Hartree-Fock 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.

Comparison of optimized effective potential with inverse Kohn-Sham method for Hartree-Fock exchange energy.

The inverse Kohn-Sham (inv-KS) density-functional theory for the electron density of the Hartree-Fock (HF) wave function was revisited within the context of the optimized effective potential (HF-OEP). First, we clarify the relationship between the inv-KS and the HF-OEP within the framework of the potential-functional theory. The similarities and the differences of the approaches are then discussed on the basis of their methodological details, which motivates comparisons of the wave function provided by each method. Next, the real-space grid implementations of the inv-KS and the HF-OEP are addressed for the comparisons. The total HF energies EHF[{φiinv-KS}] for the wave functions φiinv-KS on the effective potentials optimized by the inv-KS are computed for a set of small molecules. It is found that the mean absolute deviation (MAD) of EHF[{φiinv-KS}] from the HF energy is clearly smaller than the MAD of EHF[{φiOEP}], demonstrating that the inv-KS is advantageous in constructing the detailed structure of the exchange potential υx as compared with the HF-OEP. The inv-KS method is also applied to an ortho-benzyne radical known as a strongly correlated polyatomic molecule. It is revealed that the spin populations on the atomic sites computed by the UHF calculation can be faithfully reproduced by the wave functions on the inv-KS potential.

Contact Hole Shrinkage: Simulation Study of Resist Flow Process and Its Application to Block Copolymers.

For vertical interconnect access (VIA) in three-dimensional (3D) structure chips, including those with high bandwidth memory (HBM), shrinking contact holes (C/Hs) using the resist flow process (RFP) represents the most promising technology for low-k1 (where CD=k1λ/NA,CD is the critical dimension, λ is wavelength, and NA is the numerical aperture). This method offers a way to reduce dimensions without additional complex process steps and is independent of optical technologies. However, most empirical models are heuristic methods and use linear regression to predict the critical dimension of the reflowed structure but do not account for intermediate shapes. In this research, the resist flow process (RFP) was modeled using the evolution method, the finite-element method, machine learning, and deep learning under various reflow conditions to imitate experimental results. Deep learning and machine learning have proven to be useful for physical optimization problems without analytical solutions, particularly for regression and classification tasks. In this application, the self-assembly of cylinder-forming block copolymers (BCPs), confined in prepatterns of the resist reflow process (RFP) to produce small contact hole (C/H) dimensions, was described using the self-consistent field theory (SCFT). This research paves the way for the shrink modeling of the enhanced resist reflow process (RFP) for random contact holes (C/Hs) and the production of smaller contact holes.

Hollow ion atomic structure and X-ray emission in dense hot plasmas

Hollow ion X-ray emission is of great interest in high-energy-density research, since negligible opacity allows studies from the interior of very dense objects. In this paper, ionization potential depressions of the isoelectronic sequences for single and double K-shell vacancies are obtained from a pure ab initio multiconfiguration Hartree–Fock simulation including exact exchange terms and finite temperature dense plasma effects. It is demonstrated that the simultaneous representation of these ab initio data in the form of a map of hollow ion X-ray transition energies enables identification of important steps in the matter evolution and ionization dynamics. Mapping along the isoelectronic sequence as a function of the pumping energy of a X-ray free electron laser also enables visualization of the impact of ionization potential depression on the pathways of hollow ion formation.

Overcoming strength-ductility and functional period-degradability tradeoffs: Multifunctional biodegradable polyesters with ionic cluster skeleton and PDMS skin

To overcome the strength-ductility and functional period-degradability tradeoffs of biodegradable polyesters, we proposed a nanophase structure design strategy to construct a soft-hard dual phase in poly(butylene succinate) (PBS) copolymers by simultaneously introducing polydimethylsiloxane (PDMS) and isophthalate-5-sulfonate (SIPM) on PBS macromolecular chains to improve the stiffness, toughness, and multifunctionality. The high-resolution transmission electron microscope (HR-TEM) and energy dispersive X-Ray spectroscopy (EDX) tests indicated that PBS copolymers possessed continuous “ionic cluster skeleton” and “PDMS skin”. The hybrid liquid state theory and self-consistent field approach for copolymers further demonstrated the mechanism of microphase separation for PDMS blocks and SIPM segments. The copolymers showed excellent yield strength of 52 MPa, great impact strength of 171 J/m, large extensibility of 491 %, and long functional period in the acidic environment. The novel stereo nanostructure endowed the materials with advanced multifunctions that can be applied in various applications, such as the biomedical field and flexible electronic devices. Compared with PBS, copolymers possessed excellent in vitro attachment and proliferation of MC3T3-E1 cells. Furthermore, the synergistic effect of the ionic skeleton and PDMS skin improved the ionic conductivity of the materials. The copolymer film was innovatively employed as anion-exchanging separators of supercapacitors with acid-organized electrolytes. It has been testified that this newly designed strategy of a novel stereo nanophase structure was efficient in endowing the biodegradable polyesters with excellent combined properties and multifunctions.

Direct Unconstrained Optimization of Molecular Orbital Coefficients in Density Functional Theory.

One-electron orbitals in Kohn-Sham density functional theory (DFT) are typically constrained to be orthogonal during their variational optimization, leading to elaborate parameterization of the orbitals and complicated optimization algorithms. This work shows that orbital optimization can be performed with nonorthogonal orbitals if the DFT energy functional is augmented with a term that penalizes linearly dependent states. This approach, called variable-metric self-consistent field (VM SCF) optimization, allows us to use molecular orbital coefficients, natural descriptors of one-electron orbitals, as independent variables in a direct, unconstrained minimization, leading to very simple closed-form expressions for the electronic gradient and Hessian. It is demonstrated that efficient convergence of the VM SCF procedure can be achieved with a basic preconditioned conjugate gradient algorithm for a variety of systems, including challenging narrow-gap systems and spin-pure two-determinant states of singlet diradicals. This simple reformulation of the variational procedure can be readily extended to electron correlation methods with multiconfiguration states and to the optimization of excited-state orbitals.

Homogeneous Spaces in Hartree–Fock–Bogoliubov Theory

Homogeneous Spaces in Hartree–Fock–Bogoliubov Theory

Digital twins’ kinetics of virtual free-radical copolymerization of vinyl monomers with stable radicals. 1. Methyl methacrylate

The first experience of virtual free-radical polymerization of a set of vinyl monomers within the framework of the digital twins (DTs) concept (Polymers 2023, 15, 2999) is extended to methyl methacrylate, copolymerized with small additives of stable radicals such as fullerene C60 and TEMPO. A set of elementary reactions, covering the initial stage of the polymerization, is considered. The calculations, encompassing about 30 DTs, were carried out using a semi-empirical version of the unrestricted two-determinant Hartree-Fock approximation. The main energy and spin-density parameters of the ground state of DTs are determined. The barrier profiles of the decomposition of three DTs were calculated laying the foundation for determining the activation energy of the studied processes. The decisive role of spins in the formation of transition states is confirmed. A drastic difference in the copolymerization with fullerene C60 and TEMPO is established. The obtained virtual kinetic data are in a full consent with experimental reality.

Ionic Fragmentation of the Halothane Molecule Induced by EUV and Soft X-ray Radiation.

EUV and soft X-ray-induced photofragmentation of the halothane (CF3CHBrCl) molecule has been investigated using time-of-flight mass spectrometry in the coincidence mode (PEPICO) covering the valence region and vicinity of the bromine 3d, chlorine 2p, and carbon 1s edges. Total and partial ion yields have been recorded as a function of photon energy. At lower photon energies, the heavier singly charged molecular fragments predominate in the mass spectra. On the other hand, there is a strong tendency to the atomization of the molecule at higher photon energies. Despite the different chemical environments experienced by the two carbon atoms, weak site-specific fragmentation is observed. In addition, ab initio quantum mechanical calculations at the MP2 level and a series of computations with multiconfigurational self-consistent field have been performed to describe the inner-shell states.

Advancing Our Understanding of the Excited States of the Tantalum Anion.

Our study provides a comprehensive theoretical examination of the energy levels associated with the neutral tantalum atom and its ions in various charge states (Ta, Ta+, and Ta-), employing the multiconfiguration Dirac-Hartree-Fock (MCDHF) method, and relativistic infinite order two-component (IOTC) method with multiconfiguration complete active space self-consistent field (CASSCF) followed by the second-order single-state multireference perturbation (CASPT2) methods. The effect of spin-orbit (SO) coupling is introduced via the restricted active space state interaction (RASSI) method, utilizing atomic mean field SO integrals (AMFI). Through IOTC CASSCF/CASPT2 RASSI calculations, we determined the electron affinity (EA) of the tantalum atom to be 0.321 eV, which stands among the most accurate theoretical values achieved to date. This result closely aligns with the experimental measurement of 0.329 eV. Our investigation highlights potential discrepancies between the predicted symmetry of the excited states of the tantalum anion and experimental observations. Additionally, we calculated the bonding energies for transitions from Ta- to Ta and identified four potential bound or quasi-bound states in the tantalum anion.

The Staggered Mesh Method: Accurate Exact Exchange Toward the Thermodynamic Limit for Solids.

In periodic systems, the Hartree-Fock (HF) exchange energy exhibits the slowest convergence of all HF energy components as the system size approaches the thermodynamic limit. We demonstrate that the recently proposed staggered mesh method for Fock exchange energy [Xing, Li, and Lin, Math. Comp., 2024], which is specifically designed to sidestep certain singularities in exchange energy evaluation, can expedite the finite-size convergence rate for the exact exchange energy across a range of insulators and semiconductors when compared to the regular and truncated Coulomb methods. This remains true even for two computationally cheaper versions of this new method, which we call non-SCF and split-SCF staggered mesh. Additionally, a sequence of numerical tests on simple solids showcases the staggered mesh method's ability to improve convergence toward the thermodynamic limit for band gaps, bulk moduli, equilibrium lattice dimensions, energies, and phonon force constants.

Basis-Set Limit CCSD(T) Energies for Large Molecules with Local Natural Orbitals and Reduced-Scaling Basis-Set Corrections.

The calculation of density-based basis-set correction (DBBSC), which remedies the basis-set incompleteness (BSI) error of the correlation energy, is combined with local approximations. Aiming at large-scale applications, the procedure is implemented in our efficient local natural orbital-based coupled-cluster singles and doubles with perturbative triples [LNO-CCSD(T)] scheme. To this end, the range-separation function, which characterizes the one-electron BSI in space, is decomposed into the sum of contributions from individual localized molecular orbitals (LMOs). A compact domain is constructed around each LMO, and the corresponding contributions are evaluated only within these restricted domains. Furthermore, for the calculation of the complementary auxiliary basis set (CABS) correction, which significantly improves the Hartree-Fock (HF) energy, the local density fitting approximation is utilized. The errors arising from the local approximations are examined in detail, efficient prescreening techniques are introduced to compress the numerical quadrature used for DBBSC, and conservative default thresholds are selected for the truncation parameters. The efficiency of the DBBSC-LNO-CCSD(T) method is demonstrated through representative examples of up to 1000 atoms. Based on the numerical results, we conclude that the corrections drastically reduce the BSI error using double-ζ basis sets, often to below 1 kcal/mol compared to the reliable LNO-CCSD(T) complete basis set references, while significant improvements are also achieved with triple-ζ basis sets. Considering that the calculation of the DBBSC and CABS corrections only moderately increases the wall-clock time required for the post-HF steps in practical applications, the proposed DBBSC-LNO-CCSD(T) method offers a highly efficient and robust tool for large-scale calculations.

Multi-GPU RI-HF Energies and Analytic Gradients─Toward High-Throughput Ab Initio Molecular Dynamics.

This article presents an optimized algorithm and implementation for calculating resolution-of-the-identity Hartree-Fock (RI-HF) energies and analytic gradients using multiple graphics processing units (GPUs). The algorithm is especially designed for high throughput ab initio molecular dynamics simulations of small and medium size molecules (10-100 atoms). Key innovations of this work include the exploitation of multi-GPU parallelism and a workload balancing scheme that efficiently distributes computational tasks among GPUs. Our implementation also employs techniques for symmetry utilization, integral screening, and leveraging sparsity to optimize memory usage. Computational results show that the implementation achieves significant performance improvements, including over 3 × speedups in single GPU AIMD throughput compared to previous GPU-accelerated RI-HF and traditional HF methods. Furthermore, utilizing multiple GPUs can provide superlinear speedup when the additional aggregate GPU memory allows for the storage of decompressed three-center integrals.

Equilibrium phase behavior of gyroid-forming diblock polymer thin films.

Thin-film confinement of self-assembling block polymers results in materials with myriad potential applications-including membranes and optical devices-and provides design parameters for altering phase behavior that are not available in the bulk, namely, film thickness and preferential wetting. However, most research has been limited to lamella- and cylinder-forming polymers; three-dimensional phases, such as double gyroid (DG), have been observed in thin films, but their phase behavior under confinement is not yet well understood. We use self-consistent field theory to predict the equilibrium morphology of bulk-gyroid-forming AB diblock polymers under thin-film confinement. Phase diagrams reveal that the (211) orientation of DG, often observed in experiments, is stable between nonpreferential boundaries at thicknesses as small as 1.2 times the bulk DG lattice parameter. The (001) orientation is stable between modestly B-preferential boundaries, where B is the majority block, while a different (211)-oriented termination plane is stabilized by strongly B-preferential boundaries, neither of which has been observed experimentally. We then describe two particularly important phenomena for explaining the phase behavior of DG thin films at low film thicknesses. The first is "constructive interference," which arises when distortions due to the top and bottom boundaries overlap and is significant for certain DG orientations. The second is a symmetry-dependent, in-plane unit-cell distortion that arises because the distorted morphology near the boundary has a different preferred unit-cell size and shape than the bulk. These results provide a thermodynamic portrait of the phase behavior of DG thin films.

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.

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.

Exploring the Bonding Nature of Iron(IV)-Oxo Species through Valence Bond Calculations and Electron Density Analysis.

Compound I (Cpd I) plays a pivotal role in substrate transformations within the catalytic cycle of cytochrome P450 enzymes (P450s). A key constituent of Cpd I is the iron(IV)-oxo unit, a structural motif also found in other heme enzymes and nonheme enzymes. In this study, we performed ab initio valence bond (VB) calculations, employing the valence bond self-consistent field (VBSCF) and breathing orbital valence bond (BOVB) methods, to unveil the bonding nature of this vital "Fe(IV)═O″ unit in bioinorganic chemistry. Comparisons were drawn with the triplet O2 molecule, which shares some electronic characteristics with iron(IV)-oxo. Additionally, Cpd I models of horseradish peroxidase (HRP) and catalase (CAT) were analyzed to assess the proximal ligand effect on the electronic structure of iron(IV)-oxo. Our VB analysis underscores the significant role of noncovalent resonance effects in shaping the iron(IV)-oxo bonding. The resonance stabilization within the π and σ frameworks occurs to comparable degrees, with additional stabilization resulting from resonance between VB structures from these frameworks. Furthermore, we elucidated the substantial influence of proximal and equatorial ligands in modulating the relative significance of different VB structures. Notably, in the presence of these ligands, iron(IV)-oxo is better described as iron(III)-oxyl or iron(II)-oxygen, displaying weak covalent character but enhanced by resonance effects. Although both species exhibit diradicaloid characters, resonance stabilization in iron(IV)-oxo is weaker than in O2. Further exploration using the Laplacian of electron density shows that, unlike O2, which exhibits a charge concentration region between its two oxygen atoms, iron(IV)-oxo species display a charge depletion region.

General Condition for Polymer Cononsolvency in Binary Mixed Solvents.

Starting from a generic model based on the thermodynamics of mixing and abstracted from the chemistry and microscopic details of solution components, three consistent and complementary computational approaches are deployed to investigate the general condition for polymer cononsolvency in binary mixed solvents at the zeroth order. The study reveals χPS - χPC + χSC as the underlying universal parameter that regulates cononsolvency, where χαβ is the immiscibility parameter between the α- and β-component. Two disparate cononsolvency regimes are identified for χPS - χPC + χSC < 0 and χPS - χPC + χSC > 2, respectively, based on the behavior of the second osmotic virial coefficient at varying solvent mixture composition x C. The predicted condition is verified using self-consistent field calculations by directly examining the polymer conformational transition as a function of x C. It is further shown that in the regime χPS - χPC + χSC < 0, the reentrant polymer conformation transition is driven by maximizing the solvent-cosolvent contact, but in the regime χPS - χPC + χSC > 2, it is driven by promoting polymer and cosolvent contact. In-between the two regimes when neither effect is dominant, a monotonic response of polymer conformation to x C is observed. Effects of the mean-field approximation on the predicted condition are evaluated by comparing the mean-field calculations with computer simulations. It shows that the fluctuation effects lead to a higher threshold value of χPS - χPC + χSC in the second regime, where local enrichment of cosolvent in polymer proximity plays a critical role.

Quantum discord in zigzag graphene nanoribbons

Based on Hubbard model with the Hartree-Fock approximation, we study the properties of quantum discord (QD) between the nearest-neighbor sites A and B in zigzag graphene nanoribbons thermalized with a reservoir at temperature T. Several influences of the site position, on-site Coulomb repulsion U, temperature, and ribbon width on QD are discussed in detail. The results show that QD is robust against thermal fluctuations, and QD for the leg pairs along the zigzag chain near ribbon edges is always larger than that for the rung pairs linking two adjacent zigzag chains. QD for the rung pairs increases and then approaches to saturation as the ribbon width increases, where the velocity of saturation is strongly correlated to U. Moreover, for rung pairs the values of U at the QD peaks perform the scaling behaviors with increasing ribbon width.

Fundamental Invariant Neural Network (FI-NN) Potential Energy Surface for the OH + CH3OH Reaction with Analytical Forces.

The hydrogen abstraction reaction of OH + CH3OH plays a great role in combustion and atmospheric and interstellar chemistry and has been extensively studied theoretically and experimentally. Theoretically, the numerical gradients with respect to the Cartesian coordinates of atoms in molecular simulations on our recent potential energy surface (PES) for the title reaction trained using the permutationally invariant polynomial neural network (PIP-NN) approach hinder the extensive calculation because of the unaffordable computation cost. To address this issue, we in this work report a new full-dimensional accurate analytical PES for the title reaction using the fundamental invariant neural network (FI-NN) approach based on 140,192 points of the quality UCCSD(T)-F12a/AVTZ. Besides, the spin-orbit (SO) corrections of OH in the entrance channel were determined at the level of complete active space self-consistent field with the AVTZ basis set. As a compromise between computational cost and efficiency, the Δ-machine learning approach was employed to construct the SO-corrected PES. Based on this new FI-NN PES with analytical forces, thermal rate coefficients and various dynamic properties, including the integral cross sections, the differential cross sections, and the product energy partitioning, were determined by running a total of 5.5 million trajectories. The use of analytical gradients of the FI-NN PES accelerated simulations and about 99% of computation cost was saved, compared to that for the PIP-NN PES with numerical gradients. Such a significant acceleration is achieved mainly by replacing PIPs with FIs.