Multielectron atoms angular momentum

Now the parity of a multielectron, ionic, or atomic wave function is given by TT ( — l)/fc, where lk (the angular-momentum quantum number)... 

The Pauli Exclusion Principle states that no two electrons of any single atom may simultaneously occupy a slate described by only a single set of quantum numbers. Five such numbers arc needed to describe fully the quantum-mechanical conditions of an electron. For j-j coupling this set is generally ti. I., v. j. iij. and for l.-S it is /t. /. j. u(. nr,. From die coupling of the angular momentum associated with the latter sets a full description of the multielectron stale, described by it, L. S, J. Mis determined. 

According to the selection rules, one-photon absorption occurs only if the change in angular momentum (change in L) is +1 or -1 (Al = 1, A/ = 0, 1 (0 o 0 not allowed), AL = 0, 1, AS = 0) (Al is according to the hydrogenic atom model, whereas AL is for multielectron atoms). The selection rules allow transition in one-photon absorption only to the p states from the s ground state as a result only even-to-odd parity is allowed. 

A multielectron atom can exist in several electronic states, called microstates, which are characterized by the way the electrons are distributed among the atomic orbitals. The number of microstates for a free atom with a valence shell consisting of a set of degenerate orbitals with orbital angular momentum quantum number I housing n electrons is given by ... 

In a free multielectron atom or ion, the spin and orbital angular moments of the electrons couple to give a total angular momentum represented in the Russell-Saunders scheme by the quantum number J. Since J arises from vectorial addition of L (the total orbital quantum number) and 5 (total spin quantum number), it may take integral (or half-integral... 

To this point in the discussion of multielectron atoms, the spin and orbital angular momenta have been treated separately. In addition, the spin and orbital angular momenta couple with each other, a phenomenon known as spin-orbit coupling. In multielectron atoms, the S and L quantum numbers combine into the total angular momentum quantum number J. The quantum number J may have the following values ... 

In the Russell-Saunders approximation, the total orbital angular momentum and the total spin angular momentum of a multielectron atom can have predictable values that follow the same pattern as other angular momenta. 

The energy levels of multielectron atoms can be characterized by orbital and spin angular momentum values in the Russell-Saunders approximation. 

Results of various calculations were presented. In the orbital approximation, the energies of the orbitals in multielectron atoms depend on the angular momentum quantum number as well as on the principal quantum number, increasing as / increases. The ground state of a multielectron atom is identified by the Aufbau principle, choosing orbitals that give the lowest sum of the orbital energies consistent with the Pauli exclusion principle. 

A multielectron atom is in a state P. For this term draw a vector diagram for the maximal atomic orbital angular momentum L (Figure E7.12) and determine the angle a between this vector and the orbital angular momentum (refer to Figure 7.19)... 

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