Angular functions: Difference between revisions

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:{| border="1" cellspacing="0" cellpadding="5"
:{| 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>
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|  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 || py || <math>\sqrt{\frac{3}{4\pi}}\frac{x}{r}</math>
|  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>
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|-
|-
|  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>
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|-
|-
|  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="1" cellspacing="0" cellpadding="5"
:{| 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 px</math>
| 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>
|-
|-
|   || sp-2 || <math>\frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\rm px</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>
|-
|-
| sp2 || sp2-1 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px+\frac{1}{\sqrt 2}\rm py</math>
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|-
|-
|     || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px-\frac{1}{\sqrt 2}\rm py</math>
| sp3 || sp3-1 || <math>\frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z)</math>
|-
|-
|    || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm px</math>
|    || sp3-2 || <math>\frac{1}{2}(\rm s+\rm p_x-\rm p_y-\rm p_z)</math>
 
|-
|-
| sp3 || sp3-1 || <math>\frac{1}{2}(\rm s+\rm px+\rm py+\rm pz)</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 px-\rm py-\rm pz)</math>
|    || sp3-4 || <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 px+\rm py-\rm pz)</math>
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|style="border-bottom:1px solid"|
|-
|-
|     || sp3-4 || <math>\frac{1}{2}(\rm s-\rm px-\rm py+\rm pz)</math>
| 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 || sp3d-1 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px+\frac{1}{\sqrt 2}\rm py</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-2 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm px-\frac{1}{\sqrt 2}\rm py</math>
|      || sp3d-3 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math>
|-
|-
|      || sp3d-3 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm px</math>
|      || sp3d-4 || <math>\frac{1}{\sqrt 2}\rm p_z+\frac{1}{\sqrt 2}\rm d_{z^2}</math>
|-
|-
|      || sp3d-4 || <math>\frac{1}{\sqrt 2}\rm pz+\frac{1}{\sqrt 2}\rm dz2</math>
|      || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2}</math>
|-
|-
|     || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm pz+\frac{2}{\sqrt 2}\rm dz2</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 px-\frac{1}{\sqrt 12}\rm dz2+\frac{1}{2}\rm dx2-y2</math>
| 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 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 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 py-\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 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 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 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 pz+\frac{1}{\sqrt 3}\rm dz2</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 pz+\frac{1}{\sqrt 3}\rm dz2</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]
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