By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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169). From the given identities it follows easily that P, = oil, P, = 09-1, Pi1 Pa-1 = 0, = P , - l = 09 9 ; 0,’ = 09-1. (1 76) (177) According to Eq. (176), the Eqs. (167) and (175) are equivalent, as are Eqs. (168) and (174). C. Parametrization in Terms of Eulerian Angles The following derivations are based on operators of the kind P, which cause a positive rotation of the field. Operators of the kind P, are also used in the book by Rose (1957), 38 E. 0. Steinborn and K. Rutdenberg which contains lucid derivations, although no explicit statement that the field is rotated.
The two independent solutions Ylmand 2Zlm apparently exist because of the fact that + + &I+ I) remains unchanged if I -+ -(I + 1). (133) Special cases of the irregular solid spherical harmonics are the functions 1 rl+l y : ( 4 1 ;lr- 1 4 ) = (2n)l/2rl+1 ~ ~ ' ( 1=)const. a2 ' which had been considered in Eqs. (8) and (9) of Section A. (134) 27 Rotation and Translation of Solid Spherical Harmonics 111. Transformation of Spherical Harmonics under Rotations A. Rotations in Coordinate Space The behavior of spherical harmonics under rotations will be discussed to obtain a set of uniform formulas useful for subsequent considerations.
125), was given by Eq. (12) of Section A as well as by Eq. (43) of Section B. Because of and if E. 0. Stcinborn and K. m1 ( a i m g ? ), where the two independent solutions of Eq. ,the harmonics are centered on this origin. 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.