Three-state system quantization

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

We consider a system of two nonidentical and nonoverlapping atoms at positions ri and rj, coupled to the quantized three-dimensional electromagnetic held. The initial state of the held is the product of a single-mode coherent state of a driving laser held, and the vacuum state of the rest of the modes. Each atom is assumed to have only two levels the ground level g,) and the excited level e,)(i = 1,2), separated by an energy hot), =Ee. — Egi, and connected by an... 

In transition state theory it is assumed that a dynamical bottleneck in the interaction region controls chemical reactivity. Transition state theory relates the rate of a chemical reaction in a microcanonical ensemble to the number of energetically accessible vibrational-rotational levels of the interacting particles at the dynamical bottleneck. In spite of the success of transition state theory, direct evidence for a quantized spectrum of the transition state has been found only recently, and this evidence was found first in accurate quantum mechanical reactive scattering calculations. Quantized transition states have now been identified in accurate three-dimensional quantal calculations for 12 reactive atom-diatom systems. The systems are H + H2, D + H2, O + H2, Cl + H2, H + 02, F + H2, Cl + HC1, I + HI, I 4- DI, He + H2, Ne + H2, and O + HC1. 

The second kind of transition state resonance, as illustrated in Figure 23.20b, is known as the vibrational threshold resonance. This type of resonance corresponds to the energetic threshold for a quantized dynamical bottleneck in the transition-state region. This quasi-bound state can be characterized by two vibrational quantum numbers (for a three-atom system), corresponding to the modes of motion orthogonal to the unbound reaction coordinate. This kind of resonance has been found experimentally in... 

The essential signature of a molecule is tiiat it vibrates. For a molecule composed of N atoms, there are 3N mechanical degrees of freedom associated with the motions of the system. Three degrees of freedom are determined by the translational motions of the center of mass, and for a nonlinear molecule there are three degrees of freedom connected with the overall rotational motion of the molecule. For a macromolecule, some of the remaining 3N - 6 degrees of freedom are associated with isomerizations of the chain backbone and the side chains. Finally, there exists a set of quantized vibrational states for the molecule. If the frequencies of the vibrational states depend on the conformational state of the molecule, the measurement of the vibrational spectrum can be used to infer the conformational composition of the ensemble of macromolecules. The frequencies of quantized molecular vibrations greatly exceed the frequencies associated with isomerization of the chain backbone. 

Under a strong magnetic field the orbital motion of conduction elections is quantized and forms Landau levels. Therefore various pltysical quantities show a periodic variation with H since increasing field strength causes a sharp change in the free eneigy of the electron system when a Landau level crosses the Fermi level. In the three-dimensional system this sharp structure is observed at the extremal (maximum or minimum) cross-sectional area of the Fermi surface perpendicular to the field direction because the density of states also becomes extremal. 

Fig. 4.1 Three types of reactive resonances near the transition state region in chemical reactions, adapted from [66]. Panel (a) illustrates the case associated with a deep potential well along the reaction coordinate. The resulting bound and pie-dissociative quasi-bound states can be characterized, for a three-atom system, by three vibrational modes, (b) Threshold resonance for which only the two motions orthogontil to the unbound reaction coordinate tire quantized and thus assignable by vibrational quantum numbers. The dynamical trapped-state resonance is schematically shown in panel (c). Despite the repulsive potential energy surface along the reaction coordinate, this metastable state can be assigned by three vibrational quantum numbers...

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

The simplest procedure is to take the origin of a global I-frame so that P = 0 and linear momentum conservation forces kj = —k2. At the antipodes, kd = k2 so that the common I-frame is restricted now. The particle model in this frame becomes strongly correlated. If spin quantum state for I-frame, one corresponds to the linear superposition (a /S)[CiC2]i and the other I-frame system should display the state (a P)[c, — Ci]2, namely, an orthogonal quantum state. The quantization of three axes is fixed. Spin and space are correlated in this manner. Now, the label states (a P)[c2 — cji and (a )[C C2]2 present another set of possibilities. This is because quantum states concern possibilities. All of them must be incorporated in a base state set. At this point, classical and quantum-physical descriptions differ radically. The former case handles objects that are characterized by properties, whereas the latter handle objects that are characterized by quantum states sustained by specific materiality. 

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