Puneet Varma (Editor)

Table of spherical harmonics

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

Contents

x = r sin ⁡ θ cos ⁡ φ y = r sin ⁡ θ sin ⁡ φ z = r cos ⁡ θ

l = 0

Y 0 0 ( θ , φ ) = 1 2 1 π

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

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 ⋅ ( 2 z 2 − x 2 − 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 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

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 ) ( 4 z 2 − x 2 − y 2 ) r 3 Y 3 0 ( θ , φ ) = 1 4 7 π ⋅ ( 5 cos 3 ⁡ θ − 3 cos ⁡ θ ) = 1 4 7 π ⋅ z ( 2 z 2 − 3 x 2 − 3 y 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 ) ( 4 z 2 − x 2 − y 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

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 ) ⋅ z ⋅ ( 7 z 2 − 3 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 ) ⋅ z ⋅ ( 7 z 2 − 3 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

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

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

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

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

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

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

Real spherical harmonics

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

l = 0

Y 00 = s = Y 0 0 = 1 2 1 π

l = 1

Y 1 , − 1 = p y = i 1 2 ( Y 1 − 1 + Y 1 1 ) = 3 4 π ⋅ y r Y 1 , 0 = p z = Y 1 0 = 3 4 π ⋅ z r Y 1 , 1 = p x = 1 2 ( Y 1 − 1 − Y 1 1 ) = 3 4 π ⋅ x r

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 Y 2 , − 1 = d y z = i 1 2 ( Y 2 − 1 + Y 2 1 ) = 1 2 15 π ⋅ y z r 2 Y 2 , 0 = d z 2 = Y 2 0 = 1 4 5 π ⋅ − x 2 − y 2 + 2 z 2 r 2 Y 2 , 1 = d x z = 1 2 ( Y 2 − 1 − Y 2 1 ) = 1 2 15 π ⋅ z x r 2 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

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 ⋅ ( 3 x 2 − y 2 ) y 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 ( 4 z 2 − x 2 − y 2 ) r 3 Y 3 , 0 = f z 3 = Y 3 0 = 1 4 7 π ⋅ z ( 2 z 2 − 3 x 2 − 3 y 2 ) r 3 Y 3 , 1 = f x z 2 = 1 2 ( Y 3 − 1 − Y 3 1 ) = 1 4 21 2 π ⋅ x ( 4 z 2 − x 2 − y 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 2 − 3 y 2 ) x r 3

l = 4

Y 4 , − 4 = g x y ( x 2 − y 2 ) = i 1 2 ( Y 4 − 4 − Y 4 4 ) = 3 4 35 π ⋅ x y ( x 2 − y 2 ) r 4 Y 4 , − 3 = g z y 3 = i 1 2 ( Y 4 − 3 + Y 4 3 ) = 3 4 35 2 π ⋅ ( 3 x 2 − y 2 ) y z r 4 Y 4 , − 2 = g z 2 x y = 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 = g z 3 y = i 1 2 ( Y 4 − 1 + Y 4 1 ) = 3 4 5 2 π ⋅ y z ⋅ ( 7 z 2 − 3 r 2 ) r 4 Y 4 , 0 = g z 4 = Y 4 0 = 3 16 1 π ⋅ ( 35 z 4 − 30 z 2 r 2 + 3 r 4 ) r 4 Y 4 , 1 = g z 3 x = 1 2 ( Y 4 − 1 − Y 4 1 ) = 3 4 5 2 π ⋅ x z ⋅ ( 7 z 2 − 3 r 2 ) r 4 Y 4 , 2 = g z 2 x y = 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 = g z x 3 = 1 2 ( Y 4 − 3 − Y 4 3 ) = 3 4 35 2 π ⋅ ( x 2 − 3 y 2 ) x z r 4 Y 4 , 4 = g x 4 + y 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

References

Table of spherical harmonics Wikipedia
(Text) CC BY-SA


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