By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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I . Dejnitions Mappings and transformations in the three-dimensional coordinate space may be considered briefly. Consider an orthogonal basis spanned by unit vectors e , , e 2 , e3 which fulfill the conditions e,-ej=aij; i,j=1,2,3. , Tietz (1955), Hollingsworth (1967), Perlis (195S)l. ] Let 92 be an operator transforming every vector x into its image vector x’ = Sex. Hence, Se causes a mapping of the vector x onto the vector x’. Consider rotations about the z axis by an angle a. Positive rotations are counterclockwise if the observer looks along the z axis from its positive end toward the origin.
12a). Then, the distance side a sphere of radius A of point P from center B is greater than the distance A T over which the function is translated. In the second place, the point P may be located inside the aforementioned sphere (see Fig. 12b). Then, the distance of point P from center B is smaller than the displacement of the function from A to B. a. Distance of P greater than displacement. If c/r' < 1, Taylor's expansion of l/r is given by Rotation and Translation of Solid Spherical Harmonics 53 see Eq.
It is convenient to introduce the vectors PA = T A I R PB = ~ B I R (252) * and the following abbreviations for the sets of spherical coordinates which characterize these vectors: = (PA 9 eA ~=(P,,O,, 9 =(PB3 (PA), eS 3 (PB), b=(~B,gB,@jg). (PA), (253) In the sequel we will proceed from formulas for the translation of spherical harmonics from A to B which are derived for the case that the atomic coordinate systems having origins at A and B are in a “ lined-up” position. By rotations of these reference systems we search for generalizations of these formulas to the case that the atomic coordinate systems at the points A and E are in a “parallel, shifted” position.
Advances in Quantum Chemistry by Per-Olov Lowdin