By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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Additional resources for Advances in Quantum Chemistry
A convenient choice for specifying the location of P are spherical coordinates r, 8, cp [see Eq. (10) and Fig. 21. However, the argument of each harmonic may also be denoted by the vector r which connects the origin 0 with the field point P,directing from 0 to P, provided that the reference system is given. Then, this shorthand notation leaves no doubt about its meaning. It has been shown that there are at most (21 1) linearly independent regular solid spherical harmonics of degree I which are sufficient to construct any general regular solid spherical harmonic according to Eq.
1 36) and (137), r denotes the local vector r =xlel +xze2 +x3e3 which has components x i = (ei * r) with respect to the basis e i , i = 1,2, 3. Although, according to the aforementioned definitions, the scalar product e r will generally denote the row matrix ( x l x z x 3 ) , it is obvious that it will denote the argument ( x l , xz, x 3 ) of a function if used in the sense of Eq. (160). Besides its shortness, this way of writing functional dependences has the further advantage of expressing the fact that a simultaneous rotation of the position vector and the basis has no effect: - f ( e * r) =f(We .
D. Representation Matrices of the Rotation Group D(”(a, P, y) denotes the representation matrix of the rotation R = [u, P, y ] in the (21 1)-dimensional irreducible representation D(’) of the rotation group, if the 21 + 1 complex surface spherical harmonics I lm) = YF(8, q ) with - I I m 5 I form the basis for this representation. Because of Eqs. ),(a, /I 7) ,= (IdI Pa I Zm) - (im’ 1 e - i u L = e - i / l L y e - i y L , 1 Irn > + = e-i@m‘dU? mm (p)e-iym, (1 8 5 ) 42 E. 0. Steinborn and K. Ruedenberg since for z-axis quantization, the operator L, is diagonal with respect to the Ilm').
Advances in Quantum Chemistry by Per-Olov Lowdin