Transition probabilities rotational

We recall from our discussion (section 6.8.1) of the rigid rotor model of a diatomic molecule that the Schrodinger equation is 

E(J) = 2 J(J + l) (6J08) where, in this simple example, J takes integral values 0,1,2, etc., and /i is the reduced mass. 

The components of the dipole moment operator in polar coordinates are 

The integrals which determine the transition probabilities between states /, m and /, m of a molecule in a non-degenerate state are, from equation (6.300), 

We now make use of some relations first we have the familiar 

Next we make use of three recurrence relationships involving the associated Legendre polynomials  

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Chidsey I L and Crosley D R 1980 Calculated rotational transition probabilities for the A-X system of CH J. Quant. Spectrose. Radiat. Transfer 23 187-99... 

Table 1. Rigid-rotator transition probabilities for four...

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Under some circumstances the rotationally anisotropy may be even further simplified for T-R energy transfer of polar molecules like HF (41). To explore this quantitatively we performed additional rigid-rotator calculations in which we retained only the spherically symmetric and dipole-dipole terms of the AD potential, which yields M = 3 (see Figures 1, 3, and 4). These calculations converge more rapidly with increasing N and usually yield even less rotationally inelastic scattering. For example Table 2 compares the converged inelastic transition probabilities... 

For geometrical reasons the degree of circular polarization of the rf field is 100% only along the cylinder axis of the cavity. However, it has been shown that for samples with diameters < 6 mm the contribution of the counter-rotating component to the nuclear transition probability is less than 1%43. ... 

In a spin system with anisotropic g and A tensors, the transition probability Wba between two nuclear spin states energy levels Eb and Ea may be calculated from the coupling operator given in (3.20). For a circularly polarized rf field, B2(t)R. in (3.20) has to be replaced by Bep(t)ft with the l.h. or r.h. rotating field B t) defined in (2,1). The nuclear transition probability is then given by... 

A different situation is found for a dipolar hfs tensor. The relative nuclear transition probabilities for B0 parallel to Ap or Aj are given in Table 2.2 and 2.3. As in the case of an isotropic hfs tensor, again Beff is oriented parallel or antiparallel to B0. Since the enhancement factor E is isotropic for B0 parallel to Ay, the net circularly polarized field rotates in the same sense as the applied field. In the case of B0 parallel to Ai however, the enhancement factor E is anisotropic. This anisotropy of E is responsible for the generation of counter rotating fields which induce residual lines. 

With heating from 5 to 45°C, thermal changes in conformation in the major /3-casein are observed by spectral methods (Garnier 1966). From measurements of the optical density at 286 nm and of the specific optical rotation at 436 nm, a rapidly reversible endothermic transition (AH 30 kcal/mole) with a half-transition temperature of 23-24°C is observed. The optical rotatory dispersion data suggest a decrease in the poly-L-proline II structure (12 to 5%) and a slight increase in a-helix (11 to 16%) with increasing temperature. This transition probably occurs prior to association, since it is rapid, and the carboxyacyl derivative of the monomer, which does not polymerize with increasing temperature, also demonstrates the optical rotatory disperson thermal transition. 

For a heavier system, such as N2O + Ar, a calculation of rotational transitions and microwave or infrared line widths would follow the same course through the flow chart, as that followed above in detail for HC1 + Ar. However, at the last stage (low j, small b collisions), the number of coupled states would probably be too large for the non-perturbative, fixed classical path calculation to be practical. Then one should calculate "classical S matrices" including interference between trajectories, to cover these remaining collisions. 

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