arXiv subjects

Hugh G. A. Burton

Publications and source records attributed to Hugh G. A. Burton.

32 records · Page 2Linked to original sources

arXiv2022

Energy Landscape of State-Specific Electronic Structure Theory

State-specific approximations can provide an accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and Excited-State Mean-Field representations of singlet H$_2$ (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimisation landscape and elucidate the challenges faced by variance optimisation algorithms, including the presence of unphysical saddle points or maxima of the variance.

physics.chem-ph↗

arXiv2021

Variations of the Hartree-Fock fractional-spin error for one electron

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities, and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H$_2^+$ cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the Møller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.

physics.chem-ph↗

arXiv2021

Generalised Nonorthogonal Matrix Elements: Unifying Wick's Theorem and the Slater-Condon Rules

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder then their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantised generalised Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantised approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalised extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalised Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulae with reduced scaling, promising significant computational acceleration for methods that rely on such terms.

physics.chem-ph↗

arXiv2021

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

physics.chem-ph↗

arXiv2021

Hartree-Fock Critical Nuclear Charge in Two-Electron Atoms

Electron correlation effects play a key role in stabilising two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. Here, we investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts the ionisation of an electron. We also discover another critical nuclear charge where the closed-shell electron density detaches from the nucleus, and identify the importance of fractional spin errors and static correlation in this limit.

physics.chem-ph↗

arXiv2020

Holomorphic Density Functional Theory

Self-consistent-field (SCF) approximations formulated using Hartree-Fock (HF) or Kohn-Sham Density Functional Theory (KS-DFT) both have the potential to yield multiple solutions. However, the formal relationship between multiple solutions identified using HF or KS-DFT remains generally unknown. We investigate the connection between multiple SCF solutions for HF or KS-DFT by introducing a parametrised functional that scales between the two representations. Using the hydrogen molecule and a model of electron transfer, we continuously map multiple solutions from the HF potential to a KS-DFT description. We discover that multiple solutions can coalesce and vanish as the functional changes, forming a direct analogy with the disappearance of real HF solutions along a change in molecular structure. To overcome this disappearance of solutions, we develop a complex-analytic extension of DFT - the "holomorphic DFT" approach - that allows every SCF stationary state to be analytically continued across all molecular structures and exchange-correlation functionals.

physics.chem-ph↗

arXiv2020

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach

Nonorthogonal multireference methods can predict statically correlated adiabatic energies while providing chemical insight through the combination of diabatic reference states. However, reaching quantitative accuracy using nonorthogonal multireference expansions remains a significant challenge. In this work, we present the first rigorous perturbative correction to nonorthogonal configuration interaction, allowing the remaining dynamic correlation to be reliably computed. Our second-order "NOCI-PT2" theory exploits a zeroth-order generalised Fock Hamiltonian and builds the first-order interacting space using single and double excitations from each reference determinant. This approach therefore defines the rigorous nonorthogonal extension to conventional multireference perturbation theory. We find that NOCI-PT2 can quantitatively predict multireference potential energy surfaces and provides state-specific ground and excited states for adiabatic avoided crossings. Furthermore, we introduce an explicit imaginary-shift formalism requiring shift values that are an order of magnitude smaller than those used in conventional multireference perturbation theory.

physics.chem-ph↗

arXiv2019

$\mathcal{PT}$-Symmetry in Hartree-Fock Theory

$\mathcal{PT}$-symmetry --- invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$ --- provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. $\mathcal{PT}$-symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for $\mathcal{PT}$-symmetry in the context of electronic structure theory, and specifically, within the Hartree-Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the $\mathcal{PT}$ operator \textit{if and only if} the effective Fock Hamiltonian is $\mathcal{PT}$-symmetric, and \textit{vice versa}. By extension, if an optimal self-consistent solution is invariant under $\mathcal{PT}$, then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly $\mathcal{PT}$-symmetric Slater determinants by forming $\mathcal{PT}$ doublets (i.e. pairing each occupied orbital with its $\mathcal{PT}$-transformed analogue), allowing $\mathcal{PT}$-symmetry to be conserved throughout the self-consistent process. Finally, considering the \ce{H2} molecule as an illustrative example, we observe $\mathcal{PT}$-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are $\mathcal{PT}$-symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break $\mathcal{PT}$-symmetry.

physics.chem-ph↗

arXiv2019

A General Approach for Multireference Ground and Excited States using Non-Orthogonal Configuration Interaction

A balanced description of ground and excited states is essential for the description of many chemical processes. However, few methods can handle cases where static correlation is present, and often these scale very unfavourably with system size. Recently, multiple Hartree--Fock (HF) solutions have been proposed as a basis for non-orthogonal configuration interaction (NOCI) to provide multireference ground and excited state energies, although applications across multiple geometries have been limited by the coalescence of HF solutions. Holomorphic HF (h-HF) theory allows solutions to be analytically continued beyond the Coulson--Fischer points at which they vanish but, until now, this has only been demonstrated for small model systems. In this work, we propose a general protocol for computing NOCI ground and excited state energies using multiple HF solutions. To do so, we outline an active space variation of SCF metadynamics that allows a chemically relevant set of HF states to be identified, and describe how these states can be routinely traced across all molecular geometries by exploiting the topology of h-HF solutions in the complex plane. Finally, we illustrate our approach using the dissociation of the fluorine dimer and the pseudo-Jahn--Teller distortion of cyclobutadiene, demonstrating its applicability for multireference ground and excited states.

physics.chem-ph↗

arXiv2019

Complex Adiabatic Connection: a Hidden Non-Hermitian Path from Ground to Excited States

Processes related to electronically excited states are central in many areas of science, however accurately determining excited-state energies remains a major challenge in theoretical chemistry. Recently, higher energy stationary states of non-linear methods have themselves been proposed as approximations to excited states, although the general understanding of the nature of these solutions remains surprisingly limited. In this Letter, we present an entirely novel approach for exploring and obtaining excited stationary states by exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres on performing analytic continuations of conventional quantum chemistry methods. Considering Hartree--Fock theory as an example, we analytically continue the electron-electron interaction to expose a hidden connectivity of multiple solutions across the complex plane, revealing a close resemblance between Coulson--Fischer points and non-Hermitian degeneracies. Finally, we demonstrate how a ground-state wave function can be morphed naturally into an excited-state wave function by constructing a well-defined complex adiabatic connection.

physics.chem-ph↗

arXiv2018

Field-Programmable Gate Arrays and Quantum Monte Carlo: Power Efficient Co-processing for Scalable High-Performance Computing

Massively parallel architectures offer the potential to significantly accelerate an application relative to their serial counterparts. However, not all applications exhibit an adequate level of data and/or task parallelism to exploit such platforms. Furthermore, the power consumption associated with these forms of computation renders "scaling out" for exascale levels of performance incompatible with modern sustainable energy policies. In this work, we investigate the potential for field-programmable gate arrays (FPGAs) to feature in future exascale platforms, and their capacity to improve performance per unit power measurements for the purposes of scientific computing. We have focussed our efforts on Variational Monte Carlo, and report on the benefits of co-processing with an FPGA relative to a purely multicore system.

physics.comp-ph↗

arXiv2018

Holomorphic Hartree-Fock Theory: The Nature of Two-Electron Problems

We explore the existence and behaviour of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with $n$ basis functions is rigorously identified as $\frac{1}{2}(3^n - 1)$, proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO-3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH$^{2+}$ and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the $π$ electrons as a two-electron problem, and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.

physics.chem-ph↗

arXiv2017

Microscopic Marangoni flows cannot be predicted on the basis of pressure gradients

A concentration gradient along a fluid-fluid interface can cause flow. On a microscopic level, this so-called Marangoni effect can be viewed as being caused by a gradient in the pressures acting on the fluid elements, or as the chemical-potential gradients acting on the excess densities of different species at the interface. If the interfacial thickness can be ignored, all approaches should result in the same flow profile away from the interface. However, on a more microscopic scale, the different expressions result in different flow profiles, only one of which can be correct. Here we compare the results of direct non-equilibrium Molecular Dynamics simulations with the flows that would be generated by pressure and chemical potential gradients. We find that the approach based on the chemical potential gradients agrees with the direct simulations, whereas the calculations based on the pressure gradients do not.

cond-mat.soft↗

arXiv2015

Holomorphic Hartree-Fock theory: an inherently multireference approach

We investigate the existence of holomorphic Hartree-Fock solutions using a revised SCF algorithm. We use this algorithm to study the Hartree-Fock solutions for H$_{2}$ and H$_{4}^{2+}$ and report the emergence of holomorphic solutions at points of symmetry breaking. Finally, we find these holomorphic solutions for H$_{4}$ and use them as a basis for Non-Orthogonal Configuration Interaction at a range of rectangular geometries and show them to produce energies in good agreement with Full Configuration Interaction.

physics.chem-ph↗