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:
The and are related as follows:
where . Or equivalently,
The definition of are
(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.