Rotating wave contribution

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

Positive oo gives the so-called rotating-wave (rw) contribution whereas negative co gives the counterrotating wave (crw) contribution. 

When Eq. (73) is explicitly evaluated, we find that within the rotating wave approximation we need to calculate six pathways (Fig. 7). The contribution of three of them is the complex conjugate of the others. We therefore need to consider only the three pathways shown in Fig. 8. We then get11-12-57-64... 

This is integrated over the Q,Q2Q,-space. If the collision pair wave functions never overlap the vibration wave function Xiku(Qi>Q2>Q3 2Zu) of the QTS, there will be zero contribution to the cross section. In this case, the QTS defines the reaction domain. This is quantized by the corresponding vibration-rotation wave function. Therefore, from all possible collisions among the reactants, only those having a non-zero FC factor will contribute to the reaction rate. This is related to the steric factor, P, in elementary chemical kinetics theory. Selection rules for VR-transitions apply. The probability to find the system in one of the product channel states when starting from a QTS is controlled by the FC integral formed by the products of the type... 

The second terms in Eqs. (1.167) and (1.168) are rapidly vibrating non-resonant terms having minor contribution on average, so they are neglected in further calculations. This is called the rotating wave approximation. By solving these simultaneous equations with the initial condition of T0 = T, for / = 0, we obtain ... 

Let us now introduce the standard Born-Oppenheimer separation by which molecular states are written as a product of an electronic wavefunction depending parametrically on the nuclear coordinates J/j (Q,q) and a purely vibrational wave-function X/v(Q)- This is also done in Chapter 8, but we repeat this treatment with the aim of highlighting its consequences on the system TDSE. Once more we remind the reader that, since the focus here is on condensed-phase spectroscopy, we do not consider the role of molecular rotations, whose contributions to the absorption spectra can be detected only in high-resolution experiments. It is worthwhile to note, however, that they could be included in the time-dependent approach studying the propagation of suitable rovibronic wavepackets. 

Initially, j (the angular momentum of BC) is randomly (i.e., spherically symmetrically) distributed, whereas the orbital angular momentum of the reagent approach, L, is randomly distributed in a plane perpendicular to the initial relative velocity vector V. For simplicity, consider a reaction where many partial waves contribute, so that j L, and that, e.g., because C is a light atom, AB is formed at high rotational excitation, so that j 3> L. Conservation of total angular momentum implies that for our assiuned conditions, j L j will then be nearly parallel to L and hence confined in a plane perpendicular to the initial relative velocity. The AB product is fully aligned, with ((] k) ) 0, that is — 1. The rota-... 

An increase in co from 400 to 1600rpm tints results in a twofold increase of the signal. A deviation from linearity of a plot of z) vs. col/1 suggests some kinetic limitations. In addition, at veiy low rotation speeds (0-100 rpm), a slight upward bend is observed due to contribution by natural convection. The voltammetric wave has a sigmoidal shape for reversible systems it is identical to that common in DC polarography (described in Section 3-2), and independent of to. 

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