Zero-order eigenstates

We consider the following prototype of the mixed state representing our excited, say trans, molecule. The zero-order eigenstates are taken to consist of ... 

Besides these zeroth-order eigenfunctions, we shall also be interested in the eigenfunctions of the total Hamiltonian X In what follows, we shall assume that the effects of the perturbation A.X are small enough that we can correlate the exact gri eigenstates with their parent zero-order eigenstate q thus we can useftdly retain the quantum numbers q to define the exact eigenstates. (This is convenient but not essential). Thus after including the effects of the perturbation A.X, we have... 

The excited vibrational states can be considered as quasi-eigenstates [41]. As can be seen in the simplified scheme of Figure 2.2, these states are a result of the relatively strong coupling between a zero-order bright state (ZOBS), namely i >, with several zero-order dark states (ZODS), l > [48], that are further weakly coupled to the bath states that include a dense manifold of nearly equally coupled levels with a finite decay rate. 

J. Manz Prof. H. J. Neusser has presented to us beautiful high-resolution spectra of medium-size molecules and clusters such as benzene and C6H6 At (see current chapter). The individual lines have been assigned to individual rovibronic eigenstates of the systems, and their widths have been interpreted in terms of various intramolecular processes between zero-order states (e.g., Coriolis coupling, anharmonic couplings between bright and dark states, and so on). 

The spectra being discussed here are C2H2 A-X DF spectra, and the initial state created on the So surface corresponds to a perfectly known vibrational eigenstate of the A state (Si) surface transferred onto the X state (So) surface. However, any conceivable initial state could be expressed as a superposition of independently evolving polyads, each initially illuminated via one or more a priori known bright zero-order states. 

A time-dependent process, such as radiative absorption, internal conversion, intersystem crossing, unimolecular isomerization, or collision, may be treated in terms of a zero-order Hamiltonian H0 and a perturbation T. An unperturbed eigenstate of H0 evolves in time, since it is not an eigenstate to the perturbed Hamiltonian... 

It is much more than mere pedanticism to emphasize that the zero-order energy levels of the two subsystems described above cannot be considered to be eigenstates of the Hamiltonian of the total system. The zero-order levels of the two subsystems are degenerate or quasidegenerate, and therefore extensive configuration mixing is induced by the (small)... 

How does a relaxation process take place in a microscopic system To answer this question we amplify two comments made in the preceding section In the first place a representation of the (time-independent) eigenstates of the physical system can be displayed as a superposition of the zero-order states which correspond to the dense and to the sparse parts of the zero-order spectrum of states. In all cases considered we shall focus attention on the properties of the total system. Often, however, we find it convenient to examine the time evolution of one of the zero-order levels in the sparse set of states of one of the component subsystems of the total system. It is suggestive to then think of the remaining subsystem, with its dense manifold of states, as a reservoir. In fact we shall treat all states... 

Let us start with zero approximation states of H0 consisting of the discrete states (Xx, X2), 2(Xls X2),..., n( -i, X2) and continuum states time-independent) eigenstates of the physical system are obtained by diagonalizing the total Hamiltonian in this representation, and can be displayed as a superposition of these zero-order states. For the sake of simplicity we consider just one zero-order... 

We may represent the exact eigenstates of the Hamiltonian of our system as a superposition of the zero-order states described. If it were somehow possible to prepare our system in a zero-order state of the discrete set, we would have to regard that state as metastable or unstable. 

A correct representation of the molecular eigenstates, zero-order, Born-Oppenheimer states,6... 

The expansion coefficients representing the weights of the zero-order state molecular eigenstate 0n can be displayed in the form6... 

The properties of the metastable state resulting from the simple excitation process considered above can be further elucidated by considering the nature of the expansion of the molecular eigenstates in terms of the zero-order states (eq. (2-18)). Equation (10-4) can be rewritten in the form... 

The final state of the system, corresponding to the ground state of the molecule plus a photon, is represented by ipB = 0 k, e>. and the set 0, are the eigenfunctions of Hth while vac ) and k, e) represent the zero-photon and the one-photon eigenstates of HR, respectively. The time evolution of the amplitudes a,(t) and CE(t) can be computed from time-dependent perturbation theory. The equations of motion are determined by the energy levels of the zero-order states of Hel + HR, by the coupling matrix elements... 

We shall assume, as we did earlier, that the exact molecular eigenstate can be represented as a superposition of BO states or some other complete set of zero-order states. Suppose that the set of molecular states has the following characteristics ... 

Fig. 25. Energy-levels scheme for photodissociation, (a) Zero-order states. (6) Exact eigenstates in the absence of coupling to the continuum.

In Section XI-A we described the mixture of zero-order states that can be thought of as the generalized Fourier representation of a true eigenstate of a molecule in which a discrete zero-order state is coupled to... 

The second mode dynamics can be obtained by an expansion with respect to the eigenstates I > of a properly chosen zero order Hamiltonian h which leads to a set of equations of motion for the variables a , y, p, A, and y [18, 45, 46]... 

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