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From VASP Wiki
real spherical harmonics
l
m
Name
Y
lm
0
1
s
1
4
π
{\displaystyle \frac{1}{\sqrt{4\pi}}}
1
-1
py
3
4
π
y
r
{\displaystyle \sqrt{\frac{3}{4\pi}}\frac{y}{r}}
1
0
pz
3
4
π
z
r
{\displaystyle \sqrt{\frac{3}{4\pi}}\frac{z}{r}}
1
1
px
3
4
π
x
r
{\displaystyle \sqrt{\frac{3}{4\pi}}\frac{x}{r}}
2
-2
dxy
1
2
15
π
x
y
r
2
{\displaystyle \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{xy}{r^2}}
2
-1
dyz
1
2
15
π
y
z
r
2
{\displaystyle \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{yz}{r^2}}
2
0
dz2
1
4
5
π
3
z
2
−
r
2
r
2
{\displaystyle \frac{1}{4}\sqrt{\frac{5}{\pi}}\frac{3z^2-r^2}{r^2}}
2
1
dxz
1
2
15
π
z
x
r
2
{\displaystyle \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{zx}{r^2}}
2
2
dx2-y2
1
4
15
π
x
2
−
y
2
r
2
{\displaystyle \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2}}
3
-3
fy(3x2-y2)
1
4
35
2
π
(
3
x
2
−
y
2
)
y
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(3x^2-y^2)y}{r^3}}
3
-2
fxyz
1
2
105
π
x
y
z
r
3
{\displaystyle \frac{1}{2}\sqrt{\frac{105}{\pi}}\frac{xyz}{r^3}}
3
-1
fyz2
1
4
21
2
π
(
5
z
2
−
r
2
)
y
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)y}{r^3}}
3
0
fz3
1
4
7
π
(
5
z
2
−
3
r
2
)
z
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{7}{\pi}}\frac{(5z^2-3r^2)z}{r^3}}
3
1
fxz2
1
4
21
2
π
(
5
z
2
−
r
2
)
x
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)x}{r^3}}
3
2
fz(x2-y2)
1
4
105
π
(
x
2
−
y
2
)
z
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{105}{\pi}}\frac{(x^2-y^2)z}{r^3}}
3
3
fx(x2-3y2)
1
4
35
2
π
(
x
2
−
3
y
2
)
x
r
3
{\displaystyle \frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(x^2-3y^2)x}{r^3}}
hybrid angular functions
sp
sp-1
1
2
s
+
1
2
p
x
{\displaystyle \frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x}
sp-2
1
2
s
−
1
2
p
x
{\displaystyle \frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm p_x}
sp2
sp2-1
1
3
s
−
1
6
p
x
+
1
2
p
y
{\displaystyle \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y}
sp2-2
1
3
s
−
1
6
p
x
−
1
2
p
y
{\displaystyle \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y}
sp2-2
1
3
s
+
2
6
p
x
{\displaystyle \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x}
sp3
sp3-1
1
2
(
s
+
p
x
+
p
y
+
p
z
)
{\displaystyle \frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z)}
sp3-2
1
2
(
s
+
p
x
−
p
y
−
p
z
)
{\displaystyle \frac{1}{2}(\rm s+\rm p_x-\rm p_y-\rm p_z)}
sp3-2
1
2
(
s
−
p
x
+
p
y
−
p
z
)
{\displaystyle \frac{1}{2}(\rm s-\rm p_x+\rm p_y-\rm p_z)}
sp3-4
1
2
(
s
−
p
x
−
p
y
+
p
z
)
{\displaystyle \frac{1}{2}(\rm s-\rm p_x-\rm p_y+\rm p_z)}
sp3d
sp3d-1
1
3
s
−
1
6
p
x
+
1
2
p
y
{\displaystyle \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y}
sp3d-2
1
3
s
−
1
6
p
x
−
1
2
p
y
{\displaystyle \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y}
sp3d-3
1
3
s
+
2
6
p
x
{\displaystyle \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x}
sp3d-4
1
2
p
z
+
1
2
d
z
2
{\displaystyle \frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 2}\rm d_{z^2}}
sp3d-5
−
1
2
p
z
+
2
2
d
z
2
{\displaystyle -\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2}}
sp3d2
sp3d2-1
1
6
s
−
1
2
p
x
−
1
1
2
d
z
2
+
1
2
d
x
2
−
y
2
{\displaystyle \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_x-\frac{1}{\sqrt 12}\rm d_{z^2}+\frac{1}{2}\rm d_{x^2-y^2}}
sp3d2-2
1
6
s
+
1
2
p
x
−
1
1
2
d
z
2
+
1
2
d
x
2
−
y
2
{\displaystyle \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_x-\frac{1}{\sqrt 12}\rm d_{z^2}+\frac{1}{2}\rm d_{x^2-y^2}}
sp3d2-3
1
6
s
−
1
2
p
y
−
1
1
2
d
z
2
−
1
2
d
x
2
−
y
2
{\displaystyle \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_y-\frac{1}{\sqrt 12}\rm d_{z^2}-\frac{1}{2}\rm d_{x^2-y^2}}
sp3d2-4
1
6
s
+
1
2
p
y
−
1
1
2
d
z
2
−
1
2
d
x
2
−
y
2
{\displaystyle \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_y-\frac{1}{\sqrt 12}\rm d_{z^2}-\frac{1}{2}\rm d_{x^2-y^2}}
sp3d2-5
1
6
s
−
1
2
p
z
+
1
3
d
z
2
{\displaystyle \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}}
sp3d2-6
1
6
s
+
1
2
p
z
+
1
3
d
z
2
{\displaystyle \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}}
