球面調和関数表
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。l = 0 から l = 2 までは Chisholm (1976) 及び Blanco, Flórez & Bermejo (1996) を、l = 3 は Chisholm (1976) のみを典拠としている。
Y 9 4 ( x ) = 3 128 95095 2 π ⋅ e 4 i φ ⋅ sin 4  θ ⋅ ( 17 cos 5  θ − 10 cos 3  θ + 1 cos  θ ) {\displaystyle Y_{9}^{4}(x)={\frac {3}{128}}{\sqrt {\frac {95095}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +1\cos \theta )}
Y 9 5 ( x ) = − 3 256 2717 π ⋅ e 5 i φ ⋅ sin 5  θ ⋅ ( 85 cos 4  θ − 30 cos 2  θ + 1 ) {\displaystyle Y_{9}^{5}(x)=-{\frac {3}{256}}{\sqrt {\frac {2717}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}
Y 9 6 ( x ) = 1 128 40755 π ⋅ e 6 i φ ⋅ sin 6  θ ⋅ ( 17 cos 3  θ − 3 cos  θ ) {\displaystyle Y_{9}^{6}(x)={\frac {1}{128}}{\sqrt {\frac {40755}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}
Y 21 = d x z = 1 2 ( Y 2 − 1 − Y 2 1 ) = 1 2 15 π ⋅ z x r 2 {\displaystyle Y_{21}=d_{xz}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}-Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {zx}{r^{2}}}}
Y 22 = d x 2 − y 2 = 1 2 ( Y 2 − 2 + Y 2 2 ) = 1 4 15 π ⋅ x 2 − y 2 r 2 {\displaystyle Y_{22}=d_{x^{2}-y^{2}}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-2}+Y_{2}^{2}\right)={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {x^{2}-y^{2}}{r^{2}}}}
Y 10 5 ( x ) = − 3 256 1001 π ⋅ e 5 i φ ⋅ sin 5  θ ⋅ ( 323 cos 5  θ − 170 cos 3  θ + 15 cos  θ ) {\displaystyle Y_{10}^{5}(x)=-{\frac {3}{256}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}
Y 10 6 ( x ) = 3 1024 5005 π ⋅ e 6 i φ ⋅ sin 6  θ ⋅ ( 323 cos 4  θ − 102 cos 2  θ + 3 ) {\displaystyle Y_{10}^{6}(x)={\frac {3}{1024}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}
Y 10 7 ( x ) = − 3 512 85085 π ⋅ e 7 i φ ⋅ sin 7  θ ⋅ ( 19 cos 3  θ − 3 cos  θ ) {\displaystyle Y_{10}^{7}(x)=-{\frac {3}{512}}{\sqrt {\frac {85085}{\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}
Y 3 0 ( x ) = 1 4 7 π ⋅ ( 5 cos 3  θ − 3 cos  θ ) = 1 4 7 π ⋅ − 3 x 2 z − 3 y 2 z + 2 z 3 r 3 {\displaystyle 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 ) = − 1 8 21 π ⋅ e i φ ⋅ sin  θ ⋅ ( 5 cos 2  θ − 1 ) = − 1 8 21 π ⋅ − x 3 − i x 2 y − x y 2 + 4 x z 2 − i y 3 + 4 i y z 2 r 3 {\displaystyle Y_{3}^{1}(x)=-{\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)=-{\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot {\frac {-x^{3}-ix^{2}y-xy^{2}+4xz^{2}-iy^{3}+4iyz^{2}}{r^{3}}}}
Y 8 − 2 ( x ) = 3 128 595 π ⋅ e − 2 i φ ⋅ sin 2  θ ⋅ ( 143 cos 6  θ − 143 cos 4  θ + 33 cos 2  θ − 1 ) {\displaystyle Y_{8}^{-2}(x)={\frac {3}{128}}{\sqrt {\frac {595}{\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)}
Y 8 − 1 ( x ) = 3 64 17 2 π ⋅ e − i φ ⋅ sin  θ ⋅ ( 715 cos 7  θ − 1001 cos 5  θ + 385 cos 3  θ − 35 cos  θ ) {\displaystyle Y_{8}^{-1}(x)={\frac {3}{64}}{\sqrt {\frac {17}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )}