Single particle basis

In this section, we define some single particle basis used in the JASMINE component to write down the atomic Hamiltonian $\hat{H}_{\text{atom}}$. We set $\hbar=1$ in this note.

Spherical harmonics

The complex spherical harmonics $Y_{l}^{m}(\theta,\phi)$ are the eigenstates of operators $\hat{l}^{2}$ and $\hat{l}_{z}$,

\[\begin{equation} \hat{l}^{2}Y_{l}^{m}=l(l+1)Y_{l}^{m}, \end{equation}\]

\[\begin{equation} \hat{l}_{z}Y_{l}^{m}=mY_{l}^{m}, \end{equation}\]

where $l$ is the azimuthal quantum number ($l = 0,~1,~2,~\cdots,~n-1$), and $m$ is the magnetic quantum number ($m=-l,~-l+1,~\cdots,~l$)^[1][2]. They are defined as follows:

\[\begin{equation} Y^m_l(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P^m_l(\cos{\theta}) e^{im\phi}, \end{equation}\]

where $\theta$ is taken as the polar (colatitudinal) coordinate with $\theta \in [0,\pi]$, and $\phi$ as the azimuthal (longitudinal) coordinate with $\phi \in [0,2\pi]$, and $P^m_l(z)$ is an associated Legendre polynomial.

The spherical harmonics are orthonormal

\[\begin{equation} \int^{\pi}_{\theta = 0} \int^{2\pi}_{\phi = 0} Y^m_l(\theta,\phi) Y^{m'*}_{l'}(\theta,\phi)~d\Omega = \delta_{ll'} \delta_{mm'}, \end{equation}\]

where $\delta_{ij}$ is the Kronecker delta and $d\Omega = \sin(\theta) d\phi d\theta$.

Real Spherical Harmonics

The real spherical harmonics $Y_{lm}$ are defined as^[3]

\[\begin{gather} Y_{lm}=\begin{cases} \frac{i}{\sqrt{2}}\left(Y_{l}^{-|m|}-(-1)^{m}Y_{l}^{|m|}\right) & \text{if}\ m<0,\\ Y_{l}^{0} & \text{if}\ m=0,\\ \frac{1}{\sqrt{2}}\left(Y_{l}^{-|m|}+(-1)^{m}Y_{l}^{|m|}\right) & \text{if}\ m>0. \end{cases} \end{gather}\]

The corresponding inverse equations defining the complex spherical harmonics $Y^m_l$ in terms of the real spherical harmonics $Y_{lm}$ read:

\[\begin{gather} Y_{l}^{m}=\begin{cases} \frac{1}{\sqrt{2}}\left(Y_{l|m|}-iY_{l,-|m|}\right) & \text{if}\ m<0,\\ Y_{l0} & \text{if}\ m=0,\\ \frac{(-1)^m}{\sqrt{2}}\left(Y_{l|m|}+iY_{l,-|m|}\right) & \text{if}\ m>0. \end{cases} \end{gather}\]

The real spherical harmonics $Y_{lm}$ are sometimes known as tesseral spherical harmonics. These functions have the same orthonormality properties as the complex ones $Y_{l}^{m}$.

Spinor spherical harmonics

The spinor spherical harmonics $\Omega^l_{jm_j}(\theta,\phi)$ are eigenstates of the operators $\hat{j}^2$, $\hat{j}_z$, $\hat{l}^2$, and $\hat{s}^2$,

\[\begin{equation} \hat{j}^2 \Omega^l_{jm_j} = j (j + 1) \Omega^l_{jm_j}, \end{equation}\]

\[\begin{equation} \hat{j}_z \Omega^l_{jm_j} = m_j \Omega^l_{jm_j}, \end{equation}\]

\[\begin{equation} \hat{l}^2 \Omega^l_{jm_j} = l (l + 1) \Omega^l_{jm_j}, \end{equation}\]

\[\begin{equation} \hat{s}^2 \Omega^l_{jm_j} = s(s+1) \Omega^l_{jm_j}. \end{equation}\]

For given $j$ only two values of $l$ are possible, $l = j \pm \frac{1}{2}$, while $m_j$ assumes $2j + 1$ values ($m_j = -j,~-j + 1,~\cdots,~j - 1,~j$) ^[1][2].

For $j = l + \frac{1}{2}$, $m_j = m + \frac{1}{2}$,

\[\begin{equation} \Omega^l_{jm_j} = \sqrt{\frac{l+m+1}{2l+1}} Y^m_l \chi_{\uparrow} + \sqrt{\frac{l-m}{2l+1}} Y^{m+1}_l \chi_{\downarrow}. \end{equation}\]

For $j = l - \frac{1}{2} (l \neq 0)$, $m_j = m + \frac{1}{2}$,

\[\begin{equation} \Omega^l_{jm_j} = -\sqrt{\frac{l-m}{2l+1}} Y^m_l \chi_{\uparrow} + \sqrt{\frac{l+m+1}{2l+1}} Y^{m+1}_l \chi_{\downarrow}. \end{equation}\]

Real orbital basis

The basis functions are the real spherical harmonics $Y_{lm}(\theta,\phi)$.

For $p$ system, the basis order is^[4]

\[\begin{equation} \begin{split} & |p_y, \uparrow \rangle,~ |p_z, \uparrow \rangle,~ |p_x, \uparrow \rangle,~\\ & |p_y, \downarrow \rangle,~ |p_z, \downarrow \rangle,~ |p_x, \downarrow \rangle. \end{split} \end{equation}\]

\[\begin{equation} |p_{y}\rangle = Y_{1,-1}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right), \end{equation}\]

\[\begin{equation} |p_{z}\rangle = Y_{10}=Y_{1}^{0}, \end{equation}\]

\[\begin{equation} |p_{x}\rangle = Y_{11}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right). \end{equation}\]

For $d$ system, the basis order is^[4]

\[\begin{equation} \begin{split} & |d_{xy}, \uparrow \rangle,~ |d_{yz}, \uparrow \rangle,~ |d_{z^2}, \uparrow \rangle,~ |d_{xz}, \uparrow \rangle,~ |d_{x^2-y^2}, \uparrow \rangle,~\\ & |d_{xy}, \downarrow \rangle,~ |d_{yz}, \downarrow \rangle,~ |d_{z^2}, \downarrow \rangle,~ |d_{xz}, \downarrow \rangle,~ |d_{x^2-y^2}, \downarrow \rangle. \end{split} \end{equation}\]

\[\begin{equation} d_{xy} = Y_{2,-2}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-2}-Y_{2}^{2}\right), \end{equation}\]

\[\begin{equation} d_{yz} = Y_{2,-1}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-1}+Y_{2}^{1}\right), \end{equation}\]

\[\begin{equation} d_{z^{2}} = Y_{20}=Y_{2}^{0}, \end{equation}\]

\[\begin{equation} d_{xz} = Y_{21}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-1}-Y_{2}^{1}\right), \end{equation}\]

\[\begin{equation} d_{x^{2}-y^{2}} = Y_{22}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-2}+Y_{2}^{2}\right). \end{equation}\]

For $f$ system, the basis order is^[4]

\[\begin{equation} \begin{split} & |f_{y(3x^2-y^2)}, \uparrow \rangle,~ |f_{xyz}, \uparrow \rangle,~ |f_{yz^2}, \uparrow \rangle,~ |f_{z^3}, \uparrow \rangle,~ |f_{xz^2}, \uparrow \rangle,~ |f_{z(x^2-y^2)}, \uparrow \rangle,~ |f_{x(x^2-3y^2)}, \uparrow \rangle,\\ & |f_{y(3x^2-y^2)}, \downarrow \rangle,~ |f_{xyz}, \downarrow \rangle,~ |f_{yz^2}, \downarrow \rangle,~ |f_{z^3}, \downarrow \rangle,~ |f_{xz^2}, \downarrow \rangle,~ |f_{z(x^2-y^2)}, \downarrow \rangle,~ |f_{x(x^2-3y^2)}, \downarrow \rangle. \end{split} \end{equation}\]

\[\begin{equation} f_{y(3x^{2}-y^{2})} = Y_{3,-3}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-3}+Y_{3}^{3}\right), \end{equation}\]

\[\begin{equation} f_{xyz} = Y_{3,-2}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-2}-Y_{3}^{2}\right), \end{equation}\]

\[\begin{equation} f_{yz^{2}} = Y_{3,-1}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-1}+Y_{3}^{1}\right), \end{equation}\]

\[\begin{equation} f_{z^{3}} = Y_{30}=Y_{3}^{0}, \end{equation}\]

\[\begin{equation} f_{xz^{2}} = Y_{31}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-1}-Y_{3}^{1}\right), \end{equation}\]

\[\begin{equation} f_{z(x^{2}-y^{2})} = Y_{32}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-2}+Y_{3}^{2}\right), \end{equation}\]

\[\begin{equation} f_{x(x^{2}-3y^{2})} = Y_{33}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-3}-Y_{3}^{3}\right). \end{equation}\]

For $t_{2g}$ system, we have a $T-P$ equivalence,

\[\begin{equation} d_{xz} \rightarrow p_{y}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right), \end{equation}\]

\[\begin{equation} d_{xy} \rightarrow p_{z}=Y_{1}^{0}, \end{equation}\]

\[\begin{equation} d_{yz} \rightarrow p_{x}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right). \end{equation}\]

Complex orbital basis

It is also called the $|l^2,l_z\rangle$ basis. The basis functions are the complex spherical harmonics $Y^{m}_l(\theta,\phi)$. We just use $l$ and $m$ to label the basis functions $|l,m\rangle$.

For $p$ system ($l = 1,~m = \pm 1,~0$), the basis order is

\[\begin{equation} \begin{split} &| 1, -1, \uparrow \rangle, \\ &| 1, ~~~0, \uparrow \rangle, \\ &| 1, ~~~1, \uparrow \rangle, \\ &| 1, -1, \downarrow \rangle, \\ &| 1, ~~~0, \downarrow \rangle, \\ &| 1, ~~~1, \downarrow \rangle. \end{split} \end{equation}\]

For $d$ system ($l = 2,~m = \pm 2,~\pm 1,~0$), the basis order is

\[\begin{equation} \begin{split} &| 2, -2, \uparrow \rangle, \\ &| 2, -1, \uparrow \rangle, \\ &| 2, ~~~0, \uparrow \rangle, \\ &| 2, ~~~1, \uparrow \rangle, \\ &| 2, ~~~2, \uparrow \rangle, \\ &| 2, -2, \downarrow \rangle, \\ &| 2, -1, \downarrow \rangle, \\ &| 2, ~~~0, \downarrow \rangle, \\ &| 2, ~~~1, \downarrow \rangle, \\ &| 2, ~~~2, \downarrow \rangle. \end{split} \end{equation}\]

For $f$ system ($l = 3,~m = \pm 3,~\pm 2,~\pm 1,~0$), the basis order is

\[\begin{equation} \begin{split} &| 3, -3, \uparrow \rangle, \\ &| 3, -2, \uparrow \rangle, \\ &| 3, -1, \uparrow \rangle, \\ &| 3, ~~~0, \uparrow \rangle, \\ &| 3, ~~~1, \uparrow \rangle, \\ &| 3, ~~~2, \uparrow \rangle, \\ &| 3, ~~~3, \uparrow \rangle, \\ &| 3, -3, \downarrow \rangle, \\ &| 3, -2, \downarrow \rangle, \\ &| 3, -1, \downarrow \rangle, \\ &| 3, ~~~0, \downarrow \rangle, \\ &| 3, ~~~1, \downarrow \rangle, \\ &| 3, ~~~2, \downarrow \rangle, \\ &| 3, ~~~3, \downarrow \rangle. \end{split} \end{equation}\]

$\hat{j}^{2}-\hat{j}_{z}-\hat{l}^2-\hat{s}^2$ diagonal basis

We just use $j$ and $m_j$ to label the eigenfunctions $|j, m_j\rangle$, which are just the spinor spherical harmonics $\Omega^l_{jm_j}(\theta,\phi)$.

For $p$ system, $l = 1$, $j = \frac{1}{2}$ or $\frac{3}{2}$, the basis order is

\[\begin{equation} \begin{split} &\left|\frac{1}{2}, -\frac{1}{2}\right\rangle = -\sqrt{\frac{2}{3}}Y^{-1}_{1}\chi_{\uparrow} + \sqrt{\frac{1}{3}}Y^{ 0}_{1}\chi_{\downarrow}, \\ &\left|\frac{1}{2}, ~~~\frac{1}{2}\right\rangle = -\sqrt{\frac{1}{3}}Y^{ 0}_{1}\chi_{\uparrow} + \sqrt{\frac{2}{3}}Y^{ 1}_{1}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, -\frac{3}{2}\right\rangle = \sqrt{\frac{0}{3}}Y^{-2}_{1}\chi_{\uparrow} + \sqrt{\frac{3}{3}}Y^{-1}_{1}\chi_{\downarrow} = Y^{-1}_{1} \chi_{\downarrow}, \\ &\left|\frac{3}{2}, -\frac{1}{2}\right\rangle = \sqrt{\frac{1}{3}}Y^{-1}_{1}\chi_{\uparrow} + \sqrt{\frac{2}{3}}Y^{ 0}_{1}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, ~~~\frac{1}{2}\right\rangle = \sqrt{\frac{2}{3}}Y^{ 0}_{1}\chi_{\uparrow} + \sqrt{\frac{1}{3}}Y^{ 1}_{1}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, ~~~\frac{3}{2}\right\rangle = \sqrt{\frac{3}{3}}Y^{ 1}_{1}\chi_{\uparrow} + \sqrt{\frac{0}{3}}Y^{ 2}_{1}\chi_{\downarrow} = Y^{1}_{1}\chi_{\uparrow}. \end{split} \end{equation}\]

For $d$ system, $l = 2$, $j = \frac{3}{2}$ or $\frac{5}{2}$, the basis order is

\[\begin{equation} \begin{split} &\left|\frac{3}{2}, -\frac{3}{2}\right\rangle = -\sqrt{\frac{4}{5}}Y^{-2}_{2}\chi_{\uparrow} + \sqrt{\frac{1}{5}}Y^{-1}_{2}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, -\frac{1}{2}\right\rangle = -\sqrt{\frac{3}{5}}Y^{-1}_{2}\chi_{\uparrow} + \sqrt{\frac{2}{5}}Y^{ 0}_{2}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, ~~~\frac{1}{2}\right\rangle = -\sqrt{\frac{2}{5}}Y^{ 0}_{2}\chi_{\uparrow} + \sqrt{\frac{3}{5}}Y^{ 1}_{2}\chi_{\downarrow}, \\ &\left|\frac{3}{2}, ~~~\frac{3}{2}\right\rangle = -\sqrt{\frac{1}{5}}Y^{ 1}_{2}\chi_{\uparrow} + \sqrt{\frac{4}{5}}Y^{ 2}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, -\frac{5}{2}\right\rangle = \sqrt{\frac{0}{5}}Y^{-3}_{2}\chi_{\uparrow} + \sqrt{\frac{5}{5}}Y^{-2}_{2}\chi_{\downarrow} = Y^{-2}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, -\frac{3}{2}\right\rangle = \sqrt{\frac{1}{5}}Y^{-2}_{2}\chi_{\uparrow} + \sqrt{\frac{4}{5}}Y^{-1}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, -\frac{1}{2}\right\rangle = \sqrt{\frac{2}{5}}Y^{-1}_{2}\chi_{\uparrow} + \sqrt{\frac{3}{5}}Y^{ 0}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{1}{2}\right\rangle = \sqrt{\frac{3}{5}}Y^{ 0}_{2}\chi_{\uparrow} + \sqrt{\frac{2}{5}}Y^{ 1}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{3}{2}\right\rangle = \sqrt{\frac{4}{5}}Y^{ 1}_{2}\chi_{\uparrow} + \sqrt{\frac{1}{5}}Y^{ 2}_{2}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{5}{2}\right\rangle = \sqrt{\frac{5}{5}}Y^{ 2}_{2}\chi_{\uparrow} + \sqrt{\frac{0}{5}}Y^{ 3}_{2}\chi_{\downarrow} = Y^{2}_{2}\chi_{\uparrow}. \end{split} \end{equation}\]

For $f$ system, $l = 3$, $j = \frac{5}{2}$ or $\frac{7}{2}$, the basis order is

\[\begin{equation} \begin{split} &\left|\frac{5}{2}, -\frac{5}{2}\right\rangle = -\sqrt{\frac{6}{7}}Y^{-3}_{3}\chi_{\uparrow} + \sqrt{\frac{1}{7}}Y^{-2}_{3}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, -\frac{3}{2}\right\rangle = -\sqrt{\frac{5}{7}}Y^{-2}_{3}\chi_{\uparrow} + \sqrt{\frac{2}{7}}Y^{-1}_{3}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, -\frac{1}{2}\right\rangle = -\sqrt{\frac{4}{7}}Y^{-1}_{3}\chi_{\uparrow} + \sqrt{\frac{3}{7}}Y^{ 0}_{3}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{1}{2}\right\rangle = -\sqrt{\frac{3}{7}}Y^{ 0}_{3}\chi_{\uparrow} + \sqrt{\frac{4}{7}}Y^{ 1}_{3}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{3}{2}\right\rangle = -\sqrt{\frac{2}{7}}Y^{ 1}_{3}\chi_{\uparrow} + \sqrt{\frac{5}{7}}Y^{ 2}_{3}\chi_{\downarrow}, \\ &\left|\frac{5}{2}, ~~~\frac{5}{2}\right\rangle = -\sqrt{\frac{1}{7}}Y^{ 2}_{3}\chi_{\uparrow} + \sqrt{\frac{6}{7}}Y^{ 3}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, -\frac{7}{2}\right\rangle = \sqrt{\frac{0}{7}}Y^{-4}_{3}\chi_{\uparrow} + \sqrt{\frac{7}{7}}Y^{-3}_{3}\chi_{\downarrow} = Y^{-3}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, -\frac{5}{2}\right\rangle = \sqrt{\frac{1}{7}}Y^{-3}_{3}\chi_{\uparrow} + \sqrt{\frac{6}{7}}Y^{-2}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, -\frac{3}{2}\right\rangle = \sqrt{\frac{2}{7}}Y^{-2}_{3}\chi_{\uparrow} + \sqrt{\frac{5}{7}}Y^{-1}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, -\frac{1}{2}\right\rangle = \sqrt{\frac{3}{7}}Y^{-1}_{3}\chi_{\uparrow} + \sqrt{\frac{4}{7}}Y^{ 0}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, ~~~\frac{1}{2}\right\rangle = \sqrt{\frac{4}{7}}Y^{ 0}_{3}\chi_{\uparrow} + \sqrt{\frac{3}{7}}Y^{ 1}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, ~~~\frac{3}{2}\right\rangle = \sqrt{\frac{5}{7}}Y^{ 1}_{3}\chi_{\uparrow} + \sqrt{\frac{2}{7}}Y^{ 2}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, ~~~\frac{5}{2}\right\rangle = \sqrt{\frac{6}{7}}Y^{ 2}_{3}\chi_{\uparrow} + \sqrt{\frac{1}{7}}Y^{ 3}_{3}\chi_{\downarrow}, \\ &\left|\frac{7}{2}, ~~~\frac{7}{2}\right\rangle = \sqrt{\frac{7}{7}}Y^{ 3}_{3}\chi_{\uparrow} + \sqrt{\frac{0}{7}}Y^{ 4}_{3}\chi_{\downarrow} = Y^{3}_{3}\chi_{\uparrow}. \\ \end{split} \end{equation}\]

Natural basis

The natural basis is defined as the diagonal basis of on-site term $E_{\alpha\beta}$.

Transformation matrix from complex orbital basis to real orbital basis

For p system, the transformation matrix reads

\[\begin{equation} T = \left[ \begin{array}{ccc|ccc} \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ \end{array} \right] \end{equation}\]

For d system, the transformation matrix reads

\[T = \left[ \begin{array}{ccccc|ccccc} \frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\ \end{array} \right]\]

For f system, the transformation matrix reads

\[T= \left[ \begin{array}{ccccccc|ccccccc} \frac{i}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{i}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & -\frac{1}{\sqrt{2}} \\ \end{array} \right]\]

The following Julia script is used to construct the complex orbital basis and the real orbital basis, and the transformation matrix between them.

# To calculate the transformation matrix from the complex orbital basis
# to the real orbital basis.
function calc_matrix(l::Int64)
    println("Construct complex orbital basis for 𝑙 = $l")
    COB = [] # To save the complex orbital basis
    # m = -l, -l+1, ..., l-1, l
    mlist = collect(-l:1:l)
    for s in ("up", "down")
        for m in mlist
            push!(COB, [m, s])
        end
    end
    #
    for i in eachindex(COB)
        m = COB[i][1]
        s = COB[i][2] == "up" ? "↑" : "↓"
        println("$i -> | $l, $m, $s ⟩")
    end

    println("Construct real orbital basis for 𝑙 = $l")
    RO = [] # To save the real orbital basis basis
    ROB = [] # To save the detailed expressions for the real orbital basis
    for s in ("up", "down")
        for m in mlist
            if m < 0
                b = ["i/sqrt(2)", -abs(m), -(-1)^m, abs(m)]
            elseif m == 0
                b = [0]
            elseif m > 0
                b = ["1/sqrt(2)", -abs(m),  (-1)^m, abs(m)]
            end
            push!(RO, [m,s])
            push!(ROB, b)
        end
    end
    #
    for i in eachindex(RO)
        m = RO[i][1]
        s = RO[i][2] == "up" ? "↑" : "↓"
        println("$i -> Y_{$l,$m} χ$s")        
    end

    println("Evaluate transformation matrix for 𝑙 = $l")
    for m in eachindex(COB)
        for n in eachindex(RO)
            if COB[m][2] == RO[n][2] # their spins are the same
                # for Y_{l0} case
                if length(ROB[n]) == 1
                    if COB[m][1] == ROB[n][1]
                        println("T($m,$n) -> 1")
                    end
                # for Y_{lm} case where m /= 0
                else
                    if COB[m][1] == ROB[n][2]
                        println("T($m,$n) -> ", ROB[n][1])
                    end
                    if COB[m][1] == ROB[n][4]
                        s = ROB[n][3] < 0 ? "-" : ""
                        println("T($m,$n) -> $s", ROB[n][1])
                    end
                end
            end
        end
    end
end

Transformation matrix from complex orbital basis to $\hat{j}^{2}-\hat{j}_{z}-\hat{l}^2-\hat{s}^2$ diagonal basis

For p system, the transformation matrix reads

\[\begin{equation} T = \left[ \begin{array}{ccc|ccc} -\sqrt{\frac{2}{3}} & 0 & 0 & \sqrt{\frac{1}{3}} & 0 & 0 \\ 0 & -\sqrt{\frac{1}{3}} & 0 & 0 & \sqrt{\frac{2}{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1.0 \\ \hline 0 & 0 & 1.0 & 0 & 0 & 0 \\ \sqrt{\frac{1}{3}} & 0 & 0 & \sqrt{\frac{2}{3}} & 0 & 0 \\ 0 & \sqrt{\frac{2}{3}} & 0 & 0 & \sqrt{\frac{1}{3}} & 0 \\ \end{array} \right] \end{equation}\]

For d system, the transformation matrix reads

\[T = \left[ \begin{array}{ccccc|ccccc} -\sqrt{\frac{4}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{5}} & 0 & 0 & 0 & 0 \\ 0 & -\sqrt{\frac{3}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{2}{5}} & 0 & 0 & 0 \\ 0 & 0 & -\sqrt{\frac{2}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{3}{5}} & 0 & 0 \\ 0 & 0 & 0 & -\sqrt{\frac{1}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{4}{5}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 \\ \hline 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\frac{1}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{4}{5}} & 0 & 0 & 0 & 0 \\ 0 & \sqrt{\frac{2}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{3}{5}} & 0 & 0 & 0 \\ 0 & 0 & \sqrt{\frac{3}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{2}{5}} & 0 & 0 \\ 0 & 0 & 0 & \sqrt{\frac{4}{5}} & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{5}} & 0 \\ \end{array} \right]\]

For f system, the transformation matrix reads

\[T = \left[ \begin{array}{ccccccc|ccccccc} -\sqrt{\frac{6}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{7}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\sqrt{\frac{5}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{2}{7}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\sqrt{\frac{4}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{3}{7}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\sqrt{\frac{3}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{4}{7}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\sqrt{\frac{2}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{5}{7}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -\sqrt{\frac{1}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{6}{7}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\frac{1}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{6}{7}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sqrt{\frac{2}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{5}{7}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{\frac{3}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{4}{7}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{\frac{4}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{3}{7}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{\frac{5}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{2}{7}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{6}{7}} & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{\frac{1}{7}} & 0 \\ \end{array} \right]\]

The following Julia script is used to construct the complex orbital basis and the $j^2-j_z$ basis, and the transformation matrix between them.

# To calculate the transformation matrix from the complex orbital basis
# to the j²-jz basis.
function calc_matrix(l::Int64)
    println("Construct complex orbital basis for 𝑙 = $l")
    COB = [] # To save the complex orbital basis
    # m = -l, -l+1, ..., l-1, l
    mlist = collect(-l:1:l)
    for s in ("up", "down")
        for m in mlist
            push!(COB, [m, s])
        end
    end
    #
    for i in eachindex(COB)
        m = COB[i][1]
        s = COB[i][2] == "up" ? "↑" : "↓"
        println("$i -> | $l, $m, $s ⟩")
    end

    println("Construct j²-jz basis for 𝑙 = $l")
    JJ = []  # To save the j²-jz basis
    JJB = [] # To save the detailed expressions for the j²-jz basis
    jlist = [l-1//2, l+1//2]
    for j in jlist
        # mⱼ = -j, -j+1, ..., j-1, j
        mⱼlist = collect(-j:2//2:j)
        for mⱼ in mⱼlist
            push!(JJ, [j, mⱼ])
            m = mⱼ-1//2
            if j == l-1//2 # For j = l-1/2
                jj = ["-", (l-m)/(2*l+1), Int(m), (l+m+1)/(2*l+1), Int(m+1)]
            else           # For j = l+1/2
                jj = ["" , (l+m+1)/(2*l+1), Int(m), (l-m)/(2*l+1), Int(m+1)]
            end
            push!(JJB, jj)
        end
    end
    #
    for i in eachindex(JJ)
        j = JJ[i][1]
        mⱼ = JJ[i][2]
        print("$i -> | $j, $mⱼ ⟩ = ")

        jj = JJB[i]
        print(jj[1])
        print("sqrt(", jj[2], ")Y^{", jj[3],"}_{$l}χ↑ + ")
        print("sqrt(", jj[4], ")Y^{", jj[5],"}_{$l}χ↓\n")
    end

    println("Evaluate transformation matrix for 𝑙 = $l")
    for m in eachindex(COB)
        for n in eachindex(JJ)
            if COB[m][2] == "up"
                if COB[m][1] == JJB[n][3]
                    println("T($m,$n) -> ", JJB[n][1], "sqrt(", JJB[n][2],")")
                end
            else
                if COB[m][1] == JJB[n][5]
                    println("T($m,$n) -> ", "sqrt(", JJB[n][4],")")
                end
            end
        end
    end
end

Reference