The first order atomic fragment approach-An orbital-free implementation of density functional theory

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

Molecular Bonding in an Orbital-Free-Related Density Functional Theory

A density functional theory based on polymer self-consistent field theory is applied to systems of two atoms in order to show that this approach is capable of predicted molecular bonding. Periodic table elements from hydrogen up to neon are examined and homonuclear diatomic molecules are found to form for H₂, N₂, O₂, and F₂, in agreement with known results. The heteronuclear molecules CO and HF, which are known to exist under ambient conditions, are also found to be stable. Bond lengths for most of these molecules agree with experimental results to within less than 8%, with the exception of O₂ and F₂ which deviate more significantly. The bonding energy for H₂ is given and is within 16% of the known value, but fundamental vibrational frequencies do not agree well with experiment. The main approximations of the theory are very simple and include a Fermi-Amaldi correction to the electron-electron interaction to account for self-interactions and a basic expression for the Pauli potential to account for the exclusion principle. The self-consistent equations are solved in terms of basis functions that encode the cylindrical symmetry of diatomic molecules. Since orbitals are not used, the approach is related to orbital-free density functional theory.

Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers

This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N₂, C₂, and B₂, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.

Analysis of atomic Pauli potentials and their large-Z limit

Modeling the Pauli energy, the contribution to the kinetic energy caused by Pauli statistics, without using orbitals is the open problem of orbital-free density functional theory. An important aspect of this problem is correctly reproducing the Pauli potential, the response of the Pauli kinetic energy to a change in density. We analyze the behavior of the Pauli potential of non-relativistic neutral atoms under Lieb-Simon scaling-the process of taking nuclear charge and particle number to infinity, in which the kinetic energy tends to the Thomas-Fermi limit. We do this by mathematical analysis of the near-nuclear region and by calculating the exact orbital-dependent Pauli potential using the approach of Levy and Ouyang for closed-shell atoms out to element Z = 976. In rough analogy to Lieb and Simon's own findings for the charge density, we find that the potential does not converge smoothly to the Thomas-Fermi limit on a point-by-point basis but separates into several distinct regions of behavior. Near the nucleus, the potential approaches a constant given by the difference in energy between the lowest and highest occupied eigenvalues. We discover a transition region in the outer core where the potential deviates unexpectedly and predictably from both the Thomas-Fermi potential and the gradient expansion correction to it. These results may provide insight into the semi-classical description of Pauli statistics and new constraints to aid the improvement of orbital-free density functional theory functionals.

Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory

This work presents a method to move beyond the recently introduced atomic fragment approximation. Like the bare atomic fragment approach, the new method is an ab initio, parameter-free, orbital-free implementation of density functional theory based on the bifunctional formalism that treats the potential and the electron density as two separate variables, and provides access to the Kohn–Sham Pauli kinetic energy for an appropriately chosen Pauli potential. In the present ansatz, the molecular Pauli potential is approximated by the sum of the bare atomic fragment approach, and a so-called deformation potential that takes the interaction between the atoms into account. It is shown that this model can reproduce the bond-length contraction due to multiple bonding within the list of second-row homonuclear dimers. The present model only relies on the electron densities of the participating atoms, which themselves are represented by a simple monopole expansion. Thus, the bond-length contraction can be rationalized without referring to the angular quantum numbers of the participating atoms.

Artificial neural networks for the kinetic energy functional of non-interacting fermions

A novel approach to find the fermionic non-interacting kinetic energy functional with chemical accuracy using machine learning techniques is presented. To that extent, we apply machine learning to an intermediate quantity rather than targeting the kinetic energy directly. We demonstrate the performance of the method for three model systems containing three and four electrons. The resulting kinetic energy functional remarkably accurately reproduces self-consistently the ground state electron density and total energy of reference Kohn-Sham calculations with an error of less than 5 mHa. This development opens a new avenue to advance orbital-free density functional theory by means of machine learning.

Equilibrium Bond Lengths from Orbital-Free Density Functional Theory

This work presents an investigation to model chemical bonding in various dimers based on the atomic fragment approach. The atomic fragment approach is an ab-initio, parameter-free implementation of orbital-free density functional theory which is based on the bifunctional formalism, i.e., it uses both the density and the Pauli potential as two separate variables. While providing the exact Kohn-Sham Pauli kinetic energy when the orbital-based Kohn-Sham data are used, the bifunctional formalism allows for approximations of the functional derivative which are orbital-free. In its first implementation, the atomic fragment approach uses atoms in their ground state to model the Pauli potential. Here, it is tested how artificial closed-shell fragments with non-integer electron occupation perform regarding the prediction of bond lengths of diatomics. Such fragments can sometimes mimic the electronic structure of a molecule better than groundstate fragments. It is found that bond lengths may indeed be considerably improved in some of the tested diatomics, in accord with predictions based on the electronic structure.