  We apply the Lang-Firsov (LF) transformation to electron-boson coupled
Hamiltonians and variationally optimize the transformation parameters and
molecular orbital coefficients to determine the ground state. M\o{}ller-Plesset
(MP-$n$, with $n = 2$ and $4$) perturbation theory is then performed on top of
the optimized LF mean-field state to improve the description of
electron-electron and electron-boson correlations. The method (LF-MP) is
applied to several electron-boson coupled systems, including the
Hubbard-Holstein model, diatomic molecule dissociation ($\text{H}_2$,
$\text{HF}$), and the modification of proton transfer reactions (malonaldehyde
and aminopropenal) via the formation of polaritons in an optical cavity. We
show that with a correction for the electron-electron correlation, the method
gives quantitatively accurate energies comparable to exact diagonalization or
coupled-cluster theory. The effect of multiple photon modes, spin polarization,
and the comparison to the coherent state MP theory are also discussed.
