Table of spherical harmonics explained

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree

\ell=10

. 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

\theta

and

\varphi

$\begin\cos(\theta) & = z/r\\e^ \cdot \sin(\theta) & = (x \pm iy)/r\end$

Complex spherical harmonics

For ℓ = 0, …, 5, see.^[1]

ℓ = 0

$Y_^(\theta,\varphi)=\sqrt$

ℓ = 1

$\beginY_^(\theta,\varphi) &= & & \sqrt\cdot e^\cdot\sin\theta & &= & &\sqrt \cdot \\Y_^(\theta,\varphi) &= & & \sqrt\cdot \cos\theta & &= & &\sqrt \cdot \\Y_^(\theta,\varphi) &= &-& \sqrt\cdot e^\cdot \sin\theta & &= &-&\sqrt \cdot\end$

ℓ = 2

$\beginY_^(\theta,\varphi)&=& &\sqrt\cdot e^\cdot\sin^\theta\quad &&=& &\sqrt\cdot&\\Y_^(\theta,\varphi)&=& &\sqrt\cdot e^\cdot\sin \theta\cdot \cos\theta\quad &&=& &\sqrt\cdot&\\Y_^(\theta,\varphi)&=& &\sqrt\cdot (3\cos^\theta-1)\quad&&=& &\sqrt\cdot&\\Y_^(\theta,\varphi)&=&-&\sqrt\cdot e^\cdot\sin \theta\cdot \cos\theta\quad &&=&-&\sqrt\cdot&\\Y_^(\theta,\varphi)&=& &\sqrt\cdot e^\cdot\sin^\theta\quad &&=& &\sqrt\cdot&\end$

ℓ = 3

$\beginY_^(\theta,\varphi) &=& &\sqrt\cdot e^\cdot\sin^\theta\quad& &=& & \sqrt\cdot&\\Y_^(\theta,\varphi) &=& &\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\quad& &=& & \sqrt\cdot&\\Y_^(\theta,\varphi) &=& &\sqrt\cdot e^\cdot\sin\theta\cdot(5\cos^\theta-1)\quad& &=& &\sqrt\cdot&\\Y_^(\theta,\varphi) &=& &\sqrt\cdot(5\cos^\theta-3\cos\theta)\quad& &=& &\sqrt\cdot&\\Y_^(\theta,\varphi) &=&-&\sqrt\cdot e^ \cdot\sin\theta\cdot(5\cos^\theta-1)\quad& &=& &\sqrt\cdot&\\Y_^(\theta,\varphi) &=& &\sqrt\cdot e^ \cdot\sin^\theta\cdot\cos\theta\quad& &=& &\sqrt\cdot&\\Y_^(\theta,\varphi) &=&-&\sqrt\cdot e^ \cdot\sin^\theta\quad& &=& &\sqrt\cdot& \end$

ℓ = 4

$\beginY_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(7\cos^\theta-1)= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(7\cos^\theta-3\cos\theta)= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot(35\cos^\theta-30\cos^\theta+3)= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(7\cos^\theta-3\cos\theta)= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(7\cos^\theta-1)= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta= \frac \sqrt \cdot \frac\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta= \frac \sqrt \cdot \frac\end$

ℓ = 5

ℓ = 6

ℓ = 7

ℓ = 8

$\beginY_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(15\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(5\cos^\theta-\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(65\cos^\theta-26\cos^\theta+1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(39\cos^\theta-26\cos^\theta+3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(143\cos^\theta-143\cos^\theta+33\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(715\cos^\theta-1001\cos^\theta+385\cos^\theta-35\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot(6435\cos^\theta-12012\cos^\theta+6930\cos^\theta-1260\cos^\theta+35)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(715\cos^\theta-1001\cos^\theta+385\cos^\theta-35\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(143\cos^\theta-143\cos^\theta+33\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(39\cos^\theta-26\cos^\theta+3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(65\cos^\theta-26\cos^\theta+1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(5\cos^\theta-\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(15\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\end$

ℓ = 9

$\beginY_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(85\cos^\theta-30\cos^\theta+1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-10\cos^\theta+\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(221\cos^\theta-195\cos^\theta+39\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(221\cos^\theta-273\cos^\theta+91\cos^\theta-7\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin \theta\cdot(2431\cos^\theta-4004\cos^\theta+2002\cos^\theta-308\cos^\theta+7)\\Y_^(\theta,\varphi)&=\sqrt\cdot (12155\cos^\theta-25740\cos^\theta+18018\cos^\theta-4620\cos^\theta+315\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin \theta\cdot(2431\cos^\theta-4004\cos^\theta+2002\cos^\theta-308\cos^\theta+7)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(221\cos^\theta-273\cos^\theta+91\cos^\theta-7\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(221\cos^\theta-195\cos^\theta+39\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-10\cos^\theta+\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(85\cos^\theta-30\cos^\theta+1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(17\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\end$

ℓ = 10

$\beginY_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(19\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(19\cos^\theta-3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-102\cos^\theta+3)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-170\cos^\theta+15\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-255\cos^\theta+45\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-357\cos^\theta+105\cos^\theta-7\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(4199\cos^\theta-6188\cos^\theta+2730\cos^\theta-364\cos^\theta+7)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(4199\cos^\theta-7956\cos^\theta+4914\cos^\theta-1092\cos^\theta+63\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot(46189\cos^\theta-109395\cos^\theta+90090\cos^\theta-30030\cos^\theta+3465\cos^\theta-63)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin\theta\cdot(4199\cos^\theta-7956\cos^\theta+4914\cos^\theta-1092\cos^\theta+63\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(4199\cos^\theta-6188\cos^\theta+2730\cos^\theta-364\cos^\theta+7)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-357\cos^\theta+105\cos^\theta-7\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-255\cos^\theta+45\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-170\cos^\theta+15\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(323\cos^\theta-102\cos^\theta+3)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(19\cos^\theta-3\cos\theta)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot(19\cos^\theta-1)\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\cdot\cos\theta\\Y_^(\theta,\varphi)&=\sqrt\cdot e^\cdot\sin^\theta\end$

Visualization of complex spherical harmonics

2D polar/azimuthal angle maps

Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle,

\phi

, on the horizontal axis and the polar angle,

\theta

, on the vertical axis. 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.

Polar plots

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.

Polar plots with magnitude as radius

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.

Real spherical harmonics

For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f) is reported as well.^[2] ^[3]

For ℓ = 0, …, 3, see.^[4] ^[5]

ℓ = 0

$Y_ = s = Y_0^0 = \frac \sqrt$

ℓ = 1

$\begin Y_ & = p_y = i \sqrt \left(Y_1^ + Y_1^1 \right) = \sqrt \cdot \frac = \sqrt \sin(\theta) \sin \varphi \\ Y_ & = p_z = Y_1^0 = \sqrt \cdot \frac = \sqrt \cos(\theta) \\ Y_ & = p_x = \sqrt \left(Y_1^ - Y_1^1 \right) = \sqrt \cdot \frac = \sqrt \sin(\theta) \cos \varphi\end$

ℓ = 2

$\beginY_ & = d_ = i \sqrt \left(Y_2^ - Y_2^2\right) = \frac \sqrt \cdot \frac = \frac \sqrt \sin^\theta \sin(2\varphi) \\Y_ & = d_ = i \sqrt \left(Y_2^ + Y_2^1 \right) = \frac \sqrt \cdot \frac = \frac \sqrt \sin(2 \theta) \sin \varphi \\Y_ & = d_ = Y_2^0 = \frac \sqrt \cdot \frac = \frac \sqrt (3\cos^\theta -1)\\Y_ & = d_ = \sqrt \left(Y_2^ - Y_2^1 \right) = \frac \sqrt \cdot \frac = \frac \sqrt \sin(2 \theta) \cos \varphi\\Y_ & = d_ = \sqrt \left(Y_2^ + Y_2^2 \right) = \frac \sqrt \cdot \frac = \frac \sqrt \sin^\theta \cos(2\varphi)\end$

ℓ = 3

$\beginY_ & = f_ = i \sqrt \left(Y_3^ + Y_3^3 \right) = \frac \sqrt \cdot \frac \\Y_ & = f_ = i \sqrt \left(Y_3^ - Y_3^2 \right) = \frac \sqrt \cdot \frac \\Y_ & = f_ = i \sqrt \left(Y_3^ + Y_3^1 \right) = \frac \sqrt \cdot \frac \\Y_ & = f_ = Y_3^0 = \frac \sqrt \cdot \frac \\Y_ & = f_ = \sqrt \left(Y_3^ - Y_3^1 \right) = \frac \sqrt \cdot \frac \\Y_ & = f_ = \sqrt \left(Y_3^ + Y_3^2 \right) = \frac \sqrt \cdot \frac \\Y_ & = f_ = \sqrt \left(Y_3^ - Y_3^3 \right) = \frac \sqrt \cdot \frac\end$

ℓ = 4

$\beginY_ & = i \sqrt \left(Y_4^ - Y_4^4 \right) = \frac \sqrt \cdot \frac \\Y_ & = i \sqrt \left(Y_4^ + Y_4^3 \right) = \frac \sqrt \cdot \frac \\Y_ & = i \sqrt \left(Y_4^ - Y_4^2 \right) = \frac \sqrt \cdot \frac \\Y_ & = i \sqrt \left(Y_4^ + Y_4^1\right) = \frac \sqrt \cdot \frac \\Y_ & = Y_4^0 = \frac \sqrt \cdot \frac \\Y_ & = \sqrt \left(Y_4^ - Y_4^1 \right) = \frac \sqrt \cdot \frac \\Y_ & = \sqrt \left(Y_4^ + Y_4^2 \right) = \frac \sqrt \cdot \frac \\Y_ & = \sqrt \left(Y_4^ - Y_4^3 \right) = \frac \sqrt \cdot \frac \\Y_ & = \sqrt \left(Y_4^ + Y_4^4 \right) = \frac \sqrt \cdot \frac\end$

Visualization of real spherical harmonics

2D polar/azimuthal angle maps

Below the real spherical harmonics are represented on 2D plots with the azimuthal angle,

\phi

, on the horizontal axis and the polar angle,

\theta

, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic. Positive values are red and negative values are teal.

The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines.

Polar plots

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.

Polar plots with magnitude as radius

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.

Polar plots with amplitude as elevation

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.

External links

References

General references

See section 3 in Mathar . R. J. . 2009 . Zernike basis to cartesian transformations . . 179 . 107–120 . 0809.2368 . 2009SerAJ.179..107M . 10.2298/SAJ0979107M . 179. (see section 3.3)
For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi<nowiki>] at Wolfram Alpha], especially for specific values of l and m.

Notes and References

Book: Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols . 1988 . World Scientific Pub. . Singapore . 9971-50-107-4. D. A. Varshalovich . A. N. Moskalev . V. K. Khersonskii . 1. repr. . 155–156.
Book: Petrucci . General chemistry : principles and modern applications. . 2016 . Prentice Hall . 0133897311.
Friedman . The shapes of the f orbitals . J. Chem. Educ. . 1964 . 41 . 7 . 354.
Book: Group theoretical techniques in quantum chemistry . 1976 . Academic Press . New York . 0-12-172950-8 . C.D.H. Chisholm.
Blanco . Miguel A. . Flórez, M. . Bermejo, M. . Evaluation of the rotation matrices in the basis of real spherical harmonics . Journal of Molecular Structure: THEOCHEM . 1 December 1997 . 419 . 1–3 . 19–27 . 10.1016/S0166-1280(97)00185-1.

Table of spherical harmonics explained

Complex spherical harmonics

ℓ = 0

ℓ = 1

ℓ = 2

ℓ = 3

ℓ = 4

ℓ = 5

ℓ = 6

ℓ = 7

ℓ = 8

ℓ = 9

ℓ = 10

Visualization of complex spherical harmonics

2D polar/azimuthal angle maps

Polar plots

Polar plots with magnitude as radius

Real spherical harmonics

ℓ = 0

ℓ = 1

ℓ = 2

ℓ = 3

ℓ = 4

Visualization of real spherical harmonics

2D polar/azimuthal angle maps

Polar plots

Polar plots with magnitude as radius

Polar plots with amplitude as elevation

See also

External links

References

General references

Notes and References