Table 1.2.7.1

Coppens, P.

doi:10.1107/97809553602060000550

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.2, pp. 12-13

Table 1.2.7.1

P. Coppens^a ^*

^aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

Table 1.2.7.1 | top | pdf |
Real spherical harmonic functions (x, y, z are direction cosines)

l	Symbol	C^†	Angular function, $[c_{lmp}]$ ^‡	Normalization for wavefunctions, $[M_{lmp}]$ ^§		Normalization for density functions, $[L_{lmp}]$ ^¶
l	Symbol	C^†	Angular function, $[c_{lmp}]$ ^‡	Expression	Numerical value	Expression	Numerical value
0	00	1	1	$[(1/4\pi)^{1/2}]$	0.28209	$[1/4\pi]$	0.07958
1	$[\!\matrix{11+\cr 11-\cr 10\hfill\cr}]$	$[\!\matrix{1\cr 1\cr 1\cr}]$	$[\left.\!\matrix{x\cr y\cr z\cr}\right\}]$	$[(3/4\pi)^{1/2}]$	0.48860	$[1/\pi]$	0.31831
2	20		$[3z^{2} - 1]$	$[(5/16\pi)^{1/2}]$	0.31539	$[\displaystyle{3\sqrt{3} \over 8\pi}]$	0.20675
2	$[\!\matrix{21+\cr 21-\cr 22+\cr 22-\cr}]$	$[\!\matrix{3\cr 3\cr 6\cr 6\cr}]$	$[\left.\!\matrix{xz\cr yz\cr (x^{2} - y^{2})/2\cr xy\cr}\right\}]$	$[(15/4\pi)^{1/2}]$	1.09255		0.75
3	30		$[5z^{3} - 3z]$	$[(7/16\pi)^{1/2}]$	0.37318	$[\displaystyle{10 \over 13\pi}]$	0.24485
	$[\!\matrix{31+\cr 31-\cr}]$	$[\!\matrix{3/2\cr 3/2\cr}]$	$[\left.\!\matrix{x[5z^{2} - 1]\cr y[5z^{2} - 1]\cr}\right\}]$	$[(21/32\pi)^{1/2}]$	0.45705	$[\displaystyle\left(\hbox{ar} + {14 \over 5} - {\pi \over 4}\right)^{-1}]$ ^††	0.32033
	$[\!\matrix{32+\cr 32-\cr}]$	$[\!\matrix{15\cr 15\cr}]$	$[\left.\!\matrix{(x^{2} - y^{2})z\cr 2xyz\cr}\right\}]$	$[(105/16\pi)^{1/2}]$	1.44531	1	1
	$[\!\matrix{33+\cr 33-\cr}]$	$[\!\matrix{15\cr 15\cr}]$	$[\left.\!\matrix{x^{3} - 3xy^{2}\cr -y^{3} + 3x^{2}y\cr}\right\}]$	$[(35/32\pi)^{1/2}]$	0.59004	$[4/3\pi]$	0.42441
4	40		$[35z^{4} - 30z^{2} + 3]$	$[(9/256\pi)^{1/2}]$	0.10579	^‡‡	0.06942
	$[\!\matrix{41+\cr 41-\cr}]$	$[\!\matrix{5/2\cr 5/2\cr}]$	$[\left.\!\matrix{x[7z^{3} - 3z]\cr y[7z^{3} - 3z]\cr}\right\}]$	$[(45/32\pi)^{1/2}]$	0.66905	$[\displaystyle{735 \over 512\sqrt{7} + 196}]$	0.47400
	$[\!\matrix{42+\cr 42-\cr}]$	$[\!\matrix{15/2\cr 15/2\cr}]$	$[\left.\!\matrix{(x^{2} - y^{2})[7z^{2} - 1]\cr 2xy[7z^{2} - 1]\cr}\right\}]$	$[(45/64\pi)^{1/2}]$	0.47309	$[\displaystyle{105\sqrt{7} \over 4(136 + 28\sqrt{7})}]$	0.33059
	$[\!\matrix{43+\cr 43-\cr}]$	$[\!\matrix{105\cr 105\cr}]$	$[\left.\!\matrix{(x^{3} - 3xy^{2})z\cr (-y^{3} + 3x^{2}y)z\cr}\right\}]$	$[(315/32\pi)^{1/2}]$	1.77013		1.25
	$[\!\matrix{44+\cr 44-\cr}]$	$[\!\matrix{105\cr 105\cr}]$	$[\left.\!\matrix{x^{4} - 6x^{2}y^{2} + y^{4}\cr 4x^{3}y - 4xy^{3}\cr}\right\}]$	$[(315/256\pi)^{1/2}]$	0.62584		0.46875
5	50		$[63z^{5} - 70z^{3} - 15z]$	$[(11/256\pi)^{1/2}]$	0.11695	—	0.07674
	$[\!\matrix{51+\cr 51-\cr}]$		$[\left.\!\matrix{(21z^{4} - 14z^{2} + 1)x\cr (21z^{4} - 14z^{2} + 1)y\cr}\right\}]$	$[(165/256\pi)^{1/2}]$	0.45295	—	0.32298
	$[\!\matrix{52+\cr 52-\cr}]$		$[\left.\!\matrix{(3z^{3} - z) (x^{2} - y^{2})\cr 2xy(3z^{3} - z)\cr}\right\}]$	$[(1155/64\pi)^{1/2}]$	2.39677	—	1.68750
	$[\!\matrix{53+\cr 53-\cr}]$		$[\left.\!\matrix{(9z^{2} - 1) (x^{3} - 3xy^{2})\cr (9z^{2} - 1) (3x^{2}y - y^{3})\cr}\right\}]$	$[(385/512\pi)^{1/2}]$	0.48924	—	0.34515
	$[\!\matrix{54+\cr 54-\cr}]$		$[\left.\!\matrix{z(x^{4} - 6x^{2}y^{2} + y^{4})\cr z(4x^{3}y - 4xy^{3})\cr}\right\}]$	$[(3465/256\pi)^{1/2}]$	2.07566	—	1.50000
	$[\!\matrix{55+\cr 55-\cr}]$		$[\left.\!\matrix{x^{5} - 10x^{3}y^{2} + 5xy^{4}\cr 5x^{4} y - 10x^{2}y^{3} + y^{5}\cr}\right\}]$	$[(693/512\pi)^{1/2}]$	0.65638	—	0.50930
6	60		$[231z^{6} - 315z^{4} + 105z^{2} - 5]$	$[(13/1024\pi)^{1/2}]$	0.06357	—	0.04171
	$[\!\matrix{61+\cr 61-\cr}]$		$[\left.\!\matrix{(33z^{5} - 30z^{3} + 5z)x\cr (33z^{5} - 30z^{3} + 5z)y\cr}\right\}]$	$[(273/256\pi)^{1/2}]$	0.58262	—	0.41721
	$[\!\matrix{62+\cr 62-\cr}]$		$[\left.\!\matrix{(33z^{4} - 18z^{2} + 1) (x^{2} - y^{2})\cr 2xy (33z^{4} - 18z^{2} + 1)\cr}\right\}]$	$[(1365/2048\pi)^{1/2}]$	0.46060	—	0.32611
	$[\!\matrix{63+\cr 63-\cr}]$		$[\left.\!\matrix{(11z^{3} - 3z) (x^{3} - 3xy^{2})\cr (11z^{3} - 3z) (3x^{2}y - 3y)\cr}\right\}]$	$[(1365/512\pi)^{1/2}]$	0.92121	—	0.65132
	$[\!\matrix{64+\cr 64-\cr}]$		$[\left.\!\matrix{(11z^{2} - 1) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (11z^{2} - 1) (4x^{3}y - 4xy^{3})\cr}\right\}]$	$[(819/1024\pi)^{1/2}]$	0.50457	—	0.36104
	$[\!\matrix{65+\cr 65-\cr}]$	10395	$[\left.\!\matrix{z(x^{5} - 10x^{3}y^{2} + 5xy^{4})\cr z(5x^{4}y - 10x^{2}y^{3} + y^{5})\cr}\right\}]$	$[(9009/512\pi)^{1/2}]$	2.36662	—	1.75000
	$[\!\matrix{66+\cr 66-\cr}]$	10395	$[\left.\!\matrix{x^{6} - 15x^{4}y^{2} + 15x^{2}y^{4} - y^{6}\cr 6x^{5}y - 20x^{3}y^{3} + 6xy^{5}\cr}\right\}]$	$[(3003/2048\pi)^{1/2}]$	0.68318	—	0.54687
7	70		$[429z^{7} - 693z^{5} + 315z^{3} - 35z]$	$[(15/1024\pi)^{1/2}]$	0.06828	—	0.04480
	$[\!\matrix{71+\cr 71-\cr}]$		$[\left.\!\matrix{(429z^{6} - 495z^{4} + 135z^{2} - 5)x\cr (429z^{6} - 495z^{4} + 135z^{2} - 5)y\cr}\right\}]$	$[(105/4096\pi)^{1/2}]$	0.09033	—	0.06488
	$[\!\matrix{72+\cr 72-\cr}]$		$[\left.\!\matrix{(143z^{5} - 110z^{3} + 15z) (x^{2} - y^{2})\cr 2xy(143z^{5} - 110z^{3} + 15z)\cr}\right\}]$	$[(315/2048\pi)^{1/2}]$	0.22127	—	0.15732
	$[\!\matrix{73+\cr 73-\cr}]$		$[\left.\!\matrix{(143z^{4} - 66z^{2} + 3) (x^{3} - 3xy^{2})\cr (143z^{4} - 66z^{2} + 3) (3x^{2}y - y^{3})\cr}\right\}]$	$[(315/4096\pi)^{1/2}]$	0.15646	—	0.11092
	$[\!\matrix{74+\cr 74-\cr}]$		$[\left.\!\matrix{(13z^{3} - 3z) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (13z^{3} - 3z) (4x^{3}y - 4xy^{3})\cr}\right\}]$	$[(3465/1024\pi)^{1/2}]$	1.03783	—	0.74044
	$[\!\matrix{75+\cr 75-\cr}]$		$[\left.\!\matrix{(13z^{3} - 1) (x^{5} - 10x^{3}y^{2} + 5xy^{4})\cr (13z^{3} - 1) (5x^{4}y - 10x^{2}y^{3} + y^{5})\cr}\right\}]$	$[(3465/4096\pi)^{1/2}]$	0.51892	—	0.37723
	$[\!\matrix{76+\cr 76-\cr}]$	135135	$[\left.\!\matrix{z(x^{6} - 15x^{4}y^{2} + 15x^{2}y^{4} - y^{6})\cr z(6x^{5}y + 20x^{3}y^{3} - 6xy^{5})\cr}\right\}]$	$[(45045/2048\pi)^{1/2}]$	2.6460	—	2.00000
	$[\!\matrix{77+\cr 77-\cr}]$	135135	$[\left.\!\matrix{x^{7} - 21x^{5}y^{2} + 35x^{3}y^{4} - 7xy^{6}\cr 7x^{6}y - 35x^{4}y^{3} + 21x^{2}y^{5} - y^{7}\cr}\right\}]$	$[(6435/4096\pi)^{1/2}]$	0.70716	—	0.58205

^† Common factor such that $[C_{lm}c_{lmp} = P_{l}^{m} (\cos \theta)_{\sin m\varphi}^{\cos m\varphi}.]$
^‡ $[x = \sin \theta \cos \varphi]$ , $[y = \sin \theta \sin \varphi]$ , $[z = \cos \theta]$ .
^§As defined by $[y_{lmp} = M_{lmp}c_{lmp}]$ where $[c_{lmp}]$ are Cartesian functions.
^¶Paturle & Coppens (1988)

, as defined by $[d_{lmp} = L_{lmp}c_{lmp}]$ where $[c_{lmp}]$ are Cartesian functions.
^††ar = arctan (2).
^‡‡ $[N_{\rm ang} = \{(14A_{-}^{5} - 14A_{+}^{5} + 20A_{+}^{3} - 20A_{-}^{3} + 6A_{-} - 6A_{+}) 2\pi\}^{-1}]$ where $[A_{\pm} = [(30\pm \sqrt{480})/70]^{1/2}]$ .