Equation of State (EOS) analysis

📐 Quantum ESPRESSO `ev.x` Tutorial: EOS Fitting

This guide explains how to use Quantum ESPRESSO’s ev.x utility and an alternative Python script to fit energy vs volume or energy vs lattice parameter data using equations of state (EOS), such as the Birch-Murnaghan EOS.

📦 What is `ev.x`?

ev.x is a utility program in Quantum ESPRESSO used to fit energy–volume (or lattice parameter) data using standard EOS models to extract key material properties:

Equilibrium lattice constant
Bulk modulus (B₀)
Pressure derivative of the bulk modulus (B₀′)

Supported EOS models: - Birch 1st and 2nd order - Murnaghan - Keane

🧰 Step-by-Step: Using `ev.x`

1. 📄 Prepare Input File

Prepare a file like my_data.txt with two columns:

4.15        -563.40066682
4.16        -563.40365352
4.17        -563.40631110
4.18        -563.40864860
4.19        -563.41067215
4.20        -563.41238673
4.21        -563.41380140
4.22        -563.41492166
4.23        -563.41575443
4.24        -563.41630628
4.25        -563.41658018
4.26        -563.41658678
4.27        -563.41639698
4.28        -563.41588848
4.29        -563.41512615
4.30        -563.41411646
4.31        -563.41286694
4.32        -563.41137993
4.33        -563.40966219
4.34        -563.40771924
4.35        -563.40555615

Where each line corresponds to:

<lattice parameter or volume>         <total energy>

2. ▶️ Run `ev.x`

Launch the program and follow the prompts:

ev.x

Example session:

Lattice parameter or Volume are in (au, Ang) > Ang
Assuming Angstrom
Enter type of bravais lattice (fcc, bcc, sc, noncubic) > sc
Enter type of equation of state :
1=birch1, 2=birch2, 3=keane, 4=murnaghan > 1
Input file > my_data.txt
Minimization succeeded
Output file > birch1.out

📂 Output Files

🔹 `birch1.out`

This is a human-readable file containing:

Fitted parameters (a₀, B₀, B₀′, chi²)
Table of calculated vs fitted energies and derived pressures

Example excerpt:

# equation of state: birch 1st order.  chisq =   0.1832D-09
# a0 =  8.0419 a.u., k0 = 1511 kbar, dk0 =  3.99 d2k0 =  0.000 emin = -563.41665
# a0 =  4.25560 Ang, k0 = 151.2 GPa,  V0 =   520.09 (a.u.)^3,  V0 =   77.07 A^3 
...
Lat.Par    E_calc    E_fit    E_diff    Pressure    Enthalpy
...

🔸 `birch1.out.xml`

This XML contains the same information in machine-readable format. Key tags:

<CELL_PARAMETER_AU_A> 0.1909 0.10102 </CELL_PARAMETER_AU_A>
<BULK_MODULUS_KBAR> 4.25e+12 </BULK_MODULUS_KBAR>
<DERIVATIVE_BULK_MODULUS> 5.34 </DERIVATIVE_BULK_MODULUS>
<MINIMUM_ENERGY_RY> -74476.22 </MINIMUM_ENERGY_RY>

📐 The Birch-Murnaghan EOS Equation

The third-order Birch-Murnaghan equation is:

\[ E(V) = E_0 + \frac{9V_0 B_0}{16} \left\{ \left[ \left(\frac{V_0}{V} \right)^{2/3} - 1 \right]^3 B_0' + \left[ \left(\frac{V_0}{V} \right)^{2/3} - 1 \right]^2 \left[6 - 4 \left(\frac{V_0}{V} \right)^{2/3} \right] \right\} \]

Where: - \(E_0\): minimum energy
- \(V_0\): equilibrium volume
- \(B_0\): bulk modulus
- \(B_0'\): pressure derivative of the bulk modulus

🐍 Python Alternative to `ev.x`

You can also perform EOS fitting using the script below.

🔧 Requirements

Python
NumPy
SciPy
Matplotlib

📜 Script

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

# Define the Birch-Murnaghan EOS function
def birch_murnaghan(V, E0, B0, B0_prime, V0):
    eta = (V0 / V) ** (1/3)
    return E0 + (9 * B0 * V0 / 16) * ((eta**2 - 1)**2) * (6 + B0_prime * (eta**2 - 1) - 4 * eta**2)

# Load data from file
data = np.loadtxt('energy_vs_alat.dat')
a = data[:, 0]  # Lattice constants in Angstrom
E = data[:, 1]  # Total energies in Ry

# Convert energies from Ry to eV (1 Ry = 13.605693 eV)
E = E * 13.605693

# Convert lattice constants to volumes (assuming cubic structure)
V = a ** 3

# Initial guesses for the fitting parameters
E0_guess = np.min(E)
V0_guess = V[np.argmin(E)]
B0_guess = 1.0  # in eV/Å³
B0_prime_guess = 4.0

# Perform the curve fitting
popt, _ = curve_fit(birch_murnaghan, V, E, p0=[E0_guess, B0_guess, B0_prime_guess, V0_guess])
E0, B0, B0_prime, V0 = popt

# Generate volumes for plotting the fitted curve
V_fit = np.linspace(min(V), max(V), 100)
E_fit = birch_murnaghan(V_fit, E0, B0, B0_prime, V0)

# Plot the data and the fitted curve
plt.figure(figsize=(8, 6))
plt.plot(V, E, 'o', label='Data')
plt.plot(V_fit, E_fit, '-', label='Birch-Murnaghan Fit')
plt.xlabel('Volume (Å³)', fontsize=15)
plt.ylabel('Energy (eV)', fontsize=15)
plt.title('Birch-Murnaghan EOS Fit', fontsize=18)
plt.legend(fontsize=12)
plt.tight_layout()
plt.xticks(fontsize=14)
plt.show()

# Conversion factor from eV/Å³ to GPa
eV_per_A3_to_GPa = 160.217662

# Convert B0 to GPa and kbar
B0_GPa = B0 * eV_per_A3_to_GPa
B0_kbar = B0_GPa * 10

# Print results
print(f"Equilibrium energy (E0): {E0:.6f} eV")
print(f"Equilibrium volume (V0): {V0:.6f} Å³")
print(f"Bulk modulus (B0): {B0:.6f} eV/Å³")
print(f"Bulk modulus in GPa: {B0_GPa:.6f}")
print(f"Bulk modulus in kbar: {B0_kbar:.6f}")
print(f"Pressure derivative of B0 (B0'): {B0_prime:.6f}")

✅ Summary

Tool	EOS Types	Output Params	Best For
`ev.x`	Birch, Murnaghan, Keane	a₀, B₀, B₀′, χ²	Fast CLI-based EOS fitting
Python	Birch-Murnaghan	E₀, V₀, B₀, B₀′ (with plots)	Custom analysis & publication use

Equation of State (EOS) analysis

📐 Quantum ESPRESSO ev.x Tutorial: EOS Fitting

📦 What is ev.x?

🧰 Step-by-Step: Using ev.x