Perdew Festschrift editorial.

This Special Issue of the Journal of Chemical Physics is dedicated to the work and life of John P. Perdew. A short bio is available within the issue [J. P. Perdew, J. Chem. Phys. 160, 010402 (2024)]. Here, we briefly summarize key publications in density functional theory by Perdew and his collaborators, followed by a structured guide to the papers contributed to this Special Issue.

Investigations of the exchange energy of neutral atoms in the large-Z limit.

The non-relativistic large-Z expansion of the exchange energy of neutral atoms provides an important input to modern non-empirical density functional approximations. Recent works report results of fitting the terms beyond the dominant term, given by the local density approximation (LDA), leading to an anomalous Z ln Z term that cannot be predicted from naïve scaling arguments. Here, we provide much more detailed data analysis of the mostly smooth asymptotic trend describing the difference between exact and LDA exchange energy, the nature of oscillations across rows of the Periodic Table, and the behavior of the LDA contribution itself. Special emphasis is given to the successes and difficulties in reproducing the exchange energy and its asymptotics with existing density functional approximations.

The difference between molecules and materials: Reassessing the role of exact conditions in density functional theory.

Exact conditions have long been used to guide the construction of density functional approximations. However, hundreds of empirical-based approximations tailored for chemistry are in use, of which many neglect these conditions in their design. We analyze well-known conditions and revive several obscure ones. Two crucial distinctions are drawn: that between necessary and sufficient conditions and that between all electronic densities and the subset of realistic Coulombic ground states. Simple search algorithms find that many empirical approximations satisfy many exact conditions for realistic densities and non-empirical approximations satisfy even more conditions than those enforced in their construction. The role of exact conditions in developing approximations is revisited.

Density-Corrected Density Functional Theory for Open Shells: How to Deal with Spin Contamination.

Density functional theory (DFT) is usually used self-consistently to predict chemical properties, but the use of the Hartree-Fock (HF) density improves energetics in certain, well-characterized cases. Density-corrected (DC) DFT provides the theory behind this, but unrestricted Hartree-Fock (UHF) densities yield poor energetics in cases of strong spin contamination. Here we compare with restricted open-shell HF (ROHF) across 13 different functionals and two DC-DFT methods. For significant spin contamination, ROHF densities outperform UHF densities by as much as a factor of 3, depending on the energy functional, and ROHF-DFT improves over self-consistent DFT for most of the tested functionals. We refine the DC(HF)-DFT algorithm to use ROHF densities in cases of severe spin contamination.

Extending density functional theory with near chemical accuracy beyond pure water.

Density functional simulations of condensed phase water are typically inaccurate, due to the inaccuracies of approximate functionals. A recent breakthrough showed that the SCAN approximation can yield chemical accuracy for pure water in all its phases, but only when its density is corrected. This is a crucial step toward first-principles biosimulations. However, weak dispersion forces are ubiquitous and play a key role in noncovalent interactions among biomolecules, but are not included in the new approach. Moreover, naïve inclusion of dispersion in HF-SCAN ruins its high accuracy for pure water. Here we show that systematic application of the principles of density-corrected DFT yields a functional (HF-r2SCAN-DC4) which recovers and not only improves over HF-SCAN for pure water, but also captures vital noncovalent interactions in biomolecules, making it suitable for simulations of solutions.

Leading Correction to the Local Density Approximation for Exchange in Large-Z Atoms.

The large-Z asymptotic expansion of atomic energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. The correction for exchange is fit well with a leading ZlnZ term, and we find its coefficient numerically. The gradient expansion approximation also has such a term, but with a smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) lead to the conjecture that the coefficients are precisely 2.7 times larger than their gradient expansion counterparts, yielding an analytic expression for the exchange-energy correction which is accurate to ∼5% for all Z.

Improving Results by Improving Densities: Density-Corrected Density Functional Theory.

Density functional theory (DFT) calculations have become widespread in both chemistry and materials, because they usually provide useful accuracy at much lower computational cost than wavefunction-based methods. All practical DFT calculations require an approximation to the unknown exchange-correlation energy, which is then used self-consistently in the Kohn-Sham scheme to produce an approximate energy from an approximate density. Density-corrected DFT is simply the study of the relative contributions to the total energy error. In the vast majority of DFT calculations, the error due to the approximate density is negligible. But with certain classes of functionals applied to certain classes of problems, the density error is sufficiently large as to contribute to the energy noticeably, and its removal leads to much better results. These problems include reaction barriers, torsional barriers involving π-conjugation, halogen bonds, radicals and anions, most stretched bonds, etc. In all such cases, use of a more accurate density significantly improves performance, and often the simple expedient of using the Hartree-Fock density is enough. This Perspective explains what DC-DFT is, where it is likely to improve results, and how DC-DFT can produce more accurate functionals. We also outline challenges and prospects for the field.

How Well Does Kohn-Sham Regularizer Work for Weakly Correlated Systems?

Kohn-Sham regularizer (KSR) is a differentiable machine learning approach to finding the exchange-correlation functional in Kohn-Sham density functional theory that works for strongly correlated systems. Here we test KSR for a weak correlation. We propose spin-adapted KSR (sKSR) with trainable local, semilocal, and nonlocal approximations found by minimizing density and total energy loss. We assess the atoms-to-molecules generalizability by training on one-dimensional (1D) H, He, Li, Be, and Be2+ and testing on 1D hydrogen chains, LiH, BeH2, and helium hydride complexes. The generalization error from our semilocal approximation is comparable to other differentiable approaches, but our nonlocal functional outperforms any existing machine learning functionals, predicting ground-state energies of test systems with a mean absolute error of 2.7 mH.

Density-Corrected DFT Explained: Questions and Answers.

HF-DFT, the practice of evaluating approximate density functionals on Hartree-Fock densities, has long been used in testing density functional approximations. Density-corrected DFT (DC-DFT) is a general theoretical framework for identifying failures of density functional approximations by separating errors in a functional from errors in its self-consistent (SC) density. Most modern DFT calculations yield highly accurate densities, but important characteristic classes of calculation have large density-driven errors, including reaction barrier heights, electron affinities, radicals and anions in solution, dissociation of heterodimers, and even some torsional barriers. Here, the HF density (if not spin-contaminated) usually yields more accurate and consistent energies than those of the SC density. We use the term DC(HF)-DFT to indicate DC-DFT using HF densities only in such cases. A recent comprehensive study (J. Chem. Theory Comput. 2021, 17, 1368-1379) of HF-DFT led to many unfavorable conclusions. A reanalysis using DC-DFT shows that DC(HF)-DFT substantially improves DFT results precisely when SC densities are flawed.

Explaining and Fixing DFT Failures for Torsional Barriers.

Most torsional barriers are predicted with high accuracies (about 1 kJ/mol) by standard semilocal functionals, but a small subset was found to have much larger errors. We created a database of almost 300 carbon-carbon torsional barriers, including 12 poorly behaved barriers, that stem from the Y═C-X group, where Y is O or S and X is a halide. Functionals with enhanced exchange mixing (about 50%) worked well for all barriers. We found that poor actors have delocalization errors caused by hyperconjugation. These problematic calculations are density-sensitive (i.e., DFT predictions change noticeably with the density), and using HF densities (HF-DFT) fixes these issues. For example, conventional B3LYP performs as accurately as exchange-enhanced functionals if the HF density is used. For long-chain conjugated molecules, HF-DFT can be much better than exchange-enhanced functionals. We suggest that HF-PBE0 has the best overall performance.

Learning to Approximate Density Functionals.

Density functional theory (DFT) calculations are used in over 40,000 scientific papers each year, in chemistry, materials science, and far beyond. DFT is extremely useful because it is computationally much less expensive than ab initio electronic structure methods and allows systems of considerably larger size to be treated. However, the accuracy of any Kohn-Sham DFT calculation is limited by the approximation chosen for the exchange-correlation (XC) energy. For more than half a century, humans have developed the art of such approximations, using general principles, empirical data, or a combination of both, typically yielding useful results, but with errors well above the chemical accuracy limit (1 kcal/mol). Over the last 15 years, machine learning (ML) has made major breakthroughs in many applications and is now being applied to electronic structure calculations. This recent rise of ML begs the question: Can ML propose or improve density functional approximations? Success could greatly enhance the accuracy and usefulness of DFT calculations without increasing the cost.In this work, we detail efforts in this direction, beginning with an elementary proof of principle from 2012, namely, finding the kinetic energy of several Fermions in a box using kernel ridge regression. This is an example of orbital-free DFT, for which a successful general-purpose scheme could make even DFT calculations run much faster. We trace the development of that work to state-of-the-art molecular dynamics simulations of resorcinol with chemical accuracy. By training on ab initio examples, one bypasses the need to find the XC functional explicitly. We also discuss how the exchange-correlation energy itself can be modeled with such methods, especially for strongly correlated materials. Finally, we show how deep neural networks with differentiable programming can be used to construct accurate density functionals from very few data points by using the Kohn-Sham equations themselves as a regularizer. All these cases show that ML can create approximations of greater accuracy than humans, and is capable of finding approximations that can deal with difficult cases such as strong correlation. However, such ML-designed functionals have not been implemented in standard codes because of one last great challenge: generalization. We discuss how effortlessly human-designed functionals can be applied to a wide range of situations, and how difficult that is for ML.

Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics.

Including prior knowledge is important for effective machine learning models in physics and is usually achieved by explicitly adding loss terms or constraints on model architectures. Prior knowledge embedded in the physics computation itself rarely draws attention. We show that solving the Kohn-Sham equations when training neural networks for the exchange-correlation functional provides an implicit regularization that greatly improves generalization. Two separations suffice for learning the entire one-dimensional H_{2} dissociation curve within chemical accuracy, including the strongly correlated region. Our models also generalize to unseen types of molecules and overcome self-interaction error.

Density Sensitivity of Empirical Functionals.

Empirical fitting of parameters in approximate density functionals is common. Such fits conflate errors in the self-consistent density with errors in the energy functional, but density-corrected DFT (DC-DFT) separates these two. We illustrate with catastrophic failures of a toy functional applied to H2+ at varying bond lengths, where the standard fitting procedure misses the exact functional; Grimme's D3 fit to noncovalent interactions, which can be contaminated by large density errors such as in the WATER27 and B30 data sets; and double-hybrids trained on self-consistent densities, which can perform poorly on systems with density-driven errors. In these cases, more accurate results are found at no additional cost by using Hartree-Fock (HF) densities instead of self-consistent densities. For binding energies of small water clusters, errors are greatly reduced. Range-separated hybrids with 100% HF at large distances suffer much less from this effect.

Quantifying and Understanding Errors in Molecular Geometries.

Electronic structure calculations are ubiquitous in most branches of chemistry, but all have errors in both energies and equilibrium geometries. Quantifying errors in possibly dozens of bond angles and bond lengths is a Herculean task. A single natural measure of geometric error is introduced, the geometry energy offset (GEO). GEO links many disparate aspects of geometry errors: a new ranking of different methods, quantitative insight into errors in specific geometric parameters, and insight into trends with different methods. GEO can also reduce the cost of high-level geometry optimizations and shows when geometric errors distort the overall error of a method. Results, including some surprises, are given for both covalent and weak interactions.

Confirmation of the PPLB Derivative Discontinuity: Exact Chemical Potential at Finite Temperatures of a Model System.

The landmark 1982 work of Perdew, Parr, Levy, and Balduz (often called PPLB) laid the foundation for our modern understanding of the role of the derivative discontinuity in density functional theory, which drives much development to account for its effects. A simple model for the chemical potential at vanishing temperature played a crucial role in their argument. We investigate the validity of this model in the simplest nontrivial system to which it can be applied and which can be easily solved exactly, the Hubbard dimer. We find exact agreement in the crucial zero-temperature limit and show the model remains accurate for a significant range of temperatures. We identify how this range depends on the strength of correlations. We extend the model to approximate free energies accounting for the derivative discontinuity, a feature missing in standard semilocal approximations. We provide a correction to this approximation to yield even more accurate free energies. We discuss the relevance of these results for warm dense matter.

Quantum chemical accuracy from density functional approximations via machine learning.

Kohn-Sham density functional theory (DFT) is a standard tool in most branches of chemistry, but accuracies for many molecules are limited to 2-3 kcal â‹… mol-1 with presently-available functionals. Ab initio methods, such as coupled-cluster, routinely produce much higher accuracy, but computational costs limit their application to small molecules. In this paper, we leverage machine learning to calculate coupled-cluster energies from DFT densities, reaching quantum chemical accuracy (errors below 1 kcal â‹… mol-1) on test data. Moreover, density-based Δ-learning (learning only the correction to a standard DFT calculation, termed Δ-DFT ) significantly reduces the amount of training data required, particularly when molecular symmetries are included. The robustness of Δ-DFT  is highlighted by correcting "on the fly" DFT-based molecular dynamics (MD) simulations of resorcinol (C6H4(OH)2) to obtain MD trajectories with coupled-cluster accuracy. We conclude, therefore, that Δ-DFT  facilitates running gas-phase MD simulations with quantum chemical accuracy, even for strained geometries and conformer changes where standard DFT fails.