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:
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
z
=
r
cos
θ
Y
0
0
(
θ
,
φ
)
=
1
2
1
π
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
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
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
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
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
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
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
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
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
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
For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f, g) is reported as well.
Y
00
=
s
=
Y
0
0
=
1
2
1
π
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
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
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
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