Angular Momenta - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download Angular Momenta - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 5.2 Coupling of more than 2 angular momenta: j1, j2, j3,… 1.) Couple first two angular momenta j1 and j2 as before to generate J’ 2.) Then couple these results individually to j3 to get set of J’s 2;1;1 321 === lllExample: 1221 givetoand LllCouple ( ) 1221 0,1,2021111 L r l r l r ==→=−→+=+ ( ) ( ) ( ) 22020 1,2,3132121 0,1,2,3,4042222312 =−→+= =→=−→+= =→=−→+==+ LL r l rr ( ) ( ) ( ) 0,21,32,23,4 ×××=∴L 9 different angular momentum states, although some have the same total L values. docsity.com It should be and in fact it is possible to represent the coupled wave functions in terms of the uncoupled ones: >>=∑∑ 21 1 2 21 ,,,|,,,| 21,21 jj m m mmJ mjmjCMJjj j j jj The coefficients in this expansion are called Clebsch-Gordon coefficients or Wigner coefficients ',' 2121, 21,212121 21 2121 1 2 21 21 1 2 212121 ,;,',';',' ,;,|',';',',,,',';',' jj j j jj j j jj mm jjjj m m mm jj m m mmjjJjj C mjmjmjmjC mjmjCmjmjMJjjmjmj = = = ∑∑ ∑∑Note: Represents the degree of coupling or overlap docsity.com ( ) 0,1 2 1 2 1 2 1 2 1 2121 =−−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +=−→+= ssssSb.) Clebsch-Gordon Series: c.) Coupled wave functions: >−= >= >= >= > 1,1, 2 1, 2 1| 0,1, 2 1, 2 1| 1,1, 2 1, 2 1| 0,0, 2 1, 2 1| ,,,| 21 SMSss Singlet state, S = 0; MS = 0 Triplet State, S=1; MS=1,0,-1 docsity.com d) Start with triplet state >1,1, 2 1, 2 1| can only be formed from 2121 2 1, 2 1, 2 1, 2 1|,;,| 21 ssSss mmMmsms +=>>= Q 1 2 1, 2 1; 2 1, 2 1|1,1, 2 1, 2 1| 2 1, 2 1, 11 ==>⇒=>⇒ CC ss mm >−1,1, 2 1, 2 1| can only be formed fromSimilarly, >−−>= 2 1, 2 1, 2 1, 2 1|,;,| 21 21 ss msms 1 2 1, 2 1; 2 1, 2 1|1,1, 2 1, 2 1| 2 1, 2 1, 11 ==>⇒−−=>−⇒ −− CC ss mm docsity.com >1,1, 2 1, 2 1|>0,1, 2 1, 2 1|To get apply either lowering operator to >−1,1, 2 1, 2 1|or raising operator to Use lowering operator. Recall: −−− += 21 ˆˆˆ ssS and ( ) ( ) >−−−+>=− 1,|11,| ˆ SSSS MSMMSSMSS h ( ) >+>=∴ −−− 2 1, 2 1; 2 1, 2 1|ˆˆ1,1, 2 1, 2 1|ˆ 21 ssS ( )( ) ( )( ) >>=−>=− 0,1,2 1, 2 1|20,1, 2 1, 2 1|01211,1, 2 1, 2 1|ˆ hhSLeft-hand side: ( ) >+>>=+ −− 2 1, 2 1; 2 1, 2 1|ˆ 2 1, 2 1; 2 1, 2 1|ˆ 2 1, 2 1; 2 1, 2 1|ˆˆ 2121 ssssRight-hand side: >−+>−= >−++>−+= >−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++>−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += 2 1, 2 1; 2 1, 2 1| 2 1, 2 1; 2 1, 2 1| 2 1, 2 1; 2 1, 2 1| 4 1 4 3 2 1, 2 1; 2 1, 2 1| 4 1 4 3 2 1, 2 1; 2 1, 2 1|1 2 1 2 11 2 1 2 1 2 1, 2 1; 2 1, 2 1|1 2 1 2 11 2 1 2 1 hh hh hh docsity.com