By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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Additional info for Advances in Quantum Chemistry
Steinborn and K. Ruedenberg and according to Eq. (9,if I is replaced by I P~-~(cOS e) #-m+ 1 - m yone finally obtains I-m+k (r')k CI-m+k+ k=O 1 Pk(C0s 8'). (245) The application of the operator Om,+ produces the "solid" associated Legendre functions from the corresponding simple ones. Hence, with the substitution k - m =K, one obtains Because of r' = r < and c = r, , with 8' = 8, being the angle between r < and the z axis, there follows Although derived for positive m, this identity holds for all values of m.
If both systems coincide before the rotation is carried out, the “main transformation,” which induces P , , is described by We” = eAR= eB, ( 167) Then, the body-fixed basis vectors, collected into eB,are rotated and in a new position relative to the “resting” vectors, collected into e”: The vectors eBare expressed by the vectors e” by means of relationship Eq. (167). The inverse rotation, which is determined by W - l e A = eAR-1 = eB , (168) induces the operator P , - , which causes a backward rotation of the field (body).
167). The inverse rotation, which is determined by W - l e A = eAR-1 = eB , (168) induces the operator P , - , which causes a backward rotation of the field (body). ” The equations (167) and (168) have the advantage of being completely independent of the interpretation. They relate only the final relative position of both systems to each other without stating which reference frame has been rotated and which one was resting. A backward rotation of the body (or the shape of the function), which is defined with respect to a fixed reference system, is identical with a forward rotation of this system (which then is no longer fixed in space), if the body is kept fixed.
Advances in Quantum Chemistry by Per-Olov Lowdin