Memory kernels and effective Hamiltonians from time dependent methods. I. Predissociation with a curve crossing

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Non-Hermitian complex effective Hamiltonians resulting from bound-continuous partitioning techniques are built from time dependent methods. We treat predissociation processes with a curve crossing. The energy dependent shift and half-width matrices are obtained simultaneously by a generalization of the wave packet Golden Rule treatment, as the real and imaginary parts of the Fourier transform of a memory kernel matrix. The latter contains auto- and cross-correlation functions. They are overlap integrals among the projections on the continuum of bound states multiplied by the interchannel coupling function responsible for the predissociation. These wave packets are propagated by the propagator of the sole continuous subspace. An approximate analytical expression of this correlation matrix is established for the harmonic/linear model. The numerical method is applied to the electronic predissociation of the MgCl A 2Π state, to a Morse/exponential model and to a predissociation with two coupled repulsive decay channels. The comparison between the correlation time scales and the Golden Rule lifetimes is decisive so as to justify whether the memory kernel can be considered as an impulsive kernel. This Markovian approximation implies that the two time scales are well separated. In the energy domain, this corresponds to the introduction of a mean phenomenological effective Hamiltonian that neglects the energy variation of the discrete-continuous coupling elements. We observe that the separation of the time scales is effective for weakly open systems, but not for overlapping metastable states for which the perturbative-theory widths largely exceed the mean energy spacing. This confirms from a temporal viewpoint that a nonperturbative treatment should not neglect the energy dependence of the effective Hamiltonian, as currently assumed in the study of largely open systems.