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

Excercises

April 1st, 2007 (pdf)
April 10, 2007 (pdf)

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

Most steps of the derivation are given in the these notes.

Derive Eq. (50) using Eqs. (29) and (30)
To derive Eq. (51) first show that L=l+1
Show that the first Clebsch-Gordan coefficient in Eq. (50) must be 1 for mu=1 and m=l
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 >
Next use the recursion relations to find < 1 0 l 0 | l+1 0 >
Check Eqs. (51) and (52). Note that n!! = n (n-2) (n-4) ...
Check Eqs. (53), (54), (55), (60), and (61)
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)
If Eq. (85) holds for M=L, why must it hold for all M?
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.
Check Eqs. (88), (89), and (90).
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)