Table of spherical harmonics
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree
Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to
Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle,
, on the horizontal axis and the polar angle,
The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.
The nodal 'line of latitude' are visible as horizontal white lines.
The nodal 'line of longitude' are visible as vertical white lines.
Below the complex spherical harmonics are represented on polar plots.
The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.
Below the complex spherical harmonics are represented on polar plots.
The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.
For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f) is reported as well.
sin ( θ ) cos φ
θ sin ( 2 φ )
sin ( 2 θ ) cos φ
θ cos ( 2 φ )
{\displaystyle {\begin{aligned}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}}}={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \sin(2\varphi )\\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 {y\cdot z}{r^{2}}}={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin(2\theta )\sin \varphi \\Y_{2,0}&=d_{z^{2}}=Y_{2}^{0}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {3z^{2}-r^{2}}{r^{2}}}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}(3\cos ^{2}\theta -1)\\Y_{2,1}&=d_{xz}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}-Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {x\cdot z}{r^{2}}}={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin(2\theta )\cos \varphi \\Y_{2,2}&=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}}}={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\sin ^{2}\theta \cos(2\varphi )\end{aligned}}}
Below the real spherical harmonics are represented on 2D plots with the azimuthal angle,
, on the horizontal axis and the polar angle,
The saturation of the color at any point represents the magnitude of the spherical harmonic.
The nodal 'line of latitude' are visible as horizontal white lines.
The nodal 'line of longitude' are visible as vertical white lines.
Below the real spherical harmonics are represented on polar plots.
The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.
Below the real spherical harmonics are represented on polar plots.
The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.
Below the real spherical harmonics are represented on polar plots.
The amplitude of the spherical harmonic (magnitude and sign) at a particular polar and azimuthal angle is represented by the elevation of the plot at that point above or below the surface of a uniform sphere.
The magnitude is also represented by the saturation of the color at a given point.
The phase is represented by the hue at a given point.
