On the transition from nonadiabatic to adiabatic rate kernel: Schwinger’s stationary variational principle and Padé approximation

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For a two state system coupled to each other by a nonzero matrix element Δ and to the bath arbitrarily, the generalized master equation is derived by applying the well-known projection operator techniques to the quantum Liouville equation. The time-dependent rate kernel is expressed by an infinite summation of the perturbative terms in Fourier–Laplace space. The Schwinger’s stationary variation principle in Hilbert space is extended to Liouville space and then applied to the resummation of the rate kernel. The Cini–Fubini-type trial state vector in Liouville space is used to calculate the variational parameters. It is found that the resulting stationary value for the rate kernel in Fourier–Laplace space is given by the [N,N−1]–Padé approximants, in the N-dimensional subspace constructed by the N perturbatively expanded Liouville space vectors. The (first-order) simplest approximation satisfying the variational principle turns out to be equal to the [1,0] Padé approximant instead of the second-order Fermi golden rule expression. Two well-known approximations, the noninteracting blip approximation (NIBA) and nonadiabatic approximation, are discussed in the context of the [1,0] Padé approximants, based on the variational principle. A higher-order approximation, [2,1] Padé approximant, is also briefly discussed.