Exact factorization method for bound vibrational states: An analytical tool for accurate approximations

The Exact Factorization (XF) method represents an interesting formulation of the Schrödinger equation where subsystem wavefunctions are exactly coupled. Here, I show that the XF method can be employed as an analytical tool to study the quantum vibrational problem of bound systems. In particular, after elaborating suitable XF-based wavefunction Ansätze, the ground-state energy approximated expression for bilinearly and quartically coupled harmonic oscillators is estimated. The XF-based analytical solution is compared with adiabatic and perturbative ones, and it is usually found to be an order of magnitude more accurate than these for estimating the anharmonic and coupling correction part of the ground-state energy. This procedure will possibly increase the numerical stability and accuracy of perturbative or Hartree-product based methods when applied to bound state calculations.