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# Difference between revisions of "SCALEE"

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<math> \Delta F = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda} </math>. | <math> \Delta F = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda} </math>. | ||

− | Here <math>U_{1}(\lambda)</math> and <math>U_{0}(\lambda)</math> describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The | + | Here <math>U_{1}(\lambda)</math> and <math>U_{0}(\lambda)</math> 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 <math>\lambda</math>. The {{TAG|SCALEE}} sets the value for the coupling constant. The notation <math>\langle \ldots \rangle_{\lambda}</math> denotes an ensemble average of a system driven by the following classical Hamiltonian |

+ | <math> H_{\lambda}= \lambda H_{1} + (1-\lambda) H_{0} </math>. | ||

## Revision as of 15:15, 2 April 2020

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

.

Here and 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 . The SCALEE sets the value for the coupling constant. The notation denotes an ensemble average of a system driven by the following classical Hamiltonian

.

By default SCALEE=1 and scaling of the energies and forces via the coupling constant is carried out. To enable the scaling SCALEE<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 SCALEE1, 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. Here the Hamiltonian for a certain integration point along the thermodynamic integration pathway is given as

## Related Tags and Sections

VCAIMAGES, IMAGES, NCORE IN IMAGE1, PHON_NSTRUCT