arXiv subjects

Thomas M. Henderson

Publications and source records attributed to Thomas M. Henderson.

At least 37 records · Page 2Linked to original sources

arXiv2021

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. 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.

physics.chem-ph↗

arXiv2021

Exploring non-linear correlators on AGP

Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential ansatze in this paper. We investigate two approaches in particular. Similar to Phys. Rev. B 91, 041114(R) (2015), we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting Schrodinger's equation. The second approach is based on approximating the unitary pair-hopper ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.

cond-mat.str-el↗

arXiv2020

Geminal replacement models based on AGP

The antisymmetrized geminal power (AGP) wavefunction has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J-CI, provides excellent accuracy in energies when applied to the pairing Hamiltonian.

physics.chem-ph↗

arXiv2020

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. 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 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.

physics.chem-ph↗

arXiv2020

Correlating the Antisymmetrized Geminal Power Wave Function

Strong pairing correlations are responsible for superconductivity and off-diagonal long range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.

physics.chem-ph↗

arXiv2020

On a dual representation of the Goldstone manifold

An intrinsic wavefunction with a broken continuous symmetry can be rotated with no energy penalty leading to an infinite set of degenerate states known as a Goldstone manifold. In this work, we show that a dual representation of such manifold exists that is sampled by an infinite set of non-degenerate states. A proof that both representations are equivalent is provided. From the work of Peierls and Yoccoz (Proc. Phys. Soc. A {\bf 70}, 381 (1957)), it is known that collective states with good symmetries can be obtained from the Goldstone manifold using a generator coordinate trial wavefunction. We show that an analogous generator coordinate can be used in the dual representation; we provide numerical evidence using an intrinsic wavefunction with particle number symmetry-breaking for the electronic structure of the Be atom and one with $\hat{S}^z$ symmetry-breaking for a H$_5$ ring. We discuss how the dual representation can be used to evaluate expectation values of symmetry-projected states when the norm $|\langle Φ| \hat{P}^q | Φ\rangle|$ becomes very small.

physics.chem-ph↗

arXiv2019

Thermofield theory for finite-temperature coupled cluster

We present a coupled cluster and linear response theory to compute properties of many-electron systems at non-zero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wavefunction representation of the thermal density matrix, and extend our recently developed framework [J. Chem. Phys. 150, 154109 (2019)] to parameterize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) as well as ab-initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.

physics.chem-ph↗

arXiv2019

Geminal-Based Configuration Interaction

The antisymmetrized geminal power (AGP) wave function has a long history and considerable conceptual appeal, but in many situations its accuracy is wanting. Here, we consider a form of configuration interaction (CI) based upon the AGP wave function and taking advantage of its killing operators to construct an excitation manifold. Our geminal CI reduces to standard single-determinant--based CI in the limit in which AGP reduces to a single determinant. It substantially improves upon AGP in the reduced BCS Hamiltonian, which serves as a prototype for the kinds of strong pairing correlations relevant in Bardeen-Cooper-Schrieffer--style superconductivity. Moreover, our geminal CI naturally generalizes to add correlation to more general geminal-based wave functions than AGP.

physics.chem-ph↗

arXiv2019

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 Schrodinger 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.

physics.chem-ph↗

arXiv2018

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. 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.

physics.chem-ph↗

arXiv2018

Hartree-Fock symmetry breaking around conical intersections

We study the behavior of Hartree-Fock (HF) solutions in the vicinity of conical intersections. These are here understood as regions of a molecular potential energy surface characterized by degenerate or nearly-degenerate eigenfunctions with identical quantum numbers (point group, spin, and electron number). Accidental degeneracies between states with different quantum numbers are known to induce symmetry breaking in HF. The most common closed-shell restricted HF instability is related to singlet-triplet spin degeneracies that lead to collinear unrestricted HF (UHF) solutions. Adding geometric frustration to the mix usually results in noncollinear generalized HF (GHF) solutions, identified by orbitals that are linear combinations of up and down spins. Near conical intersections, we observe the appearance of coplanar GHF solutions that break all symmetries, including complex conjugation and time-reversal, which do not carry good quantum numbers. We discuss several prototypical examples taken from the conical intersection literature. Additionally, we utilize a recently introduced a magnetization diagnostic to characterize these solutions, as well as a solution of a Jahn-Teller active geometry of H$_8^{+2}$.

physics.chem-ph↗

arXiv2017

On the Magnetic Structure of Density Matrices

The spin structure of wave functions is reflected in the magnetic structure of the one-particle density matrix. Indeed, for single determinants we can use either one to determine the other. In this work we discuss how one can simply examine the one-particle density matrix to faithfully determine whether the spin magnetization density vector field is collinear, coplanar, or noncoplanar. For single determinants, this test suffices to distinguish collinear determinants which are eigenfunctions of $\hat{S}_{\hat{n}}$ from noncollinear determinants which are not. We also point out the close relationship between noncoplanar magnetism on the one hand and complex conjugation symmetry breaking on the other. Finally, we use these ideas to classify the various ways single determinant wave functions break and respect symmetries of the Hamiltonian in terms of their one-particle density matrix.

physics.chem-ph↗

arXiv2017

Influence of broken-pair excitations on the exact pair wavefunction

Doubly occupied configuration interaction (DOCI), the exact diagonalization of the Hamiltonian in the paired (seniority zero) sector of the Hilbert space, is a combinatorial cost wave function that can be very efficiently approximated by pair coupled cluster doubles (pCCD) at mean-field computational cost. As such, it is a very interesting candidate as a starting point for building the full configuration interaction (FCI) ground state eigenfunction belonging to all (not just paired) seniority sectors. The true seniority zero sector of FCI (referred to here as FCI${}_0$) includes the effect of coupling between all seniority sectors rather than just seniority zero, and is, in principle, different from DOCI. We here study the accuracy with which DOCI approximates FCI${}_0$. Using a set of small Hubbard lattices, where FCI is possible, we show that DOCI $\sim$ FCI${}_0$ under weak correlation. However, in the strong correlation regime, the nature of the FCI${}_0$ wavefunction can change significantly, rendering DOCI and pCCD a less than ideal starting point for approximating FCI.

physics.chem-ph↗

arXiv2017

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 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. The proposed scheme is quite general and can be easily extended to other many-body methods.

physics.chem-ph↗

arXiv2017

Spin-Projected Generalized Hartree-Fock 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. 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 particularly may offer significant improvements in the intermediate correlation regime where neither projected Hartree-Fock nor coupled cluster is particularly accurate.

physics.chem-ph↗

arXiv2017

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. In this work, we combine and try to retain the merits of these two methods by applying symmetry projection to broken symmetry coupled cluster wavefunctions. 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.

physics.chem-ph↗

arXiv2017

Projected Hartree-Fock 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 correlation 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, 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.

cond-mat.str-el↗

arXiv2017

Understanding Band Gaps of Solids in Generalized Kohn-Sham Theory

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. But the gap in the band-structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density functional theory. Here we give a simple proof of a new theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from meta-generalized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.

cond-mat.mtrl-sci↗