Recovering Unitary Invariance in Orbital-Dependent Density Functional and Wavefunction Formulations: Applications to Magnetic Molecules

The goal of deriving localized-orbitals as a means for describing transferable bonding orbitals or for more efficiently describing electronic correlation has a long history. Much of the inspiration in this area arose from a seminal paper by Edmiston and Ruedenberg[1] that derived localized orbitals based upon minimization of the magnitude of the mutual exchange, and which were observed to exhibit features that match ideas about chemical bonding. Until recently such energy-localized orbitals have been determined in both self-interaction-corrected density functional methods and multi-configurational self-consistent (MCSDF) methods, using the Jacobi-based determination of unitary matrices, originally suggested in Ref. [1]. For example, many authors, including Ref [2], have used such techniques to rigorously minimize density-functional energies based upon the localization equation that have been corrected for self-interaction error through the use of orbital-dependent corrections. In contrast to the use of such methods for qualitatively understanding chemical bonding, an issue that arises in both self-interaction-corrected formulations and in MCSCF methods is that the resulting unitary transformation between localized and canonical orbitals that leads to a variational expression for the total energy also leads to energy expressions that violate size extensivity.  To address this issue, a new way to construct energy-localized orbitals based on Fermi-Löwdin Orbitals was introduced[3]. These localized orbitals depend on a quasi-classical electronic geometry, the Fermi exchange hole, and Löwdin’s method of symmetric orthogonalization. In applications to date, the orbitals are explicitly real and unsurprisingly bear a strikingly similar resemblance to the energy localized-orbitals due to Edmiston and Ruedenberg.  Recent discussions about such orbitals have questioned  (1) whether the constraint of real rather than complex orbitals associated with the original FL formulation is a limiting factor for their use in density functional theory and (2) whether the original methods based upon the localization equations are in fact equivalent to the Fermi-Löwdin orbitals. Results on highly anionic states in the Mn₁₂-Acetate Molecular Magnet, a prototypical case where density-functional theory often fails, will be presented[4].

Views presented here are those of the author. MRP Thanks T. Baruah, T. Hahn, K.A. Jackson, D.-Y.Kao, J. Kortus, J. Peralta, J. Perdew, and R. Zope for many interesting discussions on FLOSIC.

[1]C. Edmiston and K. Ruedenberg, Localized Atomic and Molecular Orbitals, Rev. of Mod. Phys. 35 457 (1963).

[2]M.R. Pederson, R.A. Heaton, and C.C. Lin, Density functional theory with self-interaction correction: Application to the Li molecule, J. Chem. Phys. 82, 2688 (1985).

 [3] M.R. Pederson, A Ruzsinszky and J. P. Perdew, Communication: Self-Interaction Correction with Unitary Invariance in Density Functional Theory, J. of Chem. Phys., 140, 121103 (2014).

[4] J. Batool, T. Hahn and M.R. Pederson (Submitted).