Session B31: Advances in Density Functional Theory II

Self-Interaction Corrected Density Functional Approximations with Unitary Invariance: Applications to Molecules

For a system of 2N electrons, the Fermi-hole may be interpreted as the square of a normalized "Fermi orbital", $F({\bf a}) \equiv \rho_\sigma({\bf a},{\bf r})/\sqrt{\rho_\sigma({\bf a})}$. This normalized orbital captures all of the spin density at its position of definition, or descriptor, $({\bf a})$. Given a set of N quasi-classical electronic positions $({\bf a_i})$ and a spin density-matrix composed of N Kohn-Sham orbitals, the resulting set of Fermi orbitals may then be used to construct a set of localized Loewdin-orthonormalized orbitals[1]. These orbitals are explicitly a functional of the spin density and are related to the Kohn-Sham orbitals by a unitary transformation that is parametrically dependent on the set quasi-classical electronic descriptors. The construction of such localized orbitals allows for the restoration of unitary invariance into the original Perdew-Zunger self-interaction correction[2,3] and provides a possible simplification compared to the localization-equation based solution of self-interaction corrected functionals[4]. This talk will discuss the construction of this Fermi-orbital-based self-interaction corrected method and the minimization algorithm that relies upon analytical derivatives[3] of the self-interaction energy with respect to the Fermi-orbital descriptors. Recent applications to a large set of molecules including aromatic molecules, molecules with open transition-metal centers, and molecules with frustrated Kekule' structures will be discussed. Initial applications indicate improvements in atomization energies of pi-bonded systems and demonstrate the desired downward shift of orbital energies relative to their Kohn-Sham counterparts. [1]W.L.Luken and D.N. Beratan, Theo. Chim. Acta {\bf 61}, 265-281 (1982). [2]M.R. Pederson, A. Ruzsinszky and J. P. Perdew, J. Chem. Phys. ${\bf 140}$, 121103 (2014). [3]M.R. Pederson and T. Baruah, Advance in Atomic, Molecular and Optical Physics ${\bf 64}$, 153-180 (2015). [4]M. R. Pederson, R. A. Heaton, and C. C. Lin, J. Chem. Phys. ${\bf 80}$, 1972 (1984).   

Fermi-orbitals for improved electronic structure calculations on coordination complexes.

An improved density-functional formalism[1,2] proceeds by adopting the Perdew-Zunger expression for a self-interaction-corrected (SIC) density-functional energy but evaluates the total energy based on Fermi Orbitals (FOs). Each localized electron is represented by an FO, determined from the occupied Kohn-Sham orbitals and a semi-classical FO descriptor. The SIC energy is then minimized through the gradients of the energy with respect to these descriptors. In addition to providing a review of the methodology, work here identifies the need for an algorithm which thoroughly searches over initial configurations. The strategy for sampling and prioritizing initial configurations is described. Applications on coordination complexes are presented. The FO descriptors and FOs for semi-classical and quantum-mechanical understanding of bondingis discussed. Cohesive energies are improved andthe eigenvalues are shifted downward relative to the standard DFT results.Spin-dependent vibrational spectra, as a possible means for spectroscopic determination of the transition-metal moment, are also presented. [1]Pederson et al, JCP,140, 121103 (2014). [2]Baruah {\&} Pederson, AAMOPS, 64, 153-180 (2015).   

Fermi orbital self-interaction corrected electronic structure of molecules beyond local density approximation

The correction of the self-interaction error that is inherent to all standard density functional theory (DFT) calculations is an object of increasing interest. We present our results on the application of the recently developed Fermi-orbital based approach for the self-interaction correction (FO-SIC) to a set of different molecular systems [1,2]. Our study covers systems ranging from simple diatomic to large organic molecules. Our focus lies on the direct estimation of the ionization potential from orbital eigenvalues and on the ordering of electronic levels in metal-organic molecules. Further, we show that the Fermi orbital positions in structurally similar molecules appear to be transferable. [1] M. R. Pederson, A. Ruzsinszky, and J. P. Perdew, J. Chem. Phys. 140, 121103 (2014). [2] M. R. Pederson, J. Chem. Phys. 142, 064112 (2015).   

Magnetic Exchange Couplings in Transition Metal Complexes from DFT

In this talk I will review our current efforts for the evaluation of magnetic exchange couplings in transition metal complexes from density functional theory. I will focus on the performance of different DFT approximations, including a variety of hybrid density functionals, and show that hybrid density functionals containing approximately 30{\%} Hartree-Fock type exchange are in general among the best choice in terms of accuracy. I will also describe a novel computational method to evaluate exchange coupling parameters using analytic self-consistent linear response theory. This method avoids the explicit evaluation of energy differences, which can become impractical for large systems. Our approach is based on the evaluation of the transversal magnetic torque between two magnetic centers for a given spin configuration using explicit constraints of the local magnetization direction \textit{via} Lagrange multipliers. This method is applicable in combination with any modern density functional with a noncollinear spin generalization and can be utilized as a ``black-box''. I will show proof-of-concept calculations in frustrated Fe$^{\mathrm{III}}_{\mathrm{7}}$ disk-shaped clusters, and dinuclear Cu$^{\mathrm{II}}$, Fe$^{\mathrm{III}}$, and heteronuclear complexes.   

Local spin analyses using density functional theory

Local spin analysis is a valuable technique in computational investigations magnetic interactions on mono- and polynuclear transition metal complexes, which play vital roles in catalysis, molecular magnetism, artificial photosynthesis, and several other commercially important materials. The relative size and complex electronic structure of transition metal complexes often prohibits the use of multi-determinant approaches, and hence, practical calculations are often limited to single-determinant methods. Density functional theory (DFT) has become one of the most successful and widely used computational tools for the electronic structure study of complex chemical systems; transition metal complexes in particular. Within the DFT formalism, a more flexible and complete theoretical modeling of transition metal complexes can be achieved by considering noncollinear spins, in which the spin density is 'allowed to' adopt noncollinear structures in stead of being constrained to align parallel/antiparallel to a universal axis of magnetization. In this meeting, I will present local spin analyses results obtained using different DFT functionals. Local projection operators are used to decompose the expectation value $$ of the total spin operator; first introduced by Clark and Davidson.   

The Lieb-Oxfourd bound and the exchange-correlation kernel from the strictly-correlated electrons functional

I will present some recent results based on the strictly-correlated electrons (SCE) functional: 1) a rigorous method to set lower bounds to the optimal particle-number dependent constant appearing in the Lieb-Oxford bound, and 2) an investigation of exact properties in the time domain, including an analytical expression for the kernel in one-dimension, with an analysis of its behavior for the case of bond-breaking excitations.   

Exchange-correlation functionals from a local interpolation along the adiabatic connection

We use the adiabatic connection formalism to construct a density functional by doing an interpolation between the weak and the strong coupling regime. Combining the information from the two limits, we are able to construct an exchange-correlation (xc) density functional free of the bias towards weakly correlated system, which is present in the majority of approximate xc functionals. Previous attempts in doing the interpolation between the two regimes, such as the interaction strength interpolation (ISI), had a fundamental flaw: the lack of size-consistency, as the corresponding functional depends non-linearly on the global (integrated over all space) ingredients. To recover size-consistency in such a framework, we move from the global to local quantities. We use the energy densities as local quantities in the gauge of the electrostatic potential of the xc hole. We use the ``strictly-correlated electrons'' (SCE) approach to compute the energy densities in the strong-coupling limit and the Lieb maximization algorithm to extract the energy densities from the low-coupling regime. We then test the accuracy of the local interpolation schemes by using the nearly exact local energy densities. In this talk I am going to present our results with the emphasis on strongly correlated systems.   

The exact density functional for two electrons in one dimension

The exact universal density functional $F[\rho]$ is calculated for real space two-electron densities in one dimension $\rho(x)$ with a soft-Coulomb interaction. It is calculated by the Levy constrained search $F[\rho]=\min_{\Psi\rightarrow\rho}\langle\Psi|\hat{T}+\hat{V}_{ee}|\Psi\rangle$ over wavefunctions of a two-dimensional Hilbert space $\Psi(x_{1},x_{2})\rightarrow\rho(x_{1})$ and can be directly visualized. We do an approximate constrained search via density matrices and a direct approximation to natural orbitals. This allows us to make an accurate approximation to the exact functional that is calculated using a search over potentials. We investigate the exact functional and the performance of many approximations on some of the most challenging electronic structure in two-electron systems, from strongly-correlated electron transfer to the description of a localized-delocalized transition. The exact Kohn-Sham potential, $v_s(x)$, and exact Kohn-Sham eigenvalues, $\epsilon_i$, are calculated and this allows us to discuss the band-gap problem versus the perspective of the exact density functional $F[\rho]$ for all numbers of electrons. We calculate the derivative discontinuity of the exact functional in an example of a Mott-Insulator, one-dimensional stretched H$_2$.   

Landscape of the exact energy functional for a simplified universe

One of the great challenges of electronic structure theory is the quest for the exact functional of density functional theory (DFT). Its existence is proven, but it is a complicated multivariable functional that is almost impossible to conceptualize. In this talk we study the asymmetric two-site Hubbard model because it has only a two-dimensional universe of density matrices, hence the exact functional becomes a simple function of two variables whose three dimensional energy landscape can be visualized and explored. A walk on this unique landscape, tilted to an angle defined by the one-electron Hamiltonian, gives a valley whose minimum is the exact total energy. This is contrasted with the landscape of some approximate functionals, explaining their failure for electron transfer in the strongly correlated limit. We show concrete examples of pure-state density matrices that are not $v$-representable due to the underlying non-convex nature of the energy landscape. The exact functional is calculated for all numbers of electrons, including fractional, allowing the derivative discontinuity to be visualized and understood. The fundamental gap for all possible systems is obtained solely from the derivatives of the exact functional.   

Spontaneous charge carrier localization in extended one-dimensional systems

Charge carrier localization in extended atomic systems can be driven by disorder, point defects or distortions of the ionic lattice. Herein we give first-principles theoretical computational evidence that it can also appear as a purely electronic effect in otherwise perfectly ordered periodic structures and we show that electronic eigenstates can spontaneously localize upon excitation. Optimally-tuned range separated density functional calculations reveal that in trans-polyacetylene and polythiophene the hole density localizes on a length scale of several nanometers. This is due to exchange induced translational symmetry breaking of the charge density. Ionization potentials, optical absorption peaks, excitonic binding energies and the optimally-tuned range parameter itself all become independent of polymer length when it exceeds the critical localization length scale. These first-principles findings show, for the first time, that charge localization is not caused by lattice distortion but rather it is their cause, changing the physical models of polaron formation and dynamics, helping to explain experimental findings that polarons in conjugated polymers form instantaneously after exposure to ultrafast light pulses.   

Self-consistent calculation of Hubbard U parameters within linear-scaling DFT

DFT+U has proven to be a computationally efficient method for correcting for the underestimation of electron localization effects, or for the absent derivative discontinuity, inherent in conventional density functionals. Invoking an approximate interpretation of DFT+U as a corrective penalty functional for the spurious curvature of the total-energy with respect to subspace occupancy, the Hubbard U parameter may be calculated [1,2], in which case DFT+U may be considered to be fully first-principles approach. We describe our approach for computing the Hubbard U and Hund’s J parameters within ONETEP, a linear-scaling DFT code which comprises a complete DFT+U+J [3] implementation including ionic forces and a flexible choice of population analyses [4,5]. We discuss issues of charge preservation and self-consistency, and we demonstrate the capability of our method by means of numerical tests on the ground-state properties of selected molecules that present challenges for approximate DFT. [1] W. E. Pickett, et al., Phys. Rev. B, 58, 1201 (1998). [2] H. J. Kulik, et al., Phys. Rev. Lett. 97, 103001 (2006). [3] B. Himmetoglu, et al., Phys. Rev. B 84, 115108 (2011). [4] D. D. O’Regan, et al., Phys. Rev. B 85, 085107 (2012). [5] D. D. O’Regan, et al., Phys. Rev. B 83, 245124 (2011).