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By Per-Olov Lowdin

ADVANCES IN QUANTUM CHEMISTRY VOL 7.

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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. This is true since only the relative position of two reference frames is of interest; any other distinguished reference system does not exist. According to Eq. (165), the transformed function, depending on the coordinates x, y, z, is given by the original function, depending on the new coordinates 2, f, 5, which are given by Eq.

The other reference frame, which is obtained from the first one by translating it along an arbitrary direction and leaving all corresponding axes parallel to each other, has the unit vectors elB, eZB,eJB,centered at B. The point B has the spherical coordinates (R,O A R , qAR) with respect to the reference system elA,eZA,eJA,if the distance is denoted by R. It is most helpful that the “lined-up” position of the atomic coordinate systems (which is illustrated in Fig. 14) can be obtained from their ‘‘para11e1, shifted” position (shown in Fig.

In fact, the new function (Pry),taken at the point r with coordinates x , y , z relative to the reference system el, e, ,e 3 ,has (there) the value of the original function f, taken at the point (r - ro) with coordinates x - xo, y - y o , z - zo relative to the same reference frame el, e 2 , e 3 . If one thinks of the operator as referring to translations along the z axis, one has f ( x , y, z - zo>= e-roa'ay(x, y , z) (2 16) or by formal rearrangement f ( x , y , z) = egoa/azf(x,y , z - zo) (2 17) which amounts to f ( x , y, z) = ezo a'azy(x,y , z').

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