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

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

Table 1.2.7.1 

P. Coppensa*

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)

lSymbol CAngular function, [c_{lmp}]Normalization for wavefunctions, [M_{lmp}]§Normalization for density functions, [L_{lmp}]
ExpressionNumerical valueExpressionNumerical 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 [1/2] [3z^{2} - 1] [(5/16\pi)^{1/2}] 0.31539 [\displaystyle{3\sqrt{3} \over 8\pi}] 0.20675
[\!\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 [3/4] 0.75
3 30 [1/2] [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 [1/8] [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 [5/4] 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 [15/32] 0.46875
5 50 [1/8] [63z^{5} - 70z^{3} - 15z] [(11/256\pi)^{1/2}] 0.11695 0.07674
[\!\matrix{51+\cr 51-\cr}] [15/8] [\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}] [105/2] [\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}] [105/2] [\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}] [945] [\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}] [945] [\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 [1/16] [231z^{6} - 315z^{4} + 105z^{2} - 5] [(13/1024\pi)^{1/2}] 0.06357 0.04171
[\!\matrix{61+\cr 61-\cr}] [21/8] [\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}] [105/8] [\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}] [315/2] [\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}] [945/2] [\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 [1/16] [429z^{7} - 693z^{5} + 315z^{3} - 35z] [(15/1024\pi)^{1/2}] 0.06828 0.04480
[\!\matrix{71+\cr 71-\cr}] [7/16] [\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}] [63/8] [\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}] [315/8] [\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}] [3465/2] [\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}] [10395/2] [\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)[link], 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}].