Thermodynamic integration with harmonic reference: Difference between revisions

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configuration corresponding to the potential energy minimum with the  
configuration corresponding to the potential energy minimum with the  
atomic position vector <math>\mathbf{x}_0</math>,
atomic position vector <math>\mathbf{x}_0</math>,
the number of vibrational degrees of freedom <math>N_\mathrm{vib}</math>, and the angular frequency $\omega_i$ of vibrational mode <math>i</math>.
the number of vibrational degrees of freedom <math>N_\mathrm{vib}</math>, and the angular frequency <math>\omega_i</math> of vibrational mode <math>i</math>.
The
:<math>
V_{0,\mathbf{x}}(\mathbf{x}) = V_{0,\mathbf{x}}(\mathbf{x}_0) + \frac{1}{2} (\mathbf{x} - \mathbf{x}_0)^T \underline{\mathbf{H}}^\mathbf{x} (\mathbf{x} - \mathbf{x}_0)
</math>

Revision as of 07:55, 1 November 2023

The Helmholtz free energy () of a fully interacting system (1) can be expressed in terms of that of harmonic system (0) as follows

where is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)

with being the potential energy of system , is a coupling constant and is the NVT ensemble average of the system driven by the Hamiltonian

Free energy of harmonic reference system within the quasi-classical theory writes

with the electronic free energy for the configuration corresponding to the potential energy minimum with the atomic position vector , the number of vibrational degrees of freedom , and the angular frequency of vibrational mode . The