Thermodynamic integration with harmonic reference

The Helmholtz free energy ([math]\displaystyle{ A }[/math]) of a fully interacting system (1) can be expressed in terms of that of harmonic system (0) as follows

[math]\displaystyle{ A_{1} = A_{0} + \Delta A_{0\rightarrow 1} }[/math]

where [math]\displaystyle{ \Delta A_{0\rightarrow 1} }[/math] is anharmonic free energy. The latter term