Angular functions: Difference between revisions
Vaspmaster (talk | contribs) No edit summary |
No edit summary |
||
| (10 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
:{| border=" | :{| border="0" cellspacing="0" cellpadding="5" style="border:1px solid" | ||
|- | |- | ||
|+ real spherical harmonics | |+ real spherical harmonics | ||
! ''l'' | ! align="left"|''l'' | ||
! ''m'' | ! align="left"|''m'' | ||
! Name | ! align="left" width="80"|Name | ||
! ''Y<sub>lm</sub>'' | ! align="left"|''Y<sub>lm</sub>'' | ||
|- | |- | ||
| 0 || 1 || s || <math>\frac{1}{\sqrt{4\pi}}</math> | |style="border-bottom:1px solid"| | ||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| 0 || 1 || s ||<math>\frac{1}{\sqrt{4\pi}}</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |- | ||
| 1 || -1 || py || <math>\sqrt{\frac{3}{4\pi}}\frac{y}{r}</math> | | 1 || -1 || py || <math>\sqrt{\frac{3}{4\pi}}\frac{y}{r}</math> | ||
| Line 13: | Line 23: | ||
| 1 || 0 || pz || <math>\sqrt{\frac{3}{4\pi}}\frac{z}{r}</math> | | 1 || 0 || pz || <math>\sqrt{\frac{3}{4\pi}}\frac{z}{r}</math> | ||
|- | |- | ||
| 1 || 1 || | | 1 || 1 || px || <math>\sqrt{\frac{3}{4\pi}}\frac{x}{r}</math> | ||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |- | ||
| 2 || -2 || dxy || <math>\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{xy}{r^2}</math> | | 2 || -2 || dxy || <math>\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{xy}{r^2}</math> | ||
| Line 25: | Line 39: | ||
|- | |- | ||
| 2 || 2 || dx2-y2 || <math>\frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2}</math> | | 2 || 2 || dx2-y2 || <math>\frac{1}{4}\sqrt{\frac{15}{\pi}}\frac{x^2-y^2}{r^2}</math> | ||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |- | ||
| 3 || -3 || fy(3x2-y2) || <math>\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(3x^2-y^2)y}{r^3}</math> | | 3 || -3 || fy(3x2-y2) || <math>\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(3x^2-y^2)y}{r^3}</math> | ||
| Line 40: | Line 58: | ||
|- | |- | ||
| 3 || 3 || fx(x2-3y2) || <math>\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(x^2-3y^2)x}{r^3}</math> | | 3 || 3 || fx(x2-3y2) || <math>\frac{1}{4}\sqrt{\frac{35}{2\pi}}\frac{(x^2-3y^2)x}{r^3}</math> | ||
|} | |||
:{| border=" | :{| border="0" cellspacing="0" cellpadding="5" style="border:1px solid" | ||
|- | |- | ||
|+ hybrid angular functions | |+ hybrid angular functions | ||
|- | |- | ||
| sp || sp-1 || <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm | | sp | ||
|align="left" width="80"| sp-1 || <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| || sp-2 || <math>\frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm p_x</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| sp2 || sp2-1 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y</math> | |||
|- | |||
| || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y</math> | |||
|- | |||
| || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| sp3 || sp3-1 || <math>\frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z)</math> | |||
|- | |||
| || sp3-2 || <math>\frac{1}{2}(\rm s+\rm p_x-\rm p_y-\rm p_z)</math> | |||
|- | |||
| || sp3-2 || <math>\frac{1}{2}(\rm s-\rm p_x+\rm p_y-\rm p_z)</math> | |||
|- | |||
| || sp3-4 || <math>\frac{1}{2}(\rm s-\rm p_x-\rm p_y+\rm p_z)</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| sp3d || sp3d-1 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y</math> | |||
|- | |||
| || sp3d-2 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y</math> | |||
|- | |- | ||
| | | || sp3d-3 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math> | ||
|- | |||
| || sp3d-4 || <math>\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 2}\rm d_{z^2}</math> | |||
|- | |||
| || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2}</math> | |||
|- | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|style="border-bottom:1px solid"| | |||
|- | |||
| sp3d2 || sp3d2-1 || <math>\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}</math> | |||
|- | |||
| || sp3d2-2 || <math>\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}</math> | |||
|- | |||
| || sp3d2-3 || <math>\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}</math> | |||
|- | |- | ||
| | | || sp3d2-4 || <math>\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}</math> | ||
|- | |- | ||
| | | || sp3d2-5 || <math>\frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}</math> | ||
|- | |- | ||
| | | || sp3d2-6 || <math>\frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2}</math> | ||
|} | |} | ||
Latest revision as of 11:03, 21 June 2018
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 px [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 p_x }[/math] sp-2 [math]\displaystyle{ \frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm p_x }[/math] sp2 sp2-1 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y }[/math] sp2-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y }[/math] sp2-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x }[/math] sp3 sp3-1 [math]\displaystyle{ \frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z) }[/math] sp3-2 [math]\displaystyle{ \frac{1}{2}(\rm s+\rm p_x-\rm p_y-\rm p_z) }[/math] sp3-2 [math]\displaystyle{ \frac{1}{2}(\rm s-\rm p_x+\rm p_y-\rm p_z) }[/math] sp3-4 [math]\displaystyle{ \frac{1}{2}(\rm s-\rm p_x-\rm p_y+\rm p_z) }[/math] sp3d sp3d-1 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y }[/math] sp3d-2 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x-\frac{1}{\sqrt 2}\rm p_y }[/math] sp3d-3 [math]\displaystyle{ \frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x }[/math] sp3d-4 [math]\displaystyle{ \frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 2}\rm d_{z^2} }[/math] sp3d-5 [math]\displaystyle{ -\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2} }[/math] sp3d2 sp3d2-1 [math]\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} }[/math] sp3d2-2 [math]\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} }[/math] sp3d2-3 [math]\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} }[/math] sp3d2-4 [math]\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} }[/math] sp3d2-5 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s-\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2} }[/math] sp3d2-6 [math]\displaystyle{ \frac{1}{\sqrt 6}\rm s+\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 3}\rm d_{z^2} }[/math]