Quasi-energy time-dependent

The decay on a picosecond time-scale, the so-called fast band, is understood as a quasi-direct recombination process in the silicon crystallites or as an oxide-related effect [Tr2, Mgl]. This fast part of the luminescence requires an intense excitation to become sizable it then competes with non-radiative channels like Auger recombination. The observed time dependence of the slow band is explained by carrier recombination through localized states that are distributed in energy, and dimensionally disordered [Gr7]. 

The Cauchy moments have been derived in Ref. [4] for CC wavefunctions, using the time-dependent quasi-energy Lagrangian technique [I]. In Section 2.1 we recapitulate the important points of that derivation and use it in Section 2.2 to derive the CC3-specific formulas. 

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

The time-dependent wavepacket accumulates in the inner region of the PES while it oscillates back and forth in the shallow potential well as illustrated in Figure 7.8. This vibrational motion leads to an increase of the stationary wavefunction in the inner region, however, only if the energy E is in resonance with the energy of a quasi-bound level. If, on the other hand, the energy is off resonance, destructive interference of contributions belonging to different times causes cancelation of the wavefunction. 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

Molecular dynamics (MD) simulations, the most suitable theoretical tool for the investigation of internal motions, can be used to explore both equilibrium properties and time-dependent phenomena. Based on both experimental and theoretical observations two models for the internal motion of proteins have been suggested. Within the framework of the first model internal motions arise from harmonic or quasi-harmonic vibrations that occur in a single multidimensional well on the potential energy surface [4,5,6,7]. The second model assumes that motions are a superposition of oscillations within a well and... 

Keywords coupled cluster, CCSD, CC3, response theory, quasi-energy Lagrangian, time-dependent... 

The time-dependent ground-state coupled cluster wavefunction for such a system is conveniently parameterized in a form, where the oscillating phase factor caused by the so-called level-shift [43 5, 88, 89] or time-dependent quasi-energy W(r, e) (vide infra) is explicitly isolated [42 6, 90, 91] ... 

The energy, which in the time-dependent case is no longer a constant of motion, has to be generalized to the time-dependent quasi-energy ... 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

The response function of order n is then recovered by differentiating the time-averaged perturbed quasi-energy with respect to the frequency-dependent... 

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