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

− | By default {{TAG|SCALEE}}=1 and scaling of the energies and forces via the coupling constant is | + | By default {{TAG|SCALEE}}=1 and the scaling of the energies and forces via the coupling constant is internally skipped in the code. To enable the scaling {{TAG|SCALEE}}<math>\ne</math>1 has to be specified. |

Two possible options are available for the reference system: | Two possible options are available for the reference system: |

## Revision as of 06:22, 3 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 tag 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 the scaling of the energies and forces via the coupling constant is internally skipped in the code. To enable the scaling SCALEE1 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