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"
|-
|-
! ''l''
|+ real spherical harmonics
! ''m''
! align="left"|''l''
! Name
! align="left"|''m''
! ''Y<sub>lm</sub>''
! align="left" width="80"|Name
! 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 ||  0 || pz || <math>\sqrt{\frac{3}{4\pi}}\frac{z}{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 || -1 || dyz    || <math>\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{yz}{r^2}</math>
|-
|  2 ||  0 || dz2    || <math>\frac{1}{4}\sqrt{\frac{5}{\pi}}\frac{3z^2-r^2}{r^2}</math>
|-
|  2 ||  1 || dxz    || <math>\frac{1}{2}\sqrt{\frac{15}{\pi}}\frac{zx}{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 || -2 || fxyz      || <math>\frac{1}{2}\sqrt{\frac{105}{\pi}}\frac{xyz}{r^3}</math>
|-
|  3 || -1 || fyz2      || <math>\frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)y}{r^3}</math>
|-
|  3 ||  0 || fz3        || <math>\frac{1}{4}\sqrt{\frac{7}{\pi}}\frac{(5z^2-3r^2)z}{r^3}</math>
|-
|  3 ||  1 || fxz2      || <math>\frac{1}{4}\sqrt{\frac{21}{2\pi}}\frac{(5z^2-r^2)x}{r^3}</math>
|-
|  3 ||  2 || fz(x2-y2)  || <math>\frac{1}{4}\sqrt{\frac{105}{\pi}}\frac{(x^2-y^2)z}{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="0" cellspacing="0" cellpadding="5" style="border:1px solid"
|-
|+ hybrid angular functions
|-
| 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
1 -1 py
1 0 pz
1 1 px
2 -2 dxy
2 -1 dyz
2 0 dz2
2 1 dxz
2 2 dx2-y2
3 -3 fy(3x2-y2)
3 -2 fxyz
3 -1 fyz2
3 0 fz3
3 1 fxz2
3 2 fz(x2-y2)
3 3 fx(x2-3y2)


hybrid angular functions
sp sp-1
sp-2
sp2 sp2-1
sp2-2
sp2-2
sp3 sp3-1
sp3-2
sp3-2
sp3-4
sp3d sp3d-1
sp3d-2
sp3d-3
sp3d-4
sp3d-5
sp3d2 sp3d2-1
sp3d2-2
sp3d2-3
sp3d2-4
sp3d2-5
sp3d2-6
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