Orbital angular momentum expansion

The single-particle wave function for the free photoelectron may be expressed as an expansion in angular momentum partial waves characterized by an orbital angular momentum quantum number l and and associated quantum number X for the projection of l on the molecular frame (MF) z axis [22, 23, 63-66],... 

As to the evaluation of B h [as defined by (74)], there is a difference between the two variants A and B of the standard approximation. Approximation A is only used in the context of MP2-R12 as an alternative to approximation B, while in treatments on a higher level of approximation only approximation B is applied. In approximation B all those terms — that arise in the summation over the complete basis — are neglected that in atomic theory decrease as (L + l)-7 or faster, while in approximation A even terms that decrease as (L + 1) 5 are neglected, where L is the highes orbital angular momentum contained in the Cl expansion. Note that the error of a conventional Cl without rl2 terms goes as (L + 1) 3. 

Abstract - The temperature dependence of the proton nmr spectra of dithiocarbamato iron(III) complexes is markedly solvent dependent. A study is made of the temperature dependence of the nmr shifts for the N-CH2 protons in tris(N,N-dibutyldithiocar-bamato) iron(III) in acetone, benzene, carbon disulfide, chloroform, dimethyIformamide, pyridine and some mixed solvents. This contribution shall outline first how the nmr shifts may be interpreted in terms of the Fermi contact interaction and the dipolar term in the multipole expansion of the interaction of the electron orbital angular momentum and the electron spin dipol-nuclear spin angular momentum. This analysis yields a direct measure of the effect of the solvent system on the environment of the transition metal ion. The results are analysed in terms of the crystal field environment of the transition metal ion with contributions from (a) the dithiocarbamate ligand (b) the solvent molecules and (c) the interaction of the effective dipole moment of the polar solvent molecule with the transition metal ion complex. The model yields not only an explanation for the unusual nmr results but gives an insight into the solvent-solute interactions in such systems. 

PROBLEM 3.5.6. One can define linear ladder operators for angular momentum (orbital or spin) the raising operator + = Lx + iLy and the lowering operator =LX — iLy. (a) Verify that brute-force expansion yields + =... 

In theory, an infinite number of calculations for highly excited states is required to complete the expansion of the EP given by Eq. (24), since there are only a few occupied valence orbitals in neutral atoms. This difficulty also exists in the nonrelativistic case and is resolved by using the closure property of the projection operator with the assumption that radial parts of EPs are the same for all orbitals having higher angular momentum quantum numbers than are present in the core. The same approximation is applicable in the present... 

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