Atomic orbitals (AOs) and Basis sets#
Atomic orbitals for the pairing function and the Jastrow factor#
One of the most common choices for atomic orbitals in QMC is atom-centered Gaussian-type orbitals (GTOs). A primitive GTO \(\psi_{l,m,\alpha}(\mathbf{r})\) can be constructed using either solid harmonics or Cartesian polynomial basis functions.
Gaussian-Type Orbitals with Solid-Harmonic Basis#
Primitive orbitals with regular solid harmonics are given by:
\[\psi_{l,m,\alpha}(\mathbf{r}) = \mathcal{N}^{\rm solid}_{l,m,\alpha}\;\mathcal{R}_{\alpha}(\mathbf{r})\;\mathcal{S}_{l,m,\alpha}(\mathbf{r})\]
where the radial part is
\[\mathcal{R}_{\alpha}(\mathbf{r}) = e^{-Z_\alpha\,|\mathbf{r}-\mathbf{R}_\alpha|^2}\]
and the solid harmonics are
\[\mathcal{S}_{l,m,\alpha}(\mathbf{r}) = \sqrt{\frac{2l+1}{4\pi}}\;|\mathbf{r}-\mathbf{R}_\alpha|^l\;\mathcal{Y}_{l,m}(\theta_\alpha,\phi_\alpha).\]
Here \(\mathcal{Y}_{l,m}(\theta,\phi)\) are real spherical harmonics, and the Racah normalization ensures
\[\int_0^\pi\!\int_0^{2\pi} \sin\theta\,\mathcal{S}_{l,m,\alpha}^*(\mathbf{r})\,\mathcal{S}_{l,m,\alpha}(\mathbf{r})\,\mathrm{d}\phi\,\mathrm{d}\theta = \frac{4\pi}{2l+1}\;r^{2l}.\]
The normalization constant is
\[\mathcal{N}^{\rm solid}_{l,m,\alpha} = \sqrt{\frac{(2Z_\alpha/\pi)^{3/2}\,(4Z_\alpha)^l}{(2l-1)!!}}.\]
One may also write GTOs with spherical harmonics normalization
\[\psi_{l,m,\alpha}(\mathbf{r}) = \mathcal{N}^{\rm sphe}_{l,m,\alpha}\;|\mathbf{r}-\mathbf{R}_\alpha|^l\;\mathcal{R}_{\alpha}(\mathbf{r})\;\mathcal{Y}_{l,m}(\theta_\alpha,\phi_\alpha)\]
with
\[\mathcal{N}^{\rm sphe}_{l,m,\alpha} = \sqrt{\frac{2^{2l+3}(l+1)!(2Z_\alpha)^{l+3/2}}{(2l+2)!\,\sqrt{\pi}}}.\]
These two normalizations satisfy
\[\frac{\mathcal{N}^{\rm solid}_{l,m,\alpha}}{\mathcal{N}^{\rm sphe}_{l,m,\alpha}} = \sqrt{\frac{2l+1}{4\pi}}.\]
Gaussian-Type Orbitals with Cartesian Basis#
Primitive orbitals in the Cartesian basis are
\[\psi_{n_x,n_y,n_z,\alpha}(\mathbf{r}) = \mathcal{N}^{\rm cart}_{n_x,n_y,n_z,\alpha}\;e^{-Z_\alpha\,|\mathbf{r}-\mathbf{R}_\alpha|^2}\;x^{n_x}y^{n_y}z^{n_z}.\]
The normalization is
\[\mathcal{N}^{\rm cart}_{n_x,n_y,n_z,\alpha} = \sqrt{\frac{(2Z_\alpha/\pi)^{3/2}\,(4Z_\alpha)^{n_x+n_y+n_z}}{(2n_x-1)!!\,(2n_y-1)!!\,(2n_z-1)!!}} = \sqrt{\frac{(2Z_\alpha/\pi)^{3/2}\,(8Z_\alpha)^{n_x+n_y+n_z}n_x!n_y!n_z!}{(2n_x)!(2n_y)!(2n_z)!}}.\]
We define the total angular momentum as \(l=n_x+n_y+n_z\). A basis of order \(l\) includes all monomials of that degree (e.g., \(l=2\) gives \(d_{xx},d_{xy},\dots,d_{zz}\)). Note that normalization constants differ among Cartesian orbitals with the same \(l\) (except for \(s\) and \(p\) shells). For example:
\[\mathcal{N}^{\rm cart}_{2,0,0,\alpha} = \sqrt{\frac{(2Z_\alpha/\pi)^{3/2}(4Z_\alpha)^2}{3\cdot1}},\]
\[\mathcal{N}^{\rm cart}_{1,1,0,\alpha} = \sqrt{\frac{(2Z_\alpha/\pi)^{3/2}(4Z_\alpha)^2}{1\cdot1}}.\]
Practical Tip#
In jQMC (JAX), Cartesian GTOs are computationally faster than spherical ones when using jit and grad, since they avoid branching logic by varying only polynomial exponents rather than basis-function forms.
Real Spherical and Solid Harmonics#
The real spherical harmonics \(\mathcal{Y}_{l,m}(\theta,\phi)\) are built from the complex spherical harmonics \(Y_{l,m}(\theta,\phi)\) with the Condon-Shortley phase:
(1)#\[\begin{split}\mathcal{Y}_{l,m}(\theta,\phi) =
\begin{cases}
\displaystyle
\frac{1}{\sqrt{2}}\bigl(Y_{l,-|m|}(\theta,\phi) + (-1)^m\,Y_{l,|m|}(\theta,\phi)\bigr), & m>0,\\[1em]
Y_{l,0}(\theta,\phi), & m=0,\\[0.75em]
\displaystyle
\frac{i}{\sqrt{2}}\bigl(Y_{l,-|m|}(\theta,\phi) - (-1)^m\,Y_{l,|m|}(\theta,\phi)\bigr), & m<0.
\end{cases}\end{split}\]
Because the complex spherical harmonics satisfy
(2)#\[\int_{0}^{\pi}\!\!\int_{0}^{2\pi}
Y_{l',m'}^*(\theta,\phi)\,Y_{l,m}(\theta,\phi)\,
\sin\theta\,d\phi\,d\theta
= \delta_{l,l'}\,\delta_{m,m'},\]
the real ones are also orthonormal:
(3)#\[\int_{0}^{\pi}\!\!\int_{0}^{2\pi}
\mathcal{Y}_{l',m'}(\theta,\phi)\,\mathcal{Y}_{l,m}(\theta,\phi)\,
\sin\theta\,d\phi\,d\theta
= \delta_{l,l'}\,\delta_{m,m'}.\]
The spherical harmonics have singularities at the origin \((x,y,z)=(0,0,0)\). In practice one uses the regular solid harmonics centered at \(\mathbf{R}_\alpha\):
(4)#\[\mathcal{S}_{l,m,\alpha}(\mathbf{r})
= \sqrt{\frac{2l+1}{4\pi}}\;
|\mathbf{R}_\alpha-\mathbf{r}|^l\;
\mathcal{Y}_{l,m}(\theta_\alpha,\phi_\alpha).\]
Polynomial Form of Solid Harmonics#
One efficient way to compute \(\mathcal{S}_{l,m}\) is via polynomials in \((x,y,z)\). Define:
(5)#\[\begin{split}C_m(x,y) =
\begin{cases}
\displaystyle
\sum_{p=0}^{|m|}\binom{|m|}{p}x^p\,y^{|m|-p}\,\cos\!\bigl(\tfrac\pi2(|m|-p)\bigr),
& m\ge0,\\[0.75em]
\displaystyle
\sum_{p=0}^{|m|}\binom{|m|}{p}x^p\,y^{|m|-p}\,\sin\!\bigl(\tfrac\pi2(|m|-p)\bigr),
& m<0,
\end{cases}\end{split}\]
(6)#\[\gamma_{l,m,k}(z)
= (-1)^k\,2^{-l}
\,\binom{l}{k}\,\binom{2l-2k}{l}\,
\frac{(l-2k)!}{(l-2k-|m|)!},\]
(7)#\[\Gamma_{l,m}(z)
= \sum_{k=0}^{\lfloor (l-|m|)/2\rfloor}
\gamma_{l,m,k}(z)\;
|\mathbf{r}|^{2k}\;z^{\,l-2k-|m|}.\]
Then
(8)#\[\mathcal{S}_{l,m,\alpha}(\mathbf{r})
= \sqrt{\frac{(2-\delta_{m,0})(l-|m|)!}{(l+|m|)!}}\;
C_m(x-\!X_\alpha,\;y-\!Y_\alpha)\;
\Gamma_{l,m}(z-\!Z_\alpha).\]
Normalization of the Radial Part#
For a Gaussian radial factor \(\psi_{l,m,\alpha}(\mathbf{r})=\mathcal{N}_{l,\alpha}\,r^l\,e^{-Z_\alpha r^2}\,\mathcal{Y}_{l,m}(\theta,\phi)\), the overall normalization requires
(9)#\[\int_{\mathbb{R}^3}
\bigl|\psi_{l,m,\alpha}(\mathbf{r})\bigr|^2\,d\mathbf{r}
=1.\]
Separating radial and angular parts:
(10)#\[\mathcal{N}_{l,\alpha}^2
\int_{0}^{\infty}r^{2l+2}\,e^{-2Z_\alpha r^2}\,dr
=1
\;\Longrightarrow\;
\mathcal{N}_{l,\alpha}^2
\frac{(2l+2)!\,\sqrt{\pi}}{2^{2l+3}(l+1)!\,(2Z_\alpha)^{l+\frac{3}{2}}}
=1,\]
(11)#\[\int_{0}^{\pi}\!\!\int_{0}^{2\pi}
\bigl|\mathcal{Y}_{l,m}(\theta,\phi)\bigr|^2\,
\sin\theta\,d\phi\,d\theta
=1.\]
Hence the radial normalization constant is
(12)#\[\mathcal{N}_{l,\alpha}
= \sqrt{
\frac{2^{2l+3}(l+1)!\,(2Z_\alpha)^{\,l+\tfrac{3}{2}}}
{(2l+2)!\,\sqrt{\pi}}
}.\]
This factor is identical for all \((l,m)\) in the same shell.
Practical Implementation Note#
In JAX-based codes, Cartesian GTOs (polynomials in \(x,y,z\)) are often faster than spherical ones, because they avoid branching logic over \((l,m)\).
Tables of Real Solid Harmonics#
Below are the explicit real solid harmonics up to \(l=6\). Let \(r=\sqrt{x^2+y^2+z^2}\).
\(l=0\) (\(s\) orbital)#
(13)#\[Y_{0,0} = \frac{1}{2}\,\sqrt{\frac{1}{\pi}}.\]
\(l=1\) (\(p\) orbitals)#
(14)#\[\begin{split}\begin{aligned}
Y_{1,-1} &= i\sqrt{\tfrac12}\bigl(Y_1^{-1}+Y_1^1\bigr)
= \sqrt{\tfrac{3}{4\pi}}\;\frac{y}{r},\\
Y_{1,0} &= \;\;Y_1^0
= \sqrt{\tfrac{3}{4\pi}}\;\frac{z}{r},\\
Y_{1,1} &= \sqrt{\tfrac12}\bigl(Y_1^{-1}-Y_1^1\bigr)
= \sqrt{\tfrac{3}{4\pi}}\;\frac{x}{r}.
\end{aligned}\end{split}\]
\(l=2\) (\(d\) orbitals)#
(15)#\[\begin{split}\begin{aligned}
Y_{2,-2} &= i\sqrt{\tfrac12}\bigl(Y_2^{-2}-Y_2^2\bigr)
= \tfrac12\sqrt{\tfrac{15}{\pi}}\;\frac{xy}{r^2},\\
Y_{2,-1} &= i\sqrt{\tfrac12}\bigl(Y_2^{-1}+Y_2^1\bigr)
= \tfrac12\sqrt{\tfrac{15}{\pi}}\;\frac{yz}{r^2},\\
Y_{2,0} &= \;\;Y_2^0
= \tfrac14\sqrt{\tfrac{5}{\pi}}\;\frac{3z^2-r^2}{r^2},\\
Y_{2,1} &= \sqrt{\tfrac12}\bigl(Y_2^{-1}-Y_2^1\bigr)
= \tfrac12\sqrt{\tfrac{15}{\pi}}\;\frac{xz}{r^2},\\
Y_{2,2} &= \sqrt{\tfrac12}\bigl(Y_2^{-2}+Y_2^2\bigr)
= \tfrac14\sqrt{\tfrac{15}{\pi}}\;\frac{x^2-y^2}{r^2}.
\end{aligned}\end{split}\]
\(l=3\) (\(f\) orbitals)#
(16)#\[\begin{split}\begin{aligned}
Y_{3,-3} &= i\sqrt{\tfrac12}\bigl(Y_3^{-3}+Y_3^3\bigr)
= \tfrac14\sqrt{\tfrac{35}{2\pi}}\;\frac{y(3x^2-y^2)}{r^3},\\
Y_{3,-2} &= i\sqrt{\tfrac12}\bigl(Y_3^{-2}-Y_3^2\bigr)
= \tfrac12\sqrt{\tfrac{105}{\pi}}\;\frac{xyz}{r^3},\\
Y_{3,-1} &= i\sqrt{\tfrac12}\bigl(Y_3^{-1}+Y_3^1\bigr)
= \tfrac14\sqrt{\tfrac{21}{2\pi}}\;\frac{y(5z^2-r^2)}{r^3},\\
Y_{3,0} &= \;\;Y_3^0
= \tfrac14\sqrt{\tfrac{7}{\pi}}\;\frac{5z^3-3zr^2}{r^3},\\
Y_{3,1} &= \sqrt{\tfrac12}\bigl(Y_3^{-1}-Y_3^1\bigr)
= \tfrac14\sqrt{\tfrac{21}{2\pi}}\;\frac{x(5z^2-r^2)}{r^3},\\
Y_{3,2} &= \sqrt{\tfrac12}\bigl(Y_3^{-2}+Y_3^2\bigr)
= \tfrac14\sqrt{\tfrac{105}{\pi}}\;\frac{(x^2-y^2)z}{r^3},\\
Y_{3,3} &= \sqrt{\tfrac12}\bigl(Y_3^{-3}-Y_3^3\bigr)
= \tfrac14\sqrt{\tfrac{35}{2\pi}}\;\frac{x(x^2-3y^2)}{r^3}.
\end{aligned}\end{split}\]
\(l=4\) (\(g\) orbitals)#
(17)#\[\begin{split}\begin{aligned}
Y_{4,-4} &= i\sqrt{\tfrac12}\bigl(Y_4^{-4}-Y_4^4\bigr)
= \tfrac34\sqrt{\tfrac{35}{\pi}}\;\frac{xy(x^2-y^2)}{r^4},\\
Y_{4,-3} &= i\sqrt{\tfrac12}\bigl(Y_4^{-3}+Y_4^3\bigr)
= \tfrac34\sqrt{\tfrac{35}{2\pi}}\;\frac{y(3x^2-y^2)z}{r^4},\\
Y_{4,-2} &= i\sqrt{\tfrac12}\bigl(Y_4^{-2}-Y_4^2\bigr)
= \tfrac34\sqrt{\tfrac{5}{\pi}}\;\frac{xy(7z^2-r^2)}{r^4},\\
Y_{4,-1} &= i\sqrt{\tfrac12}\bigl(Y_4^{-1}+Y_4^1\bigr)
= \tfrac34\sqrt{\tfrac{5}{2\pi}}\;\frac{y(7z^3-3zr^2)}{r^4},\\
Y_{4,0} &= \;\;Y_4^0
= \tfrac{3}{16}\sqrt{\tfrac{1}{\pi}}\;\frac{35z^4-30z^2r^2+3r^4}{r^4},\\
Y_{4,1} &= \sqrt{\tfrac12}\bigl(Y_4^{-1}-Y_4^1\bigr)
= \tfrac34\sqrt{\tfrac{5}{2\pi}}\;\frac{x(7z^3-3zr^2)}{r^4},\\
Y_{4,2} &= \sqrt{\tfrac12}\bigl(Y_4^{-2}+Y_4^2\bigr)
= \tfrac38\sqrt{\tfrac{5}{\pi}}\;\frac{(x^2-y^2)(7z^2-r^2)}{r^4},\\
Y_{4,3} &= \sqrt{\tfrac12}\bigl(Y_4^{-3}-Y_4^3\bigr)
= \tfrac34\sqrt{\tfrac{35}{2\pi}}\;\frac{x(x^2-3y^2)z}{r^4},\\
Y_{4,4} &= \sqrt{\tfrac12}\bigl(Y_4^{-4}+Y_4^4\bigr)
= \tfrac{3}{16}\sqrt{\tfrac{35}{\pi}}\;\frac{x^2(x^2-3y^2)-y^2(3x^2-y^2)}{r^4}.
\end{aligned}\end{split}\]
\(l=5\) (\(h\) orbitals)#
(18)#\[\begin{split}\begin{aligned}
Y_{5,-5} &= i\sqrt{\tfrac12}\bigl(Y_5^{-5}+Y_5^5\bigr)
= \tfrac{3}{16}\sqrt{\tfrac{77}{2\pi}}\;\frac{5x^4y-10x^2y^3+y^5}{r^5},\\
Y_{5,-4} &= i\sqrt{\tfrac12}\bigl(Y_5^{-4}-Y_5^4\bigr)
= \tfrac{3}{16}\sqrt{\tfrac{385}{\pi}}\;\frac{4xyz(x^2-y^2)}{r^5},\\
Y_{5,-3} &= i\sqrt{\tfrac12}\bigl(Y_5^{-3}+Y_5^3\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{385}{2\pi}}\;\frac{-(y^3-3x^2y)(9z^2-r^2)}{r^5},\\
Y_{5,-2} &= i\sqrt{\tfrac12}\bigl(Y_5^{-2}-Y_5^2\bigr)
= \tfrac{1}{8}\sqrt{\tfrac{1155}{\pi}}\;\frac{2xy(3z^3-zr^2)}{r^5},\\
Y_{5,-1} &= i\sqrt{\tfrac12}\bigl(Y_5^{-1}+Y_5^1\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{165}{\pi}}\;\frac{y(21z^4-14z^2r^2+r^4)}{r^5},\\
Y_{5,0} &= \;\;Y_5^0
= \tfrac{1}{16}\sqrt{\tfrac{11}{\pi}}\;\frac{63z^5-70z^3r^2+15zr^4}{r^5},\\
Y_{5,1} &= \sqrt{\tfrac12}\bigl(Y_5^{-1}-Y_5^1\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{165}{\pi}}\;\frac{x(21z^4-14z^2r^2+r^4)}{r^5},\\
Y_{5,2} &= \sqrt{\tfrac12}\bigl(Y_5^{-2}+Y_5^2\bigr)
= \tfrac{1}{8}\sqrt{\tfrac{1155}{\pi}}\;\frac{(x^2-y^2)(3z^3-zr^2)}{r^5},\\
Y_{5,3} &= \sqrt{\tfrac12}\bigl(Y_5^{-3}-Y_5^3\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{385}{2\pi}}\;\frac{(x^3-3xy^2)(9z^2-r^2)}{r^5},\\
Y_{5,4} &= \sqrt{\tfrac12}\bigl(Y_5^{-4}+Y_5^4\bigr)
= \tfrac{3}{16}\sqrt{\tfrac{385}{\pi}}\;\frac{x^2z(x^2-3y^2)-y^2z(3x^2-y^2)}{r^5},\\
Y_{5,5} &= \sqrt{\tfrac12}\bigl(Y_5^{-5}-Y_5^5\bigr)
= \tfrac{3}{16}\sqrt{\tfrac{77}{2\pi}}\;\frac{x^5-10x^3y^2+5xy^4}{r^5}.
\end{aligned}\end{split}\]
\(l=6\) (\(i\) orbitals)#
(19)#\[\begin{split}\begin{aligned}
Y_{6,-6} &= i\sqrt{\tfrac12}\bigl(Y_6^{-6}-Y_6^6\bigr)
= \tfrac{1}{64}\sqrt{\tfrac{6006}{\pi}}\;\frac{6x^5y-20x^3y^3+6xy^5}{r^6},\\
Y_{6,-5} &= i\sqrt{\tfrac12}\bigl(Y_6^{-5}+Y_6^5\bigr)
= \tfrac{3}{32}\sqrt{\tfrac{2002}{\pi}}\;\frac{z(5x^4y-10x^2y^3+y^5)}{r^6},\\
Y_{6,-4} &= i\sqrt{\tfrac12}\bigl(Y_6^{-4}-Y_6^4\bigr)
= \tfrac{3}{32}\sqrt{\tfrac{91}{\pi}}\;\frac{4xy(11z^2-r^2)(x^2-y^2)}{r^6},\\
Y_{6,-3} &= i\sqrt{\tfrac12}\bigl(Y_6^{-3}+Y_6^3\bigr)
= \tfrac{1}{32}\sqrt{\tfrac{2730}{\pi}}\;\frac{-(11z^3-3zr^2)(y^3-3x^2y)}{r^6},\\
Y_{6,-2} &= i\sqrt{\tfrac12}\bigl(Y_6^{-2}-Y_6^2\bigr)
= \tfrac{1}{64}\sqrt{\tfrac{2730}{\pi}}\;\frac{2xy(33z^4-18z^2r^2+r^4)}{r^6},\\
Y_{6,-1} &= i\sqrt{\tfrac12}\bigl(Y_6^{-1}+Y_6^1\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{273}{\pi}}\;\frac{y(33z^5-30z^3r^2+5zr^4)}{r^6},\\
Y_{6,0} &= \;\;Y_6^0
= \tfrac{1}{32}\sqrt{\tfrac{13}{\pi}}\;\frac{231z^6-315z^4r^2+105z^2r^4-5r^6}{r^6},\\
Y_{6,1} &= \sqrt{\tfrac12}\bigl(Y_6^{-1}-Y_6^1\bigr)
= \tfrac{1}{16}\sqrt{\tfrac{273}{\pi}}\;\frac{x(33z^5-30z^3r^2+5zr^4)}{r^6},\\
Y_{6,2} &= \sqrt{\tfrac12}\bigl(Y_6^{-2}+Y_6^2\bigr)
= \tfrac{1}{64}\sqrt{\tfrac{2730}{\pi}}\;\frac{(x^2-y^2)(33z^4-18z^2r^2+r^4)}{r^6},\\
Y_{6,3} &= \sqrt{\tfrac12}\bigl(Y_6^{-3}-Y_6^3\bigr)
= \tfrac{1}{32}\sqrt{\tfrac{2730}{\pi}}\;\frac{(11z^3-3zr^2)(x^3-3xy^2)}{r^6},\\
Y_{6,4} &= \sqrt{\tfrac12}\bigl(Y_6^{-4}+Y_6^4\bigr)
= \tfrac{3}{32}\sqrt{\tfrac{91}{\pi}}\;\frac{(11z^2-r^2)\bigl[x^2(x^2-3y^2)+y^2(y^2-3x^2)\bigr]}{r^6},\\
Y_{6,5} &= \sqrt{\tfrac12}\bigl(Y_6^{-5}-Y_6^5\bigr)
= \tfrac{3}{32}\sqrt{\tfrac{2002}{\pi}}\;\frac{z(x^5-10x^3y^2+5xy^4)}{r^6},\\
Y_{6,6} &= \sqrt{\tfrac12}\bigl(Y_6^{-6}+Y_6^6\bigr)
= \tfrac{1}{64}\sqrt{\tfrac{6006}{\pi}}\;\frac{x^6-15x^4y^2+15x^2y^4-y^6}{r^6}.
\end{aligned}\end{split}\]
\(l\ge7\) harmonics are rarely needed in practice.