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Chemistry LibreTexts

M4: Spherical Harmonics

Represented in a system of spherical coordinates, Laplace's spherical harmonics \(Y_l^m\) are a specific set of spherical harmonics that forms an orthogonal system. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic orbital electron configurations

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

l = 2

\[Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x - iy)^2 \over r^{2}}\]

\[Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta\quad={1\over 2}\sqrt{15\over 2\pi}\cdot{(x - iy)z \over r^{2}}\]

\[Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)\quad={1\over 4}\sqrt{5\over \pi}\cdot{(2z^{2}-x^{2}-y^{2})\over r^{2}}\]

\[Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta\quad={-1\over 2}\sqrt{15\over 2\pi}\cdot{(x + iy)z \over r^{2}}\]

\[Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\quad={1\over 4}\sqrt{15\over 2\pi}\cdot{(x + iy)^2 \over r^{2}}\]

l = 3

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

l = 4

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

l = 5

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

l = 6

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

l = 7

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

l = 8

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

l = 9

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

l = 10

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

 

Real spherical harmonics

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

l = 0

\begin{align}Y_{00} & = s = Y_0^0 = \frac{1}{2} \sqrt{\frac{1}{\pi}}\end{align}

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}  \\  Y_{10} & = p_z = Y_1^0 = \sqrt{\frac{3}{4 \pi}} \cdot \frac{z}{r} \\  Y_{11} & = 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}

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}  \\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 z}{r^2} \\Y_{20} & = d_{z^2} = Y_2^0 = \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{- x^2 - y^2 + 2 z^2}{r^2}  \\Y_{21} & = d_{xz} = \sqrt{\frac{1}{2}} \left( Y_2^{- 1} - Y_2^1 \right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{z x}{r^2} \\Y_{22} & = 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}

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{\left( 3 x^2 - y^2 \right) y}{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 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 (4 z^2 - x^2 - y^2)}{r^3} \\Y_{30} & = f_{z^3} = Y_3^0 = \frac{1}{4} \sqrt{\frac{7}{\pi}} \cdot \frac{z (2 z^2 - 3 x^2 - 3 y^2)}{r^3} \\Y_{31} & = 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 (4 z^2 - x^2 - y^2)}{r^3} \\Y_{32} & = 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) z}{r^3} \\Y_{33} & = 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{\left( x^2 - 3 y^2 \right) x}{r^3}\end{align}

l = 4

\begin{align}Y_{4,-4} & = g_{xy(x^2-y^2)} = 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} & = g_{zy^3} = i \sqrt{\frac{1}{2}} \left( Y_4^{- 3} + Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(3 x^2 - y^2) yz}{r^4} \\Y_{4,-2} & = g_{z^2xy} = 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} & = g_{z^3y} = i \sqrt{\frac{1}{2}} \left( Y_4^{- 1} + Y_4^1\right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{yz \cdot (7 z^2 - 3 r^2)}{r^4} \\Y_{40} & = g_{z^4} = 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_{41} & = g_{z^3x} = \sqrt{\frac{1}{2}} \left( Y_4^{- 1} - Y_4^1 \right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{xz \cdot (7 z^2 - 3 r^2)}{r^4} \\Y_{42} & = g_{z^2xy} = \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_{43} & = g_{zx^3} = \sqrt{\frac{1}{2}} \left( Y_4^{- 3} - Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x^2 - 3 y^2) xz}{r^4} \\Y_{44} & = g_{x^4+y^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}

External links

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. p. 155-156. ISBN 9971-50-107-4.
  2. C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry. New York: Academic Press. ISBN 0-12-172950-8.
  3. 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