Thermodynamic integration calculations

VASP supports three approaches to thermodynamic integration (TI):

• TI between any two states using VCAIMAGES.
• TI with a harmonic solid or ideal gas as a reference state using SCALEE.
• TI with a harmonic solid as a reference state using TILAMBDA.

Details on choosing the ensemble size and how to perform the integration are described in the main text and especially the supplementary information of reference . Caution: the tag ISPECIAL=0 used in that reference is not valid anymore, instead the tag PHON_NSTRUCT=-1 is used.

TI using VCAIMAGES

Main tags:

• VCAIMAGES
• NCORE_IN_IMAGE1.

Special requirements:

• A parent directory with two subdirectories: 01 and 02 (these represent two images).
• All three directories must include the VCAIMAGES and NCORE_IN_IMAGE1 tags in their INCAR files, with identical values.
• Subdirectories must contain identical POSCAR and POTCAR files. Other files (e.g., KPOINTS) can differ, enabling TI between distinct calculation setups (e.g., different k-point meshes, machine learning force fields vs. ab-initio).
• The parent directory's files should match those in 01, except for the INCAR, which only needs the VCAIMAGES and VCAIMAGES entries.
• Parameters controlling atomic motion (e.g., IBRION, ISIF, POTIM, MDALGO) must be identical in both subdirectories.
• Force calculation parameters may differ.

How it works:

• Similar to nudged elastic band calculations, VASP runs multiple calculations (subdirectories 01 and 02).
• The VCAIMAGES tag linearly combines the forces:
  • Weight for 01: VCAIMAGES
  • Weight for 02: 1 - VCAIMAGES
  • Thus, VCAIMAGES must be between 0 and 1.
• NCORE_IN_IMAGE1 sets the number of cores used for 01; remaining cores are used for 02.

Warning: Both directories must contain the same positions after applying the forces and the thermostat contributions. So thermostats containing random numbers cannot be used (MD_ALGO must be 2).

TI using SCALEE

This calculaion runs in a single folder. It optionally reads in a DYNMATFULL file in the calculation directory (for more details see below). The tag SCALEE sets the coupling parameter [math]\displaystyle{ \lambda }[/math] and hence controls the Hamiltonian of the calculation. By default SCALEE=1 and the scaling of the energies and forces via the coupling constant is internally skipped in the code. To enable the scaling SCALEE[math]\displaystyle{ \lt }[/math]1 has to be specified.

Two possible options are available for the reference system:

• Ideal gas:

By default the thermodynamic integration is carried out from the ideal gas to the fully interacting case (in the case when no DYNMATFULL is present in the calculation folder). Usually the Stirling approximation is used for the free energy of the ideal gas written as

[math]\displaystyle{ F = -\frac{1}{\beta} \mathrm{ln} \left[ \frac{V^{N}}{\Alpha^{3N} N!} \right] }[/math]

where [math]\displaystyle{ V }[/math] is the volume of the system, [math]\displaystyle{ N }[/math] is the number of particles in the system and [math]\displaystyle{ \Alpha }[/math] is the de Broglie wavelength. The Stirling approximation applies in principle only in the limes of infinitely many particles. In reference the exact ideal gas equation was used since it helped to speed up the convergence of the final free energy of liquid Si with respect to the system size.

• Harmonic solid:

If the file DYNMATFULL exists in the calculation directory and SCALEE[math]\displaystyle{ \ne }[/math]1, the second order Hessian matrix is added to the force and thermodynamic integration from a harmonic model to a fully interacting system is carried out. The DYNMATFULL file stores the eigenmodes and eigenvalues from diagonalizing the dynamic matrix. This file is written by a previous calculation using the INCAR tags IBRION=6 and PHON_NSTRUCT=-1.

TI using TILAMBDA

The TI calculations in internal coordinates are performed in NVT ensemble using any thermostat available in VASP. The coupling parameter [math]\displaystyle{ \lambda }[/math] is defined by setting the parameter TILAMBDA in the INCAR file. The set of internal coordinates used in the TI calculation are defined via the ICONST file by setting the status to 3. The Hesse matrix [math]\displaystyle{ \mathbf{\underline{H}}^\mathbf{x} }[/math] is provided in the file HESSEMAT and its transformation into [math]\displaystyle{ \mathbf{\underline{H}}^\mathbf{q} }[/math] is performed by VASP. The potential energies of the system 1 and 0,[math]\displaystyle{ \mathbf{q} }[/math], needed to compute [math]\displaystyle{ \langle V_1 -V_{0,\mathbf{q}} \rangle }[/math] used as integrant in the TI expression for [math]\displaystyle{ \Delta A_{0,\mathbf{q} \rightarrow 1} }[/math], are written in the file REPORT in lines introduce by a string "e_ti>"