Advances in Quantum Chemistry by Per-Olov Lowdin

By Per-Olov Lowdin


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Advances in Quantum Chemistry


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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. 2. Regular Harmonics According to Eq. (f- m ) ! (A + m ) ! ( l - m)!

2) The new frame is rotated by -+ ef -+ about the e f axis. Then (ef” is pointing down within plane E), (unchanged), (ef’ coincides with R). , about R). Thereby ( E l A points upward within plane E ) , ef’ + SIA e;” -+ i2A = e r = SJA -eF, (in R). E. 0. Steinborn and K. Ruedenberg 56 P Fig. 13 P 5: Fig. 14 This rotation, which is-according to Eq. (182e)-denoted by R = [ q A RO ,A R , n], transforms the “parallel, shifted” position of the two coordinate systems into their “ lined-up” position. Of course, the inverse rotation R - ’ = [-n,- O A R , - q A R will ] cause the transformation of the “ lined-up” positions into the “ shifted ” ones.

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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