Advances in Quantum Chemistry by Per-Olov Lowdin

By Per-Olov Lowdin

ADVANCES IN QUANTUM CHEMISTRY VOL 7.

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0. Steinborn and K. Ruedenberg since for z-axis quantization, the operator L, is diagonal with respect to the Ilm'). The second Eulerian rotation about the e2' axis has the nondiagonal representation d,$),,,(P) = (lmrle-iBLy/ Im). (186) Of course, the component L, = (L e3) of the angular momentum operator refers to a quantization with respect to the e, axis, whereas the component L, = (L 6,) belongs to a quantization with respect to the P 3 axis. Since L, does not commute with P 3 , the function P,IIm) is not an eigenfunction of L, .

This " backward rotated" vector r" has components X,9, Z relative to the original basis. According to Eqs. (141) and (143), it must hold that r" = W-'r =9 - ' e r = eR-'r = eP, = R-'r. (148) Naturally, r" has components x, y , z relative to the basis e,", e,", e3", whence r" = W-'r = W-'er = e R - ' r = e"r, e" = eR-'. (149) Comparing the statements of paragraph (a) with those of paragraph (b), we recall the well-known fact that any coordinate transformation r ' = Rr ( 150) allows two interpretations: In apassitle interpretation (alias transformation) it means a backward rotation of the basis vectors, the vector r being fixed [cf.

W1, The operator L2was already identified as the square of the orbital angular momentum operator in Section B l , and its representation in spherical coordinates, which is to be used in Eq. (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.

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