Vibrational and rotational quantum numbers

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 analysis of spectroscopic data for bound states of diatomic molecules gives accurate potential curves if one follows the semi-classical Rydberg-Klein-Rees method. For a review of this see Ref. 126). It is sufficient to note that this gives the two values of r as a function of potential energy by considering the dependence of the total spectroscopic energy on the vibrational and rotational quantum numbers n and J. A somewhat simpler procedure, and the only one plicable to polyatomic molecule, is to use the Dunham expansion of the potential 127). 

As a consequence of the attractive region of the interaction, several bound state solutions of the radial Schrodinger equation generally exist whose wavefunctions are localized in the well region. The bound state eigenenergies are negative, discrete and will be subscripted with the vibrational and rotational quantum numbers, v and the normalization... 

Electronic Molecular Spectra.—In general the absorption and emission spectra of molecules involve change in the electronic quantum numbers as well as in the vibrational and rotational quantum numbers. These molecular spectra are complex, and their interpretation is diffi-... 

Consider the fluorescence from a molecular wavepacket excited from the ground electronic state by a short pulse of light. We assume that the initial eneigy of the molecule is EVgjg, where v, j denote, respectively, vibrational and rotational quantum numbers, with well defined magnetic quantum m,... 

Vibrational and rotational quantum numbers for two separate molecules A and B become local quantum numbers in the composite AB system, as would occur during a collision. In such a case these local... 

Results showing the dependence of the CO collision halfwidth in combustion gases on the vibrational and rotational quantum numbers are shown in Figure 6. The data were obtained with a flame temperature of 1875 K and equivalence ratios in the range 1.2 - 1.4. Although too few data points are available for a detailed analysis, it is clear that 2y decreases with increasing m and that values for 2y are nearly equal (within 5%) for ground state and excited state transitions. 

The details of the relaxation channels contributing towards the total effective cross-section of relaxation of the ground state a requires measurements with fixed vibrational and rotational quantum numbers v", J" and v J, J" of the reaction (3.1). Data on such measurements, e.g. in Na2/Na beams in collisions with noble gases can be found in the monograph [116], and those on Li2-containing vapour can be found in [306]. 

Resonantly enhanced two-photon dissociation of Na2 from a bound state of the. ground electronic state occurs [202] by initial excitation to an excited intermediate bound state Em,Jm, Mm). The latter is a superposition of states of the A1 1+ and b3Il electronic curves, a consequence of spin-orbit coupling. The continuum states reached in the two-photon excitation can have either a singlet or a triplet character, but, despite the multitude of electronic states involved in the computation reported J below, the predominant contributions to the products Na(3s) + Na(3p) and Na(3s) + Na(4s) are found to come from the 1 flg and 3 + electronic states, respectively. The resonant character of the two-photon excitation allows tire selection of a Single initial state from a thermal ensemble here results for vt = Ji — 0, where vt,./, denote the vibrational and rotational quantum numbers of the initial state, are stJjseussed. 

Until recently, few attempts have been made to extend the theory of the ammonia inversion to account for the dependence of the inversion splittings on the vibrational and rotational quantum numbers [e.g. )]. These attempts differed not only from the standard approach to the vibration—rotation problem of rigid molecules but also from the approach to the problem of nonrigid molecules with internal rotation [for example )]. 

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

To obtain the reaction attributes for a particular set of vibrational, rotational and translational energies, many trajectories were simulated at given values of N2 vibrational and rotational quantum numbers and N2-O relative translational energy. The N2 molecular orientation, vibrational phase and impact parameter were chosen randomly for each trajectory. The reaction attributes were then determined by averaging the outcomes of all collisions. The information obtained is state-specific, so for example, the energy distributions of the reactant and product molecules can be determined. The method used to calculate the vibrational and rotational state of the product molecule is outlined in Ref. 67. With the QCT approach, reaction cross sections were determined solely from the precollision state. The method knows nothing of the fluid flow environment and so... 

The determination of integral reactive cross sections, either fully state-specific in the product vibrational and rotational quantum numbers, or summed over all product states, has been discussed in a previous paper. [34] Basically, the probability for reaction at each value of the total angular momentum J is proportional to the square of the fully-state resolved S-matrix element for the transition in question. The ICS is tiien obtained by summing over all values of the total angular momentum J and multiplying by p divided by the square of the reactant wavevector. 

Fig. 9.2. Vibrational and rotational energy levels, vand Jare the vibrational and rotational quantum numbers.

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