Angular momentum, in molecules

A discussion of molecular magnetic moments due to vibrational angular momentum in molecules with degenerate vibrational states has been given by Moss, R. E., Perry, A. J. Mol. Phys. 25, 1121 (1973). 

Slater determinant, 258 Slater orbitals, 158, 159 Space groups, 285 common organic, 325 symmorphic, 288 Space symmetry operator, 286 Spectra, of crystals, 306 of organic dimers, 321 polarization of, 306 Spherical harmonics, 143ff Spin angular momentum in molecules, 188-169... 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

The same rules for the reduction of representations of groups give the following content of angular momentum in each vibrational band of bent molecules... 

Fig. 2.5. Classical and quantum calculation of the scattering of H2 by Hg in the idealization where both systems are considered to have zero angular momentum. In actuality, the undulations will be partly washed out by effects associated with the rotation of the molecule, the spin of the protons, and the energy spread in the monoenergetic beam after [30],...

Reaction 58 has been observed, and the rotational distribution measured is thermal, in marked contrast to similar measurements in H2O. Theoretical calculations suggest that this is because there is an exit valley that lies close to the bent geometry of the H2S molecule. Thus, the excited state can dissociate without producing a large amount of angular momentum in the SH fragment. 

R-type transition in spectroscopy. As a result of light absorption in this transition the difference A — J — J" between the quantum numbers of the angular momentum in excited (J ) and ground (J") state equals +1, and the angular momentum of the molecule increases. The transition with transition dipole moment d l at frequency u>o — fl corresponds to a diminution in the angular momentum of molecular rotation, and we have A = J — J" = — 1. Such a transition is called a P-type transition. 

As mentioned in the preceding section, the law of conservation of angular momentum in the process of optical pumping of molecules is reflected in the coefficients kFXk, kFxk, KA K, K Ax and CK in Eqs. (5.28), (5.29), (5.30), (5.31) and (5.32). For optical transitions between states with large angular momentum quantum numbers J, J" the explicit form of these coefficients is considerably simplified [15, 39]. These simplifications are due to the possibility of employing asymptotic expressions of the 6j-symbols presented in Appendix C. Applying Eq. (C.12) for the coefficients KA K and K Ax we obtain... 

In diatomic or linear molecules, the electronic state may still be defined in part by the orbital electronic angular momentum. In the case of more complex molecules,... 

This term describes the rotational Zeeman effect, that is, the coupling between the external field and the magnetic moment of the rotating nuclei. We note that there is no corresponding vibrational contribution since R a k is zero. The physical reason for this lack is that it is not possible to generate vibrational angular momentum in a diatomic molecule because it possesses only one, non-degenerate, vibrational mode. 

We have made use of the recipe, introduced in chapters 5 and 8, to handle the matrix elements of the total angular momentum in the molecule-fixed axis system. It is important to remember that 2 is a signed quantity. It is now worthwhile expanding the 3-j symbols in (10.147) for q = +1 we have... 

The chemical reaction is the most chemical event. The first application of symmetry considerations to chemical reactions can be attributed to Wigner and Witmer [2], The Wigner-Witmer rules are concerned with the conservation of spin and orbital angular momentum in the reaction of diatomic molecules. Although symmetry is not explicitly mentioned, it is present implicitly in the principle of conservation of orbital angular momentum. It was Emmy Noether (1882-1935), a German mathematician, who established that there was a one-to-one correspondence between symmetry and the different conservation laws [3, 4],... 

In the following table, all of the operator symbols denote the dimensionless ratio (angular momentum). (Although this is a universal practice for the quantum numbers, some authors use the operator symbols to denote angular momentum, in which case the operators would have SI units J s.) The column heading Z-axis denotes the space-fixed component, and the heading z-axis denotes the molecule-fixed component along the symmetry axis (linear or symmetric top molecules), or the axis of quantization. 

Recombination may also proceed via an electronically excited state if during the course of a bimolecular collision the system may transfer from the nonquantized part of the potential curve associated with one electronic state to a second state from which emission is allowed. This process is called preassociation or inverse predissociation, and the selection rules that control the probability of crossing in both directions are well known [109]. In such encounters total angular momentum must be conserved. For diatomic molecules, the system can pass only into the rotational level of the excited bound state which corresponds to the initial orbital angular momentum in the collision. 

Although it is impossible to formulate a definition of molecular geometry that is fully quanturn-mechanical in nature and at the same time universally applicable to all chemical species, topological analysis of the electron density leads to a rigorous statement of the dominant molecular structure for any state, spectroscopic or localized, stationary or time dependent, with zero angular momentum. In this sense, unlike geometry or shape, structure is an observable property of an isolated molecule. 

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