Angular functions: Difference between revisions
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|+ real spherical harmonics | |+ real spherical harmonics | ||
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! Name | ! Name | ||
! ''Y<sub>lm</sub>'' | ! ''Y<sub>lm</sub>'' | ||
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| 0 || 1 || s || <math>\frac{1}{\sqrt{4\pi}}</math> | | 0 || 1 || s || <math>\frac{1}{\sqrt{4\pi}}</math> | ||
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| 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 || 1 || py || <math>\sqrt{\frac{3}{4\pi}}\frac{x}{r}</math> | | 1 || 1 || py || <math>\sqrt{\frac{3}{4\pi}}\frac{x}{r}</math> | ||
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| 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> | ||
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| 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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|+ hybrid angular functions | |+ hybrid angular functions | ||
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| sp || sp-1 || <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x</math> | | sp || sp-1 || <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x</math> | ||
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| || 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 p_x</math> | ||
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| 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 || sp2-1 || <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y</math> | ||
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| || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math> | | || sp2-2 || <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math> | ||
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| sp3 || sp3-1 || <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 p_x+\rm p_y+\rm p_z)</math> | ||
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| || sp3-4 || <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> | ||
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| 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 p_x+\frac{1}{\sqrt 2}\rm p_y</math> | ||
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| || 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 p_z+\frac{2}{\sqrt 2}\rm d_{z^2}</math> | ||
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| 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 || 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> | ||
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| || 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> | | || 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> | ||
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Revision as of 16:59, 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 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]