Spherical Harmonics

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

The Questaal codes use real harmonics Ylm(r^)Y_{lm}(\hat{\mathbf{r}}), instead of the usual spherical (complex) harmonics Ylm(r^)\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:

index l m polynomial
1 0 0 1
2 1 -1 y
3 1 0 z
4 1 1 x
5 2 -2 xy
6 2 -1 yz
7 2 0 3z2−1
8 2 1 xz
9 2 2 x2y2
10 3 -3 y(3x2y2)
11 3 -2 xyz
12 3 -1 y(5z2−1)
13 3 0 z(5z2−3)
14 3 1 x(5z2−1)
15 3 2 z(x2y2)
16 3 3 x(x2−3y2)

The YlmY_{lm} and Ylm\mathrm{Y}_{lm} are related as follows:

Yl0(r^) Yl0(r^)(1)Ylm(r^)12[(1)mYlm(r^)+Ylm(r^)](2)Ylm(r^)12i[(1)mYlm(r^)Ylm(r^)].(3)\begin{array}{rlrr} {Y_{l0}}(\hat{\mathbf{r}}) & \equiv \ \mathrm{Y}_{l0}(\hat{\mathbf{r}}) & (1)\\ Y_{lm}(\hat{\mathbf{r}}) & \equiv \frac{1}{\sqrt 2}[(-1)^m{\mathrm{Y}}_{lm}(\hat{\mathbf{r}}) + {\mathrm{Y}}_{l-m}(\hat{\mathbf{r}})] & (2) \\ Y_{l - m}(\hat{\mathbf{r}}) &\equiv \frac{1}{\sqrt 2 i}[{( - 1)}^m{\mathrm{Y}_{lm}}(\hat{\mathbf{r}}) - \mathrm{Y}_{l-m}(\hat{\mathbf{r}})]. & (3) \end{array}

where m>0m>0. Or equivalently,

Yl0(r^)Yl0(r^)(4)Ylm(r^)(1)m2[Ylm(r^)+iYlm(r^)](5)Ylm(r^)12[Ylm(r^)iYlm(r^)].(6)\begin{array}{rlr} {\mathrm{Y}}_{l0}(\hat {\mathbf{r}}) & \equiv {Y_{l0}}(\hat {\mathbf{r}}) & (4)\\ {\mathrm{Y}}_{lm}(\hat {\mathbf{r}}) & \equiv \frac{( - 1)^m}{\sqrt 2}[Y_{lm}(\hat {\mathbf{r}}) + i{Y_{l - m}}(\hat {\mathbf{r}})] &(5)\\ {\mathrm{Y}}_{l-m}(\hat {\mathbf{r}}) & \equiv \frac{1}{\sqrt 2 }[{Y_{lm}}(\hat {\mathbf{r}}) - i{Y_{l - m}}(\hat {\mathbf{r}})]. & (6) \end{array}

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

Ylm(θ,ϕ)=(1)m[(2l+1)(lm)!4π(l+m)!]12Plm(cos(θ))eimϕ,(7)Plm(x)=(1x2)m/22ll!dl+mdxl+m(x21)l(8)\begin{array}{rlr} \mathrm{Y}_{lm}(\theta ,\phi ) & = {( - 1)^m}\left[ \frac{(2l + 1)(l - m)!}{4\pi (l + m)!} \right]^{\frac{1}{2}} P_l^m(\cos (\theta )){e^{im\phi }}, & (7)\\ P_l^m(x) & = \frac{(1 - {x^2})^{m/2}}{2^ll!} \frac{d^{l + m}}{d{x^{l + m}}}{({x^2} - 1)^l} & (8) \end{array}

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 PlmP_l^m differs by a phase factor (1)m(-1)^m, but his Ylm\mathrm{Y}_{lm} are the same as Eq. 7.

Wikipedia follows Jackson’s convention for PlmP_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(-1)^m. This is apparently the definition K. Haule uses in his CTQMC code.


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