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}} H ^ atom . We set ℏ = 1 \hbar=1 ℏ = 1 in this note.
Spherical harmonics
The complex spherical harmonics Y l m ( θ , ϕ ) Y_{l}^{m}(\theta,\phi) Y l m ( θ , ϕ ) are the eigenstates of operators l ^ 2 \hat{l}^{2} l ^ 2 and l ^ z \hat{l}_{z} l ^ z ,
l ^ 2 Y l m = l ( l + 1 ) Y l m , \begin{equation}
\hat{l}^{2}Y_{l}^{m}=l(l+1)Y_{l}^{m},
\end{equation} l ^ 2 Y l m = l ( l + 1 ) Y l m ,
l ^ z Y l m = m Y l m , \begin{equation}
\hat{l}_{z}Y_{l}^{m}=mY_{l}^{m},
\end{equation} l ^ z Y l m = m Y l m ,
where l l l is the azimuthal quantum number (l = 0 , 1 , 2 , ⋯ , n − 1 l = 0,~1,~2,~\cdots,~n-1 l = 0 , 1 , 2 , ⋯ , n − 1 ), and m m m is the magnetic quantum number (m = − l , − l + 1 , ⋯ , l m=-l,~-l+1,~\cdots,~l m = − l , − l + 1 , ⋯ , l ). They are defined as follows:
Y l m ( θ , ϕ ) = 2 l + 1 4 π ( l − m ) ! ( l + m ) ! P l m ( cos θ ) e i m ϕ , \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} Y l m ( θ , ϕ ) = 4 π 2 l + 1 ( l + m )! ( l − m )! P l m ( cos θ ) e im ϕ ,
where θ \theta θ is taken as the polar (colatitudinal) coordinate with θ ∈ [ 0 , π ] \theta \in [0,\pi] θ ∈ [ 0 , π ] , and ϕ \phi ϕ as the azimuthal (longitudinal) coordinate with ϕ ∈ [ 0 , 2 π ] \phi \in [0,2\pi] ϕ ∈ [ 0 , 2 π ] , and P l m ( z ) P^m_l(z) P l m ( z ) is an associated Legendre polynomial.
The spherical harmonics are orthonormal
∫ θ = 0 π ∫ ϕ = 0 2 π Y l m ( θ , ϕ ) Y l ′ m ′ ∗ ( θ , ϕ ) d Ω = δ l l ′ δ m m ′ , \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} ∫ θ = 0 π ∫ ϕ = 0 2 π Y l m ( θ , ϕ ) Y l ′ m ′ ∗ ( θ , ϕ ) d Ω = δ l l ′ δ m m ′ ,
where δ i j \delta_{ij} δ ij is the Kronecker delta and d Ω = sin ( θ ) d ϕ d θ d\Omega = \sin(\theta) d\phi d\theta d Ω = sin ( θ ) d ϕ d θ .
Real Spherical Harmonics
The real spherical harmonics Y l m Y_{lm} Y l m are defined as
Y l m = { i 2 ( Y l − ∣ m ∣ − ( − 1 ) m Y l ∣ m ∣ ) if m < 0 , Y l 0 if m = 0 , 1 2 ( Y l − ∣ m ∣ + ( − 1 ) m Y l ∣ m ∣ ) 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} Y l m = ⎩ ⎨ ⎧ 2 i ( Y l − ∣ m ∣ − ( − 1 ) m Y l ∣ m ∣ ) Y l 0 2 1 ( Y l − ∣ m ∣ + ( − 1 ) m Y l ∣ m ∣ ) if m < 0 , if m = 0 , if m > 0.
The corresponding inverse equations defining the complex spherical harmonics Y l m Y^m_l Y l m in terms of the real spherical harmonics Y l m Y_{lm} Y l m read:
Y l m = { 1 2 ( Y l ∣ m ∣ − i Y l , − ∣ m ∣ ) if m < 0 , Y l 0 if m = 0 , ( − 1 ) m 2 ( Y l ∣ m ∣ + i Y l , − ∣ 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} Y l m = ⎩ ⎨ ⎧ 2 1 ( Y l ∣ m ∣ − i Y l , − ∣ m ∣ ) Y l 0 2 ( − 1 ) m ( Y l ∣ m ∣ + i Y l , − ∣ m ∣ ) if m < 0 , if m = 0 , if m > 0.
The real spherical harmonics Y l m Y_{lm} Y l m are sometimes known as tesseral spherical harmonics. These functions have the same orthonormality properties as the complex ones Y l m Y_{l}^{m} Y l m .
Spinor spherical harmonics
The spinor spherical harmonics Ω j m j l ( θ , ϕ ) \Omega^l_{jm_j}(\theta,\phi) Ω j m j l ( θ , ϕ ) are eigenstates of the operators j ^ 2 \hat{j}^2 j ^ 2 , j ^ z \hat{j}_z j ^ z , l ^ 2 \hat{l}^2 l ^ 2 , and s ^ 2 \hat{s}^2 s ^ 2 ,
j ^ 2 Ω j m j l = j ( j + 1 ) Ω j m j l , \begin{equation}
\hat{j}^2 \Omega^l_{jm_j} = j (j + 1) \Omega^l_{jm_j},
\end{equation} j ^ 2 Ω j m j l = j ( j + 1 ) Ω j m j l ,
j ^ z Ω j m j l = m j Ω j m j l , \begin{equation}
\hat{j}_z \Omega^l_{jm_j} = m_j \Omega^l_{jm_j},
\end{equation} j ^ z Ω j m j l = m j Ω j m j l ,
l ^ 2 Ω j m j l = l ( l + 1 ) Ω j m j l , \begin{equation}
\hat{l}^2 \Omega^l_{jm_j} = l (l + 1) \Omega^l_{jm_j},
\end{equation} l ^ 2 Ω j m j l = l ( l + 1 ) Ω j m j l ,
s ^ 2 Ω j m j l = s ( s + 1 ) Ω j m j l . \begin{equation}
\hat{s}^2 \Omega^l_{jm_j} = s(s+1) \Omega^l_{jm_j}.
\end{equation} s ^ 2 Ω j m j l = s ( s + 1 ) Ω j m j l .
For given j j j only two values of l l l are possible, l = j ± 1 2 l = j \pm \frac{1}{2} l = j ± 2 1 , while m j m_j m j assumes 2 j + 1 2j + 1 2 j + 1 values (m j = − j , − j + 1 , ⋯ , j − 1 , j m_j = -j,~-j + 1,~\cdots,~j - 1,~j m j = − j , − j + 1 , ⋯ , j − 1 , j ) .
For j = l + 1 2 j = l + \frac{1}{2} j = l + 2 1 , m j = m + 1 2 m_j = m + \frac{1}{2} m j = m + 2 1 ,
Ω j m j l = l + m + 1 2 l + 1 Y l m χ ↑ + l − m 2 l + 1 Y l m + 1 χ ↓ . \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} Ω j m j l = 2 l + 1 l + m + 1 Y l m χ ↑ + 2 l + 1 l − m Y l m + 1 χ ↓ .
For j = l − 1 2 ( l ≠ 0 ) j = l - \frac{1}{2} (l \neq 0) j = l − 2 1 ( l = 0 ) , m j = m + 1 2 m_j = m + \frac{1}{2} m j = m + 2 1 ,
Ω j m j l = − l − m 2 l + 1 Y l m χ ↑ + l + m + 1 2 l + 1 Y l m + 1 χ ↓ . \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} Ω j m j l = − 2 l + 1 l − m Y l m χ ↑ + 2 l + 1 l + m + 1 Y l m + 1 χ ↓ .
Real orbital basis
The basis functions are the real spherical harmonics Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi) Y l m ( θ , ϕ ) .
For p p p system, the basis order is
∣ p y , ↑ ⟩ , ∣ p z , ↑ ⟩ , ∣ p x , ↑ ⟩ , ∣ p y , ↓ ⟩ , ∣ p z , ↓ ⟩ , ∣ p x , ↓ ⟩ . \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} ∣ p y , ↑ ⟩ , ∣ p z , ↑ ⟩ , ∣ p x , ↑ ⟩ , ∣ p y , ↓ ⟩ , ∣ p z , ↓ ⟩ , ∣ p x , ↓ ⟩ .
∣ p y ⟩ = Y 1 , − 1 = i 2 ( Y 1 − 1 + Y 1 1 ) , \begin{equation}
|p_{y}\rangle = Y_{1,-1}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right),
\end{equation} ∣ p y ⟩ = Y 1 , − 1 = 2 i ( Y 1 − 1 + Y 1 1 ) ,
∣ p z ⟩ = Y 10 = Y 1 0 , \begin{equation}
|p_{z}\rangle = Y_{10}=Y_{1}^{0},
\end{equation} ∣ p z ⟩ = Y 10 = Y 1 0 ,
∣ p x ⟩ = Y 11 = 1 2 ( Y 1 − 1 − Y 1 1 ) . \begin{equation}
|p_{x}\rangle = Y_{11}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right).
\end{equation} ∣ p x ⟩ = Y 11 = 2 1 ( Y 1 − 1 − Y 1 1 ) .
For d d d system, the basis order is
∣ d x y , ↑ ⟩ , ∣ d y z , ↑ ⟩ , ∣ d z 2 , ↑ ⟩ , ∣ d x z , ↑ ⟩ , ∣ d x 2 − y 2 , ↑ ⟩ , ∣ d x y , ↓ ⟩ , ∣ d y z , ↓ ⟩ , ∣ d z 2 , ↓ ⟩ , ∣ d x z , ↓ ⟩ , ∣ d x 2 − y 2 , ↓ ⟩ . \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} ∣ d x y , ↑ ⟩ , ∣ d yz , ↑ ⟩ , ∣ d z 2 , ↑ ⟩ , ∣ d x z , ↑ ⟩ , ∣ d x 2 − y 2 , ↑ ⟩ , ∣ d x y , ↓ ⟩ , ∣ d yz , ↓ ⟩ , ∣ d z 2 , ↓ ⟩ , ∣ d x z , ↓ ⟩ , ∣ d x 2 − y 2 , ↓ ⟩ .
d x y = Y 2 , − 2 = i 2 ( Y 2 − 2 − Y 2 2 ) , \begin{equation}
d_{xy} = Y_{2,-2}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-2}-Y_{2}^{2}\right),
\end{equation} d x y = Y 2 , − 2 = 2 i ( Y 2 − 2 − Y 2 2 ) ,
d y z = Y 2 , − 1 = i 2 ( Y 2 − 1 + Y 2 1 ) , \begin{equation}
d_{yz} = Y_{2,-1}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-1}+Y_{2}^{1}\right),
\end{equation} d yz = Y 2 , − 1 = 2 i ( Y 2 − 1 + Y 2 1 ) ,
d z 2 = Y 20 = Y 2 0 , \begin{equation}
d_{z^{2}} = Y_{20}=Y_{2}^{0},
\end{equation} d z 2 = Y 20 = Y 2 0 ,
d x z = Y 21 = 1 2 ( Y 2 − 1 − Y 2 1 ) , \begin{equation}
d_{xz} = Y_{21}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-1}-Y_{2}^{1}\right),
\end{equation} d x z = Y 21 = 2 1 ( Y 2 − 1 − Y 2 1 ) ,
d x 2 − y 2 = Y 22 = 1 2 ( Y 2 − 2 + Y 2 2 ) . \begin{equation}
d_{x^{2}-y^{2}} = Y_{22}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-2}+Y_{2}^{2}\right).
\end{equation} d x 2 − y 2 = Y 22 = 2 1 ( Y 2 − 2 + Y 2 2 ) .
For f f f system, the basis order is
∣ f y ( 3 x 2 − y 2 ) , ↑ ⟩ , ∣ f x y z , ↑ ⟩ , ∣ f y z 2 , ↑ ⟩ , ∣ f z 3 , ↑ ⟩ , ∣ f x z 2 , ↑ ⟩ , ∣ f z ( x 2 − y 2 ) , ↑ ⟩ , ∣ f x ( x 2 − 3 y 2 ) , ↑ ⟩ , ∣ f y ( 3 x 2 − y 2 ) , ↓ ⟩ , ∣ f x y z , ↓ ⟩ , ∣ f y z 2 , ↓ ⟩ , ∣ f z 3 , ↓ ⟩ , ∣ f x z 2 , ↓ ⟩ , ∣ f z ( x 2 − y 2 ) , ↓ ⟩ , ∣ f x ( x 2 − 3 y 2 ) , ↓ ⟩ . \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} ∣ f y ( 3 x 2 − y 2 ) , ↑ ⟩ , ∣ f x yz , ↑ ⟩ , ∣ f y z 2 , ↑ ⟩ , ∣ f z 3 , ↑ ⟩ , ∣ f x z 2 , ↑ ⟩ , ∣ f z ( x 2 − y 2 ) , ↑ ⟩ , ∣ f x ( x 2 − 3 y 2 ) , ↑ ⟩ , ∣ f y ( 3 x 2 − y 2 ) , ↓ ⟩ , ∣ f x yz , ↓ ⟩ , ∣ f y z 2 , ↓ ⟩ , ∣ f z 3 , ↓ ⟩ , ∣ f x z 2 , ↓ ⟩ , ∣ f z ( x 2 − y 2 ) , ↓ ⟩ , ∣ f x ( x 2 − 3 y 2 ) , ↓ ⟩ .
f y ( 3 x 2 − y 2 ) = Y 3 , − 3 = i 2 ( Y 3 − 3 + Y 3 3 ) , \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} f y ( 3 x 2 − y 2 ) = Y 3 , − 3 = 2 i ( Y 3 − 3 + Y 3 3 ) ,
f x y z = Y 3 , − 2 = i 2 ( Y 3 − 2 − Y 3 2 ) , \begin{equation}
f_{xyz} = Y_{3,-2}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-2}-Y_{3}^{2}\right),
\end{equation} f x yz = Y 3 , − 2 = 2 i ( Y 3 − 2 − Y 3 2 ) ,
f y z 2 = Y 3 , − 1 = i 2 ( Y 3 − 1 + Y 3 1 ) , \begin{equation}
f_{yz^{2}} = Y_{3,-1}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-1}+Y_{3}^{1}\right),
\end{equation} f y z 2 = Y 3 , − 1 = 2 i ( Y 3 − 1 + Y 3 1 ) ,
f z 3 = Y 30 = Y 3 0 , \begin{equation}
f_{z^{3}} = Y_{30}=Y_{3}^{0},
\end{equation} f z 3 = Y 30 = Y 3 0 ,
f x z 2 = Y 31 = 1 2 ( Y 3 − 1 − Y 3 1 ) , \begin{equation}
f_{xz^{2}} = Y_{31}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-1}-Y_{3}^{1}\right),
\end{equation} f x z 2 = Y 31 = 2 1 ( Y 3 − 1 − Y 3 1 ) ,
f z ( x 2 − y 2 ) = Y 32 = 1 2 ( Y 3 − 2 + Y 3 2 ) , \begin{equation}
f_{z(x^{2}-y^{2})} = Y_{32}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-2}+Y_{3}^{2}\right),
\end{equation} f z ( x 2 − y 2 ) = Y 32 = 2 1 ( Y 3 − 2 + Y 3 2 ) ,
f x ( x 2 − 3 y 2 ) = Y 33 = 1 2 ( Y 3 − 3 − Y 3 3 ) . \begin{equation}
f_{x(x^{2}-3y^{2})} = Y_{33}=\frac{1}{\sqrt{2}}\left(Y_{3}^{-3}-Y_{3}^{3}\right).
\end{equation} f x ( x 2 − 3 y 2 ) = Y 33 = 2 1 ( Y 3 − 3 − Y 3 3 ) .
For t 2 g t_{2g} t 2 g system, we have a T − P T-P T − P equivalence,
d x z → p y = i 2 ( Y 1 − 1 + Y 1 1 ) , \begin{equation}
d_{xz} \rightarrow p_{y}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right),
\end{equation} d x z → p y = 2 i ( Y 1 − 1 + Y 1 1 ) ,
d x y → p z = Y 1 0 , \begin{equation}
d_{xy} \rightarrow p_{z}=Y_{1}^{0},
\end{equation} d x y → p z = Y 1 0 ,
d y z → p x = 1 2 ( Y 1 − 1 − Y 1 1 ) . \begin{equation}
d_{yz} \rightarrow p_{x}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right).
\end{equation} d yz → p x = 2 1 ( Y 1 − 1 − Y 1 1 ) .
Complex orbital basis
It is also called the ∣ l 2 , l z ⟩ |l^2,l_z\rangle ∣ l 2 , l z ⟩ basis. The basis functions are the complex spherical harmonics Y l m ( θ , ϕ ) Y^{m}_l(\theta,\phi) Y l m ( θ , ϕ ) . We just use l l l and m m m to label the basis functions ∣ l , m ⟩ |l,m\rangle ∣ l , m ⟩ .
For p p p system (l = 1 , m = ± 1 , 0 l = 1,~m = \pm 1,~0 l = 1 , m = ± 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} ∣1 , − 1 , ↑ ⟩ , ∣1 , 0 , ↑ ⟩ , ∣1 , 1 , ↑ ⟩ , ∣1 , − 1 , ↓ ⟩ , ∣1 , 0 , ↓ ⟩ , ∣1 , 1 , ↓ ⟩ .
For d d d system (l = 2 , m = ± 2 , ± 1 , 0 l = 2,~m = \pm 2,~\pm 1,~0 l = 2 , m = ± 2 , ± 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} ∣2 , − 2 , ↑ ⟩ , ∣2 , − 1 , ↑ ⟩ , ∣2 , 0 , ↑ ⟩ , ∣2 , 1 , ↑ ⟩ , ∣2 , 2 , ↑ ⟩ , ∣2 , − 2 , ↓ ⟩ , ∣2 , − 1 , ↓ ⟩ , ∣2 , 0 , ↓ ⟩ , ∣2 , 1 , ↓ ⟩ , ∣2 , 2 , ↓ ⟩ .
For f f f system (l = 3 , m = ± 3 , ± 2 , ± 1 , 0 l = 3,~m = \pm 3,~\pm 2,~\pm 1,~0 l = 3 , m = ± 3 , ± 2 , ± 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} ∣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 , ↓ ⟩ .
j ^ 2 − j ^ z − l ^ 2 − s ^ 2 \hat{j}^{2}-\hat{j}_{z}-\hat{l}^2-\hat{s}^2 j ^ 2 − j ^ z − l ^ 2 − s ^ 2 diagonal basis
We just use j j j and m j m_j m j to label the eigenfunctions ∣ j , m j ⟩ |j, m_j\rangle ∣ j , m j ⟩ , which are just the spinor spherical harmonics Ω j m j l ( θ , ϕ ) \Omega^l_{jm_j}(\theta,\phi) Ω j m j l ( θ , ϕ ) .
For p p p system, l = 1 l = 1 l = 1 , j = 1 2 j = \frac{1}{2} j = 2 1 or 3 2 \frac{3}{2} 2 3 , the basis order is
∣ 1 2 , − 1 2 ⟩ = − 2 3 Y 1 − 1 χ ↑ + 1 3 Y 1 0 χ ↓ , ∣ 1 2 , 1 2 ⟩ = − 1 3 Y 1 0 χ ↑ + 2 3 Y 1 1 χ ↓ , ∣ 3 2 , − 3 2 ⟩ = 0 3 Y 1 − 2 χ ↑ + 3 3 Y 1 − 1 χ ↓ = Y 1 − 1 χ ↓ , ∣ 3 2 , − 1 2 ⟩ = 1 3 Y 1 − 1 χ ↑ + 2 3 Y 1 0 χ ↓ , ∣ 3 2 , 1 2 ⟩ = 2 3 Y 1 0 χ ↑ + 1 3 Y 1 1 χ ↓ , ∣ 3 2 , 3 2 ⟩ = 3 3 Y 1 1 χ ↑ + 0 3 Y 1 2 χ ↓ = Y 1 1 χ ↑ . \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} 2 1 , − 2 1 ⟩ = − 3 2 Y 1 − 1 χ ↑ + 3 1 Y 1 0 χ ↓ , 2 1 , 2 1 ⟩ = − 3 1 Y 1 0 χ ↑ + 3 2 Y 1 1 χ ↓ , 2 3 , − 2 3 ⟩ = 3 0 Y 1 − 2 χ ↑ + 3 3 Y 1 − 1 χ ↓ = Y 1 − 1 χ ↓ , 2 3 , − 2 1 ⟩ = 3 1 Y 1 − 1 χ ↑ + 3 2 Y 1 0 χ ↓ , 2 3 , 2 1 ⟩ = 3 2 Y 1 0 χ ↑ + 3 1 Y 1 1 χ ↓ , 2 3 , 2 3 ⟩ = 3 3 Y 1 1 χ ↑ + 3 0 Y 1 2 χ ↓ = Y 1 1 χ ↑ .
For d d d system, l = 2 l = 2 l = 2 , j = 3 2 j = \frac{3}{2} j = 2 3 or 5 2 \frac{5}{2} 2 5 , the basis order is
∣ 3 2 , − 3 2 ⟩ = − 4 5 Y 2 − 2 χ ↑ + 1 5 Y 2 − 1 χ ↓ , ∣ 3 2 , − 1 2 ⟩ = − 3 5 Y 2 − 1 χ ↑ + 2 5 Y 2 0 χ ↓ , ∣ 3 2 , 1 2 ⟩ = − 2 5 Y 2 0 χ ↑ + 3 5 Y 2 1 χ ↓ , ∣ 3 2 , 3 2 ⟩ = − 1 5 Y 2 1 χ ↑ + 4 5 Y 2 2 χ ↓ , ∣ 5 2 , − 5 2 ⟩ = 0 5 Y 2 − 3 χ ↑ + 5 5 Y 2 − 2 χ ↓ = Y 2 − 2 χ ↓ , ∣ 5 2 , − 3 2 ⟩ = 1 5 Y 2 − 2 χ ↑ + 4 5 Y 2 − 1 χ ↓ , ∣ 5 2 , − 1 2 ⟩ = 2 5 Y 2 − 1 χ ↑ + 3 5 Y 2 0 χ ↓ , ∣ 5 2 , 1 2 ⟩ = 3 5 Y 2 0 χ ↑ + 2 5 Y 2 1 χ ↓ , ∣ 5 2 , 3 2 ⟩ = 4 5 Y 2 1 χ ↑ + 1 5 Y 2 2 χ ↓ , ∣ 5 2 , 5 2 ⟩ = 5 5 Y 2 2 χ ↑ + 0 5 Y 2 3 χ ↓ = Y 2 2 χ ↑ . \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} 2 3 , − 2 3 ⟩ = − 5 4 Y 2 − 2 χ ↑ + 5 1 Y 2 − 1 χ ↓ , 2 3 , − 2 1 ⟩ = − 5 3 Y 2 − 1 χ ↑ + 5 2 Y 2 0 χ ↓ , 2 3 , 2 1 ⟩ = − 5 2 Y 2 0 χ ↑ + 5 3 Y 2 1 χ ↓ , 2 3 , 2 3 ⟩ = − 5 1 Y 2 1 χ ↑ + 5 4 Y 2 2 χ ↓ , 2 5 , − 2 5 ⟩ = 5 0 Y 2 − 3 χ ↑ + 5 5 Y 2 − 2 χ ↓ = Y 2 − 2 χ ↓ , 2 5 , − 2 3 ⟩ = 5 1 Y 2 − 2 χ ↑ + 5 4 Y 2 − 1 χ ↓ , 2 5 , − 2 1 ⟩ = 5 2 Y 2 − 1 χ ↑ + 5 3 Y 2 0 χ ↓ , 2 5 , 2 1 ⟩ = 5 3 Y 2 0 χ ↑ + 5 2 Y 2 1 χ ↓ , 2 5 , 2 3 ⟩ = 5 4 Y 2 1 χ ↑ + 5 1 Y 2 2 χ ↓ , 2 5 , 2 5 ⟩ = 5 5 Y 2 2 χ ↑ + 5 0 Y 2 3 χ ↓ = Y 2 2 χ ↑ .
For f f f system, l = 3 l = 3 l = 3 , j = 5 2 j = \frac{5}{2} j = 2 5 or 7 2 \frac{7}{2} 2 7 , the basis order is
∣ 5 2 , − 5 2 ⟩ = − 6 7 Y 3 − 3 χ ↑ + 1 7 Y 3 − 2 χ ↓ , ∣ 5 2 , − 3 2 ⟩ = − 5 7 Y 3 − 2 χ ↑ + 2 7 Y 3 − 1 χ ↓ , ∣ 5 2 , − 1 2 ⟩ = − 4 7 Y 3 − 1 χ ↑ + 3 7 Y 3 0 χ ↓ , ∣ 5 2 , 1 2 ⟩ = − 3 7 Y 3 0 χ ↑ + 4 7 Y 3 1 χ ↓ , ∣ 5 2 , 3 2 ⟩ = − 2 7 Y 3 1 χ ↑ + 5 7 Y 3 2 χ ↓ , ∣ 5 2 , 5 2 ⟩ = − 1 7 Y 3 2 χ ↑ + 6 7 Y 3 3 χ ↓ , ∣ 7 2 , − 7 2 ⟩ = 0 7 Y 3 − 4 χ ↑ + 7 7 Y 3 − 3 χ ↓ = Y 3 − 3 χ ↓ , ∣ 7 2 , − 5 2 ⟩ = 1 7 Y 3 − 3 χ ↑ + 6 7 Y 3 − 2 χ ↓ , ∣ 7 2 , − 3 2 ⟩ = 2 7 Y 3 − 2 χ ↑ + 5 7 Y 3 − 1 χ ↓ , ∣ 7 2 , − 1 2 ⟩ = 3 7 Y 3 − 1 χ ↑ + 4 7 Y 3 0 χ ↓ , ∣ 7 2 , 1 2 ⟩ = 4 7 Y 3 0 χ ↑ + 3 7 Y 3 1 χ ↓ , ∣ 7 2 , 3 2 ⟩ = 5 7 Y 3 1 χ ↑ + 2 7 Y 3 2 χ ↓ , ∣ 7 2 , 5 2 ⟩ = 6 7 Y 3 2 χ ↑ + 1 7 Y 3 3 χ ↓ , ∣ 7 2 , 7 2 ⟩ = 7 7 Y 3 3 χ ↑ + 0 7 Y 3 4 χ ↓ = Y 3 3 χ ↑ . \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} 2 5 , − 2 5 ⟩ = − 7 6 Y 3 − 3 χ ↑ + 7 1 Y 3 − 2 χ ↓ , 2 5 , − 2 3 ⟩ = − 7 5 Y 3 − 2 χ ↑ + 7 2 Y 3 − 1 χ ↓ , 2 5 , − 2 1 ⟩ = − 7 4 Y 3 − 1 χ ↑ + 7 3 Y 3 0 χ ↓ , 2 5 , 2 1 ⟩ = − 7 3 Y 3 0 χ ↑ + 7 4 Y 3 1 χ ↓ , 2 5 , 2 3 ⟩ = − 7 2 Y 3 1 χ ↑ + 7 5 Y 3 2 χ ↓ , 2 5 , 2 5 ⟩ = − 7 1 Y 3 2 χ ↑ + 7 6 Y 3 3 χ ↓ , 2 7 , − 2 7 ⟩ = 7 0 Y 3 − 4 χ ↑ + 7 7 Y 3 − 3 χ ↓ = Y 3 − 3 χ ↓ , 2 7 , − 2 5 ⟩ = 7 1 Y 3 − 3 χ ↑ + 7 6 Y 3 − 2 χ ↓ , 2 7 , − 2 3 ⟩ = 7 2 Y 3 − 2 χ ↑ + 7 5 Y 3 − 1 χ ↓ , 2 7 , − 2 1 ⟩ = 7 3 Y 3 − 1 χ ↑ + 7 4 Y 3 0 χ ↓ , 2 7 , 2 1 ⟩ = 7 4 Y 3 0 χ ↑ + 7 3 Y 3 1 χ ↓ , 2 7 , 2 3 ⟩ = 7 5 Y 3 1 χ ↑ + 7 2 Y 3 2 χ ↓ , 2 7 , 2 5 ⟩ = 7 6 Y 3 2 χ ↑ + 7 1 Y 3 3 χ ↓ , 2 7 , 2 7 ⟩ = 7 7 Y 3 3 χ ↑ + 7 0 Y 3 4 χ ↓ = Y 3 3 χ ↑ .
Natural basis
The natural basis is defined as the diagonal basis of on-site term E α β E_{\alpha\beta} E α β .
Transformation matrix from complex orbital basis to real orbital basis
For p system, the transformation matrix reads
T = [ i 2 0 1 2 0 0 0 0 1 0 0 0 0 i 2 0 − 1 2 0 0 0 0 0 0 i 2 0 1 2 0 0 0 0 1 0 0 0 0 i 2 0 − 1 2 ] \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} T = 2 i 0 2 i 0 0 0 0 1 0 0 0 0 2 1 0 − 2 1 0 0 0 0 0 0 2 i 0 2 i 0 0 0 0 1 0 0 0 0 2 1 0 − 2 1
For d system, the transformation matrix reads
T = [ i 2 0 0 0 1 2 0 0 0 0 0 0 i 2 0 1 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i 2 0 − 1 2 0 0 0 0 0 0 − i 2 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 i 2 0 0 0 1 2 0 0 0 0 0 0 i 2 0 1 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i 2 0 − 1 2 0 0 0 0 0 0 − i 2 0 0 0 1 2 ] 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] T = 2 i 0 0 0 − 2 i 0 0 0 0 0 0 2 i 0 2 i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 − 2 1 0 0 0 0 0 0 2 1 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 2 i 0 0 0 − 2 i 0 0 0 0 0 0 2 i 0 2 i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 − 2 1 0 0 0 0 0 0 2 1 0 0 0 2 1
For f system, the transformation matrix reads
T = [ i 2 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 i 2 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 i 2 0 1 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 i 2 0 − 1 2 0 0 0 0 0 0 0 0 0 0 − i 2 0 0 0 1 2 0 0 0 0 0 0 0 0 i 2 0 0 0 0 0 − 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 2 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 i 2 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 i 2 0 1 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 i 2 0 − 1 2 0 0 0 0 0 0 0 0 0 0 − i 2 0 0 0 1 2 0 0 0 0 0 0 0 0 i 2 0 0 0 0 0 − 1 2 ] 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] T = 2 i 0 0 0 0 0 2 i 0 0 0 0 0 0 0 0 2 i 0 0 0 − 2 i 0 0 0 0 0 0 0 0 0 0 2 i 0 2 i 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 2 1 0 − 2 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 2 1 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 − 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 i 0 0 0 0 0 2 i 0 0 0 0 0 0 0 0 2 i 0 0 0 − 2 i 0 0 0 0 0 0 0 0 0 0 2 i 0 2 i 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 2 1 0 − 2 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 2 1 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 − 2 1
The following Julia script is used to construct the complex orbital basis and the real orbital basis, and the transformation matrix between them.
function calc_matrix(l::Int64 )
println("Construct complex orbital basis for 𝑙 = $l " )
COB = []
mlist = collect(-l:1 :l)
for s in ("up" , "down" )
for m in mlist
push!(COB, [m, s])
end
end
for i in eachindex(COB)
m = COB[i][1 ]
s = COB[i][2 ] == "up" ? "↑" : "↓"
println("$i -> | $l , $m , $s ⟩" )
end
println("Construct real orbital basis for 𝑙 = $l " )
RO = []
ROB = []
for s in ("up" , "down" )
for m in mlist
if m < 0
b = ["i/sqrt(2)" , -abs(m), -(-1 )^m, abs(m)]
elseif m == 0
b = [0 ]
elseif m > 0
b = ["1/sqrt(2)" , -abs(m), (-1 )^m, abs(m)]
end
push!(RO, [m,s])
push!(ROB, b)
end
end
for i in eachindex(RO)
m = RO[i][1 ]
s = RO[i][2 ] == "up" ? "↑" : "↓"
println("$i -> Y_{$l ,$m } χ$s " )
end
println("Evaluate transformation matrix for 𝑙 = $l " )
for m in eachindex(COB)
for n in eachindex(RO)
if COB[m][2 ] == RO[n][2 ]
if length(ROB[n]) == 1
if COB[m][1 ] == ROB[n][1 ]
println("T($m ,$n ) -> 1" )
end
else
if COB[m][1 ] == ROB[n][2 ]
println("T($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 ^ 2 − j ^ z − l ^ 2 − s ^ 2 \hat{j}^{2}-\hat{j}_{z}-\hat{l}^2-\hat{s}^2 j ^ 2 − j ^ z − l ^ 2 − s ^ 2 diagonal basis
For p system, the transformation matrix reads
T = [ − 2 3 0 0 1 3 0 0 0 − 1 3 0 0 2 3 0 0 0 0 0 0 1.0 0 0 1.0 0 0 0 1 3 0 0 2 3 0 0 0 2 3 0 0 1 3 0 ] \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} T = − 3 2 0 0 0 3 1 0 0 − 3 1 0 0 0 3 2 0 0 0 1.0 0 0 3 1 0 0 0 3 2 0 0 3 2 0 0 0 3 1 0 0 1.0 0 0 0
For d system, the transformation matrix reads
T = [ − 4 5 0 0 0 0 1 5 0 0 0 0 0 − 3 5 0 0 0 0 2 5 0 0 0 0 0 − 2 5 0 0 0 0 3 5 0 0 0 0 0 − 1 5 0 0 0 0 4 5 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 1.0 0 0 0 0 0 1 5 0 0 0 0 4 5 0 0 0 0 0 2 5 0 0 0 0 3 5 0 0 0 0 0 3 5 0 0 0 0 2 5 0 0 0 0 0 4 5 0 0 0 0 1 5 0 ] 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] T = − 5 4 0 0 0 0 0 5 1 0 0 0 0 − 5 3 0 0 0 0 0 5 2 0 0 0 0 − 5 2 0 0 0 0 0 5 3 0 0 0 0 − 5 1 0 0 0 0 0 5 4 0 0 0 0 0 1.0 0 0 0 0 5 1 0 0 0 0 0 5 4 0 0 0 0 5 2 0 0 0 0 0 5 3 0 0 0 0 5 3 0 0 0 0 0 5 2 0 0 0 0 5 4 0 0 0 0 0 5 1 0 0 0 0 1.0 0 0 0 0 0
For f system, the transformation matrix reads
T = [ − 6 7 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 − 5 7 0 0 0 0 0 0 2 7 0 0 0 0 0 0 0 − 4 7 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 − 3 7 0 0 0 0 0 0 4 7 0 0 0 0 0 0 0 − 2 7 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 − 1 7 0 0 0 0 0 0 6 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0 1 7 0 0 0 0 0 0 6 7 0 0 0 0 0 0 0 2 7 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 4 7 0 0 0 0 0 0 0 4 7 0 0 0 0 0 0 3 7 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 2 7 0 0 0 0 0 0 0 6 7 0 0 0 0 0 0 1 7 0 ] 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] T = − 7 6 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 − 7 5 0 0 0 0 0 0 0 7 2 0 0 0 0 0 0 − 7 4 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 − 7 3 0 0 0 0 0 0 0 7 4 0 0 0 0 0 0 − 7 2 0 0 0 0 0 0 0 7 5 0 0 0 0 0 0 − 7 1 0 0 0 0 0 0 0 7 6 0 0 0 0 0 0 0 1.0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 7 6 0 0 0 0 0 0 7 2 0 0 0 0 0 0 0 7 5 0 0 0 0 0 0 7 3 0 0 0 0 0 0 0 7 4 0 0 0 0 0 0 7 4 0 0 0 0 0 0 0 7 3 0 0 0 0 0 0 7 5 0 0 0 0 0 0 0 7 2 0 0 0 0 0 0 7 6 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 1.0 0 0 0 0 0 0 0
The following Julia script is used to construct the complex orbital basis and the j 2 − j z j^2-j_z j 2 − j z basis, and the transformation matrix between them.
function calc_matrix(l::Int64 )
println("Construct complex orbital basis for 𝑙 = $l " )
COB = []
mlist = collect(-l:1 :l)
for s in ("up" , "down" )
for m in mlist
push!(COB, [m, s])
end
end
for i in eachindex(COB)
m = COB[i][1 ]
s = COB[i][2 ] == "up" ? "↑" : "↓"
println("$i -> | $l , $m , $s ⟩" )
end
println("Construct j²-jz basis for 𝑙 = $l " )
JJ = []
JJB = []
jlist = [l-1 //2 , l+1 //2 ]
for j in jlist
mⱼlist = collect(-j:2 //2 :j)
for mⱼ in mⱼlist
push!(JJ, [j, mⱼ])
m = mⱼ-1 //2
if j == l-1 //2
jj = ["-" , (l-m)/(2 *l+1 ), Int (m), (l+m+1 )/(2 *l+1 ), Int (m+1 )]
else
jj = ["" , (l+m+1 )/(2 *l+1 ), Int (m), (l-m)/(2 *l+1 ), Int (m+1 )]
end
push!(JJB, jj)
end
end
for i in eachindex(JJ)
j = JJ[i][1 ]
mⱼ = JJ[i][2 ]
print("$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