球面調和関数表

以下は球面調和関数の表である。ただし、テンプレート:Mvar と テンプレート:Mvar との関係としては

$\begin{array}{cc}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \end{array}$

である。

球面調和関数

テンプレート:Math から テンプレート:Math までは テンプレート:Harvtxt を典拠としている。また、テンプレート:Math から テンプレート:Math までの テンプレート:Mvar 形式での関数は MathWorld でも確認できる。

l = 0

$Y_0^0(x)=\frac{1}{2}\sqrt{\frac{1}{\pi }}$

l = 1

$Y_1^{-1}(x)=\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta =\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{x-iy}{r}$

$Y_1^0(x)=\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \cos \theta =\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \frac{z}{r}$

$Y_1^1(x)=-\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta =-\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{x+iy}{r}$

l = 2

$Y_2^{-2}(x)=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{x^2-2ixy-y^2}{r^2}$

$Y_2^{-1}(x)=\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot \cos \theta =\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{xz-iyz}{r^2}$

$Y_2^0(x)=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot (3{\cos }^2\theta -1)=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{-x^2-y^2+2z^2}{r^2}$

$Y_2^1(x)=-\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot \cos \theta =-\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{xz+iyz}{r^2}$

$Y_2^2(x)=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{x^2+2ixy-y^2}{r^2}$

l = 3

$Y_3^{-3}(x)=\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta =\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{x^3-3i{x}^{2}y-3x{y}^2+i{y}^3}{r^3}$

$Y_3^{-2}(x)=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{x^{2}z-2ixyz-y^{2}z}{r^3}$

$Y_3^{-1}(x)=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (5{\cos }^2\theta -1)=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{-x^3+i{x}^{2}y-x{y}^2+4x{z}^2+i{y}^3-4iy{z}^2}{r^3}$

$Y_3^0(x)=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot (5{\cos }^3\theta -3\cos \theta )=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot \frac{-3x^{2}z-3y^{2}z+2z^3}{r^3}$

$Y_3^1(x)=-\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (5{\cos }^2\theta -1)=-\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{-x^3-i{x}^{2}y-x{y}^2+4x{z}^2-i{y}^3+4iy{z}^2}{r^3}$

$Y_3^2(x)=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{x^{2}z+2ixyz-y^{2}z}{r^3}$

$Y_3^3(x)=-\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta =-\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{x^3+3i{x}^{2}y-3x{y}^2-i{y}^3}{r^3}$

l = 4

$Y_4^{-4}(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta$

$Y_4^{-3}(x)=\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta$

$Y_4^{-2}(x)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)$

$Y_4^{-1}(x)=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (7{\cos }^3\theta -3\cos \theta )$

$Y_4^0(x)=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot (35{\cos }^4\theta -30{\cos }^2\theta +3)$

$Y_4^1(x)=-\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (7{\cos }^3\theta -3\cos \theta )$

$Y_4^2(x)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)$

$Y_4^3(x)=-\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta$

$Y_4^4(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta$

l = 5

$Y_5^{-5}(x)=\frac{3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta$

$Y_5^{-4}(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^{-3}(x)=\frac{1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^{-2}(x)=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -1\cos \theta )$

$Y_5^{-1}(x)=\frac{1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (21{\cos }^4\theta -14{\cos }^2\theta +1)$

$Y_5^0(x)=\frac{1}{16}\sqrt{\frac{11}{\pi }}\cdot (63{\cos }^5\theta -70{\cos }^3\theta +15\cos \theta )$

$Y_5^1(x)=-\frac{1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (21{\cos }^4\theta -14{\cos }^2\theta +1)$

$Y_5^2(x)=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -1\cos \theta )$

$Y_5^3(x)=-\frac{1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^4(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^5(x)=-\frac{3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta$

l = 6

$Y_6^{-6}(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta$

$Y_6^{-5}(x)=\frac{3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^{-4}(x)=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^{-3}(x)=\frac{1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^{-2}(x)=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^{-1}(x)=\frac{1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )$

$Y_6^0(x)=\frac{1}{32}\sqrt{\frac{13}{\pi }}\cdot (231{\cos }^6\theta -315{\cos }^4\theta +105{\cos }^2\theta -5)$

$Y_6^1(x)=-\frac{1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )$

$Y_6^2(x)=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^3(x)=-\frac{1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^4(x)=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^5(x)=-\frac{3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^6(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta$

l = 7

$Y_7^{-7}(x)=\frac{3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta$

$Y_7^{-6}(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^{-5}(x)=\frac{3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^{-4}(x)=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^{-3}(x)=\frac{3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^{-2}(x)=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^{-1}(x)=\frac{1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)$

$Y_7^0(x)=\frac{1}{32}\sqrt{\frac{15}{\pi }}\cdot (429{\cos }^7\theta -693{\cos }^5\theta +315{\cos }^3\theta -35\cos \theta )$

$Y_7^1(x)=-\frac{1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)$

$Y_7^2(x)=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^3(x)=-\frac{3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^4(x)=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^5(x)=-\frac{3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^6(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^7(x)=-\frac{3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta$

l = 8

$Y_8^{-8}(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta$

$Y_8^{-7}(x)=\frac{3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^{-6}(x)=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^{-5}(x)=\frac{3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -1\cos \theta )$

$Y_8^{-4}(x)=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^{-3}(x)=\frac{1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^{-2}(x)=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^{-1}(x)=\frac{3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (715{\cos }^7\theta -1001{\cos }^5\theta +385{\cos }^3\theta -35\cos \theta )$

$Y_8^0(x)=\frac{1}{256}\sqrt{\frac{17}{\pi }}\cdot (6435{\cos }^8\theta -12012{\cos }^6\theta +6930{\cos }^4\theta -1260{\cos }^2\theta +35)$

$Y_8^1(x)=-\frac{3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (715{\cos }^7\theta -1001{\cos }^5\theta +385{\cos }^3\theta -35\cos \theta )$

$Y_8^2(x)=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^3(x)=-\frac{1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^4(x)=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^5(x)=-\frac{3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -1\cos \theta )$

$Y_8^6(x)=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^7(x)=-\frac{3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^8(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta$

l = 9

$Y_9^{-9}(x)=\frac{1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{-9i\phi }\cdot {\sin }^9\theta$

$Y_9^{-8}(x)=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^{-7}(x)=\frac{3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^{-6}(x)=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^{-5}(x)=\frac{3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^{-4}(x)=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +1\cos \theta )$

$Y_9^{-3}(x)=\frac{1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^{-2}(x)=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^{-1}(x)=\frac{3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (2431{\cos }^8\theta -4004{\cos }^6\theta +2002{\cos }^4\theta -308{\cos }^2\theta +7)$

$Y_9^0(x)=\frac{1}{256}\sqrt{\frac{19}{\pi }}\cdot (12155{\cos }^9\theta -25740{\cos }^7\theta +18018{\cos }^5\theta -4620{\cos }^3\theta +315\cos \theta )$

$Y_9^1(x)=-\frac{3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (2431{\cos }^8\theta -4004{\cos }^6\theta +2002{\cos }^4\theta -308{\cos }^2\theta +7)$

$Y_9^2(x)=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^3(x)=-\frac{1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^4(x)=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +1\cos \theta )$

$Y_9^5(x)=-\frac{3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^6(x)=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^7(x)=-\frac{3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^8(x)=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^9(x)=-\frac{1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{9i\phi }\cdot {\sin }^9\theta$

l = 10

$Y_{10}^{-10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi }}\cdot e^{-10i\phi }\cdot {\sin }^{10}\theta$

$Y_{10}^{-9}(x)=\frac{1}{512}\sqrt{\frac{4849845}{\pi }}\cdot e^{-9i\phi }\cdot {\sin }^9\theta \cdot \cos \theta$

$Y_{10}^{-8}(x)=\frac{1}{512}\sqrt{\frac{255255}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta \cdot (19{\cos }^2\theta -1)$

$Y_{10}^{-7}(x)=\frac{3}{512}\sqrt{\frac{85085}{\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot (19{\cos }^3\theta -3\cos \theta )$

$Y_{10}^{-6}(x)=\frac{3}{1024}\sqrt{\frac{5005}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (323{\cos }^4\theta -102{\cos }^2\theta +3)$

$Y_{10}^{-5}(x)=\frac{3}{256}\sqrt{\frac{1001}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )$

$Y_{10}^{-4}(x)=\frac{3}{256}\sqrt{\frac{5005}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (323{\cos }^6\theta -255{\cos }^4\theta +45{\cos }^2\theta -1)$

$Y_{10}^{-3}(x)=\frac{3}{256}\sqrt{\frac{5005}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (323{\cos }^7\theta -357{\cos }^5\theta +105{\cos }^3\theta -7\cos \theta )$

$Y_{10}^{-2}(x)=\frac{3}{512}\sqrt{\frac{385}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (4199{\cos }^8\theta -6188{\cos }^6\theta +2730{\cos }^4\theta -364{\cos }^2\theta +7)$

$Y_{10}^{-1}(x)=\frac{1}{256}\sqrt{\frac{1155}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (4199{\cos }^9\theta -7956{\cos }^7\theta +4914{\cos }^5\theta -1092{\cos }^3\theta +63\cos \theta )$

$Y_{10}^0(x)=\frac{1}{512}\sqrt{\frac{21}{\pi }}\cdot (46189{\cos }^{10}\theta -109395{\cos }^8\theta +90090{\cos }^6\theta -30030{\cos }^4\theta +3465{\cos }^2\theta -63)$

$Y_{10}^1(x)=-\frac{1}{256}\sqrt{\frac{1155}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (4199{\cos }^9\theta -7956{\cos }^7\theta +4914{\cos }^5\theta -1092{\cos }^3\theta +63\cos \theta )$

$Y_{10}^2(x)=\frac{3}{512}\sqrt{\frac{385}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (4199{\cos }^8\theta -6188{\cos }^6\theta +2730{\cos }^4\theta -364{\cos }^2\theta +7)$

$Y_{10}^3(x)=-\frac{3}{256}\sqrt{\frac{5005}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (323{\cos }^7\theta -357{\cos }^5\theta +105{\cos }^3\theta -7\cos \theta )$

$Y_{10}^4(x)=\frac{3}{256}\sqrt{\frac{5005}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (323{\cos }^6\theta -255{\cos }^4\theta +45{\cos }^2\theta -1)$

$Y_{10}^5(x)=-\frac{3}{256}\sqrt{\frac{1001}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )$

$Y_{10}^6(x)=\frac{3}{1024}\sqrt{\frac{5005}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (323{\cos }^4\theta -102{\cos }^2\theta +3)$

$Y_{10}^7(x)=-\frac{3}{512}\sqrt{\frac{85085}{\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot (19{\cos }^3\theta -3\cos \theta )$

$Y_{10}^8(x)=\frac{1}{512}\sqrt{\frac{255255}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta \cdot (19{\cos }^2\theta -1)$

$Y_{10}^9(x)=-\frac{1}{512}\sqrt{\frac{4849845}{\pi }}\cdot e^{9i\phi }\cdot {\sin }^9\theta \cdot \cos \theta$

$Y_{10}^{10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi }}\cdot e^{10i\phi }\cdot {\sin }^{10}\theta$

線型結合された球面調和関数

線型結合により導出される実際の電子軌道の球面調和関数。テンプレート:Math から テンプレート:Math までは テンプレート:Harvtxt 及び テンプレート:Harvtxt を、テンプレート:Math は テンプレート:Harvtxt のみを典拠としている。

l = 0

$Y_{00}=s=Y_0^0=\frac{1}{2}\sqrt{\frac{1}{\pi }}$

l = 1

$Y_{1,-1}=p_y=i\sqrt{\frac{1}{2}}(Y_1^{-1}+Y_1^1)=\sqrt{\frac{3}{4\pi }}\cdot \frac{y}{r}$

$Y_{10}=p_z=Y_1^0=\sqrt{\frac{3}{4\pi }}\cdot \frac{z}{r}$

$Y_{11}=p_x=\sqrt{\frac{1}{2}}(Y_1^{-1}-Y_1^1)=\sqrt{\frac{3}{4\pi }}\cdot \frac{x}{r}$

l = 2

$Y_{2,-2}=d_{xy}=i\sqrt{\frac{1}{2}}(Y_2^{-2}-Y_2^2)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{xy}{r^2}$

$Y_{2,-1}=d_{yz}=i\sqrt{\frac{1}{2}}(Y_2^{-1}+Y_2^1)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{yz}{r^2}$

$Y_{20}=d_{z^2}=Y_2^0=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{-x^2-y^2+2z^2}{r^2}$

$Y_{21}=d_{xz}=\sqrt{\frac{1}{2}}(Y_2^{-1}-Y_2^1)=\frac{1}{2}\sqrt{\frac{15}{\pi }}\cdot \frac{zx}{r^2}$

$Y_{22}=d_{x^2-y^2}=\sqrt{\frac{1}{2}}(Y_2^{-2}+Y_2^2)=\frac{1}{4}\sqrt{\frac{15}{\pi }}\cdot \frac{x^2-y^2}{r^2}$

l = 3

$Y_{3,-3}=f_{y(3x^2-y^2)}=i\sqrt{\frac{1}{2}}(Y_3^{-3}+Y_3^3)=\frac{1}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(3x^2-y^2)y}{r^3}$

$Y_{3,-2}=f_{xyz}=i\sqrt{\frac{1}{2}}(Y_3^{-2}-Y_3^2)=\frac{1}{2}\sqrt{\frac{105}{\pi }}\cdot \frac{xyz}{r^3}$

$Y_{3,-1}=f_{y{z}^2}=i\sqrt{\frac{1}{2}}(Y_3^{-1}+Y_3^1)=\frac{1}{4}\sqrt{\frac{21}{2\pi }}\cdot \frac{y(4z^2-x^2-y^2)}{r^3}$

$Y_{30}=f_{z^3}=Y_3^0=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot \frac{z(2z^2-3x^2-3y^2)}{r^3}$

$Y_{31}=f_{x{z}^2}=\sqrt{\frac{1}{2}}(Y_3^{-1}-Y_3^1)=\frac{1}{4}\sqrt{\frac{21}{2\pi }}\cdot \frac{x(4z^2-x^2-y^2)}{r^3}$

$Y_{32}=f_{z(x^2-y^2)}=\sqrt{\frac{1}{2}}(Y_3^{-2}+Y_3^2)=\frac{1}{4}\sqrt{\frac{105}{\pi }}\cdot \frac{(x^2-y^2)z}{r^3}$

$Y_{33}=f_{x(x^2-3y^2)}=\sqrt{\frac{1}{2}}(Y_3^{-3}-Y_3^3)=\frac{1}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x^2-3y^2)x}{r^3}$

l = 4

$Y_{4,-4}=g_{xy(x^2-y^2)}=i\sqrt{\frac{1}{2}}(Y_4^{-4}-Y_4^4)=\frac{3}{4}\sqrt{\frac{35}{\pi }}\cdot \frac{xy(x^2-y^2)}{r^4}$

$Y_{4,-3}=g_{z{y}^3}=i\sqrt{\frac{1}{2}}(Y_4^{-3}+Y_4^3)=\frac{3}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(3x^2-y^2)yz}{r^4}$

$Y_{4,-2}=g_{z^{2}xy}=i\sqrt{\frac{1}{2}}(Y_4^{-2}-Y_4^2)=\frac{3}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{xy\cdot (7z^2-r^2)}{r^4}$

$Y_{4,-1}=g_{z^{3}y}=i\sqrt{\frac{1}{2}}(Y_4^{-1}+Y_4^1)=\frac{3}{4}\sqrt{\frac{5}{2\pi }}\cdot \frac{yz\cdot (7z^2-3r^2)}{r^4}$

$Y_{40}=g_{z^4}=Y_4^0=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot \frac{(35z^4-30z^2{r}^2+3r^4)}{r^4}$

$Y_{41}=g_{z^{3}x}=\sqrt{\frac{1}{2}}(Y_4^{-1}-Y_4^1)=\frac{3}{4}\sqrt{\frac{5}{2\pi }}\cdot \frac{xz\cdot (7z^2-3r^2)}{r^4}$

$Y_{42}=g_{z^{2}xy}=\sqrt{\frac{1}{2}}(Y_4^{-2}+Y_4^2)=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x^2-y^2)\cdot (7z^2-r^2)}{r^4}$

$Y_{43}=g_{z{x}^3}=\sqrt{\frac{1}{2}}(Y_4^{-3}-Y_4^3)=\frac{3}{4}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x^2-3y^2)xz}{r^4}$

$Y_{44}=g_{x^4+y^4}=\sqrt{\frac{1}{2}}(Y_4^{-4}+Y_4^4)=\frac{3}{16}\sqrt{\frac{35}{\pi }}\cdot \frac{x^2(x^2-3y^2)-y^2(3x^2-y^2)}{r^4}$

参考文献

原論文

• テンプレート:Cite journal

書籍

• テンプレート:Cite book
• テンプレート:Cite book

関連項目

• 球面調和関数
• テンプレート:Enlink

外部リンク

• テンプレート:MathWorld