Equivalence between the Classical and Quantum IR Spectral Density Approaches of Weak H-Bonds in the Absence of Damping

Towards accurate infrared spectral density of weak H-bonds in absence of relaxation mechanisms

Following the previous theoretical developments to completely reproduce the IR spectra of weak hydrogen bond complexes within the framework of the linear response theory (LRT), the quantum theory of the high stretching mode spectral density (SD) of weak H-bonds is reconsidered. Within the LRT theory, the SD is the one sided Fourier transform of the autocorrelation function (ACF) of the high stretching mode dipole moment operator. In order to provide more accurate theoretical bandshapes, we have explored the equivalence between the SDs given in previous studies with respect to a new quantum one, and revealed that in place of the basic equations used in the precedent works for which the SD I_Old(ω)=2Re∫₀^∞G_Old(t)e^-iωtdt where the ACF G_Old(t) = ⟨μ(0)μ(t)⁺⟩ = tr {ρ {μ(0)} {μ(t)}⁺}, one can use a new expression for the SD, given by I_New(ω)=2ωRe∫₀^∞G_New(t)e^-iωtdt where G_New(t)=μ(0)μ(t)⁺=1βtrρ_B∫₀^βμ(0)μ(t+iλℏ)⁺dλ. Here ρ_B is the Boltzmann density operator, μ(0) the dipole moment operator at initial time and μ(t) the dipole moment operator at time t in the Heisenberg picture, ℏ is the Planck constant, β is the inverse of the Boltzmann factor k_BT where T is the absolute temperature and k_B the Boltzmann constant. Using this formalism, we demonstrated that the new quantum approach gives the same final SD as used by previous models, and reduces to the Franck-Condon progression appearing in the Maréchal and Witkowski's pioneering approach when the relaxation mechanisms are ignored. Results of this approach shed light on the equivalence between the quantum and classical IR SD approaches for weak H-bonds in absence of medium surroundings effect, which has been a subject of debate for decades.