Angular functions

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real spherical harmonics
l m Name Ylm
0 1 s [math]\displaystyle{ \frac{1}{\sqrt{4\pi}} }[/math]
1 -1 py [math]\displaystyle{ \sqrt{\frac{3}{4\pi}}\frac{y}{r} }[/math]
1 0 pz [math]\displaystyle{ \sqrt{\frac{3}{4\pi}}\frac{z}{r} }[/math]
1 1 py [math]\displaystyle{ \sqrt{\frac{3}{4\pi}}\frac{x}{r} }[/math]
2 -2 dxy [math]\displaystyle{ \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{xy}{r^2} }[/math]
2 -1 dyz [math]\displaystyle{ \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{yz}{r^2} }[/math]
2 0 dz2 [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{5}{\pi}}\frac{3z^2-r^2}{r^2} }[/math]
2 1 dxz [math]\displaystyle{ \frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{zx}{r^2} }[/math]
2 2 dx2-y2 [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2} }[/math]
3 -3 fy(3x2-y2) [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(3x^2-y^2)y}{r^3} }[/math]
3 -2 fxyz [math]\displaystyle{ \frac{1}{2}\sqrt{\frac{105}{\pi}}\frac{xyz}{r^3} }[/math]
3 -1 fyz2 [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)y}{r^3} }[/math]
3 0 fz3 [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{7}{\pi}}\frac{(5z^2-3r^2)z}{r^3} }[/math]
3 1 fxz2 [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)x}{r^3} }[/math]
3 2 fz(x2-y2) [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{105}{\pi}}\frac{(x^2-y^2)z}{r^3} }[/math]
3 3 fx(x2-3y2) [math]\displaystyle{ \frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(x^2-3y^2)x}{r^3} }[/math]
hybrid angular functions
sp sp-1 [math]\displaystyle{ \frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm px }[/math]
sp-2 [math]\displaystyle{ \frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm px }[/math]
sp2 sp2-1 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px+\frac{1}{\sqrt 2}\rm py }[/math]
sp2-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px-\frac{1}{\sqrt 2}\rm py }[/math]
sp2-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm px }[/math]
sp3 sp3-1 [math]\displaystyle{ \frac{1}{2}(\rm s+\rm px+\rm py+\rm pz) }[/math]
sp3-2 [math]\displaystyle{ \frac{1}{2}(\rm s+\rm px-\rm py-\rm pz) }[/math]
sp3-2 [math]\displaystyle{ \frac{1}{2}(\rm s-\rm px+\rm py-\rm pz) }[/math]
sp3-4 [math]\displaystyle{ \frac{1}{2}(\rm s-\rm px-\rm py+\rm pz) }[/math]