Vibrational angular momentum

Zhu L, Chen W, Hase W L and Kaiser E W 1993 Comparison of models for treating angular momentum in RRKM calculations with vibrator transition states. Pressure and temperature dependence of CI+C2H2 association J. Phys. Chem. 97 311-22... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

J and Vrepresent the rotational angular momentum quantum number and tire velocity of tire CO2, respectively. The hot, excited CgFg donor can be produced via absorjDtion of a 248 nm excimer-laser pulse followed by rapid internal conversion of electronic energy to vibrational energy as described above. Note tliat tire result of this collision is to... 

Eigure a shows that the eigensurfaces form an interconnected double sheet, the lower member of which has a ring of equivalent minima at r = and IV = — k. As expected angular momentum is conserved, but with the complication that it is vibronic, rather than purely vibrational in character. 

Now, we have besides the vibrational, the electronic angular momentum the latter is characterized by the quantum number A corresponding to the magnitude of its projection along the molecular axis, L. Here we shall consider A as a unsigned quantity, that is, for each A 7 0 state there will be two possible projections of the electronic angular momentum, one corresponding to A and the other to —A. The operator Lj can be written in the form... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

In the configuration space spanned by (411,1 2, 73), we may then define the vibrational angular momentum k through its classical components, that is,... 

Thus, I and m quantize the vibrational angular momentum and its z component. 

This completes our introduction to the subject of rotational and vibrational motions of molecules (which applies equally well to ions and radicals). The information contained in this Section is used again in Section 5 where photon-induced transitions between pairs of molecular electronic, vibrational, and rotational eigenstates are examined. More advanced treatments of the subject matter of this Section can be found in the text by Wilson, Decius, and Cross, as well as in Zare s text on angular momentum. 

Polarization functions are functions of a higher angular momentum than the occupied orbitals, such as adding d orbitals to carbon or / orbitals to iron. These orbitals help the wave function better span the function space. This results in little additional energy, but more accurate geometries and vibrational frequencies. 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

The elements of S-matrices are determined in the basis of orbital angular momentum l and rotational moments jt,jf of vibrational states i,f and their projections (m,m,-,m/). Both S-matrices in Eq. (4.58) have to be calculated for the same energy Ek of colliding particles. 

Valiev-Ivanov model 219, 275 vibrational broadening 123 vibrational dephasing 111, 113-15, 123 vibrational relaxation, and angular momentum relaxation 92 vibrational transition, adiabatic dephasing 92... 

There does not seem to be any selection rule such as conservation of spin or orbital angular momentum which this reaction does not satisfy. It is also not clear that overall spin conservation, for example, is necessary in efficient reactions (5, 16, 17, 20). Further, recent results (21) seem to show a greatly enhanced (20 times) reaction rate when the N2 is in an excited vibrational state (vibrational temperature 4000 °K. or about 0.3 e.v.). This suggests the presence of an activation energy or barrier. A barrier of 0.3 e.v. is consistent with the low energy variation of the measured cross-section in Figure 1. 

Figure 1. Eigenvalues of the scaled champagne bottle Hamiltonian (Eq. (2)) for p = 0.00625, in the energy, , and angular momentum, k map. The eigenvalues, represented by points, are joined (a) by lines of constant bent vibrational quantum number, vt, and (b) by lines of constant linear quantum number, v = 2vt + k. ...

---