Other density functions

The charge density P(r) and spin density Qz r) are examples of point properties or sub-observables (Hirschfelder, 1977). Their values are inferred (never directly measured) by reference to an integral such as in (5.2.20) that defines an expectation value. Thus VP r) in (5.2.20) may be interpreted as a potential-energy density , since it is the contribution per unit volume to Ve evaluated at point r and similarly, in (5.9.3), Qz(r) is interpreted as a density of spin angular momentum. 

It might have been expected that other property densities could be defined in exactly the same way. Thus, referring again to (5.2.20), 

Ihe simplest acceptable definition of a kinetic-energy density would appear to be 

This gives the usual expectation value (T) when integrated over all space, but has the conceptual advantage of being both real and positive at aJl points. 

For some purposes (5.10.1a) is not completely satisfactory it contains a term (Problem 5.19) that depends on the Laplacian of the electron density (making no reference to electronic motion), and on integrating over any finite region this term gives a contribution to (T) that depends 

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B3LYP density functional models provide somewhat better bond length results than the other density functional models, generally very close to experimental distances and to those from MP2 calculations. As with the other density functional models, the errors are largest where one (or two) second-row elements are involved. This is apparent from Figure 5-4, which compares B3LYP/6-311+G and experimental bond distances. 

Triplet oxygen deserves special attention if for no other reason that it is the only common non-closed shell molecule. As expected, the limiting (6-311+G basis set) Hartree-Fock bond distance is too short. The corresponding B3FYP model provides a bond length in nearly exact agreement with experiment, while all other density functional models and especially the MP2 model significantly overestimate the bond distance. 

Calculations have been carried out on the. S n2 reactions between chloride ion and methyl chloride or chlorosilane at several DFT levels of theory up to the OLYP/TZ2P level.102 The OLYP/TZ2P method gave much better (excellent compared with the CCDD(T)/aug-cc-pVQZ level) values than the other density functionals for both the geometry and the energies of the reactant complex and transition state for the methyl chloride reaction and the stable transition complex that forms in the chlorosilane reaction. 

The observation that bonds of all orders relate to the bonding diagram in equivalent fashion indicates that covalent bonds are conditioned by the geometry of space, rather than the geometry of electron fields, dictated by atomic orbitals or other density functions. The only special point related to electron density occurs at the junction of the attractive curves, where e = indicating that one pair of electrons mediate the covalent interaction. It is interpreted as the limiting length (dj) for first-order bonds. It is of interest to note that all known first-order bonds have d > d[. The covalence curve for the minimum ratio of x = 0.18 (for CsH) terminates at dl = d[. 

Electron density calculations have been carried out on Gp2ScR (R=H, Me, Et, Prn, vinyl, acetylide). Geometry optimizations revealed an agostic interaction for R= Et, whereas Me and Pr are bonded to the metal center without agostic interactions. Other density functional studies have been carried out on the complexes Cp2LnCl and Cp2LnX(THF) (Ln = La-Lu X = F, Cl, Br, I). In these mixed ligand complexes, Ln-Cp and Ln-THF bonds are... 

These results applying to spin states with M = S) are invariant against change of M, except for (7.5.12), which for other states of a multiplet is multiplied by a suitable factor as in (5.9.10). The other density functions defined in Chapter 5 all follow from the more general rules cited above. 

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