arXiv subjects

Kieron Burke

Publications and source records attributed to Kieron Burke.

At least 19 recordsLinked to original sources

arXiv2026

Thermal PBE in warm dense matter: Does it matter and is it accurate?

Conditional probability density functional theory has recently been used to derive the temperature dependence of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) for the exchange-correlation (XC) free energy. We implement and systematically benchmark thermal PBE within Kohn-Sham density functional theory calculations of warm dense matter. Comparisons with the local density approximation (LDA) and PBE functionals, as well as thermal LDA, show that thermal PBE significantly improves the description of warm dense matter properties, including energies, forces, pressures, and electronic charge densities. In particular, thermal PBE exhibits close agreement with path integral Monte Carlo (PIMC) reference data at negligible additional computational cost. This work demonstrates the practical utility of thermal PBE as an accurate semilocal functional for simulations in the warm dense regime.

cond-mat.mtrl-sci↗

arXiv2026

Ensemble Time-Dependent Density Functional Theory

Time-dependent density functional theory (TDDFT) is a standard approach for calculating optical excitations of molecules and solids, while ensemble DFT (EDFT) is a promising alternative under development. We introduce ensemble TDDFT (ETDDFT), a practical theory that combines the two, generalizing both; we ensemble-generalize the Gross-Kohn equation and the exchange-correlation kernel of TDDFT, and generalize EDFT to time-dependent problems. We relate coordinate scaling to the adiabatic connection. The new theory provides multiple avenues for constructing and using approximations. We illustrate these on the 2-site Hubbard model. We connect our results to the more general case of non-perturbative time-dependence.

physics.chem-ph↗

arXiv2026

Conditional probability density functional theory for solids

A recently developed approach, conditional probability density functional theory (CP-DFT), yields direct access to the exchange-correlation hole of a system, an important correlation function that is not available from any standard DFT calculation. We present the first results for extended materials with periodic boundary conditions. We demonstrate that CP-DFT works on weakly correlated materials (Na, Si). When applied to the prototypical Kagome material $CsV_3Sb_5$, we find $d$-orbital correlations that are not captured by standard DFT. Such distribution leads to a positive finding probability between two separated electrons and an enhanced charge density wave signal, suggesting a useful approach for strongly correlated systems.

cond-mat.mtrl-sci↗

arXiv2026

Enhancing molecular dynamics with equivariant machine-learned densities

Machine-learning interatomic potentials (MLIPs) have enabled molecular dynamics at near ab initio accuracy, yet remain limited to energies and forces by construction, leaving electronic observables such as dipole moments and polarizabilities inaccessible. We introduce DenSNet, a density-first approach to machine-learned electronic structure that learns the Hohenberg--Kohn map from nuclear configurations to the ground-state electron density. Our approach employs an SE(3)-equivariant neural network to predict density coefficients of a flexible atom-centered Gaussian basis, combined with a $Δ$-learning strategy that uses superposed atomic densities as a prior to accelerate training. A second equivariant network then maps the predicted density to the total energy, providing a unified framework for molecular dynamics and electronic structure. We validate DenSNet on ethanol, ethanethiol, and resorcinol, where infrared spectra from machine-learned trajectories show excellent agreement with experimental gas-phase measurements. To test scalability, we train on polythiophene oligomers with 1--6 monomers and extrapolate to chains of up to 12 monomers, generating stable long-time trajectories whose infrared spectra agree with reference density functional theory calculations. Here, we show that reinstating the electron density as the central learned quantity opens a practical route to transferable prediction of spectroscopic and electronic observables in large-scale molecular simulations.

physics.chem-ph↗

arXiv2026

Approximate normalizations for approximate density functionals

It seems self-evident that a density functional calculation should be normalized to the number of electrons in the system. We present multiple examples where the accuracy of the approximate energy is improved (sometimes greatly) by violating this basic principle. In one dimension, we explicitly derive the appropriate correction to the normalization. Beyond one dimension, Weyl asymptotics for energy levels yield these corrections for any cavity. We include examples with Coulomb potentials and the exchange energy of atoms to illustrate relevance to realistic calculations.

physics.chem-ph↗

arXiv2025

Can machines learn density functionals? Past, present, and future of ML in DFT

Density functional theory has become the world's favorite electronic structure method, and is routinely applied to both materials and molecules. Here, we review recent attempts to use modern machine-learning to improve density functional approximations. Many different researchers have tried many different approaches, but some common themes and lessons have emerged. We discuss these trends and where they might bring us in the future.

physics.comp-ph↗

arXiv2024

Exchange-correlation energy from Green's functions

DFT calculations yield useful ground-state energies and densities, while Green's function techniques (such as $GW$) are mostly used to produce spectral functions. From the Galitskii-Migdal formula, we extract the exchange-correlation of DFT directly from a Green's function. This spectral representation provides an alternative to the fluctuation-dissipation theorem of DFT, identifying distinct single-particle and many-particle contributions. Results are illustrated on the uniform electron gas and the two-site Hubbard model.

physics.chem-ph↗

arXiv2024

Orbital-free functional with sub-milliHartree errors for slabs

Using principles of asymptotic analysis, we derive the exact leading corrections to the Thomas-Fermi kinetic energy approximation for Kohn-Sham electrons for slabs. This asymptotic expansion approximation includes crucial quantum oscillations missed by standard semilocal density functionals. Because these account for the derivative discontinuity, chemical accuracy is achieved at fourth-order. The implications for both orbital-free electronic structure and exchange-correlation approximations are discussed.

cond-mat.mtrl-sci↗

arXiv2023

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 ZlnZ term that can not be predicted from naive 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.

physics.chem-ph↗

arXiv2023

Generalized Gradient Approximation Made Thermal

Using the methodology of conditional-probability density functional theory, and several mild assumptions, we calculate the temperature-dependence of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA). This numerically-defined thermal GGA reduces to the local approximation in the uniform limit and PBE at zero temperature, and can be fit reasonably accurately (within 8%) assuming the temperature-dependent enhancement is independent of the gradient. This locally thermal PBE satisfies both the coordinate-scaled correlation inequality and the concavity condition, which we prove for finite temperatures. The temperature dependence differs markedly from existing thermal GGA's.

physics.chem-ph↗

arXiv2023

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. But hundreds of empirical-based approximations tailored for chemistry are in use, many of which 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 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.

physics.chem-ph↗

arXiv2023

DC-DFT for Open Shells: How to Deal with Spin Contamination

Density functional theory (DFT) is widely used to predict chemical properties, but its accuracy is limited by functional approximations and their approximate self-consistent densities. Density-corrected DFT (DC-DFT) is the study of the errors due to densities and Hartree-Fock DFT (HF-DFT) uses HF densities to improve energetics. With increasing use of HF-DFT, the question of how to address strong spin contamination in the HF calculation becomes increasingly important. We compare two different open-shell HF densities across 13 different DFT 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, recommending ROHF-DFT in cases of severe spin contamination.

physics.chem-ph↗

arXiv2023

Exact Conditions for Ensemble Density Functional Theory

Ensemble density functional theory (EDFT) is a promising alternative to time-dependent density functional theory for computing electronic excitation energies. Using coordinate scaling, we prove several fundamental exact conditions in EDFT and illustrate them on the exact singlet bi-ensemble of the Hubbard dimer. Several approximations violate these conditions, and some ground-state conditions from quantum chemistry do not generalize to EDFT. The strong-correlation limit is derived for the dimer, revealing weight-dependent derivative discontinuities in EDFT.

cond-mat.str-el↗

arXiv2023

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 $Z \ln Z$ 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 $\sim 5\%$ for all $Z$.

physics.chem-ph↗

arXiv2023

Seven Useful Questions in Density Functional Theory

We explore a variety of unsolved problems in density functional theory, where mathematicians might prove useful. We give the background and context of the different problems, and why progress toward resolving them would help those doing computations using density functional theory. Subjects covered include the magnitude of the kinetic energy in Hartree-Fock calculations, the shape of adiabatic connection curves, using the constrained search with input densities, densities of states, the semiclassical expansion of energies, the tightness of Lieb-Oxford bounds, and how we decide the accuracy of an approximate density.

math-ph↗

arXiv2023

Can the Hartree-Fock kinetic energy exceed the exact kinetic energy?

The Hartree-Fock (HF) approximation has been an important tool for quantum-chemical calculations since its earliest appearance in the late 1920s, and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy, but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke's atom (two electrons in a parabolic well) to test the conjecture in a non-trivial case, and to parameterize the difference between density-functional and HF quantities, but find no violations of the conjecture.

physics.chem-ph↗

arXiv2023

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. Systematic application of the principles of density-corrected DFT yields a functional (HF-r$^2$SCAN-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.

physics.chem-ph↗

arXiv2022

Lies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer

Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widely appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almost all properties of the exact exchange-correlation functional are readily visualized and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC potential as correlations become very strong, the derivative discontinuity and the difference between KS gaps and true charge gaps, and how to extract optical excitations using time-dependent DFT. By the end of this text and accompanying exercises, the reader will improve their ability to both explain and visualize the concepts of DFT, as well as better understand where others may go wrong.

cond-mat.str-el↗