Rotational quantum number natural

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). 

The principal use of the nuclear symmetry character is in determining the allowed values of the rotational quantum number K of the molecule. The complete wave functions for a molecule (including the nuclear-spin function) must be either symmetric or antisymmetric in the nuclei, depending on the nature of the nuclei involved. If the nuclei have no spins, then the existent functions are of one or the other of the types listed below. 

Figure 3.1a Natural rotational quantum numbers for Hund s cases (a) and (b). Reduced term value plots for 2S (B = 1.0 cm-1) and 2nr (B = 1.0 cm-1, A = 20.0 cm-1), (a) Plot °f — BJ(J + 1) versus J(J + 1) displays case (a) limiting behavior for the 2II state at very low J. The dotted lines illustrate the B2/A corrections to the near case (a) effective B-values (See Section 3.5.4). The 2S state does not exhibit case (a) behavior even at low J (at J = 0 the limiting slopes of the 2S Fi and h l curves are —oo and +oo).

If we neglect the terms in J,.2 and J J2, and consequently the nonharmonic character and the dependence of the y s on J, the energy is resolved into a rotational component and an oscillation component of the well-known form. As a nearer approximation we have a dependence of the oscillation frequency on the rotation quantum number and also the non-harmonic character of the oscillation. Naturally our method admits of more accurate calculations of the energy, involving higher powers of J and Je. 

Unlike the case of collision-induced vibrational energy transfer, collision induced rotational energy transfer seems to be free of strong restrictions on the changes in the rotational quantum numbers. When account is taken of the spectral widths of the excitation sources used, the nature of the rotation-vibration structure in the fluorescence and absorption spectra, and the possibility of resonant ener f transfer in the collision, it is concluded that the studies of Bj aniline are the weakest, those of B2 benzene better, and those of glyoxal the best available. With this hierarchy of quality of information kept in mind, the following weaker conclusions can also be obtained from the studies cited. 

For solids and liquids, electronic absorption bands are usually broad and essentially featureless, but more information is obtainable from electronic spectra of gas-phase molecules. Transitions between two levels with long lifetimes are the most informative. Such an electronic transition for a gas-phase sample has various possible changes in vibrational and rotational quantum numbers associated with it, so that the spectrum, however it is obtained, consists of a number of vibration bands, each with rotational fine structure, together forming an electronic system of bands. The selection rules governing the changes in vibrational and rotational quantum numbers depend on the nature of the electronic transition, and they can be ascertained by analyzing the pattern and structures of the bands. 

Fig. 1.23. The hydrogen chloride vibrational-rotational band. The P and R branches are labeled. The Q branch (v ) is missing since cannot equal zero for a parallel band of a linear molecule. Each rotational line is labeled with its rotational quantum number in the ground vibrational state. In this high resolution spectrum each rotational line is a doublet since chlorine has two naturally occurring isotopes. The peaks for H Cl occur at slightly lower frequencies in cm than the peaks for H Cl [see Eq. (1.11)].

The two atomic molecule is discussed from the viewpoint of wave mechanics with the help of an approximation method. The nature of the eigenfunctions and the position of the terms, as well as the appearance of transition, are discussed. Some additional comments on the symmetry properties of the eigenfunctions, discussed in a previous paper, will be made in addition. PYom the investigation we first obtein an expression for the rotational quantum number dependency of the doublet splitting which derives from the possibility that the anguleu momentum of the electron with respect to the internuclear axis can be parallel or antiparallel. Secondly (we obtsdn] an interpretation for the deviations of single terms from the values computed from term formulae, indicated as perturbations as they have been empirically determined in many band spectra for certain combinations of electronic, vibrational and rotational quantum numbers. We finally obtain a description of the phenomenon of predissodationi discovered by Henri, namely an estimate of the lifetime of the predissociated molecule. As far as possible, the theory is compared to experiment and it is pointed out where an extension of the experimental material is desirable. 

Note that there is a quantum mechanical correction, equal to /iv/2, to the bond energy Uq (Uq has a negative sign), which is called the zero-point energy. In (4.61), n is the vibrational and j the rotational quantum number. Both n and j are natural numbers. 

The rotation-vibration interaction of Section 4.32 produces different effects in nonlinear molecules than those discussed in the previous section. In nonlinear molecules the quantum numbers are vavhvcKJM >. The connection between the group quantum numbers Ico , co2> xi > 2 -A 3/ > and the usual quantum numbers is given by Eq. (4.85). The different effect can be traced to the different nature of the rotational spectrum. In lowest order, the spectrum of a bent molecule is given by Eq. (4.107) and Figure 4.21. The rotation-vibration interaction introduces terms with selection rules... 

Excited electronic energy levels are sometimes occupied, especially when unpaired electrons are present. A set of quantum numbers describe the electronic levels, but because of the unique nature of these levels for each type of atom or molecule, it is not possible to write general expressions similar to those given earlier for translational, rotational, or vibrational levels. 

Namely, the circular orbit (l = n — 1), which is a rotating state with a nodeless radial wavefunction, corresponds to a vibrational quantum number v = 0, and the next-to-circular single-node state (l = n — 2) corresponds to v = 1,.... This theoretical possibility of large-/ circular orbits behaving like bound states in a Morse potential seems to have no other natural manifestation than in the present case of metastable exotic helium. This situation is presented in Fig. 2, where the potential as well as the wavefunctions are shown. 

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