# Browsing by Subject "Density functional theory"

###### Results Per Page

###### Sort Options

Item Open Access Theoretical and Computational Aspects of the Optimized Effective Potential Approach within Density Functional Theory(2009) Heaton-Burgess, TimThe computational success of density functional theory relies on the construction of suitable approximations to the exchange-correlation energy functional. Use of functional approximations depending explicitly upon the density alone appear unable to address all aspects of many-body interactions, such as the fundamental constraint that the ground state energy is a piecewise linear function of the total number of electrons, and the ability to model nonlocal effects. Functionals depending explicitly upon occupied and unoccupied Kohn–Sham orbitals are considered necessary to address these and other issues. This dissertation considers certain issues relevant to the successful implementation of explicitly orbital-dependent functionals through the optimized effective potential (OEP) approach, as well as extending the potential functional formalism that provides the formal basis for the OEP approach to systems in the presence of noncollinear magnetic fields.

The self-consistent implementation of orbital-dependent energy functionals is correctly done through the optimized effective potential approach—minimization of the ground state energy with respect to the Kohn–Sham potential that generates the set of orbitals employed in the energy evaluation. The focus on the potential can be problematic in finite basis set approaches as determining the exchange-correlation potential in this manner is an inverse problem which, depending upon the combination of orbital and potential basis sets employed, is often ill-posed. The ill-posed nature manifests itself as nonphysical exchange-correlation potentials and total energies. We address the problem of determining meaningful exchange-correlation potentials for arbitrary combinations of orbital and potential basis sets through an L-curve regularization approach based on biasing towards smooth potentials in the energy minimization. This approach generates physically reasonable potentials for any combination of basis sets as shown by comparisons with grid-based OEP calculations on atoms, and through direct comparison with DFT calculations employing functionals not depending on orbitals for which OEP can also be performed. This work ensures that the OEP methodology can be considered a viable many-body computational methodology.

A separate issue of our OEP implementation is that it can suffer from a lack of size-extensivity—the total energy of a system of infinitely separated monomers may not scale linearly with the total number of monomers depending upon how we construct the Kohn–Sham potential. Typically, a fixed reference potential is employed to aid in the convergence of a finite basis set expansion of the Kohn–Sham potential. This reference potential can be utilized to ensure other desirable properties of the resulting potential. In particular, it can enforce the correct asymptotic behavior. The Fermi–Amaldi potential is often used for this purpose but suffers from size-nonextensivity owing to the explicit dependence of the potential on the total number of electrons. This error is examined and shown to be rather small and rapidly approaches a limiting linear behavior. A size-extensive reference potential with the correct asymptotic behavior is suggested and examined.

We also consider a formal aspect of the potential-based approach that provides the underlying justification of the OEP methodology. The potential functional formalism of Yang, Ayers, and Wu is extended to include systems in the presence of noncollinear magnetic fields. In doing so, a solution to the nonuniqueness issue associated with mapping between potentials and wave functions in such systems is provided, and a computational implementation of the OEP in noncollinear systems is suggested.

Finally, as an example of an issue for which orbital-dependent functionals seem necessary to obtain a correct description, we consider the ground state structures of C

_{4N + 2}rings which are believed to exhibit a geometric transition from angle-alternation (N ≤ 2) to bond-alternation (N > 2). So far, no published DFT approach has been able to reproduce this behavior owing to the tendency of common density functional approximations to bias towards delocalized electron densities. Calculations are presented with the rCAM-B3LYP exchange-correlation functional that correctly predict the structural evolution of this system. This is rationalized in terms of the recently proposed delocalization error for which rCAM-B3LYP explicitly attempts to address.Item Open Access Wannier Functions and Their Role in Improving Density Functional Approximations(2021) Mahler, AaronDensity functional theory has proven to be an invaluable tool for modeling matter and chemistry . This can be seen from the fact that density functional theory papers are far and away the most cited theory from the physical sciences. While density functional theory excels at predicting total energies and equilibrium geometries, standard approximate functionals can be inadequate for determining some properties such as dissociation energies, reaction barriers, and band gaps. These deficiencies can be traced to delocalization error in density functional approximations. In finite systems, delocalization error can be attributed to the incorrect treatment of fractional electron charge whereby the total energy deviates from the correct behavior of linear interpolation between integer points. For bulk systems the delocalized nature of the orbitals results in a linear total energy at fractional charges, but the slope is incorrect due to delocalization error. Multiple methods have been proposed to fix this deficiency and produce the correct linearity condition such asthe Fermi-Löwdin orbital self-interaction correction, Koopmans-compliant functionals, the screened range-separated hybrid functional, the generalized transition state method, and the localized orbital scaling correction. All of these methods rely on spatially localized orbitals for their corrections, highlighting the importance of localized orbitals in modern density functional theory. The traditional method of obtaining localized orbitals minimizes the spatial variance, but here we explore an alternative approach that minimizes the combination of spatial and energetic variance. Minimizing the energetic variance allows for the occupied and unoccupied spaces to be considered together, a feature that is not prescribed in other localization schemes. The localization in energy results in localized orbitals that are more correlated with certain energy ranges, thereby making them more chemically relevant for the states that are associated with frontier energies. We show how these localized functions can be used in the localized orbital scaling correction to remedy many of the density functional approximation shortcomings related to delocalization error.