Lectures Notes (13) - Chem 112 Electronic Spectroscopy of...

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Chem 112: Electronic Spectroscopy of Trasition Metal Free Ions and O h Complexes. 1 Reading: Miessler and Tarr pp 379 - 408 Harris and Bertolucci pp 232 - 242 1. One-electron ions and atoms: Ψ = R(r) ⋅Θ ( θ ) ⋅Φ ( φ ) ⋅ψ s R(r) = the radial part, depends only on r Θ ( θ ) ⋅Φ ( φ ) = angular part ψ s = spin part, independent of r, θ , φ Ψ depends on 4 quantum numbers: n = principal quantum number = 1, 2, 3,… , determines R(r) l occurs in Θ ( θ ); it determines the angular momentum of the electron l = 0, 1, 2,…n - 1 m occurs in both Θ ( θ ) and Φ ( φ ); it indicates the tilt of the orbital motion with respect to some reference direction; m (= also “ m l ”) = - l , - l + 1, …, 0, 1, 2, l -1, l ; (2 l + 1) values m l = l = 2 m l = 1 m l = 0 m l = - 1 m l = - 2 ψ s (= also “ s ” = 1/2 and its projection “ m s ”) = + 1/2 , - 1/2 There is another quantum number, “ j ”. It is useful in describing the energy of the state of the ion or atom: for a one-electron system j = l + 1/2 or l - 1/2 …so long as we regard ψ s as completely independent of r, θ , and φ , these two j states have the same energy. Actually they spin and angular moments do interact, but for light atoms the magnitude of this interaction (the “spin-orbit” coupling ) is very small compared with energy differences between orbitals of different l values, so we often regard the energy level pattern as being only a function n and l .
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M S M L + 1/2 - 1/2 2 2 + 1 1 + 1 - 0 0 + 0 - -1 + -1 - 2 + - M S M L + 1/2 - 1/2 2 1 1 1 1 1 0 1 1 -1 1 1 2 1 2 2. Russell-Saunders treatment of electronic states for multi-electron (free) ions and atoms: Unfortunately, the orbital angular momentum ( l ) and spin angular momentum ( s ) do interact, but for light atoms (up to approximately the lanthanides) their coupling ("spin-orbit coupling") is small compared to energy differences for electrons with different n and/or l quantum numbers. Thus, Nature follows a scheme for electronic states that can be understood to a fairly accurate level of approximation by using a relatively simple set of realtionships designated "Russell-Saunders" or "LS" coupling, whereby the electronic state is designated by its term symbol: " 2 S +1 L ", 2 S +1 = spin multiplicity and L = degeneracy M L = Σ ( m l ) i = L , L +1, ..., 0, 1, 2,...- L. S( L = 0), P( L = 1), D( L = 2), F( L = 3), G( L = 4), H( L = 5), I( L = 6), K( L =7), etc M S = Σ ( m s ) i = S , S +1, ..., 0, 1, 2,...- S. S = 0 (singlet), S = 1/2 (doublet), S = 1 (triplet), S = 3/2 (quartet), etc. For the trivial case of one electron, for example in a d orbital, we use the following way to organize the 10 electron configurations, or "microstates" into a table for identifying the term symbol for this electronic state: i i m l = 2 m l = 1 m l = 0 m l = -1 m l = -2 m l = 2 m l = 1 m l = 0 m l = -1 m l = -2 = 2 + = 1 + = 0 + = -1 + = -2 + = 2 - = 1 - = 0 - = -1 - = -2 - S = + 1/2 S = - 1/2 -2 - 2 -2 -1 1 then listing only the number of microstates having a given set of M L and M S values:
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M S M L 1 0 -1 2 (1,1) + 1 (1,0) ++ (1,0) (0,1) + + - - (1,0) - - 0 + + + - (0,0) (1,-1) - - -1 (-1,0) ++ + + - - - -2 + M S M L 1 0 -1 2 1 1 1 2 1 0 1 3 1 -1 1 2 1 -2 1 3 When organized in this manner, we readily see that these 10 microstates span: M L = 2, 1, 0, -1, -2 L = 2, and M S = + 1/2 , - 1/2 S = 1/2 , indicating a 2 D electronic state.
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