Session A39: Density Functional Theory in Chemical Physics: Ground State DFT, Density-corrected DFT and Concepts of DFT

Unconventional Error Cancellation Explains the Success of Hartree-Fock Density Functional Theory for Barrier Heights

The barrier height controls the rate of a chemical reaction. Barrier heights are seriously underestimated by computationally-efficient semi-local density functional approximations to the exchange-correlation energy. For reaction barrier energies, the accuracy of a semi-local density functional approximation is boosted strongly by evaluating that approximation on Hartree-Fock electron densities, and not self-consistently, as known since 1992. The conventional explanation is that Hartree-Fock theory yields the more accurate density. This article presents a benchmark Kohn-Sham inversion of accurate coupled-cluster densities for the reaction H₂ + F → HHF → H + HF. We find a strong, understandable cancellation between large positive (excessively over-corrected) density-driven and large negative functional-driven errors within this Hartree-Fock density functional theory.   This work supports the conclusions reached earlier [1] for 76 barrier heights using three less reliable but useful fully-nonlocal proxies for the exact density.: the SCAN50 global hybrid, the LCωPBE long-range-corrected hybrid, and the FLOSIC self-interaction correction to the SCAN meta-generalized gradient approximation.

 [1] A.D. Kaplan, C. Shahi, R.K. Sah, P. Bhetwal, and J.P. Perdew, Understanding Density-Driven Errors for Reaction Barrier Heights, Journal of Chemical Theory and Computation 19, 532 (2023).

 

    

Density-Corrected Many-body Representations in Aqueous Phase Chemistry

Kohn-Sham density functional theory (DFT) provides an unsatisfactory description of molecular interactions in solution, where local and non-local many-body (MB) effects compete. Density-corrected DFT (DC-DFT) has resurfaced as a practical approach for accurate energies of weakly interacting systems, often misrepresented by DFT. Herein, we discuss density-corrected molecular simulation, focusing on water and ion hydration, within a general framework that combines the ansatz of the many-body expansion with DC-DFT. We show that this framework, MB-DFT(DC), accurately describes molecular interactions in aqueous systems from the dimer to the condensed phase. We analyze the individual and collective effects of density-driven errors on the description of liquid water for several functionals, and identify when density-corrected potentials should be used. We explore the predictive capability of DC-SCAN and the MB-SCAN(DC) potential, showing that our density-corrected many-body approach predicts the N-body energies of hydrated ion clusters, discerning between ion--water/water--water interactions, with size-consistency and minimal loss of accuracy relative to coupled cluster. We provide insight from simulation into how density-correction translates to accurate descriptions of the structural properties of ions and water in the condensed phase, using the MB-SCAN(DC) potential. This connection between DC-DFT with a physics-based many-body treatment of molecular interactions, provides a promising path toward efficient DFT-based simulations with chemical accuracy.   

Exchange Functionals for Particles of Arbitrary Spin

The Hartree−Fock exchange energy formula for electrons is generalized to fermions of arbitrary spin quantum number s. The explicit s-dependence of the exchange energy reveals that the spin-scaling relation for exchange functionals can be viewed as conversion between spin-unpolarized forms for the same functional appropriate for particles with s = 0 and s > 0. This implies that any exchange functional of the electron density can be easily adapted to fermions of arbitrary spin by a simple change of certain constant factors. Implications of this finding for the self-interaction error problem and for multicomponent density-functional theory are pointed out.   

The coupling-constant-averaged exchange-correlation hole of spherical atoms calculated from the effective potential derived from the coordinate-scaling relation

Density functional theory's accuracy in assessing electronic structures hinges on the exchange-correlation (XC) approximation. Fundamentally, the XC hole, describing electron-electron interaction effects around a reference electron, offers a rigorous definition for the XC approximation. This gives a unique lens to gauge the quality of various XC approximations. To compute the XC hole, an external potential (v_λ), reliant on a coupling constant (λ), is essential to ensure consistent density across the adiabatic connection. Achieving this entails optimizing the general Lieb functional against v_λ over diverse λ values. When paired with the high-caliber coupled-cluster method at λ=1 as a benchmark, this technique is termed the Lieb+CCSD(T) method, which is computationally intensive. In our research, we utilized the coordinate scaling relation of the XC potential for approximating v_λ. Subsequently, XC holes for helium and specific second-period atoms were determined using different XC potentials and juxtaposed against the precise Lieb+CCSD(T) results. Our study paves the way for using the coordinate-scaling-derived potential in systems where the Lieb+CCSD(T) approach is computationally prohibitive.   

A demonstration of the finite-temperature upside-down adiabatic connection with the asymmetric Hubbard dimer

The finite-temperature upside-down adiabatic connection is an integral expression for the exchange-correlation free energy, a quantity of interest in simulations of planetary cores and fusion experiments. This approach smoothly connects a thermal ensemble of strictly correlated electrons to one with realistic interaction strength, while requiring that the density of the system remain fixed across all interaction strengths. The asymmetric Hubbard dimer allows for an exact demonstration of finite-temperature adiabatic connections. The properties of the upside-down adiabatic connection in a variety of regimes will be presented using this exactly solvable model, along with discussion of their usefulness within the generalized thermal adiabatic connection approach.   

Correlated Orbital Theory: An alternative and complement to Kohn-Sham DFT

Coupled-cluster (CC) theory from ab initio quantum chemistry is known to provide the best answers for most computationally accessible  problems in electronic structure theory. Less appreciated, is that it is possible to develop a new correlated one-particle theory of chemistry, correlated orbital theory (COT), from CC/EOM considerations of the (ionization potential) IP-EOM-CC and (electron affinity) EA-EOM-CC. They provide a new one-particle, effective, frequency independent Hamiltonian whose occupied orbital eigenvalues are the principal IP’s for a molecule, and some of the unoccupied ones, are EA’s. This new model defines the correlated orbitals, that include the homo and lumo frontier MO’s whose eigenvalues are formally exact in the theory. Using this COT as the framework for an effective one-particle theory, one can now appeal to KS-DFT to replace the purely ab initio COT equations in application by using standard  DFT reference density gradient expressions for the exchange-correlation potential and functional, as long as the choice fulfils the requirements of the underlying COT. This route, using CAM-B3LYP as the underlying functional meant to represent COT, though practically any of them could be chosen as long as there is sufficient flexibility to guarantee our IP eigenvalue condition is adequately satisfied for the KS-DFT eigenvalues. Our objective is to create an easily applied DFT-like procedure that can successfully address the ‘Devil’s Triangle of KS DFT.”   

Optimization of kinetic energy functionals for deorbitalized exchange-correlation meta-GGAs.

Deorbitalized meta-GGA functionals for the exchange-correlation energy [1,2] have shown promise as a cost-effective alternative to conventional meta-GGAs.  These replace the noninteracting kinetic energy (KE) density used in a meta-GGA with a pure functional of the density.  The Perdew-Constantin and related KE functionals (Phys. Rev. B 75, 155109 (2007)) use the Laplacian of the density to switch from the slowly varying electron gas to the localized electron-pair limits, resulting in an accurate orbital-free model for the KE density.  However, use of the density Laplacian can create unphysical Pauli and exchange-correlation potentials that are numerically slow to converge and lead to noisy results.  We construct a measure of the noisiness of functional potentials and discuss an optimization procedure for minimizing fluctuations in the potential based on the Poisson equation, used to generate modifications of the PC and related functionals.  We find that modifications of the RPP KE functional [2] show particular promise for efficient use in meta-GGAs as measured by number of self-consistent cycles and time-per-cycle.

[1] D. Mejia-Rodriguez and S. B. Trickey Phys. Rev. A 96, 052512 (2017)

[2] A Kaplan and J Perdew Phys. Rev. Materials 6, 083803 (2022)   

Visualizing Orbital-Free Models of the Kinetic Energy Density in Solids

Meta-generalized gradient approximations (mGGAs) for the exchange-correlation (XC) energy in density functional theory (DFT) conventionally depend upon the Kohn-Sham kinetic energy density (KED). Use of the KED makes mGGAs more accurate than generalized gradient approximations (GGAs) but also more computationally expensive for applications such as ab initio molecular dynamics. Deorbitalizated mGGAs replace the KED with a pure density functional. 

Through visualization we explore how well the exact KED can be represented by a single KE mGGA functional dependent upon the scaled density, scaled density gradient and density Laplacian. We calculate the KE and electron density of representative solids with varying ionicity and atomic number using the ABINIT DFT plane-wave pseudopotential code.

For semiconductors we find a near-universal linear correlation with the density Laplacian and density gradient for regions outside the atomic bond, consistent with a modification of the second-order gradient expansion and devise a KE functional to fit these results. We finally explore how well this model extends to other classes of systems, including simple metals, where the gradient expansion should be reasonable, and transition metals.   

Analysis of Smooth and Oscillatory terms in the Large Z Exchange Expansion of Atoms

Lieb-Simon zeta-scaling, which describes the scaling of neutral closed-shell atoms as Z approaches infinity, has been used with marked success to provide theoretical constraints on density functionals and to help generate perturbative expansions with Z. It has long been shown explicitly that the LDA recovers the dominant term, of order Z^5/3, for the atomic exchange energy, but little else is known formally. To remedy this lack, we analyze OEP calculations for exchange for neutral atoms up to Z=978. Prior work shows consistent numerical and analytical evidence for an anomalous ZlnZ term as the leading order correction to LDA. [1] We report here an analysis of the LDA contribution which is complicated by a complex dependence on shell-structure. We find that it is characterized by a smooth Z^4/3 term and an oscillatory pattern with a two-row period. We propose an accurate model of the alkali earth column and qualitative model for all closed shell atoms as functions of the partial occupation of the two-row period. These results yield a stringent test for orbital free density functionals, and may also be applicable to characterize the oscillations that occur in the post-LDA data and thus be of help testing and developing functionals.

[1] Nathan Argaman, Jeremy Redd, Antonio C. Cancio, and Kieron Burke Phys. Rev. Lett. 129, 153001 (2022)   

Assessing the source of error in the Thomas-Fermi-von Weizsacker density functional

We investigate the source of error in the Thomas-Fermi-von Weizs"acker (TFW) density functional relative to Kohn-Sham density functional theory (DFT). In particular, through numerical studies on a range of materials, for a variety of crystal structures subject to strain and atomic displacements, we find that while the ground state electron density in TFW orbital-free DFT is close to the Kohn-Sham density, the corresponding energy deviates significantly from the Kohn-Sham value. We show that these differences are a consequence of the poor representation of the linear response within the TFW approximation for the electronic kinetic energy, confirming conjectures in the literature. In so doing, we find that the energy computed from a non-self-consistent Kohn-Sham calculation using the TFW electronic ground state density is in very good agreement with that obtained from the fully self-consistent Kohn-Sham solution.