By Per-Olov Lowdin
ADVANCES IN QUANTUM CHEMISTRY VOL 7.
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Extra info for Advances in Quantum Chemistry
Consideringthat x = x', y = y', z = z' + c, with respect to either the the application of the operator Q,,+(d/dz)'-"' unprimed or the primed coordinates yields the relation or by virtue of Eqs. (229) and (230) Since r' = r> and c = r< , it is expedient to write this for later purposes as if 8, denotes the angle between r> and the z axis. Since the derivation of the last identity was based upon the use of the operator Qm, + , it has been shown so far that this formula holds for nonnegative m. With the help of Eq.
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. (166) which can also be written as P,f(Rr) =f(r), According to Eq.
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,). After this first step, one has (1 82a) r = er = e'i. (2) Then, the reference system is rotated by an angle /? about the new e,' axis, yielding the new axes el",e,", e3". According to Eq. (155c), the generating operator is given by e-i@Lj (182b) = e-luL,e-iSLyeiaL, with e,'=e,, J=J, I' L, = (L e3), r = e"i. The operator is given by e-iyLz = ,-i@L- y e - i y L -z e i@L-Y ( 182c) where g3 = e3", i = 2, Lz = (L * g3), r = er = 6i.
Advances in Quantum Chemistry by Per-Olov Lowdin