結晶場基底の活性多極子と行列要素#
from IPython.display import display, Math
from multipie.group.point_group import PointGroup
from gcoreutils.nsarray import NSArray
# ==================================================
def multipole_cef():
pg = PointGroup("Oh")
bra_list = ket_list = [
"(5/2,5/2,3)",
"(5/2,3/2,3)",
"(5/2,1/2,3)",
"(5/2,-1/2,3)",
"(5/2,-3/2,3)",
"(5/2,-5/2,3)",
]
# <gamma|m> (G7,G8|+5/2,+3/2,+1/2,-1/2,-3/2,-5/2).
U = "[[0,1/sqrt(6),sqrt(5/6),0,0,0],[-sqrt(5/6),0,0,1/sqrt(6),0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,-sqrt(5/6),1/sqrt(6),0,0,0],[1/sqrt(6),0,0,sqrt(5/6),0,0]]"
U = NSArray(U)
G7 = U[:, :2]
G8 = U[:, 2:]
atomic_samb = pg.atomic_samb(bra_list, ket_list, spinful=True, u_matrix=[G7, G8])
print(f"=== (0)Γ7, (1)Γ8 in (j=5/2, l=3) ===")
for (brai, keti), d in atomic_samb.items():
print(f"--- {(brai, keti)}: ({len(d)} active atomic multipoles) ---")
for tag, mat in d.items():
display(Math(f"{tag.latex()} = {mat.latex()}"))
# ================================================== main
multipole_cef()
=== (0)Γ7, (1)Γ8 in (j=5/2, l=3) ===
--- (0, 0): (4 active atomic multipoles) ---
\[\begin{split}\displaystyle \mathbb{Q}_{0}^{(a,A_{1g})} = \begin{pmatrix} \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,0}^{(a,T_{1g})} = \begin{pmatrix} 0 & - \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 \\ - \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,1}^{(a,T_{1g})} = \begin{pmatrix} 0 & - \frac{\sqrt{2} i}{2} & 0 & 0 & 0 & 0 \\ \frac{\sqrt{2} i}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,2}^{(a,T_{1g})} = \begin{pmatrix} \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{2}}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
--- (0, 1): (16 active atomic multipoles) ---
\[\begin{split}\displaystyle \mathbb{Q}_{2,0}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & 0 & - \frac{1}{2} & 0 & 0 & 0 \\ 0 & - \frac{1}{2} & 0 & 0 & 0 & 0 \\ - \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,1}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,0}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & \frac{\sqrt{3} i}{4} & 0 & - \frac{i}{4} & 0 \\ 0 & 0 & 0 & \frac{\sqrt{3} i}{4} & 0 & - \frac{i}{4} \\ - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 \\ \frac{i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,1}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & \frac{\sqrt{3}}{4} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & - \frac{\sqrt{3}}{4} & 0 & - \frac{1}{4} \\ \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,2}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & - \frac{i}{2} \\ 0 & 0 & 0 & 0 & \frac{i}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{i}{2} & 0 & 0 & 0 & 0 \\ \frac{i}{2} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{4,0}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & \frac{i}{4} & 0 & \frac{\sqrt{3} i}{4} & 0 \\ 0 & 0 & 0 & \frac{i}{4} & 0 & \frac{\sqrt{3} i}{4} \\ - \frac{i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{i}{4} & 0 & 0 & 0 & 0 \\ - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{4,1}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & - \frac{1}{4} & 0 & \frac{\sqrt{3}}{4} & 0 \\ 0 & 0 & 0 & \frac{1}{4} & 0 & - \frac{\sqrt{3}}{4} \\ - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{4,2}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & \frac{i}{2} & 0 & 0 \\ 0 & 0 & - \frac{i}{2} & 0 & 0 & 0 \\ 0 & \frac{i}{2} & 0 & 0 & 0 & 0 \\ - \frac{i}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,0}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & - \frac{1}{4} & 0 & - \frac{\sqrt{3}}{4} & 0 \\ 0 & 0 & 0 & - \frac{1}{4} & 0 & - \frac{\sqrt{3}}{4} \\ - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 \\ - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,1}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & - \frac{i}{4} & 0 & \frac{\sqrt{3} i}{4} & 0 \\ 0 & 0 & 0 & \frac{i}{4} & 0 & - \frac{\sqrt{3} i}{4} \\ \frac{i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{i}{4} & 0 & 0 & 0 & 0 \\ - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,2}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ - \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,0}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & - \frac{\sqrt{3}}{4} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & - \frac{\sqrt{3}}{4} & 0 & \frac{1}{4} \\ - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{3}}{4} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,1}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & \frac{\sqrt{3} i}{4} & 0 & \frac{i}{4} & 0 \\ 0 & 0 & 0 & - \frac{\sqrt{3} i}{4} & 0 & - \frac{i}{4} \\ - \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\sqrt{3} i}{4} & 0 & 0 & 0 & 0 \\ - \frac{i}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{4} & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,2}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{5,0}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & - \frac{i}{2} & 0 & 0 \\ 0 & 0 & - \frac{i}{2} & 0 & 0 & 0 \\ 0 & \frac{i}{2} & 0 & 0 & 0 & 0 \\ \frac{i}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{5,1}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & \frac{i}{2} \\ 0 & 0 & 0 & 0 & \frac{i}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{i}{2} & 0 & 0 & 0 & 0 \\ - \frac{i}{2} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]
--- (1, 1): (16 active atomic multipoles) ---
\[\begin{split}\displaystyle \mathbb{Q}_{0}^{(a,A_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,0}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,1}^{(a,E_{g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{1}{2} \\ 0 & 0 & - \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,0}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{i}{2} \\ 0 & 0 & 0 & 0 & \frac{i}{2} & 0 \\ 0 & 0 & 0 & - \frac{i}{2} & 0 & 0 \\ 0 & 0 & - \frac{i}{2} & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,1}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & - \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{Q}_{2,2}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{i}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{i}{2} \\ 0 & 0 & - \frac{i}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{2} & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,0}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\sqrt{65}}{26} & 0 & \frac{\sqrt{195}}{65} \\ 0 & 0 & \frac{\sqrt{65}}{26} & 0 & \frac{\sqrt{195}}{65} & 0 \\ 0 & 0 & 0 & \frac{\sqrt{195}}{65} & 0 & \frac{9 \sqrt{65}}{130} \\ 0 & 0 & \frac{\sqrt{195}}{65} & 0 & \frac{9 \sqrt{65}}{130} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,1}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{\sqrt{65} i}{26} & 0 & \frac{\sqrt{195} i}{65} \\ 0 & 0 & \frac{\sqrt{65} i}{26} & 0 & - \frac{\sqrt{195} i}{65} & 0 \\ 0 & 0 & 0 & \frac{\sqrt{195} i}{65} & 0 & - \frac{9 \sqrt{65} i}{130} \\ 0 & 0 & - \frac{\sqrt{195} i}{65} & 0 & \frac{9 \sqrt{65} i}{130} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{1,2}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{11 \sqrt{65}}{130} & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{11 \sqrt{65}}{130} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{3 \sqrt{65}}{130} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{3 \sqrt{65}}{130} \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,0}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3 \sqrt{65}}{52} & 0 & - \frac{7 \sqrt{195}}{260} \\ 0 & 0 & \frac{3 \sqrt{65}}{52} & 0 & - \frac{7 \sqrt{195}}{260} & 0 \\ 0 & 0 & 0 & - \frac{7 \sqrt{195}}{260} & 0 & \frac{\sqrt{65}}{260} \\ 0 & 0 & - \frac{7 \sqrt{195}}{260} & 0 & \frac{\sqrt{65}}{260} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,1}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{3 \sqrt{65} i}{52} & 0 & - \frac{7 \sqrt{195} i}{260} \\ 0 & 0 & \frac{3 \sqrt{65} i}{52} & 0 & \frac{7 \sqrt{195} i}{260} & 0 \\ 0 & 0 & 0 & - \frac{7 \sqrt{195} i}{260} & 0 & - \frac{\sqrt{65} i}{260} \\ 0 & 0 & \frac{7 \sqrt{195} i}{260} & 0 & \frac{\sqrt{65} i}{260} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,2}^{(a,T_{1g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{3 \sqrt{65}}{130} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3 \sqrt{65}}{130} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{11 \sqrt{65}}{130} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{11 \sqrt{65}}{130} \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3}^{(a,A_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{i}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{i}{2} \\ 0 & 0 & - \frac{i}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{i}{2} & 0 & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,0}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\sqrt{3}}{4} & 0 & \frac{1}{4} \\ 0 & 0 & \frac{\sqrt{3}}{4} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & \frac{1}{4} & 0 & - \frac{\sqrt{3}}{4} \\ 0 & 0 & \frac{1}{4} & 0 & - \frac{\sqrt{3}}{4} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,1}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\sqrt{3} i}{4} & 0 & - \frac{i}{4} \\ 0 & 0 & - \frac{\sqrt{3} i}{4} & 0 & \frac{i}{4} & 0 \\ 0 & 0 & 0 & - \frac{i}{4} & 0 & - \frac{\sqrt{3} i}{4} \\ 0 & 0 & \frac{i}{4} & 0 & \frac{\sqrt{3} i}{4} & 0 \end{pmatrix}\end{split}\]
\[\begin{split}\displaystyle \mathbb{M}_{3,2}^{(a,T_{2g})} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & - \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \end{pmatrix}\end{split}\]