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 , instead of the usual spherical (complex) harmonics . 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 and are related as follows:

where . Or equivalently,

The definition of 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 differs by a phase factor , but his are the same as Eq. 7.

Wikipedia follows Jackson’s convention for .

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


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