arXiv subjects

Tim Gould

Publications and source records attributed to Tim Gould.

At least 19 recordsLinked to original sources

arXiv2026

Thermal Fundamental Gap Predictions in DFT via Optimally Tuned Hybrids

Predicting electronic fundamental gaps at finite temperature has remained conceptually and practically challenging. We address this in three connected steps. First, we extend generalized Kohn--Sham hybrid density functional theory to thermal ensembles, deriving a Mermin generalized Kohn--Sham framework from a thermal one-particle auxiliary system and an exact density-functional remainder. Second, via an extension of Janak's theorem that holds rigorously in this framework, we recast Hirata's thermal-quasiparticle picture as a thermal orbital gap estimator and derive a closed low-temperature form, the error of which is controlled by the derivative discontinuity. Third, because optimal tuning eliminates this error, the auxiliary orbital gap matches the interacting gap at low temperature, upgrading optimal tuning from a ground-state strategy to the governing principle -- mandatory, not optional -- for accurate finite-temperature gap predictions obtained from gaps of orbital eigenvalues within a hybrid functional framework. We present applications that validate the theory and demonstrate its consequences.

physics.chem-ph↗

arXiv2026

Quantum-Information Measure of Electron Localization

Understanding electron localization in molecules and materials plays a central role in electronic structure theory, and will increase in importance with the rise of data-driven approaches. The electron localization function (ELF) is widely used to visualize electron organization in molecules and materials, and it remains a central ingredient in modern density-functional approximations. Yet its formulation retains highly empirical elements. Here we introduce a quantum-information measure of electron localization derived from the concurrence of a correlated two-spin mixed state. This construction yields a genuine two-point localization indicator grounded in quantum-information theory, avoiding the heuristic normalization and chosen nonlinear remapping of the ELF. We show that atomic shells, covalent and ionic bonds, lone pairs, molecular dissociation, and charge-transfer processes are captured. The method is straightforward to evaluate numerically.

cond-mat.mtrl-sci↗

arXiv2025

AI4X Roadmap: Artificial Intelligence for the advancement of scientific pursuit and its future directions

Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.

physics.soc-ph↗

arXiv2025

Foundations of the ionization potential condition for localized electron removal in density functional theory

Optimal tuning of functional parameters in density functional theory approximations, based on enforcing the ionization potential theorem, has emerged as the method of choice for the non-empirical prediction of the electronic structure of finite systems. This method has recently been extended to the bulk limit, based on an ansatz that generalizes the ionization potential theorem to the removal of an electron from a localized Wannier orbital. This Wannier-localization based optimal tuning method has been shown to be highly successful for a wide range of periodic systems, accurately predicting electronic and optical properties. However, a rigorous theoretical justification for its foundational ansatz has been lacking. Here, we establish an ionization potential condition for the removal of a localized electron, by extending the piecewise linearity and Janak's theorems in density functional theory. We also provide numerical evidence supporting our theory.

physics.chem-ph↗

arXiv2025

"Ensemblization" of density functional theory

Density functional theory (DFT) has transformed our ability to investigate and understand electronic ground states. In its original formulation, however, DFT is not suited to addressing (e.g.) degenerate ground states, mixed states with different particle numbers, or excited states. All these issues can be handled, in principle exactly, via ensemble DFT (EDFT). This Perspective provides a detailed introduction to and analysis of EDFT, in an in-principle exact framework that is constructed to avoid uncontrolled errors and inconsistencies that may be associated with {\it ad hoc} extensions of conventional DFT. In particular, it focuses on the "ensemblization" of both exact and approximate density functionals, a term we coin to describe a rigorous approach that lends itself to the construction of novel approximations consistent with the general ensemble framework, yet applicable to practical problems where traditional DFT tends to fail or does not apply at all. Specifically, symmetry considerations and ensemble properties are shown to enable each other in shaping a practical DFT-based methodology that extends beyond the ground state and, in doing so, highlights the need to look outside the standard ground state Kohn-Sham treatment.

physics.chem-ph↗

arXiv2024

State-specific density functionals for excited states from ensembles

We present a first principles strategy for developing state-specific density functional approximations for excited states. We first clarify why approaches based on conventional ground state approximations miss density-driven correlations, by considering excited state physics through the lens of ensemble density functional theory. To solve this issue we gain insights on density driven correlations by exploiting the recently understood low-density limit of electrons in excited states. The theory developments are then combined to produce a proof-of-concept excited state approximation that resolves urgent paradigmatic failures (double excitations, charge transfer excitations, piecewise linearity) of existing state-of-art density-functional approaches, directly from differences of self-consistent field calculations; i.e., $\Delta$SCF. In light of its observed impressive performance, we conclude that the approach represents a major step toward unified and accurate modelling of neutral and charged excitations.

physics.chem-ph↗

arXiv2024

Stationary conditions for excited states: the surprising impact of density-driven correlations

Typical density functional theory (DFT) and approximations thereto solve the many-electron ground state problem by working from a numerically efficient non-interacting Kohn-Sham reference system; and benefit from useful minimization conditions that allow iteration (i.e. self-consistency) to the optimal energy and density. Ensembles of ground and excited states can also benefit from similar minimization conditions [see in source]. This work reveals that individual excited states also have state-specific stationary conditions, that can be deduced from the ensemble solution and apply to DFT and its interacting potential functional theory (PFT) counterpart. However, the state-specific stationary condition for the non-interacting Kohn-Sham PFT is revealed to be more complicated than the ground state problem, due in part to a contribution from density-driven correlations [Phys. Rev. Lett. 123, 016401 (2019); 124, 243001 (2020); 125, 233001 (2020)] that are neglected in ``$\Delta$SCF'' approaches. Some implications for self-consistency in exact theory and approximations are discussed.

physics.chem-ph↗

arXiv2024

Non-empirical prediction of the length-dependent ionization potential in molecular chains

The ionization potential of molecular chains is well-known to be a tunable nano-scale property that exhibits clear quantum confinement effects. State-of-the-art methods can accurately predict the ionization potential in the small molecule limit and in the solid-state limit, but for intermediate, nano-sized systems prediction of the evolution of the electronic structure between the two limits is more difficult. Recently, optimal tuning of range-separated hybrid functionals has emerged as a highly accurate method for predicting ionization potentials. This was first achieved for molecules using the ionization potential theorem (IPT) and more recently extended to solid-state systems, based on an \textit{ansatz} that generalizes the IPT to the removal of charge from a localized Wannier function. Here, we study one-dimensional molecular chains of increasing size, from the monomer limit to the infinite polymer limit using this approach. By comparing our results with other localization-based methods and where available with experiment, we demonstrate that Wannier-localization-based optimal tuning is highly accurate in predicting ionization potentials for any chain length, including the nano-scale regime.

physics.chem-ph↗

arXiv2023

Scalable molecular simulation of electrolyte solutions with quantum chemical accuracy

Electrolyte solutions play critical role in a vast range of important applications, yet an accurate and scalable method of predicting their properties without fitting to experiment has remained out of reach, despite over a century of effort. Here, we combine state-of-the-art density functional theory and equivariant neural network potentials to demonstrate this capability, reproducing key structural, thermodynamic, and kinetic properties. We show that neural network potentials (NNPs) can be recursively trained on a subset of their own output to enable coarse-grained/continuum-solvent molecular simulations that can access much longer timescales than possible with all atom simulations. We observe the surprising formation of Li cation dimers along with identical anion-anion pairing of chloride and bromide anions. Finally, we reproduce simulate the crystal phase and infinite dilution pairing free energies despite being trained only on moderate concentration solutions. This approach should be scaled to build a greatly expanded database of electrolyte solution properties than currently exists.

physics.chem-ph↗

arXiv2023

A step toward density benchmarking -- the energy-relevant "mean field error"

Since the development of generalized gradient approximations in the 1990s, approximations based on density functional theory have dominated electronic structure theory calculations. Modern approximations can yield energy differences that are precise enough to be predictive in many instances, as validated by large- and small-scale benchmarking efforts. However, assessing the quality of densities has been the subject of far less attention, in part because reliable error measures are difficult to define. To this end, this work introduces the mean-field error that directly assesses the quality of densities from approximations. The mean-field error is contextualised within existing frameworks of density functional error analysis and understanding, and shown to be part of the density-driven error. It is demonstrated on several illustrative examples. Its potential use in future benchmarking protocols is discussed, and some conclusions drawn.

physics.chem-ph↗

arXiv2023

Local density approximation for excited states

The ground state of an homogeneous electron gas is a paradigmatic state that has been used to model and predict the electronic structure of matter at equilibrium for nearly a century. For half a century, it has been successfully used to predict ground states of quantum systems via the local density approximation (LDA) of density functional theory (DFT); and systematic improvements in the form of generalized gradient approximations and evolution thereon. Here, we introduce the LDA for \emph{excited} states by considering a particular class of non-thermal ensemble states of the homogeneous electron gas. These states find sound foundation and application in ensemble-DFT -- a generalization of DFT that can deal with ground and excited states on equal footing. The ensemble-LDA is shown to successfully predict difficult low-lying excitations in atoms and molecules for which approximations based on local spin density approximation (LSDA) and time-dependent-LDA fail.

physics.chem-ph↗

arXiv2022

Effect of exchange-correlation and core-electron approximations on the calculated superconducting transition temperature of palladium hydride

Realistic prediction of the superconducting transition temperature (Tc) for PdH is a long-standing challenge, because it depends on robust calculations of the electron and phonon band structures to obtain the electron-phonon scattering matrix element. To date, first-principles calculations employing density functional theory (DFT) have been based on selected exchange-correlation and core-electron approximations. Incorporating anharmonicity produced a more realistic value of Tc that nevertheless still disagreed strongly with experiment unless adjustable parameters were introduced. Here we consider how the value of Tc predicted using DFT in the harmonic approximation depends on the DFT scheme employed. The rationale for this work is that unless the calculation of Tc within the harmonic approximation is robust, albeit incorrect, there is not a solid foundation for incorporating anharmonicity meaningfully. Six combinations of exchange-correlation approximation (LDA, PBE, PBEsol) and core-electron approximation (PAW, USPP) were tested. Following a carefully systematic methodology, the calculated Tc was found to vary by a factor exceeding two across the tested DFT schemes. This suggests strongly that "standard" DFT, even including anharmonicity, is not reliable for PdH, implying that a higher-rung method will be needed to calculate a realistic lattice constant and phonon band structure, and so predict Tc accurately.

cond-mat.mtrl-sci↗

arXiv2022

Nuclear cusps and singularities in the non-additive kinetic potential bi-functional from analytical inversion

The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} + \rho_{\rm B}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact $v^{\rm NAD}$ for the local density approximation. We examine features of $v^{\rm NAD}$ which can be used for development and testing of approximations to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.

physics.chem-ph↗

arXiv2022

Electronic excited states in extreme limits via ensemble density functionals

Density functional theory (DFT) has greatly expanded our ability to affordably compute and understand electronic ground states, by replacing intractable {\em ab initio} calculations by models based on paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited states lags behind. Here, we solve this outstanding problem by employing a generalization of density functional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all electronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show that the high-density limit connects to recent, exactly-solvable EDFT results. The low-density limit reveals an unnoticed and most unexpected result -- density functionals for strictly correlated {\em ground} states can be reused {\em directly} for excited states. Non-trivial dependence on excitation structure only shows up at third leading order. Overall, our results provide foundations for effective models of excited states that interpolate between exact low- and high-density limits, which we illustrate on the cases of singlet-singlet excitations in H$_2$ and a ring of quantum wells.

physics.chem-ph↗

arXiv2020

Combining density functional theories to correctly describe the energy, lattice structure and electronic density of functional oxide perovskites

Functional oxide perovskites are the pillar of cutting-edge technological applications. Density functional theory (DFT) simulations are the theoretical methods of choice to understand and design perovskite materials. However, tests on the reliability of DFT to describe fundamental properties of oxide perovskites are scarce and mostly ill-defined due to a lack of rigorous theoretical benchmarks for solids. Here, we present a quantum Monte Carlo benchmark study of DFT on the archetypal perovskite BaTiO$_{3}$ (BTO). It shows that no DFT approximation can simultaneously reproduce the energy, structure, and electronic density of BTO. Traditional protocols to select DFT approximations are empirical and fail to detect this shortcoming. An approach combining two different non-empirical DFT schemes, "SCAN" and "HSE06", is able to holistically describe BTO with accuracy. Combined DFT approaches should thus be considered as a promising alternative to standard methods for simulating oxide perovskites.

cond-mat.mtrl-sci↗

arXiv2020

Density driven correlations in ensemble density functional theory: insights from simple excitations in atoms

Ensemble density functional theory extends the usual Kohn-Sham machinery to quantum state ensembles involving ground- and excited states. Recent work by the authors [Phys. Rev. Lett. 119, 243001 (2017); 123, 016401 (2019)] has shown that both the Hartree-exchange and correlation energies can attain unusual features in ensembles. Density-driven(DD) correlations -- which account for the fact that pure-state densities in Kohn-Sham ensembles do not necessarily reproduce those of interacting pure states -- are one such feature. Here we study atoms (specifically $S$--$P$ and $S$--$S$ transitions) and show that the magnitude and behaviour of DD correlations can vary greatly with the variation of the orbital angular momentum of the involved states. Such estimations are obtained through an approximation for DD correlations built from relevant exact conditions Kohn-Sham inversion, and plausible assumptions for weakly correlated systems.

cond-mat.str-el↗

arXiv2019

Thermodynamics from first principles: correlations and nonextensivity

The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.

cond-mat.stat-mech↗

arXiv2019

A simple self-interaction correction to RPA-like correlation energies

The random phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of iso-electronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spin-polarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for H2 and N2, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched H2 or N2. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched H2+), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all one-electron densities, in the same simple semilocal way that the correlation energy densities of many meta-generalized gradent approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

physics.comp-ph↗