A Real vs complex spherical harmonics for hexagonal symmetry

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Orbital analysis of magnetic anisotropy energy using second order perturbation theory for hexagonal symmetry
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Are real spherical harmonics better than complex spherical harmonics for hexagonal symmetry, which are
directly associated to a finite Lz?
 
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An eigenvector of ##L_z## in position representation is a standard complex spherical harmonic, i.e.,
$$\text{Y}_{lm}=P_{lm}[\cos(\vartheta)] \exp(\mathrm{i} m \varphi).$$
Note that in spherical coordinates the position representation of ##\hat{L}_z## reads
$$\hat{L}_z=-\mathrm{i} \hbar \partial_{\varphi}.$$
 
vanhees71 said:
An eigenvector of ##L_z## in position representation is a standard complex spherical harmonic, i.e.,
$$\text{Y}_{lm}=P_{lm}[\cos(\vartheta)] \exp(\mathrm{i} m \varphi).$$
Note that in spherical coordinates the position representation of ##\hat{L}_z## reads
$$\hat{L}_z=-\mathrm{i} \hbar \partial_{\varphi}.$$

vanhees71 said:
Is there any relation between the crystal symmetry and real/complex spherical harmonics? And is there a way to judge the superiority of one over the other?
 
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