This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree [math]\displaystyle{ \ell = 10 }[/math]. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \varphi }[/math] as [math]\displaystyle{ \begin{cases} \cos(\theta) & = z/r\\\ e^{\pm i\varphi} \cdot \sin(\theta) & = (x \pm iy)/r \end{cases} }[/math]

## Contents

* 1 Complex Spherical harmonics
* 1.1 ℓ = 0
* 1.2 ℓ = 1
* 1.3 ℓ = 2
* 1.4 ℓ = 3
* 1.5 ℓ = 4
* 1.6 ℓ = 5
* 1.7 ℓ = 6
* 1.8 ℓ = 7
* 1.9 ℓ = 8
* 1.10 ℓ = 9
* 1.11 ℓ = 10
* 2 Visualization of Complex Spherical Harmonics
* 2.1 2D polar/azimuthal angle maps
* 2.2 Polar Plots
* 2.3 Polar Plots with Magnitude as Radius
* 3 Real spherical harmonics
* 3.1 ℓ = 0
* 3.2 ℓ = 1
* 3.3 ℓ = 2
* 3.4 ℓ = 3
* 3.5 ℓ = 4
* 4 Visualization of Real Spherical Harmonics
* 4.1 2D polar/azimuthal angle maps
* 4.2 Polar Plots
* 4.3 Polar Plots with Magnitude as Radius
* 4.4 Polar Plots with Amplitude as Elevation
* 5 See also
* 6 External links
* 7 References
* 7.1 Cited references
* 7.2 General references

## Complex Spherical harmonics

For ℓ = 0, …, 5, see.[1]

### ℓ = 0

[math]\displaystyle{ Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi} }[/math]

### ℓ = 1

[math]\displaystyle{ \begin{align} Y_{1}^{-1}(\theta,\varphi) &= & & {1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta & &= & &{1\over 2}\sqrt{3\over 2\pi} \cdot{(x-iy)\over r} \\\ Y_{1}^{ 0}(\theta,\varphi) &= & & {1\over 2}\sqrt{3\over \pi}\cdot \cos\theta & &= & &{1\over 2}\sqrt{3\over \pi} \cdot{z\over r} \\\ Y_{1}^{ 1}(\theta,\varphi) &= &-& {1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot \sin\theta & &= &-&{1\over 2}\sqrt{3\over 2\pi} \cdot{(x+iy)\over r} \end{align} }[/math]

### ℓ = 2

[math]\displaystyle{ \begin{align} Y_{2}^{-2}(\theta,\varphi)&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\quad &&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot{(x - iy)^2 \over r^{2}}&\\\ Y_{2}^{-1}(\theta,\varphi)&=& &{1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin \theta\cdot \cos\theta\quad &&=& &{1\over 2}\sqrt{15\over 2\pi}\cdot{(x - iy) \cdot z \over r^{2}}&\\\ Y_{2}^{ 0}(\theta,\varphi)&=& &{1\over 4}\sqrt{ 5\over \pi}\cdot (3\cos^{2}\theta-1)\quad&&=& &{1\over 4}\sqrt{ 5\over \pi}\cdot{(3z^{2}-r^{2})\over r^{2}}&\\\ Y_{2}^{ 1}(\theta,\varphi)&=&-&{1\over 2}\sqrt{15\over 2\pi}\cdot e^{ i\varphi}\cdot\sin \theta\cdot \cos\theta\quad &&=&-&{1\over 2}\sqrt{15\over 2\pi}\cdot{(x + iy) \cdot z \over r^{2}}&\\\ Y_{2}^{ 2}(\theta,\varphi)&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\quad &&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot{(x + iy)^2 \over r^{2}}& \end{align} }[/math]

### ℓ = 3

[math]\displaystyle{ \begin{align} Y_{3}^{-3}(\theta,\varphi) &=& &{1\over 8}\sqrt{ 35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\quad& &=& & {1\over 8}\sqrt{35\over \pi}\cdot{(x - iy)^{3}\over r^{3}}&\\\ Y_{3}^{-2}(\theta,\varphi) &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad& &=& & {1\over 4}\sqrt{105\over 2\pi}\cdot{(x- iy)^2 \cdot z \over r^{3}}&\\\ Y_{3}^{-1}(\theta,\varphi) &=& &{1\over 8}\sqrt{ 21\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad& &=& &{1\over 8}\sqrt{21\over \pi}\cdot{(x - iy) \cdot (5z^2- r^2)\over r^{3}}&\\\ Y_{3}^{ 0}(\theta,\varphi) &=& &{1\over 4}\sqrt{ 7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)\quad& &=& &{1\over 4}\sqrt{7\over \pi}\cdot{(5z^3 - 3zr^2)\over r^{3}}&\\\ Y_{3}^{ 1}(\theta,\varphi) &=&-&{1\over 8}\sqrt{ 21\over \pi}\cdot e^ { i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad& &=& &{-1\over 8}\sqrt{21\over \pi}\cdot{(x + iy) \cdot (5z^2 - r^2) \over r^{3}}&\\\ Y_{3}^{ 2}(\theta,\varphi) &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot e^ {2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad& &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot{(x + iy)^2 \cdot z \over r^{3}}&\\\ Y_{3}^{ 3}(\theta,\varphi) &=&-&{1\over 8}\sqrt{ 35\over \pi}\cdot e^ {3i\varphi}\cdot\sin^{3}\theta\quad& &=& &{-1\over 8}\sqrt{35\over \pi}\cdot{(x + iy)^3\over r^{3}}& \end{align} }[/math]

### ℓ = 4

[math]\displaystyle{ \begin{align} Y_{4}^{-4}(\theta,\varphi)&={ 3\over 16}\sqrt{35\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta= \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x - i y)^4}{r^4}\\\ Y_{4}^{-3}(\theta,\varphi)&={ 3\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta= \frac{3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x - i y)^3 z}{r^4}\\\ Y_{4}^{-2}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)= \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x - i y)^2 \cdot (7 z^2 - r^2)}{r^4}\\\ Y_{4}^{-1}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over \pi}\cdot e^{- i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)= \frac{3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x - i y) \cdot (7 z^3 - 3 z r^2)}{r^4}\\\ Y_{4}^{ 0}(\theta,\varphi)&={ 3\over 16}\sqrt{ 1\over \pi}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3)= \frac{3}{16} \sqrt{\frac{1}{\pi}} \cdot \frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}\\\ Y_{4}^{ 1}(\theta,\varphi)&={-3\over 8}\sqrt{ 5\over \pi}\cdot e^{ i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)= \frac{- 3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x + i y) \cdot (7 z^3 - 3 z r^2)}{r^4}\\\ Y_{4}^{ 2}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)= \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x + i y)^2 \cdot (7 z^2 - r^2)}{r^4}\\\ Y_{4}^{ 3}(\theta,\varphi)&={-3\over 8}\sqrt{35\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta= \frac{- 3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x + i y)^3 z}{r^4}\\\ Y_{4}^{ 4}(\theta,\varphi)&={ 3\over 16}\sqrt{35\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta= \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x + i y)^4}{r^4} \end{align} }[/math]

### ℓ = 5

[math]\displaystyle{ \begin{align} Y_{5}^{-5}(\theta,\varphi)&={ 3\over 32}\sqrt{ 77\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\\\ Y_{5}^{-4}(\theta,\varphi)&={ 3\over 16}\sqrt{ 385\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\\\ Y_{5}^{-3}(\theta,\varphi)&={ 1\over 32}\sqrt{ 385\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)\\\ Y_{5}^{-2}(\theta,\varphi)&={ 1\over 8}\sqrt{1155\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)\\\ Y_{5}^{-1}(\theta,\varphi)&={ 1\over 16}\sqrt{ 165\over 2\pi}\cdot e^{- i\varphi}\cdot\sin \theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)\\\ Y_{5}^{ 0}(\theta,\varphi)&={ 1\over 16}\sqrt{ 11\over \pi}\cdot (63\cos^{5}\theta-70\cos^{3}\theta+15\cos\theta)\\\ Y_{5}^{ 1}(\theta,\varphi)&={-1\over 16}\sqrt{ 165\over 2\pi}\cdot e^{ i\varphi}\cdot\sin \theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)\\\ Y_{5}^{ 2}(\theta,\varphi)&={ 1\over 8}\sqrt{1155\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)\\\ Y_{5}^{ 3}(\theta,\varphi)&={-1\over 32}\sqrt{ 385\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)\\\ Y_{5}^{ 4}(\theta,\varphi)&={ 3\over 16}\sqrt{ 385\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\\\ Y_{5}^{ 5}(\theta,\varphi)&={-3\over 32}\sqrt{ 77\over \pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta \end{align} }[/math]

### ℓ = 6

[math]\displaystyle{ \begin{align} Y_{6}^{-6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\\\ Y_{6}^{-5}(\theta,\varphi)&= {3\over 32}\sqrt{1001\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\\\ Y_{6}^{-4}(\theta,\varphi)&= {3\over 32}\sqrt{ 91\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)\\\ Y_{6}^{-3}(\theta,\varphi)&= {1\over 32}\sqrt{1365\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)\\\ Y_{6}^{-2}(\theta,\varphi)&= {1\over 64}\sqrt{1365\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)\\\ Y_{6}^{-1}(\theta,\varphi)&= {1\over 16}\sqrt{ 273\over 2\pi}\cdot e^{- i\varphi}\cdot\sin \theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)\\\ Y_{6}^{ 0}(\theta,\varphi)&= {1\over 32}\sqrt{ 13\over \pi}\cdot (231\cos^{6}\theta-315\cos^{4}\theta+105\cos^{2}\theta-5)\\\ Y_{6}^{ 1}(\theta,\varphi)&=-{1\over 16}\sqrt{ 273\over 2\pi}\cdot e^{ i\varphi}\cdot\sin \theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)\\\ Y_{6}^{ 2}(\theta,\varphi)&= {1\over 64}\sqrt{1365\over \pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)\\\ Y_{6}^{ 3}(\theta,\varphi)&=-{1\over 32}\sqrt{1365\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)\\\ Y_{6}^{ 4}(\theta,\varphi)&= {3\over 32}\sqrt{ 91\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)\\\ Y_{6}^{ 5}(\theta,\varphi)&=-{3\over 32}\sqrt{1001\over \pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\\\ Y_{6}^{ 6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot e^{ 6i\varphi}\cdot\sin^{6}\theta \end{align} }[/math]

### ℓ = 7

[math]\displaystyle{ \begin{align} Y_{7}^{-7}(\theta,\varphi)&= {3\over 64}\sqrt{ 715\over 2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\\\ Y_{7}^{-6}(\theta,\varphi)&= {3\over 64}\sqrt{5005\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta\\\ Y_{7}^{-5}(\theta,\varphi)&= {3\over 64}\sqrt{ 385\over 2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)\\\ Y_{7}^{-4}(\theta,\varphi)&= {3\over 32}\sqrt{ 385\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)\\\ Y_{7}^{-3}(\theta,\varphi)&= {3\over 64}\sqrt{ 35\over 2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)\\\ Y_{7}^{-2}(\theta,\varphi)&= {3\over 64}\sqrt{ 35\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)\\\ Y_{7}^{-1}(\theta,\varphi)&= {1\over 64}\sqrt{ 105\over 2\pi}\cdot e^{- i\varphi}\cdot\sin \theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)\\\ Y_{7}^{ 0}(\theta,\varphi)&= {1\over 32}\sqrt{ 15\over \pi}\cdot (429\cos^{7}\theta-693\cos^{5}\theta+315\cos^{3}\theta-35\cos\theta)\\\ Y_{7}^{ 1}(\theta,\varphi)&=-{1\over 64}\sqrt{ 105\over 2\pi}\cdot e^{ i\varphi}\cdot\sin \theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)\\\ Y_{7}^{ 2}(\theta,\varphi)&= {3\over 64}\sqrt{ 35\over \pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)\\\ Y_{7}^{ 3}(\theta,\varphi)&=-{3\over 64}\sqrt{ 35\over 2\pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)\\\ Y_{7}^{ 4}(\theta,\varphi)&= {3\over 32}\sqrt{ 385\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)\\\ Y_{7}^{ 5}(\theta,\varphi)&=-{3\over 64}\sqrt{ 385\over 2\pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)\\\ Y_{7}^{ 6}(\theta,\varphi)&= {3\over 64}\sqrt{5005\over \pi}\cdot e^{ 6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta\\\ Y_{7}^{ 7}(\theta,\varphi)&=-{3\over 64}\sqrt{ 715\over 2\pi}\cdot e^{ 7i\varphi}\cdot\sin^{7}\theta \end{align} }[/math]

### ℓ = 8

[math]\displaystyle{ \begin{align} Y_{8}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\\\ Y_{8}^{-7}(\theta,\varphi)&={ 3\over 64}\sqrt{12155\over 2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta\\\ Y_{8}^{-6}(\theta,\varphi)&={ 1\over 128}\sqrt{7293\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)\\\ Y_{8}^{-5}(\theta,\varphi)&={ 3\over 64}\sqrt{17017\over 2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-\cos\theta)\\\ Y_{8}^{-4}(\theta,\varphi)&={ 3\over 128}\sqrt{1309\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)\\\ Y_{8}^{-3}(\theta,\varphi)&={ 1\over 64}\sqrt{19635\over 2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)\\\ Y_{8}^{-2}(\theta,\varphi)&={ 3\over 128}\sqrt{595\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)\\\ Y_{8}^{-1}(\theta,\varphi)&={ 3\over 64}\sqrt{17\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)\\\ Y_{8}^{ 0}(\theta,\varphi)&={ 1\over 256}\sqrt{17\over \pi}\cdot(6435\cos^{8}\theta-12012\cos^{6}\theta+6930\cos^{4}\theta-1260\cos^{2}\theta+35)\\\ Y_{8}^{ 1}(\theta,\varphi)&={-3\over 64}\sqrt{17\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)\\\ Y_{8}^{ 2}(\theta,\varphi)&={ 3\over 128}\sqrt{595\over \pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)\\\ Y_{8}^{ 3}(\theta,\varphi)&={-1\over 64}\sqrt{19635\over 2\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)\\\ Y_{8}^{ 4}(\theta,\varphi)&={ 3\over 128}\sqrt{1309\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)\\\ Y_{8}^{ 5}(\theta,\varphi)&={-3\over 64}\sqrt{17017\over 2\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-\cos\theta)\\\ Y_{8}^{ 6}(\theta,\varphi)&={ 1\over 128}\sqrt{7293\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)\\\ Y_{8}^{ 7}(\theta,\varphi)&={-3\over 64}\sqrt{12155\over 2\pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta\\\ Y_{8}^{ 8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta \end{align} }[/math]

### ℓ = 9

[math]\displaystyle{ \begin{align} Y_{9}^{-9}(\theta,\varphi)&={ 1\over 512}\sqrt{230945\over \pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta\\\ Y_{9}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{230945\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta\\\ Y_{9}^{-7}(\theta,\varphi)&={ 3\over 512}\sqrt{ 13585\over \pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)\\\ Y_{9}^{-6}(\theta,\varphi)&={ 1\over 128}\sqrt{ 40755\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)\\\ Y_{9}^{-5}(\theta,\varphi)&={ 3\over 256}\sqrt{ 2717\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)\\\ Y_{9}^{-4}(\theta,\varphi)&={ 3\over 128}\sqrt{ 95095\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)\\\ Y_{9}^{-3}(\theta,\varphi)&={ 1\over 256}\sqrt{ 21945\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)\\\ Y_{9}^{-2}(\theta,\varphi)&={ 3\over 128}\sqrt{ 1045\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)\\\ Y_{9}^{-1}(\theta,\varphi)&={ 3\over 256}\sqrt{ 95\over 2\pi}\cdot e^{- i\varphi}\cdot\sin \theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)\\\ Y_{9}^{ 0}(\theta,\varphi)&={ 1\over 256}\sqrt{ 19\over \pi}\cdot (12155\cos^{9}\theta-25740\cos^{7}\theta+18018\cos^{5}\theta-4620\cos^{3}\theta+315\cos\theta)\\\ Y_{9}^{ 1}(\theta,\varphi)&={-3\over 256}\sqrt{ 95\over 2\pi}\cdot e^{ i\varphi}\cdot\sin \theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)\\\ Y_{9}^{ 2}(\theta,\varphi)&={ 3\over 128}\sqrt{ 1045\over \pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)\\\ Y_{9}^{ 3}(\theta,\varphi)&={-1\over 256}\sqrt{ 21945\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)\\\ Y_{9}^{ 4}(\theta,\varphi)&={ 3\over 128}\sqrt{ 95095\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)\\\ Y_{9}^{ 5}(\theta,\varphi)&={-3\over 256}\sqrt{ 2717\over \pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)\\\ Y_{9}^{ 6}(\theta,\varphi)&={ 1\over 128}\sqrt{ 40755\over \pi}\cdot e^{ 6i\varphi}\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)\\\ Y_{9}^{ 7}(\theta,\varphi)&={-3\over 512}\sqrt{ 13585\over \pi}\cdot e^{ 7i\varphi}\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)\\\ Y_{9}^{ 8}(\theta,\varphi)&={ 3\over 256}\sqrt{230945\over 2\pi}\cdot e^{ 8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta\\\ Y_{9}^{ 9}(\theta,\varphi)&={-1\over 512}\sqrt{230945\over \pi}\cdot e^{ 9i\varphi}\cdot\sin^{9}\theta \end{align} }[/math]

### ℓ = 10

[math]\displaystyle{ \begin{align} Y_{10}^{-10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\cdot e^{-10i\varphi}\cdot\sin^{10}\theta\\\ Y_{10}^{- 9}(\theta,\varphi)&={1\over 512}\sqrt{4849845\over \pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta\\\ Y_{10}^{- 8}(\theta,\varphi)&={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)\\\ Y_{10}^{- 7}(\theta,\varphi)&={3\over 512}\sqrt{85085\over \pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)\\\ Y_{10}^{- 6}(\theta,\varphi)&={3\over 1024}\sqrt{5005\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)\\\ Y_{10}^{- 5}(\theta,\varphi)&={3\over 256}\sqrt{1001\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)\\\ Y_{10}^{- 4}(\theta,\varphi)&={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2}\theta-1)\\\ Y_{10}^{- 3}(\theta,\varphi)&={3\over 256}\sqrt{5005\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}\theta-7\cos\theta)\\\ Y_{10}^{- 2}(\theta,\varphi)&={3\over 512}\sqrt{385\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730\cos^{4}\theta-364\cos^{2}\theta+7)\\\ Y_{10}^{- 1}(\theta,\varphi)&={1\over 256}\sqrt{1155\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5}\theta-1092\cos^{3}\theta+63\cos\theta)\\\ Y_{10}^{ 0}(\theta,\varphi)&={1\over 512}\sqrt{21\over \pi}\cdot(46189\cos^{10}\theta-109395\cos^{8}\theta+90090\cos^{6}\theta-30030\cos^{4}\theta+3465\cos^{2}\theta-63)\\\ Y_{10}^{ 1}(\theta,\varphi)&={-1\over 256}\sqrt{1155\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5}\theta-1092\cos^{3}\theta+63\cos\theta)\\\ Y_{10}^{ 2}(\theta,\varphi)&={3\over 512}\sqrt{385\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730\cos^{4}\theta-364\cos^{2}\theta+7)\\\ Y_{10}^{ 3}(\theta,\varphi)&={-3\over 256}\sqrt{5005\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}\theta-7\cos\theta)\\\ Y_{10}^{ 4}(\theta,\varphi)&={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2}\theta-1)\\\ Y_{10}^{ 5}(\theta,\varphi)&={-3\over 256}\sqrt{1001\over \pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)\\\ Y_{10}^{ 6}(\theta,\varphi)&={3\over 1024}\sqrt{5005\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)\\\ Y_{10}^{ 7}(\theta,\varphi)&={-3\over 512}\sqrt{85085\over \pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)\\\ Y_{10}^{ 8}(\theta,\varphi)&={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)\\\ Y_{10}^{ 9}(\theta,\varphi)&={-1\over 512}\sqrt{4849845\over \pi}\cdot e^{9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta\\\ Y_{10}^{ 10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\cdot e^{10i\varphi}\cdot\sin^{10}\theta \end{align} }[/math]

## Visualization of Complex Spherical Harmonics

### 2D polar/azimuthal angle maps

Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, [math]\displaystyle{ \phi }[/math], on the horizontal axis and the polar angle, [math]\displaystyle{ \theta }[/math], on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.

Visual Array of Complex Spherical Harmonics Represented as 2D Theta/Phi Maps

### Polar Plots

Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.

Visual Array of Complex Spherical Harmonics Represented with Polar Plot

### Polar Plots with Magnitude as Radius

Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.

Visual Array of Complex Spherical Harmonics Represented with Polar Plot with Magnitude Mapped to Radius

## Real spherical harmonics

For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f) is reported as well.[2][3]

For ℓ = 0, …, 3, see.[4][5]

### ℓ = 0

[math]\displaystyle{ Y_{00} = s = Y_0^0 = \frac{1}{2} \sqrt{\frac{1}{\pi}} }[/math]

### ℓ = 1

[math]\displaystyle{ \begin{align} Y_{1,-1} & = p_y = i \sqrt{\frac{1}{2}} \left( Y_1^{- 1} + Y_1^1 \right) = \sqrt{\frac{3}{4 \pi}} \cdot \frac{y}{r} \\\ Y_{1,0} & = p_z = Y_1^0 = \sqrt{\frac{3}{4 \pi}} \cdot \frac{z}{r} \\\ Y_{1,1} & = p_x = \sqrt{\frac{1}{2}} \left( Y_1^{- 1} - Y_1^1 \right) = \sqrt{\frac{3}{4 \pi}} \cdot \frac{x}{r} \end{align} }[/math]

### ℓ = 2

[math]\displaystyle{ \begin{align} Y_{2,-2} & = d_{xy} = i \sqrt{\frac{1}{2}} \left( Y_2^{- 2} - Y_2^2\right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{x y}{r^2} \\\ Y_{2,-1} & = d_{yz} = i \sqrt{\frac{1}{2}} \left( Y_2^{- 1} + Y_2^1 \right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{y \cdot z}{r^2} \\\ Y_{2,0} & = d_{z^2} = Y_2^0 = \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{3z^2 - r^2}{r^2} \\\ Y_{2,1} & = d_{xz} = \sqrt{\frac{1}{2}} \left( Y_2^{- 1} - Y_2^1 \right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{x \cdot z}{r^2} \\\ Y_{2,2} & = d_{x^2-y^2} = \sqrt{\frac{1}{2}} \left( Y_2^{- 2} + Y_2^2 \right) = \frac{1}{4} \sqrt{\frac{15}{\pi}} \cdot \frac{x^2 - y^2 }{r^2} \end{align} }[/math]

### ℓ = 3

[math]\displaystyle{ \begin{align} Y_{3,-3} & = f_{y(3x^2-y^2)} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 3} + Y_3^3 \right) = \frac{1}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{y \left( 3 x^2 - y^2 \right)}{r^3} \\\ Y_{3,-2} & = f_{xyz} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 2} - Y_3^2 \right) = \frac{1}{2} \sqrt{\frac{105}{\pi}} \cdot \frac{xy \cdot z}{r^3} \\\ Y_{3,-1} & = f_{yz^2} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 1} + Y_3^1 \right) = \frac{1}{4} \sqrt{\frac{21}{2 \pi}} \cdot \frac{y \cdot (5 z^2 - r^2)}{r^3} \\\ Y_{3,0} & = f_{z^3} = Y_3^0 = \frac{1}{4} \sqrt{\frac{7}{\pi}} \cdot \frac{5 z^3 - 3 z r^2}{r^3} \\\ Y_{3,1} & = f_{xz^2} = \sqrt{\frac{1}{2}} \left( Y_3^{- 1} - Y_3^1 \right) = \frac{1}{4} \sqrt{\frac{21}{2 \pi}} \cdot \frac{x \cdot (5 z^2 - r^2)}{r^3} \\\ Y_{3,2} & = f_{z(x^2-y^2)} = \sqrt{\frac{1}{2}} \left( Y_3^{- 2} + Y_3^2 \right) = \frac{1}{4} \sqrt{\frac{105}{\pi}} \cdot \frac{\left( x^2 - y^2 \right) \cdot z}{r^3} \\\ Y_{3,3} & = f_{x(x^2-3y^2)} = \sqrt{\frac{1}{2}} \left( Y_3^{- 3} - Y_3^3 \right) = \frac{1}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{x \left( x^2 - 3 y^2 \right)}{r^3} \end{align} }[/math]

### ℓ = 4

[math]\displaystyle{ \begin{align} Y_{4,-4} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 4} - Y_4^4 \right) = \frac{3}{4} \sqrt{\frac{35}{\pi}} \cdot \frac{xy \left( x^2 - y^2 \right)}{r^4} \\\ Y_{4,-3} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 3} + Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{y (3 x^2 - y^2) \cdot z}{r^4} \\\ Y_{4,-2} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 2} - Y_4^2 \right) = \frac{3}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{xy \cdot (7 z^2 - r^2)}{r^4} \\\ Y_{4,-1} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 1} + Y_4^1\right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{y \cdot (7 z^3 - 3 z r^2)}{r^4} \\\ Y_{4,0} & = Y_4^0 = \frac{3}{16} \sqrt{\frac{1}{\pi}} \cdot \frac{35 z^4 - 30 z^2 r^2 + 3 r^4}{r^4} \\\ Y_{4,1} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 1} - Y_4^1 \right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{x \cdot (7 z^3 - 3 z r^2)}{r^4} \\\ Y_{4,2} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 2} + Y_4^2 \right) = \frac{3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x^2 - y^2) \cdot (7 z^2 - r^2)}{r^4} \\\ Y_{4,3} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 3} - Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{x(x^2 - 3 y^2) \cdot z}{r^4} \\\ Y_{4,4} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 4} + Y_4^4 \right) = \frac{3}{16} \sqrt{\frac{35}{\pi}} \cdot \frac{x^2 \left( x^2 - 3 y^2 \right) - y^2 \left( 3 x^2 - y^2 \right)}{r^4} \end{align} }[/math]

## Visualization of Real Spherical Harmonics

### 2D polar/azimuthal angle maps

Below the real spherical harmonics are represented on 2D plots with the azimuthal angle, [math]\displaystyle{ \phi }[/math], on the horizontal axis and the polar angle, [math]\displaystyle{ \theta }[/math], on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.

Visual Array of Real Spherical Harmonics Represented as 2D Theta/Phi Maps

### Polar Plots

Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.

Visual Array of Real Spherical Harmonics Represented with Polar Plot

### Polar Plots with Magnitude as Radius

Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.

Visual Array of Real Spherical Harmonics Represented with Polar Plot with Magnitude Mapped to Radius

### Polar Plots with Amplitude as Elevation

Below the real spherical harmonics are represented on polar plots. The amplitude of the spherical harmonic (magnitude and sign) at a particular polar and azimuthal angle is represented by the elevation of the plot at that point above or below the surface of a uniform sphere. The magnitude is also represented by the saturation of the color at a given point. The phase is represented by the hue at a given point.

Visual Array of Real Spherical Harmonics Represented with Polar Plot with Amplitude Mapped to Elevation and Saturation

## See also

* Spherical harmonics

## External links

* Spherical Harmonics at MathWorld
* Spherical Harmonics 3D representation

## References

### Cited references

1. ↑ D. A. Varshalovich; A. N. Moskalev; V. K. Khersonskii (1988). Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols (1. repr. ed.). Singapore: World Scientific Pub.. pp. 155–156. ISBN 9971-50-107-4.
2. ↑ Petrucci (2016). General chemistry : principles and modern applications.. [Place of publication not identified]: Prentice Hall. ISBN 0133897311.
3. ↑ Friedman (1964). "The shapes of the f orbitals". J. Chem. Educ. 41 (7): 354.
4. ↑ C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry. New York: Academic Press. ISBN 0-12-172950-8. https://archive.org/details/grouptheoretical00chis.
5. ↑ Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 December 1997). "Evaluation of the rotation matrices in the basis of real spherical harmonics". Journal of Molecular Structure: THEOCHEM 419 (1–3): 19–27. doi:10.1016/S0166-1280(97)00185-1.

### General references

* See section 3 in Mathar, R. J. (2009). "Zernike basis to cartesian transformations". Serbian Astronomical Journal 179 (179): 107–120. doi:10.2298/SAJ0979107M. Bibcode: 2009SerAJ.179..107M. (see section 3.3)
* For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi] at Wolfram Alpha, especially for specific values of l and m.

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