Correlation Energy Research Articles

Forcing broken symmetry to recover static electronic correlation: the H2 molecule test case

ABSTRACT A self-consistent field method based on a density matrix functional theory scheme is presented to compute the potential energy curve of the hydrogen molecule. A functional to recover the so-called cumulant correlation energy is introduced. The form of such functional is very simple being dependent explicitly only on the square of the overlap matrix element between the unrestricted non-orthogonal occupied orbitals of the calculation. The functional is a sum of two contributions referring to non-dynamic and dynamic correlation. The agreement with the full-CI potential energy curve is within chemical accuracy.

Dynamic mask generation based on peak to correlation energy ratio for light reflection and shadow in PIV images

The performance of particle image velocimetry (PIV) measurements directly depends on the identification and removal of noisy regions in the field of view. Regions in the image with shadows and light reflections directly affect the PIV correlation and consequently deteriorate the vector field. This work proposes a new approach for mask generation based on peak correlation energy (PCE) ratio to remove the noisy effects of light reflection and shadow in PIV images. This approach considers that it is possible to relate low and high PCE values to noise and signal, respectively. Based on this, the PCE values are organized in a probability density function and a threshold is automatically estimated to separate the two regions. Furthermore, a mask correction step is proposed in the present method. The efficacy of this method is demonstrated through cases using both synthetic and experimental images, showcasing its potential to improve PIV analysis accuracy.

Making quantum chemistry compressive and expressive: Toward practical ab‐initio simulation

AbstractAb‐initio quantum chemistry simulations are essential for understanding electronic structure of molecules and materials in almost all areas of chemistry. A broad variety of electronic structure theories and implementations has been developed in the past decades to hopefully solve the many‐body Schrödinger equation in an approximate manner on modern computers. In this review, we present recent progress in advancing low‐rank electronic structure methodologies that rely on the wavefunction sparsity and compressibility to select the important subset of electronic configurations for both weakly and strongly correlated molecules. Representative chemistry applications that require the many‐body treatment beyond traditional density functional approximations are discussed. The low‐rank electronic structure theories have further prompted us to highlight compressive and expressive principles that are useful to catalyze idea of quantum learning models. The intersection of the low‐rank correlated feature design and the modern deep neural network learning provides new feasibilities to predict chemically accurate correlation energies of unknown molecules that are not represented in the training dataset. The results by others and us are discussed to reveal that the electronic feature sets from an extremely low‐rank correlation representation, which is very poor for explicit energy computation, are however sufficiently expressive for capturing and transferring electron correlation patterns across distinct molecular compositions, bond types and geometries.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Software > Simulation Methods

Modeling Multi-Step Organic Reactions: Can Density Functional Theory Deliver Misleading Chemistry?

Many organic reactions are characterized by a complex mechanism with a variety of transition states and intermediates of different chemical natures. Their correct and accurate theoretical characterization critically depends on the accuracy of the computational method used. In this work, we study a complex ambimodal cycloaddition with five transition states, two intermediates, and three products, and we ask whether density functional theory (DFT) can provide a correct description of this type of complex and multifaceted reaction. Our work fills a gap in that most systematic benchmarks of DFT for chemical reactions have considered much simpler reactions. Our results show that many density functionals not only lead to seriously large errors but also differ from one another in predicting whether the reaction is ambimodal. Only a few of the available functionals provide a balanced description of the complex and multifaceted reactions. The parameters varied in the tested functionals are the ingredients, the treatment of medium-range and nonlocal correlation energy, and the inclusion of Hartree-Fock exchange. These results show a clear need for more benchmarks on the mechanisms of large molecules in complex reactions.

Correlation consistent basis sets designed for density functional theory: Third-row atoms (Ga-Br).

The correlation consistent basis sets (cc-pVnZ with n = D, T, Q, 5) for the Ga-Br elements have been redesigned, tuning the sets for use for density functional approximations. Steps to redesign these basis sets for an improved correlation energy recovery and efficiency include truncation of higher angular momentum functions, recontraction of basis set coefficients, and reoptimization of basis set exponents. These redesigned basis sets are compared with conventional cc-pVnZ basis sets and other basis sets, which are, in principle, designed to achieve systematic improvement with respect to increasing basis set size. The convergence of atomic energies, bond lengths, bond dissociation energies, and enthalpies of formation to the Kohn-Sham limit is improved relative to other basis sets where convergence to the Kohn-Sham limit is typically not observed.

Perturbative variational quantum algorithms for material simulations

Reducing circuit depth is essential for implementing quantum simulations of electronic structure on near-term quantum devices. In this work, we propose a variational quantum eigensolver (VQE) based perturbation theory (PT) algorithm to accurately simulate electron correlation of periodic materials with shallow ansatz circuits, which are generated from adaptive derivative-assembled pseudo-trotter or qubit-excitation-based VQE calculations using a loose convergence criteria. Here, the major part of the electron correlation is described using the VQE ansatz circuit and the remaining correlation energy is described by either multireference or similarity transformation-based PT. Numerical results demonstrate that the new algorithms are able to accurately describe electron correlation of the LiH crystal with only one circuit parameter, in contrast with ~30 parameters required in the adaptive VQE to achieve the same accuracy. Meanwhile, for fixed-depth ansätze, e.g. unitary coupled cluster, we demonstrate that the VQE-based PT provides an appealing scheme to improve their accuracy.

Determination of notch stress intensity factors for V-notched specimens under mode I loading using the 3D-Digital image correlation and strain energy approach

In this paper, the fracture behaviour of Single Edge Notched Tension Specimen (SENT) is investigated by using the three-dimensional digital image correlation (3D-DIC) method and the mode I notch stress intensity factors (NSIFs) for the SENT specimen is derived. Initially, the concept of NSIFs and different experimental methods are reviewed. This is followed by a review of the theory of the DIC method. Then, the step-by-step process for setting up a 3D-DIC full-field experimental system for the SENT specimens tested is described. In the experiments, the SENT specimens were tested with pinned and clamped boundary conditions (BCs) and different notch initial opening angles. Furthermore, the analytical stress fields around the sharp V-notches are presented. Combining with the strain energy approach (SEA), the experimental strain fields are analysed to calculate the mode I NSIFs. Moreover, the presented numerical mode I NSIF values for the SENT model are validated by the available results in the literature. Finally, the experimental results obtained from the 3D-DIC method are compared with the numerical results from finite element analysis (FEA) to verify the feasibility and accuracy of the 3D-DIC method. The effect of different BCs of pinned and clamped ends is discussed.

Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins "Conjecture".

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

The Role of Bond Functions in Describing Intermolecular Electron Correlation for Van der Waals Dimers: A Study of (CH4)2 and Ne2.

We present a study of the intermolecular interactions in van der Waals complexes of methane and neon dimers within the framework of the CCSD method. This approach was implemented and applied to calculate and examine the behavior of the contracted two-particle reduced density matrix (2-RDM). It was demonstrated that the region near the minimum of the two-particle density matrix correlation part, corresponding to the primary bulk of the Coulomb hole contribution, exerts a significant influence on the dispersion interaction energetics of the studied systems. As a result, the bond functions approach was applied to improve the convergence performance for the intermolecular correlation energy results with respect to the size of the atomic basis. For this, substantial acceleration was achieved by introducing an auxiliary basis of bond functions centered on the minima of the 2-RDM. For both methane and neon dimers, this general conclusion was confirmed with a series of CCSD calculations for the 2-RDM and the correlation energies.

The Relationship of Postprandial Plasma Glucose Versus Fasting Plasma Glucose in the Early Morning, Carbohydrates and Sugar Intake Amount, and Post-Meal Walking Steps using both Statistical Correlation and Viscoplastic Energy Model of GH-Method: Mathphysical Medicine (No. 954, VMT #353, 11/15/2023)

The Relationship of Postprandial Plasma Glucose Versus Fasting Plasma Glucose in the Early Morning, Carbohydrates and Sugar Intake Amount, and Post-Meal Walking Steps using both Statistical Correlation and Viscoplastic Energy Model of GH-Method: Mathphysical Medicine (No. 954, VMT #353, 11/15/2023)

Theoretical study of susceptibility and effective mass in semiconductor quantum well-wires

The susceptibility and effective mass of electrons and hole vary as functions of carrier density and temperature in parabolic lateral barrier quantum-well wires (QWWRs). To investigate these properties, we utilize the leading-order dynamically screened Coulomb interaction, wherein the electron self-energy is calculated within the plasmon-pole approximation (PPA). All findings for GaAs QWWRs reveal a substantial difference between susceptibility and effective mass at low and high temperatures for varying carrier densities. These results contribute to illuminate of how Coulomb interactions and exchange–correlation energies influence the susceptibility and effective mass of GaAs semiconductors across a broad range of temperature and carrier density conditions.

Small-Occupation Density Functional Correlation Energy Correction to Wave Function Approximations.

In this work, we introduce a novel hybrid approach, termed WFT-soDFT, designed to seamlessly incorporate DFT correlation into wave function ansatzes. This is achieved through a partitioning of the orbital space, distinguishing between large and small natural occupation numbers associated with wave function theory (WFT) and DFT correlation, respectively. The method uses a novel criterion for partitioning the orbital space and mapping the electron density in natural orbitals with a small occupation with the correlation energy of fast electrons within the homogeneous electron gas. Central to our approach is the introduction of a separation parameter ν, the choice of the WFT approach, and the correlation functional. Here, we combine the RASCI wave function with hole and particle truncation with a local density correlation functional to only account for small-occupation correlation energy. We investigate the performance of the method in the study of small but challenging chemical systems, for which WFT-soDFT demonstrates notable improvements over pristine wave function calculations. These findings collectively highlight the potential of the WFT-soDFT approach as a computationally affordable strategy to improve the accuracy of WFT electronic structure calculations.

Block-Correlated Coupled Cluster Theory with up to Four-Pair Correlation for Accurate Static Correlation of Strongly Correlated Systems.

A block-correlated coupled cluster method with up to four-pair correlation based on the generalized valence bond wave function (GVB-BCCC4) is first implemented, which offers an alternative method for electronic structure calculations of strongly correlated systems. We developed some techniques to derive a set of compact and cost-effective equations for GVB-BCCC4, which include the definition of n-block (n = 1-4) Hamiltonian matrices, the combination of excitation operators, and the definition of independent amplitudes. We then applied the GVB-BCCC4 method to investigate several potential energy surfaces of strongly correlated systems with singlet ground states. Our calculations demonstrate that the GVB-BCCC4 method can provide nearly exact static correlation energies as the density matrix renormalization group method (on the basis of the same GVB orbitals). This work highlights the significance of four-pair correlation in quantitative descriptions of static correlation energy for strongly correlated systems.

Finite-size effects in periodic coupled cluster calculations

We provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Given exact Hartree-Fock orbitals and their corresponding orbital energies, we demonstrate that the correlation energy obtained from the approximate CCD method, after a finite number of fixed-point iterations over the amplitude equation, exhibits a finite-size error scaling as O(Nk−13). Here Nk is the number of discretization points in the Brillouin zone and characterizes the system size. Under additional assumptions ensuring the convergence of the fixed-point iterations, we demonstrate that the CCD correlation energy also exhibits a finite-size error scaling as O(Nk−13). Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculation can be boosted to O(Nk−1) with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order Møller-Plesset perturbation theory (MP3) for periodic systems.

Development of a machine learning finite-range nonlocal density functional.

Kohn-Sham density functional theory has been the most popular method in electronic structure calculations. To fulfill the increasing accuracy requirements, new approximate functionals are needed to address key issues in existing approximations. It is well known that nonlocal components are crucial. Current nonlocal functionals mostly require orbital dependence such as in Hartree-Fock exchange and many-body perturbation correlation energy, which, however, leads to higher computational costs. Deviating from this pathway, we describe functional nonlocality in a new approach. By partitioning the total density to atom-centered local densities, a many-body expansion is proposed. This many-body expansion can be truncated at one-body contributions, if a base functional is used and an energy correction is approximated. The contribution from each atom-centered local density is a single finite-range nonlocal functional that is universal for all atoms. We then use machine learning to develop this universal atom-centered functional. Parameters in this functional are determined by fitting to data that are produced by high-level theories. Extensive tests on several different test sets, which include reaction energies, reaction barrier heights, and non-covalent interaction energies, show that the new functional, with only the density as the basic variable, can produce results comparable to the best-performing double-hybrid functionals, (for example, for the thermochemistry test set selected from the GMTKN55 database, BLYP based machine learning functional gives a weighted total mean absolute deviations of 3.33kcal/mol, while DSD-BLYP-D3(BJ) gives 3.28kcal/mol) with a lower computational cost. This opens a new pathway to nonlocal functional development and applications.

The Energy of Data and Distance Correlation

The Energy of Data and Distance Correlation

On the notion of strong correlation in electronic structure theory.

Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples - hydrogen systems (Hn, n = 2, 4, 6), Be2, H-He-H, and benzene - and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of "correlation energy" and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird's rule.

Behavior of the continuum coupling correlation energy in the vicinity of the particle emission threshold - Gamow shell model study

The Gamow shell model provides the open quantum system formulation of nuclear shell model. In the coupled-channel representation, Gamow shell model provides the unified theory of nuclear structure and reactions which is well suited for the study of resonances and clustering. In this work, we apply this approach to study the continuum-coupling correlation energy for selected near-threshold states of 7Li, 7Be, using a translationally invariant Hamiltonian with an effective finite-range two-body interaction.

Spinless formulation of linearized adiabatic connection approximation and its comparison with the second order N-electron valence state perturbation theory.

The adiabatic connection (AC) approximation, along with its linearized variant AC0, was introduced as a method of obtaining dynamic correlation energy. When using a complete active space self-consistent field (CASSCF) wave function as a reference, the AC0 approximation is considered one of the most efficient multi-reference perturbation theories. It only involves the use of 1st- and 2nd-order reduced density matrices. However, some numerical results have indicated that the excitation energies predicted by AC0 are not as reliable as those from the second-order N-electron valence state perturbation theory (NEVPT2). In this study, we develop a spinless formulation of AC0 based on the Dyall Hamiltonian and provide a detailed comparison between AC0 and NEVPT2 approaches. We demonstrate the components within the correlation energy expressions that are common to both methods and those unique to either AC0 or NEVPT2. We investigate the role of the terms exclusive to NEVPT2 and explore the possibility of enhancing AC0's performance in this regard.

Dynamical electron correlation and the chemical bond. III. Covalent bonds in the A2 molecules (A = C–F)

The behavior of the dynamical electron correlation energy is remarkably complex at short internuclear distances: ΔEDEC(ΔR) = EDEC(ΔR) − EDEC(R = ∞) with ΔR = R − Re.