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# SCALEE

SCALEE = [real]
Default: SCALEE = 1

Description: This tag specifies the coupling parameter of the energies and forces between a fully interacting system and a reference system.

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

The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system

$F_{1}=F_{0}+\Delta F_{0\rightarrow 1}$ .

Using thermodynamic integration the free energy difference between the two systems is written as

$\Delta F_{0\rightarrow 1}=\int \limits _{0}^{1}d\lambda \langle U_{1}(\lambda )-U_{0}(\lambda )\rangle _{\lambda }$ .

Here $U_{1}(\lambda )$ and $U_{0}(\lambda )$ describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter $\lambda$ . It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation $\langle \ldots \rangle _{\lambda }$ denotes an ensemble average of a system driven by the following classical Hamiltonian

$H_{\lambda }=\lambda H_{1}+(1-\lambda )H_{0}$ .

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$\neq$ 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 .

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

$F=-{\frac {1}{\beta }}\mathrm {ln} \left[{\frac {V^{N}}{\mathrm {A} ^{3N}N!}}\right]$ whre $V$ is the volume of the system, $N$ is the number of particles in the system and $\mathrm {A}$ 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$\neq$ 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.