Pure density functionals

As readers of this volume are also aware, the best of both approaches have been blended together with the result that many computations are now performed by a careful mixture of wavefunction and density approaches within the same computations (Hehre et al., 1986). But the unfortunate fact is that, as yet, there is really no such thing as a pure density functional method for performing calculations. The philosophical appeal of a universal solution for all the atoms in the periodic table based on observable electron density, rather than fictional orbitals, has not yet borne fruit.21,22... 

Regarding the height of the insertion barrier, the situation is much more controversial, since pure density functionals and some MP2 calculations suggest that this a barrierless reaction, or it occurs with a negligible barrier. HF, hybrid density functionals and several post-HF calculations, instead, suggest a barrier in the range of 5-10 kcal/mol, roughly. 

Both Hartree-F ock and density functional models actually formally scale as the fourth power of the number of basis functions. In practice, however, both scale as the cube or even lower power. Semi-empirical models appear to maintain a cubic dependence. Pure density functional models (excluding hybrid models such as B3LYP which require the Hartree-F ock exchange) can be formulated to scale linearly for sufficiently large systems. MP2 models scale formally as the fifth power of the number of basis functions, and this dependence does not diminish significantly with increasing number of basis functions. 

Vibrational frequencies measured in IR experiments can be used as a probe of the metal—ligand bond strength and hence for the variation of the electronic structure due to metal—radical interactions. Theoretical estimations of the frequencies are obtained from the molecular Hessian, which can be straightforwardly calculated after a successful geometry optimization. Pure density functionals usually give accurate vibrational frequencies due to an error cancellation resulting from the neglect of... 

Also pure density-functional methods combined with plane-wave basis sets and ultrasoft pseudopotentials [58] were used in our studies of extended systems [59]. The computational efficiency of these methods enables larger systems and to some extent dynamical processes to be studied. Generalized-gradient approximation (GGA) or spin-polarized GGA DFT functionals [60, 61] were employed in the electronic structure calculations. 

The ionization energy of a molecule can be regarded as the energy required to remove an electron from its HOMO. How then would a pure density functional theory, with no orbitals, be able to calculate ionization energy ... 

The possibility of N2 coordination to up to four (six) iron atoms has been proposed by Dance on the basis of restricted frozen-core Kohn-Sham calculations on a FeMoco model (32,33). It was found that a binding mode intermediate between p Vn and p4,r 2 coordination to be most stable. However, these propositions are not necessarily the final answer since open-shell states are most likely to become important and exact exchange was not present in the density functional chosen to cure the singlet preference of the pure density functional (cf. discussion in the Appendix). 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

This follows directly from the Hohenberg-Kohn theorem applied to a noninteracting system. The density n uniquely determines the Kohn-Sham potential vs (up to a constant) and therefore the also the orbitals (up to a phase factor) and eigenvalues (up to constant). The arbitrariness with respect to a constant shift and with respect to the phase factor cancels out in the energy expression and therefore the zth-order energy becomes a pure density functional. We therefore have the following series of... 

The problem of a pure density functional theory lies in the fact that the exact forms of T[p] and Exc[p] in terms of p(r) are unknown at the moment [18,19], and the approximations, which are generally based on the generalized gradient expansion, do not provide the accuracy required for chemical properties, such as the bond energies, mainly... 

On the other hand, the pure density functional theory provides a conceptual framework that has proven to be very useful to establish expressions that are closely related with chemical concepts, and to rationalize, through the values associated with them, the behavior of a wide variety of chemical species under different circumstances [20]. In order to calculate the values of these quantities, it has become common practice to make use of conventional molecular orbital theory [21-26]. Thus, this procedure allows one to transform the relevant information contained in the wavefunction, into chemically meaningful results, through the bridge provided by pure density functional theory. 

A third important development concerns the application of DFT to diradicals. As discussed in the section on this type of molecules, one of the main limitations of density functional theory is its ability to deal properly with singlet diradicals. Interestingly, Schreiner has found that pure density functionals such as BLYP perform much better than hybrid functionals (i.e., B3LYP) in calculations on dehydrobenzene singlet diradicals which are intermediates in the Bergman cyclization of enediynes. Apparently, the admixture of Hartree-Fock density appears to constitute a disadvantage in such cases. ... 

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