球面調和関数表
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)=\frac{1}{128}\sqrt{\frac{40755}{\pi}\cdot e^{6i\varphi}\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\varphi}\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\varphi}\cdot\sin^{8}\theta\cdot\cos\theta
Y_{9}^{9}(x)=-\frac{1}{512}\sqrt{\frac{230945}{\pi}\cdot e^{9i\varphi}\cdot\sin^{9}\theta
Y_{10}^{-10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi}\cdot e^{-10i\varphi}\cdot\sin^{10}\theta
Y_{10}^{-9}(x)=\frac{1}{512}\sqrt{\frac{4849845}{\pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta
Y_{10}^{-8}(x)=\frac{1}{512}\sqrt{\frac{255255}{2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)
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_{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}^{-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}^{-4}(x)=\frac{3}{256}\sqrt{\frac{5005}{2\pi}\cdot e^{-4i\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\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\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)
Y_{10}^{9}(x)=-\frac{1}{512}\sqrt{\frac{4849845}{\pi}\cdot e^{9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta
Y_{10}^{10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi}\cdot e^{10i\varphi}\cdot\sin^{10}\theta
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  及び  を、l  3 は  のみを典拠としている。
Y_{00}=s=Y_0^0=\frac{1}{2}\sqrt{\frac{1}{\pi}
Y_{1,-1}=p_y=i\sqrt{\frac{1}{2}\left(Y_1^{-1}+Y_1^1\right)=\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}\left(Y_1^{-1}-Y_1^1\right)=\sqrt{\frac{3}{4\pi}\cdot\frac{x}{r}
Y_{2,-2}=d_{xy}=i\sqrt{\frac{1}{2}\left(Y_2^{-2}-Y_2^2\right)=\frac{1}{2}\sqrt{\frac{15}{\pi}\cdot\frac{xy}{r^2}
Y_{2,-1}=d_{yz}=i\sqrt{\frac{1}{2}\left(Y_2^{-1}+Y_2^1\right)=\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}\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}=\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_{3,-3}=f_{y(3x^2-y^2)}=i\sqrt{\frac{1}{2}\left(Y_3^{-3}+Y_3^3\right)=\frac{1}{4}\sqrt{\frac{35}{2\pi}\cdot\frac{\left(3x^2-y^2\right)y}{r^3}
Y_{3,-2}=f_{xyz}=i\sqrt{\frac{1}{2}\left(Y_3^{-2}-Y_3^2\right)=\frac{1}{2}\sqrt{\frac{105}{\pi}\cdot\frac{xyz}{r^3}
Y_{3,-1}=f_{yz^2}=i\sqrt{\frac{1}{2}\left(Y_3^{-1}+Y_3^1\right)=\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_{xz^2}=\sqrt{\frac{1}{2}\left(Y_3^{-1}-Y_3^1\right)=\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}\left(Y_3^{-2}+Y_3^2\right)=\frac{1}{4}\sqrt{\frac{105}{\pi}\cdot\frac{\left(x^2-y^2\right)z}{r^3}
Y_{33}=f_{x(x^2-3y^2)}=\sqrt{\frac{1}{2}\left(Y_3^{-3}-Y_3^3\right)=\frac{1}{4}\sqrt{\frac{35}{2\pi}\cdot\frac{\left(x^2-3y^2\right)x}{r^3}
Y_{4,-4}=g_{xy(x^2-y^2)}=i\sqrt{\frac{1}{2}\left(Y_4^{-4}-Y_4^4\right)=\frac{3}{4}\sqrt{\frac{35}{\pi}\cdot\frac{xy\left(x^2-y^2\right)}{r^4}
Y_{4,-3}=g_{zy^3}=i\sqrt{\frac{1}{2}\left(Y_4^{-3}+Y_4^3\right)=\frac{3}{4}\sqrt{\frac{35}{2\pi}\cdot\frac{(3x^2-y^2)yz}{r^4}
Y_{4,-2}=g_{z^2xy}=i\sqrt{\frac{1}{2}\left(Y_4^{-2}-Y_4^2\right)=\frac{3}{4}\sqrt{\frac{5}{\pi}\cdot\frac{xy\cdot(7z^2-r^2)}{r^4}
Y_{4,-1}=g_{z^3y}=i\sqrt{\frac{1}{2}\left(Y_4^{-1}+Y_4^1\right)=\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^2r^2+3r^4)}{r^4}
Y_{41}=g_{z^3x}=\sqrt{\frac{1}{2}\left(Y_4^{-1}-Y_4^1\right)=\frac{3}{4}\sqrt{\frac{5}{2\pi}\cdot\frac{xz\cdot(7z^2-3r^2)}{r^4}
Y_{42}=g_{z^2xy}=\sqrt{\frac{1}{2}\left(Y_4^{-2}+Y_4^2\right)=\frac{3}{8}\sqrt{\frac{5}{\pi}\cdot\frac{(x^2-y^2)\cdot(7z^2-r^2)}{r^4}
Y_{43}=g_{zx^3}=\sqrt{\frac{1}{2}\left(Y_4^{-3}-Y_4^3\right)=\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}\left(Y_4^{-4}+Y_4^4\right)=\frac{3}{16}\sqrt{\frac{35}{\pi}\cdot\frac{x^2\left(x^2-3y^2\right)-y^2\left(3x^2-y^2\right)}{r^4}