Zero-order states

If only zero-order states from the same polyad are conpled together, this constitutes a fantastic simplification in the Flamiltonian. Enonnons compntational economies result in fitting spectra, becanse the spectroscopic Flamiltonian is block diagonal in the polyad nnmber. That is, only zero-order states within blocks with the same polyad number are coupled the resulting small matrix diagonalization problem is vastly simpler than diagonalizing a matrix with all the zero-order states conpled to each other. 

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

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. 

What is a polyad A polyad is a subset of the zero-order states within a specifiable region of Evib (typically a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyad quantum numbers. For the C2H2S0 state, a procedure proposed by Kellman [9, 10] identifies the three polyad quantum numbers... 

The time-dependent probability of finding the system in the kth zero-order state belonging to the [n,nTes,/] polyad is... 

Figure 7. Survival (Py) and transfer (Py) probabilities for the C2H2 [1, 11,0] polyad. These P , t) and Pjj t) curves were computed using the eigenvectors and eigenvalues of the Hp[ j y0j polyad matrix. The ZOBS is (0, 1,0, 8°, 0°)° and the four panels correspond to (0, 1, 0, 8°, 0°)° upper left, the first-tier zero-order states (0, 1,0, 6+2, 2 2)0 upper right and (0, 1, 0, 6°, 2°)° lower left, and a far-edge dark state (1, 0, 0, 0°, 6°)° lower right. The four panels are repeated, at time scales increasing in factor of 10 or 100 steps (a) 0-4 ps (b) 0-40 ps (c) 0-400 ps (d) 0-40 ns.

Lacking a trick to accomplish unzipping in the time domain, the dynamics uncovered in a time-domain experiment would appear far more complicated. Even if unzipping were possible, it would be necessary to record the intrapolyad dynamics of several zero-order states to obtain sufficient information to characterize a polyad Heff. Without such a characterization, it would be impossible to make even qualitative predictions of the dynamics in another region of Evib or of another class of probe-able zero-order state. 

R. W. Field Each acetylene polyad contains zero-order states that are easily accessible via plausible direct or multiple-resonance A XAU — X Franck-Condon pumping schemes. Each Vib 16,000... 

The Ay state arising from 3B1 and the Ax state arising from 1A1 may now mix, since the noncrossing rule applies.128 Thus in the presence of spin interactions there is an avoided crossing as indicated in Figure 5. This noncrossing implies that in the isolated CH2 molecule there is a finite transition probability between the nonstationary zero-order states. If we... 

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. 

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