Spherical Harmonics
This section provides documentation for the evaluation of complex and real spherical harmonics and solid harmonics, including gradient and Laplacian calculations.
- Associated Legendre Polynomials
- Complex Spherical Harmonics
- Complex Solid Harmonics
- Real Spherical Harmonics
- Real Solid Harmonics
Associated Legendre Polynomials
Associated Legendre polynomials $P_{\ell}^m$ are closely related to the spherical harmonics, $P_{\ell}^m$ of degree $\ell$ and order $m\geq 0$ are defined as (in the phase convention of Condon and Shortley)
\[ P_{\ell}^m(x) = \frac{(-1)^m}{2^{\ell}\ell!}(1-x^2)^{m/2}\frac{\mathrm{d}^{\ell+m}}{\mathrm{d}x^{\ell+m}}(x^2-1)^{\ell}.\]
The negative order can be related to the corresponding positive order via a proportionality constant that involves only $\ell$ and $m$,
\[P_{\ell}^{-m}(x) = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^m(x). \]
The associated Legendre polynomials are orthogonal on the interval $-1\leq x\leq 1$ in the sense that
\[\int_{-1}^{1} P_{k}^m(x) P_{\ell}^m(x) \mathrm{d}x = \frac{2}{2\ell+1}\frac{(\ell+m)!}{(\ell-m)!}\delta_{k\ell}. \]
In alp.jl, Polynomials4ML utilizes the following normalization for the associated Legendre polynomials,
\[\bar{P}_{\ell}^{m}(x) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{2\pi (\ell+m)!}}P_{\ell}^m, \qquad m\geq 0\]
and one can generate a data structure as
ALPs = ALPolynomials(maxL::Integer, T::Type=Float64)where maxL specifies the maximum degree of the polynomials.
The associated Legendre polynomials allow for
P = evaluate(basis, X)
P, dP = evaluate_ed(basis, X)X is a point in spherical coordinates, P and dp stores P_l^m(X.cosθ) and dP_l^m(X.cosθ). Specifically, only non-negative m terms are stored, and are arranged in $\ell$-major order. To retrieve the specific values of P_l^m and dP_l^m for given indices (l, m), one can use
index_p(l,m)The algorithm for computing associated Legendre polynomials is based on Dusson(2022) eq.(A.7), where A_l^m, B_l^m, C_l^m can be found in Limpanuparb(2014) eq.(7)-(14).
Condon-Shortley Sign Convention
There are two sign conventions for associated Legendre polynomials.
- Include the Condon-Shortley phase factor:
\[P_{\ell}^m(x) = \frac{(-1)^m}{2^{\ell}\ell!}(1-x^2)^{m/2}\frac{\mathrm{d}^{\ell+m}}{\mathrm{d}x^{\ell+m}}(x^2-1)^{\ell}.\]
- Exclude the Condon-Shortley phase factor:
\[P_{\ell}^m(x) = \frac{1}{2^{\ell}\ell!}(1-x^2)^{m/2}\frac{\mathrm{d}^{\ell+m}}{\mathrm{d}x^{\ell+m}}(x^2-1)^{\ell}.\]
One possible way to distinguish the two conventions is
\[P_{\ell m}(x) = (-1)^m P_{\ell}^m(x). \]
The Condon-Shortley sign convention enables us to establish the following relationships between spherical harmonics and angular momentum ladder operators
\[Y_{\ell}^m(\theta, \varphi) = A_{\ell m}\hat{L}_-^{\ell-m}Y_{\ell}^{\ell}(\theta, \varphi), \]
\[Y_{\ell}^m(\theta, \varphi) = A_{\ell, -m}\hat{L}_+^{\ell+m}Y_{\ell}^{-\ell}(\theta, \varphi), \]
with all positive constants $A_{\ell m} = \sqrt{\frac{(\ell+m)!}{(2\ell)!(\ell+m)!}}$. Ignoring the Condon-Shortley phase would introduce signs into the $A_{\ell m}$. It's only a sign convention.
Including the factor of $(-1)^m$ and written in terms $x=\cos\theta$, the first few associated Legendre polynomials are
| $m\backslash\ell$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 3 | $-15\sin^3\theta$ | |||
| 2 | $3\sin^2\theta$ | $15\cos\theta\sin^2\theta$ | ||
| 1 | $-\sin\theta$ | $-3\sin\theta\cos\theta$ | $-\frac{3}{2}(5\cos^2\theta - 1)\sin\theta$ | |
| 0 | $1$ | $\cos\theta$ | $\frac{1}{2}(3\cos^2\theta - 1)$ | $\frac{1}{2}\cos\theta(5\cos^2\theta-3)$ |
Complex Spherical Harmonics
In cylm.jl, Polynomials4ML utilizes orthonormalized complex spherical harmonics that includes the Condon-Shortley phase, defined as
\[ Y_{\ell}^m(\theta, \varphi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}P_{\ell}^m(\cos \theta)\mathrm{e}^{\mathrm{i}m \varphi}. \]
The normalization in complex spherical harmonics is chosen to satisfy
\[ \int_0^{2\pi}\int_0^{\pi}Y_{k}^m(\theta, \varphi)\bar{Y}_{\ell}^n(\theta, \varphi)\sin \theta \mathrm{d}\theta\mathrm{d}\varphi =\delta_{k\ell}\delta_{mn}.\]
Orthonormalized complex spherical harmonics that include the Condon-Shortley phase up to degree $\ell = 3$ are
| $m\backslash\ell$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 3 | $-\frac{1}{8}\sqrt{\frac{35}{\pi}}\cdot \mathrm{e}^{3\mathrm{i}\varphi}\cdot \sin^3\theta$ | |||
| 2 | $\frac{1}{4}\sqrt{\frac{15}{2\pi}}\cdot \mathrm{e}^{2\mathrm{i}\varphi}\cdot \sin^2\theta$ | $\frac{1}{4}\sqrt{\frac{105}{2\pi}}\cdot \mathrm{e}^{2\mathrm{i}\varphi}\cdot \sin^2\theta\cdot \cos\theta$ | ||
| 1 | $-\frac{1}{2}\sqrt{\frac{3}{2\pi}}\cdot \mathrm{e}^{\mathrm{i}\varphi}\cdot \sin\theta$ | $-\frac{1}{2}\sqrt{\frac{15}{2\pi}}\cdot \mathrm{e}^{\mathrm{i}\varphi}\cdot \sin\theta \cdot \cos\theta$ | $-\frac{1}{8}\sqrt{\frac{21}{\pi}}\cdot \mathrm{e}^{\mathrm{i}\varphi}\cdot \sin\theta \cdot (5\cos^2\theta-1)$ | |
| 0 | $\frac{1}{2}\sqrt{\frac{1}{\pi}}$ | $\frac{1}{2}\sqrt{\frac{3}{\pi}}\cdot \cos \theta$ | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\cdot (3\cos^2\theta - 1)$ | $\frac{1}{4}\sqrt{\frac{7}{\pi}}\cdot (5\cos^3\theta - 3\cos\theta)$ |
| -1 | $\frac{1}{2}\sqrt{\frac{3}{2\pi}}\cdot \mathrm{e}^{-\mathrm{i}\varphi}\cdot \sin\theta$ | $\frac{1}{2}\sqrt{\frac{15}{2\pi}}\cdot \mathrm{e}^{-\mathrm{i}\varphi}\cdot \sin\theta \cdot \cos\theta$ | $\frac{1}{8}\sqrt{\frac{21}{\pi}}\cdot \mathrm{e}^{-\mathrm{i}\varphi}\cdot \sin\theta \cdot (5\cos^2\theta-1)$ | |
| -2 | $\frac{1}{4}\sqrt{\frac{15}{2\pi}}\cdot \mathrm{e}^{-2\mathrm{i}\varphi}\cdot \sin^2\theta$ | $\frac{1}{4}\sqrt{\frac{105}{2\pi}}\cdot \mathrm{e}^{-2\mathrm{i}\varphi}\cdot \sin^2\theta\cdot \cos\theta$ | ||
| -3 | $\frac{1}{8}\sqrt{\frac{35}{\pi}}\cdot \mathrm{e}^{-3\mathrm{i}\varphi}\cdot \sin^3\theta$ |
To generate the complex spherical harmonics $Y_{\ell}^m$ with normalized associated Legendre polynomials $\bar{P}_{\ell}^m$, the formulas can be rewritten as
\[\begin{cases} Y_{\ell}^0(\theta, \varphi) = \sqrt{\frac{1}{2}}\bar{P}_{\ell}^0(\cos \theta)\\ Y_{\ell}^m(\theta, \varphi) = \sqrt{\frac{1}{2}}\bar{P}_{\ell}^m(\cos \theta)\mathrm{e}^{\mathrm{i}m \varphi}\\ Y_{\ell}^{-m}(\theta, \varphi) = (-1)^m\cdot \sqrt{\frac{1}{2}} \bar{P}_{\ell}^m(\cos \theta)\mathrm{e}^{-\mathrm{i}m \varphi} \end{cases}, \qquad m>0.\]
To evaluate the gradients of the spherical harmonics $\nabla Y_{\ell}^m$, one need to convert a gradient with respect to spherical coordinates to a gradient with respect to cartesian coordinates,
\[\begin{cases} (\frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z}) = (-\frac{\sin\varphi}{r\sin \theta}, \frac{\cos\varphi}{r\sin \theta}, 0)\\ (\frac{\partial \theta}{\partial x}, \frac{\partial \theta}{\partial y}, \frac{\partial \theta}{\partial z}) = (\frac{\cos \varphi \cos \theta}{r}, \frac{\sin \varphi \cos \theta}{r}, -\frac{\sin \theta}{r}) \end{cases}.\]
Therefore, the gradient of $Y_{\ell}^m$ can be expressed as,
\[\nabla Y_{\ell}^m = \frac{\mathrm{i}m P_{\ell}^m \mathrm{e}^{\mathrm{i}m\varphi}}{r\sin \theta}\begin{bmatrix} -\sin \varphi \\\cos \varphi \\0 \end{bmatrix} + \frac{\partial_{\theta}P_{\ell}^m \mathrm{e}^{\mathrm{i}m\varphi}}{r}\begin{bmatrix} \cos \varphi \cos \theta\\\sin \varphi\cos \theta \\-\sin\theta \end{bmatrix}.\]
For the sake of simplicity, we incorporated the coefficient in of $P_{\ell}^m$ into the term $P_{\ell}^m$ itself.
To ensure numerically stable evaluation of gradients near $\sin \theta = 0$, we compute $P_{\ell}^m/\sin \theta$ instead of $P_{\ell}^m$. We refer to section A.1 of Dusson(2022) for detailed discussion.
We can further compute $\nabla^2 Y_{\ell}^m$ as,
\[\nabla^2 Y_{\ell m} = \left(\frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r} - \frac{L^2}{r^2} \right)Y_{\ell}^m = -\frac{\ell(\ell+1)}{r^2}Y_{\ell}^{m}.\]
One can generate a data structure as
cylm = CYlmBasis(maxL::Integer, T::Type=Float64)The complex spherical harmonics allow for
P = evaluate(basis, X)
P, dP = evaluate_ed(basis, X)To retrieve the specific values of Y_l^m and dY_l^m for given indices (l, m), one can use
index_y(l,m)Alternative normalizations conventions
Here, we provide a list of alternative normalizations conventions for complex spherical harmonics,
- Schmidt semi-normalized (Racah's normalization)
\[C_{\ell}^m(\theta, \varphi) = \sqrt{\frac{4\pi}{2\ell + 1}}Y_{\ell}^m(\theta, \varphi) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}}P_{\ell}^m(\cos \theta)\mathrm{e}^{\mathrm{i}m \varphi}, \]
with
\[\int_0^{2\pi}\int_0^{\pi}C_{k}^m(\theta, \varphi)\bar{C}_{\ell}^n(\theta, \varphi)\sin \theta \mathrm{d}\theta\mathrm{d}\varphi = \frac{4\pi}{2\ell + 1}\delta_{k\ell}\delta_{mn}. \]
In this normalization, $C_0^0(\theta, \varphi)$ is equal to $1$.
- 4π-normalized
\[\mathscr{Y}_{\ell}^m (\theta, \varphi) = \sqrt{4\pi}Y_{\ell}^m(\theta, \varphi) = \sqrt{(2\ell+1)\frac{(l-m)!}{(l+m)!}}P_{\ell}^m(\cos \theta)\mathrm{e}^{\mathrm{i}m \varphi}, \]
with
\[\int_0^{2\pi}\int_0^{\pi}\mathscr{Y}_{k}^m(\theta, \varphi)\bar{\mathscr{Y}}_{\ell}^n(\theta, \varphi)\sin \theta \mathrm{d}\theta\mathrm{d}\varphi = 4\pi\delta_{k\ell}\delta_{mn}. \]
Complex Solid Harmonics
In crlm.jl, Polynomials4ML utilizes orthonormalized complex solid harmonics defined as
\[ \gamma_{\ell}^m(r, \theta, \varphi) = r^{\ell}Y_{\ell}^m(\theta, \varphi). \]
$\gamma_{\ell}^m$'s are orthogonal is the sense that
\[ \int_0^{2\pi}\int_0^{\pi}\gamma_{k}^m(\theta, \varphi)\bar{\gamma}_{\ell}^n(\theta, \varphi)\sin \theta \mathrm{d}\theta\mathrm{d}\varphi =\delta_{k\ell}\delta_{mn}r^{k+\ell}.\]
The evaluation of solid harmonics can be obtained from the spherical harmonics by a simple scaling with $r^{\ell}$. To evaluate the gradients of the solid harmonics, $\nabla \gamma_{\ell}^m$, the following expressions are used,
\[\begin{cases} (\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial r}{\partial z}) = (\sin \theta \cos \varphi,\sin\theta\sin \varphi, \cos \theta)\\ (\frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z}) = (-\frac{\sin\varphi}{r\sin \theta}, \frac{\cos\varphi}{r\sin \theta}, 0)\\ (\frac{\partial \theta}{\partial x}, \frac{\partial \theta}{\partial y}, \frac{\partial \theta}{\partial z}) = (\frac{\cos \varphi \cos \theta}{r}, \frac{\sin \varphi \cos \theta}{r}, -\frac{\sin \theta}{r}) \end{cases}.\]
Therefore, the gradient of $\gamma_{\ell}^m$ can be expressed as,
\[\nabla \gamma_{\ell}^m = \frac{\ell r^{\ell} P_{\ell}^m \mathrm{e}^{\mathrm{i}m\varphi}}{r}\begin{bmatrix} \sin \theta \cos \varphi\\ \sin\theta\sin \varphi\\ \cos \theta \end{bmatrix}+ \frac{\mathrm{i}m P_{\ell}^m \mathrm{e}^{\mathrm{i}m\varphi}}{r\sin \theta}\begin{bmatrix} -\sin \varphi \\\cos \varphi \\0 \end{bmatrix} + \frac{\partial_{\theta}P_{\ell}^m \mathrm{e}^{\mathrm{i}m\varphi}}{r}\begin{bmatrix} \cos \varphi \cos \theta\\\sin \varphi\cos \theta \\-\sin\theta \end{bmatrix}.\]
Similarly, we incorporated the coefficient in of $P_{\ell}^m$ into the term $P_{\ell}^m$ itself. We can further compute $\nabla^2 \gamma_{\ell}^m$ as,
\[\nabla^2 r^{\ell}Y_{\ell m} = \left(\frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r} - \frac{L^2}{r^2} \right)r^{\ell}Y_{\ell}^m = \frac{Y_{\ell m}}{r^2}\frac{\partial}{\partial r}r^2\frac{\partial r^{\ell}}{\partial r} - \frac{r^{\ell}L^2 Y_{\ell}^{m}}{r^2} = 0,\]
that is, the solid harmonics are solutions to Laplace's equation.
Real Spherical Harmonics
In rylm.jl, Polynomials4ML utilizes orthonormalized real spherical harmonics that exclude the Condon-Shortley phase.
- Include the Condon-Shortley phase factor:
\[Y_{\ell m}(\theta, \varphi) = \begin{cases} \frac{\mathrm{i}}{\sqrt{2}}(Y_{\ell}^m - (-1)^m Y_{\ell}^{-m}) & m < 0\\ Y_{\ell}^0 & m = 0 \\ \frac{1}{\sqrt{2}}(Y_{\ell}^{-m} + (-1)^m Y_{\ell}^{m}) & m > 0 \end{cases} = \begin{cases} (-1)^m \bar{P}_{\ell}^{|m|}(\cos \theta)\sin(|m|\varphi) & m < 0\\ \frac{1}{\sqrt{2}} \bar{P}_{\ell}^0(\cos \theta) & m = 0 \\ (-1)^m \bar{P}_{\ell}^{m}(\cos \theta)\cos(m\varphi) & m > 0 \end{cases}\]
- Exclude the Condon-Shortley phase factor:
\[Y_{\ell m}(\theta, \varphi) = \begin{cases} -\bar{P}_{\ell}^{|m|}(\cos \theta)\sin(|m|\varphi) & m < 0\\ \frac{1}{\sqrt{2}} \bar{P}_{\ell}^0(\cos \theta) & m = 0 \\ \bar{P}_{\ell}^{m}(\cos \theta)\cos(m\varphi) & m > 0 \end{cases}\]
Orthonormalized real spherical harmonics that employ the Condon-Shortley phase up to degree $\ell = 3$ are
| $m\backslash\ell$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 3 | $\frac{1}{4}\sqrt{\frac{35}{2\pi}}\cdot \frac{x(x^2-3y^2)}{r^3}$ | |||
| 2 | $\frac{1}{4}\sqrt{\frac{15}{\pi}}\cdot \frac{x^2-y^2}{r^2}$ | $\frac{1}{4}\sqrt{\frac{105}{\pi}}\cdot \frac{(x^2-y^2)z}{r^3}$ | ||
| 1 | $\sqrt{\frac{3}{4\pi}}\cdot \frac{x}{r}$ | $\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot \frac{zx}{r^2}$ | $\frac{1}{4}\sqrt{\frac{21}{2\pi}}\cdot \frac{x(5z^2-r^2)}{r^3}$ | |
| 0 | $\frac{1}{2}\sqrt{\frac{1}{\pi}}$ | $\sqrt{\frac{3}{4\pi}}\cdot \frac{z}{r}$ | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\cdot \frac{3z^2-r^2}{r^2}$ | $\frac{1}{4}\sqrt{\frac{7}{\pi}}\cdot \frac{z(5z^2-3r^2)}{r^3}$ |
| -1 | $\sqrt{\frac{3}{4\pi}}\cdot \frac{y}{r}$ | $\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot \frac{yz}{r^2}$ | $\frac{1}{4}\sqrt{\frac{21}{2\pi}}\cdot \frac{y(5z^2-r^2)}{r^3}$ | |
| -2 | $\frac{1}{2}\sqrt{\frac{15}{\pi}}\cdot \frac{xy}{r^2}$ | $\frac{1}{2}\sqrt{\frac{105}{\pi}}\cdot \frac{xyz}{r^2}$ | ||
| -3 | $\frac{1}{4}\sqrt{\frac{35}{2\pi}}\cdot \frac{(3x^2-y^2)y}{r^3}$ |
Real Solid Harmonics
In rrlm.jl, Polynomials4ML utilizes Schmidt semi-normalized real solid harmonics that exclude the Condon-Shortley phase.
- Include the Condon-Shortley phase factor:
\[S_{\ell m}(r, \theta, \varphi) = \begin{cases} \frac{\mathbb{i}}{\sqrt{2}}\left(C_{\ell, m}-(-1)^m C_{\ell,-m} \right) & m < 0\\ C_{10} & m = 0 \\ \frac{1}{\sqrt{2}}\left(C_{\ell, -m}+(-1)^m C_{\ell,m}\right) & m > 0 \end{cases} = \begin{cases} (-1)^m \sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{|m|}(\cos \theta)\sin(|m|\varphi) & m < 0\\ \sqrt{\frac{2\pi}{2l+1}}\bar{P}_{\ell}^0(\cos \theta) & m = 0 \\ (-1)^m \sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{m}(\cos \theta)\cos(m\varphi) & m > 0 \end{cases},\]
where
\[C_{\ell, m}(r, \theta, \varphi) = \sqrt{\frac{4\pi}{2\ell + 1}}\gamma_{\ell}^m(\theta, \varphi), \]
with
\[\int_0^{2\pi}\int_0^{\pi}C_{k,m}(r, \theta, \varphi)\bar{C}_{\ell, n}(r, \theta, \varphi)\sin \theta \mathrm{d}\theta\mathrm{d}\varphi = \frac{4\pi}{2\ell + 1}\delta_{k\ell}\delta_{mn} r^{k+\ell}. \]
- Exclude the Condon-Shortley phase factor:
\[S_{\ell m}(r, \theta, \varphi) = \begin{cases} -\sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{|m|}(\cos \theta)\sin(|m|\varphi) & m < 0\\ \sqrt{\frac{2\pi}{2l+1}}\bar{P}_{\ell}^0(\cos \theta) & m = 0 \\ \sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{m}(\cos \theta)\cos(m\varphi) & m > 0 \end{cases}\]
Schmidt semi-normalized real spherical harmonics that employ the Condon-Shortley phase up to degree $\ell = 3$ are
| $m\backslash\ell$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 3 | $\frac{1}{2}\sqrt{\frac{5}{2}}(x^2-3y^2)x$ | |||
| 2 | $\frac{1}{2}\sqrt{3}(x^2-y^2)$ | $\frac{1}{2}\sqrt{15}(x^2-y^2)z$ | ||
| 1 | $x$ | $\sqrt{3}xz$ | $\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)x$ | |
| 0 | $1$ | $z$ | $\frac{1}{2}(3z^2-r^2)$ | $\frac{1}{2}(5z^2-3r^2)z$ |
| -1 | $y$ | $\sqrt{3}yz$ | $\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)y$ | |
| -2 | $\sqrt{3}xy$ | $\sqrt{15}xyz$ | ||
| -3 | $\frac{1}{2}\sqrt{\frac{5}{2}}(3x^2-y^2)y$ |
References
- Dusson, G., Bachmayr, M., Csányi, G., Drautz, R., Etter, S., van der Oord, C., & Ortner, C. (2022). Atomic cluster expansion: Completeness, efficiency and stability. Journal of Computational Physics, 454, 110946.
- Helgaker, T., Jorgensen, P., & Olsen, J. (2013). Molecular electronic-structure theory. John Wiley & Sons.
- Limpanuparb, T., & Milthorpe, J. (2014). Associated Legendre polynomials and spherical harmonics computation for chemistry applications. arXiv preprint arXiv:1410.1748.
- Wieczorek, M. A., & Meschede, M. (2018). SHTools: Tools for working with spherical harmonics. Geochemistry, Geophysics, Geosystems, 19(8), 2574-2592.