Transformation properties of angular momentum and multipole operators

The k-component of total angular momentum, $\hat{I}_k$, transforms obviously as k-antipseudocovariant under D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, and its transformation rules can be easily read off from Table 3.

 

Table 5: Symmetry properties of electric multipole operators $\hat{Q}_{\lambda\mu}$ with respect to operators of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups. The results of the symmetry operator $\hat{U}^\dagger\hat{Q}_{\lambda\mu}\hat{U}$are given for three spatial directions k=x, y, z. Where applicable, the upper part of the Table gives expressions in terms of changed signs of magnetic components, and the lower part gives the equivalent expressions in terms of the complex conjugation.

k	$\hat{R}_{k}$		$\hat{R}^T_{k}$		$\hat{S}_{k}$		$\hat{S}^T_{k}$
x	$(-1)^{\lambda}$	$\hat{Q}_{\lambda,-\mu}$	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda\mu}$		$\hat{Q}_{\lambda,-\mu}$	$(-1)^{\mu}$	$\hat{Q}_{\lambda\mu}$
y	$(-1)^{\lambda-\mu}$	$\hat{Q}_{\lambda,-\mu}$	$(-1)^{\lambda}$	$\hat{Q}_{\lambda\mu}$	$(-1)^{-\mu}$	$\hat{Q}_{\lambda,-\mu}$		$\hat{Q}_{\lambda\mu}$
z	$(-1)^{\mu}$	$\hat{Q}_{\lambda\mu}$		$\hat{Q}_{\lambda,-\mu}$	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda\mu}$	$(-1)^{\lambda}$	$\hat{Q}_{\lambda,-\mu}$
x	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda \mu}^*$	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda\mu}$	$(-1)^{\mu}$	$\hat{Q}_{\lambda \mu}^*$	$(-1)^{\mu}$	$\hat{Q}_{\lambda\mu}$
y	$(-1)^{\lambda}$	$\hat{Q}_{\lambda \mu}^*$	$(-1)^{\lambda}$	$\hat{Q}_{\lambda\mu}$		$\hat{Q}_{\lambda \mu}^*$		$\hat{Q}_{\lambda\mu}$
z	$(-1)^{\mu}$	$\hat{Q}_{\lambda\mu}$	$(-1)^{\mu}$	$\hat{Q}_{\lambda \mu}^*$	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda\mu}$	$(-1)^{\lambda+\mu}$	$\hat{Q}_{\lambda \mu}^*$

For $\lambda$ even (odd), the electric multipole operators $\hat{Q}_{\lambda\mu}$ are even (odd), respectively, under the action of the inversion, and are all even with respect to the time reversal, i.e.,

   

$$\hat{P}^{\dagger}\hat{Q}_{\lambda\mu}\hat{P} (-1)^{\lambda}\hat{Q}_{\lambda\mu} ,$$ = (76)

$$\hat{T}^{\dagger}\hat{Q}_{\lambda\mu}\hat{T} \hat{Q}_{\lambda\mu}^* .$$ = (77)

The magnetic multipole operators $\hat{M}_{\lambda\mu}$ have opposite transformation properties,

   

$$\hat{P}^{\dagger}\hat{M}_{\lambda\mu}\hat{P} - (-1)^{\lambda}\hat{M}_{\lambda\mu} ,$$ = (78)

$$\hat{T}^{\dagger}\hat{M}_{\lambda\mu}\hat{T} - \hat{M}_{\lambda\mu}^* .$$ = (79)

Table 5 gives transformation properties[21] of $\hat{Q}_{\lambda\mu}$ with respect to operators of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups, other than $\hat{T}$ and $\hat{P}$. One may note that the electric multipole operators are invariant with respect to the $\hat{S}^T_{y}$ symmetry. This is of course a consequence of the standard phase convention for the rotational irreducible tensor operators[21,22],

   

$$\hat{Q}_{\lambda\mu}^* (-1)^{-\mu}\hat{Q}_{\lambda,-\mu} ,$$ = (80)

$$\hat{M}_{\lambda\mu}^* (-1)^{-\mu}\hat{M}_{\lambda,-\mu} ,$$ = (81)

which ensures that the antilinear operator $\hat{S}^T_{y}$ acts as an identity upon any irreducible spherical tensor operator.