# Spherical Harmonics

Questaal objects with basis sets defined by spherical harmonics, such as LMTOs or smooth Hankel functions, order the harmonics as m=−l, −l+1, … 0, … l.

The Questaal codes use real harmonics $Y_{lm}(\hat{\mathbf{r}})$, instead of the usual spherical (complex) harmonics $\mathrm{Y}_{lm}(\hat\mathbf{r})$. The Ylm are functions of solid angle, while Ylmrl are polynomials in x, y, and z. These polynomials (apart from a normalization) are ordered as follows for l=0…3:

indexlmpolynomial
1001
21-1y
310z
411x
52-2xy
62-1yz
7203z2−1
821xz
922x2y2
103-3y(3x2y2)
113-2xyz
123-1y(5z2−1)
1330z(5z2−3)
1431x(5z2−1)
1532z(x2y2)
1633x(x2−3y2)

The $Y_{lm}$ and $\mathrm{Y}_{lm}$ are related as follows:

where $m>0$. Or equivalently,

The definition of $\mathrm{Y}_{lm}(\hat\mathbf{r})$ are

See
(1) A.R.Edmonds, Angular Momentum in quantum Mechanics, Princeton University Press, 1960
(2) M.E.Rose, Elementary Theory of angular Momentum, John Wiley & Sons, INC. 1957
(3) Jackson, Electrodynamics.

Definitions (7) and (8) of spherical harmonics are the same in these books. Jackson’s definition of $P_l^m$ differs by a phase factor $(-1)^m$, but his $\mathrm{Y}_{lm}$ are the same as Eq. 7.

Wikipedia follows Jackson’s convention for $P_l^m$.

Wikipedia (wiki/Spherical_harmonics) refer to a “quantum mechanics” defnition of spherical harmonics (following Messiah; Tannoudji). It differs from Jackson by a factor $(-1)^m$. This is apparently the definition K. Haule uses in his CTQMC code.

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