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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 reference [1] (caution: the tag ISPECIAL=0 used in that reference is not valid anymore, instead the tag PHON_NSTRUCT=-1 is used).

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

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

Here ${\displaystyle U_{1}(\lambda )}$ and ${\displaystyle U_{0}(\lambda )}$ 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 ${\displaystyle \lambda }$. The tag SCALEE sets the value for the coupling constant. The notation ${\displaystyle \langle \ldots \rangle _{\lambda }}$ denotes an ensemble average of a system driven by the following classical Hamiltonian

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

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${\displaystyle \neq }$1 has to be specified.

Two possible options are available for the reference system:

• Ideal gas:

Usually the thermodynamic integration is carried out from the ideal gas to the liquid state.

• Harmonic solid:

If the file DYNMATFULL exists in the calculation directory and SCALEE${\displaystyle \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.