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 can be determined by means of thermodynamic integration

[math]\displaystyle{ \Delta A_{0\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_0 \rangle_\lambda }[/math]

with [math]\displaystyle{ V_i }[/math] being the potential energy of system [math]\displaystyle{ i }[/math], [math]\displaystyle{ \lambda }[/math] is a coupling constant and [math]\displaystyle{ \langle\cdots\rangle_\lambda }[/math] is the NVT ensemble average of the system driven by the Hamiltonian

[math]\displaystyle{ \mathcal{H}_\lambda = \lambda \mathcal{H}_1 + (1-\lambda)\mathcal{H}_0, }[/math]