Single particle basis

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


Spherical harmonics

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

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

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

where ll is the azimuthal quantum number (l=0, 1, 2, , n1l = 0,~1,~2,~\cdots,~n-1), and mm is the magnetic quantum number (m=l, l+1, , lm=-l,~-l+1,~\cdots,~l)[1][2]. They are defined as follows:

Ylm(θ,ϕ)=2l+14π(lm)!(l+m)!Plm(cosθ)eimϕ,\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 θ\theta is taken as the polar (colatitudinal) coordinate with θ[0,π]\theta \in [0,\pi], and ϕ\phi as the azimuthal (longitudinal) coordinate with ϕ[0,2π]\phi \in [0,2\pi], and Plm(z)P^m_l(z) is an associated Legendre polynomial.

The spherical harmonics are orthonormal

θ=0πϕ=02πYlm(θ,ϕ)Ylm(θ,ϕ) dΩ=δllδmm,\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 δij\delta_{ij} is the Kronecker delta and dΩ=sin(θ)dϕdθd\Omega = \sin(\theta) d\phi d\theta.


Real Spherical Harmonics

The real spherical harmonics YlmY_{lm} are defined as[3]

Ylm={i2(Ylm(1)mYlm)if m<0,Yl0if m=0,12(Ylm+(1)mYlm)if m>0.\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 YlmY^m_l in terms of the real spherical harmonics YlmY_{lm} read:

Ylm={12(YlmiYl,m)if m<0,Yl0if m=0,(1)m2(Ylm+iYl,m)if m>0.\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 YlmY_{lm} are sometimes known as tesseral spherical harmonics. These functions have the same orthonormality properties as the complex ones YlmY_{l}^{m}.


Spinor spherical harmonics

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

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

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

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

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

For given jj only two values of ll are possible, l=j±12l = j \pm \frac{1}{2}, while mjm_j assumes 2j+12j + 1 values (mj=j, j+1, , j1, jm_j = -j,~-j + 1,~\cdots,~j - 1,~j) [1][2].

For j=l+12j = l + \frac{1}{2}, mj=m+12m_j = m + \frac{1}{2},

Ωjmjl=l+m+12l+1Ylmχ+lm2l+1Ylm+1χ.\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=l12(l0)j = l - \frac{1}{2} (l \neq 0), mj=m+12m_j = m + \frac{1}{2},

Ωjmjl=lm2l+1Ylmχ+l+m+12l+1Ylm+1χ.\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 Ylm(θ,ϕ)Y_{lm}(\theta,\phi).

For pp system, the basis order is[4]

py,, pz,, px,, py,, pz,, px,.\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}

py=Y1,1=i2(Y11+Y11),\begin{equation} |p_{y}\rangle = Y_{1,-1}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right), \end{equation}

pz=Y10=Y10,\begin{equation} |p_{z}\rangle = Y_{10}=Y_{1}^{0}, \end{equation}

px=Y11=12(Y11Y11).\begin{equation} |p_{x}\rangle = Y_{11}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right). \end{equation}

For dd system, the basis order is[4]

dxy,, dyz,, dz2,, dxz,, dx2y2,, dxy,, dyz,, dz2,, dxz,, dx2y2,.\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}

dxy=Y2,2=i2(Y22Y22),\begin{equation} d_{xy} = Y_{2,-2}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-2}-Y_{2}^{2}\right), \end{equation}

dyz=Y2,1=i2(Y21+Y21),\begin{equation} d_{yz} = Y_{2,-1}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-1}+Y_{2}^{1}\right), \end{equation}

dz2=Y20=Y20,\begin{equation} d_{z^{2}} = Y_{20}=Y_{2}^{0}, \end{equation}

dxz=Y21=12(Y21Y21),\begin{equation} d_{xz} = Y_{21}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-1}-Y_{2}^{1}\right), \end{equation}

dx2y2=Y22=12(Y22+Y22).\begin{equation} d_{x^{2}-y^{2}} = Y_{22}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-2}+Y_{2}^{2}\right). \end{equation}

For ff system, the basis order is[4]

fy(3x2y2),, fxyz,, fyz2,, fz3,, fxz2,, fz(x2y2),, fx(x23y2),,fy(3x2y2),, fxyz,, fyz2,, fz3,, fxz2,, fz(x2y2),, fx(x23y2),.\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}

fy(3x2y2)=Y3,3=i2(Y33+Y33),\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}

fxyz=Y3,2=i2(Y32Y32),\begin{equation} f_{xyz} = Y_{3,-2}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-2}-Y_{3}^{2}\right), \end{equation}

fyz2=Y3,1=i2(Y31+Y31),\begin{equation} f_{yz^{2}} = Y_{3,-1}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-1}+Y_{3}^{1}\right), \end{equation}

fz3=Y30=Y30,\begin{equation} f_{z^{3}} = Y_{30}=Y_{3}^{0}, \end{equation}

fxz2=Y31=12(Y31Y31),\begin{equation} f_{xz^{2}} = Y_{31}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-1}-Y_{3}^{1}\right), \end{equation}

fz(x2y2)=Y32=12(Y32+Y32),\begin{equation} f_{z(x^{2}-y^{2})} = Y_{32}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-2}+Y_{3}^{2}\right), \end{equation}

fx(x23y2)=Y33=12(Y33Y33).\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 t2gt_{2g} system, we have a TPT-P equivalence,

dxzpy=i2(Y11+Y11),\begin{equation} d_{xz} \rightarrow p_{y}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right), \end{equation}

dxypz=Y10,\begin{equation} d_{xy} \rightarrow p_{z}=Y_{1}^{0}, \end{equation}

dyzpx=12(Y11Y11).\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 l2,lz|l^2,l_z\rangle basis. The basis functions are the complex spherical harmonics Ylm(θ,ϕ)Y^{m}_l(\theta,\phi). We just use ll and mm to label the basis functions l,m|l,m\rangle.

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

1,1,,1,   0,,1,   1,,1,1,,1,   0,,1,   1,.\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 dd system (l=2, m=±2, ±1, 0l = 2,~m = \pm 2,~\pm 1,~0), the basis order is

2,2,,2,1,,2,   0,,2,   1,,2,   2,,2,2,,2,1,,2,   0,,2,   1,,2,   2,.\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 ff system (l=3, m=±3, ±2, ±1, 0l = 3,~m = \pm 3,~\pm 2,~\pm 1,~0), the basis order is

3,3,,3,2,,3,1,,3,   0,,3,   1,,3,   2,,3,   3,,3,3,,3,2,,3,1,,3,   0,,3,   1,,3,   2,,3,   3,.\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}


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

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

For pp system, l=1l = 1, j=12j = \frac{1}{2} or 32\frac{3}{2}, the basis order is

12,12=23Y11χ+13Y10χ,12,   12=13Y10χ+23Y11χ,32,32=03Y12χ+33Y11χ=Y11χ,32,12=13Y11χ+23Y10χ,32,   12=23Y10χ+13Y11χ,32,   32=33Y11χ+03Y12χ=Y11χ.\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 dd system, l=2l = 2, j=32j = \frac{3}{2} or 52\frac{5}{2}, the basis order is

32,32=45Y22χ+15Y21χ,32,12=35Y21χ+25Y20χ,32,   12=25Y20χ+35Y21χ,32,   32=15Y21χ+45Y22χ,52,52=05Y23χ+55Y22χ=Y22χ,52,32=15Y22χ+45Y21χ,52,12=25Y21χ+35Y20χ,52,   12=35Y20χ+25Y21χ,52,   32=45Y21χ+15Y22χ,52,   52=55Y22χ+05Y23χ=Y22χ.\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 ff system, l=3l = 3, j=52j = \frac{5}{2} or 72\frac{7}{2}, the basis order is

52,52=67Y33χ+17Y32χ,52,32=57Y32χ+27Y31χ,52,12=47Y31χ+37Y30χ,52,   12=37Y30χ+47Y31χ,52,   32=27Y31χ+57Y32χ,52,   52=17Y32χ+67Y33χ,72,72=07Y34χ+77Y33χ=Y33χ,72,52=17Y33χ+67Y32χ,72,32=27Y32χ+57Y31χ,72,12=37Y31χ+47Y30χ,72,   12=47Y30χ+37Y31χ,72,   32=57Y31χ+27Y32χ,72,   52=67Y32χ+17Y33χ,72,   72=77Y33χ+07Y34χ=Y33χ.\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 EαβE_{\alpha\beta}.


Transformation matrix from complex orbital basis to real orbital basis

For p system, the transformation matrix reads

T=[i2012000010000i2012000000i2012000010000i2012]\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=[i200012000000i201200000000100000000i2012000000i2000120000000000i200012000000i201200000000100000000i2012000000i200012]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=[i2000001200000000i2000120000000000i20120000000000001000000000000i20120000000000i20001200000000i2000001200000000000000i2000001200000000i2000120000000000i20120000000000001000000000000i20120000000000i20001200000000i20000012]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 j^2j^zl^2s^2\hat{j}^{2}-\hat{j}_{z}-\hat{l}^2-\hat{s}^2 diagonal basis

For p system, the transformation matrix reads

T=[2300130001300230000001.0001.00001300230002300130]\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=[4500001500000350000250000025000035000001500004500000000001.000001.000000150000450000025000035000003500002500000450000150]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=[67000000170000000570000002700000004700000037000000037000000470000000270000005700000001700000067000000000000001.00000001.00000000170000006700000002700000057000000037000000470000000470000003700000005700000027000000067000000170]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 j2jzj^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

  • 1曾谨言, 量子力学(卷1, 第四版), 科学出版社, 2007。
  • 2D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
  • 3See https://handwiki.org/wiki/Physics:Spherical_harmonics.
  • 4The orbital orders are consistent with the definition of local basis used by VASP (see https://www.vasp.at/wiki/index.php/LOCPROJ), and the definition in HandWiki (see https://handwiki.org/wiki/Table_of_spherical_harmonics).