Event Information
Theoretical and Computational Aspects of the Optimized Effective Potential Approach Within Density Functional Theory
- Abstract:
The computational success of density functional theory relies on the construction of suitable approximations to the exchange-correlation energy functional. A popular approach incorporates an implicit dependence on the density by explicitly including Kohn-Sham orbitals. This thesis 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 OEP approach is an inverse problem that is often ill-posed. This 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 is able to generate 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 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 the OEP implementation we employ 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. This error is examined and shown to be rather small and rapidly approaches a limiting linear behavior. A size-extensive construction of the Kohn-Sham potential is suggested and examined.
We 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 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-dependant functionals seem necessary to obtain a correct description, we consider the ground state structures of carbon 4N+2 rings which are believed to exhibit a geometric transition from angle alternation (N ? 2) to bond-alternation (N > 2). So far no functional theory 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.
Ph.D. Dissertation Defense
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