Thermodynamic integration between machine-learned force fields

Calculating the free energy difference [math]\displaystyle{ \Delta A }[/math] between two species can be difficult, especially if the structural changes between the species are significant. For this, we can use thermodynamic integration (TI), integrating over a coupling parameter λ, designating one species as the interacting system ([math]\displaystyle{ \lambda = 1 }[/math]) and the other species as the non-interacting system ([math]\displaystyle{ \lambda = 0 }[/math]):

[math]\displaystyle{ \Delta A = \int_0^1 \langle U_1 - U_0 \rangle_{\lambda} d\lambda }[/math]

where U₁ and U₀ are the potential energies of the interacting and non-interacting systems, respectively.

Introduction

In this how-to, we will go over the example of an Fe²⁺/Fe³⁺ electrochemical half-cell . A molecular dynamics (MD) calculation is performed in parallel using TI. Since this requires many steps, we use two machine-learned force fields (MLFFs) to speed this up. We can check the accuracy of this later by performing another TI from the MLFF to a DFT functional.

During the Fe²⁺/Fe³⁺ half-cell reaction, one electron is transferred from the reservoir. The Helmholtz free energy difference ΔA is:

[math]\displaystyle{ \Delta A = \int_0^1 \langle U_1 - U_0 \rangle_{\lambda} d\lambda - \mu n }[/math]

where [math]\displaystyle{ \langle U_1 - U_0 \rangle }[/math] is the potential energy difference between an interacting system ([math]\displaystyle{ \lambda=1 }[/math]) and non-interacting system ([math]\displaystyle{ \lambda=0 }[/math]) taken from thermodynamic integration with coupling parameter [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ \mu }[/math] is the reference chemical potential, and [math]\displaystyle{ n }[/math] is the number of electrons involved in the reaction.

There is an accompanying how-to on calculating the reference chemical potential. In this how-to, you will learn how to calculate the remaining [math]\displaystyle{ \int_0^1 \langle U_1 - U_0 \rangle_{\lambda} d\lambda }[/math] term, then add this to [math]\displaystyle{ \mu }[/math] to calculate the free energy difference [math]\displaystyle{ \Delta A }[/math].

Procedure and input

Two MD calculations need to be run in parallel between two different systems: Fe³⁺ in 64 H₂O (Fe3P_64H2O) and Fe²⁺ in 64 H₂O (Fe2P_64H2O). I.e., [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math] and [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math].

Procedure

The procedure is split into two parts, the calculation part:

0. Obtain an initial structure
1. Prepare the directory structure for the TI
2. TI calculation

and the post-processing:

3. Extracting and averaging the energies
4. Integrate to obtain the free energy

Input files

The POSCAR files can be found in the calculating the electrochemical potential how-to page.

The Gamma-point only is used for the KPOINTS file:

Gamma-point only
 0
Monkhorst Pack
 1 1 1
 0 0 0

The INCAR files are provided in the text and discussed there. Finally, standard POTCAR files are used: PAW_PBE H 15Jun2001, PAW_PBE O 08Apr2002, and PAW_PBE Fe_sv 23Jul2007.

Calculation (MLFF:MLFF)

The first step is thermodynamic integration between two species: Fe³⁺ and Fe²⁺ using MLFFs. The procedure is as follows:

Step 0: Obtain initial POSCARs for Fe³⁺ and Fe²⁺

A starting structure for the MD simulations in TI should be carefully chosen. In Ref. , a homemade MD simulation program was used to anneal the two systems: [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{3+} }[/math] and [math]\displaystyle{ [\mathrm{Fe}(\mathrm{H}_2\mathrm{O})_n]^{2+} }[/math] from 1000 K to 400 K in a 1 ns NVT ensemble MD simulation. We begin with the final structure from each of those two simulations.

Tip: You can learn about running an MD simulation in VASP in the MD calculations page or the first part of the MD tutorials.

Step 1: Preparing the directories

To perform TI, you need to use the VCAIMAGES tag. This requires a parent directory from which the TI is run, containing the following INCAR file:

# TI settings
VCAIMAGES = 0.25
NCORE_IN_IMAGE1 = 12

# MD settings
IBRION = 0
ISYM = 0
NSW = 100000
POTIM = 1.0
TEBEG = 298
TEEND = 298

MDALGO = 2
ISIF = 2
SMASS = 0

POMASS = 2.0 16.0 55.847
RANDOM_SEED =         248489752                0                0

# General settings
ML_ESTBLOCK = 100                    # only write to OUTCAR every 100 ionic steps

IMAGE_1 {
#Machine learning
ML_LMLFF = .TRUE.                    # switches on machine learning
ML_MODE = run
}

IMAGE_2 {
#Machine learning
ML_LMLFF = .TRUE.                    # switches on machine learning
ML_MODE = run
}

Important: NCORE_IN_IMAGE1 sets the number of cores used by the first image; the second image uses the remaining cores. VCAIMAGES sets the λ parameter for the thermodynamic integration. In this case, [math]\displaystyle{ \lambda = 0.25 }[/math].

The VCAIMAGES tag runs calculations in two image directories 01 and 02, which contain the two "non-interacting" λ=0 amd "interacting" λ=1 systems, respectively. In this case, Fe³⁺ (the oxidised state Ox) and Fe²⁺ (the reduced state Red). Since MLFFs are used, they must contain the ML_FF's trained for the Ox and Red systems, respectively. Make sure to place identical POSCAR, POTCAR, and KPOINTS files in each of these image directories, as well as their respective MLFFs.

Tip: You can link the ML_FF files to the directory where they were refitted using: ln -s ${ML_FFN_directory}/ML_FFN ML_FF

Step 2: Molecular dynamics calculation

Set up the TI calculations for different λ values defined by VCAIMAGES (e.g., 0.0, 0.25, 0.5, 0.75, and 1.0). This will require 5 separate directories, e.g.:

lambda_0p0  lambda_0p25  lambda_0p5  lambda_0p75  lambda_1p0

for 0.0, 0.25, 0.5, 0.75, and 1.0, respectively. The calculation should be submitted from the parent directory, similar to for NEB. These will each run two parallel MD calculations for the value of λ defined in VCAIMAGES.

Important: We have used the Nosé-Hoover thermostat (MDALGO = 2). The trajectories in 01 and 02 directories need to be identical, so you cannot use the Langevin thermostat as it introduces random numbers.

Post-processing

Step 3: Extracting and averaging the energies

Each of these calculations will output the energy for each MD step in the following format:

free  energy ML TOTEN  =      -953.27166392 eV

You should take the values for the 01 and 02 directories separately, e.g., by grepping for the energies and temperatures in each of the sub-directories:

grep "free  energy" 01/OUTCAR | awk '{print $5}' > free_E.dat
grep "free  energy ML TOTEN  =" 02/OUTCAR | awk '{print $6}' >> free_E.dat
grep temperature 01/OUTCAR | awk '{print $6}' > T.dat

then calculate the two ensemble averages, before taking the difference between the two. We exclude the first 20000 MD steps to allow time for equilibration. You can do this with the following script, which will plot the probability vs potential energy, the MD step number vs the potential energy, and the MD step number vs the temperature:

from py4vasp import plot
import plotly.graph_objects as go
import numpy as np
import os 
import matplotlib.pyplot as plt
from scipy.stats import gaussian_kde

def delta_A(path, directory, lower, upper, step):
    number_str = directory.split("_")[1] 
    number_str = number_str.replace("p", ".")
    Lambda = float(number_str)
    print(path+directory)
    
    data = np.genfromtxt(path + str(directory) + "/free_E.dat", dtype=None, encoding=None)
    length=len(data)
    print(length)
    
    U_ox, U_red = data[lower:upper:step], data[int(length/2)+lower:int(length/2)+upper:step]
    n_ox, n_red = range(lower, upper+1, step), range(int(length/2)+lower,int(length/2)+upper+1, step)
    print(n_ox, n_red)
    print(len(U_ox), len(U_red))
    U_red_av, U_ox_av = np.average(U_red), np.average(U_ox)
    #U_1_0 =  U_red - U_ox
    U_1_0_av = U_red_av - U_ox_av
    #U_1_0_av = np.average(U_1_0)
    return(Lambda, (U_1_0_av), (U_red-U_ox))

def diff_cutoff(path, lower, upper, step):
    files = [d for d in os.listdir(path) if d.startswith("lambda_")]

    lambdas, A = [], []
    for a in ['lambda_0p0', 'lambda_0p25', 'lambda_0p5', 'lambda_0p75', 'lambda_1p0']:
        print(files)
        temp1, temp2, data = delta_A(path, a, lower, upper, step)
        lambdas.append(temp1)
        A.append(temp2)
        print(lambdas, A)
        U = data
        # Plot first graph
        kde = gaussian_kde(data)
        x = np.linspace(min(data), max(data), 20)
        ax1.plot(x, kde(x), '-', linewidth=2, alpha= 0.5, label=a)
        ax1.set_title('P vs. U')
        ax1.set_xlabel(r'$\Delta U_{ML}$')
        ax1.set_ylabel(r'$P(\Delta U_{ML})$')
        ax1.legend()       

        # Plot second graph
        ax2.set_title('MD step vs. U')
        ax2.set_xlabel(r'$\Delta U_{ML}$')
        ax2.set_ylabel('MD step')
        ax2.plot(U, range(lower,upper,step), '-', linewidth=2, alpha= 0.5, label=a)        

        # Plot T
        data = np.genfromtxt(path + str(a) + "/T.dat", dtype=None, encoding=None)
        ax3.plot(data, range(len(data)), '-', linewidth=2, alpha= 0.5, label=a)
        ax3.set_title('MD step vs. T')
        ax3.set_xlabel('T')
        ax3.set_ylabel('MD step')
        ax3.legend()  
        
    return(lambdas, A)

path = "$PATH_TO_TI_MLFF_MLFF_DIRECTORIES/"

l_cutoff, A_cutoff = [], []

# Create a figure with 1 row and 2 columns
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(10, 4))  # 1 row, 2 columns

lambdas, A = diff_cutoff(path, 20000, 100000, 1)
l_cutoff.append(lambdas)
A_cutoff.append(A)

print((A_cutoff))
print((l_cutoff))

ax1.legend()
ax2.legend()

# Adjust layout so titles/labels don't overlap
plt.tight_layout()
plt.savefig("TI_mlff_mlff.png")

Figure 4. Probability vs potential energy (cf. Supplementary Figure 8), the MD step number vs the potential energy, and the MD step number vs the temperature for between Fe²⁺ (λ = 0) and Fe³⁺ (λ = 1).

Step 4: Integrate to obtain the free energy

You can then plot the free energy against the lambda values; ideally, it should be almost linear:

Figure 5. Free energy difference ΔA for thermodynamic integration using a parameter λ between Fe³⁺ (λ = 0) and Fe²⁺ (λ = 1) (cf. Supplementary Figure 8).

With the free energy for each individual λ, you can integrate over them to obtain the free energy of the TI.

[math]\displaystyle{ \Delta A = \int_0^1 \langle U_1 - U_0 \rangle_{\lambda} d\lambda - ne\Delta \bar{\phi} }[/math]

from scipy.integrate import simpson
print('Simpson: ' + str(simpson(A_cutoff, l_cutoff)[0]))

This gave a value of: -1.272 eV. This is almost identical to the literature -1.260 obtained for Std. POTCARs provided by R. Jinnouchi (Ref. uses GW POTCARs). Adding this to the value of [math]\displaystyle{ e \Delta \bar{\phi} }[/math] from the previous how-to gives [math]\displaystyle{ \Delta A = -4.98 \: \mathrm{ eV} }[/math]. Considering that the redox potential [math]\displaystyle{ U_{redox} = -\Delta A /e }[/math], [math]\displaystyle{ U_{redox} }[/math] can be calculated as:

[math]\displaystyle{ U_{redox} = -\Delta A/e = -(-1.27 - (3.71))/1 = 4.98 \: \mathrm{ eV} }[/math].

Comparing this to the literature value of 4.95 eV (for Std. POTCAR; cf. Supplementary Table 6 for GW POTCAR), we are in reasonable agreement despite taking additional approximations.

Recommendations and advice

• Make sure to carefully check that you are using the correct ML_FF files for each directory. If you mix them up, then you can always switch λ in the post-processing.
• If you see that during the MD simulation (in the TI), U drifts far from the average, this is an indication that a chemical change has happened. Check to see if an Fe-O bond has formed. This should not happen and is an indication that your force field is unstable.

Related tags and articles

Tags

• VCAIMAGES, NCORE_IN_IMAGE1

Theory

• Thermodynamic integration

How-tos

• Construction:Thermodynamic integration: redox potential
• Construction:Vacuum reference
• Construction:Thermodynamic integration between MLFF and GGAs
• Construction:Thermodynamic integration between density functionals