Full Domain Hash

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree l = 10. Some of these formulas give the "Cartesian" version. This assumes x, y, z, and r are related to $\theta$ and $\phi$ through the usual spherical-to-Cartesian coordinate transformation:

$\begin{array}{rl}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \end{array}$

Spherical harmonics

l = 0

$Y_0^0(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{1}{\pi }}$

l = 1

$\begin{array}{rl}{Y}_1^{-1}(\theta ,\phi )& =\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta =\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{(x-iy)}{r}\\ Y_1^0(\theta ,\phi )& =\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \cos \theta =\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \frac{z}{r}\\ Y_1^1(\theta ,\phi )& =\frac{-1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta =\frac{-1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{(x+iy)}{r}\end{array}$

l = 2

$Y_2^{-2}(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x-iy{)}^2}{r^2}$

$Y_2^{-1}(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot \cos \theta =\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x-iy)z}{r^2}$

$Y_2^0(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot (3{\cos }^2\theta -1)=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{(2z^2-x^2-y^2)}{r^2}$

$Y_2^1(\theta ,\phi )=\frac{-1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot \cos \theta =\frac{-1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x+iy)z}{r^2}$

$Y_2^2(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x+iy{)}^2}{r^2}$

l = 3

$Y_3^{-3}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta =\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x-iy{)}^3}{r^3}$

$Y_3^{-2}(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{(x-iy{)}^{2}z}{r^3}$

$Y_3^{-1}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (5{\cos }^{}\theta -1)=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{(x-iy)(4z^2-x^2-y^2)}{r^3}$

$Y_3^0(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot (5{\cos }^3\theta -3\cos \theta )=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot \frac{z(2z^2-3x^2-3y^2)}{r^3}$

$Y_3^1(\theta ,\phi )=\frac{-1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (5{\cos }^{}\theta -1)=\frac{-1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{(x+iy)(4z^2-x^2-y^2)}{r^3}$

$Y_3^2(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{(x+iy{)}^{2}z}{r^3}$

$Y_3^3(\theta ,\phi )=\frac{-1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta =\frac{-1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x+iy{)}^3}{r^3}$

l = 4

$Y_4^{-4}(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta =\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x-iy{)}^4}{r^4}$

$Y_4^{-3}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta =\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x-iy{)}^{3}z}{r^4}$

$Y_4^{-2}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot \frac{(x-iy{)}^2\cdot (7z^2-r^2)}{r^4}$

$Y_4^{-1}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (7{\cos }^{}\theta -3\cos \theta )=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x-iy)\cdot z\cdot (7z^2-3r^2)}{r^4}$

$Y_4^0(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot (35{\cos }^4\theta -30{\cos }^2\theta +3)=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot \frac{(35z^4-30z^2{r}^2+3r^4)}{r^4}$

$Y_4^1(\theta ,\phi )=\frac{-3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (7{\cos }^{}\theta -3\cos \theta )=\frac{-3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x+iy)\cdot z\cdot (7z^2-3r^2)}{r^4}$

$Y_4^2(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot \frac{(x+iy{)}^2\cdot (7z^2-r^2)}{r^4}$

$Y_4^3(\theta ,\phi )=\frac{-3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta =\frac{-3}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x+iy{)}^{3}z}{r^4}$

$Y_4^4(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta =\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x+iy{)}^4}{r^4}$

l = 5

$Y_5^{-5}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta$

$Y_5^{-4}(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^{-3}(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^{-2}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -\cos \theta )$

$Y_5^{-1}(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (21{\cos }^{}\theta -14{\cos }^{}\theta +1)$

$Y_5^0(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{11}{\pi }}\cdot (63{\cos }^5\theta -70{\cos }^3\theta +15\cos \theta )$

$Y_5^1(\theta ,\phi )=\frac{-1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (21{\cos }^{}\theta -14{\cos }^{}\theta +1)$

$Y_5^2(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -\cos \theta )$

$Y_5^3(\theta ,\phi )=\frac{-1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^4(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^5(\theta ,\phi )=\frac{-3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta$

l = 6

$Y_6^{-6}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta$

$Y_6^{-5}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^{-4}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^{-3}(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^{-2}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^{-1}(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (33{\cos }^{}\theta -30{\cos }^{}\theta +5\cos \theta )$

$Y_6^0(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{13}{\pi }}\cdot (231{\cos }^6\theta -315{\cos }^4\theta +105{\cos }^2\theta -5)$

$Y_6^1(\theta ,\phi )=\frac{-1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (33{\cos }^{}\theta -30{\cos }^{}\theta +5\cos \theta )$

$Y_6^2(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^3(\theta ,\phi )=\frac{-1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^4(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^5(\theta ,\phi )=\frac{-3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^6(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta$

l = 7

$Y_7^{-7}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta$

$Y_7^{-6}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^{-5}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^{-4}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^{-3}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^{-2}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^{-1}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (429{\cos }^{}\theta -495{\cos }^{}\theta +135{\cos }^{}\theta -5)$

$Y_7^0(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{15}{\pi }}\cdot (429{\cos }^7\theta -693{\cos }^5\theta +315{\cos }^3\theta -35\cos \theta )$

$Y_7^1(\theta ,\phi )=\frac{-1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (429{\cos }^{}\theta -495{\cos }^{}\theta +135{\cos }^{}\theta -5)$

$Y_7^2(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^3(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^4(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^5(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^6(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^7(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta$

l = 8

$Y_8^{-8}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta$

$Y_8^{-7}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^{-6}(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^{-5}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -\cos \theta )$

$Y_8^{-4}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^{-3}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^{-2}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^{-1}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (715{\cos }^{}\theta -1001{\cos }^{}\theta +385{\cos }^{}\theta -35\cos \theta )$

$Y_8^0(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{17}{\pi }}\cdot (6435{\cos }^8\theta -12012{\cos }^6\theta +6930{\cos }^4\theta -1260{\cos }^2\theta +35)$

$Y_8^1(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (715{\cos }^{}\theta -1001{\cos }^{}\theta +385{\cos }^{}\theta -35\cos \theta )$

$Y_8^2(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^3(\theta ,\phi )=\frac{-1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^4(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^5(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -\cos \theta )$

$Y_8^6(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^7(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^8(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta$

l = 9

$Y_9^{-9}(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{-9i\phi }\cdot {\sin }^9\theta$

$Y_9^{-8}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^{-7}(\theta ,\phi )=\frac{3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^{-6}(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^{-5}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^{-4}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +\cos \theta )$

$Y_9^{-3}(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^{-2}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^{-1}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (2431{\cos }^{}\theta -4004{\cos }^{}\theta +2002{\cos }^{}\theta -308{\cos }^{}\theta +7)$

$Y_9^0(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{19}{\pi }}\cdot (12155{\cos }^9\theta -25740{\cos }^7\theta +18018{\cos }^5\theta -4620{\cos }^3\theta +315\cos \theta )$

$Y_9^1(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (2431{\cos }^{}\theta -4004{\cos }^{}\theta +2002{\cos }^{}\theta -308{\cos }^{}\theta +7)$

$Y_9^2(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^3(\theta ,\phi )=\frac{-1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^4(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +\cos \theta )$

$Y_9^5(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^6(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^7(\theta ,\phi )=\frac{-3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^8(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^9(\theta ,\phi )=\frac{-1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{9i\phi }\cdot {\sin }^9\theta$

l = 10

$Y_{10}^{-10}(\theta ,\phi )=\frac{1}{1024}\sqrt{\frac{969969}{\pi }}\cdot e^{-10i\phi }\cdot {\sin }^{10}\theta$

$Y_{10}^{-9}(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{4849845}{\pi }}\cdot e^{-9i\phi }\cdot {\sin }^9\theta \cdot \cos \theta$

$Y_{10}^{-8}(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{255255}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta \cdot (19{\cos }^2\theta -1)$

$Y_{10}^{-7}(\theta ,\phi )=\frac{3}{512}\sqrt{\frac{85085}{\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot (19{\cos }^3\theta -3\cos \theta )$

$Y_{10}^{-6}(\theta ,\phi )=\frac{3}{1024}\sqrt{\frac{5005}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (323{\cos }^4\theta -102{\cos }^2\theta +3)$

$Y_{10}^{-5}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{1001}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )$

$Y_{10}^{-4}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{5005}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (323{\cos }^6\theta -255{\cos }^4\theta +45{\cos }^2\theta -1)$

$Y_{10}^{-3}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{5005}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (323{\cos }^7\theta -357{\cos }^5\theta +105{\cos }^3\theta -7\cos \theta )$

$Y_{10}^{-2}(\theta ,\phi )=\frac{3}{512}\sqrt{\frac{385}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (4199{\cos }^8\theta -6188{\cos }^6\theta +2730{\cos }^4\theta -364{\cos }^2\theta +7)$

$Y_{10}^{-1}(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{1155}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (4199{\cos }^{}\theta -7956{\cos }^{}\theta +4914{\cos }^{}\theta -1092{\cos }^{}\theta +63\cos \theta )$

$Y_{10}^0(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{21}{\pi }}\cdot (46189{\cos }^{10}\theta -109395{\cos }^8\theta +90090{\cos }^6\theta -30030{\cos }^4\theta +3465{\cos }^2\theta -63)$

$Y_{10}^1(\theta ,\phi )=\frac{-1}{256}\sqrt{\frac{1155}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (4199{\cos }^{}\theta -7956{\cos }^{}\theta +4914{\cos }^{}\theta -1092{\cos }^{}\theta +63\cos \theta )$

$Y_{10}^2(\theta ,\phi )=\frac{3}{512}\sqrt{\frac{385}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (4199{\cos }^8\theta -6188{\cos }^6\theta +2730{\cos }^4\theta -364{\cos }^2\theta +7)$

$Y_{10}^3(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{5005}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (323{\cos }^7\theta -357{\cos }^5\theta +105{\cos }^3\theta -7\cos \theta )$

$Y_{10}^4(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{5005}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (323{\cos }^6\theta -255{\cos }^4\theta +45{\cos }^2\theta -1)$

$Y_{10}^5(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{1001}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )$

$Y_{10}^6(\theta ,\phi )=\frac{3}{1024}\sqrt{\frac{5005}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (323{\cos }^4\theta -102{\cos }^2\theta +3)$

$Y_{10}^7(\theta ,\phi )=\frac{-3}{512}\sqrt{\frac{85085}{\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot (19{\cos }^3\theta -3\cos \theta )$

$Y_{10}^8(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{255255}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta \cdot (19{\cos }^2\theta -1)$

$Y_{10}^9(\theta ,\phi )=\frac{-1}{512}\sqrt{\frac{4849845}{\pi }}\cdot e^{9i\phi }\cdot {\sin }^9\theta \cdot \cos \theta$

$Y_{10}^{10}(\theta ,\phi )=\frac{1}{1024}\sqrt{\frac{969969}{\pi }}\cdot e^{10i\phi }\cdot {\sin }^{10}\theta$

Real spherical harmonics

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

l = 0

$\begin{array}{rl}{Y}_{00}& =s=Y_0^0=\frac{1}{2}\sqrt{\frac{1}{\pi }}\end{array}$

l = 1

$\begin{array}{rl}{Y}_{1,-1}& =p_y=i\sqrt{\frac{1}{2}}(Y_1^{-1}+Y_1^1)=\sqrt{\frac{3}{4\pi }}\cdot \frac{y}{r}\\ Y_{10}& =p_z=Y_1^0=\sqrt{\frac{3}{4\pi }}\cdot \frac{z}{r}\\ Y_{11}& =p_x=\sqrt{\frac{1}{2}}(Y_1^{-1}-Y_1^1)=\sqrt{\frac{3}{4\pi }}\cdot \frac{x}{r}\end{array}$

l = 2

$\begin{array}{rl}{Y}_{2,-2}& =d_{xy}=i\sqrt{\frac{1}{2}}(Y_2^{-2}-Y_2^2)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{xy}{r^2}\\ Y_{2,-1}& =d_{yz}=i\sqrt{\frac{1}{2}}(Y_2^{-1}+Y_2^1)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{yz}{r^2}\\ Y_{20}& =d_{z^2}=Y_2^0=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{-x^2-y^2+2z^2}{r^2}\\ Y_{21}& =d_{xz}=\sqrt{\frac{1}{2}}(Y_2^{-1}-Y_2^1)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{zx}{r^2}\\ Y_{22}& =d_{x^2-y^2}=\sqrt{\frac{1}{2}}(Y_2^{-2}+Y_2^2)=\frac{1}{4}\sqrt{\frac{15}{\pi }}\cdot \frac{x^2-y^2}{r^2}\end{array}$

l = 3

$\begin{array}{rl}{Y}_{3,-3}& =f_{y(3x^2-y^2)}=i\sqrt{\frac{1}{2}}(Y_3^{-3}+Y_3^3)=\frac{1}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(3x^2-y^2)y}{r^3}\\ Y_{3,-2}& =f_{xyz}=i\sqrt{\frac{1}{2}}(Y_3^{-2}-Y_3^2)=\frac{1}{2}\sqrt{\frac{105}{\pi }}\cdot \frac{xyz}{r^3}\\ Y_{3,-1}& =f_{y{z}^2}=i\sqrt{\frac{1}{2}}(Y_3^{-1}+Y_3^1)=\frac{1}{4}\sqrt{\frac{21}{2\pi }}\cdot \frac{y(4z^2-x^2-y^2)}{r^3}\\ Y_{30}& =f_{z^3}=Y_3^0=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot \frac{z(2z^2-3x^2-3y^2)}{r^3}\\ Y_{31}& =f_{x{z}^2}=\sqrt{\frac{1}{2}}(Y_3^{-1}-Y_3^1)=\frac{1}{4}\sqrt{\frac{21}{2\pi }}\cdot \frac{x(4z^2-x^2-y^2)}{r^3}\\ Y_{32}& =f_{z(x^2-y^2)}=\sqrt{\frac{1}{2}}(Y_3^{-2}+Y_3^2)=\frac{1}{4}\sqrt{\frac{105}{\pi }}\cdot \frac{(x^2-y^2)z}{r^3}\\ Y_{33}& =f_{x(x^2-3y^2)}=\sqrt{\frac{1}{2}}(Y_3^{-3}-Y_3^3)=\frac{1}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x^2-3y^2)x}{r^3}\end{array}$

l = 4

$\begin{array}{rl}{Y}_{4,-4}& =g_{xy(x^2-y^2)}=i\sqrt{\frac{1}{2}}(Y_4^{-4}-Y_4^4)=\frac{3}{4}\sqrt{\frac{35}{\pi }}\cdot \frac{xy(x^2-y^2)}{r^4}\\ Y_{4,-3}& =g_{z{y}^3}=i\sqrt{\frac{1}{2}}(Y_4^{-3}+Y_4^3)=\frac{3}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(3x^2-y^2)yz}{r^4}\\ Y_{4,-2}& =g_{z^{2}xy}=i\sqrt{\frac{1}{2}}(Y_4^{-2}-Y_4^2)=\frac{3}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{xy\cdot (7z^2-r^2)}{r^4}\\ Y_{4,-1}& =g_{z^{3}y}=i\sqrt{\frac{1}{2}}(Y_4^{-1}+Y_4^1)=\frac{3}{4}\sqrt{\frac{5}{2\pi }}\cdot \frac{yz\cdot (7z^2-3r^2)}{r^4}\\ Y_{40}& =g_{z^4}=Y_4^0=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot \frac{(35z^4-30z^2{r}^2+3r^4)}{r^4}\\ Y_{41}& =g_{z^{3}x}=\sqrt{\frac{1}{2}}(Y_4^{-1}-Y_4^1)=\frac{3}{4}\sqrt{\frac{5}{2\pi }}\cdot \frac{xz\cdot (7z^2-3r^2)}{r^4}\\ Y_{42}& =g_{z^{2}xy}=\sqrt{\frac{1}{2}}(Y_4^{-2}+Y_4^2)=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x^2-y^2)\cdot (7z^2-r^2)}{r^4}\\ Y_{43}& =g_{z{x}^3}=\sqrt{\frac{1}{2}}(Y_4^{-3}-Y_4^3)=\frac{3}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x^2-3y^2)xz}{r^4}\\ Y_{44}& =g_{x^4+y^4}=\sqrt{\frac{1}{2}}(Y_4^{-4}+Y_4^4)=\frac{3}{16}\sqrt{\frac{35}{\pi }}\cdot \frac{x^2(x^2-3y^2)-y^2(3x^2-y^2)}{r^4}\end{array}$

See also

• Spherical harmonics

External links

• Spherical Harmonic at MathWorld

References

Cited references

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

General references

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