By Per-Olov Lowdin

ISBN-10: 008058232X

ISBN-13: 9780080582320

ISBN-10: 0120348071

ISBN-13: 9780120348077

ADVANCES IN QUANTUM CHEMISTRY VOL 7.

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Again, simple tests can be made by means of thep functions given in Eq. (169). From the given identities it follows easily that P, = oil, P, = 09-1, Pi1 Pa-1 = 0, = P , - l = 09 9 ; 0,’ = 09-1. (1 76) (177) According to Eq. (176), the Eqs. (167) and (175) are equivalent, as are Eqs. (168) and (174). C. Parametrization in Terms of Eulerian Angles The following derivations are based on operators of the kind P, which cause a positive rotation of the field. Operators of the kind P, are also used in the book by Rose (1957), 38 E.

D. Representation Matrices of the Rotation Group D(”(a, P, y) denotes the representation matrix of the rotation R = [u, P, y ] in the (21 1)-dimensional irreducible representation D(’) of the rotation group, if the 21 + 1 complex surface spherical harmonics I lm) = YF(8, q ) with - I I m 5 I form the basis for this representation. Because of Eqs. ),(a, /I 7) ,= (IdI Pa I Zm) - (im’ 1 e - i u L = e - i / l L y e - i y L , 1 Irn > + = e-i@m‘dU? mm (p)e-iym, (1 8 5 ) 42 E. 0. Steinborn and K. Ruedenberg since for z-axis quantization, the operator L, is diagonal with respect to the Ilm').

36) the identity r'P,"'(C)cos rncp = (- 1)" ( I + m)! 2 . i-- ( I -2MI- m - 1) Z I - m - 2 ttl (2m + 2) - m - 3) + 2( I- -4 m)(2m +( I2)(2m + 4) * * * z1-m-4(~s)z - + ... * from which Eq. (225) immediately follows. The following relationships, which may also be deduced from the foregoing, will prove useful later. For m > 0, two operators shall be defined by ant,+= (a/aO"' + (alatl)"' Qm,- = [(apt)"- (a/i7q)m]i. Then, because of the following relationship holds (if 12 m 2 0 ) as well as (if 1 - m 2 rn 2 0) The factor cosmcp applies for + index, the factor sin rncp applies for - index.

### Advances in Quantum Chemistry by Per-Olov Lowdin

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