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Henderson, Thomas M.

Publications and source records attributed to Henderson, Thomas M..

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Correlated pair ansatz with a binary tree structure

We develop an efficient algorithm to implement the recently introduced binary tree state (BTS) ansatz on a classical computer. BTS allows a simple approximation to permanents arising from the computationally intractable antisymmetric product of interacting geminals and respects size-consistency. We show how to compute BTS overlap and reduced density matrices efficiently. We also explore two routes for developing correlated BTS approaches: Jastrow coupled cluster on BTS and linear combinations of BT states. Here, the resulting methods show great promise in benchmark applications to the reduced Bardeen–Cooper–Schrieffer Hamiltonian and the one-dimensional XXZ Heisenberg Hamiltonian.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Hartree–Fock–Bogoliubov theory for number-parity-violating fermionic Hamiltonians

It is usually asserted that physical Hamiltonians for fermions must contain an even number of fermion operators. This is indeed true in electronic structure theory. However, when the Jordan–Wigner (JW) transformation is used to map physical spin Hamiltonians to Hamiltonians of spinless fermions, terms that contain an odd number of fermion operators may appear. The resulting fermionic Hamiltonian thus does not have number parity symmetry and requires wave functions that do not have this symmetry either. In this work, we discuss the extension of standard Hartree–Fock–Bogoliubov (HFB) theory to the number-parity-nonconserving case. These ideas had appeared in the literature before but, perhaps for lack of practical applications, had, to the best of our knowledge, never been employed. We here present a useful application for this more general HFB theory based on coherent states of the SO(2M + 1) Lie group, where M is the number of orbitals. Here, we also show how using these unusual mean-field states can provide significant improvements when studying the JW transformation of chemically relevant spin Hamiltonians.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Restoring permutational invariance in the Jordan–Wigner transformation

The Jordan–Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labelling of the spins. One consequence of this dependence is that approximate solutions of a Jordan–Wigner-transformed Hamiltonian may depend on the (physically inconsequential) labelling of the spins. In this work, we turn to an extended Jordan–Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree–Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan–Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. Here, we show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and J 1 –J 2 Heisenberg Hamiltonians.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

Symmetry-projected cluster mean-field theory applied to spin systems

We introduce S z spin-projection based on cluster mean-field theory and apply it to the ground state of strongly correlated spin systems. In cluster mean-fields, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is S z symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all S z sectors, breaking S z symmetry. In addition, a projection scheme is used to restore global S z , which gives rise to the S z spin-projected generalized cluster mean-field (S z GcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D J 1 – J 2 and XXZ Heisenberg models. Furthermore, our results indicate that the new methods (GcMF and S z GcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Thermofield Theory for Finite-Temperature Electronic Structure

Wave function methods have offered a robust, systematically improvable means to study ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogues of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space, which is a formidable task. Besides, excited-state theories are generally not as well studied as ground-state ones. In this mini-review, we present an overview of a finite-temperature wave function formalism based on thermofield dynamics to overcome these difficulties. Thermofield dynamics allows us to map the equilibrium thermal density matrix to a pure state, i.e., a single wave function, albeit in an expanded Hilbert space. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we have developed a procedure to generalize ground-state wave function theories to finite temperatures. As explicit examples, we highlight formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of Fermions in the grand-canonical ensemble. To assess the quality of these approximations, we also show benchmark studies for the one-dimensional Hubbard model, while comparing against exact results. We will see that the thermal methods perform similarly to their ground-state counterparts, while merely adding a prefactor to the asymptotic computational cost. Furthermore, they also inherit all the properties, good or bad, from the ground-state methods, signifying the robustness of our formalism and the scope for future development.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Thermal coupled cluster theory for SU(2) systems

Coupled cluster (CC) has established itself as a powerful theory to study correlated quantum many-body systems. Finite-temperature generalizations of CC theory have attracted considerable interest and have been shown to work as nicely as the ground-state theory. However, most of these recent developments address only fermionic or bosonic systems. The distinct structure of the su(2) algebra requires the development of a similar thermal CC theory for spin degrees of freedom. In this paper, we provide a formulation of our thermofield-inspired thermal CC for SU(2) systems. Here, we apply the thermal CC to the Lipkin-Meshkov-Glick system as well as the one-dimensional transverse field Ising model as benchmark applications to highlight the accuracy of thermal CC in the study of finite-temperature phase diagrams in SU(2) systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Construction of linearly independent non-orthogonal AGP states

We show how to construct a linearly independent set of antisymmetrized geminal power (AGP) states, which allows us to rewrite our recently introduced geminal replacement models as linear combinations of non-orthogonal AGPs. Furthermore, this greatly simplifies the evaluation of matrix elements and permits us to introduce an AGP-based selective configuration interaction method, which can reach arbitrary excitation levels relative to a reference AGP, balancing accuracy and cost as we see fit.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems

We present a wave function representation for the canonical ensemble thermal density matrix by projecting the thermofield double state against the desired number of particles. Furthermore, the resulting canonical thermal state obeys an imaginary-time evolution equation. Starting with the mean-field approximation, where the canonical thermal state becomes an antisymmetrized geminal power (AGP) wave function, we explore two different schemes to add correlation: by number-projecting a correlated grand-canonical thermal state and by adding correlation to the number-projected mean-field state. As benchmark examples, we use number-projected configuration interaction and an AGP-based perturbation theory to study the hydrogen molecule in a minimal basis and the six-site Hubbard model.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Thermofield theory for finite-temperature quantum chemistry

Thermofield dynamics has proven to be a very useful theory in high-energy physics, particularly since it permits the treatment of both time- and temperature-dependence on an equal footing. We here show that it also has an excellent potential for studying thermal properties of electronic systems in physics and chemistry. We describe a general framework for constructing finite temperature correlated wave function methods typical of ground state methods. We then introduce two distinct approaches to the resulting imaginary time Schrödinger equation, which we refer to as fixed-reference and covariant methods. As an example, we derive the two corresponding versions of thermal configuration interaction theory and apply them to the Hubbard model, while comparing with exact benchmark results.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Polynomial-product states: A symmetry-projection-based factorization of the full coupled cluster wavefunction in terms of polynomials of double excitations

Our goal is to remedy the failure of symmetry-adapted coupled-cluster theory in the presence of strong correlation. Previous work along these lines has taken us from a diagram-level analysis of the coupled-cluster equations to an understanding of the collective modes which can occur in various channels of the coupled-cluster equations to the exploration of non-exponential wavefunctions in efforts to combine coupled-cluster theory with symmetry projection. In this manuscript, we extend these efforts by introducing a new, polynomial product wavefunction ansatz that incorporates information from symmetry projection into standard coupled-cluster theory in a way that attempts to mitigate the effects of the lack of size extensivity and size consistency characteristic of symmetry-projected methods. Finally, we describe the new approach in detail within the context of our previous efforts, explore some illustrative calculations, and consider one route for reducing the computational cost of the new method.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection

Methods which aim at universal applicability must be able to describe both weak and strong electronic correlation with equal facility. Such methods are in short supply. The combination of symmetry projection for strong correlation and coupled cluster theory for weak correlation offers tantalizing promise to account for both on an equal footing. In order to do so, however, the coupled cluster portion of the wave function must be optimized in the presence of the symmetry projection. Finally, this paper discusses how this may be accomplished, and shows the importance of doing so for both the Hubbard model Hamiltonian and the molecular Hamiltonian, all with a computational scaling comparable to that of traditional coupled cluster theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Tensor-structured coupled cluster theory

We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on the decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and canonical polyadic decomposition. While the original theory scales as O ( N 6 ) with respect to the number of basis functions, we demonstrate numerically that we achieve sub-millihartree difference from the original theory with O ( N 4 ) scaling. This is accomplished by solving directly for the factors that decompose the cluster operator. Here, the proposed scheme is quite general and can be easily extended to other many-body methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Projected coupled cluster theory

Coupled cluster theory is the method of choice for weakly correlated systems. But in the strongly correlated regime, it faces a symmetry dilemma, where it either completely fails to describe the system or has to artificially break certain symmetries. On the other hand, projected Hartree-Fock theory captures the essential physics of many kinds of strong correlations via symmetry breaking and restoration. Here, we combine and try to retain the merits of these two methods by applying symmetry projection to broken symmetry coupled cluster wave functions. The non-orthogonal nature of states resulting from the application of symmetry projection operators furnishes particle-hole excitations to all orders, thus creating an obstacle for the exact evaluation of overlaps. Here we provide a solution via a disentanglement framework theory that can be approximated rigorously and systematically. Results of projected coupled cluster theory are presented for molecules and the Hubbard model, showing that spin projection significantly improves unrestricted coupled cluster theory while restoring good quantum numbers. The energy of projected coupled cluster theory reduces to the unprojected one in the thermodynamic limit, albeit at a much slower rate than projected Hartree-Fock.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Spin-projected generalized Hartree-Fock method as a polynomial of particle-hole excitations

The past several years have seen renewed interest in the use of symmetry-projected Hartree-Fock for the description of strong correlations. Unfortunately, these symmetry-projected mean-field methods do not adequately account for dynamic correlation. Presumably, this shortcoming could be addressed if one could combine symmetry-projected Hartree-Fock with a many-body method such as coupled-cluster theory, but this is by no means straightforward because the two techniques are formulated in very different ways. However, we have recently shown that the singlet S 2 -projected unrestricted Hartree-Fock wave function can in fact be written in a coupled-cluster-like wave function. That is, the spin-projected unrestricted Hartree-Fock wave function can be written as a polynomial of a double-excitation operator acting on some closed-shell reference determinant. Here, we extend this result and show that the spin-projected generalized Hartree-Fock wave function (which has both S 2 and S z projection) is likewise a polynomial of low-order excitation operators acting on a closed-shell determinant and provide a closed-form expression for the resulting polynomial coefficients. The spin projection of the generalized Hartree-Fock wave function introduces connected triple and quadruple excitations which are absent when spin-projecting an unrestricted Hartree-Fock determinant. We include a few preliminary applications of the combination of this spin-projected Hartree-Fock and coupled-cluster theory to the Hubbard Hamiltonian and comment on generalizations of the methodology. Results here are not for production level, but a similarity-transformed theory that combines the two offers the promise of being accurate for both weak and strong correlation, and may offer significant improvements in the intermediate correlation regime where neither projected Hartree-Fock nor coupled cluster is particularly accurate.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2017

Projected Hartree-Fock theory as a polynomial of particle-hole excitations and its combination with variational coupled cluster theory

Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly correlated systems. Coupled cluster theory, in contrast, does the opposite. It therefore seems natural to combine the two so as to describe both strong and weak correlations with high accuracy in a relatively black-box manner. Combining the two approaches, however, is made more difficult by the fact that the two techniques are formulated very differently. In earlier work, we showed how to write spin-projected Hartree-Fock in a coupled-cluster-like language. Here in this work, we fill in the gaps in that earlier work. Further, we combine projected Hartree-Fock and coupled cluster theory in a variational formulation and show how the combination performs for the description of the Hubbard Hamiltonian and for several small molecular systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗