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
th
{\displaystyle \theta }
ph
{\displaystyle \varphi }
{
cos
(
th
)
=
z
/
r
e
+-
i
ph
sin
(
th
)
=
(
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 [ edit ] For l = 0, ..., 5, see.[ 1]
Y
0
0
(
th
,
ph
)
=
1
2
1
p
{\displaystyle Y_{0}^{0}(\theta ,\varphi )={1 \over 2}{\sqrt {1 \over \pi }}}
Y
1
-
1
(
th
,
ph
)
=
1
2
3
2
p
e
-
i
ph
sin
th
=
1
2
3
2
p
(
x
-
i
y
)
r
Y
1
0
(
th
,
ph
)
=
1
2
3
p
cos
th
=
1
2
3
p
z
r
Y
1
1
(
th
,
ph
)
=
-
1
2
3
2
p
e
i
ph
sin
th
=
-
1
2
3
2
p
(
x
+
i
y
)
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}}}
Y
2
-
2
(
th
,
ph
)
=
1
4
15
2
p
e
-
2
i
ph
sin
2
th
=
1
4
15
2
p
(
x
-
i
y
)
2
r
2
Y
2
-
1
(
th
,
ph
)
=
1
2
15
2
p
e
-
i
ph
sin
th
cos
th
=
1
2
15
2
p
(
x
-
i
y
)
z
r
2
Y
2
0
(
th
,
ph
)
=
1
4
5
p
(
3
cos
2
th
-
1
)
=
1
4
5
p
(
3
z
2
-
r
2
)
r
2
Y
2
1
(
th
,
ph
)
=
-
1
2
15
2
p
e
i
ph
sin
th
cos
th
=
-
1
2
15
2
p
(
x
+
i
y
)
z
r
2
Y
2
2
(
th
,
ph
)
=
1
4
15
2
p
e
2
i
ph
sin
2
th
=
1
4
15
2
p
(
x
+
i
y
)
2
r
2
{\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}}}
Y
3
-
3
(
th
,
ph
)
=
1
8
35
p
e
-
3
i
ph
sin
3
th
=
1
8
35
p
(
x
-
i
y
)
3
r
3
Y
3
-
2
(
th
,
ph
)
=
1
4
105
2
p
e
-
2
i
ph
sin
2
th
cos
th
=
1
4
105
2
p
(
x
-
i
y
)
2
z
r
3
Y
3
-
1
(
th
,
ph
)
=
1
8
21
p
e
-
i
ph
sin
th
(
5
cos
2
th
-
1
)
=
1
8
21
p
(
x
-
i
y
)
(
5
z
2
-
r
2
)
r
3
Y
3
0
(
th
,
ph
)
=
1
4
7
p
(
5
cos
3
th
-
3
cos
th
)
=
1
4
7
p
(
5
z
3
-
3
z
r
2
)
r
3
Y
3
1
(
th
,
ph
)
=
-
1
8
21
p
e
i
ph
sin
th
(
5
cos
2
th
-
1
)
=
-
1
8
21
p
(
x
+
i
y
)
(
5
z
2
-
r
2
)
r
3
Y
3
2
(
th
,
ph
)
=
1
4
105
2
p
e
2
i
ph
sin
2
th
cos
th
=
1
4
105
2
p
(
x
+
i
y
)
2
z
r
3
Y
3
3
(
th
,
ph
)
=
-
1
8
35
p
e
3
i
ph
sin
3
th
=
-
1
8
35
p
(
x
+
i
y
)
3
r
3
{\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}}}
Y
4
-
4
(
th
,
ph
)
=
3
16
35
2
p
e
-
4
i
ph
sin
4
th
=
3
16
35
2
p
(
x
-
i
y
)
4
r
4
Y
4
-
3
(
th
,
ph
)
=
3
8
35
p
e
-
3
i
ph
sin
3
th
cos
th
=
3
8
35
p
(
x
-
i
y
)
3
z
r
4
Y
4
-
2
(
th
,
ph
)
=
3
8
5
2
p
e
-
2
i
ph
sin
2
th
(
7
cos
2
th
-
1
)
=
3
8
5
2
p
(
x
-
i
y
)
2
(
7
z
2
-
r
2
)
r
4
Y
4
-
1
(
th
,
ph
)
=
3
8
5
p
e
-
i
ph
sin
th
(
7
cos
3
th
-
3
cos
th
)
=
3
8
5
p
(
x
-
i
y
)
(
7
z
3
-
3
z
r
2
)
r
4
Y
4
0
(
th
,
ph
)
=
3
16
1
p
(
35
cos
4
th
-
30
cos
2
th
+
3
)
=
3
16
1
p
(
35
z
4
-
30
z
2
r
2
+
3
r
4
)
r
4
Y
4
1
(
th
,
ph
)
=
-
3
8
5
p
e
i
ph
sin
th
(
7
cos
3
th
-
3
cos
th
)
=
-
3
8
5
p
(
x
+
i
y
)
(
7
z
3
-
3
z
r
2
)
r
4
Y
4
2
(
th
,
ph
)
=
3
8
5
2
p
e
2
i
ph
sin
2
th
(
7
cos
2
th
-
1
)
=
3
8
5
2
p
(
x
+
i
y
)
2
(
7
z
2
-
r
2
)
r
4
Y
4
3
(
th
,
ph
)
=
-
3
8
35
p
e
3
i
ph
sin
3
th
cos
th
=
-
3
8
35
p
(
x
+
i
y
)
3
z
r
4
Y
4
4
(
th
,
ph
)
=
3
16
35
2
p
e
4
i
ph
sin
4
th
=
3
16
35
2
p
(
x
+
i
y
)
4
r
4
{\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}}}
Y
5
-
5
(
th
,
ph
)
=
3
32
77
p
e
-
5
i
ph
sin
5
th
Y
5
-
4
(
th
,
ph
)
=
3
16
385
2
p
e
-
4
i
ph
sin
4
th
cos
th
Y
5
-
3
(
th
,
ph
)
=
1
32
385
p
e
-
3
i
ph
sin
3
th
(
9
cos
2
th
-
1
)
Y
5
-
2
(
th
,
ph
)
=
1
8
1155
2
p
e
-
2
i
ph
sin
2
th
(
3
cos
3
th
-
cos
th
)
Y
5
-
1
(
th
,
ph
)
=
1
16
165
2
p
e
-
i
ph
sin
th
(
21
cos
4
th
-
14
cos
2
th
+
1
)
Y
5
0
(
th
,
ph
)
=
1
16
11
p
(
63
cos
5
th
-
70
cos
3
th
+
15
cos
th
)
Y
5
1
(
th
,
ph
)
=
-
1
16
165
2
p
e
i
ph
sin
th
(
21
cos
4
th
-
14
cos
2
th
+
1
)
Y
5
2
(
th
,
ph
)
=
1
8
1155
2
p
e
2
i
ph
sin
2
th
(
3
cos
3
th
-
cos
th
)
Y
5
3
(
th
,
ph
)
=
-
1
32
385
p
e
3
i
ph
sin
3
th
(
9
cos
2
th
-
1
)
Y
5
4
(
th
,
ph
)
=
3
16
385
2
p
e
4
i
ph
sin
4
th
cos
th
Y
5
5
(
th
,
ph
)
=
-
3
32
77
p
e
5
i
ph
sin
5
th
{\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}}}
Y
6
-
6
(
th
,
ph
)
=
1
64
3003
p
e
-
6
i
ph
sin
6
th
Y
6
-
5
(
th
,
ph
)
=
3
32
1001
p
e
-
5
i
ph
sin
5
th
cos
th
Y
6
-
4
(
th
,
ph
)
=
3
32
91
2
p
e
-
4
i
ph
sin
4
th
(
11
cos
2
th
-
1
)
Y
6
-
3
(
th
,
ph
)
=
1
32
1365
p
e
-
3
i
ph
sin
3
th
(
11
cos
3
th
-
3
cos
th
)
Y
6
-
2
(
th
,
ph
)
=
1
64
1365
p
e
-
2
i
ph
sin
2
th
(
33
cos
4
th
-
18
cos
2
th
+
1
)
Y
6
-
1
(
th
,
ph
)
=
1
16
273
2
p
e
-
i
ph
sin
th
(
33
cos
5
th
-
30
cos
3
th
+
5
cos
th
)
Y
6
0
(
th
,
ph
)
=
1
32
13
p
(
231
cos
6
th
-
315
cos
4
th
+
105
cos
2
th
-
5
)
Y
6
1
(
th
,
ph
)
=
-
1
16
273
2
p
e
i
ph
sin
th
(
33
cos
5
th
-
30
cos
3
th
+
5
cos
th
)
Y
6
2
(
th
,
ph
)
=
1
64
1365
p
e
2
i
ph
sin
2
th
(
33
cos
4
th
-
18
cos
2
th
+
1
)
Y
6
3
(
th
,
ph
)
=
-
1
32
1365
p
e
3
i
ph
sin
3
th
(
11
cos
3
th
-
3
cos
th
)
Y
6
4
(
th
,
ph
)
=
3
32
91
2
p
e
4
i
ph
sin
4
th
(
11
cos
2
th
-
1
)
Y
6
5
(
th
,
ph
)
=
-
3
32
1001
p
e
5
i
ph
sin
5
th
cos
th
Y
6
6
(
th
,
ph
)
=
1
64
3003
p
e
6
i
ph
sin
6
th
{\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}}}
Y
7
-
7
(
th
,
ph
)
=
3
64
715
2
p
e
-
7
i
ph
sin
7
th
Y
7
-
6
(
th
,
ph
)
=
3
64
5005
p
e
-
6
i
ph
sin
6
th
cos
th
Y
7
-
5
(
th
,
ph
)
=
3
64
385
2
p
e
-
5
i
ph
sin
5
th
(
13
cos
2
th
-
1
)
Y
7
-
4
(
th
,
ph
)
=
3
32
385
2
p
e
-
4
i
ph
sin
4
th
(
13
cos
3
th
-
3
cos
th
)
Y
7
-
3
(
th
,
ph
)
=
3
64
35
2
p
e
-
3
i
ph
sin
3
th
(
143
cos
4
th
-
66
cos
2
th
+
3
)
Y
7
-
2
(
th
,
ph
)
=
3
64
35
p
e
-
2
i
ph
sin
2
th
(
143
cos
5
th
-
110
cos
3
th
+
15
cos
th
)
Y
7
-
1
(
th
,
ph
)
=
1
64
105
2
p
e
-
i
ph
sin
th
(
429
cos
6
th
-
495
cos
4
th
+
135
cos
2
th
-
5
)
Y
7
0
(
th
,
ph
)
=
1
32
15
p
(
429
cos
7
th
-
693
cos
5
th
+
315
cos
3
th
-
35
cos
th
)
Y
7
1
(
th
,
ph
)
=
-
1
64
105
2
p
e
i
ph
sin
th
(
429
cos
6
th
-
495
cos
4
th
+
135
cos
2
th
-
5
)
Y
7
2
(
th
,
ph
)
=
3
64
35
p
e
2
i
ph
sin
2
th
(
143
cos
5
th
-
110
cos
3
th
+
15
cos
th
)
Y
7
3
(
th
,
ph
)
=
-
3
64
35
2
p
e
3
i
ph
sin
3
th
(
143
cos
4
th
-
66
cos
2
th
+
3
)
Y
7
4
(
th
,
ph
)
=
3
32
385
2
p
e
4
i
ph
sin
4
th
(
13
cos
3
th
-
3
cos
th
)
Y
7
5
(
th
,
ph
)
=
-
3
64
385
2
p
e
5
i
ph
sin
5
th
(
13
cos
2
th
-
1
)
Y
7
6
(
th
,
ph
)
=
3
64
5005
p
e
6
i
ph
sin
6
th
cos
th
Y
7
7
(
th
,
ph
)
=
-
3
64
715
2
p
e
7
i
ph
sin
7
th
{\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}}}
Y
8
-
8
(
th
,
ph
)
=
3
256
12155
2
p
e
-
8
i
ph
sin
8
th
Y
8
-
7
(
th
,
ph
)
=
3
64
12155
2
p
e
-
7
i
ph
sin
7
th
cos
th
Y
8
-
6
(
th
,
ph
)
=
1
128
7293
p
e
-
6
i
ph
sin
6
th
(
15
cos
2
th
-
1
)
Y
8
-
5
(
th
,
ph
)
=
3
64
17017
2
p
e
-
5
i
ph
sin
5
th
(
5
cos
3
th
-
cos
th
)
Y
8
-
4
(
th
,
ph
)
=
3
128
1309
2
p
e
-
4
i
ph
sin
4
th
(
65
cos
4
th
-
26
cos
2
th
+
1
)
Y
8
-
3
(
th
,
ph
)
=
1
64
19635
2
p
e
-
3
i
ph
sin
3
th
(
39
cos
5
th
-
26
cos
3
th
+
3
cos
th
)
Y
8
-
2
(
th
,
ph
)
=
3
128
595
p
e
-
2
i
ph
sin
2
th
(
143
cos
6
th
-
143
cos
4
th
+
33
cos
2
th
-
1
)
Y
8
-
1
(
th
,
ph
)
=
3
64
17
2
p
e
-
i
ph
sin
th
(
715
cos
7
th
-
1001
cos
5
th
+
385
cos
3
th
-
35
cos
th
)
Y
8
0
(
th
,
ph
)
=
1
256
17
p
(
6435
cos
8
th
-
12012
cos
6
th
+
6930
cos
4
th
-
1260
cos
2
th
+
35
)
Y
8
1
(
th
,
ph
)
=
-
3
64
17
2
p
e
i
ph
sin
th
(
715
cos
7
th
-
1001
cos
5
th
+
385
cos
3
th
-
35
cos
th
)
Y
8
2
(
th
,
ph
)
=
3
128
595
p
e
2
i
ph
sin
2
th
(
143
cos
6
th
-
143
cos
4
th
+
33
cos
2
th
-
1
)
Y
8
3
(
th
,
ph
)
=
-
1
64
19635
2
p
e
3
i
ph
sin
3
th
(
39
cos
5
th
-
26
cos
3
th
+
3
cos
th
)
Y
8
4
(
th
,
ph
)
=
3
128
1309
2
p
e
4
i
ph
sin
4
th
(
65
cos
4
th
-
26
cos
2
th
+
1
)
Y
8
5
(
th
,
ph
)
=
-
3
64
17017
2
p
e
5
i
ph
sin
5
th
(
5
cos
3
th
-
cos
th
)
Y
8
6
(
th
,
ph
)
=
1
128
7293
p
e
6
i
ph
sin
6
th
(
15
cos
2
th
-
1
)
Y
8
7
(
th
,
ph
)
=
-
3
64
12155
2
p
e
7
i
ph
sin
7
th
cos
th
Y
8
8
(
th
,
ph
)
=
3
256
12155
2
p
e
8
i
ph
sin
8
th
{\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}}}
Y
9
-
9
(
th
,
ph
)
=
1
512
230945
p
e
-
9
i
ph
sin
9
th
Y
9
-
8
(
th
,
ph
)
=
3
256
230945
2
p
e
-
8
i
ph
sin
8
th
cos
th
Y
9
-
7
(
th
,
ph
)
=
3
512
13585
p
e
-
7
i
ph
sin
7
th
(
17
cos
2
th
-
1
)
Y
9
-
6
(
th
,
ph
)
=
1
128
40755
p
e
-
6
i
ph
sin
6
th
(
17
cos
3
th
-
3
cos
th
)
Y
9
-
5
(
th
,
ph
)
=
3
256
2717
p
e
-
5
i
ph
sin
5
th
(
85
cos
4
th
-
30
cos
2
th
+
1
)
Y
9
-
4
(
th
,
ph
)
=
3
128
95095
2
p
e
-
4
i
ph
sin
4
th
(
17
cos
5
th
-
10
cos
3
th
+
cos
th
)
Y
9
-
3
(
th
,
ph
)
=
1
256
21945
p
e
-
3
i
ph
sin
3
th
(
221
cos
6
th
-
195
cos
4
th
+
39
cos
2
th
-
1
)
Y
9
-
2
(
th
,
ph
)
=
3
128
1045
p
e
-
2
i
ph
sin
2
th
(
221
cos
7
th
-
273
cos
5
th
+
91
cos
3
th
-
7
cos
th
)
Y
9
-
1
(
th
,
ph
)
=
3
256
95
2
p
e
-
i
ph
sin
th
(
2431
cos
8
th
-
4004
cos
6
th
+
2002
cos
4
th
-
308
cos
2
th
+
7
)
Y
9
0
(
th
,
ph
)
=
1
256
19
p
(
12155
cos
9
th
-
25740
cos
7
th
+
18018
cos
5
th
-
4620
cos
3
th
+
315
cos
th
)
Y
9
1
(
th
,
ph
)
=
-
3
256
95
2
p
e
i
ph
sin
th
(
2431
cos
8
th
-
4004
cos
6
th
+
2002
cos
4
th
-
308
cos
2
th
+
7
)
Y
9
2
(
th
,
ph
)
=
3
128
1045
p
e
2
i
ph
sin
2
th
(
221
cos
7
th
-
273
cos
5
th
+
91
cos
3
th
-
7
cos
th
)
Y
9
3
(
th
,
ph
)
=
-
1
256
21945
p
e
3
i
ph
sin
3
th
(
221
cos
6
th
-
195
cos
4
th
+
39
cos
2
th
-
1
)
Y
9
4
(
th
,
ph
)
=
3
128
95095
2
p
e
4
i
ph
sin
4
th
(
17
cos
5
th
-
10
cos
3
th
+
cos
th
)
Y
9
5
(
th
,
ph
)
=
-
3
256
2717
p
e
5
i
ph
sin
5
th
(
85
cos
4
th
-
30
cos
2
th
+
1
)
Y
9
6
(
th
,
ph
)
=
1
128
40755
p
e
6
i
ph
sin
6
th
(
17
cos
3
th
-
3
cos
th
)
Y
9
7
(
th
,
ph
)
=
-
3
512
13585
p
e
7
i
ph
sin
7
th
(
17
cos
2
th
-
1
)
Y
9
8
(
th
,
ph
)
=
3
256
230945
2
p
e
8
i
ph
sin
8
th
cos
th
Y
9
9
(
th
,
ph
)
=
-
1
512
230945
p
e
9
i
ph
sin
9
th
{\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}}}
Y
10
-
10
(
th
,
ph
)
=
1
1024
969969
p
e
-
10
i
ph
sin
10
th
Y
10
-
9
(
th
,
ph
)
=
1
512
4849845
p
e
-
9
i
ph
sin
9
th
cos
th
Y
10
-
8
(
th
,
ph
)
=
1
512
255255
2
p
e
-
8
i
ph
sin
8
th
(
19
cos
2
th
-
1
)
Y
10
-
7
(
th
,
ph
)
=
3
512
85085
p
e
-
7
i
ph
sin
7
th
(
19
cos
3
th
-
3
cos
th
)
Y
10
-
6
(
th
,
ph
)
=
3
1024
5005
p
e
-
6
i
ph
sin
6
th
(
323
cos
4
th
-
102
cos
2
th
+
3
)
Y
10
-
5
(
th
,
ph
)
=
3
256
1001
p
e
-
5
i
ph
sin
5
th
(
323
cos
5
th
-
170
cos
3
th
+
15
cos
th
)
Y
10
-
4
(
th
,
ph
)
=
3
256
5005
2
p
e
-
4
i
ph
sin
4
th
(
323
cos
6
th
-
255
cos
4
th
+
45
cos
2
th
-
1
)
Y
10
-
3
(
th
,
ph
)
=
3
256
5005
p
e
-
3
i
ph
sin
3
th
(
323
cos
7
th
-
357
cos
5
th
+
105
cos
3
th
-
7
cos
th
)
Y
10
-
2
(
th
,
ph
)
=
3
512
385
2
p
e
-
2
i
ph
sin
2
th
(
4199
cos
8
th
-
6188
cos
6
th
+
2730
cos
4
th
-
364
cos
2
th
+
7
)
Y
10
-
1
(
th
,
ph
)
=
1
256
1155
2
p
e
-
i
ph
sin
th
(
4199
cos
9
th
-
7956
cos
7
th
+
4914
cos
5
th
-
1092
cos
3
th
+
63
cos
th
)
Y
10
0
(
th
,
ph
)
=
1
512
21
p
(
46189
cos
10
th
-
109395
cos
8
th
+
90090
cos
6
th
-
30030
cos
4
th
+
3465
cos
2
th
-
63
)
Y
10
1
(
th
,
ph
)
=
-
1
256
1155
2
p
e
i
ph
sin
th
(
4199
cos
9
th
-
7956
cos
7
th
+
4914
cos
5
th
-
1092
cos
3
th
+
63
cos
th
)
Y
10
2
(
th
,
ph
)
=
3
512
385
2
p
e
2
i
ph
sin
2
th
(
4199
cos
8
th
-
6188
cos
6
th
+
2730
cos
4
th
-
364
cos
2
th
+
7
)
Y
10
3
(
th
,
ph
)
=
-
3
256
5005
p
e
3
i
ph
sin
3
th
(
323
cos
7
th
-
357
cos
5
th
+
105
cos
3
th
-
7
cos
th
)
Y
10
4
(
th
,
ph
)
=
3
256
5005
2
p
e
4
i
ph
sin
4
th
(
323
cos
6
th
-
255
cos
4
th
+
45
cos
2
th
-
1
)
Y
10
5
(
th
,
ph
)
=
-
3
256
1001
p
e
5
i
ph
sin
5
th
(
323
cos
5
th
-
170
cos
3
th
+
15
cos
th
)
Y
10
6
(
th
,
ph
)
=
3
1024
5005
p
e
6
i
ph
sin
6
th
(
323
cos
4
th
-
102
cos
2
th
+
3
)
Y
10
7
(
th
,
ph
)
=
-
3
512
85085
p
e
7
i
ph
sin
7
th
(
19
cos
3
th
-
3
cos
th
)
Y
10
8
(
th
,
ph
)
=
1
512
255255
2
p
e
8
i
ph
sin
8
th
(
19
cos
2
th
-
1
)
Y
10
9
(
th
,
ph
)
=
-
1
512
4849845
p
e
9
i
ph
sin
9
th
cos
th
Y
10
10
(
th
,
ph
)
=
1
1024
969969
p
e
10
i
ph
sin
10
th
{\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}}}
Visualization of complex spherical harmonics [ edit ] [ edit ] Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle,
ph
{\displaystyle \phi }
th
{\displaystyle \theta }
The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines.
Visual Array of Complex Spherical Harmonics Represented as 2D Theta/Phi Maps 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 [ edit ] 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 [ edit ] 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]
Y
0
,
0
=
s
=
Y
0
0
=
1
2
1
p
{\displaystyle Y_{0,0}=s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}
Y
1
,
-
1
=
p
y
=
i
1
2
(
Y
1
-
1
+
Y
1
1
)
=
3
4
p
y
r
=
3
4
p
sin
(
th
)
sin
(
ph
)
Y
1
,
0
=
p
z
=
Y
1
0
=
3
4
p
z
r
=
3
4
p
cos
(
th
)
Y
1
,
1
=
p
x
=
1
2
(
Y
1
-
1
-
Y
1
1
)
=
3
4
p
x
r
=
3
4
p
sin
(
th
)
cos
(
ph
)
{\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}}}
Y
2
,
-
2
=
d
x
y
=
i
1
2
(
Y
2
-
2
-
Y
2
2
)
=
1
2
15
p
x
y
r
2
=
1
4
15
p
sin
2
(
th
)
sin
(
2
ph
)
Y
2
,
-
1
=
d
y
z
=
i
1
2
(
Y
2
-
1
+
Y
2
1
)
=
1
2
15
p
y
z
r
2
=
1
4
15
p
sin
(
2
th
)
sin
(
ph
)
Y
2
,
0
=
d
z
2
=
Y
2
0
=
1
4
5
p
3
z
2
-
r
2
r
2
=
1
4
5
p
(
3
cos
2
(
th
)
-
1
)
Y
2
,
1
=
d
x
z
=
1
2
(
Y
2
-
1
-
Y
2
1
)
=
1
2
15
p
x
z
r
2
=
1
4
15
p
sin
(
2
th
)
cos
(
ph
)
Y
2
,
2
=
d
x
2
-
y
2
=
1
2
(
Y
2
-
2
+
Y
2
2
)
=
1
4
15
p
x
2
-
y
2
r
2
=
1
4
15
p
sin
2
(
th
)
cos
(
2
ph
)
{\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}}}
Y
3
,
-
3
=
f
y
(
3
x
2
-
y
2
)
=
i
1
2
(
Y
3
-
3
+
Y
3
3
)
=
1
4
35
2
p
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
p
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
p
y
(
5
z
2
-
r
2
)
r
3
Y
3
,
0
=
f
z
3
=
Y
3
0
=
1
4
7
p
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
p
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
p
(
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
p
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{\s qrt {\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}}}
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{\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 [ edit ] [ edit ] Below the real spherical harmonics are represented on 2D plots with the azimuthal angle,
ph
{\displaystyle \phi }
th
{\displaystyle \theta }
The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines.
Visual Array of Real Spherical Harmonics Represented as 2D Theta/Phi Maps 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 [ edit ] 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 [ edit ] 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
^ 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.; Florez, 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 .
