Advances in Quantum Chemistry by Per-Olov Lowdin

By Per-Olov Lowdin


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234) may be expressed as Using the definition given by Eq. (131) and putting FA = f j B = cp, as it is required by the problem considered here (see Fig. 1l), one finally obtains the normalization constants are those of Eq. (59). Fig. II 51 E. 0. Steinborn and K. Ruedenberg 2. Irregular Harmonics Now the translation of an irregular solid spherical harmonic along the z axis from the original center A to a new center B will be considered. Such an operation produces infinite series expansions as the irregular harmonics contain negative powers of the length of the local vector.

234) is finite and, therefore, the remainder vanishes, it should hold for all values of zo . Hobson (1931) was the first to derive this important relationship. However, his derivation is much more cumbersome than the one given here. With the help of Eqs. (85)-(87) one shows that Eq. (234) holds for all m ; hence, it may also be written for m = I rn I . For the purpose of a uniform notation, the result of Eq. (234) may be expressed as Using the definition given by Eq. (131) and putting FA = f j B = cp, as it is required by the problem considered here (see Fig.

Translation Operator The derivation of the transformation of solid spherical harmonics under translations is facilitated by discussing first the translation of general functions, and by deriving some auxiliary identities which will be needed in the sequel. The transformation of functions under translations could be discussed in just the same way as their behavior under rotations, provided that some characteristicfeatures are observed which follow from the parallel translation of basis systems. However, because of the restricted space available, we will refrain from observing a strict analogy, especially as far as the discussion of the active and passive interpretation is concerned.

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