# 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 *Y _{lm}* are functions of solid angle, while

*Y*are polynomials in

_{lm}r^{l}*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 | 3z^{2}−1 |

8 | 2 | 1 | xz |

9 | 2 | 2 | x^{2}−y^{2} |

10 | 3 | -3 | y(3x^{2}−y^{2}) |

11 | 3 | -2 | xyz |

12 | 3 | -1 | y(5z^{2}−1) |

13 | 3 | 0 | z(5z^{2}−3) |

14 | 3 | 1 | x(5z^{2}−1) |

15 | 3 | 2 | z(x^{2}−y^{2}) |

16 | 3 | 3 | x(x^{2}−3y^{2}) |

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.