipy.

# initial guess for the volume and c/a
x0 = (24, 1.7)

# objective function
def energy_per_atom(x):
    vol_per_atom, c_over_a = x
    # generate lattice constant, cell vectors, and ion positions
    a = ((2*vol_per_atom) / (np.sqrt(3)/2*c_a_ratio))**(1/3)
    cell_vectors = np.array([[   a,              0,           0],
                             [-a/2, np.sqrt(3)/2*a,           0],
                             [   0,              0, c_over_a*a]])
    frac_coords = np.array([[  0,   0,   0],
                            [2/3, 1/3, 1/2]])
    xyz_coords = (cell_vectors.T).dot(frac_coords.T).T
    # create a profess system
    energy_cutoff = 600
    system = (
        profess.System.create(cell_vectors, energy_cutoff, ['a','ev'])
        .add_ions('potentials/mg.gga.recpot', xyz_coords, 'a')
        .add_electrons()
        .add_wang_teter_functional()
        .add_hartree_functional()
        .add_perdew_burke_ernzerhof_functional() 
        .add_ion_electron_functional()
        .add_ion_ion_interaction()
    )
    # return the energy per atom
    return system.minimize_energy().energy('ev')/2

# solve with scipy's Nelder-Mead simplex algorithm
result = scipy.optimize.minimize(energy_per_atom, x0, method='Nelder-Mead')
print('Relaxed volume per atom:  {:5.2f} A3'.format(result.x[0]))
print('Relaxed c/a:              {:5.2f}'.format(result.x[1]))

Relaxed volume per atom:  23.05 A3
Relaxed c/a:               1.63