New algorithms for iterative matrix‐free eigensolvers in quantum chemistry

A Jacobian-free pseudo-arclength continuation method for phase transitions in inhomogeneous thermodynamic systems

Developing phase diagrams for inhomogeneous systems in thermodynamics is difficult, in part, due to the large phase space and the possibility of unstable and metastable solutions arising from first-order phase transitions. Pseudo-arclength continuation (PAC) is a method that allows one to trace out stable and unstable solutions of nonlinear systems. Typically, PAC utilizes the Jacobian in order to implement Newton (or quasi-Newton) steps. In this work, we present a Jacobian-free PAC method that is amenable to the usual workflows in inhomogeneous thermodynamics. We demonstrate our method in systems that have first-order phase transitions, including a novel example of polyelectrolyte complex coacervation in confinement, where multiple surface phase transitions occur and can overlap with one another.

Approximate Exponential Integrators for Time-Dependent Equation-of-Motion Coupled Cluster Theory

With a growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schrödinger equation is of keen interest in modern electronic structure theory. In this work, we present two approaches for the formation of the quantum propagator for time-dependent equation-of-motion coupled cluster theory based on the Chebyshev and Arnoldi expansions of the complex, nonhermitian matrix exponential, respectively. The proposed algorithms are compared with the short-iterative Lanczos method of Cooper et al. [J. Phys. Chem. A 2021 125, 5438-5447], the fourth-order Runge-Kutta method, and exact dynamics for a set of small but challenging test problems. For each of the cases studied, both of the proposed integration schemes demonstrate superior accuracy and efficiency relative to the reference simulations.

Time-Dependent Density Functional Theory for X-ray Absorption Spectra: Comparing the Real-Time Approach to Linear Response

We simulate X-ray absorption spectra at elemental K-edges using time-dependent density functional theory (TDDFT) in both its conventional linear-response implementation and its explicitly time-dependent or "real-time" formulation. Real-time TDDFT simulations enable broadband spectra calculations without the need to invoke frozen occupied orbitals ("core/valence separation"), but we find that these spectra are often contaminated by transitions to the continuum that originate from lower-energy core and semicore orbitals. This problem becomes acute in triple-ζ basis sets, although it is sometimes sidestepped in double-ζ basis sets. Transitions to the continuum acquire surprisingly large dipole oscillator strengths, leading to spectra that are difficult to interpret. Meaningful spectra can be recovered by means of a filtering technique that decomposes the spectrum into contributions from individual occupied orbitals, and the same procedure can be used to separate L- and K-edge spectra arising from different elements within a given molecule. In contrast, conventional linear-response TDDFT requires core/valence separation but is free of these artifacts. It is also significantly more efficient than the real-time approach, even when hundreds of individual states are needed to reproduce near-edge absorption features and even when Padé approximants are used to reduce the real-time simulations to just 2-4 fs of time propagation. Despite the cost, the real-time approach may be useful to examine the validity of the core/valence separation approximation.

Synergy of semiempirical models and machine learning in computational chemistry

Catalyzed by enormous success in the industrial sector, many research programs have been exploring data-driven, machine learning approaches. Performance can be poor when the model is extrapolated to new regions of chemical space, e.g., new bonding types, new many-body interactions. Another important limitation is the spatial locality assumption in model architecture, and this limitation cannot be overcome with larger or more diverse datasets. The outlined challenges are primarily associated with the lack of electronic structure information in surrogate models such as interatomic potentials. Given the fast development of machine learning and computational chemistry methods, we expect some limitations of surrogate models to be addressed in the near future; nevertheless spatial locality assumption will likely remain a limiting factor for their transferability. Here, we suggest focusing on an equally important effort-design of physics-informed models that leverage the domain knowledge and employ machine learning only as a corrective tool. In the context of material science, we will focus on semi-empirical quantum mechanics, using machine learning to predict corrections to the reduced-order Hamiltonian model parameters. The resulting models are broadly applicable, retain the speed of semiempirical chemistry, and frequently achieve accuracy on par with much more expensive ab initio calculations. These early results indicate that future work, in which machine learning and quantum chemistry methods are developed jointly, may provide the best of all worlds for chemistry applications that demand both high accuracy and high numerical efficiency.

Two Algorithms for Excited-State Quantum Solvers: Theory and Application to EOM-UCCSD

Near-term quantum devices promise to revolutionize quantum chemistry, but simulations using the current noisy intermediate-scale quantum (NISQ) devices are not practical due to their high susceptibility to errors. This motivated the design of NISQ algorithms leveraging classical and quantum resources. While several developments have shown promising results for ground-state simulations, extending the algorithms to excited states remains challenging. This paper presents two cost-efficient excited-state algorithms inspired by the classical Davidson algorithm. We implemented the Davidson method into the quantum self-consistent equation-of-motion unitary coupled-cluster (q-sc-EOM-UCC) excited-state method adapted for quantum hardware. The circuit strategies for generating desired excited states are discussed, implemented, and tested. We demonstrate the performance and accuracy of the proposed algorithms (q-sc-EOM-UCC/Davidson and its variational variant) by simulations of H2, H4, LiH, and H2O molecules. Similar to the classical Davidson scheme, q-sc-EOM-UCC/Davidson algorithms are capable of targeting a small number of excited states of the desired character.

Libkrylov: A modular open-source software library for extremely large on-the-fly matrix computations

We present the design and implementation of libkrylov, an open-source library for solving matrix-free eigenvalue, linear, and shifted linear equations using Krylov subspace methods. The primary objectives of libkrylov are flexible API design and modular structure, which enables integration with specialized matrix-vector evaluation "engines." Libkrylov features pluggable preconditioning, orthonormalization, and tunable convergence control. Diagonal (conjugate gradient, CG), Davidson, and Jacobi-Davidson preconditioners are available, along with orthonormal and nonorthonormal (nKs) schemes. All functionality of libkrylov is exposed via Fortran and C application programming interfaces (APIs). We illustrate the performance of libkrylov for eigenvalue calculations arising in time-dependent density functional theory (TDDFT) in the Tamm-Dancoff approximation (TDA) and discuss the convergence behavior as a function of preconditioning and orthonormalization methods.

Dynamic Precision Approach for Accelerating Large-Scale Eigenvalue Solvers in Electronic Structure Calculations on Graphics Processing Units

Single precision (SP) arithmetic can be greatly accelerated as compared to double precision (DP) arithmetic on graphics processing units (GPUs). However, the use of SP in the whole process of electronic structure calculations is inappropriate for the required accuracy. We propose a 3-fold dynamic precision approach for accelerated calculations but still with the accuracy of DP. Here, SP, DP, and mixed precision are dynamically switched during an iterative diagonalization process. We applied this approach to the locally optimal block preconditioned conjugate gradient method to accelerate a large-scale eigenvalue solver for the Kohn-Sham equation. We determined a proper threshold for switching each precision scheme by examining the convergence pattern on the eigenvalue solver only with the kinetic energy operator of the Kohn-Sham Hamiltonian. As a result, we achieved up to 8.53× and 6.60× speedups for band structure and self-consistent field calculations, respectively, for test systems under various boundary conditions on NVIDIA GPUs.

Accurate Prediction of Vertical Ionization Potentials and Electron Affinities from Spin-Component Scaled CC2 and ADC(2) Models

The CC2 and ADC(2) wave function models and their spin-component scaled modifications are adopted for predicting vertical ionization potentials (VIPs) and electron affinities (VEAs). The ionic solutions are obtained as electronic excitations in the continuum orbital formalism, making possible the use of existing, widespread quantum chemistry codes with minimal modifications, in full consistency with the treatment of charge transfer excitations. The performance of different variants is evaluated via benchmark calculations on various sets from previous works, containing small- and medium-sized systems, including the nucleobases. It is shown that with the spin-scaled approximate methods, in particular the scaled opposite-spin variant of the ADC(2) method, the accuracy of EOM-CCSD is achievable at a fraction of the computational cost, also outperforming many common electron propagator approaches.

Cherry-Picking Resolvents: Recovering the Valence Contribution in X-ray Two-Photon Absorption within the Core-Valence-Separated Equation-of-Motion Coupled-Cluster Response Theory

Calculations of first-order response wave functions in the X-ray regime often diverge within correlated frameworks such as equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), a consequence of the coupling with the valence ionization continuum. Here, we extend our strategy of introducing a hierarchy of approximations to the EOM-EE-CCSD resolvent (or, inversely, the model Hamiltonian) involved in the response equations for the calculation of X-ray two-photon absorption (X2PA) cross sections. We exploit the frozen-core core-valence separation (fc-CVS) scheme to first decouple the core and valence Fock spaces, followed by a separate approximate treatment of the valence resolvent. We demonstrate the robust convergence of X-ray response calculations within this framework and compare X2PA spectra of small benchmark molecules with the previously reported density functional theory results.

Relativistic EOM-CCSD for Core-Excited and Core-Ionized State Energies Based on the Four-Component Dirac-Coulomb(-Gaunt) Hamiltonian

We report an implementation of the core-valence separation approach to the four-component relativistic Hamiltonian-based equation-of-motion coupled-cluster with singles and doubles theory (CVS-EOM-CCSD) for the calculation of relativistic core-ionization potentials and core-excitation energies. With this implementation, which is capable of exploiting double group symmetry, we investigate the effects of the different CVS-EOM-CCSD variants and the use of different Hamiltonians based on the exact two-component (X2C) framework on the energies of different core-ionized and -excited states in halogen- (CH₃I, HX, and X^-, X = Cl-At) and xenon-containing (Xe, XeF₂) species. Our results show that the X2C molecular mean-field approach [Sikkema, J.; J. Chem. Phys. 2009, 131, 124116], based on four-component Dirac-Coulomb mean-field calculations (²DC^M), is capable of providing core excitations and ionization energies that are nearly indistinguishable from the reference four-component energies for up to and including fifth-row elements. We observe that two-electron integrals over the small-component basis sets lead to non-negligible contributions to core binding energies for the K and L edges for atoms such as iodine or astatine and that the approach based on Dirac-Coulomb-Gaunt mean-field calculations (²DCG^M) are significantly more accurate than X2C calculations for which screened two-electron spin-orbit interactions are included via atomic mean-field integrals.

Probing Basis Set Requirements for Calculating Core Ionization and Core Excitation Spectra Using Correlated Wave Function Methods

We investigate the basis set requirements for the accurate calculation of core excitations and core ionizations using correlated wave functions of coupled cluster type and linear response methods for describing the excitation. When a core excitation is described as an energy difference calculated using density functional theory, the basis set can be tailored to provide a balanced description of the reference- and excited-hole states. When the core excitation process is described by coupled cluster linear response methods, however, the basis set requirements are somewhat different. A systematic study of the sensitivity of the result to the basis set parameters suggests that a relatively large set of s- and p-type basis functions in combination with a careful selection of valence and core polarization functions is required. Based on these results, we propose a hierarchical sequence of basis sets, denoted ccX-nZ (n = D, T, Q, 5) for the atoms B-Ne, which are suitable for the calculation of core excitations by the correlated wave function linear response and equation-of-motion methods. The ccX-nZ series provides lower basis set errors for a given cardinal number or number of basis functions than other existing basis sets. For large systems, the ccX-nZ basis sets can be combined with the standard basis sets by placing the ccX-nZ only on the atoms where core excitations are of interest, but the accuracy of such mixed basis sets appears to be system-dependent.

Accelerated multimodel Newton-type algorithms for faster convergence of ground and excited state coupled cluster equations

We introduce a multimodel approach to solve coupled cluster equations, employing a quasi-Newton algorithm for the ground state and an Olsen algorithm for the excited states. In these algorithms, both of which can be viewed as Newton algorithms, the Jacobian matrix of a lower level coupled cluster model is used in Newton equations associated with the target model. Improvements in convergence then imply savings for sufficiently large molecular systems, since the computational cost of macroiterations scales more steeply with system size than the cost of microiterations. The multimodel approach is suitable when there is a lower level Jacobian matrix that is much more accurate than the zeroth order approximation. Applying the approach to the CC3 equations, using the CCSD approximation of the Jacobian, we show that the time spent to determine the ground and valence excited states can be significantly reduced. We also find improved convergence for core excited states, indicating that similar savings will be obtained with an explicit implementation of the core-valence separated CCSD Jacobian transformation.

Dyson orbitals within the fc-CVS-EOM-CCSD framework: theory and application to X-ray photoelectron spectroscopy of ground and excited states

Ionization energies and Dyson orbitals within frozen-core core–valence separated equation-of-motion coupled cluster singles and doubles (fc-CVS-EOM-CCSD) enable efficient and reliable calculations of standard XPS and of UV-pump/XPS probe spectra.

How to stay out of trouble in RIXS calculations within equation-of-motion coupled-cluster damped response theory? Safe hitchhiking in the excitation manifold by means of core–valence separation

We present a novel approach with robust convergence of the response equations for computing resonant inelastic X-ray scattering (RIXS) cross sections within the equation-of-motion coupled-cluster (EOM-CC) framework.

Approximate Green's Function Coupled Cluster Method Employing Effective Dimension Reduction

The Green's function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structure of molecular and periodic systems, especially when electrons of the system are strongly correlated. However, in order for the GFCC to become a method that may be routinely used in the electronic structure calculations, robust numerical techniques and approximations must be employed to reduce its extremely high computational overhead. In our recent studies, it has been demonstrated that the GFCC equations can be solved directly in the frequency domain using iterative linear solvers, which can be easily distributed in a massively parallel environment. In the present work, we demonstrate a successful application of model-order-reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires high-resolution descriptions of spectral function, instead of solving the GFCC linear equation of full dimension for every single frequency point of interest, an efficiently solvable linear system model of a reduced dimension may be built upon projecting the original GFCC linear system onto a subspace. From this reduced order model is obtained a reasonable approximation to the full dimensional GFCC linear equations in both interpolative and extrapolative spectral regions. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some selected frequencies within the spectral region of interest. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The method is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems such as carbon monoxide, 1,3-butadiene, benzene, and adenine. To reach the same level of accuracy as that of the original GFCC method, the application of MOR in the GFCC method is able to significantly lower the original computational cost for the aforementioned molecules in designated frequency regimes. As a byproduct, the reduced order model obtained by this method is found to provide a high-quality initial guess, which improves the convergence rate for the existing iterative linear solver.

New and Efficient Equation-of-Motion Coupled-Cluster Framework for Core-Excited and Core-Ionized States

We present a fully analytical implementation of the core-valence separation (CVS) scheme for the equation-of-motion (EOM) coupled-cluster singles and doubles (CCSD) method for calculations of core-level states. Inspired by the CVS idea as originally formulated by Cederbaum, Domcke, and Schirmer, pure valence excitations are excluded from the EOM target space and the frozen-core approximation is imposed on the reference-state amplitudes and multipliers. This yields an efficient, robust, practical, and numerically balanced EOM-CCSD framework for calculations of excitation and ionization energies as well as state and transition properties (e.g., spectral intensities, natural transition, and Dyson orbitals) from both the ground and excited states. The errors in absolute excitation/ionization energies relative to the experimental reference data are on the order of 0.2-3.0 eV, depending on the K-edge considered and on the basis set used, and the shifts are systematic for each edge. Compared to a previously proposed CVS scheme where CVS was applied as a posteriori projection only during the solution of the EOM eigenvalue equations, the new scheme is computationally cheaper. It also achieves better cancellation of errors, yielding similar spectral profiles but with absolute core excitation and ionization energies that are systematically closer to the corresponding experimental data. Among the presented results are calculations of transient-state X-ray absorption spectra, relevant for interpretation of UV-pump/X-ray probe experiments.

Green's function coupled cluster formulations utilizing extended inner excitations

In this paper, we analyze new approximations of the Green's function coupled cluster (GFCC) method where locations of poles are improved by extending the excitation level of inner auxiliary operators. These new GFCC approximations can be categorized as the GFCC-i(n, m) method, where the excitation level of the inner auxiliary operators (m) used to describe the ionization potential and electron affinity effects in the N - 1 and N + 1 particle spaces is higher than the excitation level (n) used to correlate the ground-state coupled cluster wave function for the N-electron system. Furthermore, we reveal the so-called "n + 1" rule in this category [or the GFCC-i(n, n + 1) method], which states that in order to maintain size-extensivity of the Green's function matrix elements, the excitation level of inner auxiliary operators X _p (ω) and Y _q (ω) cannot exceed n + 1. We also discuss the role of the moments of coupled cluster equations that in a natural way assures these properties. Our implementation in the present study is focused on the first approximation in this GFCC category, i.e., the GFCC-i(2,3) method. As our first practice, we use the GFCC-i(2,3) method to compute the spectral functions for the N₂ and CO molecules in the inner and outer valence regimes. In comparison with the Green's function coupled cluster singles, doubles results, the computed spectral functions from the GFCC-i(2,3) method exhibit better agreement with the experimental results and other theoretical results, particularly in terms of providing higher resolution of satellite peaks and more accurate relative positions of these satellite peaks with respect to the main peak positions.

Green's Function Coupled-Cluster Approach: Simulating Photoelectron Spectra for Realistic Molecular Systems

In this paper, we present an efficient implementation for the analytical energy-dependent Green's function coupled-cluster with singles and doubles (GFCCSD) approach with our first practice being computing spectral functions of realistic molecular systems. Because of its algebraic structure, the presented method is highly scalable and is capable of computing spectral function for a given molecular system in any energy region. Several typical examples have been given to demonstrate its capability of computing spectral functions not only in the valence band but also in the core-level energy region. Satellite peaks have been observed in the inner valence band and core-level energy region where a many-body effect becomes significant and the single particle picture of ionization often breaks down. The accuracy test has been carried out by extensively comparing the computed spectral functions by our GFCCSD method with experimental photoelectron spectra as well as the theoretical ionization potentials obtained from other methods. It turns out the GFCCSD method is able to provide a qualitative or semiquantitative level of description of ionization processes in both the core and valence regimes. To significantly improve the GFCCSD results for the main ionic states, a larger basis set can usually be employed, whereas the improvement of the GFCCSD results for the satellite states needs higher-order many-body terms to be included in the GFCC implementation.

Relativistic time-dependent density functional theories

The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed.

Model Order Reduction Algorithm for Estimating the Absorption Spectrum

The ab initio description of the spectral interior of the absorption spectrum poses both a theoretical and computational challenge for modern electronic structure theory. Due to the often spectrally dense character of this domain in the quantum propagator's eigenspectrum for medium-to-large sized systems, traditional approaches based on the partial diagonalization of the propagator often encounter oscillatory and stagnating convergence. Electronic structure methods which solve the molecular response problem through the solution of spectrally shifted linear systems, such as the complex polarization propagator, offer an alternative approach which is agnostic to the underlying spectral density or domain location. This generality comes at a seemingly high computational cost associated with solving a large linear system for each spectral shift in some discretization of the spectral domain of interest. In this work, we present a novel, adaptive solution to this high computational overhead based on model order reduction techniques via interpolation. Model order reduction reduces the computational complexity of mathematical models and is ubiquitous in the simulation of dynamical systems and control theory. The efficiency and effectiveness of the proposed algorithm in the ab initio prediction of X-ray absorption spectra is demonstrated using a test set of challenging water clusters which are spectrally dense in the neighborhood of the oxygen K-edge. On the basis of a single, user defined tolerance we automatically determine the order of the reduced models and approximate the absorption spectrum up to the given tolerance. We also illustrate that, for the systems studied, the automatically determined model order increases logarithmically with the problem dimension, compared to a linear increase of the number of eigenvalues within the energy window. Furthermore, we observed that the computational cost of the proposed algorithm only scales quadratically with respect to the problem dimension.