1

1 Properties of Spherical Harmonics

1.1 Repetition

qf, In the lecture the spherical harmonicsY,, m() were introduced as the ˆ ˆ2 of angular momentum operatorslz andl in spherical coordinates. We found that

ˆ qf, " qf, lzY,, m()= mY,, m() [1.1] and

ˆ2 qf, ",,() qf, l Y,, m()= + 1 Y,, m(). [1.2]

The spherical harmonics can be defined as

qf, ⋅⋅m q imf Y,, m()= N,, m P, ()cos e [1.3] where, is the of the orbital angular momentum andm the magnetic quantum number. There are analytical definitions for the normalization m q factorN,, m and the associated Legendre PolynomialsP, ()cos that allow the calculations of the spherical harmonics.

The spherical harmonics for, = 0, 1, and 2 are given by

qf, 1 Y00, ()= ------[1.4] 4p

1 3 Y , ()qf, = ------cosq 10 2 p [1.5] −1 3 ±if Y , ± ()qf, = +------sinqe 11 2 2p 2 Chapter 1

1 5 2 Y , ()qf, = ------()3cos q Ð 1 20 4 p

−1 15 ±if Y , ± ()qf, = +------sinqcosqe . [1.6] 21 2 2p

1 15 2 ±2if Y , ± ()qf, = ------sin qe 22 4 2p qf, qf, Note, that the sign of the functionsY11, ± () andY21, ± () is defined differently than in the script of the lecture. The definition here is in agreement with most of the literature on spherical harmonics.

1.2 Graphical Representation of Spherical Harmonics

The spherical harmonics are often represented graphically since their linear combinations correspond to the angular functions of orbitals. Figure 1.1a shows a plot of the spherical harmonics where the phase is color coded. One can clearly see that qf, Y,, m()is symmetric for a rotation about the z axis. The linear combinations ⁄ ()qf, ()m qf, qf, ()f qf, 12Y,, m()+ Ð1 Y,, Ðm()= 2 Y,, m()cos m ,Y,, 0() and ⁄ ()qf, ()m qf, qf, ()f Ði 2 Y,, m()Ð Ð1 Y,, Ðm()= 2 Y,, m()sin m are always real and have the form of typical atomic orbitals that are often shown.

1.3 Properties of Spherical Harmonics

There are some important properties of spherical harmonics that simplify working with them.

1.3.1 and Normalization

The spherical harmonics are normalized and orthogonal, i.e.,

2p p * Y, , ()qf, Y, , ()qf, sinq()dq df = d , d, , , [1.7] ∫ ∫ 1 m1 2 m2 m1 m2 1 2 0 0 where the is defined as Properties of Spherical Harmonics 3

a) m = Ð4 m = Ð3 m = Ð2 m = Ð1 m = 0 m = 1 m = 2 m = 3 m = 4 -π , = 0

-π/2

, = 1

0

, = 2 π/2

, = 3 π

, = 4

b) m = 4 m = 3 m = 2 m = 1 m = 0 m = 1 m = 2 m = 3 m = 4 -π , = 0

-π/2

, = 1

0

, = 2 π/2

, = 3 π

, = 4

Figure 1.1: Graphical Representation of the Spherical Harmonics qf, a) Plot of the spherical harmonicsY,, m() where the phase of the function is color coded. qf, Note thatY,, m() is always axially symmetric with respect to a rotation about the z axis since it depends only on the angleq . The phase of the function changes with a periodicity of ⁄ ()qf, ()m qf, qf, ()f m . b) The linear combinations12Y,, m()+ Ð1 Y,, Ðm()= 2 Y,, m()cos m , qf, ⁄ ()qf, ()m qf, qf, ()f Y,, 0(), andÐi 2 Y,, m()Ð Ð1 Y,, Ðm()= 2 Y,, m()sin m are all real and show only a phase of 0 (positive) andp (negative) and correspond to the typical orbital shapes. 4 Chapter 1

⎧0 ab≠ d , = ⎨ . [1.8] ab ⎩1 ab=

They form a complete basis set of the of square-integrable functions, i.e., every such function can be expressed as a of spherical harmonics

∞ , qf, qf, f ()= ∑ ∑ f ,, mY,, m(). [1.9] , = 0 m = Ð,

The coefficientsf ,, m can be calculated as

2p p qf, * qf, q()q f f ,, m = ∫ ∫ f ()Y,, m()sin d d . [1.10] 0 0

1.3.2 Product of Two Spherical Harmonics

Since the spherical harmonics form a set, the product of two spherical harmonics can again be expressed in spherical harmonics. Let us first look at a simple example

1 3 1 3 3 2 Y , ()qf, ⋅ Y , ()qf, ==------cosq ⋅ ------cosq ------cos q . [1.11] 10 10 2 p 2 p 4p

Comparing this to the spherical harmonics of Eqs. [1.4]-[1.6] it is immediately clear qf, qf, that we need the functionsY20, () andY00, () to express the product. We can make an Ansatz

qf, ⋅ qf, qf, qf, Y10, ()Y10, ()= c00, Y00, ()+ c20, Y20, () [1.12] which leads to

------3 2q ------1 - --1- 5()2q pcos = c00, + c20, p--- 3cos Ð 1 4 4p 4 . [1.13] ------1 - --1- 5 --3- 5 2q = c00, Ð c20, p--- + c20, p--- cos 4p 4 4

From this it is immediately clear that Properties of Spherical Harmonics 5

p ------3 ⋅ --4- 1 c20, ==p ------[1.14] 4 3 5 5p and

------1 ---1- 5 ⋅ p 1 c00, ==p--- 4 ------. [1.15] 5p4 4p

For a general product this is of course more complicated but there are a few simple rules for the general productY, , ()qf, ⋅ Y, , ()qf, . Since the dependence 1 m1 2 m2 onf is always given byexp()imf , it is immediately clear that the product function has to have the magnetic quantum numberMm= 1 + m2 . Using similar arguments, the orbital angular quantum number can be limited to the range , , ≤≤, , 1 Ð 2 L 1 + 2 . In principle, it is not important to know these restrictions since the Clebsch-Gordan coefficients (or the Wigner 3j symbols) will do the selection automatically.

The product can in general be written as the following linear combination

(), (), () 2 1 + 1 2 2 + 1 2L + 1 Y, , ()qf, ⋅ Y, , ()qf, = ∑ ------1 m1 2 m2 p LM, 4 [1.16] ⎛⎞, , ⎛⎞, , 1 2 L * L × ⎜⎟Y , ()qf, ⎜⎟1 2 ⎝⎠LM ⎝⎠ m1 m2 M 000 where the Wigner 3j symbols are related to the Racah or Clebsch-Gordan coefficients by

⎛⎞, , L , Ð , Ð 1 ⎜⎟1 2 = ()Ð1 1 2 M------c(), ,,,,,m , m LMÐ . [1.17] ⎝⎠ ()1 1 2 2 m1 m2 M 2L + 1

The Wigner 3j symbols or the Clebsch-Gordan coefficients can be found in tables in books about angular momentum or calculated using programs like Matlab, Macsyma, or Mathematica. Written with Clebsch-Gordan coefficients we obtain for Eq. [1.16] 6 Chapter 1

(), (), 2 1 + 1 2 2 + 1 Y, , ()qf, ⋅ Y, , ()qf, = ∑ ------Y , ()qf, 1 m1 2 m2 p()+ LM LM, 4 2L 1 . [1.18] × (), ,,,,,, (), ,,, ,,, c 1 m1 2 m2 LMc 1 0 2 0 L 0

To calculate the coefficients for the above example, we need the Wigner 3j symbols for

⎛⎞ ⎛⎞ ⎛⎞ 1 2 ⎜⎟110 = Ð------⎜⎟111 = 0 ⎜⎟112 = ------[1.19] ⎝⎠000 3 ⎝⎠000 ⎝⎠000 15 and obtain

2 ()⎛⎞2 qf, ⋅ qf, 2L + 1 ⎜⎟11L qf, Y10, ()Y10, ()= ∑ 3 ------p YL, 0() 4 ⎝⎠000 L = 0 . [1.20] 1 qf, qf, 1 qf, = ------Y00, ()++0Y10, ()------Y20, () 4p 5p

In the same way more complex products can be calculated and decomposed in the spherical harmonics. This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. We will discuss this in more detail in an exercise.

1.3.3 Addition Theorem of Spherical Harmonics

The spherical harmonics obey an addition theorem that can often be used to simplify expressions

, * 2, + 1 ∑ Y,, ()q , f Y,, ()q , f = ------P,()cosw [1.21] m 1 1 m 2 2 4p m = Ð, wherew omega describes the angle between two unit vectors oriented at the polar ()q , f ()q , f coordinates1 1 and2 2 with w q q q q ()f f cos = cos 1 cos 2 + sin 1 sin 2 cos 1 Ð 2 . [1.22] Properties of Spherical Harmonics 7

1.3.4 Integrals Over Spherical Harmonics

The integration over the product of three spherical harmonics can be simplified using the product rule of Eq. [1.16] and the orthogonality of Eq. [1.7]. This leads to

2p p

Y, , ()qf, Y, , ()qf, Y, , ()qf, sinq()dq df ∫ ∫ 1 m1 2 m2 3 m3 0 0

()2, + 1 ()2, + 1 ()2L + 1 ⎛⎞, , L ⎛⎞, , L = ∑ ------1 2 ⎜⎟1 2 ⎜⎟1 2 4p ⎝⎠⎝⎠ LM, m1 m2 M 000 [1.23] 2p p * × Y , ()qf, Y, , ()qf, sinq()dq df ∫ ∫ LM 3 m3 0 0 ()2, + 1 ()2, + 1 ()2, + 1 ⎛⎞, , , ⎛⎞, , , = ------1 2 3 ⎜⎟1 2 3 ⎜⎟1 2 3 4p ⎝⎠⎝⎠ m1 m2 m3 000 a simple expression involving only a normalization constant and two Wigner 3j symbols.

1.4 Literature

(1) D. M. Brink, G. R. Satchler, Angular Momentum, third edition, Clarendon Press, 1993.

(2) A. R. Edmonds, Angular Momentum in , Princeton University Press, 1960.

(3) M. E. Rose, Elementary Theory of Angular Momentum, John Wiley & Sons Inc., New York, 1957. 8 Chapter 1