Zero-order states, superposition

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. 

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... 

In our analysis, at t = 0 the metastable state of the system will be taken to be the zero-order state (Xb X2), which state is one from the set which defines the manifold of the sparse part of our system. Of course, 0i(Xx, X2) can be represented in terms of the superposition eq. (2-16). The state function at the time t can then be displayed as a time-... 

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. 

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 ... 

A short pulse excites a coherent superposition of the eigenstates ip+ and ip which result from the interaction between the zero-order states ipi and ip2. This coherent superposition exhibits quantum beats at frequency and decay rate, respectively, wqb = (e+ - e )/fi and rQB/fi = (T+ + T )/ h,... 

The properties of the dissociative states of a molecule with the spectrum described above were studied by Rice, McLaughlin, and Jortner. They represented the true eigenstates of the molecule in the absence of a radiation field as a superposition of zero-order states (which need not be BO states but which are the eigenstates of a zero-order separable Hamiltonian). Suppose... 

For these stationary states the system cannot be defined as being in either well, and there is no time evolution that is, the wave function is spread over both wells. If the excitation is sufficiently broad to excite both and i /, which are separated by an amount AE, a nonstationary superposition (or zero-order) state will be prepared. At t = 0 this state is found by inverting Eq. (4.18) and is given by... 

If the superposition state is written as a zero-order state instead of a linear combination of eigenstates, the probability amplitude is given by Eq. (4.17), so that the absorption spectrum becomes... 

Kosloff, 1994) have also been used to find the complex energies for compound-state resonances. A localized wave packet [i.e., a coherent superposition state, Eq. (4.7)], P(O) is initially placed in the bound region of the potential energy surface and propagated in time to give ( l (0) (r)), which is C t) in Eq. (4.16). If I (O) is a superposition of resonant states, it can be considered a zero-order state (see chapter 4) and can be written as... 

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

A very peculiar dynamics has been revealed in the Ca(OH)2 crystal by means of inelastic neutron scattering technique [26]. It has been found that anharmonic terms must be included, which mix the vibrational states of the OH and lattice modes. In particularly, the lattice modes have successfully been represented as the superposition of oxygen and proton synchronous oscillators, and it appears that the proton bending mode Eu is strongly coupled to the lattice modes. The contribution of the proton harmonic wave functions has been taken as the zero-order approximation. 

In 90° twisted ethylene, there are two zwitterionic singlet states, Z and V [36, 38-40], which in Cg symmetry are described in zero order by the superpositions. 

The case of fhe ground state, S, was already discussed in Section 5, with numerical results for different values of Z. Bofh fhe H-F sea and the Fermi-sea zero-order wavefunctions are described by a superposition of fhe ls 2s IS and W2p SACs. 

In certain problems of excited states, depending on the electronic structures of the moieties involved and on the type of superpositions that may be relevant in zero order, actual or effective projections are implemented via the appropriate choice of the functions spaces. 

The concept of intramolecular vibrational energy redistribution (IVR) can be formulated from both time-dependent and time-independent viewpoints (Li et al., 1992 Sibert et al., 1984a). IVR is often viewed as an explicitly time-dependent phenomenon, in which a nonstationary superposition state, as described above, is initially prepared and evolves in time. Energy flows out of the initially excited zero-order mode, which may be localized in one part of the molecule, to other zero-order modes and, consequently, other parts of the molecule. However, delocalized zero-order modes are also possible. The nonstationary state initially prepared is often referred to as the bright state, as it carries oscillator strength for the spectroscopic transition of interest, and IVR results in the flow of amplitude into the manifold of so-called dark states that are not excited directly. It is of interest to understand what physical interactions couple different zero-order modes, allowing energy to flow between them. A particular type of superposition state that has received considerable study are A/-H local modes (overtones), where M is a heavy atom (Child and Halonen, 1984 Hayward and Henry, 1975 Watson et al., 1981). 

As described at the beginning of section 4.3.3 (p. 77), the Fourier transform of the frequency spectrum gives the survival probability for the initially prepared zero-order (i.e., superposition) state, i). A limited Fourier transform of /(co) in Eq. (4.29) can be defined as... 

From this, the molecular resonant states v> and their complex energies v = 6v - 7v/2, can be simply derived by diagonalizing the matrix (164). v> corresponds to the superposition of zero-order decaying radiant and nonradiant states. 

Following here the treatment of Ref [11], the molecular eigenstates are defined as a superposition of zero-order BO states by the Ansatz... 

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