Table of spherical harmonics

Complex spherical harmonics

For ℓ = 0, …, 5, see.

''ℓ'' = 0

Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}

''ℓ'' = 1

\begin{align} Y_{1}^{-1}(\theta,\varphi) &= & & {1\over 2}\sqrt{3\over 2\pi}\cdot \mathrm 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 \mathrm e^{i\varphi}\cdot \sin\theta & &= &-&{1\over 2}\sqrt{3\over 2\pi} \cdot{(x+iy)\over r} \end{align}

''ℓ'' = 2

\begin{align} Y_{2}^{-2}(\theta,\varphi)&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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}

''ℓ'' = 3

\begin{align} Y_{3}^{-3}(\theta,\varphi) &=& &{1\over 8}\sqrt{ 35\over \pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^ {3i\varphi}\cdot\sin^{3}\theta\quad& &=& &{-1\over 8}\sqrt{35\over \pi}\cdot{(x + iy)^3\over r^{3}}& \end{align}

''ℓ'' = 4

\begin{align} Y_{4}^{-4}(\theta,\varphi) &=& &{ 3\over 16} \sqrt{35\over 2\pi} \cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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}

''ℓ'' = 5

\begin{align} Y_{5}^{-5}(\theta,\varphi)&={ 3\over 32}\sqrt{ 77\over \pi}\cdot \mathrm e^{-5i\varphi}\cdot\sin^{5}\theta\ Y_{5}^{-4}(\theta,\varphi)&={ 3\over 16}\sqrt{ 385\over 2\pi}\cdot \mathrm e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\ Y_{5}^{-3}(\theta,\varphi)&={ 1\over 32}\sqrt{ 385\over \pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\ Y_{5}^{ 5}(\theta,\varphi)&={-3\over 32}\sqrt{ 77\over \pi}\cdot \mathrm e^{ 5i\varphi}\cdot\sin^{5}\theta \end{align}

''ℓ'' = 6

\begin{align} Y_{6}^{-6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot \mathrm e^{-6i\varphi}\cdot\sin^{6}\theta\ Y_{6}^{-5}(\theta,\varphi)&= {3\over 32}\sqrt{1001\over \pi}\cdot \mathrm e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\ Y_{6}^{-4}(\theta,\varphi)&= {3\over 32}\sqrt{ 91\over 2\pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{ 5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\ Y_{6}^{ 6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot \mathrm e^{ 6i\varphi}\cdot\sin^{6}\theta \end{align}

''ℓ'' = 7

\begin{align} Y_{7}^{-7}(\theta,\varphi)&= {3\over 64}\sqrt{ 715\over 2\pi}\cdot \mathrm e^{-7i\varphi}\cdot\sin^{7}\theta\ Y_{7}^{-6}(\theta,\varphi)&= {3\over 64}\sqrt{5005\over \pi}\cdot \mathrm e^{-6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta\ Y_{7}^{-5}(\theta,\varphi)&= {3\over 64}\sqrt{ 385\over 2\pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{ 6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta\ Y_{7}^{ 7}(\theta,\varphi)&=-{3\over 64}\sqrt{ 715\over 2\pi}\cdot \mathrm e^{ 7i\varphi}\cdot\sin^{7}\theta \end{align}

''ℓ'' = 8

\begin{align} Y_{8}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\cdot \mathrm e^{-8i\varphi}\cdot\sin^{8}\theta\ Y_{8}^{-7}(\theta,\varphi)&={ 3\over 64}\sqrt{12155\over 2\pi}\cdot \mathrm e^{-7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta\ Y_{8}^{-6}(\theta,\varphi)&={ 1\over 128}\sqrt{7293\over \pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta\ Y_{8}^{ 8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\cdot \mathrm e^{8i\varphi}\cdot\sin^{8}\theta \end{align}

''ℓ'' = 9

\begin{align} Y_{9}^{-9}(\theta,\varphi)&={ 1\over 512}\sqrt{230945\over \pi}\cdot \mathrm e^{-9i\varphi}\cdot\sin^{9}\theta\ Y_{9}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{230945\over 2\pi}\cdot \mathrm e^{-8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta\ Y_{9}^{-7}(\theta,\varphi)&={ 3\over 512}\sqrt{ 13585\over \pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{ 8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta\ Y_{9}^{ 9}(\theta,\varphi)&={-1\over 512}\sqrt{230945\over \pi}\cdot \mathrm e^{ 9i\varphi}\cdot\sin^{9}\theta \end{align}

''ℓ'' = 10

\begin{align} Y_{10}^{-10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\cdot \mathrm e^{-10i\varphi}\cdot\sin^{10}\theta\ Y_{10}^{- 9}(\theta,\varphi)&={1\over 512}\sqrt{4849845\over \pi}\cdot \mathrm e^{-9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta\ Y_{10}^{- 8}(\theta,\varphi)&={1\over 512}\sqrt{255255\over 2\pi}\cdot \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm 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 \mathrm e^{9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta\ Y_{10}^{ 10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\cdot \mathrm e^{10i\varphi}\cdot\sin^{10}\theta \end{align}

Visualization of complex spherical harmonics

2D polar/azimuthal angle maps

Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, \phi, on the horizontal axis and the polar angle, \theta, 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.

The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines.

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.

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.

Real spherical harmonics

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

For ℓ = 0, …, 3, see.

''ℓ'' = 0

Y_{0,0} = s = Y_0^0 = \frac{1}{2} \sqrt{\frac{1}{\pi}}

''ℓ'' = 1

\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} = \sqrt{\frac{3}{4 \pi}} \sin( \theta) \sin (\varphi) \ Y_{1,0} & = p_z = Y_1^0 = \sqrt{\frac{3}{4 \pi}} \cdot \frac{z}{r} = \sqrt{\frac{3}{4 \pi}} \cos( \theta) \ 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} = \sqrt{\frac{3}{4 \pi}} \sin( \theta) \cos (\varphi) \end{align}

''ℓ'' = 2

\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} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin^{2}(\theta) \sin(2\varphi) \ 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} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin(2 \theta) \sin (\varphi) \ Y_{2,0} & = d_{z^2} = Y_2^0 = \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{3z^2 - r^2}{r^2} = \frac{1}{4} \sqrt{\frac{5}{\pi}} (3\cos^{2}(\theta) -1)\ 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} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin(2 \theta) \cos (\varphi)\ 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} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin^{2}(\theta) \cos(2\varphi) \end{align}

''ℓ'' = 3

\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}

''ℓ'' = 4

\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}

Visualization of real spherical harmonics

2D polar/azimuthal angle maps

Below the real spherical harmonics are represented on 2D plots with the azimuthal angle, \phi, on the horizontal axis and the polar angle, \theta, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic. Positive values are red and negative values are teal.

The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines.

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.

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.

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.

References

Cited references

General references

• See section 3 in {{cite journal
• For complex spherical harmonics, see also [SphericalHarmonicYl,m,theta,phi at Wolfram Alpha], especially for specific values of l and m.

References

1. (1988). "Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols". World Scientific Pub..
2. (2016). "General chemistry : principles and modern applications.". Prentice Hall.
3. (1964). "The shapes of the f orbitals". J. Chem. Educ..
4. C.D.H. Chisholm. (1976). "Group theoretical techniques in quantum chemistry". Academic Press.
5. Blanco, Miguel A.. (1 December 1997). "Evaluation of the rotation matrices in the basis of real spherical harmonics". Journal of Molecular Structure: THEOCHEM.