Correlation Energy Contributions Research Articles

Unveiling hidden dynamic correlations in CASSCF correlation energies by Hartree-Fock nodes.

We have recently introduced an original method for sharply partitioning the correlation energy into dynamic and non-dynamic contributions. This method is based on the node of the Hartree-Fock (HF) Slater determinant and the stochastic projector fixed-node diffusion Monte Carlo (FNDMC) method [Šulka et al., J. Chem. Theory Comput. 19, 8147 (2023)]. This approach addresses the challenge of dissecting correlation energy in quantum chemistry. Here, we present the first application of this technique to explore CASSCF correlation energy contributions in selected molecular systems such as BH, FH, F2, and H2-H2. The results show that correlation energies derived from the full-valence active space CASSCF method, often believed to describe mostly non-dynamic correlation effects, contain an extraneous, unwanted, system-dependent component that belongs to the dynamic correlation energy. The findings suggest that the new HF-node/FNDMC-based electron correlation energy decomposition method provides a useful complementary tool, enabling the detection of inherent challenges in distinguishing between dynamic and non-dynamic contributions to correlation energies within methods where precise dissection of these effects is not possible.

CO adsorption on Pt(111) studied by periodic coupled cluster theory.

We present an application of periodic coupled-cluster theory to the calculation of CO adsorption energies on the Pt(111) surface for different adsorption sites. The calculations employ a range of recently developed theoretical and computational methods. In particular, we use a recently introduced coupled-cluster ansatz, denoted as CCSD(cT), to compute correlation energies of the metallic Pt surface with and without adsorbed CO molecules. The convergence of Hartree-Fock adsorption energy contributions with respect to randomly shifted k-meshes is discussed. Recently introduced basis set incompleteness error corrections make it possible to achieve well-converged correlation energy contributions to the adsorption energies. We show that CCSD(cT) theory predicts the correct order of adsorption energies for the considered adsorption sites. Furthermore, we find that binding of the CO molecule to the top and fcc site is dominated by Hartree-Fock and correlation energy contributions, respectively.

Investigating the Basis Set Convergence of Diagrammatically Decomposed Coupled-Cluster Correlation Energy Contributions for the Uniform Electron Gas.

We investigate the convergence of coupled-cluster (CC) correlation energies and related quantities with respect to the employed basis set size for the uniform electron gas (UEG) to gain a better understanding of the basis set incompleteness error (BSIE). To this end, coupled-cluster doubles (CCD) theory is applied to the three-dimensional UEG for a range of densities, basis set sizes, and electron numbers. We present a detailed analysis of individual diagrammatically decomposed contributions to the amplitudes at the level of CCD theory. In particular, we show that only two terms from the amplitude equations contribute to the asymptotic large-momentum behavior of the transition structure factor, corresponding to the cusp region at short interelectronic distances. However, due to the coupling present in the amplitude equations, all decomposed correlation energy contributions show the same asymptotic convergence behavior to the complete basis set limit. These findings provide an additional rationale for the success of a recently proposed correction to the BSIE of CC theory. Lastly, we examine the BSIE in the CCD plus perturbative triples [CCD(T)] method, as well as in the newly proposed CCD plus complete perturbative triples [CCD(cT)] method.

Surface science using coupled cluster theory via local Wannier functions and in-RPA-embedding: The case of water on graphitic carbon nitride.

A first-principles study of the adsorption of a single water molecule on a layer of graphitic carbon nitride is reported employing an embedding approach for many-electron correlation methods. To this end, a plane-wave based implementation to obtain intrinsic atomic orbitals and Wannier functions for arbitrary localization potentials is presented. In our embedding scheme, the localized occupied orbitals allow for a separate treatment of short-range and long-range correlation contributions to the adsorption energy by a fragmentation of the simulation cell. In combination with unoccupied natural orbitals, the coupled cluster ansatz with single, double, and perturbative triple particle-hole excitation operators is used to capture the correlation in local fragments centered around the adsorption process. For the long-range correlation, a seamless embedding into the random phase approximation yields rapidly convergent adsorption energies with respect to the local fragment size. Convergence of computed binding energies with respect to the virtual orbital basis set is achieved employing a number of recently developed techniques. Moreover, we discuss fragment size convergence for a range of approximate many-electron perturbation theories. The obtained benchmark results are compared to a number of density functional calculations.

Dual XH-π Interaction of Hexafluoroisopropanol with Arenes.

The dual XH (OH and CH) hydrogen-bond-donating property of 1,1,1,3,3,3-hexafluoroisopropanol (HFIP) and the strong dual XH–π interaction with arenes were firstly disclosed by theoretical studies. Here, the high accuracy post-Hartree–Fock methods, CCSD(T)/CBS, reveal the interaction energy of HFIP/benzene complex (−7.22 kcal/mol) and the contribution of the electronic correlation energy in the total interaction energy. Strong orbital interaction between HFIP and benzene was found by using the DFT method in this work to disclose the dual XH–π intermolecular orbital interaction of HFIP with benzene-forming bonding and antibonding orbitals resulting from the orbital symmetry of HFIP. The density of states and charge decomposition analyses were used to investigate the orbital interactions. Isopropanol (IP), an analogue of HFIP, and chloroform (CHCl3) were studied to compare them with the classical OH–π, and non-classical CH–π interactions. In addition, the influence of the aggregating effect of HFIP, and the numbers of substituted methyl groups in benzene rings were also studied. The interaction energies of HFIP with the selected 24 common organic compounds were calculated to understand the role of HFIP as solvent or additive in organic transformation in a more detailed manner. A single-crystal X-ray diffraction study of hexafluoroisopropyl benzoate further disclosed and confirmed that the CH of HFIP shows the non-classical hydrogen-bond-donating behavior.

Fast Estimation of Møller-Plesset Correlation Energies Based on Atomic Contributions.

Dynamic correlation plays an important role in the accurate calculation of chemical compounds such as the description of equilibrium structures in chemical systems. A model for the fast estimation of dynamic correlation energy is introduced in this work. This model is based on the idea of decomposition of the contribution of dynamic correlation energy calculated by nth order Møller-Plesset perturbation (MPn) theory with respect to atomic regions. Multiple levels of theory, including MP2, MP2.5, and MP4, are used as the reference, and the corresponding correlation energy densities are calculated. The proposed model is concise, fast, and promising for practical use, such as the prediction of reaction energies. It can also work as a baseline model or pretrained model for follow-up studies of machine learning.

Energy-based automatic determination of buffer region in the divide-and-conquer second-order Møller-Plesset perturbation theory.

In the linear‐scaling divide‐and‐conquer (DC) electronic structure method, each subsystem is calculated together with the neighboring buffer region, the size of which affects the energy error introduced by the fragmentation in the DC method. The DC self‐consistent field calculation utilizes a scheme to automatically determine the appropriate buffer region that is as compact as possible for reducing the computational time while maintaining acceptable accuracy (J. Comput. Chem. 2018, 39, 909). To extend the automatic determination scheme of the buffer region to the DC second‐order Møller–Plesset perturbation (MP2) calculation, a scheme for estimating the subsystem MP2 correlation energy contribution from each atom in the buffer region is proposed. The estimation is based on the atomic orbital Laplace MP2 formalism. Based on this, an automatic buffer determination scheme for the DC‐MP2 calculation is constructed and its performance for several types of systems is assessed.

Intermolecular interaction energies from fourth order many-body perturbation theory. Impact of individual electron correlation contributions.

The performance of Møller-Plesset perturbation theory methods for describing intermolecular interaction energies has been investigated with the focus on illuminating the impact of individual electron correlation energy contributions in fourth order. It is shown that a physically meaningful decomposition of the fourth order correlation energy can be obtained by grouping individual correlation energy terms that share the same diagrammatic loop structure. This decomposition of the fourth order singles (S), doubles (D), triples (T), and quadruples (Q) terms revealed that individual terms from each excitation class can have a huge impact on the energy that is much larger than the total fourth order correlation contribution. A partial summation of S, D, and Q terms has been derived that can reproduce the full fourth order interaction energies with a good accuracy and which does not include the computationally expensive triples energy term.

A systematic electronic structure study of the O–O bond dissociation energy of hydrogen peroxide and the electron affinity of the hydroxyl radical

Hydroxyl radical reduction and peroxide bond breaking in hydrogen peroxide are reactions involved in various processes such as the Fenton reaction, which has applications as e.g. groundwater remediation. Here, we study these two reactions from a thermodynamical point of view through the bond dissociation energy (BDE) of the O–O bond in hydrogen peroxide and the electron affinity (EA) of the hydroxyl radical. High-level ab-initio calculations at the complete basis set (CBS) limit were carried out, and the performance of different DFT-based methods was addressed by following a specific classification on the basis of the Jacob’s ladder in combination with various Pople’s basis sets. The ab-initio calculations at the CBS limit are in agreement with experimental reference data and identify a significant contribution of the electron correlation energy to the BDE and EA. The studied DFT-based methods were able to reproduce the ab-initio reference values, although no functional was particularly detected as the best for both reactions. The inclusion of certain percentage of Hartree–Fock (HF) exchange in DFT functionals leads in most cases to smaller BDE and EA values, which might be related to the poor description of the two reactions by the HF method. Considering the computational cost, DFT methods provide better BDE and EA values than HF methods with an accuracy comparable to the MP2 or CCSD level of theory. Additionally, the quality of the hydrogen peroxide, hydroxyl radical and hydroxyl anion structures obtained from these functionals was compared to experimental reference data. In general, bond lengths were well reproduced and the errors in the angles were between one and two degrees with some systematic trend with respect to the basis set’s size. From our results we conclude that DFT methods present a computationally less expensive alternative to describe these two reactions that play a role in the Fenton reaction. The benchmark that is carried out in this study provides a systematic validation of various approximated $$E_{xc}[\rho ]$$ functionals combined with different basis sets, which could serve as a stepping-stone for future research on the Fenton reaction.

Machine Learning of Dynamic Electron Correlation Energies from Topological Atoms.

We present an innovative method for predicting the dynamic electron correlation energy of an atom or a bond in a molecule utilizing topological atoms. Our approach uses the machine learning method Kriging (Gaussian Process Regression with a non-zero mean function) to predict these dynamic electron correlation energy contributions. The true energy values are calculated by partitioning the MP2 two-particle density-matrix via the Interacting Quantum Atoms (IQA) procedure. To our knowledge, this is the first time such energies have been predicted by a machine learning technique. We present here three important proof-of-concept cases: the water monomer, the water dimer, and the van der Waals complex H2···He. These cases represent the final step toward the design of a full IQA potential for molecular simulation. This final piece will enable us to consider situations in which dispersion is the dominant intermolecular interaction. The results from these examples suggest a new method by which dispersion potentials for molecular simulation can be generated.

Structural, magnetic, dielectric and mechanical properties of (Ba,Sr)MnO3 ceramics

Ceramic samples, produced by conventional sintering method in ambient air, 6H-SrMnO3 (6H-SMO), 15R-BaMnO3 (15R-BMO), 4H-Ba0.5Sr0.5MnO3 (4H-BSMO) were studied. In the XRD measurements for SMO the new anomalies of the lattice parameters at 600–800K range and the increasing of thermal expansion coefficients with a clear maximum in a vicinity at 670K were detected. The Ne´el phase transition for BSMO was observed at TN=250K in magnetic measurements and its trace was detected in dielectric, FTIR, DSC, and DMA experiments. The enthalpy and entropy changes of the phase transition for BSMO at TN were determined as 17.5J/mol and 70mJ/K mol, respectively. The activation energy values and relaxation times characteristic for relaxation processes were determined from the Arrhenius law. Results of ab initio simulations showed that the contribution of the exchange correlation energy to the total energy is about 30%.

Lattice energies of molecular solids from the random phase approximation with singles corrections.

We use the random phase approximation (RPA) method with the singles correlation energy contributions to calculate lattice energies of ten molecular solids. While RPA gives too weak binding, underestimating the reference data by 13.7% on average, much improved results are obtained when the singles are included at the GW singles excitations (GWSE) level, with average absolute difference to the reference data of only 3.7%. Consistently with previous results, we find a very good agreement with the reference data for hydrogen bonded systems, while the binding is too weak for systems where dispersion forces dominate. In fact, the overall accuracy of the RPA+GWSE method is similar to an estimated accuracy of the reference data.

Electron Correlation Effects in All-Metal Aromatic Clusters: A Quantum Monte Carlo Study.

The electron correlation effects on the atomic and electronic structure of a few multiaromatic clusters XAl3– (X = Si, Ge, and Sn) are investigated using the diffusion quantum Monte Carlo method (DMC). We found that the vertical detachment energies are in very good agreement with available photoelectron spectroscopy data. The total binding energy of the clusters is dominated by the electron correlation energy contribution with about 55% of its total values. However, the binding energy gained in adding the dopant X into the Al3– unit to form the cluster XAl3– has been almost equally distributed among the contributions from Hartree–Fock (HF) and correlation energies. The resonance energy is found to be about 100 kcal/mol, which is roughly five times that of the organic aromatic compounds such as benzene. On the basis of some thermodynamical extremum principles we found that the order of decreasing stability of the clusters is SiAl3– ≈ GeAl3– > SnAl3–, the electron correlation impacting more the most stable ...

Singles correlation energy contributions in solids.

The random phase approximation to the correlation energy often yields highly accurate results for condensed matter systems. However, ways how to improve its accuracy are being sought and here we explore the relevance of singles contributions for prototypical solid state systems. We set out with a derivation of the random phase approximation using the adiabatic connection and fluctuation dissipation theorem, but contrary to the most commonly used derivation, the density is allowed to vary along the coupling constant integral. This yields results closely paralleling standard perturbation theory. We re-derive the standard singles of Görling-Levy perturbation theory [A. Görling and M. Levy, Phys. Rev. A 50, 196 (1994)], highlight the analogy of our expression to the renormalized singles introduced by Ren and coworkers [Phys. Rev. Lett. 106, 153003 (2011)], and introduce a new approximation for the singles using the density matrix in the random phase approximation. We discuss the physical relevance and importance of singles alongside illustrative examples of simple weakly bonded systems, including rare gas solids (Ne, Ar, Xe), ice, adsorption of water on NaCl, and solid benzene. The effect of singles on covalently and metallically bonded systems is also discussed.

Alternative formulation of explicitly correlated third-order Møller–Plesset perturbation theory

The second-order wave operator in the explicitly correlated wave function theory has been newly defined as an extension of the conventional s- and p-wave (SP) ansatz (also referred to as the FIXED amplitude ansatz) based on the linked-diagram theorem. The newly defined second-order wave operator has been applied to the calculation of the F12 correction to the third-order many-body perturbation (MP3) energy. In addition to this new wave operator, the F12 correction with the conventional first-order wave operator has been derived and calculated. Among three components of the MP3 correlation energy, the particle ladder contribution, which has shown the slowest convergence with respect to the basis set size, is fairly ameliorated by employing these F12 corrections. Both the newly defined and conventional formalisms of the F12 corrections exhibit a similar recovery of over 90% of the complete basis set limit of the particle ladder contribution of the MP3 correlation energy with a triple-zeta quality basis set for the neon atom, while the amount is about 75% without the F12 correction. The corrections to the ring term are small but the corrected energy has shown similar recovery as the particle ladder term. The hole ladder term has shown a rapid convergence even without the F12 corrections. Owing to these balanced recoveries, the deviation of the total MP3 correlation energy from the complete basis set limit has been calculated to be about 1 kcal/mol with the triple-zeta quality basis set, which is more than five times smaller than the error without the F12 correction.

A quantum chemical calculation on Fe(CO)5 revealing the operation of the Dewar–Chatt–Duncanson model

A nonstandard computational scheme has been applied to calculate Fe(CO)(5) with the aim to illustrate the operation of the Dewar-Chatt-Duncanson model by computation. A full configuration interaction (CI) calculation in an active space has been performed. The active space is built from naturally localized molecular orbitals (NLMOs) localized in bond regions or forming lone pairs. For selecting this active space, Weinhold's perturbation theory formulated in the natural bond orbital (NBO) space has been applied. Bonding, lone pair, and antibond NBOs exhibiting large interaction energies serve to define the active space. The actually applied active space, however, comprises NLMOs that are close in shape to the NBOs indicated by perturbation theory. Thus, a CI calculation with localized orbitals has been performed meeting the classical reasoning of chemists that is often based on local bonding concepts. The computational scheme yields the Lewis structure for Fe(CO)(5) whose energy is identical to the Hartree-Fock energy. The Lewis energy comprises CO → Fe σ-electron transfer (ET) and CO ← Fe electron back donation (BD). This Lewis energy gets lowered by localized correlation energy contributions caused by ET processes where electrons are back donated from the Fe d-lone pairs into the CO ligands. Thus, electron correlation within the selected active space is dominated by electron BD. Energies and electron populations of the NBOs support the notion that electrons are preferentially back donated into the equatorial CO ligands. Weights of local Slater determinants, determining the correlation energy, also point to a predominant BD into the equatorial CO ligands. Correlation energy increments resulting from electron BD into single antibond orbitals of the CO ligands have been calculated. These energy quantities also demonstrate that BD into the equatorial CO ligands is more energy lowering than BD into the axial CO ligands.

Assessment of correlation energies based on the random-phase approximation

The random-phase approximation to the ground state correlation energy (RPA) in combination with exact exchange (EX) has brought the Kohn–Sham (KS) density functional theory one step closer towards a universal, ‘general purpose first-principles method’. In an effort to systematically assess the influence of several correlation energy contributions beyond RPA, this paper presents dissociation energies of small molecules and solids, activation energies for hydrogen transfer and non-hydrogen transfer reactions, as well as reaction energies for a number of common test sets. We benchmark EX + RPA and several flavors of energy functionals going beyond it: second-order screened exchange (SOSEX), single-excitation (SE) corrections, renormalized single-excitation (rSE) corrections and their combinations. Both the SE correction and the SOSEX contribution to the correlation energy significantly improve on the notorious tendency of EX + RPA to underbind. Surprisingly, activation energies obtained using EX + RPA based on a KS reference alone are remarkably accurate. RPA + SOSEX + rSE provides an equal level of accuracy for reaction as well as activation energies and overall gives the most balanced performance, because of which it can be applied to a wide range of systems and chemical reactions.

Local explicitly correlated second- and third-order Møller–Plesset perturbation theory with pair natural orbitals

We present an algorithm for computing explicitly correlated second- and third-order Møller-Plesset energies near the basis set limit for large molecules with a cost that scales formally as N(4) with system size N. This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The N(4) scaling does not rely upon the a priori definition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an aug-cc-pVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs.

High-density limit of the two-dimensional electron liquid with Rashba spin-orbit coupling

We discuss by analytic means the theory of the high-density limit of the unpolarized two-dimensional electron liquid in the presence of Rashba or Dresselhaus spin-orbit coupling. A generalization of the ring-diagram expansion is performed. We find that in this regime the spin-orbit coupling leads to small changes of the exchange and correlation energy contributions, while modifying also, via repopulation of the momentum states, the noninteracting energy. As a result, the leading corrections to the chirality and total energy of the system stem from the Hartree-Fock contributions. The final results are found to be vanishing to lowest order in the spin-orbit coupling, in agreement with a general property valid to every order in the electron-electron interaction. We also show that recent quantum Monte Carlo data in the presence of Rashba spin-orbit coupling are well understood by neglecting corrections to the exchange-correlation energy, even at low density values.

Third-order corrections to random-phase approximation correlation energies

Several random-phase approximation (RPA) correlation methods were compared in third order of perturbation theory. While all of the considered approaches are exact in second order of perturbation theory, it is found that their corresponding third-order correlation energy contributions strongly differ from the exact third-order correlation energy contribution due to missing interactions of the particle-particle-hole-hole type. Thus a simple correction method is derived which makes the different RPA methods also exact to third-order of perturbation theory. By studying the reaction energies of 16 chemical reactions for 21 small organic molecules and intermolecular interaction energies of 23 intermolecular complexes comprising weakly bound and hydrogen-bridged systems, it is found that the third-order correlation energy correction considerably improves the accuracy of RPA methods if compared to coupled-cluster singles doubles with perturbative triples as a reference.