Appendix J Phase convention of real spherical harmonics

In quantum chemistry applications, an Condon-Shortley phase can be added to the real spherical harmonics (RSH) based on the developers’ choice.

If, the commonly used Condon-Shortley phase is added to the RSH, the definition of RSH becomes

\displaystyle Y_{l}^{m}=\left\{\begin{array}[]{ll}(-1)^{m}\sqrt{2}K_{l}^{m}% \cos(m\phi)P_{l}^{m}(\cos\theta),&m>0\\ (-1)^{m}\sqrt{2}K_{l}^{m}\sin(|m|\phi)P_{l}^{|m|}(\cos\theta),&m<0\\ K_{l}^{m}P_{l}^{m}(\cos\theta),&m=0\end{array}\right.

(J.4)

where $P_{l}^{m}$ is the associated Legendre polynomials without the Condon-Shortley phase

P_{l}^{m}(\cos\theta)=\left(\sin\theta\right)^{m}\cfrac{d^{m}}{d(\cos\theta)^{% m}}\left(P_{l}(\cos\theta)\right),

(J.5)

and the coefficient

K_{l}^{m}=\sqrt{\cfrac{(2l+1)(l-|m|)!}{4\pi(l+|m|)!}}

(J.6)

The RSH in the FHI-aims code adopt a partly Condon-Shortley phase. Thus, the definition of RSH becomes

\displaystyle Y_{l}^{m}=\left\{\begin{array}[]{ll}(-1)^{m}\sqrt{2}K_{l}^{m}% \cos(m\phi)P_{l}^{m}(\cos\theta),&m>0\\ \sqrt{2}K_{l}^{m}\sin(|m|\phi)P_{l}^{|m|}(\cos\theta),&m<0\\ K_{l}^{m}P_{l}^{m}(\cos\theta),&m=0\end{array}\right.

(J.10)

The code related to these definition can be found in file basis_sets/increment_ylm_original.f90 or basis_sets/SHEval.f90. In some subroutines, the real spherical harmonics are generated using external/ylm_real.f90. The definition of RSH therein seems a bit different

\displaystyle Y_{l}^{m}=\left\{\begin{array}[]{ll}\sqrt{2}\mathrm{Re}\tilde{Y}% ^{m}_{l},&m>0\\ -\sqrt{2}\mathrm{Im}\tilde{Y}^{m}_{l},&m<0\\ \tilde{Y}^{m}_{l},&m=0\end{array}\right.

(J.14)

where $\tilde{Y}^{m}_{l}$ is the complex spherical harmonics (CSH)

\tilde{Y}^{m}_{l}(\theta,\phi)=(-1)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l% +m)!}}P^{m}_{l}(\cos\theta)e^{im\phi}

(J.15)

It can be shown that the two conventions are equivalent. it is trivial for $m=0$ . For $m>0$ ,

	$\displaystyle(-1)^{m}\sqrt{2}K_{l}^{m}\cos(m\phi)P_{l}^{m}(\cos\theta)=$	$\displaystyle(-1)^{m}\sqrt{2}\mathrm{Re}\left[e^{im\phi}\right]K_{l}^{m}P_{l}^% {m}(\cos\theta)$
	$\displaystyle=$	$\displaystyle\sqrt{2}\mathrm{Re}\left[(-1)^{m}e^{im\phi}K_{l}^{m}P_{l}^{m}(% \cos\theta)\right]$
	$\displaystyle=$	$\displaystyle\sqrt{2}\mathrm{Re}\tilde{Y}_{l}^{m}$

For $m<0$

		$\displaystyle\sqrt{2}K_{l}^{m}\sin(\|m\|\phi)P_{l}^{\|m\|}(\cos\theta)$
	$\displaystyle=$	$\displaystyle\sqrt{2}\mathrm{Im}\left[e^{i\|m\|\phi}\right]K_{l}^{m}P_{l}^{\|m\|}(% \cos\theta)$
	$\displaystyle=$	$\displaystyle\sqrt{2}\mathrm{Im}\left[-e^{im\phi}\right]\sqrt{\cfrac{(2l+1)(l+% m)!}{4\pi(l-m)!}}P_{l}^{-m}(\cos\theta)$
	$\displaystyle=$	$\displaystyle-\sqrt{2}\mathrm{Im}\left[e^{im\phi}\sqrt{\cfrac{(2l+1)(l-m)!}{4% \pi(l+m)!}}(-1)^{m}P_{l}^{m}(\cos\theta)\right]$
	$\displaystyle=$	$\displaystyle-\sqrt{2}\mathrm{Im}\tilde{Y}_{l}^{m}$

where in the third equality we have used the following property

P_{l}^{-m}=(-1)^{m}\frac{(l-m)!}{(l+m)!}P_{l}^{m}.

(J.16)

With the property of CSH

\tilde{Y}_{l}^{-m}=(-1)^{m}\tilde{Y}_{l}^{m*},

(J.17)

we can derive another equivalent expression from Eq. (J.14)

\displaystyle Y_{l}^{m}=\left\{\begin{array}[]{ll}\frac{1}{\sqrt{2}}\left[% \tilde{Y}^{m}_{l}+(-1)^{m}\tilde{Y}^{-m}_{l}\right],&m>0\\ \frac{i}{\sqrt{2}}\left[\tilde{Y}^{m}_{l}-(-1)^{m}\tilde{Y}^{-m}_{l}\right],&m% <0\\ \tilde{Y}^{m}_{l},&m=0\end{array}\right.

(J.21)

leading to the representation matrix of FHI-aims RSH in CSH. This can be useful to connect between RSHs from FHI-aims and other codes.