Angular functions

real spherical harmonics

l	m	Name	Ylm
0	1	s	$\frac{1}{\sqrt{4\pi}}$
1	-1	py	$\sqrt{\frac{3}{4\pi}}\frac{y}{r}$
1	0	pz	$\sqrt{\frac{3}{4\pi}}\frac{z}{r}$
1	1	py	$\sqrt{\frac{3}{4\pi}}\frac{x}{r}$
2	-2	dxy	$\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{xy}{r^2}$
2	-1	dyz	$\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{yz}{r^2}$
2	0	dz2	$\frac{1}{4}\sqrt{\frac{5}{\pi}}\frac{3z^2-r^2}{r^2}$
2	1	dxz	$\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{zx}{r^2}$
2	2	dx2-y2	$\frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2}$
3	-3	fy(3x2-y2)	$\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(3x^2-y^2)y}{r^3}$
3	-2	fxyz	$\frac{1}{2}\sqrt{\frac{105}{\pi}}\frac{xyz}{r^3}$
3	-1	fyz2	$\frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)y}{r^3}$
3	0	fz3	$\frac{1}{4}\sqrt{\frac{7}{\pi}}\frac{(5z^2-3r^2)z}{r^3}$
3	1	fxz2	$\frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)x}{r^3}$
3	2	fz(x2-y2)	$\frac{1}{4}\sqrt{\frac{105}{\pi}}\frac{(x^2-y^2)z}{r^3}$
3	3	fx(x2-3y2)	$\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(x^2-3y^2)x}{r^3}$

hybrid angular functions

sp	sp-1	$\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x$
	sp-2	$\frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm p_x$
sp2	sp2-1	$\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y$
	sp2-2	$\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y$
	sp2-2	$\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x$
sp3	sp3-1	$\frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z)$
	sp3-2	$\frac{1}{2}(\rm s+\rm p_x-\rm p_y-\rm p_z)$
	sp3-2	$\frac{1}{2}(\rm s-\rm p_x+\rm p_y-\rm p_z)$
	sp3-4	$\frac{1}{2}(\rm s-\rm p_x-\rm p_y+\rm p_z)$
sp3d	sp3d-1	$\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y$
	sp3d-2	$\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y$
	sp3d-3	$\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x$
	sp3d-4	$\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 2}\rm d_{z^2}$
	sp3d-5	$-\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2}$
sp3d2	sp3d2-1	$\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_x-\frac{1}{\sqrt 12}\rm d_{z^2}+\frac{1}{2}\rm d_{x^2-y^2}$
	sp3d2-2	$\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_x-\frac{1}{\sqrt 12}\rm d_{z^2}+\frac{1}{2}\rm d_{x^2-y^2}$
	sp3d2-3	$\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_y-\frac{1}{\sqrt 12}\rm d_{z^2}-\frac{1}{2}\rm d_{x^2-y^2}$
	sp3d2-4	$\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_y-\frac{1}{\sqrt 12}\rm d_{z^2}-\frac{1}{2}\rm d_{x^2-y^2}$
	sp3d2-5	$\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}$
	sp3d2-6	$\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}$