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Tabla de armónicos esféricos

Esta es una tabla de armónicos esféricos ortonormalizados que emplean la fase de Condon-Shortley hasta el grado ℓ = 10 {\displaystyle \ell =10} Algunas de estas fórmulas se expr...

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Esta es una tabla de armónicos esféricos ortonormalizados que emplean la fase de Condon-Shortley hasta el gradoℓ=10{\displaystyle \ell =10}Algunas de estas fórmulas se expresan en términos de la expansión cartesiana de los armónicos esféricos en polinomios en x , y , z y r . Para los fines de esta tabla, es útil expresar las transformaciones habituales de esféricas a cartesianas que relacionan estos componentes cartesianos conθ{\displaystyle \theta }yφ{\displaystyle \varphi }como

{porque⁡(θ)=z/rmi±iφ⋅pecado⁡(θ)=(incógnita±iy)/r{\displaystyle {\begin{cases}\cos(\theta )&=z/r\\e^{\pm i\varphi }\cdot \sin(\theta )&=(x\pm iy)/r\end{cases}}}

Armónicos esféricos complejos

Para ℓ = 0, …, 5, véase. [ 1 ]

ℓ = 0

Y00(θ,φ)=121π{\displaystyle Y_{0}^{0}(\theta,\varphi)={1 \over 2}{\sqrt {1 \over \pi }}}

ℓ = 1

Y1−1(θ,φ)=1232π⋅mi−iφ⋅pecado⁡θ=1232π⋅(incógnita−iy)rY10(θ,φ)=123π⋅porque⁡θ=123π⋅zrY11(θ,φ)=−1232π⋅miiφ⋅pecado⁡θ=−1232π⋅(incógnita+iy)r{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 2

Y2−2(θ,φ)=14152π⋅mi−2iφ⋅pecado2⁡θ=14152π⋅(incógnita−iy)2r2Y2−1(θ,φ)=12152π⋅mi−iφ⋅pecado⁡θ⋅porque⁡θ=12152π⋅(incógnita−iy)⋅zr2Y20(θ,φ)=145π⋅(3porque2⁡θ−1)=145π⋅(3z2−r2)r2Y21(θ,φ)=−12152π⋅miiφ⋅pecado⁡θ⋅porque⁡θ=−12152π⋅(incógnita+iy)⋅zr2Y22(θ,φ)=14152π⋅mi2iφ⋅pecado2⁡θ=14152π⋅(incógnita+iy)2r2{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 3

Y3−3(θ,φ)=1835π⋅mi−3iφ⋅pecado3⁡θ=1835π⋅(incógnita−iy)3r3Y3−2(θ,φ)=141052π⋅mi−2iφ⋅pecado2⁡θ⋅porque⁡θ=141052π⋅(incógnita−iy)2⋅zr3Y3−1(θ,φ)=1821π⋅mi−iφ⋅pecado⁡θ⋅(5porque2⁡θ−1)=1821π⋅(incógnita−iy)⋅(5z2−r2)r3Y30(θ,φ)=147π⋅(5porque3⁡θ−3porque⁡θ)=147π⋅(5z3−3zr2)r3Y31(θ,φ)=−1821π⋅miiφ⋅pecado⁡θ⋅(5porque2⁡θ−1)=−1821π⋅(incógnita+iy)⋅(5z2−r2)r3Y32(θ,φ)=141052π⋅mi2iφ⋅pecado2⁡θ⋅porque⁡θ=141052π⋅(incógnita+iy)2⋅zr3Y33(θ,φ)=−1835π⋅mi3iφ⋅pecado3⁡θ=−1835π⋅(incógnita+iy)3r3{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 4

Y4−4(θ,φ)=316352π⋅mi−4iφ⋅pecado4⁡θ=316352π⋅(incógnita−iy)4r4Y4−3(θ,φ)=3835π⋅mi−3iφ⋅pecado3⁡θ⋅porque⁡θ=3835π⋅(incógnita−iy)3zr4Y4−2(θ,φ)=3852π⋅mi−2iφ⋅pecado2⁡θ⋅(7porque2⁡θ−1)=3852π⋅(incógnita−iy)2⋅(7z2−r2)r4Y4−1(θ,φ)=385π⋅mi−iφ⋅pecado⁡θ⋅(7porque3⁡θ−3porque⁡θ)=385π⋅(incógnita−iy)⋅(7z3−3zr2)r4Y40(θ,φ)=3161π⋅(35porque4⁡θ−30porque2⁡θ+3)=3161π⋅(35z4−30z2r2+3r4)r4Y41(θ,φ)=−385π⋅miiφ⋅pecado⁡θ⋅(7porque3⁡θ−3porque⁡θ)=−385π⋅(incógnita+iy)⋅(7z3−3zr2)r4Y42(θ,φ)=3852π⋅mi2iφ⋅pecado2⁡θ⋅(7porque2⁡θ−1)=3852π⋅(incógnita+iy)2⋅(7z2−r2)r4Y43(θ,φ)=−3835π⋅mi3iφ⋅pecado3⁡θ⋅porque⁡θ=−3835π⋅(incógnita+iy)3zr4Y44(θ,φ)=316352π⋅mi4iφ⋅pecado4⁡θ=316352π⋅(incógnita+iy)4r4{\displaystyle {\begin{aligned}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-iy)^{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-iy)^{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-iy)^{2}\cdot (7z^{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-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 \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+iy)\cdot (7z^{3}-3zr^{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+iy)^{2}\cdot (7z^{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+iy)^{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+iy)^{4}}{r^{4}}}\end{aligned}}}

ℓ = 5

Y5−5(θ,φ)=33277π⋅mi−5iφ⋅pecado5⁡θY5−4(θ,φ)=3163852π⋅mi−4iφ⋅pecado4⁡θ⋅porque⁡θY5−3(θ,φ)=132385π⋅mi−3iφ⋅pecado3⁡θ⋅(9porque2⁡θ−1)Y5−2(θ,φ)=1811552π⋅mi−2iφ⋅pecado2⁡θ⋅(3porque3⁡θ−porque⁡θ)Y5−1(θ,φ)=1161652π⋅mi−iφ⋅pecado⁡θ⋅(21porque4⁡θ−14porque2⁡θ+1)Y50(θ,φ)=11611π⋅(63porque5⁡θ−70porque3⁡θ+15porque⁡θ)Y51(θ,φ)=−1161652π⋅miiφ⋅pecado⁡θ⋅(21porque4⁡θ−14porque2⁡θ+1)Y52(θ,φ)=1811552π⋅mi2iφ⋅pecado2⁡θ⋅(3porque3⁡θ−porque⁡θ)Y53(θ,φ)=−132385π⋅mi3iφ⋅pecado3⁡θ⋅(9porque2⁡θ−1)Y54(θ,φ)=3163852π⋅mi4iφ⋅pecado4⁡θ⋅porque⁡θY55(θ,φ)=−33277π⋅mi5iφ⋅pecado5⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 6

Y6−6(θ,φ)=1643003π⋅mi−6iφ⋅pecado6⁡θY6−5(θ,φ)=3321001π⋅mi−5iφ⋅pecado5⁡θ⋅porque⁡θY6−4(θ,φ)=332912π⋅mi−4iφ⋅pecado4⁡θ⋅(11porque2⁡θ−1)Y6−3(θ,φ)=1321365π⋅mi−3iφ⋅pecado3⁡θ⋅(11porque3⁡θ−3porque⁡θ)Y6−2(θ,φ)=1641365π⋅mi−2iφ⋅pecado2⁡θ⋅(33porque4⁡θ−18porque2⁡θ+1)Y6−1(θ,φ)=1162732π⋅mi−iφ⋅pecado⁡θ⋅(33porque5⁡θ−30porque3⁡θ+5porque⁡θ)Y60(θ,φ)=13213π⋅(231porque6⁡θ−315porque4⁡θ+105porque2⁡θ−5)Y61(θ,φ)=−1162732π⋅miiφ⋅pecado⁡θ⋅(33porque5⁡θ−30porque3⁡θ+5porque⁡θ)Y62(θ,φ)=1641365π⋅mi2iφ⋅pecado2⁡θ⋅(33porque4⁡θ−18porque2⁡θ+1)Y63(θ,φ)=−1321365π⋅mi3iφ⋅pecado3⁡θ⋅(11porque3⁡θ−3porque⁡θ)Y64(θ,φ)=332912π⋅mi4iφ⋅pecado4⁡θ⋅(11porque2⁡θ−1)Y65(θ,φ)=−3321001π⋅mi5iφ⋅pecado5⁡θ⋅porque⁡θY66(θ,φ)=1643003π⋅mi6iφ⋅pecado6⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 7

Y7−7(θ,φ)=3647152π⋅mi−7iφ⋅pecado7⁡θY7−6(θ,φ)=3645005π⋅mi−6iφ⋅pecado6⁡θ⋅porque⁡θY7−5(θ,φ)=3643852π⋅mi−5iφ⋅pecado5⁡θ⋅(13porque2⁡θ−1)Y7−4(θ,φ)=3323852π⋅mi−4iφ⋅pecado4⁡θ⋅(13porque3⁡θ−3porque⁡θ)Y7−3(θ,φ)=364352π⋅mi−3iφ⋅pecado3⁡θ⋅(143porque4⁡θ−66porque2⁡θ+3)Y7−2(θ,φ)=36435π⋅mi−2iφ⋅pecado2⁡θ⋅(143porque5⁡θ−110porque3⁡θ+15porque⁡θ)Y7−1(θ,φ)=1641052π⋅mi−iφ⋅pecado⁡θ⋅(429porque6⁡θ−495porque4⁡θ+135porque2⁡θ−5)Y70(θ,φ)=13215π⋅(429porque7⁡θ−693porque5⁡θ+315porque3⁡θ−35porque⁡θ)Y71(θ,φ)=−1641052π⋅miiφ⋅pecado⁡θ⋅(429porque6⁡θ−495porque4⁡θ+135porque2⁡θ−5)Y72(θ,φ)=36435π⋅mi2iφ⋅pecado2⁡θ⋅(143porque5⁡θ−110porque3⁡θ+15porque⁡θ)Y73(θ,φ)=−364352π⋅mi3iφ⋅pecado3⁡θ⋅(143porque4⁡θ−66porque2⁡θ+3)Y74(θ,φ)=3323852π⋅mi4iφ⋅pecado4⁡θ⋅(13porque3⁡θ−3porque⁡θ)Y75(θ,φ)=−3643852π⋅mi5iφ⋅pecado5⁡θ⋅(13porque2⁡θ−1)Y76(θ,φ)=3645005π⋅mi6iφ⋅pecado6⁡θ⋅porque⁡θY77(θ,φ)=−3647152π⋅mi7iφ⋅pecado7⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 8

Y8−8(θ,φ)=3256121552π⋅mi−8iφ⋅pecado8⁡θY8−7(θ,φ)=364121552π⋅mi−7iφ⋅pecado7⁡θ⋅porque⁡θY8−6(θ,φ)=11287293π⋅mi−6iφ⋅pecado6⁡θ⋅(15porque2⁡θ−1)Y8−5(θ,φ)=364170172π⋅mi−5iφ⋅pecado5⁡θ⋅(5porque3⁡θ−porque⁡θ)Y8−4(θ,φ)=312813092π⋅mi−4iφ⋅pecado4⁡θ⋅(65porque4⁡θ−26porque2⁡θ+1)Y8−3(θ,φ)=164196352π⋅mi−3iφ⋅pecado3⁡θ⋅(39porque5⁡θ−26porque3⁡θ+3porque⁡θ)Y8−2(θ,φ)=3128595π⋅mi−2iφ⋅pecado2⁡θ⋅(143porque6⁡θ−143porque4⁡θ+33porque2⁡θ−1)Y8−1(θ,φ)=364172π⋅mi−iφ⋅pecado⁡θ⋅(715porque7⁡θ−1001porque5⁡θ+385porque3⁡θ−35porque⁡θ)Y80(θ,φ)=125617π⋅(6435porque8⁡θ−12012porque6⁡θ+6930porque4⁡θ−1260porque2⁡θ+35)Y81(θ,φ)=−364172π⋅miiφ⋅pecado⁡θ⋅(715porque7⁡θ−1001porque5⁡θ+385porque3⁡θ−35porque⁡θ)Y82(θ,φ)=3128595π⋅mi2iφ⋅pecado2⁡θ⋅(143porque6⁡θ−143porque4⁡θ+33porque2⁡θ−1)Y83(θ,φ)=−164196352π⋅mi3iφ⋅pecado3⁡θ⋅(39porque5⁡θ−26porque3⁡θ+3porque⁡θ)Y84(θ,φ)=312813092π⋅mi4iφ⋅pecado4⁡θ⋅(65porque4⁡θ−26porque2⁡θ+1)Y85(θ,φ)=−364170172π⋅mi5iφ⋅pecado5⁡θ⋅(5porque3⁡θ−porque⁡θ)Y86(θ,φ)=11287293π⋅mi6iφ⋅pecado6⁡θ⋅(15porque2⁡θ−1)Y87(θ,φ)=−364121552π⋅mi7iφ⋅pecado7⁡θ⋅porque⁡θY88(θ,φ)=3256121552π⋅mi8iφ⋅pecado8⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 9

Y9−9(θ,φ)=1512230945π⋅mi−9iφ⋅pecado9⁡θY9−8(θ,φ)=32562309452π⋅mi−8iφ⋅pecado8⁡θ⋅porque⁡θY9−7(θ,φ)=351213585π⋅mi−7iφ⋅pecado7⁡θ⋅(17porque2⁡θ−1)Y9−6(θ,φ)=112840755π⋅mi−6iφ⋅pecado6⁡θ⋅(17porque3⁡θ−3porque⁡θ)Y9−5(θ,φ)=32562717π⋅mi−5iφ⋅pecado5⁡θ⋅(85porque4⁡θ−30porque2⁡θ+1)Y9−4(θ,φ)=3128950952π⋅mi−4iφ⋅pecado4⁡θ⋅(17porque5⁡θ−10porque3⁡θ+porque⁡θ)Y9−3(θ,φ)=125621945π⋅mi−3iφ⋅pecado3⁡θ⋅(221porque6⁡θ−195porque4⁡θ+39porque2⁡θ−1)Y9−2(θ,φ)=31281045π⋅mi−2iφ⋅pecado2⁡θ⋅(221porque7⁡θ−273porque5⁡θ+91porque3⁡θ−7porque⁡θ)Y9−1(θ,φ)=3256952π⋅mi−iφ⋅pecado⁡θ⋅(2431porque8⁡θ−4004porque6⁡θ+2002porque4⁡θ−308porque2⁡θ+7)Y90(θ,φ)=125619π⋅(12155porque9⁡θ−25740porque7⁡θ+18018porque5⁡θ−4620porque3⁡θ+315porque⁡θ)Y91(θ,φ)=−3256952π⋅miiφ⋅pecado⁡θ⋅(2431porque8⁡θ−4004porque6⁡θ+2002porque4⁡θ−308porque2⁡θ+7)Y92(θ,φ)=31281045π⋅mi2iφ⋅pecado2⁡θ⋅(221porque7⁡θ−273porque5⁡θ+91porque3⁡θ−7porque⁡θ)Y93(θ,φ)=−125621945π⋅mi3iφ⋅pecado3⁡θ⋅(221porque6⁡θ−195porque4⁡θ+39porque2⁡θ−1)Y94(θ,φ)=3128950952π⋅mi4iφ⋅pecado4⁡θ⋅(17porque5⁡θ−10porque3⁡θ+porque⁡θ)Y95(θ,φ)=−32562717π⋅mi5iφ⋅pecado5⁡θ⋅(85porque4⁡θ−30porque2⁡θ+1)Y96(θ,φ)=112840755π⋅mi6iφ⋅pecado6⁡θ⋅(17porque3⁡θ−3porque⁡θ)Y97(θ,φ)=−351213585π⋅mi7iφ⋅pecado7⁡θ⋅(17porque2⁡θ−1)Y98(θ,φ)=32562309452π⋅mi8iφ⋅pecado8⁡θ⋅porque⁡θY99(θ,φ)=−1512230945π⋅mi9iφ⋅pecado9⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

ℓ = 10

Y10−10(θ,φ)=11024969969π⋅mi−10iφ⋅pecado10⁡θY10−9(θ,φ)=15124849845π⋅mi−9iφ⋅pecado9⁡θ⋅porque⁡θY10−8(θ,φ)=15122552552π⋅mi−8iφ⋅pecado8⁡θ⋅(19porque2⁡θ−1)Y10−7(θ,φ)=351285085π⋅mi−7iφ⋅pecado7⁡θ⋅(19porque3⁡θ−3porque⁡θ)Y10−6(θ,φ)=310245005π⋅mi−6iφ⋅pecado6⁡θ⋅(323porque4⁡θ−102porque2⁡θ+3)Y10−5(θ,φ)=32561001π⋅mi−5iφ⋅pecado5⁡θ⋅(323porque5⁡θ−170porque3⁡θ+15porque⁡θ)Y10−4(θ,φ)=325650052π⋅mi−4iφ⋅pecado4⁡θ⋅(323porque6⁡θ−255porque4⁡θ+45porque2⁡θ−1)Y10−3(θ,φ)=32565005π⋅mi−3iφ⋅pecado3⁡θ⋅(323porque7⁡θ−357porque5⁡θ+105porque3⁡θ−7porque⁡θ)Y10−2(θ,φ)=35123852π⋅mi−2iφ⋅pecado2⁡θ⋅(4199porque8⁡θ−6188porque6⁡θ+2730porque4⁡θ−364porque2⁡θ+7)Y10−1(θ,φ)=125611552π⋅mi−iφ⋅pecado⁡θ⋅(4199porque9⁡θ−7956porque7⁡θ+4914porque5⁡θ−1092porque3⁡θ+63porque⁡θ)Y100(θ,φ)=151221π⋅(46189porque10⁡θ−109395porque8⁡θ+90090porque6⁡θ−30030porque4⁡θ+3465porque2⁡θ−63)Y101(θ,φ)=−125611552π⋅miiφ⋅pecado⁡θ⋅(4199porque9⁡θ−7956porque7⁡θ+4914porque5⁡θ−1092porque3⁡θ+63porque⁡θ)Y102(θ,φ)=35123852π⋅mi2iφ⋅pecado2⁡θ⋅(4199porque8⁡θ−6188porque6⁡θ+2730porque4⁡θ−364porque2⁡θ+7)Y103(θ,φ)=−32565005π⋅mi3iφ⋅pecado3⁡θ⋅(323porque7⁡θ−357porque5⁡θ+105porque3⁡θ−7porque⁡θ)Y104(θ,φ)=325650052π⋅mi4iφ⋅pecado4⁡θ⋅(323porque6⁡θ−255porque4⁡θ+45porque2⁡θ−1)Y105(θ,φ)=−32561001π⋅mi5iφ⋅pecado5⁡θ⋅(323porque5⁡θ−170porque3⁡θ+15porque⁡θ)Y106(θ,φ)=310245005π⋅mi6iφ⋅pecado6⁡θ⋅(323porque4⁡θ−102porque2⁡θ+3)Y107(θ,φ)=−351285085π⋅mi7iφ⋅pecado7⁡θ⋅(19porque3⁡θ−3porque⁡θ)Y108(θ,φ)=15122552552π⋅mi8iφ⋅pecado8⁡θ⋅(19porque2⁡θ−1)Y109(θ,φ)=−15124849845π⋅mi9iφ⋅pecado9⁡θ⋅porque⁡θY1010(θ,φ)=11024969969π⋅mi10iφ⋅pecado10⁡θ{\displaystyle {\begin{aligned}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{aligned}}}

Visualización de armónicos esféricos complejos

Mapas de ángulo polar/azimutal en 2D

A continuación se representan los armónicos esféricos complejos en gráficos 2D con el ángulo azimutal,ϕ{\displaystyle \phi }, en el eje horizontal y el ángulo polar,θ{\displaystyle \theta }, en el eje vertical. La saturación del color en cualquier punto representa la magnitud del armónico esférico y el tono representa la fase.

Las líneas nodales de latitud se visualizan como líneas blancas horizontales. Las líneas nodales de longitud se visualizan como líneas blancas verticales.

Representación visual de armónicos esféricos complejos como mapas theta/phi bidimensionales.

Diagramas polares

A continuación se representan los armónicos esféricos complejos en diagramas polares. La magnitud del armónico esférico en ángulos polares y azimutales específicos se representa mediante la saturación del color en ese punto, y la fase se representa mediante el tono en ese punto.

Representación visual de armónicos esféricos complejos mediante un diagrama polar.

Gráficos polares con magnitud como radio

A continuación se representan los armónicos esféricos complejos en diagramas polares. La magnitud del armónico esférico en ángulos polares y azimutales específicos se representa mediante el radio del diagrama en ese punto, y la fase se representa mediante el tono en ese punto.

Representación visual de armónicos esféricos complejos mediante un gráfico polar con magnitud mapeada al radio.

Armónicos esféricos reales

Para cada armónico esférico real, también se informa el símbolo del orbital atómico correspondiente ( s , p , d , f ). [ 2 ] [ 3 ]

Para ℓ = 0, …, 3, véase. [ 4 ] [ 5 ]

ℓ = 0

Y0,0=s=Y00=121π{\displaystyle Y_{0,0}=s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}

ℓ = 1

Y1,−1=pagy=i12(Y1−1+Y11)=34π⋅yr=34πpecado⁡(θ)pecado⁡(φ)Y1,0=pagz=Y10=34π⋅zr=34πporque⁡(θ)Y1,1=pagincógnita=12(Y1−1−Y11)=34π⋅incógnitar=34πpecado⁡(θ)porque⁡(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

Y2,−2=dincógnitay=i12(Y2−2−Y22)=1215π⋅incógnitayr2=1415πpecado2⁡(θ)pecado⁡(2φ)Y2,−1=dyz=i12(Y2−1+Y21)=1215π⋅y⋅zr2=1415πpecado⁡(2θ)pecado⁡(φ)Y2,0=dz2=Y20=145π⋅3z2−r2r2=145π(3porque2⁡(θ)−1)Y2,1=dincógnitaz=12(Y2−1−Y21)=1215π⋅incógnita⋅zr2=1415πpecado⁡(2θ)porque⁡(φ)Y2,2=dincógnita2−y2=12(Y2−2+Y22)=1415π⋅incógnita2−y2r2=1415πpecado2⁡(θ)porque⁡(2φ){\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

Y3,−3=Fy(3incógnita2−y2)=i12(Y3−3+Y33)=14352π⋅y(3incógnita2−y2)r3Y3,−2=Fincógnitayz=i12(Y3−2−Y32)=12105π⋅incógnitay⋅zr3Y3,−1=Fyz2=i12(Y3−1+Y31)=14212π⋅y⋅(5z2−r2)r3Y3,0=Fz3=Y30=147π⋅5z3−3zr2r3Y3,1=Fincógnitaz2=12(Y3−1−Y31)=14212π⋅incógnita⋅(5z2−r2)r3Y3,2=Fz(incógnita2−y2)=12(Y3−2+Y32)=14105π⋅(incógnita2−y2)⋅zr3Y3,3=Fincógnita(incógnita2−3y2)=12(Y3−3−Y33)=14352π⋅incógnita(incógnita2−3y2)r3{\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

Y4,−4=i12(Y4−4−Y44)=3435π⋅incógnitay(incógnita2−y2)r4Y4,−3=i12(Y4−3+Y43)=34352π⋅y(3incógnita2−y2)⋅zr4Y4,−2=i12(Y4−2−Y42)=345π⋅incógnitay⋅(7z2−r2)r4Y4,−1=i12(Y4−1+Y41)=3452π⋅y⋅(7z3−3zr2)r4Y4,0=Y40=3161π⋅35z4−30z2r2+3r4r4Y4,1=12(Y4−1−Y41)=3452π⋅incógnita⋅(7z3−3zr2)r4Y4,2=12(Y4−2+Y42)=385π⋅(incógnita2−y2)⋅(7z2−r2)r4Y4,3=12(Y4−3−Y43)=34352π⋅incógnita(incógnita2−3y2)⋅zr4Y4,4=12(Y4−4+Y44)=31635π⋅incógnita2(incógnita2−3y2)−y2(3incógnita2−y2)r4{\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}}}

Visualización de armónicos esféricos reales

Mapas de ángulo polar/azimutal en 2D

A continuación se representan los armónicos esféricos reales en gráficos 2D con el ángulo azimutal,ϕ{\displaystyle \phi }, en el eje horizontal y el ángulo polar,θ{\displaystyle \theta }, en el eje vertical. La saturación del color en cualquier punto representa la magnitud del armónico esférico. Los valores positivos son rojos y los negativos son verde azulado.

Las líneas nodales de latitud se visualizan como líneas blancas horizontales. Las líneas nodales de longitud se visualizan como líneas blancas verticales.

Representación visual de armónicos esféricos reales como mapas theta/phi bidimensionales.

Diagramas polares

A continuación se representan los armónicos esféricos reales en diagramas polares. La magnitud del armónico esférico en ángulos polares y azimutales específicos se representa mediante la saturación del color en ese punto, y la fase se representa mediante el tono en ese punto.

Representación visual de armónicos esféricos reales mediante un diagrama polar.

Gráficos polares con magnitud como radio

A continuación se representan los armónicos esféricos reales en diagramas polares. La magnitud del armónico esférico en ángulos polares y azimutales específicos se representa mediante el radio del diagrama en ese punto, y la fase se representa mediante el tono en ese punto.

Representación visual de armónicos esféricos reales mediante un diagrama polar con la magnitud mapeada al radio.

Gráficos polares con amplitud como elevación

A continuación se muestran los armónicos esféricos reales representados en diagramas polares. La amplitud del armónico esférico (magnitud y signo) en un ángulo polar y azimutal determinado se representa mediante la elevación del diagrama en ese punto, por encima o por debajo de la superficie de una esfera uniforme. La magnitud también se representa mediante la saturación del color en un punto dado. La fase se representa mediante el tono en un punto dado.

Representación visual de armónicos esféricos reales mediante un diagrama polar con amplitud mapeada a elevación y saturación.

Véase también

  • Armónicos esféricos en MathWorld
  • Representación 3D de armónicos esféricos

Referencias

Referencias citadas

  1. ↑ DA Varshalovich; AN Moskalev; VK Khersonskii (1988). Teoría cuántica del momento angular  : tensores irreducibles, armónicos esféricos, coeficientes de acoplamiento vectorial, símbolos 3nj (1.ª ed. reimpr  .). Singapur: World Scientific Pub. pp. 155–156 . ISBN  9971-50-107-4.
  2. ↑ Petrucci (2016). Química general : principios y aplicaciones modernas . Prentice Hall. ISBN  0133897311.
  3. ↑ Friedman (1964). "Las formas de los orbitales f". J. Chem. Educ . 41 (7): 354.
  4. ↑ CDH Chisholm (1976). Técnicas de teoría de grupos en química cuántica . Nueva York: Academic Press. ISBN 0-12-172950-8.
  5. ↑ Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 de diciembre de 1997). "Evaluación de las matrices de rotación en la base de armónicos esféricos reales". Journal of Molecular Structure: THEOCHEM . 419 ( 1–3 ): 19–27 . doi : 10.1016/S0166-1280(97)00185-1 .

Referencias generales

  • Véase la sección 3 en Mathar, RJ (2009). "Base de Zernike para transformaciones cartesianas". Revista Astronómica Serbia . 179 (179): 107– 120. arXiv : 0809.2368 . Bibcode : 2009SerAJ.179..107M . doi : 10.2298/SAJ0979107M .(véase la sección 3.3)
  • Para armónicos esféricos complejos, consulte también SphericalHarmonicY[l,m,theta,phi ] en Wolfram Alpha , especialmente para valores específicos de l y m.