Angular functions - Revision history

Mani at 11:03, 21 June 2018

2018-06-21T11:03:46Z

← Older revision		Revision as of 11:03, 21 June 2018
Line 23:		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 \|\| 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"\|

Vaspmaster at 17:53, 13 January 2017

2017-01-13T17:53:14Z

← Older revision		Revision as of 17:53, 13 January 2017
Line 65:		Line 65:
	\|+ hybrid angular functions		\|+ hybrid angular functions
	\|-		\|-
	\| sp \|\| sp-1 \|\| <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm p_x</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"\|

Vaspmaster at 17:33, 13 January 2017

2017-01-13T17:33:00Z

← Older revision		Revision as of 17:33, 13 January 2017
Line 4:		Line 4:
	! align="left"\|''l''		! align="left"\|''l''
	! align="left"\|''m''		! align="left"\|''m''
	! align="left"\|Name		! align="left" width="80"\|Name
	! align="left"\|''Y<sub>lm</sub>''		! align="left"\|''Y<sub>lm</sub>''
	\|-		\|-

Vaspmaster at 17:22, 13 January 2017

2017-01-13T17:22:36Z

← Older revision		Revision as of 17:22, 13 January 2017
Line 58:		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="0" cellspacing="0" cellpadding="5" style="border:1px solid"		:{\| border="0" cellspacing="0" cellpadding="5" style="border:1px solid"

Vaspmaster at 17:22, 13 January 2017

2017-01-13T17:22:07Z

← Older revision		Revision as of 17:22, 13 January 2017
Line 2:		Line 2:
	\|-		\|-
	\|+ real spherical harmonics		\|+ real spherical harmonics
	! ''l''		! align="left"\|''l''
	! ''m''		! align="left"\|''m''
	! Name		! align="left"\|Name
	! ''Y<sub>lm</sub>''		! align="left"\|''Y<sub>lm</sub>''
	\|-		\|-
	\|style="border-bottom:1px solid"\|		\|style="border-bottom:1px solid"\|

Vaspmaster at 17:08, 13 January 2017

2017-01-13T17:08:48Z

← Older revision		Revision as of 17:08, 13 January 2017
Line 12:		Line 12:
	\|style="border-bottom:1px solid"\|		\|style="border-bottom:1px solid"\|
	\|-		\|-
	\| 0 \|\| 1 \|\| s \|\| <math>\frac{1}{\sqrt{4\pi}}</math>		\| 0 \|\| 1 \|\| s \|\|<math>\frac{1}{\sqrt{4\pi}}</math>
	\|-		\|-
	\|style="border-bottom:1px solid"\|		\|style="border-bottom:1px solid"\|

Vaspmaster at 16:59, 13 January 2017

2017-01-13T16:59:19Z

← Older revision		Revision as of 16:59, 13 January 2017
Line 1:		Line 1:
	:{\| border="1" cellspacing="0" cellpadding="5"		:{\| border="0" cellspacing="0" cellpadding="5" style="border:1px solid"
	\|-		\|-
	\|+ real spherical harmonics		\|+ real spherical harmonics
Line 6:		Line 6:
	! Name		! Name
	! ''Y<sub>lm</sub>''		! ''Y<sub>lm</sub>''
			\|-
			\|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>		\| 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 14:		Line 24:
	\|-		\|-
	\| 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>
			\|-
			\|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 43:		Line 61:
	\|}		\|}

	:{\| 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 p_x</math>		\| sp \|\| 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 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 \|\| sp2-1 \|\| <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y</math>
Line 57:		Line 84:
	\|-		\|-
	\| \|\| 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>
			\|-
			\|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 \|\| sp3-1 \|\| <math>\frac{1}{2}(\rm s+\rm p_x+\rm p_y+\rm p_z)</math>
Line 66:		Line 97:
	\|-		\|-
	\| \|\| 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>
			\|-
			\|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 \|\| sp3d-1 \|\| <math>\frac{1}{\sqrt 3}\rm s-\frac{1}{\sqrt 6}\rm p_x+\frac{1}{\sqrt 2}\rm p_y</math>
Line 77:		Line 112:
	\|-		\|-
	\| \|\| 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>
			\|-
			\|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 \|\| 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>
Line 90:		Line 129:
	\|-		\|-
	\| \|\| 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>

	\|}		\|}

Vaspmaster at 16:01, 13 January 2017

2017-01-13T16:01:46Z

Vaspmaster at 15:49, 13 January 2017

2017-01-13T15:49:17Z

@@ Line 77: / Line 77: @@
 |-
 |      || sp3d-5 || <math>-\frac{1}{\sqrt 2}\rm pz+\frac{2}{\sqrt 2}\rm dz2</math>
 |}

Vaspmaster at 15:41, 13 January 2017

2017-01-13T15:41:14Z

@@ Line 66: / Line 66: @@
 |-
 |     || sp3-4 || <math>\frac{1}{2}(\rm s-\rm px-\rm py+\rm pz)</math>
 |}

← Older revision		Revision as of 17:33, 13 January 2017
Line 4:		Line 4:
	! align="left"\|''l''		! align="left"\|''l''
	! align="left"\|''m''		! align="left"\|''m''
	! align="left"\|Name		! align="left" width="80"\|Name
	! align="left"\|''Y<sub>lm</sub>''		! align="left"\|''Y<sub>lm</sub>''
	\|-		\|-

← Older revision		Revision as of 16:01, 13 January 2017
Line 47:		Line 47:
	\|+ hybrid angular functions		\|+ hybrid angular functions
	\|-		\|-
	\| sp \|\| sp-1 \|\| <math>\frac{1}{\sqrt 2}\rm s+\frac{1}{\sqrt 2}\rm px</math>		\| sp \|\| sp-1 \|\| <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>		\| \|\| sp-2 \|\| <math>\frac{1}{\sqrt 2}\rm s-\frac{1}{\sqrt 2}\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>		\| 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 px-\frac{1}{\sqrt 2}\rm py</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 px</math>		\| \|\| sp2-2 \|\| <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm p_x</math>

	\|-		\|-
	\| sp3 \|\| sp3-1 \|\| <math>\frac{1}{2}(\rm s+\rm px+\rm py+\rm pz)</math>		\| 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 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>		\| \|\| sp3-2 \|\| <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 px-\rm py+\rm pz)</math>		\| \|\| sp3-4 \|\| <math>\frac{1}{2}(\rm s-\rm p_x-\rm p_y+\rm p_z)</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 \|\| 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 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-3 \|\| <math>\frac{1}{\sqrt 3}\rm s+\frac{2}{\sqrt 6}\rm px</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 pz+\frac{1}{\sqrt 2}\rm ~~dz2~~</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 pz+\frac{2}{\sqrt 2}\rm ~~dz2~~</math>		\| \|\| sp3d-5 \|\| <math>-\frac{1}{\sqrt 2}\rm p_z+\frac{2}{\sqrt 2}\rm d_{z^2}</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 \|\| 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>

	\|}		\|}