Self-correlation correction

The PKZB self-correlation correction to the PBE GGA has a remarkable feature Under uniform scaling to the low-density or strongly-interacting limit (see (1.115)), it yields essentially correct correlation energies while LSD and GGA yield correlation energies that are much too negative [109]. 

Encl[p] is the non-classical contribution to the electron-electron interaction containing all the effects of self-interaction correction, exchange and Coulomb correlation described previously. It will come as no surprise that finding explicit expressions for the yet unknown functionals, i. e. T[p] and Encl[p], represents the major challenge in density functional theory and a large fraction of this book will be devoted to that problem. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Equation (96) shows that the effective KS potential may be simply obtained by adding to the standard KS potential of the isolated solute, an electrostatic correction which turns out to be the RE potential Or, and the exchange- correlation correction 8vxc. It is worth mentioning here, that Eq (96) is formally equivalent to the effective Fock operator correction bfteffi defined in the context of the self consistent reaction field (SCRF) theory [2,3,14] within the HF theory, the exchange contribution is exactly self-contained in Or, whereas correlation effects are completely neglected. As a result, within the HF theory 8v = Or, as expected. 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

Successful density functional approximations such as the PW91 GGA or the self-interaction correction (SIC) [57] to LSD recover [19] LSD values for the on-top hole density and cusp. The weighted density approximation (WDA) [41,42], which recovers the LSD exchange hole density but not the LSD correlation hole density [19] in the limit u -> 0, needs improvement in this respect. 

DPT schemes, which allow to calculate the electron affinities of atoms are based on the exact [59,60] and generalized (local) [61,62] exchange self-interaction-corrected (SIC) density functionals, treating the correlation separately in some approximation. Having better asymptotic behavior than GGA s, like in the improved SIC-LSD methods, one should obtain more... 

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn-Sham formalism (Perdew and Zunger 1981 KUmmel and Perdew 2003 Grafenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 

Finally, in careful comparative studies of the molecular electron densities generated by HF, correlated ab initio, and pure, self-interaction corrected, and hybrid DFT calculations, Cremer et al have made a very interesting observation [72, 73]. They found that the pure DFT generated densities differed from those obtained with accurate ab initio methods in a particular way, and that both hybrid, and self-interaction corrected DFT methods, yielded densities closer to the correct ones. Based on this observation, they suggested that mixing in of exact exchange in hybrid functionals serves as a proxy for the self-interaction correction. 

In practice, the Schrodinger equation with the Hamiltonian of Eq. (1-173) is first solved within the self-consistent field approximation252, leading to the so-called SCRF free energy of solvation, AGSCRF. If the correlation corrections are included, e.g. via the MP2 approach255,256, we get the MP2-SCRF free energy of solvation... 

It can be seen that the relaxation correction decreases the Koopmans estimate, whereas the difference in the two correlation corrections increases the estimate. The errors are thus, to a certain extent, self-compensating. Transition metal d orbitals are often associated with much greater relaxation corrections than ligand orbitals, and this can lead to an inversion of ordering between an orbital energy calculation and the associated PE bands. Thus, it is not uncommon to And from a calculation a filled metal orbital to be more stable than a ligand orbital but to give rise to the first IE of a molecule. 

As for the bulk modulus, the LDA well represents its trend across the 4f series, although the absolute errors are quite large (7 A careful analysis of the bulk modulus for the mixed valent metal TmSe indicates that the LDA error in that case can be corrected if the f levels are shifted down 40 mRy (4). This problem is connected to the fact that the LDA does progressively worse for higher angular momentum states (due to probable misrepresentation of the shape of the exchange-correlation hole). Self-interaction corrections may be able to explain this error (9-1OL... 

These cases can be contrasted by most uranium intermetallics, which have Fermi surfaces in good agreement with LDA calculations which treat the f electrons as band states (IQ. In the one case where a mixed valent Fermi surface is known (CeSno), it is also in excellent agreement with an LDA f band calculation (T8-19L with a mass renormalization of five due to a self-energy correction resulting from virtual spin fluctuation excitations (2Q). Notice the different dynamic correlations used to explain the mass renormalizations in the f core and f band cases. 

Firstly, the essential correctness of the tube picture has only recently been established in a remarkable series of experiments. The complex monomer diffusive self-correlation predicted has now been seen in field-gradient NMR. Reptative motion across an interface was the only successful explanation of time-resolved neutron reflectivity. Neutron Spin Echo (NSE) can now be extended in time sufficiently to identify the tube diameter directly. A series of massive many-chain numerical simulations have shown tube-like constraints with sizes identical to those obtained by rheology via the plateau modulus Go and NSE). 

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