By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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Extra info for Advances in Quantum Chemistry
Regular Harmonics According to Eq. (f- m ) ! (A + m ) ! ( l - m)!
For example: The three real p functions are given by Py = [y/rl, P z = [zlrl. ( 169) This covariant set transforms under 9, if the operator Pa (not 0,) is chosen, like Px = uxiri =~ [x/rI, 1 ,P,WI = [f/ri, pLo[ziri= [wi. (170) Rotation and Translation of Solid Spherical Harmonics 37 These relations are in agreement with their geometrical meaning. The operator 0, has the effect (o,f)(e,40) =f(4 q + 4 7 0,m =f(Rr), O,f(R-’r) =f(r). Hence, 0, causes a forward rotation of a certain reference system, leaving the resting field unchanged.
K+rn’-m)! , the integer K assumes all values for which the factorial arguments are greater than or equal to zero. ,m,being more suitable for numerical calculations, is with Again, the sum is over all those values of the integer K for which the factorial arguments are nonnegative: max(0, m - m’) < K I min(l - m’,I + m). ) are important; however, they will not be given here. 45 Rotation and Translation of Solid Spherical Harmonics In the sequel, elements of the rotation matrix with one index being zero will often be needed.
Advances in Quantum Chemistry by Per-Olov Lowdin