Correlation Energy Research Articles

Evaluation of Coulomb integrals with various energy operators to estimate the correlation energy in electronic structure calculations for molecules

Using energy operators RC1-nRD1-m, RC1-nr12-m, and r12-nr13-m with small (n, m) values is fundamental in electronic structure calculations. Analytical integrations of the cases (n, m) = (1, 0) and (0, 1) are based on the Laplace transformation with the integrand exp(-a2t2), the other cases are based on the Laplace transformation with the integrand exp(-a2t) and the two-dimensional version of the Boys function. These analytic expressions, with Gaussian function integrands, are useful for manipulation with higher moments of interelectronic distances, for example, in correlation calculations. The equations derived help to evaluate the one-, two-, and three-electron Coulomb integrals, òρ(1)RC1-nRD1-mdr1, òρ(1)ρ(2)RC1-nr12-mdr1dr2, and òρ(1)ρ(2)ρ(3)r12-nr13-mdr1dr2dr3, wherein ρ(i) is the one-electron density describing the electron clouds in molecules, solids, or any media or ensemble of materials. Analytical solutions to integrals are more useful than numeric solutions; however, the former is not available in many cases. We evaluate these integrals numerically, even more so, the òf(ρ(1))dr1 to òf(ρ(1),ρ(2),ρ(3))dr1dr2dr3 with the analytical function f. For this task, the commonly used density functional theory numerical integration scheme has been elaborated to 6 and 9 dimensions via Descartes product. More importantly, this numerical integration scheme works not only for Gaussian type but also for Slaterian types. Analogy is commented on in terms of the powerful empirical correction between quantum potential energy correction and the empirically corrected Newton’s universal law of gravity in the explanation of dark matter and energy, as well as its relation to Hartree-Fock and Kohn-Sham formalisms.

Enhancement of energy decomposition analysis in fragment molecular orbital calculations.

Energy decomposition analysis is one of the most attractive features of fragment molecular orbital (FMO) calculations from the point of view of practical applications. Here we report some enhancements for PIEDA in the ABINIT-MP program. One is a separation of the dispersion-type stabilization from the electron correlation energy, traditionally referred to as the "dispersion interaction" (DI). Another is an alternative evaluation of the electrostatic (ES) interaction using the restrained electrostatic potential (RESP) charges. The GA:CT stacked base pair and the Trp-Cage miniprotein were used as illustrative examples.

Structural, electronic, magnetic, and thermoelectric properties of full-Heusler alloys A2CoS (A=Cu, Zn) for spintronic application via conceptual DFT study

Half metallic ferromagnets play a pivotal role in spintronic applications. Our study focuses on theoretically predicted half metallic ferromagnetic compounds, determined through density functional theory calculations utilizing a first principles approach. The calculations involved the use of Perdew-Burke-Ernzerhof (GGA) and GGA + U exchange correlation energies. Structural analysis revealed the ferromagnetic nature, while electronic properties confirmed the half metallic behavior of these compounds-specifically, Cu2CoS and Zn2CoS. Our findings suggest that within a certain lattice parameter range, both compounds exhibit ferromagnetic behavior. However, beyond this range, Cu2CoS and Zn2CoS do not demonstrate ferromagnetism. Furthermore, our interest lies in examining their thermoelectric behavior using semi-classical Boltzmann transport theory. These results offer valuable insights for leveraging these materials in spintronics applications.

Adiabatic connection interaction strength interpolation method made accurate for the uniform electron gas.

The adiabatic connection interaction strength interpolation (ISI)-like method provides a high-level expression for the correlation energy, being, in principle, exact not only in the weak-interaction limit, where it recovers the second-order Görling-Levy perturbation term, but also in the strong-interaction limit that is described by the strictly correlated electron approach. In this work, we construct a genISI functional made accurate for the uniform electron gas, a solid-state physics paradigm that is a very difficult test for ISI-like correlation functionals. We assess the genISI functional for various jellium spheres with the number of electrons Z ≤ 912 and for the non-relativistic noble atoms with Z ≤ 290. For the jellium clusters, the genISI is remarkably accurate, while for the noble atoms, it shows a good performance, similar to other ISI-like methods. Then, the genISI functional can open the path using the ISI-like method in solid-state calculations.

Modulated transport and magnetic behavior in antiferromagnetic NdNiO3/SrIrO3 bilayers

Antiferromagnetic spintronics is intrigued due to its unique properties that could break through the restrictions in ferromagnets. A 3d/5d transition metal oxide heterostructure is a good platform in antiferromagnetic spintronics research since the strong spin–orbit coupling in 5d oxides may bring about delicate interaction with the correlation energy and motivate unconventional phenomena. Here, the transport and magnetic characters of bilayers composed of antiferromagnetic 3d perovskite NdNiO3 and 5d perovskite SrIrO3 were investigated. The decreased metal–insulator transition and Néel temperature associated with suppressed negative magnetoresistance, emerged spin-glass like phenomenon, and the humped nonlinear Hall effect were observed in NdNiO3/SrIrO3 bilayers, which were absent in NdNiO3 and SrIrO3 pure films. It suggests the important role of interfacial interaction between NdNiO3 and SrIrO3 in modulating heterostructure transport and magnetic behavior and also manifests that complex magnetic structures might be realized in NdNiO3/SrIrO3 bilayers.

Second quantization of many-body dispersion interactions for chemical and biological systems

The many-body dispersion (MBD) framework is a successful approach for modeling the long-range electronic correlation energy and optical response of systems with thousands of atoms. Inspired by field theory, here we develop a second-quantized MBD formalism (SQ-MBD) that recasts a system of atomic quantum Drude oscillators in a Fock-space representation. SQ-MBD provides: (i) tools for projecting observables (interaction energy, transition multipoles, polarizability tensors) on coarse-grained representations of the atomistic system ranging from single atoms to large structural motifs, (ii) a quantum-information framework to analyze correlations and (non)separability among fragments in a given molecular complex, and (iii) a path toward the applicability of the MBD framework to molecular complexes with even larger number of atoms. The SQ-MBD approach offers conceptual insights into quantum fluctuations in molecular systems and enables direct coupling of collective plasmon-like MBD degrees of freedom with arbitrary environments, providing a tractable computational framework to treat dispersion interactions and polarization response in intricate systems.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals.

Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.

Doubles Connected Moments Expansion: A Tractable Approximate Horn-Weinstein Approach for Quantum Chemistry.

Ab initio methods based on the second-order and higher connected moments, or cumulants, of a reference function have seen limited use in the determination of correlation energies of chemical systems over the years. Moment-based methods have remained unattractive relative to more ubiquitous methods, such as perturbation theory and coupled cluster theory, due in part to the intractable cost of assembling moments of high-order and poor performance of low-order expansions. Many of the traditional quantum chemical methodologies can be recast as a selective summation of perturbative contributions to their energy; using this familiar structure as a guide in selecting terms, we develop a scheme to approximate connected moments limited to double excitations. The tractable Doubles Connected Moments [DCM(N)] approximation is developed and tested against a multitude of common single-reference methods to determine its efficacy in the determination of the correlation energy of model systems and small molecules. The DCM(N) sequence of energies exhibits smooth convergence toward limiting values in the range of N = 11-14, with compute costs that scale as a noniterative O(M6) with molecule size, M. Numerical tests on correlation energy recovery for 55 small molecules comprising the G1 test set in the cc-pVDZ basis show that DCM(N) strongly outperforms MP2 and even CCD with a Hartree-Fock reference. When using an approximate Brueckner reference from orbital-optimized (oo) MP2, the resulting oo:DCM(N) energies converge to values more accurate than CCSD for 49 of 55 molecules. The qualitative success of the method in regions where strong correlation effects begin to dominate, even while maintaining spin purity, suggests this may be a good starting point in the development of methodologies for the description of strongly correlated or spin-contaminated systems while maintaining a tractable single-reference formalism.

Plane Waves Versus Correlation-Consistent Basis Sets: A Comparison of MP2 Non-Covalent Interaction Energies in the Complete Basis Set Limit.

Second-order Møller-Plesset perturbation theory (MP2) is the most expedient wave function-based method for considering electron correlation in quantum chemical calculations and, as such, provides a cost-effective framework to assess the effects of basis sets on correlation energies, for which the complete basis set (CBS) limit can commonly only be obtained via extrapolation techniques. Software packages providing MP2 energies are commonly based on atom-centered bases with innate issues related to possible basis set superposition errors (BSSE), especially in the case of weakly bonded systems. Here, we present noncovalent interaction energies in the CBS limit, free of BSSE, for 20 dimer systems of the S22 data set obtained via a highly parallelized MP2 implementation in the plane-wave pseudopotential molecular dynamics package CPMD. The specificities related to plane waves for accurate and efficient calculations of gas-phase energies are discussed, and results are compared to the localized (aug-)cc-pV[D,T,Q,5]Z correlation-consistent bases as well as their extrapolated CBS estimates. We find that the BSSE-corrected aug-cc-pV5Z basis can provide MP2 energies highly consistent with the CBS plane wave values with a minimum mean absolute deviation of ∼0.05 kcal/mol without the application of any extrapolation scheme. In addition, we tested the performance of 13 different extrapolation schemes and found that the X-3 expression applied to the (aug-)cc-pVXZ bases provides the smallest deviations against CBS plane wave values if the extrapolation sequence is composed of points D and T, while performs slightly better for TQ and Q5 extrapolations. Also, we propose as a reliable alternative to extrapolate total energies from the DTQ, TQ5, or DTQ5 data points. In spite of the general good agreement between the values obtained from the two types of basis sets, it is noticed that differences between plane waves and (aug-)cc-pVXZ basis sets, extrapolated or not, tend to increase with the number of electrons, thus raising the question of whether these discrepancies could indeed limit the attainable accuracy for localized bases in the limit of large systems.

Application of density functional theory for evaluating the mechanical properties and structural stability of dental implant materials

BackgroundTitanium is a commonly used material for dental implants owing to its excellent biocompatibility, strength-to-weight ratio, corrosion resistance, lightweight nature, hypoallergenic properties, and ability to promote tissue adhesion. However, alternative materials, such as titanium alloys (Ti–Al-2 V) and zirconia, are available for dental implant applications. This study discusses the application of Density Functional Theory (DFT) in evaluating dental implant materials' mechanical properties and structural stability, with a specific focus on titanium (Ti) metal. It also discusses the electronic band structures, dynamic stability, and surface properties. Furthermore, it presents the mechanical properties of Ti metal, Ti–Al-2 V alloy, and zirconia, including the stiffness matrices, average properties, and elastic moduli. This research comprehensively studies Ti metal's mechanical properties, structural stability, and surface properties for dental implants.MethodsWe used computational techniques, such as the CASTEP code based on DFT, GGA within the PBE scheme for evaluating electronic exchange–correlation energy, and the BFGS minimization scheme for geometry optimization. The results provide insights into the structural properties of Ti, Ti–Al-2 V, and zirconia, including their crystal structures, space groups, and atomic positions. Elastic properties, Fermi surface analysis, and phonon studies were conducted to evaluate the tensile strength, yield strength, ductility, elastic modulus, Poisson's ratio, hardness, fatigue resistance, and corrosion resistance.ResultsThe findings were compared with those of Ti–Al-2 V and zirconia to assess the advantages and limitations of each material for dental implant applications. This study demonstrates the application of DFT in evaluating dental implant materials, focusing on titanium, and provides valuable insights into their mechanical properties, structural stability, and surface characteristics.ConclusionsThe findings of this study contribute to the understanding of dental implant material behavior and aid in the design of improved materials with long-term biocompatibility and stability in the oral environment.

Unitary coupled-cluster for quantum computation of molecular properties in a strong magnetic field.

In truncated coupled-cluster (CC) theories, non-variational and/or generally complex ground-state energies can occur. This is due to the non-Hermitian nature of the similarity transformed Hamiltonian matrix in combination with CC truncation. For chemical problems that deal with real-valued Hamiltonian matrices, complex CC energies rarely occur. However, for complex-valued Hamiltonian matrices, such as those that arise in the presence of strong magnetic fields, complex CC energies can be regularly observed unless certain symmetry conditions are fulfilled. Therefore, in the presence of magnetic fields, it is desirable to pursue CC methods that are guaranteed to give upper-bound, real-valued energies. In this work, we present the first application of unitary CC to chemical systems in a strong magnetic field. This is achieved utilizing the variational quantum eigensolver algorithm applied to the unitary coupled-cluster singles and doubles (UCCSD) method. We benchmark the method on the H2 molecule in a strong magnetic field and then calculate UCCSD energies for the H4 molecule as a function of both geometry and field angle. We show that while standard CCSD can yield generally complex energies that are not an upper-bound to the true energy, UCCSD always results in variational and real-valued energies. We also show that the imaginary components of the CCSD energy are largest in the strongly correlated region. Last, the UCCSD calculations capture a large percentage of the correlation energy.

Impact analysis of mergers and acquisitions on the performance of China's new energy industries

The development of new energy industries is crucial to a nation's energy transition to carbon neutrality, while photovoltaic, wind power, lithium battery, and new energy vehicles industries have been active in the capital markets. We choose mergers and acquisitions (M&A) activities in such industries in China to investigate how they affect firms' short-run market performance and long-run innovation. Using data gathered from 2012 to 2020, we evaluate the degree of stock price volatility with the Carhart four-factor model combined with event study analysis. Changes in innovation after M&A activity has occurred are evaluated using the PSM-DID model. Empirical analysis shows that M&A could have a positive, short-run impact on stock prices of new energy industries, while the impact of technology M&A on sub-industries is not consistent. The cumulative average abnormal returns (CAAR) of the wind power and photovoltaic industries turn from negative to positive, and the correlative new energy vehicles and lithium battery industries reach the highest returns in a window period with similar trend lines. Furthermore, technology M&A positively affects the innovation performance of new energy firms, and this effect lasts three to four years. In contrast to prior industry-wide and conclusion-inconsistent M&A studies, we concentrate on new energy industries and prove the definite positive effect of technology M&A for the firms.

Entropy is a good approximation to the electronic (static) correlation energy

For an electronic system, given a mean field method and a distribution of orbital occupation numbers that are close to the natural occupations of the correlated system, we provide formal evidence and computational support to the hypothesis that the entropy (or more precisely −σS, where σ is a parameter and S is the entropy) of such a distribution is a good approximation to the correlation energy. Underpinning the formal evidence are mild assumptions: the correlation energy is strictly a functional of the occupation numbers, and the occupation numbers derive from an invertible distribution. Computational support centers around employing different mean field methods and occupation number distributions (Fermi–Dirac, Gaussian, and linear), for which our claims are verified for a series of pilot calculations involving bond breaking and chemical reactions. This work establishes a formal footing for those methods employing entropy as a measure of electronic correlation energy (e.g., i-DMFT [Wang and Baerends, Phys. Rev. Lett. 128, 013001 (2022)] and TAO-DFT [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)]) and sets the stage for the widespread use of entropy functionals for approximating the (static) electronic correlation.

Construction of SrTiO3/TiO2 heterojunction by in-situ ion exchange method to enhance photochemical performance

Construction of heterojunctions has been shown to be one of the most promising methods for the preparation of highly efficient catalysts, because it is feasible and effective for the separation of electron-hole pairs, prolonging the lifetime of photogenerated electrons. In this study, TiO2 was grown on fluorine-doped tin oxide glass (FTO), and then a strontium source was added to obtain heterojunctions between SrTiO3 and TiO2 using in-situ ion exchange hydrothermal method. The in-situ grown SrTiO3/TiO2 (TST) composite has a better absorption performance, a narrower band gap, and a lower carrier recombination rate compared to the pure TiO2 sample, which improves the photoelectrochemical performance. The photocurrent density of TST-10 (hydrothermally treated for 10 h) has the largest value of 0.06 mA·cm−2, which is about three times higher than that of pure TiO2. Then, the first-principles density functional theory (DFT) calculation was used to further explore the photocatalytic mechanism of TST. The exchange correlation energy was based on the Generalised Gradient Approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) functional calculation method to study the electronic and optical properties of SrTiO3/TiO2 heterostructures. The results show that the carrier transfer rate and the optical absorption coefficient of the material increase significantly after the establishment of the in-situ heterojunction. This study offers insight into the heterostructures’ interface significance on photocatalytic activity.

REAlgo: Rapid and efficient algorithm for estimating MP2/CCSD energy gradients for large molecular clusters.

This work reports the development of an algorithm for rapid and efficient evaluation of energy gradients for large molecular clusters employing correlated methods viz. second-order Møller-Plesset perturbation theory (MP2) theory and couple cluster singles and doubles (CCSD). The procedure segregates the estimation of Hartree-Fock (HF) and correlation components. The HF energy and gradients are obtained by performing a full calculation. The correlation energy is approximated as the corresponding two-body interaction energy. Correlation gradients for each monomer are approximated from the respective monomer-centric fragments comprising its immediate neighbours. The programmed algorithm is explored for the geometry optimization of large molecular clusters using the BERNY optimizer as implemented in the Gaussian suite of software. The accuracy and efficacy of the method are critically probed for a variety of large molecular clusters containing up to 3000 basis functions, in particular large water clusters. The CCSD level geometry optimization of molecular clusters containing ∼800 basis functions employing a modest hardware is also reported.

Electronic structure and molecular properties of nitisinone and mesotrione in water

ContextNitisinone is a medium-sized organic molecule that is used in treating hereditary tyrosinemia type 1 (HT-1). The structurally analogous mesotrione, however, is used as a pesticide/herbicide. What molecular properties are responsible for the similarity/dissimilarity of these molecules is investigated here. The solvent effect reduces the electron affinity to rather negative values and causes the negative electron affinity which manifests itself in a very high positive absolute reduction potential.MethodsB3LYP method was utilized for a geometry optimization of nitisinone and mesotrione in their neural and ionized (L0, L+, L−) forms of 6 structures. The calculations were conducted in water as a solvent using conductor-like polarizable continuum model (CPCM), nitisinone also in vacuo. The complete vibrational analysis at the true energy minimum allows evaluating the thermodynamic functions with focus to the zero-point energy and overall entropic term. The change of the Gibbs energy on reductions and/or oxidation facilitates evaluating the absolute reduction and absolute oxidation potentials. Also, DLPNO-CCSD(T) method that involves the major part of the correlation energy has been applied to nitisinone and mesotrione and their molecular ions.

Dynamic and Nondynamic Electron Correlation Energy Decomposition Based on the Node of the Hartree-Fock Slater Determinant.

Distinguishing between dynamic and nondynamic electron correlation energy is a fundamental concept in quantum chemistry. It can be challenging to make a clear distinction between the two types of correlation energy or to determine their actual contributions in specific cases using wave function theory. This is because both single-reference and multireference methods cover both types of correlation energy to some extent. Fixed-node diffusion quantum Monte Carlo (FNDMC) accurately covers dynamic correlations, but it is limited in overall accuracy by the node of the trial wave function. We introduce a methodology for partitioning an exact electron correlation energy into its dynamic and nondynamic components. This is accomplished by restricting a ground-state solution from sharing its node with a spin-restricted Hartree-Fock Slater determinant. The FNDMC method is used as a tool to conveniently project out a lowest-energy state obeying such a boundary condition. The proposed approach provides an unambiguous and useful procedure for separating electron correlation energy, as demonstrated on multiple systems, including the He atom, bond breaking of H2, the parametric H2-H2 system, the Be-Ne atomic series with low- and high-spin states for C, N, and O atoms, and small molecules such as BH, HF, and CO at both equilibrium and elongated configurations, respectively.

Optimizing the regularization in size-consistent second-order Brillouin-Wigner perturbation theory.

Despite its simplicity and relatively low computational cost, second-order Møller-Plesset perturbation theory (MP2) is well-known to overbind noncovalent interactions between polarizable monomers and some organometallic bonds. In such situations, the pairwise-additive correlation energy expression in MP2 is inadequate. Although energy-gap dependent amplitude regularization can substantially improve the accuracy of conventional MP2 in these regimes, the same regularization parameter worsens the accuracy for small molecule thermochemistry and density-dependent properties. Recently, we proposed a repartitioning of Brillouin-Wigner perturbation theory that is size-consistent to second order (BW-s2), and a free parameter (α) was set to recover the exact dissociation limit of H2 in a minimal basis set. Alternatively α can be viewed as a regularization parameter, where each value of α represents a valid variant of BW-s2, which we denote as BW-s2(α). In this work, we semi-empirically optimize α for noncovalent interactions, thermochemistry, alkane conformational energies, electronic response properties, and transition metal datasets, leading to improvements in accuracy relative to the abinitio parameterization of BW-s2 and MP2. We demonstrate that the optimal α parameter (α = 4) is more transferable across chemical problems than energy-gap-dependent regularization parameters. This is attributable to the fact that the BW-s2(α) regularization strength depends on all of the information encoded in the t amplitudes rather than just orbital energy differences. While the computational scaling of BW-s2(α) is iterative O(N5), this effective and transferable approach to amplitude regularization is a promising route to incorporate higher-order correlation effects at second-order cost.

Energy levels of carbon dangling-bond center (PbC center) at 4H-SiC(0001)/SiO2 interface

The electric properties of the carbon dangling-bond (PbC) center at a thermally oxidized 4H-SiC(0001)/SiO2 interface are investigated. We experimentally and theoretically determine the energy levels of the associated interface states to estimate the impacts of the PbC center on power device operations. By combining electrically detected magnetic resonance spectroscopy and capacitance–voltage measurements, the two PbC electronic levels [(0/−) and (+/0)] are determined as ∼1.2 and 0.6 eV from the valence band maximum, respectively. The effective correlation energy of the PbC center is 0.6 eV, which is 1.5 times larger than that of the silicon dangling-bond (Pb) center at Si/SiO2 interfaces. Our first-principles calculations confirm that the electronic levels of PbC are similar to experimental values. Considering these energy levels, the PbC center must impact both p- and n-channel devices, which is closely related to previously reported channel features.

Lead-free silver-indium based halide double perovskites for energy harvesting applications

Herein, density functional theory (DFT) is utilized to analyze physical characteristics of two novel halide double perovskites (HDPs) Na2AgInBr6 and K2AgInBr6. Exchange and correlation energies were estimated via employing generalized gradient approximation (GGA) along with modified Becke-Johnson (mBJ) potential. Geometry optimization predicted nonmagnetic ground state for both HDPs with cubic structure having Fm3m space group. Computed formation enthalpies (ΔHf) and tolerance factors (τG) revealed the thermodynamically stability of both compounds in cubic phase. Na2AgInBr6 is regarded as brittle while K2AgInBr6 is ductile in nature. Na2AgInBr6 exhibited a band gap (Eg) of 1.39 eV while K2AgInBr6 has a Eg of 1.41 eV. Density of states analysis reveals that 4d of Ag- and 4p of Br states are majorly contributing in the higher valence band while In-5s and 4p of Br states are present in lower conduction band and all these states define the semiconductive nature of these systems. Both HDPs are optically active in visible light region suggesting the suitability for solar cell applications. Thermoelectric efficiency is also computed in terms of figure of merit (ZT(e)) whose values for both compounds are 0.7. Based on these outcomes, both HDPs can be considered as highly suitable candidates for energy harvesting applications.