Thermodynamic integration calculations

A detailed description of calculations using thermodynamic integration within VASP is given in the supplemental information of reference ^[1] (caution: the tag ISPECIAL=0 used in that reference is not valid anymore, instead the tag PHON_NSTRUCT=-1 is used).

The tag SCALEE sets the coupling parameter $\lambda$ 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 $\ne$ 1 has to be specified. A VASP calculation outputs the integrand for a given coupling constant at every molecular dynamics step. How to choose the ensemble size and carry out the integration is described in the main text and especially in the supplementary information of reference ^[1].

To carry out thermodyanamic integration calculations the calculation needs to be split into two images via the tag VCAIMAGES. One also needs to set how many cores are used for each image vial NCORE_IN_IMAGE1.

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

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

where $V$ is the volume of the system, $N$ is the number of particles in the system and $\Alpha$ is the de Broglie wavelength. The Stirling approximation applies in principle only in the limes of infinitely many particles. In reference ^[1] 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 $\ne$ 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.

Alternative way of thermodynamic integration for harmonic solid

The TI calculations in internal coordinates are performed in NVT ensemble using any thermostat available in VASP. The coupling parameter $\lambda$ 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 $\mathbf{\underline{H}}^\mathbf{x}$ is provided in the file HESSEMAT and its transformation into $\mathbf{\underline{H}}^\mathbf{q}$ is performed by VASP. The potential energies of the system 1 and 0, $\mathbf{q}$ , needed to compute $\langle V_1 -V_{0,\mathbf{q}} \rangle$ used as integrant in the TI expression for $\Delta A_{0,\mathbf{q} \rightarrow 1}$ , are written in the file REPORT in lines introduce by a string "e_ti>"