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{{Short description|Mathematical table}}This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree ell = 10. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in 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 theta and varphi as begin{cases}cos(theta) & = z/re^{pm ivarphi} cdot sin(theta) & = (x pm iy)/rend{cases}

Complex spherical harmonics

For â„“ = 0, …, 5, see.BOOK, Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, 1988, World Scientific Pub., Singapore, 9971-50-107-4, D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, 1. repr., 155–156, â„“ 0“>

â„“ 0

Y_{0}^{0}(theta,varphi)={1over 2}sqrt{1over pi}â„“ 1“>

â„“ 1

Y_{1}^{-1}(theta,varphi) &= & & {1over 2}sqrt{3over 2pi}cdot e^{-ivarphi}cdotsintheta & &= & &{1over 2}sqrt{3over 2pi} cdot{(x-iy)over r} Y_{1}^{ 0}(theta,varphi) &= & & {1over 2}sqrt{3over pi}cdot costheta & &= & &{1over 2}sqrt{3over pi} cdot{zover r} Y_{1}^{ 1}(theta,varphi) &= &-& {1over 2}sqrt{3over 2pi}cdot e^{ivarphi}cdot sintheta & &= &-&{1over 2}sqrt{3over 2pi} cdot{(x+iy)over r}end{align} â„“ 2“>

â„“ 2

Y_{2}^{-2}(theta,varphi)&=& &{1over 4}sqrt{15over 2pi}cdot e^{-2ivarphi}cdotsin^{2}thetaquad &&=& &{1over 4}sqrt{15over 2pi}cdot{(x - iy)^2 over r^{2}}&Y_{2}^{-1}(theta,varphi)&=& &{1over 2}sqrt{15over 2pi}cdot e^{-ivarphi}cdotsin thetacdot costhetaquad &&=& &{1over 2}sqrt{15over 2pi}cdot{(x - iy) cdot z over r^{2}}&Y_{2}^{ 0}(theta,varphi)&=& &{1over 4}sqrt{ 5over pi}cdot (3cos^{2}theta-1)quad&&=& &{1over 4}sqrt{ 5over pi}cdot{(3z^{2}-r^{2})over r^{2}}&Y_{2}^{ 1}(theta,varphi)&=&-&{1over 2}sqrt{15over 2pi}cdot e^{ ivarphi}cdotsin thetacdot costhetaquad &&=&-&{1over 2}sqrt{15over 2pi}cdot{(x + iy) cdot z over r^{2}}&Y_{2}^{ 2}(theta,varphi)&=& &{1over 4}sqrt{15over 2pi}cdot e^{ 2ivarphi}cdotsin^{2}thetaquad &&=& &{1over 4}sqrt{15over 2pi}cdot{(x + iy)^2 over r^{2}}&end{align}â„“ 3“>

â„“ 3

Y_{3}^{-3}(theta,varphi) Y_{3}^{-2}(theta,varphi) Y_{3}^{-1}(theta,varphi) Y_{3}^{ 0}(theta,varphi) Y_{3}^{ 1}(theta,varphi) Y_{3}^{ 2}(theta,varphi) Y_{3}^{ 3}(theta,varphi) â„“ 4“>

â„“ 4

Y_{4}^{-4}(theta,varphi)&={ 3over 16}sqrt{35over 2pi}cdot e^{-4ivarphi}cdotsin^{4}theta= frac{3}{16} sqrt{frac{35}{2 pi}} cdot frac{(x - i y)^4}{r^4}Y_{4}^{-3}(theta,varphi)&={ 3over 8}sqrt{35over pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdotcostheta= frac{3}{8} sqrt{frac{35}{pi}} cdot frac{(x - i y)^3 z}{r^4}Y_{4}^{-2}(theta,varphi)&={ 3over 8}sqrt{ 5over 2pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(7cos^{2}theta-1)= frac{3}{8} sqrt{frac{5}{2 pi}} cdot frac{(x - i y)^2 cdot (7 z^2 - r^2)}{r^4}Y_{4}^{-1}(theta,varphi)&={ 3over 8}sqrt{ 5over pi}cdot e^{- ivarphi}cdotsinthetacdot(7cos^{3}theta-3costheta)= frac{3}{8} sqrt{frac{5}{pi}} cdot frac{(x - i y) cdot (7 z^3 - 3 z r^2)}{r^4}Y_{4}^{ 0}(theta,varphi)&={ 3over 16}sqrt{ 1over pi}cdot(35cos^{4}theta-30cos^{2}theta+3)= frac{3}{16} sqrt{frac{1}{pi}} cdot frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}Y_{4}^{ 1}(theta,varphi)&={-3over 8}sqrt{ 5over pi}cdot e^{ ivarphi}cdotsinthetacdot(7cos^{3}theta-3costheta)= frac{- 3}{8} sqrt{frac{5}{pi}} cdot frac{(x + i y) cdot (7 z^3 - 3 z r^2)}{r^4}Y_{4}^{ 2}(theta,varphi)&={ 3over 8}sqrt{ 5over 2pi}cdot e^{ 2ivarphi}cdotsin^{2}thetacdot(7cos^{2}theta-1)= frac{3}{8} sqrt{frac{5}{2 pi}} cdot frac{(x + i y)^2 cdot (7 z^2 - r^2)}{r^4}Y_{4}^{ 3}(theta,varphi)&={-3over 8}sqrt{35over pi}cdot e^{ 3ivarphi}cdotsin^{3}thetacdotcostheta= frac{- 3}{8} sqrt{frac{35}{pi}} cdot frac{(x + i y)^3 z}{r^4}Y_{4}^{ 4}(theta,varphi)&={ 3over 16}sqrt{35over 2pi}cdot e^{ 4ivarphi}cdotsin^{4}theta= frac{3}{16} sqrt{frac{35}{2 pi}} cdot frac{(x + i y)^4}{r^4}end{align} â„“ 5“>

â„“ 5

Y_{5}^{-5}(theta,varphi)&={ 3over 32}sqrt{ 77over pi}cdot e^{-5ivarphi}cdotsin^{5}thetaY_{5}^{-4}(theta,varphi)&={ 3over 16}sqrt{ 385over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdotcosthetaY_{5}^{-3}(theta,varphi)&={ 1over 32}sqrt{ 385over pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(9cos^{2}theta-1)Y_{5}^{-2}(theta,varphi)&={ 1over 8}sqrt{1155over 2pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(3cos^{3}theta-costheta)Y_{5}^{-1}(theta,varphi)&={ 1over 16}sqrt{ 165over 2pi}cdot e^{- ivarphi}cdotsin thetacdot(21cos^{4}theta-14cos^{2}theta+1)Y_{5}^{ 0}(theta,varphi)&={ 1over 16}sqrt{ 11over pi}cdot (63cos^{5}theta-70cos^{3}theta+15costheta)Y_{5}^{ 1}(theta,varphi)&={-1over 16}sqrt{ 165over 2pi}cdot e^{ ivarphi}cdotsin thetacdot(21cos^{4}theta-14cos^{2}theta+1)Y_{5}^{ 2}(theta,varphi)&={ 1over 8}sqrt{1155over 2pi}cdot e^{ 2ivarphi}cdotsin^{2}thetacdot(3cos^{3}theta-costheta)Y_{5}^{ 3}(theta,varphi)&={-1over 32}sqrt{ 385over pi}cdot e^{ 3ivarphi}cdotsin^{3}thetacdot(9cos^{2}theta-1)Y_{5}^{ 4}(theta,varphi)&={ 3over 16}sqrt{ 385over 2pi}cdot e^{ 4ivarphi}cdotsin^{4}thetacdotcosthetaY_{5}^{ 5}(theta,varphi)&={-3over 32}sqrt{ 77over pi}cdot e^{ 5ivarphi}cdotsin^{5}theta â„“ 6“>

â„“ 6

Y_{6}^{-6}(theta,varphi)&= {1over 64}sqrt{3003over pi}cdot e^{-6ivarphi}cdotsin^{6}thetaY_{6}^{-5}(theta,varphi)&= {3over 32}sqrt{1001over pi}cdot e^{-5ivarphi}cdotsin^{5}thetacdotcosthetaY_{6}^{-4}(theta,varphi)&= {3over 32}sqrt{ 91over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdot(11cos^{2}theta-1)Y_{6}^{-3}(theta,varphi)&= {1over 32}sqrt{1365over pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(11cos^{3}theta-3costheta)Y_{6}^{-2}(theta,varphi)&= {1over 64}sqrt{1365over pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(33cos^{4}theta-18cos^{2}theta+1)Y_{6}^{-1}(theta,varphi)&= {1over 16}sqrt{ 273over 2pi}cdot e^{- ivarphi}cdotsin thetacdot(33cos^{5}theta-30cos^{3}theta+5costheta)Y_{6}^{ 0}(theta,varphi)&= {1over 32}sqrt{ 13over pi}cdot (231cos^{6}theta-315cos^{4}theta+105cos^{2}theta-5)Y_{6}^{ 1}(theta,varphi)&=-{1over 16}sqrt{ 273over 2pi}cdot e^{ ivarphi}cdotsin thetacdot(33cos^{5}theta-30cos^{3}theta+5costheta)Y_{6}^{ 2}(theta,varphi)&= {1over 64}sqrt{1365over pi}cdot e^{ 2ivarphi}cdotsin^{2}thetacdot(33cos^{4}theta-18cos^{2}theta+1)Y_{6}^{ 3}(theta,varphi)&=-{1over 32}sqrt{1365over pi}cdot e^{ 3ivarphi}cdotsin^{3}thetacdot(11cos^{3}theta-3costheta)Y_{6}^{ 4}(theta,varphi)&= {3over 32}sqrt{ 91over 2pi}cdot e^{ 4ivarphi}cdotsin^{4}thetacdot(11cos^{2}theta-1)Y_{6}^{ 5}(theta,varphi)&=-{3over 32}sqrt{1001over pi}cdot e^{ 5ivarphi}cdotsin^{5}thetacdotcosthetaY_{6}^{ 6}(theta,varphi)&= {1over 64}sqrt{3003over pi}cdot e^{ 6ivarphi}cdotsin^{6}theta â„“ 7“>

â„“ 7

Y_{7}^{-7}(theta,varphi)&= {3over 64}sqrt{ 715over 2pi}cdot e^{-7ivarphi}cdotsin^{7}thetaY_{7}^{-6}(theta,varphi)&= {3over 64}sqrt{5005over pi}cdot e^{-6ivarphi}cdotsin^{6}thetacdotcosthetaY_{7}^{-5}(theta,varphi)&= {3over 64}sqrt{ 385over 2pi}cdot e^{-5ivarphi}cdotsin^{5}thetacdot(13cos^{2}theta-1)Y_{7}^{-4}(theta,varphi)&= {3over 32}sqrt{ 385over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdot(13cos^{3}theta-3costheta)Y_{7}^{-3}(theta,varphi)&= {3over 64}sqrt{ 35over 2pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(143cos^{4}theta-66cos^{2}theta+3)Y_{7}^{-2}(theta,varphi)&= {3over 64}sqrt{ 35over pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(143cos^{5}theta-110cos^{3}theta+15costheta)Y_{7}^{-1}(theta,varphi)&= {1over 64}sqrt{ 105over 2pi}cdot e^{- ivarphi}cdotsin thetacdot(429cos^{6}theta-495cos^{4}theta+135cos^{2}theta-5)Y_{7}^{ 0}(theta,varphi)&= {1over 32}sqrt{ 15over pi}cdot (429cos^{7}theta-693cos^{5}theta+315cos^{3}theta-35costheta)Y_{7}^{ 1}(theta,varphi)&=-{1over 64}sqrt{ 105over 2pi}cdot e^{ ivarphi}cdotsin thetacdot(429cos^{6}theta-495cos^{4}theta+135cos^{2}theta-5)Y_{7}^{ 2}(theta,varphi)&= {3over 64}sqrt{ 35over pi}cdot e^{ 2ivarphi}cdotsin^{2}thetacdot(143cos^{5}theta-110cos^{3}theta+15costheta)Y_{7}^{ 3}(theta,varphi)&=-{3over 64}sqrt{ 35over 2pi}cdot e^{ 3ivarphi}cdotsin^{3}thetacdot(143cos^{4}theta-66cos^{2}theta+3)Y_{7}^{ 4}(theta,varphi)&= {3over 32}sqrt{ 385over 2pi}cdot e^{ 4ivarphi}cdotsin^{4}thetacdot(13cos^{3}theta-3costheta)Y_{7}^{ 5}(theta,varphi)&=-{3over 64}sqrt{ 385over 2pi}cdot e^{ 5ivarphi}cdotsin^{5}thetacdot(13cos^{2}theta-1)Y_{7}^{ 6}(theta,varphi)&= {3over 64}sqrt{5005over pi}cdot e^{ 6ivarphi}cdotsin^{6}thetacdotcosthetaY_{7}^{ 7}(theta,varphi)&=-{3over 64}sqrt{ 715over 2pi}cdot e^{ 7ivarphi}cdotsin^{7}thetaend{align}â„“ 8“>

â„“ 8

Y_{8}^{-8}(theta,varphi)&={ 3over 256}sqrt{12155over 2pi}cdot e^{-8ivarphi}cdotsin^{8}thetaY_{8}^{-7}(theta,varphi)&={ 3over 64}sqrt{12155over 2pi}cdot e^{-7ivarphi}cdotsin^{7}thetacdotcosthetaY_{8}^{-6}(theta,varphi)&={ 1over 128}sqrt{7293over pi}cdot e^{-6ivarphi}cdotsin^{6}thetacdot(15cos^{2}theta-1)Y_{8}^{-5}(theta,varphi)&={ 3over 64}sqrt{17017over 2pi}cdot e^{-5ivarphi}cdotsin^{5}thetacdot(5cos^{3}theta-costheta)Y_{8}^{-4}(theta,varphi)&={ 3over 128}sqrt{1309over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdot(65cos^{4}theta-26cos^{2}theta+1)Y_{8}^{-3}(theta,varphi)&={ 1over 64}sqrt{19635over 2pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(39cos^{5}theta-26cos^{3}theta+3costheta)Y_{8}^{-2}(theta,varphi)&={ 3over 128}sqrt{595over pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(143cos^{6}theta-143cos^{4}theta+33cos^{2}theta-1)Y_{8}^{-1}(theta,varphi)&={ 3over 64}sqrt{17over 2pi}cdot e^{-ivarphi}cdotsinthetacdot(715cos^{7}theta-1001cos^{5}theta+385cos^{3}theta-35costheta)Y_{8}^{ 0}(theta,varphi)&={ 1over 256}sqrt{17over pi}cdot(6435cos^{8}theta-12012cos^{6}theta+6930cos^{4}theta-1260cos^{2}theta+35)Y_{8}^{ 1}(theta,varphi)&={-3over 64}sqrt{17over 2pi}cdot e^{ivarphi}cdotsinthetacdot(715cos^{7}theta-1001cos^{5}theta+385cos^{3}theta-35costheta)Y_{8}^{ 2}(theta,varphi)&={ 3over 128}sqrt{595over pi}cdot e^{2ivarphi}cdotsin^{2}thetacdot(143cos^{6}theta-143cos^{4}theta+33cos^{2}theta-1)Y_{8}^{ 3}(theta,varphi)&={-1over 64}sqrt{19635over 2pi}cdot e^{3ivarphi}cdotsin^{3}thetacdot(39cos^{5}theta-26cos^{3}theta+3costheta)Y_{8}^{ 4}(theta,varphi)&={ 3over 128}sqrt{1309over 2pi}cdot e^{4ivarphi}cdotsin^{4}thetacdot(65cos^{4}theta-26cos^{2}theta+1)Y_{8}^{ 5}(theta,varphi)&={-3over 64}sqrt{17017over 2pi}cdot e^{5ivarphi}cdotsin^{5}thetacdot(5cos^{3}theta-costheta)Y_{8}^{ 6}(theta,varphi)&={ 1over 128}sqrt{7293over pi}cdot e^{6ivarphi}cdotsin^{6}thetacdot(15cos^{2}theta-1)Y_{8}^{ 7}(theta,varphi)&={-3over 64}sqrt{12155over 2pi}cdot e^{7ivarphi}cdotsin^{7}thetacdotcosthetaY_{8}^{ 8}(theta,varphi)&={ 3over 256}sqrt{12155over 2pi}cdot e^{8ivarphi}cdotsin^{8}thetaend{align}â„“ 9“>

â„“ 9

Y_{9}^{-9}(theta,varphi)&={ 1over 512}sqrt{230945over pi}cdot e^{-9ivarphi}cdotsin^{9}thetaY_{9}^{-8}(theta,varphi)&={ 3over 256}sqrt{230945over 2pi}cdot e^{-8ivarphi}cdotsin^{8}thetacdotcosthetaY_{9}^{-7}(theta,varphi)&={ 3over 512}sqrt{ 13585over pi}cdot e^{-7ivarphi}cdotsin^{7}thetacdot(17cos^{2}theta-1)Y_{9}^{-6}(theta,varphi)&={ 1over 128}sqrt{ 40755over pi}cdot e^{-6ivarphi}cdotsin^{6}thetacdot(17cos^{3}theta-3costheta)Y_{9}^{-5}(theta,varphi)&={ 3over 256}sqrt{ 2717over pi}cdot e^{-5ivarphi}cdotsin^{5}thetacdot(85cos^{4}theta-30cos^{2}theta+1)Y_{9}^{-4}(theta,varphi)&={ 3over 128}sqrt{ 95095over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdot(17cos^{5}theta-10cos^{3}theta+costheta)Y_{9}^{-3}(theta,varphi)&={ 1over 256}sqrt{ 21945over pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(221cos^{6}theta-195cos^{4}theta+39cos^{2}theta-1)Y_{9}^{-2}(theta,varphi)&={ 3over 128}sqrt{ 1045over pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(221cos^{7}theta-273cos^{5}theta+91cos^{3}theta-7costheta)Y_{9}^{-1}(theta,varphi)&={ 3over 256}sqrt{ 95over 2pi}cdot e^{- ivarphi}cdotsin thetacdot(2431cos^{8}theta-4004cos^{6}theta+2002cos^{4}theta-308cos^{2}theta+7)Y_{9}^{ 0}(theta,varphi)&={ 1over 256}sqrt{ 19over pi}cdot (12155cos^{9}theta-25740cos^{7}theta+18018cos^{5}theta-4620cos^{3}theta+315costheta)Y_{9}^{ 1}(theta,varphi)&={-3over 256}sqrt{ 95over 2pi}cdot e^{ ivarphi}cdotsin thetacdot(2431cos^{8}theta-4004cos^{6}theta+2002cos^{4}theta-308cos^{2}theta+7)Y_{9}^{ 2}(theta,varphi)&={ 3over 128}sqrt{ 1045over pi}cdot e^{ 2ivarphi}cdotsin^{2}thetacdot(221cos^{7}theta-273cos^{5}theta+91cos^{3}theta-7costheta)Y_{9}^{ 3}(theta,varphi)&={-1over 256}sqrt{ 21945over pi}cdot e^{ 3ivarphi}cdotsin^{3}thetacdot(221cos^{6}theta-195cos^{4}theta+39cos^{2}theta-1)Y_{9}^{ 4}(theta,varphi)&={ 3over 128}sqrt{ 95095over 2pi}cdot e^{ 4ivarphi}cdotsin^{4}thetacdot(17cos^{5}theta-10cos^{3}theta+costheta)Y_{9}^{ 5}(theta,varphi)&={-3over 256}sqrt{ 2717over pi}cdot e^{ 5ivarphi}cdotsin^{5}thetacdot(85cos^{4}theta-30cos^{2}theta+1)Y_{9}^{ 6}(theta,varphi)&={ 1over 128}sqrt{ 40755over pi}cdot e^{ 6ivarphi}cdotsin^{6}thetacdot(17cos^{3}theta-3costheta)Y_{9}^{ 7}(theta,varphi)&={-3over 512}sqrt{ 13585over pi}cdot e^{ 7ivarphi}cdotsin^{7}thetacdot(17cos^{2}theta-1)Y_{9}^{ 8}(theta,varphi)&={ 3over 256}sqrt{230945over 2pi}cdot e^{ 8ivarphi}cdotsin^{8}thetacdotcosthetaY_{9}^{ 9}(theta,varphi)&={-1over 512}sqrt{230945over pi}cdot e^{ 9ivarphi}cdotsin^{9}thetaend{align}â„“ 10“>

â„“ 10

Y_{10}^{-10}(theta,varphi)&={1over 1024}sqrt{969969over pi}cdot e^{-10ivarphi}cdotsin^{10}thetaY_{10}^{- 9}(theta,varphi)&={1over 512}sqrt{4849845over pi}cdot e^{-9ivarphi}cdotsin^{9}thetacdotcosthetaY_{10}^{- 8}(theta,varphi)&={1over 512}sqrt{255255over 2pi}cdot e^{-8ivarphi}cdotsin^{8}thetacdot(19cos^{2}theta-1)Y_{10}^{- 7}(theta,varphi)&={3over 512}sqrt{85085over pi}cdot e^{-7ivarphi}cdotsin^{7}thetacdot(19cos^{3}theta-3costheta)Y_{10}^{- 6}(theta,varphi)&={3over 1024}sqrt{5005over pi}cdot e^{-6ivarphi}cdotsin^{6}thetacdot(323cos^{4}theta-102cos^{2}theta+3)Y_{10}^{- 5}(theta,varphi)&={3over 256}sqrt{1001over pi}cdot e^{-5ivarphi}cdotsin^{5}thetacdot(323cos^{5}theta-170cos^{3}theta+15costheta)Y_{10}^{- 4}(theta,varphi)&={3over 256}sqrt{5005over 2pi}cdot e^{-4ivarphi}cdotsin^{4}thetacdot(323cos^{6}theta-255cos^{4}theta+45cos^{2}theta-1)Y_{10}^{- 3}(theta,varphi)&={3over 256}sqrt{5005over pi}cdot e^{-3ivarphi}cdotsin^{3}thetacdot(323cos^{7}theta-357cos^{5}theta+105cos^{3}theta-7costheta)Y_{10}^{- 2}(theta,varphi)&={3over 512}sqrt{385over 2pi}cdot e^{-2ivarphi}cdotsin^{2}thetacdot(4199cos^{8}theta-6188cos^{6}theta+2730cos^{4}theta-364cos^{2}theta+7)Y_{10}^{- 1}(theta,varphi)&={1over 256}sqrt{1155over 2pi}cdot e^{-ivarphi}cdotsinthetacdot(4199cos^{9}theta-7956cos^{7}theta+4914cos^{5}theta-1092cos^{3}theta+63costheta)Y_{10}^{ 0}(theta,varphi)&={1over 512}sqrt{21over pi}cdot(46189cos^{10}theta-109395cos^{8}theta+90090cos^{6}theta-30030cos^{4}theta+3465cos^{2}theta-63)Y_{10}^{ 1}(theta,varphi)&={-1over 256}sqrt{1155over 2pi}cdot e^{ivarphi}cdotsinthetacdot(4199cos^{9}theta-7956cos^{7}theta+4914cos^{5}theta-1092cos^{3}theta+63costheta)Y_{10}^{ 2}(theta,varphi)&={3over 512}sqrt{385over 2pi}cdot e^{2ivarphi}cdotsin^{2}thetacdot(4199cos^{8}theta-6188cos^{6}theta+2730cos^{4}theta-364cos^{2}theta+7)Y_{10}^{ 3}(theta,varphi)&={-3over 256}sqrt{5005over pi}cdot e^{3ivarphi}cdotsin^{3}thetacdot(323cos^{7}theta-357cos^{5}theta+105cos^{3}theta-7costheta)Y_{10}^{ 4}(theta,varphi)&={3over 256}sqrt{5005over 2pi}cdot e^{4ivarphi}cdotsin^{4}thetacdot(323cos^{6}theta-255cos^{4}theta+45cos^{2}theta-1)Y_{10}^{ 5}(theta,varphi)&={-3over 256}sqrt{1001over pi}cdot e^{5ivarphi}cdotsin^{5}thetacdot(323cos^{5}theta-170cos^{3}theta+15costheta)Y_{10}^{ 6}(theta,varphi)&={3over 1024}sqrt{5005over pi}cdot e^{6ivarphi}cdotsin^{6}thetacdot(323cos^{4}theta-102cos^{2}theta+3)Y_{10}^{ 7}(theta,varphi)&={-3over 512}sqrt{85085over pi}cdot e^{7ivarphi}cdotsin^{7}thetacdot(19cos^{3}theta-3costheta)Y_{10}^{ 8}(theta,varphi)&={1over 512}sqrt{255255over 2pi}cdot e^{8ivarphi}cdotsin^{8}thetacdot(19cos^{2}theta-1)Y_{10}^{ 9}(theta,varphi)&={-1over 512}sqrt{4849845over pi}cdot e^{9ivarphi}cdotsin^{9}thetacdotcosthetaY_{10}^{ 10}(theta,varphi)&={1over 1024}sqrt{969969over pi}cdot e^{10ivarphi}cdotsin^{10}thetaend{align}

Visualization of complex spherical harmonics

2D polar/azimuthal angle maps

Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, phi, on the horizontal axis and the polar angle, theta, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.(File:Complex Spherical Harmonics Figure Table Complex 2D.png|thumb|center|upright=3|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.(File:Complex Spherical Harmonics Figure Table Complex Polar Plot.gif|thumb|center|upright=3|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.(File:Complex Spherical Harmonics Figure Table Complex Radial Magnitude.gif|thumb|center|upright=3|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.BOOK, Petrucci, General chemistry : principles and modern applications., 2016, Prentice Hall, 0133897311, JOURNAL, Friedman, The shapes of the f orbitals, J. Chem. Educ., 1964, 41, 7, 354, For â„“ = 0, …, 3, see.BOOK, Group theoretical techniques in quantum chemistry, 1976, Academic Press, New York, 0-12-172950-8, C.D.H. Chisholm, JOURNAL, Blanco, Miguel A., Flórez, M., Bermejo, M., Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure: THEOCHEM, 1 December 1997, 419, 1–3, 19–27, 10.1016/S0166-1280(97)00185-1, â„“ 0“>

â„“ 0

Y_{00} = s = Y_0^0 = frac{1}{2} sqrt{frac{1}{pi}}â„“ 1“>

â„“ 1

begin{align} end{align}â„“ 2“>

â„“ 2

Y_{2,-2} & = d_{xy} = i sqrt{frac{1}{2}} left( Y_2^{- 2} - Y_2^2right) = frac{1}{2} sqrt{frac{15}{pi}} cdot frac{x y}{r^2} = frac{1}{4} sqrt{frac{15}{pi}} sin^{2}theta sin(2varphi) 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}} (3cos^{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 varphiY_{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(2varphi)end{align}â„“ 3“>

â„“ 3

Y_{3,-3} & = f_{y(3x^2-y^2)} = i sqrt{frac{1}{2}} left( Y_3^{- 3} + Y_3^3 right) = frac{1}{4} sqrt{frac{35}{2 pi}} cdot frac{y left( 3 x^2 - y^2 right)}{r^3} Y_{3,-2} & = f_{xyz} = i sqrt{frac{1}{2}} left( Y_3^{- 2} - Y_3^2 right) = frac{1}{2} sqrt{frac{105}{pi}} cdot frac{xy cdot z}{r^3} Y_{3,-1} & = f_{yz^2} = i sqrt{frac{1}{2}} left( Y_3^{- 1} + Y_3^1 right) = frac{1}{4} sqrt{frac{21}{2 pi}} cdot frac{y cdot (5 z^2 - r^2)}{r^3} Y_{3,0} & = f_{z^3} = Y_3^0 = frac{1}{4} sqrt{frac{7}{pi}} cdot frac{5 z^3 - 3 z r^2}{r^3} Y_{3,1} & = f_{xz^2} = sqrt{frac{1}{2}} left( Y_3^{- 1} - Y_3^1 right) = frac{1}{4} sqrt{frac{21}{2 pi}} cdot frac{x cdot (5 z^2 - r^2)}{r^3} Y_{3,2} & = f_{z(x^2-y^2)} = sqrt{frac{1}{2}} left( Y_3^{- 2} + Y_3^2 right) = frac{1}{4} sqrt{frac{105}{pi}} cdot frac{left( x^2 - y^2 right) cdot z}{r^3} Y_{3,3} & = f_{x(x^2-3y^2)} = sqrt{frac{1}{2}} left( Y_3^{- 3} - Y_3^3 right) = frac{1}{4} sqrt{frac{35}{2 pi}} cdot frac{x left( x^2 - 3 y^2 right)}{r^3}end{align}â„“ 4“>

â„“ 4

Y_{4,-4} & = i sqrt{frac{1}{2}} left( Y_4^{- 4} - Y_4^4 right) = frac{3}{4} sqrt{frac{35}{pi}} cdot frac{xy left( x^2 - y^2 right)}{r^4} Y_{4,-3} & = i sqrt{frac{1}{2}} left( Y_4^{- 3} + Y_4^3 right) = frac{3}{4} sqrt{frac{35}{2 pi}} cdot frac{y (3 x^2 - y^2) cdot z}{r^4} Y_{4,-2} & = i sqrt{frac{1}{2}} left( Y_4^{- 2} - Y_4^2 right) = frac{3}{4} sqrt{frac{5}{pi}} cdot frac{xy cdot (7 z^2 - r^2)}{r^4} Y_{4,-1} & = i sqrt{frac{1}{2}} left( Y_4^{- 1} + Y_4^1right) = frac{3}{4} sqrt{frac{5}{2 pi}} cdot frac{y cdot (7 z^3 - 3 z r^2)}{r^4} Y_{4,0} & = Y_4^0 = frac{3}{16} sqrt{frac{1}{pi}} cdot frac{35 z^4 - 30 z^2 r^2 + 3 r^4}{r^4} Y_{4,1} & = sqrt{frac{1}{2}} left( Y_4^{- 1} - Y_4^1 right) = frac{3}{4} sqrt{frac{5}{2 pi}} cdot frac{x cdot (7 z^3 - 3 z r^2)}{r^4} Y_{4,2} & = sqrt{frac{1}{2}} left( Y_4^{- 2} + Y_4^2 right) = frac{3}{8} sqrt{frac{5}{pi}} cdot frac{(x^2 - y^2) cdot (7 z^2 - r^2)}{r^4} Y_{4,3} & = sqrt{frac{1}{2}} left( Y_4^{- 3} - Y_4^3 right) = frac{3}{4} sqrt{frac{35}{2 pi}} cdot frac{x(x^2 - 3 y^2) cdot z}{r^4} Y_{4,4} & = sqrt{frac{1}{2}} left( Y_4^{- 4} + Y_4^4 right) = frac{3}{16} sqrt{frac{35}{pi}} cdot frac{x^2 left( x^2 - 3 y^2 right) - y^2 left( 3 x^2 - y^2 right)}{r^4}end{align}

Visualization of real spherical harmonics

2D polar/azimuthal angle maps

Below the real spherical harmonics are represented on 2D plots with the azimuthal angle, phi, on the horizontal axis and the polar angle, theta, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.(File:Real Spherical Harmonics Figure Table Complex 2D.png|thumb|center|upright=3|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.(File:Real Spherical Harmonics Figure Table Complex Polar Plot.gif|thumb|center|upright=3|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.(File:Real Spherical Harmonics Figure Table Complex Radial Magnitude.gif|thumb|center|upright=3|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.(File:Sph harm table real bumpy.gif|thumb|center|upright=3|Visual Array of Real Spherical Harmonics Represented with Polar Plot with Amplitude Mapped to Elevation and Saturation)

See also

• Spherical harmonics

External links

• Spherical Harmonics at MathWorld
• Spherical Harmonics 3D representation

References

Cited references

{{reflist}}

General references

• See section 3 in JOURNAL

, (see section 3.3)

• For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi at Wolfram Alpha], especially for specific values of l and m.