A critical assessment of the Self-Interaction Corrected Local Density Functional method and its algorithmic implementation

We calculate the electronic structure of several atoms and small molecules by direct minimization of the Self-Interaction Corrected Local Density Approximation (SIC-LDA) functional. To do this we first derive an expression for the gradient of this functional under the constraint that the orbitals be orthogonal and show that previously given expressions do not correctly incorporate this constraint. In our atomic calculations the SIC-LDA yields total energies, ionization energies and charge densities that are superior to results obtained with the Local Density Approximation (LDA). However, for molecules SIC-LDA gives bond lengths and reaction energies that are inferior to those obtained from LDA. The nonlocal BLYP functional, which we include as a representative GGA functional, outperforms both LDA and SIC-LDA for all ground state properties we considered.Comment: 14 pages, 5 figure

Consistent LDA'+DMFT approach to electronic structure of transition metal oxides: charge transfer insulators and correlated metals

We discuss the recently proposed LDA'+DMFT approach providing consistent parameter free treatment of the so called double counting problem arising within the LDA+DMFT hybrid computational method for realistic strongly correlated materials. In this approach the local exchange-correlation portion of electron-electron interaction is excluded from self consistent LDA calculations for strongly correlated electronic shells, e.g. d-states of transition metal compounds. Then the corresponding double counting term in LDA+DMFT Hamiltonian is consistently set in the local Hartree (fully localized limit - FLL) form of the Hubbard model interaction term. We present the results of extensive LDA'+DMFT calculations of densities of states, spectral densities and optical conductivity for most typical representatives of two wide classes of strongly correlated systems in paramagnetic phase: charge transfer insulators (MnO, CoO and NiO) and strongly correlated metals (SrVO3 and Sr2RuO4). It is shown that for NiO and CoO systems LDA'+DMFT qualitatively improves the conventional LDA+DMFT results with FLL type of double counting, where CoO and NiO were obtained to be metals. We also include in our calculations transition metal 4s-states located near the Fermi level missed in previous LDA+DMFT studies of these monooxides. General agreement with optical and X-ray experiments is obtained. For strongly correlated metals LDA$^\prime$+DMFT results agree well with earlier LDA+DMFT calculations and existing experiments. However, in general LDA'+DMFT results give better quantitative agreement with experimental data for band gap sizes and oxygen states positions, as compared to the conventional LDA+DMFT.Comment: 13 pages, 11 figures, 1 table. In v2 there some additional clarifications are include

Correlated metals and the LDA+U method

While LDA+U method is well established for strongly correlated materials with well localized orbitals, its application to weakly correlated metals is questionable. By extending the LDA Stoner approach onto LDA+U, we show that LDA+U enhances the Stoner factor, while reducing the density of states. Arguably the most important correlation effects in metals, fluctuation-induced mass renormalization and suppression of the Stoner factor, are missing from LDA+U. On the other hand, for {\it moderately} correlated metals LDA+U may be useful. With this in mind, we derive a new version of LDA+U that is consistent with the Hohenberg-Kohn theorem and can be formulated as a constrained density functional theory. We illustrate all of the above on concrete examples, including the controversial case of magnetism in FeAl.Comment: Substantial changes. In particular, examples of application of the proposed functional are adde

Do unbalanced data have a negative effect on LDA?

For two-class discrimination, Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562] claimed that, when covariance matrices of the two classes were unequal, a (class) unbalanced data set had a negative effect on the performance of linear discriminant analysis (LDA). Through re-balancing 10 real-world data sets, Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562] provided empirical evidence to support the claim using AUC (Area Under the receiver operating characteristic Curve) as the performance metric. We suggest that such a claim is vague if not misleading, there is no solid theoretical analysis presented in Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562], and AUC can lead to a quite different conclusion from that led to by misclassification error rate (ER) on the discrimination performance of LDA for unbalanced data sets. Our empirical and simulation studies suggest that, for LDA, the increase of the median of AUC (and thus the improvement of performance of LDA) from re-balancing is relatively small, while, in contrast, the increase of the median of ER (and thus the decline in performance of LDA) from re-balancing is relatively large. Therefore, from our study, there is no reliable empirical evidence to support the claim that a (class) unbalanced data set has a negative effect on the performance of LDA. In addition, re-balancing affects the performance of LDA for data sets with either equal or unequal covariance matrices, indicating that having unequal covariance matrices is not a key reason for the difference in performance between original and re-balanced data

Potential Energy Landscape of the Apparent First-Order Phase Transition between Low-Density and High-Density Amorphous Ice

The potential energy landscape (PEL) formalism is a valuable approach within statistical mechanics for describing supercooled liquids and glasses. Here we use the PEL formalism and computer simulations to study the pressure-induced transformations between low-density amorphous ice (LDA) and high-density amorphous ice (HDA) at different temperatures. We employ the ST2 water model for which the LDA-HDA transformations are remarkably sharp, similar to what is observed in experiments, and reminiscent of a first-order phase transition. Our results are consistent with the view that LDA and HDA configurations are associated with two distinct regions (megabasins) of the PEL that are separated by a potential energy barrier. At higher temperature, we find that low-density liquid (LDL) configurations are located in the same megabasin as LDA, and that high-density liquid (HDL) configurations are located in the same megabasin as HDA. We show that the pressure-induced LDL-HDL and LDA-HDA transformations occur along paths that interconnect these two megabasins, but that the path followed by the liquid is different than the path followed by the amorphous solid. At higher pressure, we also study the liquid-to-ice-VII first-order phase transition, and find that the behavior of the PEL properties across this transition are qualitatively similar to the changes found during the LDA-HDA transformation. This similarity supports the interpretation that the LDA-HDA transformation is a first-order-like phase transition between out-of-equilibrium states.Comment: 29 pages, 8 figure