Application to orbital angular momentum

We now apply the results of the quantum-mechanical treatment of generalized angular momentum to the case of orbital angular momentum. The orbital angular momentum operator L, defined in Section 5.1, is identified with the operator J of Section 5.2. Likewise, the operators I , L, Ly, and are identified with J, Jx, Jy, and Jz, respectively. The parameter j of Section 5.2 is denoted by I when applied to orbital angular momentum. The simultaneous eigenfunctions of P and are denoted by Im), so that we have 

These coordinates are defined over the following intervals 

The volume element dr = dv dj dz becomes dr = sin OdrdOdtp in spherical polar coordinates. 

Substitution of these three expressions into equations (5.7) gives 

Since the variable r does not appear in any of these operators, their eigenfunctions are independent of r and are functions only of the variables 6 and tp. The simultaneous eigenfunctions Im) of I and L will now be denoted by the function YimiO, tp) so as to acknowledge explicitly their dependence on the angles 6 and tp. 

By squaring each of the operators Lx, Ly, Lz and adding, we find that L2 is given in spherical polar coordinates by 

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Application to orbital angular momentum We then operate on this result with L+ to obtain... 

Nitrosyl complexes of both S = and S = are common. The S = nitrosyl complexes or iron and copper have slightly anisotropic g tensors (Ag/g < 2) with the anisotropy provided primarily by contributions of orbital angular momentum from the metal d orbitals. As an example, we consider the model proposed by Kon and Katakoa (1969) for ferroprotoheme-NO complexes, which is also applicable to the ferrous-nitrosyl complexes of heme proteins. In the complexes studied by these workers, the axial ligands are a nitrogenous base and NO. They proposed that the unpaired electron resides primarily in the metal d z orbital. The spin-orbit coupling would then mix contributions from the d, and dy orbitals. 

It has been pointed out above that two electrons in the Is orbital must have their spins opposed, and hence give rise to the singlet state So, with no spin or orbital angular momentum, and hence with no magnetic moment. Similarly it is found that a completed subshell of electrons, such as six electrons occupying the three 2p orbitals, must have S — 0 and L = 0, corresponding to the Russell-Saunders term symbol lS0 such a completed subshell has spherical symmetry and zero magnetic moment. The application of the Pauli exclusion prin-... 

Bonifacic and Huzinaga[3] use explicit core orbital projection operators, while orbital angular momentum projection operators are used by Goddard, Kahn and Melius[4], by Barthelat and Durand[5] and others. Explicit core orbital projection operators can, in the full basis set, be viewed as shift operators which ensure that the first root in the Fock matrix really corresponds to a valence orbital. However, in applications the basis set is always modified and the role of the core orbital projection operators thus partly changes. 

Frank s research career began at a time when Solid State Physics was a new topic in Physics. He contributed very much to what became an all-embracing topic in both basic physics and in its numerous applications in chemistry as well as physics in areas now frequently described as Condensed Matter . He pointed out how random strains can force a dynamic Jahn-Teller system to reflect distorted static behaviour in certain cases particularly in EPR spectra. This involved considerations of the orbital angular momentum of the magnetic ions present in these systems and he showed how the Ham Effect , as it became known, could explain why the electronic angular momentum could be quenched in many systems. 

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],... 

The fact that an electron in an orbital produces a magnetic field proportional to its angular momentum has already been discussed. The application... 

For nonlinear polyatomic molecules, no orbital angular-momentum operator commutes with the electronic Hamiltonian, and the angular-momentum classification of electronic terms cannot be used. Operators that do commute with the electronic Hamiltonian are the symmetry operators Or of the molecule (Section 12.1), and the electronic states of polyatomic molecules are classified according to the behavior of the electronic wave function on application of these operators. Consider H2O as an example. 

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