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Chemistry LibreTexts

M4: Spherical Harmonics

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    Nov 3, 2020
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Represented in a system of spherical coordinates, Laplace's spherical harmonics Yml are a specific set of spherical harmonics that forms an orthogonal system. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic orbital electron configurations

l = 0

Y00(θ,φ)=12√1π

l = 1

Y−11(θ,φ)=12√32π⋅e−iφ⋅sinθ=12√32π⋅(x−iy)rY01(θ,φ)=12√3π⋅cosθ=12√3π⋅zrY11(θ,φ)=−12√32π⋅eiφ⋅sinθ=−12√32π⋅(x+iy)r

l = 2

Y−22(θ,φ)=14√152π⋅e−2iφ⋅sin2θ=14√152π⋅(x−iy)2r2

Y−12(θ,φ)=12√152π⋅e−iφ⋅sinθ⋅cosθ=12√152π⋅(x−iy)zr2

Y02(θ,φ)=14√5π⋅(3cos2θ−1)=14√5π⋅(2z2−x2−y2)r2

Y12(θ,φ)=−12√152π⋅eiφ⋅sinθ⋅cosθ=−12√152π⋅(x+iy)zr2

Y22(θ,φ)=14√152π⋅e2iφ⋅sin2θ=14√152π⋅(x+iy)2r2

l = 3

Y−33(θ,φ)=18√35π⋅e−3iφ⋅sin3θ=18√35π⋅(x−iy)3r3
Y−23(θ,φ)=14√1052π⋅e−2iφ⋅sin2θ⋅cosθ=14√1052π⋅(x−iy)2zr3
Y−13(θ,φ)=18√21π⋅e−iφ⋅sinθ⋅(5cos2θ−1)=18√21π⋅(x−iy)(4z2−x2−y2)r3
Y03(θ,φ)=14√7π⋅(5cos3θ−3cosθ)=14√7π⋅z(2z2−3x2−3y2)r3
Y13(θ,φ)=−18√21π⋅eiφ⋅sinθ⋅(5cos2θ−1)=−18√21π⋅(x+iy)(4z2−x2−y2)r3
Y23(θ,φ)=14√1052π⋅e2iφ⋅sin2θ⋅cosθ=14√1052π⋅(x+iy)2zr3
Y33(θ,φ)=−18√35π⋅e3iφ⋅sin3θ=−18√35π⋅(x+iy)3r3

l = 4

Y−44(θ,φ)=316√352π⋅e−4iφ⋅sin4θ=316√352π⋅(x−iy)4r4
Y−34(θ,φ)=38√35π⋅e−3iφ⋅sin3θ⋅cosθ=38√35π⋅(x−iy)3zr4
Y−24(θ,φ)=38√52π⋅e−2iφ⋅sin2θ⋅(7cos2θ−1)=38√52π⋅(x−iy)2⋅(7z2−r2)r4
Y−14(θ,φ)=38√5π⋅e−iφ⋅sinθ⋅(7cos3θ−3cosθ)=38√5π⋅(x−iy)⋅z⋅(7z2−3r2)r4
Y04(θ,φ)=316√1π⋅(35cos4θ−30cos2θ+3)=316√1π⋅(35z4−30z2r2+3r4)r4
Y14(θ,φ)=−38√5π⋅eiφ⋅sinθ⋅(7cos3θ−3cosθ)=−38√5π⋅(x+iy)⋅z⋅(7z2−3r2)r4
Y24(θ,φ)=38√52π⋅e2iφ⋅sin2θ⋅(7cos2θ−1)=38√52π⋅(x+iy)2⋅(7z2−r2)r4
Y34(θ,φ)=−38√35π⋅e3iφ⋅sin3θ⋅cosθ=−38√35π⋅(x+iy)3zr4
Y44(θ,φ)=316√352π⋅e4iφ⋅sin4θ=316√352π⋅(x+iy)4r4

l = 5

Y−55(θ,φ)=332√77π⋅e−5iφ⋅sin5θ
Y−45(θ,φ)=316√3852π⋅e−4iφ⋅sin4θ⋅cosθ
Y−35(θ,φ)=132√385π⋅e−3iφ⋅sin3θ⋅(9cos2θ−1)
Y−25(θ,φ)=18√11552π⋅e−2iφ⋅sin2θ⋅(3cos3θ−cosθ)
Y−15(θ,φ)=116√1652π⋅e−iφ⋅sinθ⋅(21cos4θ−14cos2θ+1)
Y05(θ,φ)=116√11π⋅(63cos5θ−70cos3θ+15cosθ)
Y15(θ,φ)=−116√1652π⋅eiφ⋅sinθ⋅(21cos4θ−14cos2θ+1)
Y25(θ,φ)=18√11552π⋅e2iφ⋅sin2θ⋅(3cos3θ−cosθ)
Y35(θ,φ)=−132√385π⋅e3iφ⋅sin3θ⋅(9cos2θ−1)
Y45(θ,φ)=316√3852π⋅e4iφ⋅sin4θ⋅cosθ
Y55(θ,φ)=−332√77π⋅e5iφ⋅sin5θ

l = 6

Y−66(θ,φ)=164√3003π⋅e−6iφ⋅sin6θ
Y−56(θ,φ)=332√1001π⋅e−5iφ⋅sin5θ⋅cosθ
Y−46(θ,φ)=332√912π⋅e−4iφ⋅sin4θ⋅(11cos2θ−1)
Y−36(θ,φ)=132√1365π⋅e−3iφ⋅sin3θ⋅(11cos3θ−3cosθ)
Y−26(θ,φ)=164√1365π⋅e−2iφ⋅sin2θ⋅(33cos4θ−18cos2θ+1)
Y−16(θ,φ)=116√2732π⋅e−iφ⋅sinθ⋅(33cos5θ−30cos3θ+5cosθ)
Y06(θ,φ)=132√13π⋅(231cos6θ−315cos4θ+105cos2θ−5)
Y16(θ,φ)=−116√2732π⋅eiφ⋅sinθ⋅(33cos5θ−30cos3θ+5cosθ)
Y26(θ,φ)=164√1365π⋅e2iφ⋅sin2θ⋅(33cos4θ−18cos2θ+1)
Y36(θ,φ)=−132√1365π⋅e3iφ⋅sin3θ⋅(11cos3θ−3cosθ)
Y46(θ,φ)=332√912π⋅e4iφ⋅sin4θ⋅(11cos2θ−1)
Y56(θ,φ)=−332√1001π⋅e5iφ⋅sin5θ⋅cosθ
Y66(θ,φ)=164√3003π⋅e6iφ⋅sin6θ"mwe-math-fallback-png-inline tex"

l = 7

Y−77(θ,φ)=364√7152π⋅e−7iφ⋅sin7θ
Y−67(θ,φ)=364√5005π⋅e−6iφ⋅sin6θ⋅cosθ
Y−57(θ,φ)=364√3852π⋅e−5iφ⋅sin5θ⋅(13cos2θ−1)
Y−47(θ,φ)=332√3852π⋅e−4iφ⋅sin4θ⋅(13cos3θ−3cosθ)
Y−37(θ,φ)=364√352π⋅e−3iφ⋅sin3θ⋅(143cos4θ−66cos2θ+3)
Y−27(θ,φ)=364√35π⋅e−2iφ⋅sin2θ⋅(143cos5θ−110cos3θ+15cosθ)
Y−17(θ,φ)=164√1052π⋅e−iφ⋅sinθ⋅(429cos6θ−495cos4θ+135cos2θ−5)
Y07(θ,φ)=132√15π⋅(429cos7θ−693cos5θ+315cos3θ−35cosθ)
Y17(θ,φ)=−164√1052π⋅eiφ⋅sinθ⋅(429cos6θ−495cos4θ+135cos2θ−5)
Y27(θ,φ)=364√35π⋅e2iφ⋅sin2θ⋅(143cos5θ−110cos3θ+15cosθ)
Y37(θ,φ)=−364√352π⋅e3iφ⋅sin3θ⋅(143cos4θ−66cos2θ+3)
Y47(θ,φ)=332√3852π⋅e4iφ⋅sin4θ⋅(13cos3θ−3cosθ)
Y57(θ,φ)=−364√3852π⋅e5iφ⋅sin5θ⋅(13cos2θ−1)
Y67(θ,φ)=364√5005π⋅e6iφ⋅sin6θ⋅cosθ
Y77(θ,φ)=−364√7152π⋅e7iφ⋅sin7θ

l = 8

Y−88(θ,φ)=3256√121552π⋅e−8iφ⋅sin8θ
Y−78(θ,φ)=364√121552π⋅e−7iφ⋅sin7θ⋅cosθ
Y−68(θ,φ)=1128√7293π⋅e−6iφ⋅sin6θ⋅(15cos2θ−1)
Y−58(θ,φ)=364√170172π⋅e−5iφ⋅sin5θ⋅(5cos3θ−cosθ)
Y−48(θ,φ)=3128√13092π⋅e−4iφ⋅sin4θ⋅(65cos4θ−26cos2θ+1)
Y−38(θ,φ)=164√196352π⋅e−3iφ⋅sin3θ⋅(39cos5θ−26cos3θ+3cosθ)
Y−28(θ,φ)=3128√595π⋅e−2iφ⋅sin2θ⋅(143cos6θ−143cos4θ+33cos2θ−1)
Y−18(θ,φ)=364√172π⋅e−iφ⋅sinθ⋅(715cos7θ−1001cos5θ+385cos3θ−35cosθ)
Y08(θ,φ)=1256√17π⋅(6435cos8θ−12012cos6θ+6930cos4θ−1260cos2θ+35)
Y18(θ,φ)=−364√172π⋅eiφ⋅sinθ⋅(715cos7θ−1001cos5θ+385cos3θ−35cosθ)
Y28(θ,φ)=3128√595π⋅e2iφ⋅sin2θ⋅(143cos6θ−143cos4θ+33cos2θ−1)
Y38(θ,φ)=−164√196352π⋅e3iφ⋅sin3θ⋅(39cos5θ−26cos3θ+3cosθ)
Y48(θ,φ)=3128√13092π⋅e4iφ⋅sin4θ⋅(65cos4θ−26cos2θ+1)
Y58(θ,φ)=−364√170172π⋅e5iφ⋅sin5θ⋅(5cos3θ−cosθ)
Y68(θ,φ)=1128√7293π⋅e6iφ⋅sin6θ⋅(15cos2θ−1)
Y78(θ,φ)=−364√121552π⋅e7iφ⋅sin7θ⋅cosθ
Y88(θ,φ)=3256√121552π⋅e8iφ⋅sin8θ

l = 9

Y−99(θ,φ)=1512√230945π⋅e−9iφ⋅sin9θ
Y−89(θ,φ)=3256√2309452π⋅e−8iφ⋅sin8θ⋅cosθ
Y−79(θ,φ)=3512√13585π⋅e−7iφ⋅sin7θ⋅(17cos2θ−1)
Y−69(θ,φ)=1128√40755π⋅e−6iφ⋅sin6θ⋅(17cos3θ−3cosθ)
Y−59(θ,φ)=3256√2717π⋅e−5iφ⋅sin5θ⋅(85cos4θ−30cos2θ+1)
Y−49(θ,φ)=3128√950952π⋅e−4iφ⋅sin4θ⋅(17cos5θ−10cos3θ+cosθ)
Y−39(θ,φ)=1256√21945π⋅e−3iφ⋅sin3θ⋅(221cos6θ−195cos4θ+39cos2θ−1)
Y−29(θ,φ)=3128√1045π⋅e−2iφ⋅sin2θ⋅(221cos7θ−273cos5θ+91cos3θ−7cosθ)
Y−19(θ,φ)=3256√952π⋅e−iφ⋅sinθ⋅(2431cos8θ−4004cos6θ+2002cos4θ−308cos2θ+7)
Y09(θ,φ)=1256√19π⋅(12155cos9θ−25740cos7θ+18018cos5θ−4620cos3θ+315cosθ)
Y19(θ,φ)=−3256√952π⋅eiφ⋅sinθ⋅(2431cos8θ−4004cos6θ+2002cos4θ−308cos2θ+7)
Y29(θ,φ)=3128√1045π⋅e2iφ⋅sin2θ⋅(221cos7θ−273cos5θ+91cos3θ−7cosθ)
Y39(θ,φ)=−1256√21945π⋅e3iφ⋅sin3θ⋅(221cos6θ−195cos4θ+39cos2θ−1)
Y49(θ,φ)=3128√950952π⋅e4iφ⋅sin4θ⋅(17cos5θ−10cos3θ+cosθ)
Y59(θ,φ)=−3256√2717π⋅e5iφ⋅sin5θ⋅(85cos4θ−30cos2θ+1)
Y69(θ,φ)=1128√40755π⋅e6iφ⋅sin6θ⋅(17cos3θ−3cosθ)
Y79(θ,φ)=−3512√13585π⋅e7iφ⋅sin7θ⋅(17cos2θ−1)
Y89(θ,φ)=3256√2309452π⋅e8iφ⋅sin8θ⋅cosθ
Y99(θ,φ)=−1512√230945π⋅e9iφ⋅sin9θ

l = 10

Y−1010(θ,φ)=11024√969969π⋅e−10iφ⋅sin10θ
Y−910(θ,φ)=1512√4849845π⋅e−9iφ⋅sin9θ⋅cosθ
Y−810(θ,φ)=1512√2552552π⋅e−8iφ⋅sin8θ⋅(19cos2θ−1)
Y−710(θ,φ)=3512√85085π⋅e−7iφ⋅sin7θ⋅(19cos3θ−3cosθ)
Y−610(θ,φ)=31024√5005π⋅e−6iφ⋅sin6θ⋅(323cos4θ−102cos2θ+3)
Y−510(θ,φ)=3256√1001π⋅e−5iφ⋅sin5θ⋅(323cos5θ−170cos3θ+15cosθ)
Y−410(θ,φ)=3256√50052π⋅e−4iφ⋅sin4θ⋅(323cos6θ−255cos4θ+45cos2θ−1)
Y−310(θ,φ)=3256√5005π⋅e−3iφ⋅sin3θ⋅(323cos7θ−357cos5θ+105cos3θ−7cosθ)
Y−210(θ,φ)=3512√3852π⋅e−2iφ⋅sin2θ⋅(4199cos8θ−6188cos6θ+2730cos4θ−364cos2θ+7)
Y−110(θ,φ)=1256√11552π⋅e−iφ⋅sinθ⋅(4199cos9θ−7956cos7θ+4914cos5θ−1092cos3θ+63cosθ)
Y010(θ,φ)=1512√21π⋅(46189cos10θ−109395cos8θ+90090cos6θ−30030cos4θ+3465cos2θ−63)
Y110(θ,φ)=−1256√11552π⋅eiφ⋅sinθ⋅(4199cos9θ−7956cos7θ+4914cos5θ−1092cos3θ+63cosθ)
Y210(θ,φ)=3512√3852π⋅e2iφ⋅sin2θ⋅(4199cos8θ−6188cos6θ+2730cos4θ−364cos2θ+7)
Y310(θ,φ)=−3256√5005π⋅e3iφ⋅sin3θ⋅(323cos7θ−357cos5θ+105cos3θ−7cosθ)
Y410(θ,φ)=3256√50052π⋅e4iφ⋅sin4θ⋅(323cos6θ−255cos4θ+45cos2θ−1)
Y510(θ,φ)=−3256√1001π⋅e5iφ⋅sin5θ⋅(323cos5θ−170cos3θ+15cosθ)
Y610(θ,φ)=31024√5005π⋅e6iφ⋅sin6θ⋅(323cos4θ−102cos2θ+3)
Y710(θ,φ)=−3512√85085π⋅e7iφ⋅sin7θ⋅(19cos3θ−3cosθ)
Y810(θ,φ)=1512√2552552π⋅e8iφ⋅sin8θ⋅(19cos2θ−1)
Y910(θ,φ)=−1512√4849845π⋅e9iφ⋅sin9θ⋅cosθ
Y1010(θ,φ)=11024√969969π⋅e10iφ⋅sin10θ

Real spherical harmonics

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

l = 0

Y00=s=Y00=12√1π

l = 1

Y1,−1=py=i√12(Y−11+Y11)=√34π⋅yrY10=pz=Y01=√34π⋅zrY11=px=√12(Y−11−Y11)=√34π⋅xr

l = 2

Y2,−2=dxy=i√12(Y−22−Y22)=12√15π⋅xyr2Y2,−1=dyz=i√12(Y−12+Y12)=12√15π⋅yzr2Y20=dz2=Y02=14√5π⋅−x2−y2+2z2r2Y21=dxz=√12(Y−12−Y12)=12√15π⋅zxr2Y22=dx2−y2=√12(Y−22+Y22)=14√15π⋅x2−y2r2

l = 3

Y3,−3=fy(3x2−y2)=i√12(Y−33+Y33)=14√352π⋅(3x2−y2)yr3Y3,−2=fxyz=i√12(Y−23−Y23)=12√105π⋅xyzr3Y3,−1=fyz2=i√12(Y−13+Y13)=14√212π⋅y(4z2−x2−y2)r3Y30=fz3=Y03=14√7π⋅z(2z2−3x2−3y2)r3Y31=fxz2=√12(Y−13−Y13)=14√212π⋅x(4z2−x2−y2)r3Y32=fz(x2−y2)=√12(Y−23+Y23)=14√105π⋅(x2−y2)zr3Y33=fx(x2−3y2)=√12(Y−33−Y33)=14√352π⋅(x2−3y2)xr3

l = 4

Y4,−4=gxy(x2−y2)=i√12(Y−44−Y44)=34√35π⋅xy(x2−y2)r4Y4,−3=gzy3=i√12(Y−34+Y34)=34√352π⋅(3x2−y2)yzr4Y4,−2=gz2xy=i√12(Y−24−Y24)=34√5π⋅xy⋅(7z2−r2)r4Y4,−1=gz3y=i√12(Y−14+Y14)=34√52π⋅yz⋅(7z2−3r2)r4Y40=gz4=Y04=316√1π⋅(35z4−30z2r2+3r4)r4Y41=gz3x=√12(Y−14−Y14)=34√52π⋅xz⋅(7z2−3r2)r4Y42=gz2xy=√12(Y−24+Y24)=38√5π⋅(x2−y2)⋅(7z2−r2)r4Y43=gzx3=√12(Y−34−Y34)=34√352π⋅(x2−3y2)xzr4Y44=gx4+y4=√12(Y−44+Y44)=316√35π⋅x2(x2−3y2)−y2(3x2−y2)r4

External links

References

Cited references
  1. D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii (1988). Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols (1. repr. ed.). Singapore: World Scientific Pub. p. 155-156. ISBN 9971-50-107-4.
  2. C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry. New York: Academic Press. ISBN 0-12-172950-8.
  3. Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 December 1997). "Evaluation of the rotation matrices in the basis of real spherical harmonics". Journal of Molecular Structure: THEOCHEM 419 (1–3): 19–27. doi:10.1016/S0166-1280(97)00185-1.
General references

M4: Spherical Harmonics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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