# 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:

indexlmpolynomialSpherical coordinates
10011
21-1y$-\frac{i}{2} \sin\theta\left[ e^{i\phi} - e^{-i\phi} \right]$
310z$\cos\theta$
411x$+\frac{1}{2} \sin\theta\left[ e^{i\phi} + e^{-i\phi} \right]$
52-2xy$-\frac{i}{2}\sin^2\theta\left[ e^{2i\phi} - e^{-2i\phi} \right]$
62-1yz$-\frac{i}{2}\sin\theta\cos\theta\left[ e^{i\phi} - e^{-i\phi} \right]$
7203z2−1$3\,\cos^2\theta-1$
821xz$+\frac{1}{2}\sin^2\theta\cos^2\theta\left[ e^{i\phi} + e^{-i\phi} \right]$
922x2y2$+\frac{1}{2}\sin^2\theta\left[ e^{2i\phi} + e^{-2i\phi} \right]$
103-3y(3x2y2)$-\frac{i}{2}\sin^3\theta\left[ e^{3i\phi} - e^{-3i\phi} \right]$
113-2xyz$-\frac{i}{4}\sin^2 \theta \cos\theta\left[ e^{2i\phi} - e^{-2i\phi} \right]$
123-1y(5z2−1)$-\frac{i}{2} \sin\theta(5\cos^2\theta-1)\left[ e^{i\phi} - e^{-i\phi} \right]$
1330z(5z2−3)$\cos\theta(5\cos^2\theta-3)$
1431x(5z2−1)$+\frac{1}{2} \sin\theta(5\cos^2\theta-1)\left[ e^{i\phi} + e^{-i\phi} \right]$
1532z(x2y2)$+\frac{1}{2}\sin^2\theta\cos\theta\left[ e^{2i\phi} + e^{-2i\phi} \right]$
1633x(x2−3y2)$+\frac{1}{2}\sin^3\theta\left[ e^{3i\phi} + e^{-3i\phi} \right]$

The $Y_{lm}$ and $\mathrm{Y}_{lm}$ are related as follows, using Jackson conventions (see below)

where $m>0$. The inverse operation is

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

The $(-m)$ and $m$ functions are related by [see Jackson (3.51) and (3.53)]

$P_l^{-m}(x) = (-1)^m \frac{(l - m)!}{(l + m)!} P_l^m(x)$   and   $Y_{l,-m}(\hat {\mathbf{r}}) = (-1)^m Y^{*}_{l\,m}(\hat {\mathbf{r}})$

Explicit expressions for spherical harmonics for l = 0 and 1

indexlmpolynomial$\mathrm{Y}_{l\,m}(\theta,\phi)$
100$\sqrt\frac{1}{4\pi}$$\sqrt\frac{1}{4\pi}$
21-1$\sqrt\frac{3}{4\pi}\,y$$+\sqrt\frac{3}{8\pi} \sin\theta e^{-i\phi}$
310$\sqrt\frac{3}{4\pi}\,z$$+\sqrt\frac{3}{4\pi} \cos\theta$
411$\sqrt\frac{3}{4\pi}\,x$$-\sqrt\frac{3}{8\pi} \sin\theta e^{i\phi}$

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, 1957.
(3) Jackson, Electrodynamics.

Definitions (7) and (8) of spherical harmonics are the same as in Jackson. Jackson’s definition differs from that of Edmonds and Rose, by a phase factor $(-1)^m$. This phase is sometimes referred to as the “Condon–Shortley phase.” Wikipedia follows Jackson’s convention for $P_l^m$. Some authors, e.g. Abramowitz and Stegun omit the Condon–Shortley phase.

Wikipedia also refers to Jackson’s definition as the “standard definition,” and refers to the definition used here as a “quantum mechanics” defnition of spherical harmonics (e.g. Messiah), which omits the Condon–Shortley phase in both the $P_l^{m}$ and the $Y_{l\,m}$. Since it is omitted in both places, “standard definition” and “quantum mechanics definition” are identical for the $Y_{l\,m}$ , byt differ by the Condon–Shortley phase for the $P_l^{m}$. This is apparently also the definition K. Haule uses in his CTQMC code.

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