Quantum numbers rotational

Regardless of the nature of the intramolecular dynamics of the reactant A, there are two constants of the motion in a nnimolecular reaction, i.e. the energy E and the total angular momentum j. The latter ensures the rotational quantum number J is fixed during the nnimolecular reaction and the quantum RRKM rate constant is specified as k E, J). 

Thomas and exponential P(lc) are more pronounced for the model in which the rotational quantum number K is treated as adiabatic than the one with Kactive. 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

We will now consider the consequences of these mles in the simple case of FI2. In this molecule both whatever the value of v, and in the ground electronic state, are symmetric to nuclear exchange so we need consider only the behaviour of lAr A - Since / = i for FI, ij/ and therefore i/ r A rnust be antisymmetric to nuclear exchange. It can be shown that, for even values of the rotational quantum number J, ij/ is symmetric (x) to exchange and, for odd values of J, j/ is antisymmetric a) to exchange, as shown in Figure 5.18. 

The origin of the rotational structure of the isotropic Q-branch (Av = 0, Aj = 0) is connected with the dependence of the vibrational transition frequency shift on rotational quantum number j [121, 126]... 

Factorization of the impact operator (5.1) greatly reduces the computational effort required in any variant of IOS [197]. When the inequality (5.43) holds, this factorization is acceptable but only for purely non-adiabatic relaxation as is J-diffusion. Though N2-Ar collisions are mostly non-adiabatic, it would still be better to account for adiabatic-ity, which becomes more significant the higher the rotational quantum number j. 

The spectrum calculated in the secular non-adiabatic approximation reproduces some special peculiarities of the spectra observed. In following papers (see Table 7.1) for diatomic molecules the dependence of the resolved spectra components on the rotational quantum number was described. As an example, the experimental dependence %(j) = Tj+1. /r is shown in Fig. 7.5. 

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

In order to evaluate the above expression, solutions were found for the Schrodinger equation using the Morse potential for rotational quantum number i not equal to zero ... 

The experimental figures, with one exception, were obtained from oscillation-rotation spectra with the use of integral rotational quantum numbers by Kratzer, Z. f. Physik, vol. 3, p. 289 (1920). The second figure for hydrogen chloride was calculated by Colby, Astrophys. Journ., vol. 58, p. 303 (1923), from the same data, with the use of half quantum numbers, and by Czerny,... 

Initial conditions for the total molecular wavefunction with n = I (including electronic, vibrational and rotational quantum numbers) can be imposed by adding elementary solutions obtained for each set of initial nuclear variables, keeping in mind that the xi and 5 depend parametrically on the initial variables... 

A typical initial condition in ordinary wave packet dynamics is an incoming Gaussian wave packet consistent with particular diatomic vibrational and rotational quantum numbers. In the present case, of course, one has two diatomics and with the rotational basis representation of Eq. (30) one would have, for the full complex wave packet. 

The H2O molecules are cooled in a supersonic expansion to a rotational temperature of 10K before photodissociation. The evidence for pathway competition is an odd-even intensity alteration in the OH product state distribution for rotational quantum numbers V = 33 45. This intensity alternation is attributed to quantum mechanical interference due to the N-dependent phase shifts that arise as the population passes through the two different conical intersections. 

It is customary to use the index J for the rotational quantum number. Equation (10.33) then becomes... 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Fig. 12. Partitionings of hydrogen fragment translational energy distribution into three components. The solid line denotes the contribution from H2S — 8H(,4 "S+ ) + H which yields a resolved structure with a rovibrational state assignment on the top. The dotted line denotes the contribution of hydrogen from the SH(442 +) —> S(3P) + H reaction, which is a reflection of the solid curve but the structure is smeared out. The corresponding rotational quantum numbers of the parent molecule SI I (A 2>l 1 ) l =0 is marked on the bottom. The remaining part of the P(E) spectrum is represented by the square-like dashed curve.

P(Jjz) in equation 15 is the probability density of the CH3Br total rotational angular momentum quantum number, j, and rotational quantum number, jz. It is given by,... 

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