Hybridizational invariance

When the Roothaan equations (13.157) [or (13.179)] are solved exactly, the canonical MOs and the calculated values of molecular properties do not change if one changes the orientation of the coordinate axes the calculated values are said to be rota-tionally invariant Likewise, the results do not change if each basis AO on a particular atom is replaced by a linear combination of the basis AOs on that atom, and the results are hybridizntioruilly invariant When approximations are made in solving the Hartree-Fock-Roothaan equations, rotational and hybridizational invariance may not hold. 

To maintain rotational and hybridizational invariance when the ZDO approximation is used, the CNDO method makes the additional approximation that the electron repulsion integral y = rr tt) depends only on which atoms the AOs /, and /, are centered on and does not depend on the nature of the orbitals and Let the notation ff, indicate that the valence AOs f, and /, are centered on atoms A and B, respectively, where A and B might be the same atom or different atoms. The CNDO method takes b) = = Tab for all valence AOs f, on A and all valence AOs... 

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