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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  (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

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

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

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

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.

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 (from a previous calculation using PHON_NSTRUCT=-1) 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.