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Requests for technical support from the VASP group should be posted in the
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Angular functions
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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 {1}}2}}{\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 {1}}2}}{\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 {1}}2}}{\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 {1}}2}}{\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}}}}}}}}}
