Mathematical table
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ℓ = 10 {\displaystyle \ell =10} x , y , z , and r . For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to θ {\displaystyle \theta } φ {\displaystyle \varphi }
{ cos ( θ ) = z / r e ± i φ ⋅ sin ( θ ) = ( x ± i y ) / r {\displaystyle {\begin{cases}\cos(\theta )&=z/r\\e^{\pm i\varphi }\cdot \sin(\theta )&=(x\pm iy)/r\end{cases}}} Complex spherical harmonics For ℓ = 0, …, 5, see.[1]
ℓ = 0
Y 0 0 ( θ , φ ) = 1 2 1 π {\displaystyle Y_{0}^{0}(\theta ,\varphi )={1 \over 2}{\sqrt {1 \over \pi }}} ℓ = 1
Y 1 − 1 ( θ , φ ) = 1 2 3 2 π ⋅ e − i φ ⋅ sin θ = 1 2 3 2 π ⋅ ( x − i y ) r Y 1 0 ( θ , φ ) = 1 2 3 π ⋅ cos θ = 1 2 3 π ⋅ z r Y 1 1 ( θ , φ ) = − 1 2 3 2 π ⋅ e i φ ⋅ sin θ = − 1 2 3 2 π ⋅ ( x + i y ) r {\displaystyle {\begin{aligned}Y_{1}^{-1}(\theta ,\varphi )&=&&{1 \over 2}{\sqrt {3 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta &&=&&{1 \over 2}{\sqrt {3 \over 2\pi }}\cdot {(x-iy) \over r}\\Y_{1}^{0}(\theta ,\varphi )&=&&{1 \over 2}{\sqrt {3 \over \pi }}\cdot \cos \theta &&=&&{1 \over 2}{\sqrt {3 \over \pi }}\cdot {z \over r}\\Y_{1}^{1}(\theta ,\varphi )&=&-&{1 \over 2}{\sqrt {3 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta &&=&-&{1 \over 2}{\sqrt {3 \over 2\pi }}\cdot {(x+iy) \over r}\end{aligned}}} ℓ = 2
Y 2 − 2 ( θ , φ ) = 1 4 15 2 π ⋅ e − 2 i φ ⋅ sin 2 θ = 1 4 15 2 π ⋅ ( x − i y ) 2 r 2 Y 2 − 1 ( θ , φ ) = 1 2 15 2 π ⋅ e − i φ ⋅ sin θ ⋅ cos θ = 1 2 15 2 π ⋅ ( x − i y ) ⋅ z r 2 Y 2 0 ( θ , φ ) = 1 4 5 π ⋅ ( 3 cos 2 θ − 1 ) = 1 4 5 π ⋅ ( 3 z 2 − r 2 ) r 2 Y 2 1 ( θ , φ ) = − 1 2 15 2 π ⋅ e i φ ⋅ sin θ ⋅ cos θ = − 1 2 15 2 π ⋅ ( x + i y ) ⋅ z r 2 Y 2 2 ( θ , φ ) = 1 4 15 2 π ⋅ e 2 i φ ⋅ sin 2 θ = 1 4 15 2 π ⋅ ( x + i y ) 2 r 2 {\displaystyle {\begin{aligned}Y_{2}^{-2}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {15 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \quad &&=&&{1 \over 4}{\sqrt {15 \over 2\pi }}\cdot {(x-iy)^{2} \over r^{2}}&\\Y_{2}^{-1}(\theta ,\varphi )&=&&{1 \over 2}{\sqrt {15 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot \cos \theta \quad &&=&&{1 \over 2}{\sqrt {15 \over 2\pi }}\cdot {(x-iy)\cdot z \over r^{2}}&\\Y_{2}^{0}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {5 \over \pi }}\cdot (3\cos ^{2}\theta -1)\quad &&=&&{1 \over 4}{\sqrt {5 \over \pi }}\cdot {(3z^{2}-r^{2}) \over r^{2}}&\\Y_{2}^{1}(\theta ,\varphi )&=&-&{1 \over 2}{\sqrt {15 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot \cos \theta \quad &&=&-&{1 \over 2}{\sqrt {15 \over 2\pi }}\cdot {(x+iy)\cdot z \over r^{2}}&\\Y_{2}^{2}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {15 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \quad &&=&&{1 \over 4}{\sqrt {15 \over 2\pi }}\cdot {(x+iy)^{2} \over r^{2}}&\end{aligned}}} ℓ = 3
Y 3 − 3 ( θ , φ ) = 1 8 35 π ⋅ e − 3 i φ ⋅ sin 3 θ = 1 8 35 π ⋅ ( x − i y ) 3 r 3 Y 3 − 2 ( θ , φ ) = 1 4 105 2 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ cos θ = 1 4 105 2 π ⋅ ( x − i y ) 2 ⋅ z r 3 Y 3 − 1 ( θ , φ ) = 1 8 21 π ⋅ e − i φ ⋅ sin θ ⋅ ( 5 cos 2 θ − 1 ) = 1 8 21 π ⋅ ( x − i y ) ⋅ ( 5 z 2 − r 2 ) r 3 Y 3 0 ( θ , φ ) = 1 4 7 π ⋅ ( 5 cos 3 θ − 3 cos θ ) = 1 4 7 π ⋅ ( 5 z 3 − 3 z r 2 ) r 3 Y 3 1 ( θ , φ ) = − 1 8 21 π ⋅ e i φ ⋅ sin θ ⋅ ( 5 cos 2 θ − 1 ) = − 1 8 21 π ⋅ ( x + i y ) ⋅ ( 5 z 2 − r 2 ) r 3 Y 3 2 ( θ , φ ) = 1 4 105 2 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ cos θ = 1 4 105 2 π ⋅ ( x + i y ) 2 ⋅ z r 3 Y 3 3 ( θ , φ ) = − 1 8 35 π ⋅ e 3 i φ ⋅ sin 3 θ = − 1 8 35 π ⋅ ( x + i y ) 3 r 3 {\displaystyle {\begin{aligned}Y_{3}^{-3}(\theta ,\varphi )&=&&{1 \over 8}{\sqrt {35 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \quad &&=&&{1 \over 8}{\sqrt {35 \over \pi }}\cdot {(x-iy)^{3} \over r^{3}}&\\Y_{3}^{-2}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {105 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta \quad &&=&&{1 \over 4}{\sqrt {105 \over 2\pi }}\cdot {(x-iy)^{2}\cdot z \over r^{3}}&\\Y_{3}^{-1}(\theta ,\varphi )&=&&{1 \over 8}{\sqrt {21 \over \pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)\quad &&=&&{1 \over 8}{\sqrt {21 \over \pi }}\cdot {(x-iy)\cdot (5z^{2}-r^{2}) \over r^{3}}&\\Y_{3}^{0}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {7 \over \pi }}\cdot (5\cos ^{3}\theta -3\cos \theta )\quad &&=&&{1 \over 4}{\sqrt {7 \over \pi }}\cdot {(5z^{3}-3zr^{2}) \over r^{3}}&\\Y_{3}^{1}(\theta ,\varphi )&=&-&{1 \over 8}{\sqrt {21 \over \pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)\quad &&=&&{-1 \over 8}{\sqrt {21 \over \pi }}\cdot {(x+iy)\cdot (5z^{2}-r^{2}) \over r^{3}}&\\Y_{3}^{2}(\theta ,\varphi )&=&&{1 \over 4}{\sqrt {105 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta \quad &&=&&{1 \over 4}{\sqrt {105 \over 2\pi }}\cdot {(x+iy)^{2}\cdot z \over r^{3}}&\\Y_{3}^{3}(\theta ,\varphi )&=&-&{1 \over 8}{\sqrt {35 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \quad &&=&&{-1 \over 8}{\sqrt {35 \over \pi }}\cdot {(x+iy)^{3} \over r^{3}}&\end{aligned}}} ℓ = 4
Y 4 − 4 ( θ , φ ) = 3 16 35 2 π ⋅ e − 4 i φ ⋅ sin 4 θ = 3 16 35 2 π ⋅ ( x − i y ) 4 r 4 Y 4 − 3 ( θ , φ ) = 3 8 35 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ cos θ = 3 8 35 π ⋅ ( x − i y ) 3 z r 4 Y 4 − 2 ( θ , φ ) = 3 8 5 2 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 7 cos 2 θ − 1 ) = 3 8 5 2 π ⋅ ( x − i y ) 2 ⋅ ( 7 z 2 − r 2 ) r 4 Y 4 − 1 ( θ , φ ) = 3 8 5 π ⋅ e − i φ ⋅ sin θ ⋅ ( 7 cos 3 θ − 3 cos θ ) = 3 8 5 π ⋅ ( x − i y ) ⋅ ( 7 z 3 − 3 z r 2 ) r 4 Y 4 0 ( θ , φ ) = 3 16 1 π ⋅ ( 35 cos 4 θ − 30 cos 2 θ + 3 ) = 3 16 1 π ⋅ ( 35 z 4 − 30 z 2 r 2 + 3 r 4 ) r 4 Y 4 1 ( θ , φ ) = − 3 8 5 π ⋅ e i φ ⋅ sin θ ⋅ ( 7 cos 3 θ − 3 cos θ ) = − 3 8 5 π ⋅ ( x + i y ) ⋅ ( 7 z 3 − 3 z r 2 ) r 4 Y 4 2 ( θ , φ ) = 3 8 5 2 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 7 cos 2 θ − 1 ) = 3 8 5 2 π ⋅ ( x + i y ) 2 ⋅ ( 7 z 2 − r 2 ) r 4 Y 4 3 ( θ , φ ) = − 3 8 35 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ cos θ = − 3 8 35 π ⋅ ( x + i y ) 3 z r 4 Y 4 4 ( θ , φ ) = 3 16 35 2 π ⋅ e 4 i φ ⋅ sin 4 θ = 3 16 35 2 π ⋅ ( x + i y ) 4 r 4 {\displaystyle {\begin{aligned}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-iy)^{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-iy)^{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-iy)^{2}\cdot (7z^{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-iy)\cdot (7z^{3}-3zr^{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 {(35z^{4}-30z^{2}r^{2}+3r^{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+iy)\cdot (7z^{3}-3zr^{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+iy)^{2}\cdot (7z^{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+iy)^{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+iy)^{4}}{r^{4}}}\end{aligned}}} ℓ = 5
Y 5 − 5 ( θ , φ ) = 3 32 77 π ⋅ e − 5 i φ ⋅ sin 5 θ Y 5 − 4 ( θ , φ ) = 3 16 385 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ cos θ Y 5 − 3 ( θ , φ ) = 1 32 385 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 9 cos 2 θ − 1 ) Y 5 − 2 ( θ , φ ) = 1 8 1155 2 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 3 cos 3 θ − cos θ ) Y 5 − 1 ( θ , φ ) = 1 16 165 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 21 cos 4 θ − 14 cos 2 θ + 1 ) Y 5 0 ( θ , φ ) = 1 16 11 π ⋅ ( 63 cos 5 θ − 70 cos 3 θ + 15 cos θ ) Y 5 1 ( θ , φ ) = − 1 16 165 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 21 cos 4 θ − 14 cos 2 θ + 1 ) Y 5 2 ( θ , φ ) = 1 8 1155 2 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 3 cos 3 θ − cos θ ) Y 5 3 ( θ , φ ) = − 1 32 385 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 9 cos 2 θ − 1 ) Y 5 4 ( θ , φ ) = 3 16 385 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ cos θ Y 5 5 ( θ , φ ) = − 3 32 77 π ⋅ e 5 i φ ⋅ sin 5 θ {\displaystyle {\begin{aligned}Y_{5}^{-5}(\theta ,\varphi )&={3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \\Y_{5}^{-4}(\theta ,\varphi )&={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta \\Y_{5}^{-3}(\theta ,\varphi )&={1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)\\Y_{5}^{-2}(\theta ,\varphi )&={1 \over 8}{\sqrt {1155 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -\cos \theta )\\Y_{5}^{-1}(\theta ,\varphi )&={1 \over 16}{\sqrt {165 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)\\Y_{5}^{0}(\theta ,\varphi )&={1 \over 16}{\sqrt {11 \over \pi }}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )\\Y_{5}^{1}(\theta ,\varphi )&={-1 \over 16}{\sqrt {165 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)\\Y_{5}^{2}(\theta ,\varphi )&={1 \over 8}{\sqrt {1155 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -\cos \theta )\\Y_{5}^{3}(\theta ,\varphi )&={-1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)\\Y_{5}^{4}(\theta ,\varphi )&={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta \\Y_{5}^{5}(\theta ,\varphi )&={-3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \end{aligned}}} ℓ = 6
Y 6 − 6 ( θ , φ ) = 1 64 3003 π ⋅ e − 6 i φ ⋅ sin 6 θ Y 6 − 5 ( θ , φ ) = 3 32 1001 π ⋅ e − 5 i φ ⋅ sin 5 θ ⋅ cos θ Y 6 − 4 ( θ , φ ) = 3 32 91 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ ( 11 cos 2 θ − 1 ) Y 6 − 3 ( θ , φ ) = 1 32 1365 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 11 cos 3 θ − 3 cos θ ) Y 6 − 2 ( θ , φ ) = 1 64 1365 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 33 cos 4 θ − 18 cos 2 θ + 1 ) Y 6 − 1 ( θ , φ ) = 1 16 273 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 33 cos 5 θ − 30 cos 3 θ + 5 cos θ ) Y 6 0 ( θ , φ ) = 1 32 13 π ⋅ ( 231 cos 6 θ − 315 cos 4 θ + 105 cos 2 θ − 5 ) Y 6 1 ( θ , φ ) = − 1 16 273 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 33 cos 5 θ − 30 cos 3 θ + 5 cos θ ) Y 6 2 ( θ , φ ) = 1 64 1365 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 33 cos 4 θ − 18 cos 2 θ + 1 ) Y 6 3 ( θ , φ ) = − 1 32 1365 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 11 cos 3 θ − 3 cos θ ) Y 6 4 ( θ , φ ) = 3 32 91 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ ( 11 cos 2 θ − 1 ) Y 6 5 ( θ , φ ) = − 3 32 1001 π ⋅ e 5 i φ ⋅ sin 5 θ ⋅ cos θ Y 6 6 ( θ , φ ) = 1 64 3003 π ⋅ e 6 i φ ⋅ sin 6 θ {\displaystyle {\begin{aligned}Y_{6}^{-6}(\theta ,\varphi )&={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \\Y_{6}^{-5}(\theta ,\varphi )&={3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta \\Y_{6}^{-4}(\theta ,\varphi )&={3 \over 32}{\sqrt {91 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)\\Y_{6}^{-3}(\theta ,\varphi )&={1 \over 32}{\sqrt {1365 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )\\Y_{6}^{-2}(\theta ,\varphi )&={1 \over 64}{\sqrt {1365 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)\\Y_{6}^{-1}(\theta ,\varphi )&={1 \over 16}{\sqrt {273 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )\\Y_{6}^{0}(\theta ,\varphi )&={1 \over 32}{\sqrt {13 \over \pi }}\cdot (231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5)\\Y_{6}^{1}(\theta ,\varphi )&=-{1 \over 16}{\sqrt {273 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )\\Y_{6}^{2}(\theta ,\varphi )&={1 \over 64}{\sqrt {1365 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)\\Y_{6}^{3}(\theta ,\varphi )&=-{1 \over 32}{\sqrt {1365 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )\\Y_{6}^{4}(\theta ,\varphi )&={3 \over 32}{\sqrt {91 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)\\Y_{6}^{5}(\theta ,\varphi )&=-{3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta \\Y_{6}^{6}(\theta ,\varphi )&={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \end{aligned}}} ℓ = 7
Y 7 − 7 ( θ , φ ) = 3 64 715 2 π ⋅ e − 7 i φ ⋅ sin 7 θ Y 7 − 6 ( θ , φ ) = 3 64 5005 π ⋅ e − 6 i φ ⋅ sin 6 θ ⋅ cos θ Y 7 − 5 ( θ , φ ) = 3 64 385 2 π ⋅ e − 5 i φ ⋅ sin 5 θ ⋅ ( 13 cos 2 θ − 1 ) Y 7 − 4 ( θ , φ ) = 3 32 385 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ ( 13 cos 3 θ − 3 cos θ ) Y 7 − 3 ( θ , φ ) = 3 64 35 2 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 143 cos 4 θ − 66 cos 2 θ + 3 ) Y 7 − 2 ( θ , φ ) = 3 64 35 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 143 cos 5 θ − 110 cos 3 θ + 15 cos θ ) Y 7 − 1 ( θ , φ ) = 1 64 105 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 429 cos 6 θ − 495 cos 4 θ + 135 cos 2 θ − 5 ) Y 7 0 ( θ , φ ) = 1 32 15 π ⋅ ( 429 cos 7 θ − 693 cos 5 θ + 315 cos 3 θ − 35 cos θ ) Y 7 1 ( θ , φ ) = − 1 64 105 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 429 cos 6 θ − 495 cos 4 θ + 135 cos 2 θ − 5 ) Y 7 2 ( θ , φ ) = 3 64 35 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 143 cos 5 θ − 110 cos 3 θ + 15 cos θ ) Y 7 3 ( θ , φ ) = − 3 64 35 2 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 143 cos 4 θ − 66 cos 2 θ + 3 ) Y 7 4 ( θ , φ ) = 3 32 385 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ ( 13 cos 3 θ − 3 cos θ ) Y 7 5 ( θ , φ ) = − 3 64 385 2 π ⋅ e 5 i φ ⋅ sin 5 θ ⋅ ( 13 cos 2 θ − 1 ) Y 7 6 ( θ , φ ) = 3 64 5005 π ⋅ e 6 i φ ⋅ sin 6 θ ⋅ cos θ Y 7 7 ( θ , φ ) = − 3 64 715 2 π ⋅ e 7 i φ ⋅ sin 7 θ {\displaystyle {\begin{aligned}Y_{7}^{-7}(\theta ,\varphi )&={3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \\Y_{7}^{-6}(\theta ,\varphi )&={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta \\Y_{7}^{-5}(\theta ,\varphi )&={3 \over 64}{\sqrt {385 \over 2\pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)\\Y_{7}^{-4}(\theta ,\varphi )&={3 \over 32}{\sqrt {385 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (13\cos ^{3}\theta -3\cos \theta )\\Y_{7}^{-3}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over 2\pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (143\cos ^{4}\theta -66\cos ^{2}\theta +3)\\Y_{7}^{-2}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )\\Y_{7}^{-1}(\theta ,\varphi )&={1 \over 64}{\sqrt {105 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)\\Y_{7}^{0}(\theta ,\varphi )&={1 \over 32}{\sqrt {15 \over \pi }}\cdot (429\cos ^{7}\theta -693\cos ^{5}\theta +315\cos ^{3}\theta -35\cos \theta )\\Y_{7}^{1}(\theta ,\varphi )&=-{1 \over 64}{\sqrt {105 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)\\Y_{7}^{2}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )\\Y_{7}^{3}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {35 \over 2\pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (143\cos ^{4}\theta -66\cos ^{2}\theta +3)\\Y_{7}^{4}(\theta ,\varphi )&={3 \over 32}{\sqrt {385 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (13\cos ^{3}\theta -3\cos \theta )\\Y_{7}^{5}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {385 \over 2\pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)\\Y_{7}^{6}(\theta ,\varphi )&={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta \\Y_{7}^{7}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \end{aligned}}} ℓ = 8
Y 8 − 8 ( θ , φ ) = 3 256 12155 2 π ⋅ e − 8 i φ ⋅ sin 8 θ Y 8 − 7 ( θ , φ ) = 3 64 12155 2 π ⋅ e − 7 i φ ⋅ sin 7 θ ⋅ cos θ Y 8 − 6 ( θ , φ ) = 1 128 7293 π ⋅ e − 6 i φ ⋅ sin 6 θ ⋅ ( 15 cos 2 θ − 1 ) Y 8 − 5 ( θ , φ ) = 3 64 17017 2 π ⋅ e − 5 i φ ⋅ sin 5 θ ⋅ ( 5 cos 3 θ − cos θ ) Y 8 − 4 ( θ , φ ) = 3 128 1309 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ ( 65 cos 4 θ − 26 cos 2 θ + 1 ) Y 8 − 3 ( θ , φ ) = 1 64 19635 2 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 39 cos 5 θ − 26 cos 3 θ + 3 cos θ ) Y 8 − 2 ( θ , φ ) = 3 128 595 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 143 cos 6 θ − 143 cos 4 θ + 33 cos 2 θ − 1 ) Y 8 − 1 ( θ , φ ) = 3 64 17 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 715 cos 7 θ − 1001 cos 5 θ + 385 cos 3 θ − 35 cos θ ) Y 8 0 ( θ , φ ) = 1 256 17 π ⋅ ( 6435 cos 8 θ − 12012 cos 6 θ + 6930 cos 4 θ − 1260 cos 2 θ + 35 ) Y 8 1 ( θ , φ ) = − 3 64 17 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 715 cos 7 θ − 1001 cos 5 θ + 385 cos 3 θ − 35 cos θ ) Y 8 2 ( θ , φ ) = 3 128 595 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 143 cos 6 θ − 143 cos 4 θ + 33 cos 2 θ − 1 ) Y 8 3 ( θ , φ ) = − 1 64 19635 2 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 39 cos 5 θ − 26 cos 3 θ + 3 cos θ ) Y 8 4 ( θ , φ ) = 3 128 1309 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ ( 65 cos 4 θ − 26 cos 2 θ + 1 ) Y 8 5 ( θ , φ ) = − 3 64 17017 2 π ⋅ e 5 i φ ⋅ sin 5 θ ⋅ ( 5 cos 3 θ − cos θ ) Y 8 6 ( θ , φ ) = 1 128 7293 π ⋅ e 6 i φ ⋅ sin 6 θ ⋅ ( 15 cos 2 θ − 1 ) Y 8 7 ( θ , φ ) = − 3 64 12155 2 π ⋅ e 7 i φ ⋅ sin 7 θ ⋅ cos θ Y 8 8 ( θ , φ ) = 3 256 12155 2 π ⋅ e 8 i φ ⋅ sin 8 θ {\displaystyle {\begin{aligned}Y_{8}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \\Y_{8}^{-7}(\theta ,\varphi )&={3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta \\Y_{8}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)\\Y_{8}^{-5}(\theta ,\varphi )&={3 \over 64}{\sqrt {17017 \over 2\pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -\cos \theta )\\Y_{8}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {1309 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)\\Y_{8}^{-3}(\theta ,\varphi )&={1 \over 64}{\sqrt {19635 \over 2\pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )\\Y_{8}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {595 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)\\Y_{8}^{-1}(\theta ,\varphi )&={3 \over 64}{\sqrt {17 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )\\Y_{8}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {17 \over \pi }}\cdot (6435\cos ^{8}\theta -12012\cos ^{6}\theta +6930\cos ^{4}\theta -1260\cos ^{2}\theta +35)\\Y_{8}^{1}(\theta ,\varphi )&={-3 \over 64}{\sqrt {17 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )\\Y_{8}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {595 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)\\Y_{8}^{3}(\theta ,\varphi )&={-1 \over 64}{\sqrt {19635 \over 2\pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )\\Y_{8}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {1309 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)\\Y_{8}^{5}(\theta ,\varphi )&={-3 \over 64}{\sqrt {17017 \over 2\pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -\cos \theta )\\Y_{8}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)\\Y_{8}^{7}(\theta ,\varphi )&={-3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta \\Y_{8}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \end{aligned}}} ℓ = 9
Y 9 − 9 ( θ , φ ) = 1 512 230945 π ⋅ e − 9 i φ ⋅ sin 9 θ Y 9 − 8 ( θ , φ ) = 3 256 230945 2 π ⋅ e − 8 i φ ⋅ sin 8 θ ⋅ cos θ Y 9 − 7 ( θ , φ ) = 3 512 13585 π ⋅ e − 7 i φ ⋅ sin 7 θ ⋅ ( 17 cos 2 θ − 1 ) Y 9 − 6 ( θ , φ ) = 1 128 40755 π ⋅ e − 6 i φ ⋅ sin 6 θ ⋅ ( 17 cos 3 θ − 3 cos θ ) Y 9 − 5 ( θ , φ ) = 3 256 2717 π ⋅ e − 5 i φ ⋅ sin 5 θ ⋅ ( 85 cos 4 θ − 30 cos 2 θ + 1 ) Y 9 − 4 ( θ , φ ) = 3 128 95095 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ ( 17 cos 5 θ − 10 cos 3 θ + cos θ ) Y 9 − 3 ( θ , φ ) = 1 256 21945 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 221 cos 6 θ − 195 cos 4 θ + 39 cos 2 θ − 1 ) Y 9 − 2 ( θ , φ ) = 3 128 1045 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 221 cos 7 θ − 273 cos 5 θ + 91 cos 3 θ − 7 cos θ ) Y 9 − 1 ( θ , φ ) = 3 256 95 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 2431 cos 8 θ − 4004 cos 6 θ + 2002 cos 4 θ − 308 cos 2 θ + 7 ) Y 9 0 ( θ , φ ) = 1 256 19 π ⋅ ( 12155 cos 9 θ − 25740 cos 7 θ + 18018 cos 5 θ − 4620 cos 3 θ + 315 cos θ ) Y 9 1 ( θ , φ ) = − 3 256 95 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 2431 cos 8 θ − 4004 cos 6 θ + 2002 cos 4 θ − 308 cos 2 θ + 7 ) Y 9 2 ( θ , φ ) = 3 128 1045 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 221 cos 7 θ − 273 cos 5 θ + 91 cos 3 θ − 7 cos θ ) Y 9 3 ( θ , φ ) = − 1 256 21945 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 221 cos 6 θ − 195 cos 4 θ + 39 cos 2 θ − 1 ) Y 9 4 ( θ , φ ) = 3 128 95095 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ ( 17 cos 5 θ − 10 cos 3 θ + cos θ ) Y 9 5 ( θ , φ ) = − 3 256 2717 π ⋅ e 5 i φ ⋅ sin 5 θ ⋅ ( 85 cos 4 θ − 30 cos 2 θ + 1 ) Y 9 6 ( θ , φ ) = 1 128 40755 π ⋅ e 6 i φ ⋅ sin 6 θ ⋅ ( 17 cos 3 θ − 3 cos θ ) Y 9 7 ( θ , φ ) = − 3 512 13585 π ⋅ e 7 i φ ⋅ sin 7 θ ⋅ ( 17 cos 2 θ − 1 ) Y 9 8 ( θ , φ ) = 3 256 230945 2 π ⋅ e 8 i φ ⋅ sin 8 θ ⋅ cos θ Y 9 9 ( θ , φ ) = − 1 512 230945 π ⋅ e 9 i φ ⋅ sin 9 θ {\displaystyle {\begin{aligned}Y_{9}^{-9}(\theta ,\varphi )&={1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \\Y_{9}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta \\Y_{9}^{-7}(\theta ,\varphi )&={3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)\\Y_{9}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{-5}(\theta ,\varphi )&={3 \over 256}{\sqrt {2717 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{-3}(\theta ,\varphi )&={1 \over 256}{\sqrt {21945 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{-1}(\theta ,\varphi )&={3 \over 256}{\sqrt {95 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {19 \over \pi }}\cdot (12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )\\Y_{9}^{1}(\theta ,\varphi )&={-3 \over 256}{\sqrt {95 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{3}(\theta ,\varphi )&={-1 \over 256}{\sqrt {21945 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{5}(\theta ,\varphi )&={-3 \over 256}{\sqrt {2717 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{7}(\theta ,\varphi )&={-3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)\\Y_{9}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta \\Y_{9}^{9}(\theta ,\varphi )&={-1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \end{aligned}}} ℓ = 10
Y 10 − 10 ( θ , φ ) = 1 1024 969969 π ⋅ e − 10 i φ ⋅ sin 10 θ Y 10 − 9 ( θ , φ ) = 1 512 4849845 π ⋅ e − 9 i φ ⋅ sin 9 θ ⋅ cos θ Y 10 − 8 ( θ , φ ) = 1 512 255255 2 π ⋅ e − 8 i φ ⋅ sin 8 θ ⋅ ( 19 cos 2 θ − 1 ) Y 10 − 7 ( θ , φ ) = 3 512 85085 π ⋅ e − 7 i φ ⋅ sin 7 θ ⋅ ( 19 cos 3 θ − 3 cos θ ) Y 10 − 6 ( θ , φ ) = 3 1024 5005 π ⋅ e − 6 i φ ⋅ sin 6 θ ⋅ ( 323 cos 4 θ − 102 cos 2 θ + 3 ) Y 10 − 5 ( θ , φ ) = 3 256 1001 π ⋅ e − 5 i φ ⋅ sin 5 θ ⋅ ( 323 cos 5 θ − 170 cos 3 θ + 15 cos θ ) Y 10 − 4 ( θ , φ ) = 3 256 5005 2 π ⋅ e − 4 i φ ⋅ sin 4 θ ⋅ ( 323 cos 6 θ − 255 cos 4 θ + 45 cos 2 θ − 1 ) Y 10 − 3 ( θ , φ ) = 3 256 5005 π ⋅ e − 3 i φ ⋅ sin 3 θ ⋅ ( 323 cos 7 θ − 357 cos 5 θ + 105 cos 3 θ − 7 cos θ ) Y 10 − 2 ( θ , φ ) = 3 512 385 2 π ⋅ e − 2 i φ ⋅ sin 2 θ ⋅ ( 4199 cos 8 θ − 6188 cos 6 θ + 2730 cos 4 θ − 364 cos 2 θ + 7 ) Y 10 − 1 ( θ , φ ) = 1 256 1155 2 π ⋅ e − i φ ⋅ sin θ ⋅ ( 4199 cos 9 θ − 7956 cos 7 θ + 4914 cos 5 θ − 1092 cos 3 θ + 63 cos θ ) Y 10 0 ( θ , φ ) = 1 512 21 π ⋅ ( 46189 cos 10 θ − 109395 cos 8 θ + 90090 cos 6 θ − 30030 cos 4 θ + 3465 cos 2 θ − 63 ) Y 10 1 ( θ , φ ) = − 1 256 1155 2 π ⋅ e i φ ⋅ sin θ ⋅ ( 4199 cos 9 θ − 7956 cos 7 θ + 4914 cos 5 θ − 1092 cos 3 θ + 63 cos θ ) Y 10 2 ( θ , φ ) = 3 512 385 2 π ⋅ e 2 i φ ⋅ sin 2 θ ⋅ ( 4199 cos 8 θ − 6188 cos 6 θ + 2730 cos 4 θ − 364 cos 2 θ + 7 ) Y 10 3 ( θ , φ ) = − 3 256 5005 π ⋅ e 3 i φ ⋅ sin 3 θ ⋅ ( 323 cos 7 θ − 357 cos 5 θ + 105 cos 3 θ − 7 cos θ ) Y 10 4 ( θ , φ ) = 3 256 5005 2 π ⋅ e 4 i φ ⋅ sin 4 θ ⋅ ( 323 cos 6 θ − 255 cos 4 θ + 45 cos 2 θ − 1 ) Y 10 5 ( θ , φ ) = − 3 256 1001 π ⋅ e 5 i φ ⋅ sin 5 θ ⋅ ( 323 cos 5 θ − 170 cos 3 θ + 15 cos θ ) Y 10 6 ( θ , φ ) = 3 1024 5005 π ⋅ e 6 i φ ⋅ sin 6 θ ⋅ ( 323 cos 4 θ − 102 cos 2 θ + 3 ) Y 10 7 ( θ , φ ) = − 3 512 85085 π ⋅ e 7 i φ ⋅ sin 7 θ ⋅ ( 19 cos 3 θ − 3 cos θ ) Y 10 8 ( θ , φ ) = 1 512 255255 2 π ⋅ e 8 i φ ⋅ sin 8 θ ⋅ ( 19 cos 2 θ − 1 ) Y 10 9 ( θ , φ ) = − 1 512 4849845 π ⋅ e 9 i φ ⋅ sin 9 θ ⋅ cos θ Y 10 10 ( θ , φ ) = 1 1024 969969 π ⋅ e 10 i φ ⋅ sin 10 θ {\displaystyle {\begin{aligned}Y_{10}^{-10}(\theta ,\varphi )&={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta \\Y_{10}^{-9}(\theta ,\varphi )&={1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta \\Y_{10}^{-8}(\theta ,\varphi )&={1 \over 512}{\sqrt {255255 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)\\Y_{10}^{-7}(\theta ,\varphi )&={3 \over 512}{\sqrt {85085 \over \pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )\\Y_{10}^{-6}(\theta ,\varphi )&={3 \over 1024}{\sqrt {5005 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)\\Y_{10}^{-5}(\theta ,\varphi )&={3 \over 256}{\sqrt {1001 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )\\Y_{10}^{-4}(\theta ,\varphi )&={3 \over 256}{\sqrt {5005 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)\\Y_{10}^{-3}(\theta ,\varphi )&={3 \over 256}{\sqrt {5005 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (323\cos ^{7}\theta -357\cos ^{5}\theta +105\cos ^{3}\theta -7\cos \theta )\\Y_{10}^{-2}(\theta ,\varphi )&={3 \over 512}{\sqrt {385 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (4199\cos ^{8}\theta -6188\cos ^{6}\theta +2730\cos ^{4}\theta -364\cos ^{2}\theta +7)\\Y_{10}^{-1}(\theta ,\varphi )&={1 \over 256}{\sqrt {1155 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (4199\cos ^{9}\theta -7956\cos ^{7}\theta +4914\cos ^{5}\theta -1092\cos ^{3}\theta +63\cos \theta )\\Y_{10}^{0}(\theta ,\varphi )&={1 \over 512}{\sqrt {21 \over \pi }}\cdot (46189\cos ^{10}\theta -109395\cos ^{8}\theta +90090\cos ^{6}\theta -30030\cos ^{4}\theta +3465\cos ^{2}\theta -63)\\Y_{10}^{1}(\theta ,\varphi )&={-1 \over 256}{\sqrt {1155 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (4199\cos ^{9}\theta -7956\cos ^{7}\theta +4914\cos ^{5}\theta -1092\cos ^{3}\theta +63\cos \theta )\\Y_{10}^{2}(\theta ,\varphi )&={3 \over 512}{\sqrt {385 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (4199\cos ^{8}\theta -6188\cos ^{6}\theta +2730\cos ^{4}\theta -364\cos ^{2}\theta +7)\\Y_{10}^{3}(\theta ,\varphi )&={-3 \over 256}{\sqrt {5005 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (323\cos ^{7}\theta -357\cos ^{5}\theta +105\cos ^{3}\theta -7\cos \theta )\\Y_{10}^{4}(\theta ,\varphi )&={3 \over 256}{\sqrt {5005 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)\\Y_{10}^{5}(\theta ,\varphi )&={-3 \over 256}{\sqrt {1001 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )\\Y_{10}^{6}(\theta ,\varphi )&={3 \over 1024}{\sqrt {5005 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)\\Y_{10}^{7}(\theta ,\varphi )&={-3 \over 512}{\sqrt {85085 \over \pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )\\Y_{10}^{8}(\theta ,\varphi )&={1 \over 512}{\sqrt {255255 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)\\Y_{10}^{9}(\theta ,\varphi )&={-1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta \\Y_{10}^{10}(\theta ,\varphi )&={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta \end{aligned}}} Visualization of complex spherical harmonics 2D polar/azimuthal angle maps Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, ϕ {\displaystyle \phi } θ {\displaystyle \theta }
Visual Array of Complex Spherical Harmonics Represented as 2D Theta/Phi Maps Polar plots Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.
Visual Array of Complex Spherical Harmonics Represented with Polar Plot Polar plots with magnitude as radius Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.
Visual Array of Complex Spherical Harmonics Represented with Polar Plot with Magnitude Mapped to Radius Real spherical harmonics For each real spherical harmonic, the corresponding atomic orbital symbol (s , p , d , f ) is reported as well.[2] [3]
For ℓ = 0, …, 3, see.[4] [5]
ℓ = 0
Y 00 = s = Y 0 0 = 1 2 1 π {\displaystyle Y_{00}=s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}} ℓ = 1
Y 1 , − 1 = p y = i 1 2 ( Y 1 − 1 + Y 1 1 ) = 3 4 π ⋅ y r = 3 4 π sin ( θ ) sin φ Y 1 , 0 = p z = Y 1 0 = 3 4 π ⋅ z r = 3 4 π cos ( θ ) Y 1 , 1 = p x = 1 2 ( Y 1 − 1 − Y 1 1 ) = 3 4 π ⋅ x r = 3 4 π sin ( θ ) cos φ {\displaystyle {\begin{aligned}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{aligned}}} ℓ = 2
Y 2 , − 2 = d x y = i 1 2 ( Y 2 − 2 − Y 2 2 ) = 1 2 15 π ⋅ x y r 2 = 1 4 15 π sin 2 θ sin ( 2 φ ) Y 2 , − 1 = d y z = i 1 2 ( Y 2 − 1 + Y 2 1 ) = 1 2 15 π ⋅ y ⋅ z r 2 = 1 4 15 π sin ( 2 θ ) sin φ Y 2 , 0 = d z 2 = Y 2 0 = 1 4 5 π ⋅ 3 z 2 − r 2 r 2 = 1 4 5 π ( 3 cos 2 θ − 1 ) Y 2 , 1 = d x z = 1 2 ( Y 2 − 1 − Y 2 1 ) = 1 2 15 π ⋅ x ⋅ z r 2 = 1 4 15 π sin ( 2 θ ) cos φ Y 2 , 2 = d x 2 − y 2 = 1 2 ( Y 2 − 2 + Y 2 2 ) = 1 4 15 π ⋅ x 2 − y 2 r 2 = 1 4 15 π sin 2 θ cos ( 2 φ ) {\displaystyle {\begin{aligned}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 {xy}{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{aligned}}} ℓ = 3
Y 3 , − 3 = f y ( 3 x 2 − y 2 ) = i 1 2 ( Y 3 − 3 + Y 3 3 ) = 1 4 35 2 π ⋅ y ( 3 x 2 − y 2 ) r 3 Y 3 , − 2 = f x y z = i 1 2 ( Y 3 − 2 − Y 3 2 ) = 1 2 105 π ⋅ x y ⋅ z r 3 Y 3 , − 1 = f y z 2 = i 1 2 ( Y 3 − 1 + Y 3 1 ) = 1 4 21 2 π ⋅ y ⋅ ( 5 z 2 − r 2 ) r 3 Y 3 , 0 = f z 3 = Y 3 0 = 1 4 7 π ⋅ 5 z 3 − 3 z r 2 r 3 Y 3 , 1 = f x z 2 = 1 2 ( Y 3 − 1 − Y 3 1 ) = 1 4 21 2 π ⋅ x ⋅ ( 5 z 2 − r 2 ) r 3 Y 3 , 2 = f z ( x 2 − y 2 ) = 1 2 ( Y 3 − 2 + Y 3 2 ) = 1 4 105 π ⋅ ( x 2 − y 2 ) ⋅ z r 3 Y 3 , 3 = f x ( x 2 − 3 y 2 ) = 1 2 ( Y 3 − 3 − Y 3 3 ) = 1 4 35 2 π ⋅ x ( x 2 − 3 y 2 ) r 3 {\displaystyle {\begin{aligned}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(3x^{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 (5z^{2}-r^{2})}{r^{3}}}\\Y_{3,0}&=f_{z^{3}}=Y_{3}^{0}={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {5z^{3}-3zr^{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 (5z^{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}-3y^{2}\right)}{r^{3}}}\end{aligned}}} ℓ = 4
Y 4 , − 4 = i 1 2 ( Y 4 − 4 − Y 4 4 ) = 3 4 35 π ⋅ x y ( x 2 − y 2 ) r 4 Y 4 , − 3 = i 1 2 ( Y 4 − 3 + Y 4 3 ) = 3 4 35 2 π ⋅ y ( 3 x 2 − y 2 ) ⋅ z r 4 Y 4 , − 2 = i 1 2 ( Y 4 − 2 − Y 4 2 ) = 3 4 5 π ⋅ x y ⋅ ( 7 z 2 − r 2 ) r 4 Y 4 , − 1 = i 1 2 ( Y 4 − 1 + Y 4 1 ) = 3 4 5 2 π ⋅ y ⋅ ( 7 z 3 − 3 z r 2 ) r 4 Y 4 , 0 = Y 4 0 = 3 16 1 π ⋅ 35 z 4 − 30 z 2 r 2 + 3 r 4 r 4 Y 4 , 1 = 1 2 ( Y 4 − 1 − Y 4 1 ) = 3 4 5 2 π ⋅ x ⋅ ( 7 z 3 − 3 z r 2 ) r 4 Y 4 , 2 = 1 2 ( Y 4 − 2 + Y 4 2 ) = 3 8 5 π ⋅ ( x 2 − y 2 ) ⋅ ( 7 z 2 − r 2 ) r 4 Y 4 , 3 = 1 2 ( Y 4 − 3 − Y 4 3 ) = 3 4 35 2 π ⋅ x ( x 2 − 3 y 2 ) ⋅ z r 4 Y 4 , 4 = 1 2 ( Y 4 − 4 + Y 4 4 ) = 3 16 35 π ⋅ x 2 ( x 2 − 3 y 2 ) − y 2 ( 3 x 2 − y 2 ) r 4 {\displaystyle {\begin{aligned}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(3x^{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 (7z^{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 (7z^{3}-3zr^{2})}{r^{4}}}\\Y_{4,0}&=Y_{4}^{0}={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot {\frac {35z^{4}-30z^{2}r^{2}+3r^{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 (7z^{3}-3zr^{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 (7z^{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}-3y^{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}-3y^{2}\right)-y^{2}\left(3x^{2}-y^{2}\right)}{r^{4}}}\end{aligned}}} Visualization of real spherical harmonics 2D polar/azimuthal angle maps Below the real spherical harmonics are represented on 2D plots with the azimuthal angle, ϕ {\displaystyle \phi } θ {\displaystyle \theta }
Visual Array of Real Spherical Harmonics Represented as 2D Theta/Phi Maps Polar plots Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.
Visual Array of Real Spherical Harmonics Represented with Polar Plot Polar plots with magnitude as radius Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.
Visual Array of Real Spherical Harmonics Represented with Polar Plot with Magnitude Mapped to Radius Polar plots with amplitude as elevation Below the real spherical harmonics are represented on polar plots. The amplitude of the spherical harmonic (magnitude and sign) at a particular polar and azimuthal angle is represented by the elevation of the plot at that point above or below the surface of a uniform sphere. The magnitude is also represented by the saturation of the color at a given point. The phase is represented by the hue at a given point.
Visual Array of Real Spherical Harmonics Represented with Polar Plot with Amplitude Mapped to Elevation and Saturation See also External links Spherical Harmonics at MathWorld Spherical Harmonics 3D representation References Cited references ^ D. A. Varshalovich; A. N. Moskalev; V. K. Khersonskii (1988). Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols (1. repr. ed.). Singapore: World Scientific Pub. pp. 155–156. ISBN 9971-50-107-4 . ^ Petrucci (2016). General chemistry : principles and modern applications . Prentice Hall. ISBN 0133897311 . ^ Friedman (1964). "The shapes of the f orbitals". J. Chem. Educ . 41 (7): 354. ^ C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry . New York: Academic Press. ISBN 0-12-172950-8 . ^ Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 December 1997). "Evaluation of the rotation matrices in the basis of real spherical harmonics". Journal of Molecular Structure: THEOCHEM . 419 (1–3): 19–27. doi:10.1016/S0166-1280(97)00185-1. General references See section 3 in Mathar, R. J. (2009). "Zernike basis to cartesian transformations". Serbian Astronomical Journal . 179 (179): 107–120. arXiv:0809.2368 . Bibcode:2009SerAJ.179..107M. doi:10.2298/SAJ0979107M. For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi] at Wolfram Alpha, especially for specific values of l and m. 
