# Angular momentum theory

Lecturer: Gerrit C. Groenenboom

### date, time, room, topic

 1: Thursday Apr 17, 2008 13:45-15:45 HG03.632 Zare 1.1, 1.2, 1.3 2: Thursday Apr 24, 2008 13:45-15:45 HG02.702 canceled 3: Tuesday Apr 29, 2008 13:45-15:45 HG03.054 4: Friday May 2, 2008 13:45-15:45 HG03.044 Zare 2.1 and 2.2 5: Thursday May 8, 2008 13:45-15:45 HG01.057 Zare 3.1-3.4 6: Thursday May 15, 2008 13:45-15:45 HG03.632 Zare 3.5 7: Thursday May 22, 2008 13:45-15:45 HG02.702 Zare 3.7, 3.8, and 3.9 8: Thursday May 29, 2008 13:45-15:45 HG03.632 Zare 4.1 9: Thursday Jun 5, 2008 13:45-15:45 HG03.082 Zare 5.1 and 5.2 10: Thursday Jun 12, 2008 13:45-15:45 HG03.082 Zare 5.3 and 5.4 11: Thursday Jun 19, 2008 13:45-15:45 HG03.082 12: Thursday Jun 26, 2008 13:45-15:45 HG03.082 13: Thursday Jul 3, 2008 13:45-15:45 HG03.082 14: Friday Aug 22, 2008 13:45-15:45 HG03.054 Multipole expansion 15: Tuesday Aug 26, 2008 13:45-15:45 HG03.054 Alignment parameters 16: Wednesday Sep 24, 2008 13:45-15:45 HG Mo and Suzuki paper, photodissociation 17: (Wednesday) (Oct 1), 2008 (15:45-17:00) (HG 2.702) Op verzoek naar vrijdag middag verplaatst

### Hints for the derivation of the multipole expansion of the Coulomb operator

Most steps of the derivation are given in the these notes.
1. Derive Eq. (50) using Eqs. (29) and (30)
2. To derive Eq. (51) first show that L=l+1
3. Show that the first Clebsch-Gordan coefficient in Eq. (50) must be 1 for mu=1 and m=l
4. To derive the expression for the second CG coefficient, use the CG recursion relations to find an explicit expression for < 1 1 l m | l+1 m+1 >
5. Next use the recursion relations to find < 1 0 l 0 | l+1 0 >
6. Check Eqs. (51) and (52). Note that n!! = n (n-2) (n-4) ...
7. Check Eqs. (53), (54), (55), (60), and (61)
8. Derive Eq. (85) for M=L starting with Eq. (60)
• use binomial expression for (x+y)^l for x=R_1 and y=r_1
• to simplify the result use: (2n)! = (2n)!! (2n-1)!! = 2^n n! (2n-1)!! (check this)
9. If Eq. (85) holds for M=L, why must it hold for all M?
10. Equation (87) follows easily from the generating function of the Legendre polynomials. Alternatively:
• Assume the function can be expanded in products of spherical harmonics of the polar angles of the vectors r and R and coefficients that only depend on then lengths of the vectors.
• Require the function to be invariant under simultaneous rotation of the vectors, and use the explicit expression for < l1 m1 l2 m2 | 0 0 >, Eq. (35), and the spherical harmonic addition theorem to write the function as a Legendre expansion.
• Show that P_l(1)=1
• Show that 1/(1-x)=1+x+x^2+x^3+...
• To find the expansion coefficients consider the case where the angle between the vectors is zero.
11. Check Eqs. (88), (89), and (90).
12. Introduce the multipole operators [Eq. (78)] to write down the multipole expansion of the Coulomb interaction between two molecules.

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Last updated: March 26, 2007, by Gerrit C. Groenenboom, (e-mail: gerritg at theochem.ru.nl)