Download Spin Orbit Coupling - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 7.9: Spin-orbit coupling: SLJsSL i i i i rrrrr l rr +=== ∑∑ Russell-Saunders coupling assumes that coupling interactions between the electrons are stronger than between individual spin angular momenta and orbital angular momenta. ( )∑∑ +== i ii i i sjJ r l rrr = j-j couplingOtherwise: docsity.com Russell-Saunders Term Symbols: used to designate electron energy states in multielectron atoms J S L12 + a) Total Orbital Angular Momentum The degeneracy of 2S+1L energy level gLS = (2S+1)(2L+1) K K KJIHGFDPS L = = 876543210 Compare with H-atom: K Kl kjihgfdps= = 876543210 Similar: In H-atom these are states which correspond to a unique one-electron configuration. In multielectron atoms one electron configuration can give rise to many terms. docsity.com Wave functions describing a system of bosons must be symmetric with respect to interchange of any pair of bosons. The implication of this is that any and all wave functions can have the same set of quantum numbers. Consider two non-equivalent p electrons: configuration = (np1)(n’p1). Since n ≠ n’, the Pauli principle is not a factor here. Therefore we can use what we’ve learnt from the Clebsch-Gordon series to derive terms and levels. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00000 11010 10101 0,1,21111 20202 1,2,31212 1,3)12( 0,1 2 1 2 1 2 1 2 1 0,1,21111 2 1,1; 2 1,1 2121 2121 2211 =−→+= =−→+= =−→+= =−→+= =−→+= =−→+=−→+= =+ =−→⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +=−→+= =−→+=−→+= ==== SLSLJ S ssssS L ss llll ll docsity.com Terms and levels are: 0 1 1 1 2 1 1 3 0,1,2 3 1,2,3 3 SPD SPD Six terms, 10 levels for one electron configuration. docsity.com Now consider two equivalent p electrons: electron configuration is (np)2 Here the Pauli Exclusion Principle is important. To see why set up a table as shown: docsity.com Given that we can determine terms from a configuration, how do we order them by energy? Two options: 1.) Do a high level quantum mechanical calculation: involved 2.) Make an intelligent guess. We can do this by invoking Hund’s Rules Very approximate and only valid for ground state configurations. Also presumes L-S coupling is valid Warning: Hund’s Rule #1: States of highest spin multiplicity (2S+1) lies lowest in energy. The explanation of the rule lies in the effects of the spin-spin interaction. docsity.com versus -same orbital -electrons closer together -more e- - e- repulsion -different spatially oriented orbitals -electrons are farther apart -less e- - e- repulsion S = 0 S = 1 docsity.com Hund’s Rule #2: For a given S (or 2S+1) the state of highest L lies lowest in energy. +Z e+Z e versus e - e - e - e - e - e - e - e - -higher L; that is electrons “orbit” in same direction and keep out of each other’s way. -minimizes e- -e- repulsions. -lower L; that is, some electrons “orbit” in the opposite direction to others and therefore can not avoid each other -increased e- - e- repulsions. docsity.com
