Symmetric Nonlocal Weighted Density Approximations from the Exchange-Correlation Hole of the Uniform Electron Gas

Tilted-Plane Structure of the Energy of Finite Quantum Systems

The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related to static correlation error, in approximate density functional theory. The magnetic analog of Koopmans' theorem in density functional theory is also derived. Moving to fractional electron count, the tilted-plane condition is derived, lifting certain assumptions in previous works. This generalization of the flat-plane condition characterizes the total energy surface of a finite system for all values of electron count N and magnetization M. This result is used in combination with tabulated spectroscopic data to show the flat-plane structure of the oxygen atom, among others. We find that derivative discontinuities with respect to electron count sometimes occur at noninteger values. A diverse set of tilted-plane structures is shown to occur in d-orbital subspaces, depending on chemical coordination. General occupancy-based total-energy expressions are demonstrated thereby to be necessarily dependent on the symmetry-imposed degeneracies.

Spin-Polarized Conceptual Density Functional Theory from the Convex Hull

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., {N, S} vs {Nα, Nβ}) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions {P+(r), P-(r)} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Computational investigation of structural, electronic, and spectroscopic properties of Ni and Zn metalloporphyrins with varying anchoring groups

Porphyrins are prime candidates for a host of molecular electronics applications. Understanding the electronic structure and the role of anchoring groups on porphyrins is a prerequisite for researchers to comprehend their role in molecular devices at the molecular junction interface. Here, we use the density functional theory approach to investigate the influence of anchoring groups on Ni and Zn diphenylporphyrin molecules. The changes in geometry, electronic structure, and electronic descriptors were evaluated. There are minimal changes observed in geometry when changing the metal from Ni to Zn and the anchoring group. However, we find that the distribution of electron density changes when changing the anchoring group in the highest occupied and lowest unoccupied molecular orbitals. This has a direct effect on electronic descriptors such as global hardness, softness, and electrophilicity. Additionally, the optical spectra of both Ni and Zn diphenylporphyrin molecules exhibit either blue or red shifts when changing the anchoring group. These results indicate the importance of the anchoring group on the electronic structure and optical properties of porphyrin molecules.

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized

Description of many-electron systems with a fractional electron number (Ntot) and fractional spin (Mtot) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both Ntot and Mtot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant Ntot, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.

The factorization ansatz for non-local approximations to the exchange-correlation hole

Among the various types of approximations to the exchange-correlation energy (E_XC), the completely non-local approach is one of the lesser explored approximation schemes. It has not yet reached the predictive power of the widely used generalized gradient approximations, meta-generalized gradient approximations, hybrids, etc. In non-local functionals pursued here, the electron density at every point in space is employed to express the exchange-correlation energy per particle ϵ_XC(r) at a given position r. Here, we use the non-local, spherical-averaged density ρ(r,u)=∫dΩ_u4πρ(r+u) as a starting point to construct approximate exchange-correlation holes through the factorization ansatz ρ_XC(r, u) = f(r, u)ρ(r, u). We present upper and lower bounds to the exchange energy per particle ϵ_X(r) in terms of ρ(r, u). The factor f(r, u) is then designed to satisfy various conditions that represent important exchange and correlation effects. We assess the resulting approximations and find that the complex, oscillatory structure of ρ(r, u) makes the construction of a corresponding f(r, u) very challenging. This conclusion, identifying the main issue of the non-local approximation, is supported by a detailed analysis of the resulting exchange-correlation holes.

Perspective: Chemical Information Encoded in Electron Density

In this perspective, we review the chemical information encoded in electron density and other ingredients used in semilocal functionals. This information is usually looked at from the functional point of view: the exchange density or the enhancement factor are discussed in terms of the reduced density gradient. However, what parts of a molecule do these 3D functions represent? We look at these quantities in real space, aiming to understand the electronic structure information they encode and provide an insight from the quantum chemical topology (QCT). Generalized gradient approximations (GGAs) provide information about the presence of chemical interactions, whereas meta-GGAs can differentiate between the different bonding types. By merging these two techniques, we show new insight into the failures of semilocal functionals owing to three main errors: fractional charges, fractional spins, and non-covalent interactions. We build on simple models. We also analyze the delocalization error in hydrogen chains, showing the ability of QCT to reveal the delocalization error introduced by semilocal functionals. Then, we show how the analysis of localization can help understand the fractional spin error in alkali atoms, and how it can be used to correct it. Finally, we show that the poor description of GGAs of isodesmic reactions in alkanes is due to 1,3-interactions.

A Diagonally Updated Limited-Memory Quasi-Newton Method for the Weighted Density Approximation

We propose a limited-memory quasi-Newton method using the bad Broyden update and apply it to the nonlinear equations that must be solved to determine the effective Fermi momentum in the weighted density approximation for the exchange energy density functional. This algorithm has advantages for nonlinear systems of equations with diagonally dominant Jacobians, because it is easy to generalize the method to allow for periodic updates of the diagonal of the Jacobian. Systematic tests of the method for atoms show that one can determine the effective Fermi momentum at thousands of points in less than fifteen iterations.

A new exchange–correlation functional free of delocalization and static correlation errors

A combination of weighted density approximation and classical mapping leads to a new exchange–correlation energy free of delocalization and static correlation errors in Kohn–Sham density functional theory.