Angular functions: Difference between revisions
Vaspmaster (talk | contribs) No edit summary |
Vaspmaster (talk | contribs) No edit summary |
||
| Line 77: | Line 77: | ||
|- | |- | ||
| || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm pz+\frac{2}{\sqrt 2}\rm dz2</math> | | || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm pz+\frac{2}{\sqrt 2}\rm dz2</math> | ||
|- | |||
| sp3d2 || sp3d2-1 || <math>\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm px-\frac{1}{\sqrt 12}\rm dz2+\frac{1}{2}\rm dx2-y2</math> | |||
|- | |||
| || sp3d2-2 || <math>\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm px-\frac{1}{\sqrt 12}\rm dz2+\frac{1}{2}\rm dx2-y2</math> | |||
|- | |||
| || sp3d2-3 || <math>\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm py-\frac{1}{\sqrt 12}\rm dz2-\frac{1}{2}\rm dx2-y2</math> | |||
|- | |||
| || sp3d2-4 || <math>\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm py-\frac{1}{\sqrt 12}\rm dz2-\frac{1}{2}\rm dx2-y2</math> | |||
|- | |||
| || sp3d2-5 || <math>\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 3}\rm dz2</math> | |||
|- | |||
| || sp3d2-6 || <math>\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 3}\rm dz2</math> | |||
|} | |} | ||
Revision as of 15:49, 13 January 2017
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] sp3d sp3d-1 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px+\frac{1}{\sqrt 2}\rm py }[/math] sp3d-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px-\frac{1}{\sqrt 2}\rm py }[/math] sp3d-3 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm px }[/math] sp3d-4 [math]\displaystyle{ \frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 2}\rm dz2 }[/math] sp3d-5 [math]\displaystyle{ -\frac{1}{\sqrt 2}\rm pz+\frac{2}{\sqrt 2}\rm dz2 }[/math] sp3d2 sp3d2-1 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm px-\frac{1}{\sqrt 12}\rm dz2+\frac{1}{2}\rm dx2-y2 }[/math] sp3d2-2 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm px-\frac{1}{\sqrt 12}\rm dz2+\frac{1}{2}\rm dx2-y2 }[/math] sp3d2-3 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm py-\frac{1}{\sqrt 12}\rm dz2-\frac{1}{2}\rm dx2-y2 }[/math] sp3d2-4 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm py-\frac{1}{\sqrt 12}\rm dz2-\frac{1}{2}\rm dx2-y2 }[/math] sp3d2-5 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 3}\rm dz2 }[/math] sp3d2-6 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 3}\rm dz2 }[/math]