# Calculating Delta-values¶

In this tutorial we compare the equation-of-state (EOS) calculated for 7 FCC metals using values from EMT, WIEN2k and experiment. Each EOS is described by three parameters:

• volume per atom

• bulk-modulus

• pressure derivative of bulk-modulus

Differences between two EOS’es can be measured by a single $$\Delta$$ value defined as:

$\sqrt{\frac{\int_{V_a}^{V_b} (E_1(V) - E_2(V))^2 dV} {V_b - V_a}},$

where $$E_n(V)$$ is the energy per atom as a function of volume. The $$\Delta$$ value can be calculated using the ase.utils.deltacodesdft.delta() function:

ase.utils.deltacodesdft.delta(v1: float, B1: float, Bp1: float, v2: float, B2: float, Bp2: float, symmetric=True) [source]

Calculate Delta-value between two equation of states.

Parameters:
• v1 (float) – Volume per atom.

• v2 (float) – Volume per atom.

• B1 (float) – Bulk-modulus (in eV/Ang^3).

• B2 (float) – Bulk-modulus (in eV/Ang^3).

• Bp1 (float) – Pressure derivative of bulk-modulus.

• Bp2 (float) – Pressure derivative of bulk-modulus.

• symmetric (bool) – Default is to calculate a symmetric delta.

Returns:

delta – Delta value in eV/atom.

Return type:

float

We get the WIEN2k and experimental numbers from the DeltaCodesDFT ASE-collection and we calculate the EMT EOS using this script:

from ase.calculators.emt import EMT
from ase.collections import dcdft
from ase.io import Trajectory

for symbol in ['Al', 'Ni', 'Cu', 'Pd', 'Ag', 'Pt', 'Au']:
traj = Trajectory(f'{symbol}.traj', 'w')
for s in range(94, 108, 2):
atoms = dcdft[symbol]
atoms.set_cell(atoms.cell * (s / 100)**(1 / 3), scale_atoms=True)
atoms.calc = EMT()
atoms.get_potential_energy()
traj.write(atoms)


And fit to a Birch-Murnaghan EOS:

import json
from pathlib import Path
from typing import Tuple

from ase.eos import EquationOfState as EOS

def fit(symbol: str) -> Tuple[float, float, float, float]:
V = []
E = []
V.append(atoms.get_volume() / len(atoms))
E.append(atoms.get_potential_energy() / len(atoms))
eos = EOS(V, E, 'birchmurnaghan')
eos.fit(warn=False)
e0, B, Bp, v0 = eos.eos_parameters
return e0, v0, B, Bp

data = {}  # Dict[str, Dict[str, float]]
for path in Path().glob('*.traj'):
symbol = path.stem
e0, v0, B, Bp = fit(symbol)
data[symbol] = {'emt_energy': e0,
'emt_volume': v0,
'emt_B': B,
'emt_Bp': Bp}

Path('fit.json').write_text(json.dumps(data))


Result for Pt:

Volumes in Ang^3:

 # symbol emt exp wien2k Al 15.93 16.27 16.48 Ni 10.60 10.81 10.89 Cu 11.57 11.65 11.95 Pd 14.59 14.56 15.31 Ag 16.77 16.85 17.85 Pt 15.08 15.02 15.64 Au 16.68 16.82 17.97

Bulk moduli in GPa:

 # symbol emt exp wien2k Al 39.70 77.14 78.08 Ni 176.23 192.46 200.37 Cu 134.41 144.28 141.33 Pd 180.43 187.19 168.63 Ag 100.06 105.71 90.15 Pt 278.67 285.51 248.71 Au 174.12 182.01 139.11

Pressure derivative of bulk-moduli:

 # symbol emt exp wien2k Al 2.72 4.45 4.57 Ni 3.76 4.00 5.00 Cu 4.21 4.88 4.86 Pd 5.17 5.00 5.56 Ag 4.75 4.72 5.42 Pt 5.31 5.18 5.46 Au 5.46 6.40 5.76

Now, we can calculate $$\Delta$$ between EMT and WIEN2k for Pt:

>>> from ase.utils.deltacodesdft import delta
>>> from ase.units import kJ
>>> delta(15.08, 278.67 * 1e-24 * kJ, 5.31,
...       15.64, 248.71 * 1e-24 * kJ, 5.46)
0.03205389052984122


Here are all the values (in meV/atom) calculated with the script below:

 # symbol emt-exp emt-wien2k exp-wien2k Al 5.9 8.6 3.6 Ni 8.6 12.5 3.7 Cu 2.7 11.9 9.5 Pd 1.0 27.6 29.0 Ag 1.9 22.4 21.3 Pt 3.5 32.2 35.9 Au 5.9 43.7 39.4
import json
from pathlib import Path

import matplotlib.pyplot as plt
import numpy as np

from ase.collections import dcdft
from ase.eos import birchmurnaghan
from ase.units import kJ
from ase.utils.deltacodesdft import delta

# Insert values from experiment and WIEN2k:
for symbol in data:
dcdft_dct = dcdft.data[symbol]
dcdft_dct['exp_B'] *= 1e-24 * kJ
dcdft_dct['wien2k_B'] *= 1e-24 * kJ
data[symbol].update(dcdft_dct)

for name in ['volume', 'B', 'Bp']:
with open(name + '.csv', 'w') as f:
print('# symbol, emt, exp, wien2k', file=f)
for symbol, dct in data.items():
values = [dct[code + '_' + name]
for code in ['emt', 'exp', 'wien2k']]
if name == 'B':
values = [val * 1e24 / kJ for val in values]
print(f'{symbol},',
', '.join(f'{value:.2f}' for value in values),
file=f)

with open('delta.csv', 'w') as f:
print('# symbol, emt-exp, emt-wien2k, exp-wien2k', file=f)
for symbol, dct in data.items():
# Get v0, B, Bp:
emt, exp, wien2k = ((dct[code + '_volume'],
dct[code + '_B'],
dct[code + '_Bp'])
for code in ['emt', 'exp', 'wien2k'])
print(f'{symbol},',
'{:.1f}, {:.1f}, {:.1f}'.format(delta(*emt, *exp) * 1000,
delta(*emt, *wien2k) * 1000,
delta(*exp, *wien2k) * 1000),
file=f)

if symbol == 'Pt':
va = min(emt[0], exp[0], wien2k[0])
vb = max(emt[0], exp[0], wien2k[0])
v = np.linspace(0.94 * va, 1.06 * vb)
for (v0, B, Bp), code in [(emt, 'EMT'),
(exp, 'experiment'),
(wien2k, 'WIEN2k')]:
plt.plot(v, birchmurnaghan(v, 0.0, B, Bp, v0), label=code)
e0 = dct['emt_energy']
V = []
E = []