How to Calculate Free Energy in LAMMPS: A Comprehensive Guide

Free energy calculations are a crucial aspect of molecular simulations, as they provide valuable insights into the thermodynamic properties of a system. In the context of the popular molecular dynamics simulation software LAMMPS (Large-Scale Atomic/Molecular Massively Parallel Simulator), there are two main methods for calculating free energy: the Free Energy Perturbation (FEP) method and the Thermodynamic Integration (TI) method. This comprehensive guide will delve into the details of each method, providing you with the necessary tools and understanding to perform accurate free energy calculations in LAMMPS.

Free Energy Perturbation (FEP) Method

The FEP method involves perturbing the potential energy of a system in small increments and calculating the change in free energy for each increment. The change in free energy for a single increment can be calculated using the following formula:

ΔA = -kBT ln< exp(-βΔU)>

where:
– ΔA is the change in free energy
– kB is the Boltzmann constant
– T is the temperature
– ΔU is the change in potential energy for the increment
– β = 1 / (kBT)
– The angle brackets denote an ensemble average

To perform FEP calculations in LAMMPS, you can use the compute fep command. This command calculates the exponential Boltzmann term and the potential energy difference for each increment, which you can then use to calculate the change in free energy.

Measurable and Quantifiable Details for FEP Calculations in LAMMPS

1. Potential Energy Difference (ΔU): The potential energy difference for each increment can be calculated using the pair_style and pair_coeff commands, which define the interatomic potential and its parameters.

2. Ensemble Average (< exp(-βΔU) >): The ensemble average of the exponential Boltzmann term can be calculated using the fix ave/time command, which allows you to time-average a quantity over the simulation.

3. Change in Free Energy (ΔA): The change in free energy for each increment can be calculated using the formula provided above.

4. Perturbation Increments: The size and number of perturbation increments can be controlled using the compute fep command, allowing you to fine-tune the resolution of the free energy calculation.

5. Convergence Criteria: You can monitor the convergence of the FEP calculation by tracking the change in free energy as a function of the number of increments or the simulation time.

6. Error Estimation: The statistical uncertainty in the FEP calculation can be estimated using techniques such as block averaging or the Bennet Acceptance Ratio method.

Thermodynamic Integration (TI) Method

The TI method involves integrating the derivative of the free energy with respect to a coupling parameter over a range of values for the coupling parameter. The derivative of the free energy with respect to the coupling parameter can be calculated using the following formula:

∂A/∂λ = -kBT < ∂U/∂λ >

where:
– ∂A/∂λ is the derivative of the free energy with respect to the coupling parameter λ
– kB is the Boltzmann constant
– T is the temperature
– ∂U/∂λ is the derivative of the potential energy with respect to the coupling parameter λ
– The angle brackets denote an ensemble average

To perform TI calculations in LAMMPS, you can use the compute ti command. This command calculates the derivative of the potential energy with respect to the coupling parameter for each increment, which you can then use to integrate the derivative of the free energy.

Measurable and Quantifiable Details for TI Calculations in LAMMPS

1. Derivative of Potential Energy (∂U/∂λ): The derivative of the potential energy with respect to the coupling parameter λ for each increment can be calculated using the compute ti command.

2. Ensemble Average (< ∂U/∂λ >): The ensemble average of the derivative of the potential energy can be calculated using the fix ave/time command.

3. Integral of the Derivative of Free Energy: The integral of the derivative of the free energy with respect to the coupling parameter can be calculated using numerical integration techniques, such as the trapezoidal rule or Simpson’s rule.

4. Coupling Parameter Range: The range of values for the coupling parameter λ can be controlled using the compute ti command, allowing you to explore the free energy landscape in detail.

5. Convergence Criteria: You can monitor the convergence of the TI calculation by tracking the change in the integrated free energy as a function of the number of increments or the simulation time.

6. Error Estimation: The statistical uncertainty in the TI calculation can be estimated using techniques such as block averaging or the Multistate Bennett Acceptance Ratio (MBAR) method.

Additional LAMMPS Tools for Free Energy Calculations

In addition to the compute fep and compute ti commands, LAMMPS provides several other tools that can be useful for free energy calculations:

1. fix ave/time: This command allows you to time-average a quantity over the simulation, which is essential for calculating ensemble averages in both FEP and TI methods.

2. thermo_style custom: This command allows you to customize the output of thermodynamic quantities, such as potential energy, temperature, and pressure, which can be useful for monitoring the progress of your free energy calculations.

3. variable: This command allows you to define and manipulate variables, which can be useful for storing and processing the results of your free energy calculations.

4. dump: This command allows you to output the state of your simulation at regular intervals, which can be useful for post-processing and analysis of your free energy calculations.

By leveraging these tools and the detailed information provided in this guide, you can perform accurate and reliable free energy calculations in LAMMPS, unlocking a deeper understanding of the thermodynamic properties of your molecular systems.

Reference:

1. LAMMPS tutorial on free energy calculations: https://lammpstutorials.github.io/sphinx/build/html/tutorials/level3/free-energy-calculation.html
2. LAMMPS documentation on the compute fep command: https://docs.lammps.org/compute_fep.html
3. LAMMPS documentation on the compute ti command: https://docs.lammps.org/compute_ti.html