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

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

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

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