Stochastic theory of line shape and relaxation

This paper is a generalization of Kubo’s stochastic theory of spectroscopic line shapes which appeared in 1962 under the same title. Using a stochastic, Hermitian Hamiltonian to phenomenologically incorporate absorber-perturber interactions resulting from molecular collisions, we are able to reproduce Kubo’s derivation of the simple Lorentzian line shape: I(ω)=${\pi }^{\mathrm{-}1}$Γ[(ω-${\omega }_0$ ${)}^2$+Γ ${}_{}{}^2{}_{}{}^{}$ ${]}^{\mathrm{-}1}$ in which the width Γ is related to the mean-square strength, $Q_3$, of a diagonal stochastic Hamiltonian by Γ=2$Q_3$. Our generalization of this line shape is I(ω)=(1/π){1+(1-ω/Ω)[(${\mathrm{\Gamma }}_2$ ${\mathrm{-}\mathit{\Gamma }}_1$)/2Γ]}Γ/[(ω-Ω${)}^2$+Γ ${}_{}{}^2{}_{}{}^{}$ ${]}^{\mathrm{-}1}$, where ${\Gamma }_1$ and ${\Gamma }_2$ measure the degree of anisotropy in the off-diagonal part of the stochastic Hamiltonian. With perfect isotropy their contribution vanishes, but we are still left with additional corrections to the simple Lorentzian: Γ=2Q(1+${\mathrm{\omega }}_0^2$/${\mathrm{\epsilon }}^2$ ${)}^{\mathrm{-}1}$+ 2${\mathrm{Q}}_3$ and Ω=${\omega }_0$+Δ, where Δ=2Q(${\mathrm{\omega }}_0$/ε)(1+${\mathrm{\omega }}_0^2$/ε ${}_{}{}^2{}_{}{}^{}$ ${)}^{\mathrm{-}1}$. Our result does reduce to Kubo’s in the absence of off-diagonal stochastic terms. The parameter ε measures the non-Markovianness of the stochastic Hamiltonian and yields white noise in the limit ε→∞. Our results are valid for this limit regime. The frequency shift Δ is a measure of the non-Markovianness.