Advances in Quantum Chemistry by Per-Olov Lowdin

By Per-Olov Lowdin


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Ruedenberg 40 Since the origin is fixed and pure rotations are considered (implying r = T), the final position 6 of the reference frame may be obtained from the original position e of the reference system with the help of the following three rotations by Eulerian angles u, B, y: (1) The original reference system e l , e 2 , e3 is rotated by an angle c( about the e3 axis. The new position is given by the basis vectors e,', e21, e3'. The operator P9, which is associated with this rotation, is given by exp( - iuL,).

Then, the rotated reference system can be discussed against the body-fixed system which now is fixed in space. This description of the situation is, of course, equivalent to saying 0, effects a backward rotation of the field (the body-fixed reference system) if this rotation is defined with respect to another reference frame fixed in space. For both interpretations, the main transformation is given by WeB= eBR = eA. ( 174) 0,-, belongs to the inverse rotation of a certain (so-called “nonbody-fixed”) reference system, if the rotation is defined with respect to a reference system which is embedded in the resting field (body).

151), its active interpretation being formulated by Eq. (163a), its passive one by Eq. (163b). If the operators P or 0, defined in Eq. (159), are applied to a given function f, they will generate new functions which will be denoted by (Pf)or (Of),respectively, or shorter as Pf or Of. If the operator P , of Eq. (159) is used in connection with the “main transformation ” characterized by 9, then we have ( P 9 f ) ( e ,cp) =f(& cp - 4, ( 164) if W causes positive rotations about the z axis. For general rotations, there follows Psrf(r)=f(R-’r), r’ = R - l r , (1 65) r’ = Rr.

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