Table of spherical harmonics#Real spherical harmonics

For ℓ = 0, …, 5, see.{{cite book|title=Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols | year=1988 | publisher=World Scientific Pub. | location=Singapore | isbn=9971-50-107-4|author1=D. A. Varshalovich | author2=A. N. Moskalev |author3=V. K. Khersonskii |edition=1. repr. | pages=155–156}}

$$Y\_\{0\}^\{0\}(\backslash theta,\backslash varphi)=\{1\backslash over\; 2\}\backslash sqrt\{1\backslash over\; \backslash pi\}$$

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

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

File:Complex Spherical Harmonics Figure Table Complex 2D.png

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.

File:Complex Spherical Harmonics Figure Table Complex Polar Plot.gif

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.

File:Complex Spherical Harmonics Figure Table Complex Radial Magnitude.gif

For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f) is reported as well.{{cite book |last1=Petrucci |title=General chemistry : principles and modern applications. |date=2016 |publisher=Prentice Hall |isbn=0133897311}}{{cite journal |last1=Friedman |title=The shapes of the f orbitals |journal=J. Chem. Educ. |date=1964 |volume=41 |issue=7 |page=354}}

For ℓ = 0, …, 3, see.{{cite book | title=Group theoretical techniques in quantum chemistry | year=1976 | publisher=Academic Press | location=New York | isbn=0-12-172950-8 | author=C.D.H. Chisholm}}{{cite journal | last=Blanco | first=Miguel A. | author2=Flórez, M. |author3=Bermejo, M. | title=Evaluation of the rotation matrices in the basis of real spherical harmonics | journal=Journal of Molecular Structure: THEOCHEM | date=1 December 1997 | volume=419 | issue=1–3 | pages=19–27 | doi=10.1016/S0166-1280(97)00185-1}}

$$Y\_\{00\}\; =\; s\; =\; Y\_0^0\; =\; \backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash frac\{1\}\{\backslash pi\}\}$$

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

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}

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

File:Real Spherical Harmonics Figure Table Complex 2D.png

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.

File:Real Spherical Harmonics Figure Table Complex Polar Plot.gif

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.

File:Real Spherical Harmonics Figure Table Complex Radial Magnitude.gif

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.

File:Sph harm table real bumpy.gif

• Spherical harmonics

• [http://mathworld.wolfram.com/SphericalHarmonic.html Spherical Harmonics] at MathWorld
• [https://www.quantum-physics.polytechnique.fr/sphericalHarmonics.php?lang=1 Spherical Harmonics 3D representation]

{{reflist}}

• See section 3 in {{cite journal

|last1=Mathar |first1=R. J.

|year=2009

|title=Zernike basis to cartesian transformations

|journal=Serbian Astronomical Journal

|volume=179 |pages=107–120

|arxiv = 0809.2368

|bibcode=2009SerAJ.179..107M

|doi=10.2298/SAJ0979107M

|issue=179

}} (see section 3.3)

• For complex spherical harmonics, see also [http://www.wolframalpha.com/input/?i=SphericalHarmonicY%5Bl,m,theta,phi%5D SphericalHarmonicY[l,m,theta,phi] at Wolfram Alpha], especially for specific values of l and m.

{{DEFAULTSORT:Table Of Spherical Harmonics}}

Category:Special hypergeometric functions