Thermodynamic integration with harmonic reference: Difference between revisions

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# compute <math>\omega_i</math> in vibrational analysis
# compute <math>\omega_i</math> in vibrational analysis
# use the data obtained in the point 2 to determine <math>\underline{\mathbf{H}}^\mathbf{x}</math> that defines the harmonic forcefield
# use the data obtained in the point 2 to determine <math>\underline{\mathbf{H}}^\mathbf{x}</math> that defines the harmonic forcefield
# perform NVT MD simulations for several values of  <math>\lambda \in <0,1></math> and determined <math>\langle V_1 -V_{0,\mathbf{x}}  \rangle</math>
# perform NVT MD simulations for several values of  <math>\lambda \in \langle0,1\rangle</math> and determined <math>\langle V_1 -V_{0,\mathbf{x}}  \rangle</math>
#
#

Revision as of 08:10, 1 November 2023

The Helmholtz free energy () of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (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 obtained using the Hesse matrix . Finally, the harmonic potential energy is expressed as

Thus, a conventional TI calculation consists of the following steps:

  1. determine and in structural relaxation
  2. compute in vibrational analysis
  3. use the data obtained in the point 2 to determine that defines the harmonic forcefield
  4. perform NVT MD simulations for several values of and determined