# Kieron Burke's research while affiliated with University of California and other places

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## Publications (329)

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...

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...

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...

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 Har...

Using principles of asymptotic analysis, we derive the exact leading correction 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, ch...

Exact conditions have long been used to guide the construction of density functional approximations. Nowadays hundreds of approximations are in common use, many of which neglect these conditions in their design. We analyze several well-known exact conditions and revive several obscure ones. Two crucial distinctions are drawn: that between necessary...

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 biosimulat...

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...

The Hartree-Fock (HF) approximation has been an important tool for quantumchemical 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 theexact kinetic energy, but no proof of this conjecture exists,d...

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant terms...

In recent years, we have been witnessing a paradigm shift in computational materials science. In fact, traditional methods, mostly developed in the second half of the XXth century, are being complemented, extended, and sometimes even completely replaced by faster, simpler, and often more accurate approaches. The new approaches, that we collectively...

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 Har...

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 biosimulat...

We present conditional probability (CP) density functional theory (DFT) as a formally exact theory. In essence, CP-DFT determines the ground-state energy of a system by finding the CP density from a series of independent Kohn-Sham (KS) DFT calculations. By directly calculating CP densities, we bypass the need for an approximate XC energy functional...

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 exis...

Over the past decade machine learning has made significant advances in approximating density functionals, but whether this signals the end of human-designed functionals remains to be seen. Ryan Pederson, Bhupalee Kalita and Kieron Burke discuss the rise of machine learning for functional design.

Over the past decade machine learning has made significant advances in approximating density functionals, but whether this signals the end of human-designed functionals remains to be seen. Ryan Pederson, Bhupalee Kalita and Kieron Burke discuss the rise of machine learning for functional design. Over the past decade machine learning has made signif...

Conditional-probability density functional theory (CP-DFT) is a formally exact method for finding correlation energies from Kohn-Sham DFT without evaluating an explicit energy functional. We present details on how to generate accurate exchange-correlation energies for the ground-state uniform gas. We also use the exchange hole in a CP antiparallel...

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant terms...

The large-$Z$ asymptotic expansion of atomic exchange energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. We find that the necessary correction is fit well with a leading $Z \ln Z$ term, and find its coefficient numerically. The gradient expansion approximation al...

We present conditional probability (CP) density functional theory (DFT) as a formally exact theory. In essence, CP-DFT determines the ground-state energy of a system by finding the CP density from a series of independent Kohn-Sham (KS) DFT calculations. By directly calculating CP densities, we bypass the need for an approximate XC energy functional...

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 pr...

The importance of the Lieb-Simon proof of the relative exactness of Thomas-Fermi theory in the large-Z limit to modern density functional theory (DFT) is explored. The principle, that there is a specific semiclassical limit in which functionals become local, implies that there exist well-defined leading functional corrections to local approximation...

Conditional-probability density functional theory (CP-DFT) is a formally exact method for finding correlation energies from Kohn-Sham DFT without evaluating an explicit energy functional. We present details on how to generate accurate exchange-correlation energies for the ground-state uniform gas. We also use the exchange hole in a CP antiparallel...

Machine learning has now become an integral part of research and innovation. The field of machine learning density functional theory has continuously expanded over the years while making several noticeable advances. We briefly discuss the status of this field and point out some current and future challenges. We also talk about how state-of-the-art...

Kohn-Sham regularizer (KSR) is a machine learning approach that optimizes a physics-informed exchange-correlation functional within a differentiable Kohn-Sham density functional theory framework. We evaluate the generalizability of KSR by training on atomic systems and testing on molecules at equilibrium. We propose a spin-polarized version of KSR...

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-co...

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 for...

We argue that the success of DFT can be understood in terms of a semiclassical expansion around a very specific limit. This limit was identified long ago by Lieb and Simon for the total electronic energy of a system. This is a universal limit of all electronic structure: atoms, molecules, and solids. For the total energy, Thomas-Fermi theory become...

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. Funct...

Most torsional barriers are predicted to high accuracy (about 1kJ/mol) by standard semilocal functionals, but a small subset has been found to have much larger errors. We create a database of almost 300 carbon-carbon torsional barriers, including 12 poorly behaved barriers, all stemming from Y=C-X group, where X is O or S, and Y is a halide. Functi...

ConspectusDensity 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...

Classical turning surfaces of Kohn–Sham potentials separate classically allowed regions (CARs) from classically forbidden regions (CFRs). They are useful for understanding many chemical properties of molecules but need not exist in solids, where the density never decays to zero. At equilibrium geometries, we find that CFRs are absent in perfect met...

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 e...

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 f...

Density functional calculations can fail for want of an accurate exchange-correlation approximation. The energy can instead be extracted from a sequence of density functional calculations of conditional probabilities (CP DFT). Simple CP approximations yield usefully accurate results for two-electron ions, the hydrogen dimer, and the uniform gas at...

Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10⁻³⁴, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x² and x⁴) beyond this level of accuracy using a basis of several h...

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...

Over the last decade, we have witnessed the emergence of ever more machine learning applications in all aspects of the chemical sciences. Here, we highlight specific achievements of machine learning models in the field of computational chemistry by considering selected studies of electronic structure, interatomic potentials, and chemical compound s...

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 dispar...

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 ⋅ 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...

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...

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-Z expansions for the corresponding energies of neutr...

Empirical fitting of parameters in approximate density functionals is commonplace. Such database fits conflate errors in the self-consistent density with errors in the energy functional, but density-corrected DFT (DC-DFT) separates these two. Three examples both show the pitfalls and how they can be avoided: Catastrophic failures in a toy example,...

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 dispar...

Kohn-Sham inversion, that is, the finding of the exact Kohn-Sham potential for a given density, is difficult in localized basis sets. We study the precision and reliability of several inversion schemes, finding estimates of density-driven errors at a useful level of accuracy. In typical cases of substantial density-driven errors, HF-DFT is almost a...

Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order $10^{-34}$, but few non-trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of $x^2$ and $x^4$ ) beyond this level of accuracy using a basis...

Modern density functional approximations achieve moderate accuracy at low computational cost for many electronic structure calculations. A mathematical framework for analyzing asymptotic behavior unites both corrections to the gradient expansion of DFT and hyperasymptotics of sums. Simple examples are given for the model problem of orbital-free DFT...

The landmark 1982 paper 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...

Classical turning surfaces of Kohn-Sham potentials, separating classically-allowed regions (CARs) from classically-forbidden regions (CFRs), provide a useful and rigorous approach to understanding many chemical properties of molecules. Here we calculate such surfaces for several paradigmatic solids. Our study of perfect crystals at equilibrium geom...

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-$Z$ expansions for the corresponding energies of neu...

Accurate ground-state energies are the focus of most electronic structure calculations. Such energies can, in principle, be extracted from a sequence of density functional calculations of conditional probabilities (CP-DFT), without approximating the energy directly. Simple CP approximations yield usefully accurate results for a broad range of syste...

Second order M\o ller-Plesset perturbation theory (MP2) approximates the exact Hartree-Fock (HF) adiabatic connection (AC) curve by a straight line. Thus by using the deviation of the exact curve from the linear behaviour, we construct an indicator for the accuracy of MP2. We then use an interpolation along the HF AC to transform the exact form of...

Kohn-Sham inversion, that is, the finding of the exact Kohn-Sham potential for a given density, is difficult in localized basis sets. We study the precision and reliability of several inversion schemes, finding estimates of density-driven errors at a useful level of accuracy. In typical cases of substantial density-driven errors, HF-DFT is almost a...

A mathematical framework is constructed for the sum of the lowest N eigenvalues of a potential. Exactness is illustrated on several one-dimensional systems (harmonic oscillator, particle in a box, and Poschl–Teller well). Semiclassical expansion yields the leading corrections for finite systems, identifying the error in common gradient expansions i...

Sums of the N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regime N ≫ 1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levels N + 1, N + 2…. For the harmonic oscillator and the particle in a box, the formula is exac...

Second order M{\o}ller-Plesset perturbation theory (MP2) approximates the exact Hartree-Fock (HF) adiabatic connection (AC) curve by a straight line. Thus by using the deviation of the exact curve from the linear behaviour, we construct an indicator for the accuracy of MP2. We then use an interpolation along the HF AC to transform the exact form of...

Density-corrected density functional theory (DC-DFT) is enjoying substantial success in improving semilocal DFT calculations in a wide variety of chemical problems. This paper provides the formal theoretical framework and assumptions for the analysis of any functional minimization with an approximate functional. We generalize DC-DFT to allow compar...

Density-corrected density functional theory (DC-DFT) is enjoying substantial success in improving semilocal DFT calculations in a wide variety of chemical problems. This paper provides the formal theoretical framework and assumptions for the analysis of any functional minimization with an approximate functional. We generalize DC-DFT to allow compar...

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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/mol 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....

Dispersion corrections of various kinds usually improve DFT energetics of weak non-covalent interactions. But in some cases involving molecules or halides, especially those with σ-hole interactions, the density-driven errors of uncorrected DFT are larger than the dispersion corrections. In these abnormal situations, HF-DFT (using Hartree-Fock densi...

The Kohn-Sham scheme of density functional theory is one of the most widely used methods to solve electronic structure problems for a vast variety of atomistic systems across different scientific fields. While the method is fast relative to other first principles methods and widely successful, the computational time needed is still not negligible,...

Density functional theory (DFT) has become the most popular approach to electronic structure across disciplines, especially in material and chemical sciences. In 2016, at least 30,000 papers used DFT to make useful predictions or give insight into an enormous diversity of scientific problems, ranging from battery development to solar cell efficienc...

We argue that any general mathematical measure of density error, no matter how reasonable, is too arbitrary to be of universal use. However the energy functional itself provides a universal relevant measure of density errors. For the self-consistent density of any Kohn-Sham calculation with an approximate functional, the theory of density corrected...

The recent use of a new ensemble in density functional theory (DFT) to produce direct corrections to the Kohn-Sham transitions yields the elusive double excitations that are missed by time-dependent DFT (TDDFT) with the standard adiabatic approximation. But accuracies are lower than for single excitations, and formal arguments about TDDFT suggest t...

We argue that any general mathematical measure of density error, no matter how reasonable, is too arbitrary to be of universal use. However the energy functional itself provides a universal relevant measure of density errors. For the self-consistent density of any Kohn-Sham calculation with an approximate functional, the theory of density-corrected...

For quantum fermion problems, many accurate solvers are limited by the temperature regime in which they can be usefully applied. The Mermin theorem implies the uniqueness of an effective potential from which both the exact density and free energy at a target temperature can be found, via a calculation at a different, reference temperature. We deriv...

A survey of the contributions to the Special Topic on Data-enabled Theoretical Chemistry is given, including a glossary of relevant machine learning terms.

Historical methods of functional development in density functional theory have often been guided by analytic conditions that constrain the exact functional one is trying to approximate. Recently, machine-learned functionals have been created by interpolating the results from a small number of exactly solved systems to unsolved systems that are simi...

The recent use of a new ensemble in density functional theory (DFT) to yield direct corrections to the Kohn-Sham transitions yields the elusive double excitations that are missed by time-dependent DFT with the standard adiabatic approximation. But accuracies are lower than for single excitations, and formal arguments suggest that direct corrections...

An introduction to the current state of the art in data-enabled theoretical chemistry is given. It includes a glossary of relevant machine learning terms, plus a survey of the papers in the Journal of Chemical Physics Special Topic on Data-enabled Theoretical Chemistry.

All-electron fixed-node diffusion Monte Carlo provides benchmark spin gaps for four Fe(II) octahedral complexes. Standard quantum chemical methods (semilocal DFT and CCSD(T)) fail badly for the energy difference between their high- and low-spin states. Density-corrected DFT is both significantly more accurate and reliable, and yields a consistent p...

The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. A simple interpolation between care...

We propose a general method for constructing system-dependent basis functions for correlated quantum calculations. Our construction combines features from several traditional approaches: plane waves, localized basis functions, and wavelets. In a one-dimensional mimic of Coulomb systems, it requires only 2–3 basis functions per electron to achieve h...

We consider the implications of the Lieb-Simon limit for correlation in density functional theory. In this limit, exemplified by the scaling of neutral atoms to large atomic number, local density approximation (LDA) becomes relatively exact, and the leading correction to this limit for correlation has recently been determined for neutral atoms. We...

Last year, at least 30,000 scientific papers used the Kohn-Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields, ranging from materials science to biochemistry to astrophysics. Machine learning holds the promise of learning the kinetic energy functional via examples, by-passing the n...