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, using Jackson conventions (see below)
where . The inverse operation is
The definition of are
The and functions are related by [see Jackson (3.51) and (3.53)]
Explicit expressions for spherical harmonics for l = 0 and 1
(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 . This phase is sometimes referred to as the “Condon–Shortley phase.” Wikipedia follows Jackson’s convention for . 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 and the . Since it is omitted in both places, “standard definition” and “quantum mechanics definition” are identical for the , byt differ by the Condon–Shortley phase for the . This is apparently also the definition K. Haule uses in his CTQMC code.