Atomic Orbital Explorer
Atomic Orbital 3D Viewer
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Hydrogenic atom wavefunction:
\(\psi_{nlm}\left(r,\theta,\phi\right)=R_{nl}\left(r\right)\cdot Y_l^m\left(\theta,\phi\right)\)Radial Part \(R_{nl}\left(r\right)\):
\(R_{10}\left(r\right)=2\left(\frac{Z_{eff}}{a_o}\right)^{\frac{3}{2}}\cdot e^{-\frac{Z_{eff} \cdot r}{a_o}}\)
\(R_{20}\left(r\right)=\frac{1}{2\sqrt{2}}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{3}{2}}\left(-\frac{Z_{eff} \cdot r}{a_o}+2\right)\cdot e^{-\frac{Z_{eff} \cdot r}{2a_o}}\)
\(R_{21}\left(r\right)=\frac{1}{2\sqrt{6}}\left(\frac{Z_{eff}}{a_o}\right)^{\frac{5}{2}}\cdot r\cdot e^{-\frac{Z_{eff} \cdot r}{2a_o}}\)
\(R_{30}\left(r\right)=\frac{2}{81\sqrt{3}}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{3}{2}}\cdot \left(2\frac{Z_{eff}^2\cdot r^2}{a_o}-18\frac{Z_{eff} \cdot r}{a_o}+27\right)\cdot e^{-\frac{Z_{eff}\cdot r}{3a_o}}\)
\(R_{31}\left(r\right)=\frac{4}{81\sqrt{6}}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{5}{2}}r\cdot\left(-\frac{Z_{eff}\cdot r}{a_o}+6\right)\cdot e^{-\frac{Z_{eff} \cdot r}{3a_o}}\)
\(R_{32}\left(r\right)=\frac{4}{81\sqrt{30}}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{7}{2}}\cdot r^2\cdot e^{-\frac{Z_{eff}\cdot r}{3a_o}}\)
\(R_{40}\left(r\right)=\frac{1}{768}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{3}{2}}\cdot \left(-\frac{Z_{eff}^3\cdot r^3}{a^3_o}+24\frac{Z_{eff}^2\cdot r^2}{a^2_o}-144\frac{Z_{eff}\cdot r}{a_o}+192\right)\cdot e^{-\frac{Z_{eff}\cdot r}{4a_o}}\)
\(R_{41}\left(r\right)=\frac{\sqrt{15}}{3840} \left(\frac{Z_{eff}}{a_0}\right)^{\frac{5}{2}}\cdot r\cdot\left(\frac{Z_{eff}^2\cdot r^2}{a^2_o}-20\frac{Z_{eff}\cdot r}{a_o}+80\right)\cdot e^{-\frac{Z_{eff}\cdot r}{4a_o}}\)
\(R_{42}\left(r\right)=\frac{\sqrt{5}}{3840}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{7}{2}}\cdot r^2\cdot\left(-\frac{Z_{eff}\cdot r}{a_o}+12\right)\cdot e^{-\frac{Z_{eff}\cdot r}{4a_o}}\)
\(R_{43}\left(r\right)=\frac{\sqrt{35}}{26880}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{9}{2}}\cdot r^3\cdot e^{-\frac{Z_{eff}\cdot r}{4a_o}}\)
\(R_{50}\left(r\right)=\frac{4\sqrt{5}}{234375}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{3}{2}}\cdot \left(\frac{Z_{eff}^4\cdot r^4}{a^4_o}-50\frac{Z_{eff}^3\cdot r^3}{a^3_o}+750\frac{Z_{eff}^2\cdot r^2}{a^2_o}-3750\frac{Z_{eff}\cdot r}{a_o}+\frac{9375}{2}\right)\cdot e^{-\frac{Z_{eff}\cdot r}{5a_o}}\)
\(R_{51}\left(r\right)=\frac{4\sqrt{30}}{703125}\left(\frac{Z_{eff}}{a_0}\right)^{\frac{5}{2}}\cdot r\cdot\left(-\frac{Z_{eff}^3\cdot r^3}{a^3_o}+45\frac{Z_{eff}^2\cdot r^2}{a^2_o}-\frac{1125}{2}\frac{Z_{eff}\cdot r}{a_o}+1875\right)\cdot e^{-\frac{Z_{eff}\cdot r}{5a_o}}\)
\(R_{nl}\left(r\right)=\sqrt{\frac{\left(n-l-1\right)!}{2n\left(n+l\right)!}}\left(\frac{2}{n}\right)^{l+\frac{3}{2}}\cdot\left(\frac{Z}{a_o}\right)^{l+\frac{3}{2}}\cdot r^l\cdot L_{n-l-1}^{2l+1}\left(\frac{2r\cdot Z}{na_o}\right)\cdot e^{-\frac{Zr}{na_o}}\)
Angular Part \(Y_l^m\left(\theta,\phi\right)\):
\(Y_0^0\left(\theta,\phi\right)=\frac{1}{2\sqrt{\pi}}\)
\(Y_1^{-1}=\sqrt{\frac{3}{8\pi}}\sin\theta e^{-i\phi}\)
\(Y_1^0\left(\theta,\phi\right)=\sqrt{\frac{3}{4\pi}}\cos\theta\)
\(Y_1^1\left(\theta,\phi\right)=-\sqrt{\frac{3}{8\pi}}\sin\theta e^{-i\phi}\)
\(Y_2^{-2}\left(\theta,\phi\right)=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{-2i\phi}\)
\(Y_2^{-1}\left(\theta,\phi\right)=\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{-i\phi}\)
\(Y_2^0\left(\theta,\phi\right)=\sqrt{\frac{5}{16\pi}}\left(3\cos^2\theta-1\right)\)
\(Y_2^1\left(\theta,\phi\right)=-\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{i\phi}\)
\(Y_2^{2}\left(\theta,\phi\right)=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{2i\phi}\)
\(Y_3^{-3}=\frac{1}{8}\sqrt{\frac{35}{\pi}}\sin^3\theta e^{-3i\phi}\)
\(Y_3^{-2}=\frac{1}{4}\sqrt{\frac{105}{2\pi}}\sin^2\theta\cos\theta e^{-2i\phi}\)
\(Y_3^{-1}=\frac{1}{8}\sqrt{\frac{21}{\pi}}\left(5\cos^2\theta-1\right)\sin\theta e^{-i\phi}\)
\(Y_3^0=\frac{1}{4}\sqrt{\frac{7}{\pi}}\left(5\cos^3\theta-3\cos\theta\right)\)
\(Y_3^1=-\frac{1}{8}\sqrt{\frac{21}{\pi}}\left(5\cos^2\theta-1\right)\sin\theta e^{i\phi}\)
\(Y_3^2=\frac{1}{4}\sqrt{\frac{105}{2\pi}}\sin^2\theta\cos\theta e^{2i\phi}\)
\(Y_3^3=-\frac{1}{8}\sqrt{\frac{35}{\pi}}\sin^3\theta e^{3i\phi}\)
\(Y_4^{-4}=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\sin^4\theta\cdot e^{-4i\phi}\)
\(Y_4^{-3}=\frac{3}{8}\sqrt{\frac{35}{pi}}\sin^3\theta\cos\theta\cdot e^{-3i\phi}\)
\(Y_4^{-2}=\frac{3}{8}\sqrt{\frac{5}{2\pi}}\left(7\cos^2\theta-1\right)\sin^2\theta\cdot e^{-2i\phi}\)
\(Y_4^{-1}=\frac{3}{8}\sqrt{\frac{5}{\pi}}\left(7\cos^3\theta-3\cos\theta\right)\sin2\theta\cdot e^{-i\phi}\)
\(Y_4^0=\frac{3}{16}\sqrt{\frac{1}{\pi}}\left(35\cos^4\theta-30\cos^2\theta+3\right)\)
\(Y_4^{1}=-\frac{3}{8}\sqrt{\frac{5}{\pi}}\left(7\cos^3\theta-3\cos\theta\right)\sin2\theta\cdot e^{i\phi}\)
\(Y_4^{2}=\frac{3}{8}\sqrt{\frac{5}{2\pi}}\left(7\cos^2\theta-1\right)\sin^2\theta\cdot e^{2i\phi}\)
\(Y_4^{3}=-\frac{3}{8}\sqrt{\frac{35}{pi}}\sin^3\theta\cos\theta\cdot e^{3i\phi}\)
\(Y_4^{4}=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\sin^4\theta\cdot e^{4i\phi}\)
*Note - Wavefunctions with m valued other than 0 are complex (eimφ part). In chemistry we examine the real orbitals, which are the linear combinations of the complex orbitals. The real orbitals are labeled by its orientation in Cartesian coordinate. For example, φ211 (m =1) and φ21-1 (m =-1) orbitals are complex, but (1/√2)(φ211 + φ21-1 ) is real and is called 2px orbital as it orients along x-axis when plotted in Cartesian coordinate. In the drop-down menu, “2px [m=+-1(+)]” means: real orbital 2px is the normalized sum of 2p orbitals with m=1 and -1.