Electron Correlation Energy 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.

Revealing the regularities of electron correlation energies associated with valence electrons in atoms in the first three rows of the periodic table

Although correlation energy in an atom can be decomposed into different components such as inter- and intra-orbital pair-correlation energies (PCE), it is generally believed that the PCEs in different atoms cannot be the same. In this work, we find that when the correlation energy is defined as the difference between the exact ground-state energy and the unrestricted Hartree–Fock energy, the PCEs associated with the valence electrons of the atoms in the same row of the periodic table have the same values. For two specific orbitals, the inter-orbital correlation energy is the same between two electrons of parallel or anti-parallel spins.

Dynamical Electron Correlation and the Chemical Bond. IV. Covalent Bonds in A2 Molecules (A = N-As and F-Br).

In a series of recent papers, we investigated the effect of dynamical electron correlation on the potential energy curves and spectroscopic constants of several diatomic molecules, including the simple diatomic hydrides (AH) and the more complex diatomic fluorides (AF) and homonuclear diatomic molecules (A2) with A = B-F (AF) or A = C-F (A2), respectively. Our goal was to understand the dependence of the dynamical electron correlation energy, EDEC, on the internuclear distance, R, and quantify how dynamical electron correlation influences the spectroscopic constants (De, Re, and ωe) of these molecules. At large R, we found that the magnitude of EDEC(R) had a simple dependence on R, with EDEC(R) increasing nearly exponentially with decreasing R. However, as R continued to decrease, there were significant variations in EDEC(R). These variations led to differing changes in the predicted spectroscopic constants of the molecules. In many molecules, the changes in EDEC(R) could be correlated with changes in the underlying spin-coupled generalized valence bond wave function, either in the orbitals or the spin-coupling coefficients. In the current paper, we extend these studies to higher main group elements, comparing the effects of EDEC(R) on P2 and As2 versus N2, and on Cl2 and Br2 versus F2. We find that there are significant differences between the effects of dynamical electron correlation on the molecules in the first and subsequent rows of the periodic table.

Ground state properties of the screened helium atom under harmonic confinement

In this contribution, we show electronic property results for the ground state of the helium atom screened by a Debye plasma under an isotropic harmonic confinement as a function of the screening length (λD) and the angular frequency (ω0). The associate Hamiltonian is solved by expanding the wave function in the Hylleraas basis to incorporate the electron correlation in a simple manner. Furthermore, in order to compare with other findings the Hartree–Fock approach is implemented using a numerical grid method. The main results show that the total energy reaches the ionization threshold by the action of the electron coupling with the plasma medium and the strength of the harmonic confinement. The electron–electron average distance shows the compression of the system as the Debye plasma and the harmonic interactions become strongest (low λD and large ω0 values). We found acceptable results for the electron correlation energy when separate interactions are considered. In the absence of both interactions, the findings are in excellent agreement with other theoretical results.

Linear Scaling Incremental Scheme for Correlation Energies with Embedding Generated Virtuals.

A novel incremental scheme is presented including an incremental expansion of the virtual space for the calculation of electron correlation energies, which is compatible with any size-extensive correlation method and scales asymptotically linear for large molecules. The performance is studied for organic molecules, water clusters, and a La(III)-water complex, where the compatibility with pseudopotentials is also examined. The computational requirements are already reduced tremendously for medium-sized water clusters and hydrocarbons with respect to the canonical CCSD as well as the ordinary incremental scheme references. Correlation energies within chemical accuracy have been observed for all studied systems. The novelty of the method is that relatively small virtual spaces are used in combination with tuples of localized occupied spaces. The corresponding orthonormal occupied and virtual orbitals are obtained from QM/QM embedding calculations and can thus be used with standard quantum chemistry codes for correlation calculations. It is presented how relevant virtual spaces are selected and the correlation energies are linked in the new virtual space expansion.

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.

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.

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.

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.

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.

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.

Fitting Potential Energy Surfaces by Learning the Charge Density Matrix with Permutationally Invariant Polynomials.

The electronic energy in the Hartree-Fock (HF) theory is the trace of the product of the charge density matrix (CDM) with the one-electron and two-electron matrices represented in an atomic orbital basis, where the two-electron matrix is also a function of the same CDM. In this work, we examine a formalism of analytic representation of a generic molecular potential energy surface (PES) as a sum of a linearly parameterized HF and a correction term, the latter formally representing the electron correlation energy, also linearly parameterized, by expressing the elements of CDM using permutationally invariant polynomials (PIPs). We show on a variety of numerical examples, ranging from exemplary two-electron systems HeH+ and H3+ to the more challenging cases of methanium (CH5+) fragmentation and high-energy tautomerization of formamide to formimidic acid that such a formulation requires significantly fewer, 10-20% of PIPs, to accomplish the same accuracy of the fit as the conventional representation at practically the same computational cost.

Orbital-optimized pair-correlated electron simulations on trapped-ion quantum computers

Variational quantum eigensolvers (VQE) are among the most promising approaches for solving electronic structure problems on near-term quantum computers. A critical challenge for VQE in practice is that one needs to strike a balance between the expressivity of the VQE ansatz versus the number of quantum gates required to implement the ansatz, given the reality of noisy quantum operations on near-term quantum computers. In this work, we consider an orbital-optimized pair-correlated approximation to the unitary coupled cluster with singles and doubles (uCCSD) ansatz and report a highly efficient quantum circuit implementation for trapped-ion architectures. We show that orbital optimization can recover significant additional electron correlation energy without sacrificing efficiency through measurements of low-order reduced density matrices (RDMs). In the dissociation of small molecules, the method gives qualitatively accurate predictions in the strongly-correlated regime when running on noise-free quantum simulators. On IonQ’s Harmony and Aria trapped-ion quantum computers, we run end-to-end VQE algorithms with up to 12 qubits and 72 variational parameters—the largest full VQE simulation with a correlated wave function on quantum hardware. We find that even without error mitigation techniques, the predicted relative energies across different molecular geometries are in excellent agreement with noise-free simulators.

Optical spectra of EGFR inhibitor AG-1478 for benchmarking DFT functionals

Optical spectroscopy (UV–vis and fluorescence spectroscopy) is sensitive to the chemical environment and conformation of fluorophores and therefore, serves as an ideal probe for the conformation and solvent responses. Tyrosine kinase inhibitors (TKI) such as AG-1478 of epidermal growth factor receptor when containing a quinazolinamine scaffold are fluorophores. It is, however, very important to benchmark density functional theory (DFT) method against optical spectral measurements, when time-dependent DFT is applied. In this study, the performance of up to 22 DFT functionals is benchmarked with respect to the measured optical spectra of AG-1478 in dimethyl sulfoxide (DMSO) solvent. It is discovered when combined with the 6–311++G(d, p) basis set, there are top seven functionals; B3PW91, B3LYP, B3P86, PBE1PBE, APFD, HSEH1PBE, and N12SX DFT-VXC functionals are identified as the top performers. Becke’s three-parameter exchange functional (B3) tends to generate accurate optical spectra to form the best three functionals, B3LYP, B3PW91 and B3P86. Specifically, B3PW91 was recommended for studying the optical properties of 4-quinazolinamine TKIs, B3LYP was found to be excellent for absorption spectrum, while B3P86 was identified as the best for emission spectrum. Any further corrections to B3LYP, such as CAM-B3LYP, LC-B3LYP, and B3LYP-D3 result in larger errors in the optical spectra of AG-1478 in DMSO solvent. These best three (B3Vc) functionals are reliable tools for optical properties of the TKIs and therefore the design of new agents with larger Stokes shift for medical image applications. To obtain reliable optical spectra for this class of 4-quinazolinamine based TKIs, it is important to include the electron correlation energy.

Engineering metal oxidation using epitaxial strain.

The oxides of platinum group metals are promising for future electronics and spintronics due to the delicate interplay of spin-orbit coupling and electron correlation energies. However, their synthesis as thin films remains challenging due to their low vapour pressures and low oxidation potentials. Here we show how epitaxial strain can be used as a control knob to enhance metal oxidation. Using Ir as an example, we demonstrate the use of epitaxial strain in engineering its oxidation chemistry, enabling phase-pure Ir or IrO2 films despite using identical growth conditions. The observations are explained using a density-functional-theory-based modified formation enthalpy framework, which highlights the important role of metal-substrate epitaxial strain in governing the oxide formation enthalpy. We also validate the generality of this principle by demonstrating epitaxial strain effect on Ru oxidation. The IrO2 films studied in our work further revealed quantum oscillations, attesting to the excellent film quality. The epitaxial strain approach we present could enable growth of oxide films of hard-to-oxidize elements using strain engineering.

Periodic Bootstrap Embedding.

Bootstrap embedding (BE) is a recently developed electronic structure method that has shown great success at treating electron correlation in molecules. Here, we extend BE to treat surfaces and solids where the wave function is represented in periodic boundary conditions using reciprocal space sums (i.e., k-point sampling). The major benefit of this approach is that the resulting fragment Hamiltonians carry no explicit dependence on the reciprocal space sums, allowing one to apply traditional nonperiodic electronic structure codes to the fragments even though the entire system requires careful consideration of periodic boundary conditions. Using coupled cluster singles and doubles (CCSD) as an example method to solve the fragment Hamiltonians, we present minimal basis set CCSD-in-HF results on 1D conducting polymers. We show that periodic BE-CCSD can typically recover ∼99.9% of the electron correlation energy. We further demonstrate that periodic BE-CCSD is feasible even for complex donor-acceptor polymers of interest to organic solar cells─despite the fact that the monomers are sufficiently large that even a Γ-point periodic CCSD calculation is prohibitive. We conclude that BE is a promising new tool for applying molecular electronic structure tools to solids and interfaces.

Porting fragmentation methods to GPUs using an OpenMP API: Offloading the resolution-of-the-identity second-order Møller-Plesset perturbation method.

Using an OpenMP Application Programming Interface, the resolution-of-the-identity second-order Møller-Plesset perturbation (RI-MP2) method has been off-loaded onto graphical processing units (GPUs), both as a standalone method in the GAMESS electronic structure program and as an electron correlation energy component in the effective fragment molecular orbital (EFMO) framework. First, a new scheme has been proposed to maximize data digestion on GPUs that subsequently linearizes data transfer from central processing units (CPUs) to GPUs. Second, the GAMESS Fortran code has been interfaced with GPU numerical libraries (e.g., NVIDIA cuBLAS and cuSOLVER) for efficient matrix operations (e.g., matrix multiplication, matrix decomposition, and matrix inversion). The standalone GPU RI-MP2 code shows an increasing speedup of up to 7.5× using one NVIDIA V100 GPU with one IBM 42-core P9 CPU for calculations on fullerenes of increasing size from 40 to 260 carbon atoms using the 6-31G(d)/cc-pVDZ-RI basis sets. A single Summit node with six V100s can compute the RI-MP2 correlation energy of a cluster of 175 water molecules using the correlation consistent basis sets cc-pVDZ/cc-pVDZ-RI containing 4375 atomic orbitals and 14 700 auxiliary basis functions in ∼0.85h. In the EFMO framework, the GPU RI-MP2 component shows near linear scaling for a large number of V100s when computing the energy of an 1800-atom mesoporous silica nanoparticle in a bath of 4000 water molecules. The parallel efficiencies of the GPU RI-MP2 component with 2304 and 4608 V100s are 98.0% and 96.1%, respectively.

Wire-width and electron-density dependence of the crossover in the peak of the static structure factor from 2kF → 4kF in one-dimensional paramagnetic electron gases

We use the variational quantum Monte Carlo (VMC) method to study the wire-width $(b)$ and electron-density $({r}_{\text{s}})$ dependences of the ground-state properties of quasi-one-dimensional paramagnetic electron fluids. The onset of a quasi-Wigner crystal phase is known to depend on electron density and the crossover occurs in the low density regime. We study the effect of wire width on the crossover of the dominant peak in the static structure factor from $k=2{k}_{\text{F}}$ to $k=4{k}_{\text{F}}$. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2{k}_{\text{F}}$ to $4{k}_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the interplay of both ${r}_{\text{s}}$ and $b<{r}_{\text{s}}$. The finite wire-width correlation effect is reflected in the peak height of the charge and spin structure factors. We fit the dominant peaks of the charge and spin structure factors assuming fit functions based on our finite wire-width theory and clues from bosonization, resulting in a good fit of the VMC data. The pronounced peaks in the charge and spin structure factors at $4{k}_{\text{F}}$ and $2{k}_{\text{F}}$, respectively, indicate the complete decoupling of the charge and spin degrees of freedom. Furthermore, the wire-width dependence of the electron correlation energy and the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ is found to be significant.

Coulson–Fischer Wave Function on Self-consistent Field and Perturbation Correction

We propose a computational method to obtain Coulson–Fischer (CF) wave functions using the conventional Gaussian-type orbital function as the basis sets and the self-consistent field method. Because the obtained wave functions are similar to unrestricted Hartree–Fock functions, the electron correlation energies can be estimated by the perturbation correction using our CF wave function. Comparing the corrected total energy with the result via the full configuration interaction confirms the usability of our method.

First-principles calculations to investigate structural, elastic, electronic, optical, and magnetic properties of Hg2WO4 for photocatalytic applications

In the present work, structural, elastic, electronic, optical, and magnetic properties of Hg2WO4 are investigated using CASTEP simulation tool. Generalized Gradient Approximation (GGA) along with Perdew-Burke-Ernzerhof (PBE) are employed for the calculation of electronic exchange and correlation energy. In structural analysis, Hg2WO4 is found to be stable with C-centered monoclinic structure having space group C2/c. Material’s ductile nature is confirmed with Pugh’s approximation where calculated B/G ratio is greater than 0.75 and Frantsevich’s rule where v is greater than 0.26. The calculations of electronic properties show that Hg2WO4 has indirect band gap of 2.14 eV. While the optical study reveals that Hg2WO4 has a good photosensitivity in visible light region. Furthermore, the spin-polarized treatment shows that Hg2WO4 is antiferromagnetic in nature. The current findings can be of great importance for the development of efficient photocatalytic materials for water splitting and associated applications.