Structural manifestation of the delocalization error of density functional approximations: C(4N+2) rings and C(20) bowl, cage, and ring isomers.

The ground state structure of C(4N+2) rings is believed to exhibit a geometric transition from angle alternation (N < or = 2) to bond alternation (N > 2). All previous density functional theory (DFT) studies on these molecules have failed to reproduce this behavior by predicting either that the transition occurs at too large a ring size, or that the transition leads to a higher symmetry cumulene. Employing the recently proposed perspective of delocalization error within DFT we rationalize this failure of common density functional approximations (DFAs) and present calculations with the rCAM-B3LYP exchange-correlation functional that show an angle-to-bond-alternation transition between C(10) and C(14). The behavior exemplified here manifests itself more generally as the well known tendency of DFAs to bias toward delocalized electron distributions as favored by Huckel aromaticity, of which the C(4N+2) rings provide a quintessential example. Additional examples are the relative energies of the C(20) bowl, cage, and ring isomers; we show that the results from functionals with minimal delocalization error are in good agreement with CCSD(T) results, in contrast to other commonly used DFAs. An unbiased DFT treatment of electron delocalization is a key for reliable prediction of relative stability and hence the structures of complex molecules where many structure stabilization mechanisms exist.

Optimized effective potentials from arbitrary basis sets.

We investigate the use of a regularized optimized effective potential (OEP) energy functional and L-curve procedure [T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007)] for determining physically meaningful OEPs from arbitrary combinations of finite orbital and potential basis sets. The important issue of the manner in which the optimal regularization parameter is determined from the L-curve perspective is reconsidered with the introduction of a rigorous measure of the quality of the potential generated-that being, the extent to which the Ghosh-Parr exchange energy virial relation is satisfied along the L-curve. This approach yields nearly identical potentials to our previous work employing a minimum derivative condition, however, gives rise to slightly lower exact-exchange total energies. We observe that the ground-state energy and orbital energies obtained from this approach, either with balanced or unbalanced basis sets, yield meaningful potentials and energies which are in good comparison to other (a priori balanced) finite basis OEP calculations and experimental ionization potentials. As such, we believe that the regularized OEP functional approach provides a computationally robust method to address the numerical stability issues of this often ill-posed problem.

Size extensivity of the direct optimized effective potential method.

We investigate the size extensivity of the direct optimized effective potential procedure of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)]. The choice of reference potential within the finite basis construction of the local Kohn-Sham potential can lead to a method that is not size extensive. Such a situation is encountered when one employs the Fermi-Amaldi potential, which is often used to enforce the correct asymptotic behavior of the exact exchange-correlation potential. The size extensivity error with the Fermi-Amaldi reference potential is shown to behave linearly with the number of electrons in the limit of an infinite number of well separated monomers. In practice, the error tends to be rather small and rapidly approaches the limiting linear behavior. Moreover, with a flexible enough potential basis set, the error can be decreased significantly. We also consider one possible reference potential, constructed from the van Leeuwen-Baerends potential, which provides a size extensive implementation while also enforcing the correct asymptotic behavior.

Optimized effective potentials from electron densities in finite basis sets.

The Wu-Yang method for determining the optimized effective potential (OEP) and implicit density functionals from a given electron density is revisited to account for its ill-posed nature, as recently done for the direct minimization method for OEP's from a given orbital functional [T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007)]. To address the issues on the general validity and practical applicability of methods that determine the Kohn-Sham (local) multiplicative potential in a finite basis expansion, a new functional is introduced as a regularized version of the original work of Wu and Yang. It is shown that the unphysical, highly oscillatory potentials that can be obtained when unbalanced basis sets are used are the controllable manifestation of the ill-posed nature of the problem. The new method ensures that well behaved potentials are obtained for arbitrary basis sets.

Optimized effective potentials in finite basis sets.

The finite basis optimized effective potential (OEP) method within density functional theory is examined as an ill-posed problem. It is shown that the generation of nonphysical potentials is a controllable manifestation of the use of unbalanced, and thus unsuitable, basis sets. A modified functional incorporating a regularizing smoothness measure of the OEP is introduced. This provides a condition on balanced basis sets for the potential, as well as a method to determine the most appropriate OEP and energy from calculations performed with any finite basis set.

Spin-potential functional formalism for current-carrying noncollinear magnetic systems.

We develop a formalism dual to spin-current-density functional theory (CDFT) where minimization with respect to the scalar and vector spin potentials is used. In this way we circumvent the issues surrounding the nonuniqueness of the mapping between spin potentials and ground-state wave functions, and the v representability issue of current-density functionals. The approach applied within the Kohn-Sham formalism provides the foundations for the optimized effective potential method for CDFT.