Third-order electron density

Here, the induced dipoles depend on the electric field from the part of the third-order electron density which depends on lower order parameters (Eq. (110)). The transformed electric field 1 P 1-"2-"3 jS/ thus, defined through... 

As discussed earlier, we cannot only derive first-order electron densities, but also we can extend them to higher order electron densities. We have used the second-order electron density p(ri,r2) in lieu of the first-order electron density on several occasions in molecular quantum similarity, because the second-order electron density is in fact the lowest order density where electron correlation becomes apparent. It has been used extensively by Ponec et al. "- in the study of similarity in pericyclic reactions where the second-order electron density offers important advantages over the first-order electron density. In another contribution, Ponec et al. went to the third-order electron density. Again, most of the discussion relating to molecular quantum similarity indices and molecular alignment is also applicable to higher order electron densities, replacing where necessary the first-order electron density by, for example, the second-order electron density. 

The third term of Eq (54) is the electronic Hartree potential, whereas the fourth one represents the exchange-correlation potential. This last term is usually obtained from a model exchange-correlation energy functional xc[pl To a first order approximation, the effective KS potential compatible with the electron density p f) given in Eq (51) may be written as ... 

Upon comparison of the k< m exchange rate of the Tc(V) system with that of the Re(V), the significant increase in reactivity (ca. 3 orders of magnitude) is very prominent and not necessarily indicative of an associative activation. It is, however, possible that the Tc(V) hydroxo complex might be very reactive via an associative pathway, since it is known that the Tc(V) center much more readily accepts electron density than does the corresponding Re(V) complexes (55). The greater ease by which coordination sphere expansion can occur in third-row d-series transition elements such as W(IV) and Re(V) (not very easily... 

This expression is actually sometimes used to define the first and second order density matrix. It is anyway useful to know that the first order density matrix elements are equal to the coefficients for the corresponding one-electron integrals in the energy expression and similarly for the second order density matrix elements. The definition of the third order density matrix is,... 

Finally, a third dithiolene ligand model has been utilized with success in order to understand the electronic structure and spectroscopy of a number of oxo-molybdenum mono- and bis(dithiolenes) (23). This dithiolene ligand bonding description utilizes the symmetric and antisymmetric out-of-plane Sop p 7t orbitals, in addition to the corresponding in-plane symmetric and antisymmetric S p p orbitals. Ab initio and density functional theory (DFT) calculations have been performed on the simple dithiolene dianion, [S2C2H2]2 (23), in order to illustrate the details of this four orbital model and electron density contours of the four MOs are presented in Fig. 4. These calculations result in an isolated set of four filled dithiolene orbitals, and these are the ligand... 

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