arXiv subjects

Paul W. Ayers

Publications and source records attributed to Paul W. Ayers.

At least 19 recordsLinked to original sources

arXiv2026

PyCDFT: A Python-scriptable library for analytical evaluation of orbital conceptual density (matrix) functional theory

Conceptual density functional theory (CDFT) defines chemical reactivity descriptors as derivatives of the electronic energy with respect to the number of electrons and the external potential, or combinations thereof; in practice, however, researchers almost always replace these derivatives with finite-difference and frontier-orbital approximations, and no common software exists for their analytical evaluation. We present PyCDFT, to our knowledge, the first standardized, open-source, and scriptable code that analytically computes the descriptors of conceptual density (matrix) functional theory for ground and excited states up to second order, including the orbital hardness, the Fukui function and Fukui matrix, and the linear response function, from a single converged mean-field wavefunction imported from virtually any electronic-structure package. The descriptors are obtained from matrix-free, preconditioned Krylov-subspace solutions of the coupled-perturbed self-consistent-field equations for both spin-unpolarized and spin-polarized references, with the work distributed over MPI ranks and all data flowing through a single HDF5 checkpoint file that supports restart, post-processing, and the export of real-space descriptors as cube files for visualization. The design delegates integrals, grids, and exchange-correlation kernels to PySCF and Libxc, keeping the library compact, user-friendly, and interoperable. Worked examples on the ground state of H$_2$O and a broken-symmetry $Δ$SCF excited state of NH$_3$ show that a complete second-order CDFT analysis requires only a few lines of user code.

physics.chem-ph↗

arXiv2026

Second-Derivative-Corrected FANPT for Robust Continuation of Nonlinear Wavefunction Equations

We present a second-derivative-corrected extension of the Flexible Ansatz for N-body Perturbation Theory (FANPT) for solving nonlinear Flexible Ansatz for N-body Configuration Interaction (FANCI) wavefunction equations. The original quasilinear FANPT approximation neglects second- and higher-order derivatives of the determinant overlap with respect to wavefunction parameters. In this work, we retain the overlap Hessian and neglect only third- and higher-order parameter derivatives, thereby including the leading nonlinear response of the wavefunction ansatz while preserving the same response-matrix structure used in the original FANPT formulation. The resulting additional terms enter only through the constant vector of the response equations and are implemented for coupled-cluster wavefunctions in the FanPy/FANCI framework. The new approximation is tested on the Lithium Hydride molecule and the insertion of Beryllium into H$_2$ using seniority-restricted coupled-cluster wavefunctions in the STO-6G basis. The main advantage of the second-derivative correction is that it provides a better initial guess for solving the projected FANCI equations at the next point along the adiabatic connection. This improvement is diagnosed by comparing the FANPT-propagated parameters with the independently optimized FANCI parameters at the same value of $λ$. For nonlinear coupled-cluster ansatzes, especially when larger $λ$-steps are used, the corrected approximation reduces the same-$λ$ parameter deviations and suppresses large parameter-space excursions. These results show that the leading nonlinear overlap correction improves the reliability of FANPT as a continuation strategy for nonlinear wavefunction equations.

physics.chem-ph↗

arXiv2026

Fanpy 2.0: Wavefunction Implementation and Analysis Tools for Flexible Ansatz Design

Fanpy is a Python library for developing new wavefunction methods. It enables users to quickly convert mathematical expressions into working code through a modular design based on the Flexible Ansatz for N-electron Configuration Interaction (FANCI) theory. This architecture facilitates a straightforward extension of the codebase. Here we present version 2.0 of the Fanpy package. This release includes several new wavefunction implementations, including coupled- cluster-inspired geminal approaches. A new analysis module enables a more detailed inspection of computational results and lays the groundwork for future features. In addition, the PySCF interface has been redesigned, and an interface to the PyCI package has been introduced to offload computationally expensive components. Finally, we introduce an improved software development environment, including automated testing and issue tracking.

physics.chem-ph↗

arXiv2026

Towards more accurate natural orbital functional approximations: including 4-index cumulant contributions

Accurate modeling of bond breaking remains a central challenge for reduced density matrix functional theory (RDMFT). Although some modern functionals can yield reasonably accurate dissociation energies, they often fail to reproduce key properties of the dissociated fragments, such as a vanishing fragment population covariance (also known as the delocalization index) and the correct total spin angular momentum of each fragment (local spin). In this work, we revisit the construction of natural orbital functionals by correcting the cumulant contribution produced by the PNOF5 functional. Our method enforces known contributions of the cumulant to local spin fragments and the delocalization index at the dissociation limit. We obtain the closest cumulant consistent with these physically motivated constraints and subsequently purify the corresponding one- and two-electron reduced density matrices by imposing the standard $P, Q, \text{and } G$ $N$-representability conditions. The resulting functional yields improved behavior in strongly correlated regimes. Benchmarking on the dissociation of the singlet states of \ce{N2}, \ce{NO+}, \ce{O2}, \ce{S2}, and \ce{CO} shows that in the dissociation regime the energies computed from the updated cumulant exactly reproduce the complete active space self-consistent field (CASSCF) energies. We further analyze the limitations of the approach and identify scenarios in which the current approach performs poorly. This work provides a pathway for systematically improving natural orbital functionals to achieve reliable bond-breaking calculations within RDMFT.

physics.chem-ph↗

arXiv2026

Seniority-zero Quadratic Canonical Transformation Theory

We propose a method to solve the Schrödinger equation for systems with static/strong electron correlation using Hamiltonian transformations. Building on our previous work on seniority-zero canonical transformation theory, which seeks a unitary transformation that maps the Hamiltonian into the seniority-zero space, this method presents an alternative way of evaluating the Baker--Campbell--Hausdorff (BCH) expansion based on quadratic canonical transformation theory. The extension aims to relax the small-generator constraint by allowing approximate four-body contributions in the expansion, thus expanding the class of excitations previously allowed in SZ-LCT, where only approximate three-body operators were retained. Numerical tests reveal that the seniority-zero quadratic canonical transformation method (SZ-QCT) delivers good accuracy, with most errors within chemical accuracy. In particular, SZ-QCT shows sub-millihartree errors in cases where larger generators are needed to recover the residual dynamic correlation. The computational scaling of SZ-QCT is the same as that of SZ-LCT, $\mathcal{O}(N^8/n_c)$, where $n_c$ is the number of cores available for the computation

physics.chem-ph↗

arXiv2026

Seniority-Zero Canonical Transformation Theory: Reducing Truncation Error with Late Truncation

We show how to add the effects of residual electron correlation to a reference seniority-zero wavefunction by making a unitary transformation of the true electronic Hamiltonian into seniority-zero form. The transformation is treated via the Baker Campbell Hausdorff (BCH) expansion and the seniority-zero structure of the reference is exploited to evaluate the first three commutators exactly; the remaining contributions are handled with a recursive commutator approximation, as is typical in canonical transformation methods. By choosing a seniority-zero reference and using parallel computation, this method is practical for small- to medium-sized systems. Numerical tests show high accuracy, with errors $\sim 10^{-4}$ Hartree.

physics.chem-ph↗

arXiv2026

Seniority-zero Linear Canonical Transformation Theory

We propose a method to solve the electronic Schrödinger equation for strongly correlated systems by applying a unitary transformation to reduce the complexity of the physical Hamiltonian. In particular, we seek a transformation that maps the Hamiltonian into the seniority-zero space: seniority-zero wavefunctions are computationally simpler, but still capture strong correlation within electron pairs. The unitary rotation is evaluated using the Baker Campbell Hausdorff (BCH) expansion, truncated to two-body operators through the operator decomposition strategy of canonical transformation (CT) theory, which rewrites higher-rank terms approximately in terms of one- and two-body operators. Unlike conventional approaches to CT theory, the generator is chosen to minimize the size of non-seniority-zero elements of the transformed Hamiltonian. Numerical tests reveal that this Seniority-zero Linear Canonical Transformation (SZ-LCT) method delivers highly accurate results, usually with submilliHartree error. The effective computational scaling of SZ-LCT is $\mathcal{O}(N^8/n_c)$ , where $n_c$ is the number of cores available for the computation.

physics.chem-ph↗

arXiv2026

Necessary and sufficient conditions for the N-representability of functionals of the one-electron reduced density matrix

We establish necessary and sufficient conditions for the N-representability of the universal one-electron reduced density matrix functional. Functionals satisfying these conditions are guaranteed to yield variational upper bounds on the true energy in one-electron reduced density matrix functional theory, regardless of the strength of the interparticle repulsion. Conversely, any functional violating these conditions will necessarily underestimate the true energy for certain systems. These exact constraints impose a stringent restriction on density matrix functional approximations, as many existing functionals-including the Hartree-Fock functional-appear to violate them. This mathematical formalism, therefore, can guide the development of new approximate functionals and numerical algorithms.

quant-ph↗

arXiv2026

Paying attention to long-range electron correlation: a size-independent deep-learning approach to predicting molecules' electronic energies from one- and two-electron integrals

We propose a descriptor for molecular electronic structure that is based solely on the one- and two-electron integrals but is translationally, rotationally, and unitarily invariant. Then, directly exploiting size consistency, we train and fine tune a neural network to predict the energies of strongly-correlated systems, specifically hydrogen clusters. We use an attention mechanism to formulate a size-independent approach that uses and preserves size-consistency. Therefore, training on few-electron systems can guide predictions for systems with more electrons. Our results are more accurate than alternative geometry-based machine-learning models.

quant-ph↗

arXiv2025

Optimizing and benchmarking the computation of the permanent of general matrices

Evaluating the permanent of a matrix is a fundamental computation that emerges in many domains, including traditional fields like computational complexity theory, graph theory, many-body quantum theory and emerging disciplines like machine learning and quantum computing. While conceptually simple, evaluating the permanent is extremely challenging: no polynomial-time algorithm is available (unless $\textsc{P} = \textsc{NP}$). To the best of our knowledge there is no publicly available software that automatically uses the most efficient algorithm for computing the permanent. In this work we designed, developed, and investigated the performance of our software package which evaluates the permanent of an arbitrary rectangular matrix, supporting three algorithms generally regarded as the fastest while giving the exact solution (the straightforward combinatoric algorithm, the Ryser algorithm, and the Glynn algorithm) and, optionally, automatically switching to the optimal algorithm based on the type and dimensionality of the input matrix. To do this, we developed an extension of the Glynn algorithm to rectangular matrices. Our free and open-source software package is distributed via Github, at https://github.com/theochem/matrix-permanent.

quant-ph↗

arXiv2025

A new family of seniority-restricted coupled cluster methods

We introduce a novel class of coupled cluster (CC) methods that leverage the seniority concept to enhance efficiency and accuracy in electronic structure calculations. While existing approaches, such as the pair coupled cluster doubles (pCCD) method, are limited to seniority-zero (Ω = 0) wavefunctions,1 we propose a more flexible framework: seniority-restricted coupled cluster (sr-CC). This new methodology selectively constrains the seniority sectors accessible through excitation operators in the cluster expansion, enabling a more systematic exploration of electron correlation effects. By balancing computational cost and accuracy, sr-CC provides a promising pathway for advancing electronic structure theory, particularly in strongly correlated systems.

physics.chem-ph↗

arXiv2024

Singlet Geminal Wavefunctions

Wavefunction forms based on products of electron pairs are usually constructed as closed-shell singlets, which is insufficient when the molecular state has a nonzero spin or when the chemistry is determined by $d$- or $f-$electrons. A set of two-electron forms are considered as explicit couplings of second-quantized operators to open-shell singlets. Geminal wavefunctions are constructed and their structure is elaborated. Numerical results for small model systems clearly demonstrate improvement over closed-shell singlet pairs.

physics.chem-ph↗

arXiv2024

Foundation for the ΔSCF Approach in Density Functional Theory

We extend ground-state density-functional theory to excited states and provide the theoretical formulation for the widely used $ΔSCF$ method for calculating excited-state energies and densities. As the electron density alone is insufficient to characterize excited states, we formulate excited-state theory using the defining variables of a noninteracting reference system, namely (1) the excitation quantum number $n_{s}$ and the potential $w_{s}(\mathbf{r})$ (excited-state potential-functional theory, $n$PFT), (2) the noninteracting wavefunction $Φ$ ($Φ$-functional theory, $Φ$FT), or (3) the noninteracting one-electron reduced density matrix $γ_{s}(\mathbf{r},\mathbf{r}')$ (density-matrix-functional theory, $γ_{s}$FT). We show the equivalence of these three sets of variables and their corresponding energy functionals. Importantly, the ground and excited-state exchange-correlation energy use the \textit{same} universal functional, regardless of whether $\left(n_{s},w_{s}(\boldsymbol{r})\right)$, $Φ$, or $γ_{s}(\mathbf{r},\mathbf{r}')$ is selected as the fundamental descriptor of the system. We derive the excited-state (generalized) Kohn-Sham equations. The minimum of all three functionals is the ground-state energy and, for ground states, they are all equivalent to the Hohenberg-Kohn-Sham method. The other stationary points of the functionals provide the excited-state energies and electron densities, establishing the foundation for the $ΔSCF$ method.

physics.chem-ph↗

arXiv2023

Coupled Cluster-Inspired Geminal Wavefunctions

Electron pairs have an illustrious history in chemistry, from powerful concepts to understanding structural stability and reactive changes, to the promise of serving as building blocks of quantitative descriptions of the electronic structure of complex molecules and materials. However, traditionally, two-electron wavefunctions (geminals), have not enjoyed the popularity and widespread use of the more standard single-particle methods. This has changed recently, with a renewed interest in the development of geminal wavefunctions as an alternative to describing strongly-correlated phenomena. Hence, there is a need to find geminal methods that are accurate, computationally-tractable, and do not demand significant input from the user (particularly, via cumbersome and often ill-behaved orbital optimization steps). Here we propose new families of geminal wavefunctions, inspired by the pair Coupled Cluster (pCCD) ansatz. We present a new hierarchy of two-electron wavefunctions that extend the one-reference orbital idea to other geminals. Moreover, we show how to incorporate single-like excitations in this framework, without leaving the quasiparticle picture. We explore the role of imposing seniority restrictions on these wavefunctions, and benchmark these new methods on model strongly-correlated systems.

physics.chem-ph↗

arXiv2023

Flexible Ansatz for N-body Perturbation Theory

We propose a new Perturbation Theory framework that can be used to help with the projective solution of the Schrödinger equation for arbitrary wavefunctions. This Flexible Ansatz for N-body Perturbation Theory (FANPT) is based on our previously proposed Flexible Ansatz for N-body Configuration Interaction (FANCI). We derive recursive FANPT expressions including arbitrary orders in the perturbation hierarchy. We show that the FANPT equations are well-behaved across a wide range of conditions, including static correlation-dominated configurations and highly non-linear wavefunctions.

physics.chem-ph↗

arXiv2023

Inductive Proof of Borchardt's Theorem

We provide an inductive proof of Borchardt's theorem for calculating the permanent of a Cauchy matrix via the determinants of auxiliary matrices. This result has implications for antisymmetric products of interacting geminals (APIG), and suggests that the restriction of the APIG coefficients to Cauchy form (typically called APr2G) is special in its tractability.

physics.chem-ph↗

arXiv2023

A Size-Consistent Wave-function Ansatz Built from Statistical Analysis of Orbital Occupations

Direct approaches to the quantum many-body problem suffer from the so-called "curse of dimensionality": the number of parameters needed to fully specify the exact wavefunction grows exponentially with increasing system size. This motivates the develop of accurate, but approximate, ways to parametrize the wavefunction, including methods like couple cluster theory and correlator product states (CPS). Recently, there has been interest in approaches based on machine learning both direct applications of neural network architecture and the combinations of conventional wavefunction parametrizations with various Boltzmann machines. While all these methods can be exact in principle, they are usually applied with only a polynomial number of parameters, limiting their applicability. This research's objective is to present a fresh approach to wavefunction parametrization that is size-consistent, rapidly convergent, and robust numerically. Specifically, we propose a hierarchical ansatz that converges rapidly (with respect to the number of least-squares optimization). The general utility of this approach is verified by applying it to uncorrelated, weakly-correlated, and strongly-correlated systems, including small molecules and the one-dimensional Hubbard model.

quant-ph↗

arXiv2022

Bivariational Principle for an Antisymmetrized Product of Nonorthogonal Geminals Appropriate for Strong Electron Correlation

We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the same type as Richardson-Gaudin (RG) states, but which are not eigenvectors of a model Hamiltonian which would allow for more freedom in the mean-field. The general idea is to work with the same state in a primal picture in terms of pairs, and in a dual picture in terms of pair-holes. This leads to an asymmetric energy expression which may be optimized bivariationally, and is strictly variational when the two representations are consistent. The general approach may be useful in other contexts, such as for computationally feasible variational coupled-cluster methods.

physics.chem-ph↗