Quantum chemical equations particle basis functions

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these... 

CNDO/S electron densities of complex allenes predicted with these parameters on the basis of Equations 29 and 30 agree well with those obtained from direct quantum-chemical CNDO/S computations (cf. Table 14). With respect to the CNDO/S level most pair-terms (t, y, v ) are small compared with the one-particle functions. Therefore, for numerical purposes they seem to be not so relevant as, for instance, in case of the C chemical shifts of the allenic carbon atoms (Section II.C.2) which are described by approximation ansatze similar to Equations 29 and 30. In the light of the discussions in Section 11. A the preceding findings concerning the pair-terms must be viewed as accidental and cannot be predicted a priori. For instance, the neglection of the pair-terms may no longer be justified, if one uses a different computational procedure for the evaluation of the electron densities (such as the INDO or MlNDO/3 scheme). 

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