Introduction

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. The spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics.

Table of spherical harmonics

A more comploete list of spherical harmonics can be found here.

Complex spherical harmonics $Y_l^m(\theta, \phi)$

• $l=0$

\[Y_0^0(\theta, \phi) = {1\over2}\sqrt{1\over\pi}\]

• $l=1$

\[\begin{alignat*}{2} Y_1^{-1} &= {1\over2} \sqrt{3\over2\pi} \cdot e^{-i\phi} \cdot \sin{\theta} & &= {1\over2} \sqrt{3\over2\pi} \cdot {(x-iy) \over r} \\[6pt] Y_1^{0} &= {1\over2} \sqrt{3\over\pi} \cdot \cos{\theta} & &= {1\over2} \sqrt{3\over\pi} \cdot {z \over r} \\[6pt] Y_1^{1} &= -{1\over2} \sqrt{3\over2\pi} \cdot e^{i\phi} \cdot \sin{\theta} & &= -{1\over2} \sqrt{3\over2\pi} \cdot {(x+iy) \over r} \\[6pt] \end{alignat*}\]

• $l=2$

\[\begin{alignat*}{2} Y_2^{-2} &= {1\over4} \sqrt{15\over2\pi} \cdot e^{-i2\phi} \cdot \sin^2{\theta} & &= {1\over4} \sqrt{15\over2\pi} \cdot {(x - iy)^2 \over r^2} \\[6pt] Y_2^{-1} &= {1\over2} \sqrt{15\over2\pi} \cdot e^{-i\phi} \cdot \sin{\theta} \cos\theta & &= {1\over2} \sqrt{15\over2\pi} \cdot {(x - iy) z \over r^2} \\[6pt] Y_2^{0} &= {1\over4} \sqrt{5\over\pi} \cdot (3\cos^2\theta - 1) & &= {1\over4} \sqrt{5\over\pi} \cdot {(3z^2 - r^2) \over r^2} \\[6pt] Y_2^{1} &=-{1\over2} \sqrt{15\over2\pi} \cdot e^{i\phi} \cdot \sin{\theta} \cos\theta & &=-{1\over2} \sqrt{15\over2\pi} \cdot {(x + iy) z \over r^2} \\[6pt] Y_2^{2} &= {1\over4} \sqrt{15\over2\pi} \cdot e^{-i2\phi} \cdot \sin^2{\theta} & &= {1\over4} \sqrt{15\over2\pi} \cdot {(x + iy)^2 \over r^2} \end{alignat*}\]

• $l=3$

\[\begin{alignat*}{2} Y_3^{-3} &= {1\over8} \sqrt{35\over\pi} \cdot e^{-i3\phi} \cdot \sin^3{\theta} & &= {1\over8} \sqrt{35\over\pi} \cdot {(x - iy)^3 \over r^3} \\[6pt] Y_3^{-2} &= {1\over4} \sqrt{105\over2\pi} \cdot e^{-i2\phi} \cdot \sin^2{\theta} \cos\theta & &= {1\over4} \sqrt{105\over2\pi} \cdot {(x - iy)^2z \over r^3} \\[6pt] Y_3^{-1} &= {1\over8} \sqrt{21\over\pi} \cdot e^{-i\phi} \cdot \sin{\theta} (5\cos^2\theta - 1) & &= {1\over8} \sqrt{21\over\pi} \cdot {(x - iy) (5z^2-r^2) \over r^3} \\[6pt] Y_3^{0} &= {1\over4} \sqrt{7\over\pi} \cdot (5\cos^3\theta - 3\cos\theta) & &= {1\over4} \sqrt{7\over\pi} \cdot {z(5z^2 - 3r^2) \over r^3} \\[6pt] Y_3^{1} &=-{1\over8} \sqrt{21\over\pi} \cdot e^{i\phi} \cdot \sin{\theta} (5\cos^2\theta - 1) & &=-{1\over8} \sqrt{21\over\pi} \cdot {(x + iy)(5z^2-r^2) \over r^3} \\[6pt] Y_3^{2} &= {1\over4} \sqrt{105\over2\pi} \cdot e^{i2\phi} \cdot \sin^2{\theta} \cos\theta & &= {1\over4} \sqrt{105\over2\pi} \cdot {(x + iy)^2z \over r^2} \\[6pt] Y_3^{3} &=-{1\over8} \sqrt{35\over\pi} \cdot e^{i3\phi} \cdot \sin^3{\theta} & &=-{1\over8} \sqrt{35\over\pi} \cdot {(x + iy)^3 \over r^2} \end{alignat*}\]

Real spherical harmonics $Y_{lm}(\theta, \phi)$

• $l=0$

\[Y_{00} = s = Y_0^0 = {1\over2}\sqrt{1\over\pi}\]

• $l=1$

\[\begin{align*} Y_{1,-1} &= p_y = i\sqrt{1\over2}(Y_1^{-1} + Y_1^{1}) = \sqrt{3\over4\pi}\cdot{y\over r} \\[6pt] Y_{1,0} &= p_z = Y_1^0 = \sqrt{3\over4\pi}\cdot{z\over r} \\[6pt] Y_{1,1} &= p_x = \sqrt{1\over2}(Y_1^{-1} + Y_1^{1}) = \sqrt{3\over4\pi}\cdot{x\over r} \end{align*}\]

• $l=2$

\[\begin{align*} Y_{2,-2} &= d_{xy} = i\sqrt{1\over2}(Y_2^{-2} - Y_2^{2}) = {1\over2}\sqrt{15\over\pi}\cdot{xy\over r^2} \\[6pt] Y_{2,-1} &= d_{yz} = i\sqrt{1\over2}(Y_2^{-1} + Y_2^{1}) = {1\over2}\sqrt{15\over\pi}\cdot{yz\over r^2} \\[6pt] Y_{2,0} &= d_{z^2} = Y_2^0 = {1\over4}\sqrt{5\over\pi}\cdot {-x^2-y^2+2z^2\over r^2} \\[6pt] Y_{2,1} &= d_{xz} = \sqrt{1\over2}(Y_2^{-1} - Y_2^{1}) = {1\over2}\sqrt{15\over\pi}\cdot{xz\over r^2} \\[6pt] Y_{2,2} &= d_{x^2-y^2} = \sqrt{1\over2}(Y_2^{-2} + Y_2^{2}) = {1\over4}\sqrt{15\over\pi}\cdot{x^2 - y^2\over r^2} \end{align*}\]

• $l=3$

\[\begin{align*} Y_{3,-3} &= f_{y(3x^2-y^2)} = i\sqrt{1\over2}(Y_3^{-3} + Y_3^{3}) = {1\over4}\sqrt{35\over2\pi}\cdot{(3x^2-y^2)y\over r^3} \\[6pt] Y_{3,-2} &= f_{xyz} = i\sqrt{1\over2}(Y_3^{-2} - Y_3^{2}) = {1\over2}\sqrt{105\over\pi}\cdot{xyz\over r^3} \\[6pt] Y_{3,-1} &= f_{yz^2} = i\sqrt{1\over2}(Y_3^{-1} + Y_3^{1}) = {1\over4}\sqrt{21\over2\pi}\cdot{y(4z^2-x^2-y^2)\over r^3} \\[6pt] Y_{3,0} &= f_{z^3} = Y_3^0 = {1\over4}\sqrt{7\over\pi}\cdot{z(2z^2-3x^2-3y^2)\over r^3} \\[6pt] Y_{3,1} &= f_{xz^2} = \sqrt{1\over2}(Y_3^{-1} - Y_3^{1}) = {1\over4}\sqrt{21\over2\pi}\cdot{x(4z^2-x^2-y^2)\over r^3} \\[6pt] Y_{3,2} &= f_{z(x^2-y^2)} = \sqrt{1\over2}(Y_3^{-2} + Y_3^{2}) = {1\over2}\sqrt{105\over\pi}\cdot{(x^2-y^2)z\over r^3} \\[6pt] Y_{3,3} &= f_{x(x^2-3y^2)} = \sqrt{1\over2}(Y_3^{-3} - Y_3^{3}) = {1\over4}\sqrt{35\over2\pi}\cdot{(x^2-3y^2)x\over r^3} \end{align*}\]

Conversion matrix

• Complex to real

\[\begin{align*} Y_{lm} &= \begin{cases} {i\over\sqrt{2}} [Y_l^m - (-1)^mY_l^{-m}], & \quad\text{if } m < 0 \\[6pt] Y_l^0, & \quad\text{if } m = 0 \\[6pt] {1\over\sqrt{2}} [Y_l^{-m} + (-1)^mY_l^{m}], & \quad\text{if } m > 0 \end{cases} \\[12pt] &= \begin{cases} {i\over\sqrt{2}} [Y_l^{-|m|} - (-1)^mY_l^{|m|}], & \quad\text{if } m < 0 \\[6pt] Y_l^0, & \quad\text{if } m = 0 \\[6pt] {1\over\sqrt{2}} [Y_l^{-|m|} + (-1)^mY_l^{|m|}], & \quad\text{if } m > 0 \end{cases} \\[12pt] &= \begin{cases} \sqrt{2}(-1)^m \Im[Y_l^{|m|}], & \quad\text{if } m < 0 \\[6pt] Y_l^0, & \quad\text{if } m = 0 \\[6pt] \sqrt{2}(-1)^m \Re[Y_l^{m}], & \quad\text{if } m > 0 \\ \end{cases} \\ \end{align*}\]

• Real to complex

\[\begin{equation*} Y_l^m = \begin{cases} {1\over\sqrt{2}} (Y_{l|m|} - i Y_{l,-|m|}), & \quad\text{if } m < 0 \\[6pt] Y_{l0}, & \quad\text{if } m = 0 \\[6pt] {(-1)^m\over\sqrt{2}} (Y_{lm} + iY_{l,-m}), & \quad\text{if } m > 0 \end{cases} \end{equation*}\]

Spherical harmonics visualization

The complex spherical harmonics can be computed by scipy.special.sph_harm. I also wrote a small helping script to convert from the complex spherical harmonics to real ones. For example, with my little script

from sph_harm import show_sph_harm

# available plotting methods are 'mpl', 'mayavi' and 'plotly'
show_sph_harm(l=2, m=1, real=True, plot='mpl')

shows the real spherical harmonics $Y_{lm}(\theta, \phi)$ with $l=2, m=1$ on a sphere, the resulting figure

Figure. Visual representations of the real spherical harmonics on a sphere with radius 1.0.

Or we can use another method to better visualize the nodes.

from sph_harm import show_sph_harm

# available plotting methods are 'mpl', 'mayavi' and 'plotly'
show_sph_harm(l=2, m=1, real=True, use_sphere=False, plot='mpl')

Figure. Visual representations of the real spherical harmonics. The distance of the surface from the origin indicate the absolute value of $Y_{lm}(\theta, \phi)$ in angular direction $(\theta, \phi)$, i.e. $|Y_{lm}(\theta, \phi)|$. The color represents the value of $Y_{lm}(\theta, \phi)$.

Below, I used Plotly to list the real spherical harmonics up to $l = 3$