Cluster perturbation theory. II. Excitation energies for a coupled cluster target state

Cluster perturbation theory. XI. Excitation-energy series using a variational excitation-energy function

Traditionally, excitation energies in coupled-cluster (CC) theory have been calculated by solving the CC Jacobian eigenvalue equation. However, based on our recent work [Jørgensen et al., Sci. Adv. 10, eadn3454 (2024)], we propose a reformulation of the calculation of excitation energies where excitation energies are determined as a conventional molecular property. To this end, we introduce an excitation-energy function that depends on the CC Jacobian and the right and left eigenvectors for the Jacobian eigenvalue problem. This excitation-energy function is variational with respect to the right and left eigenvectors but not with respect to the cluster amplitudes. Instead, the cluster amplitudes satisfy the cluster-amplitude equations, and we set up an excitation-energy Lagrangian by adding to the excitation-energy function the cluster-amplitude equations with an undetermined multiplier for each cluster-amplitude constraint. The excitation-energy Lagrangian is variational in all its parameters. Based on the variational property of the Lagrangian, we have determined two quadratically convergent excitation-energy series: the total-order cluster-perturbation (tCP) and variational cluster-perturbation (vCP) excitation-energy series. Calculations of the excitation energies of three small molecules have shown that the vCP series is to be preferred over the tCP series. The test calculations have been carried out for CPS(D) expansions [targeting the CC singles-and-doubles (CCSD) wave function from the CC singles wave function] and the CPSD(T) expansion [targeting the CC singles-doubles-triples (CCSDT) wave function from the CCSD wave function]. For the S(D) and SD(T) orbital excitation space calculations, we obtain in the second vCP iteration excitation energies with a mean deviation from CCSD excitation energies of about 0.04 eV for the S(D) orbital spaces, and for the SD(T) orbital space calculation, we obtain a mean deviation from the CCSDT excitation energies of 0.001 eV.

Cluster Perturbation Theory for Core Excited States and Core Ionization Potentials Using Core-Valence Separation

The development of accurate and fast computational procedures for the ab initio calculation of X-ray spectroscopies is paramount to facilitate theoretical analysis of modern X-ray experiments on molecules. Herein, we present the extension of Cluster Perturbation theory to comprehend the calculation of core excited states and core ionization potentials using the core-valence separation approximation, which has seen widespread success for various quantum chemistry methods. We derive the theoretical framework for introducing core-valence separation into Cluster Perturbation series for excitation energies and display the performance of the methodology in S(D) orbital excitation spaces. The obtained core excitation energies on a test set of medium sized organic molecules show that carbon, nitrogen, and oxygen K-edge excitation energies can be determined with errors below 2 eV relative to the CCSD reference results using the developed CPS(D) excitation energy models which can be used for systems way beyond the reach of conventional CCSD.

Cluster perturbation theory. X. A parallel implementation of Lagrangian perturbation series for the coupled cluster singles and doubles ground-state energy through fifth order

We describe an efficient implementation of cluster perturbation and Møller-Plesset Lagrangian energy series through the fifth order that targets the coupled cluster singles and doubles energy utilizing the resolution of the identity approximation. We illustrate the computational performance of the implementation by performing ground state energy calculations on systems with up to 1200 basis functions using a single node and by comparison to conventional coupled cluster singles and doubles calculations. We further show that our hybrid message passing interface/open multiprocessing parallel implementation that also utilizes graphical processing units can be used to obtain fifth order energies on systems with almost 1200 basis functions with a 90 min "time to solution" running on Frontier at Oak Ridge National Laboratory.

A variational reformulation of molecular properties in electronic-structure theory

Conventional quantum-mechanical calculations of molecular properties, such as dipole moments and electronic excitation energies, give errors that depend linearly on the error in the wave function. An exception is the electronic energy, whose error depends quadratically on the error in wave function. We here describe how all properties may be calculated with a quadratic error, by setting up a variational Lagrangian for the property of interest. Because the construction of the Lagrangian is less expensive than the calculation of the wave function, this approach substantially improves the accuracy of quantum-chemical calculations without increasing cost. As illustrated for excitation energies, this approach enables the accurate calculation of molecular properties for larger systems, with a short time-to-solution and in a manner well suited for modern computer architectures.

Cluster perturbation theory IX: Perturbation series for the coupled cluster singles and doubles ground state energy

In this paper, we develop and analyze a number of perturbation series that target the coupled cluster singles and doubles (CCSD) ground state energy. We show how classical Møller-Plesset perturbation theory series can be restructured to target the CCSD energy based on a reference CCS calculation and how the corresponding cluster perturbation series differs from the classical Møller-Plesset perturbation series. Subsequently, we reformulate these series using the coupled cluster Lagrangian framework to obtain series, where fourth and fifth order energies are determined only using parameters through second order. To test the methods, we perform a series of test calculations on molecular photoswitches of both total energies and reaction energies. We find that the fifth order reaction energies are of CCSD quality and that they are of comparable accuracy to state-of-the-art approximations to the CCSD energy based on local pair natural orbitals. The advantage of the present approach over local correlation methods is the absence of user defined threshold parameters for neglecting or approximating contributions to the correlation energy. Fixed threshold parameters lead to discontinuous energy surfaces, although this effect is often small enough to be ignored, but the present approach has a differentiable energy that will facilitate derivation and implementation of gradients and higher derivatives. A further advantage is that the calculation of the perturbation correction is non-iterative and can, therefore, be calculated in parallel, leading to a short time-to-solution.

Tensor Hypercontraction of Cluster Perturbation Theory: Quartic Scaling Perturbation Series for the Coupled Cluster Singles and Doubles Ground-State Energies

Even though cluster perturbation theory has been shown to be a robust noniterative alternative to coupled cluster theory, it is still plagued by high order polynomial computational scaling and the storage of higher order tensors. We present a proof-of-concept strategy for implementing a cluster perturbation theory ground-state energy series for the coupled cluster singles and doubles energy with N4 computational scaling using tensor hypercontraction (THC). The reduction in computational scaling by two orders is achieved by decomposing two electron repulsion integrals, doubles amplitudes and multipliers, as well as selected double intermediates to the THC format. Using the outlined strategy, we showcase that the THC pilot implementations retain numerical accuracy to within 1 kcal/mol relative to corresponding conventional and density fitting implementations, and we empirically verify the N4 scaling.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].

Coupled cluster theory on modern heterogeneous supercomputers

This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.

Cluster perturbation theory. VIII. First order properties for a coupled cluster state

We have extended cluster perturbation (CP) theory to comprehend the calculation of first order properties (FOPs). We have determined CP FOP series where FOPs are determined as a first energy derivative and also where the FOPs are determined as a generalized expectation value of the external perturbation operator over the coupled cluster state and its biorthonormal multiplier state. For S(D) orbital excitation spaces, we find that the CP series for FOPs that are determined as a first derivative in general in second order have errors of a few per cent in the singles and doubles correlation contribution relative to the targeted coupled cluster (CC) results. For a SD(T) orbital excitation space, we find that the CP series for FOPs determined as a generalized expectation value in second order have errors of about ten percent in the triples correlation contribution relative to the targeted CC results. These second order models therefore constitute viable alternatives for determining high quality FOPs.

Cluster Perturbation Theory. VII. The convergence of Cluster Perturbation Expansions

The convergence of the recently developed cluster perturbation CP expansions (Pawlowski et al, J. Chem. Phys. 150 134108(2019)) is analyzed with the double purpose of developing the mathematical tools and concepts needed to describe these expansions at general order and to identify the factors that define the rate of convergence of CP series. To this end, the CP energy, amplitude, and Lagrangian multiplier equations as functions of the perturbation strength are developed. By determining the critical points, defined as the perturbation strengths for which the Jacobian become singular, the rate of convergence as well as the intruder and critical states are determined for five simple molecules: BH, CO, H2O, NH3, and HF. To describe the patterns of convergence for these expansions at orders lower than the high-order asymptotic limit, a model is developed, where the perturbation corrections arise from two critical points. It is shown that this model allows rationalization of the behavior of the perturbation corrections at much lower order than required for the onset of the asymptotic convergence. For the H2O, CO, and HF molecules, the pattern and rate of convergence is defined by critical states where the Fock-operator underestimates the excitation energies, whereas the pattern and rate of convergence for BH is defined by critical states where the Fock-operator overestimates the excitation energy. For the NH3 molecule, both forms of critical points are required to describe the convergence behavior up to at least order 25.

Cluster perturbation theory. VI. Ground-state energy series using the Lagrangian

In this paper, we derive alternative cluster perturbation series to the energy cluster perturbation (ECP) series of Paper I. The ECP series were derived using the standard coupled cluster energy framework. Here, we use the coupled cluster Lagrangian framework to derive the Lagrangian cluster perturbation (LCP) series and show that a slightly modified order concept means that the ECP and the LCP series become identical. Using the Lagrangian, we also derive a perturbation series where total cluster amplitudes and multipliers are determined through the same orders as dictated by the 2n+1/2n+2 rule, the VCP series. The VCP energies have errors that are bilinear in the errors of the total cluster amplitudes and multipliers. Test calculations have been performed for S(D) and SD(T) orbital excitation spaces. The convergence of the test calculations can be divided into two groups. The first group contains molecules that exhibit a slow monotonic geometric convergence pattern in both orbital spaces. The second group, containing the rest of the molecules, exhibits a fast low order convergence. For the S(D) calculations, the deviations in fourth order from the coupled cluster singles and doubles energy are below 1.3 per cent for the LCP series and 0.3 per cent for the VCP series, respectively. For the SD(T) calculations, the second group hardly shows any difference between the low order convergence of the three series ECP, LCP, and LCP and at fourth order the deviations from the triples correlation energy are less than 1 per cent.

Near-Exact CCSDT Energetics from Rank-Reduced Formalism Supplemented by Non-iterative Corrections

We introduce a non-iterative energy correction, added on top of the rank-reduced coupled-cluster method with single, double, and triple substitutions, that accounts for excitations excluded from the parent triple excitation subspace. The formula for the correction is derived by employing the coupled-cluster Lagrangian formalism, with an additional assumption that the parent excitation subspace is closed under the action of the Fock operator. Owing to the rank-reduced form of the triple excitation amplitudes tensor, the computational cost of evaluating the correction scales as N⁷, where N is the system size. The accuracy and computational efficiency of the proposed method is assessed for both total and relative correlation energies. We show that the non-iterative correction can fulfill two separate roles. If the accuracy level of a fraction of kJ/mol is sufficient for a given system, the correction significantly reduces the dimension of the parent triple excitation subspace needed in the iterative part of the calculations. Simultaneously, it enables reproducing the exact CCSDT results to an accuracy level below 0.1 kJ/mol, with a larger, yet still reasonable, dimension of the parent excitation subspace. This typically can be achieved at a computational cost only several times larger than required for the CCSD(T) method. The proposed method retains the black-box features of the single-reference coupled-cluster theory; the dimension of the parent excitation subspace remains the only additional parameter that has to be specified.

A new generation of diagonal self-energies for the calculation of electron removal energies

A new generation of diagonal self-energy approximations in ab initio electron propagator theory for the calculation of electron removal energies of molecules and molecular ions has been derived from an intermediately normalized, Hermitized super-operator metric. These methods and widely used antecedents such as the outer valence Green's function and the approximately renormalized partial third order method are tested with respect to a dataset of vertical ionization energies generated with a valence, triple-ζ, correlation-consistent basis set and a converged series of many-body calculations whose accuracy approaches that of full configuration interaction. Several modifications of the diagonal second-order self-energy, a version of G₀W₀ theory based on Tamm-Dancoff excitations and several non-diagonal self-energies are also included in the tests. All new methods employ canonical Hartree-Fock orbitals. No adjustable or empirical parameters appear. A hierarchy of methods with optimal accuracy for a given level of computational efficiency is established. Several widely used diagonal self-energy methods are rendered obsolete by the new hierarchy whose members, in order of increasing accuracy, are (1) the opposite-spin non-Dyson diagonal second-order or os-nD-D2, (2) the approximately renormalized third-order quasiparticle or Q3+, (3) the renormalized third-order quasiparticle or RQ3, (4) the approximately renormalized linear third-order or L3+, and (5) the renormalized linear third-order or RL3 self-energies.

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.

Cluster perturbation theory. IV. Convergence of cluster perturbation series for energies and molecular properties

The theoretical foundation has been developed for establishing whether cluster perturbation (CP) series for the energy, molecular properties, and excitation energies are convergent or divergent and for using a two-state model to describe the convergence rate and convergence patterns of the higher-order terms in the CP series. To establish whether the perturbation series are convergent or divergent, a fictitious system is introduced, for which the perturbation is multiplied by a complex scaling parameter z. The requirement for convergent perturbation series becomes that the energy or molecular property, including an excitation energy, for the fictitious system is an analytic, algebraic function of z that has no singularities when the norm |z| is smaller than one. Examples of CP series for the energy and molecular properties, including excitation energies, are also presented, and the two-state model is used for the interpretation of the convergence rate and the convergence patterns of the higher-order terms in these series. The calculations show that the perturbation series effectively become a two-state model at higher orders.