content_image/353801.jpg content_image/353799.jpg content_image/353800.jpg content_image/353798.jpg content_image/353797.jpg content_image/353796.jpg content_image/353794.jpg content_image/353795.jpg
The next step consists of the trivial integration in Eq. (A.6) over Fourier components, which yields 

for λ$_{1}$ + 2 g = 0 and 

otherwise. In the last two expressions which give the zero-th order free energy F $_{0}$, we have defined the pseudo-phonon frequencies in the form 

and an additional parameter 

The calculation of the first order correction to free energy 

<img src='content_image/353801.jpg'>

<img src='content_image/353799.jpg'>

<img src='content_image/353800.jpg'>

<img src='content_image/353798.jpg'>

<img src='content_image/353797.jpg'>

proceeds in a similar way. The usual trick consists in expanding the on-site and inter-site potential in Fourier basis, 

<img src='content_image/353796.jpg'>

Then we evaluate the average potentials per particle 〈 V ( x$_{k}$ ) 〉 and 〈 V ( x$_{k}$ ) 〉 in (A.15) by integrating over Fourier components and finally invert the fourier transforms of Eq. (A.16). Consequently, the average value of the potential energies can be expressed in the form of smeared-out potentials, 

<img src='content_image/353794.jpg'>

where we have defined two parameters ρ and γ which correspond to lattice displacement and two-point correlation function, that is 

<img src='content_image/353795.jpg'>

