Thermodynamic integration with harmonic reference

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

$A_{1}=A_{0}+\Delta A_{0\rightarrow 1}$

where $\Delta A_{0\rightarrow 1}$ is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)

$\Delta A_{0\rightarrow 1}=\int _{0}^{1}d\lambda \langle V_{1}-V_{0}\rangle _{\lambda }$

with $V_{i}$ being the potential energy of system $i$, $\lambda$ is a coupling constant and $\langle \cdots \rangle _{\lambda }$ is the NVT ensemble average of the system driven by the Hamiltonian

${\mathcal {H}}_{\lambda }=\lambda {\mathcal {H}}_{1}+(1-\lambda ){\mathcal {H}}_{0}$

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

Failed to parse (unknown function "\vct"): {\displaystyle A_{0,\vct{x}} = A_\mathrm{el}(\vct{x}_0) - k_\mathrm{B} T \sum_{i = 1}^{N_\mathrm{vib}} \ln \frac{k_\mathrm{B} T}{\hbar \omega_i} }

with the electronic free energy Failed to parse (unknown function "\vct"): {\displaystyle A_\mathrm{el}(\vct{x}_0)} for the configuration corresponding to the potential energy minimum with the atomic position vector Failed to parse (unknown function "\vct"): {\displaystyle \vct{x}_0} , the number of vibrational degrees of freedom $N_{\mathrm {vib} }$, and the angular frequency $\omega_i$ of vibrational mode $i$.