Hartree-Fock Wave Functions Research Articles

Capturing Strong Correlation in Molecules with Phaseless Auxiliary-Field Quantum Monte Carlo Using Generalized Hartree-Fock Trial Wave Functions.

Generalized Hartree-Fock (GHF) is a long-established electronic structure method that can lower the energy (compared to spin-restricted variants) by breaking physical wave function symmetries, namely and . After an exposition of GHF theory, we assess the use of GHF trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC-G) calculations of strongly correlated molecular systems including symmetrically stretched hydrogen rings, carbon dioxide, and dioxygen. Imaginary time propagation is able to restore symmetry and yields energies of comparable or better accuracy than CCSD(T) with unrestricted HF and GHF references, and consistently smooth dissociation curves─a remarkable result given the relative scalability of ph-AFQMC-G to larger system sizes. The present exploration of model strongly correlated systems marks a promising starting point for future studies of more chemically relevant molecules, and demonstrates that ph-AFQMC-G provides a highly accurate (and, in contrast to active-space-based trials, relatively black box and always size-consistent) description of challenging systems exhibiting, e.g., antiferromagnetic coupling and/or geometric spin frustration.

Theoretical study of differential cross sections for the ionization of helium by fast proton impact

Abstract We present the angular distribution of the ejected electron for single ionization of He by fast proton impact. A four-body formalism of the three-Coulomb wave is applied to calculate the triple differential cross sections at several impact energies in the scattering, perpendicular and azimuthal planes. Moreover, the three-body formalism of three-Coulomb, two-Coulomb and first Born approximation models has also been used to study the many-body effect on electron emission and the validity of the models. In the three-Coulomb wave model, the final state wave function incorporates distortion due to the three-body mutual Coulombic interaction. In this formalism, we use an uncorrelated and correlated Born initial state, which consists of a plane wave for the incoming projectile times a two-electron bound state wavefunction of the helium atom representing the 1s2(1S) state. But, in the case of the three-body formalism, the initial state wavefunction consists of a long-range Coulomb distortion for the incoming projectile and one active electron of the He atom described by the Roothaan–Hartree–Fock wavefunction. The structure with a single or two peaks with unequal intensity is observed in the angular distributions of the triple differential cross sections for the different kinematic conditions. In addition, the influence of static electron correlations is investigated using different bound state wavefunctions for the ground state of the He target. In the four-body formalism, the present computations are very fast by reducing a nine-dimensional integral to a two-dimensional real integral. Despite the simplicity and speed of the proposed quadrature, the comparison shows that the obtained results are in reasonable agreement with the experiment and are compatible with those of other theories.

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.

Particle-Breaking Unrestricted Hartree-Fock Theory for Open Molecular Systems.

We recently introduced the particle-breaking restricted Hartree-Fock (PBRHF) model, a mean-field approach to address the fractional charging of molecules when they interact with an electronic environment. In this paper, we present an extension of the model referred to as particle-breaking unrestricted Hartree-Fock (PBUHF). The unrestricted formulation contains odd-electron states necessary for a realistic description of fractional charging. Within the PBUHF parametrization, we use two-body operators as they yield convenient operator transformations. However, two-body operators can change only the particle number by two. Therefore, we include noninteracting zero-energy bath orbitals to generate a linear combination of even and odd electron states. Depending on whether the occupied or virtual orbitals of a molecule interact with the environment, the average number of electrons is either decreased or increased. Without interaction, PBUHF reduces to the unrestricted Hartree-Fock wave function.

TurboGenius: Python suite for high-throughput calculations of ab initio quantum Monte Carlo methods

TurboGenius is an open-source Python package designed to fully control ab initio quantum Monte Carlo (QMC) jobs using a Python script, which allows one to perform high-throughput calculations combined with TurboRVB [Nakano et al. J. Phys. Chem. 152, 204121 (2020)]. This paper provides an overview of the TurboGenius package and showcases several results obtained in a high-throughput mode. For the purpose of performing high-throughput calculations with TurboGenius, we implemented another open-source Python package, TurboWorkflows, that enables one to construct simple workflows using TurboGenius. We demonstrate its effectiveness by performing (1) validations of density functional theory (DFT) and QMC drivers as implemented in the TurboRVB package and (2) benchmarks of Diffusion Monte Carlo (DMC) calculations for several datasets. For (1), we checked inter-package consistencies between TurboRVB and other established quantum chemistry packages. By doing so, we confirmed that DFT energies obtained by PySCF are consistent with those obtained by TurboRVB within the local density approximation (LDA) and that Hartree–Fock (HF) energies obtained by PySCF and Quantum Package are consistent with variational Monte Carlo energies obtained by TurboRVB with the HF wavefunctions. These validation tests constitute a further reliability check of the TurboRVB package. For (2), we benchmarked the atomization energies of the Gaussian-2 set, the binding energies of the S22, A24, and SCAI sets, and the equilibrium lattice parameters of 12 cubic crystals using DMC calculations. We found that, for all compounds analyzed here, the DMC calculations with the LDA nodal surface give satisfactory results, i.e., consistent either with high-level computational or with experimental reference values.

Systematic determination of coupling constants in spin clusters from broken-symmetry mean-field solutions.

Quantum-chemical calculations aimed at deriving magnetic coupling constants in exchange-coupled spin clusters commonly utilize a broken-symmetry (BS) approach. This involves calculating several distinct collinear spin configurations, predominantly by density-functional theory. The energies of these configurations are interpreted in terms of the Heisenberg model, H̃=∑i<jJijs̃i⋅s̃j, to determine coupling constants Jij for spin pairs. However, this energy-based procedure has inherent limitations, primarily in its inability to provide information on isotropic spin interactions beyond those included in the Heisenberg model. Biquadratic exchange or multi-center terms, for example, are usually inaccessible and hence assumed to be negligible. The present work introduces a novel approach employing BS mean-field solutions, specifically Hartree-Fock wave functions, for the construction of effective spin Hamiltonians. This expanded method facilitates the extraction of a broader range of coupling parameters by considering not only the energies, but also Hamiltonian and overlap elements between different BS states. We demonstrate how comprehensive s=12 Hamiltonians, including multi-center terms, can be straightforwardly constructed from a complete set of BS solutions. The approach is exemplified for small clusters within the context of the half-filled single-band Hubbard model. This allows to contrast the current strategy against exact results, thereby offering an enriched understanding of the spin-Hamiltonian construction from BS solutions.

Symmetry-Projected Nuclear-Electronic Hartree-Fock: Eliminating Rotational Energy Contamination.

We present a symmetry projection technique for enforcing rotational and parity symmetries in nuclear-electronic Hartree-Fock wave functions, which treat electrons and nuclei on equal footing. The molecular Hamiltonian obeys rotational and parity inversion symmetries, which are, however, broken by expanding in Gaussian basis sets that are fixed in space. We generate a trial wave function with the correct symmetry properties by projecting the wave function onto representations of the three-dimensional rotation group, i.e., the special orthogonal group in three dimensions SO(3). As a consequence, the wave function becomes an eigenfunction of the angular momentum operator which (i) eliminates the contamination of the ground-state wave function by highly excited rotational states arising from the broken rotational symmetry and (ii) enables the targeting of specific rotational states of the molecule. We demonstrate the efficiency of the symmetry projection technique by calculating the energies of the low-lying rotational states of the H2 and H3+ molecules.

Accuracy of expectation values of one‐electron operators obtained from Hartree–Fock wavefunctions expanded using Lambda functions

AbstractThe accuracy of the expectation values () of one‐electron operators is examined using Hartree–Fock wavefunctions expanded using functions. In this expansion, 150 terms, then 149, 148, and 147 terms are used for the s‐, p‐, d‐, and f‐symmetries, respectively. The systems investigated are He–Ne and the Group 18 atoms of Ar–Og. The one‐electron properties investigated are the cusp condition (CC), the electron density at the nucleus (), and (). Convergence of is examined by increasing the number of expansion terms () up to the given limit (150). The number of significant figures (SF) of is counted by comparing the calculated value at =150 () with the extrapolated value . For He, the SF of CC is found to be 26. For the atoms under consideration, the SF of CC is approximately half that of the total energy (TE). The SFs of expectation values of the other properties are also smaller than for the TE.

A geminal theory based on the generalized electron pairing.

As in the hierarchy of the Hartree-Fock theory in terms of spin symmetry, the extension of spin functions in a theory of two-electron units or geminal, was developed in this study. A trial wave function is constructed as an antisymmetrized product of geminals, in which singlet and triplet two-electron functions are fully mixed. We present a variational optimization method for this generalized pairing wave function in the strong orthogonality condition. The present method is considered an extension of the antisymmetrized product of strongly orthogonal geminals or perfect pairing generalized valence bond methods, maintaining the compactness of the trial wave function. The obtained broken-symmetry solutions were similar to the unrestricted Hartree-Fock wave functions in terms of spin contamination while giving lower energy due to the inclusion of the electron correlation effect in geminals. The degeneracy of the obtained broken-symmetry solutions in the Sz space is reported for the tested four-electron systems.

Introducing MPEC: Massively parallel electron correlation.

We have developed a new program for carrying out improved internally contracted Multi-reference Configuration Interaction Singles and Doubles (i2cMRCISD) calculations. It is designed from the ground up to be used on distributed memory parallel computers. Tests show good scaling properties with the number of cores per node and the number nodes. This program features Gaussian basis sets with ℓ > 6; scalar special relativity via the spin-free method; convergence to C∞v, D∞v, or spherical electronic states; special code to determine Rydberg orbitals; both uncontracted and contracted MRCISD wavefunctions; one and two electron properties, including full spin-orbit matrix elements with the Breit interaction; analytic calculation of Born-Oppenheimer diagonal correction for multi-configuration Hartree-Fock wavefunctions; and analytic calculation of second order Born-Oppenheimer corrections for Hartree-Fock wavefunctions. The program can be obtained from software.nasa.gov.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems.

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

Synthesis, crystallographic, quantum chemical and molecular docking studies of 4-benzoyl-1,2,3,11a-tetrahydro-6H-chromeno[3,2-e]imidazo[1,2-a]pyridin-6-one

The synthesis of pyridine derivatives is an active area of research in the development of new drugs. In view of the above importance, a novel compound with fused rings of derivatives of biologically active imidazole, pyridine and chromanone was synthesized. The crystal structure was analysed by single crystal X-ray diffraction method. The quantum chemical parameters of the molecule in the gaseous state were calculated by semi-empirical method using restricted Hartree-Fock wavefunction. The physicochemical properties such as druglikeness and bioactive scores were predicted to understand the drug transport and receptor interactions. Molecular docking studies were performed to analyze the binding of this compound with various bacterial and fungal pathogens. This study points out the potential of this compound to act as an antimicrobial agent. Comparison with related compounds shows that the addition of more number of biologically active groups enhances the activity.

Consequences of approximating electron correlation effects

ABSTRACT The effect of approximating electron correlation in few-electron systems is investigated using three wavefunctions: the many body wavefunction referred to as fully-correlated (FC), the Hartree Fock wavefunction (HF), and the Colle and Salvetti Jastrow-style wavefunction (CS) used in the derivation of the popular Lee, Yang and Parr (LYP) correlation functional. The electron correlation energy, static and dynamic Coulomb holes, various expectation values, and radial energy distributions (energy densities) are considered. It is shown that whilst the CS wavefunction can capture the behaviour of the wavefunction at the singularities of the Coulomb potential corresponding to the coalescence of two particles and decreases the area of the primary static Coulomb hole by nearly 30% compared to HF for He and Li + , it is unable to accurately model expectation values that depend on the whole of space and key features of the electron density behaviour in dynamic Coulomb holes and the energy density behaviour in energy distributions. These issues are amplified when considering the hydride ion and a system with the critical nuclear charge for binding.

Many recent density functionals are numerically ill-behaved.

Most computational studies in chemistry and materials science are based on the use of density functional theory. Although the exact density functional is unknown, several density functional approximations (DFAs) offer a good balance of affordable computational cost and semi-quantitative accuracy for applications. The development of DFAs still continues on many fronts, and several new DFAs aiming for improved accuracy are published every year. However, the numerical behavior of these DFAs is an often-overlooked problem. In this work, we look at all 592 DFAs for three-dimensional systems available in Libxc 5.2.2 and examine the convergence of the density functional total energy based on tabulated atomic Hartree-Fock wave functions. We show that several recent DFAs, including the celebrated SCAN family of functionals, show impractically slow convergence with typically used numerical quadrature schemes, making these functionals unsuitable both for routine applications and high-precision studies, as thousands of radial quadrature points may be required to achieve sub-μEh accurate total energies for these functionals, while standard quadrature grids like the SG-3 grid only contain O(100) radial quadrature points. These results are both a warning to users to always check the sufficiency of the quadrature grid when adopting novel functionals, as well as a guideline to the theory community to develop better-behaved density functionals.

Theoretical Investigation on C11H8 Bicyclic Carbene and Allene Isomers

Recent studies postulate that the presence of polycyclic aromatic hydrocarbons (PAHs) in the interstellar medium (ISM) could have been formed through resonance-stabilized arylcarbene intermediates. However, identifying most of these reactive intermediates is very challenging experimentally due to their metastability and other experimental constrains. Thus, computational studies that cover the thermodynamic versus kinetic stability of various possible structures would be beneficial for successfully identifying new molecules either in the laboratory and/or in the ISM. In this paper, more than four hundred C11H8 carbene isomers have been theoretically investigated employing density functional theory (DFT). Hybrid density functionals B3LYP and ωB97XD with 6-311 + G (d,p) basis set have been used for singlet electronic states, whereas, triplet spin states were optimized at the same level using an unrestricted Hartree-Fock wavefunction. Although the skeletal structures of C11H8 can be categorized into monocyclic, bicyclic, tricyclic, tetracyclic and acyclic isomers, bicyclic carbenes have shown better stability due to the presence of resonance stabilized azulenyl/naphthyl rings. In this category, some isomers (1-, 2-, 5- and 6-azulenylcarbenes and 1- and 2-naphthylcarbenes) have also been detected recently in the laboratory and simple aromatic carbenes such as cyclopropenylidene and its homologues are detected in the ISM. Thus, we have systematically investigated the energetic and spectroscopic properties of resonance stabilized 5-, 6-, 7- and 8-membered ring containing bicyclic isomers of C11H8 and the fingerprint regions of the infrared spectrum for each class of these bicyclic compounds.

Multicomponent Quantum Mechanics/Molecular Mechanics Study of Hydrated Positronium.

We propose a model for solvated positronium (Ps) atoms in water, based on the sequential quantum mechanics/molecular mechanics (s-QM/MM) protocol. We developed a Lennard-Jones force field to account for Ps-water interactions in the MM step. The repulsive term was obtained from a previously reported model for the solvated electron, while the dispersion constant was derived from the Slater-Kirkwood formula. The force field was employed in classical Monte Carlo (MC) simulations to generate Ps-solvent configurations in the NpT ensemble, while the quantum properties were computed with the any-particle molecular orbital method in the subsequent QM step. Our approach is general, as it can be applied to other liquids and materials. One basically needs to describe the solvated electron in the environment of interest to obtain the Ps solvation model. The thermodynamical properties computed from the MC simulations point out similarities between the solvation of Ps and noble gas atoms, hydrophobic solutes that form clathrate structures. We performed convergence tests for the QM step, with particular attention to the choice of basis set and expansion centers for the positronic and electronic subsystems. Our largest model was composed of the Ps atom and 22 water molecules in the QM region, corresponding to the first solvation shell, surrounded by 128 molecules described as point charges. The mean electronic and positronic vertical detachment energies were (4.73 ± 0.04) eV and (5.33 ± 0.04) eV, respectively. The latter estimates were computed with Koopmans' theorem corrected by second-order self-energies, for a set of statistically uncorrelated MC configurations. While the Hartree-Fock wave functions do not properly account for the annihilation rates, they were useful for numerical tests, pointing out that annihilation is more sensitive to the choice of basis sets and expansion centers than the detachment energies. We further explored a model with reduced solute cavity size by changing the Ps-solvent force field. Although the pick-off annihilation lifetimes were affected by the cavity size, essentially the same conclusions were drawn from both models.

Proposal for measuring line profiles and angular distributions of the radiative-electron-capture transition in low-energy ion-atom collisions: Beyond the impulse approximation

We propose to measure line profiles and angular distributions of the radiative-electron-capture transition in collisions of sub-MeV/$u {\mathrm{Ar}}^{18+}$ with helium. We estimate the line profile and the angular distribution using hydrogenlike and Hartree-Fock wave functions and show that the asymmetry of the line profile and the deviation of the angular distribution from the ${sin}^{2}{\ensuremath{\theta}}_{\mathrm{t}}$ law are magnified in the sub-MeV/$u$ region. Finite detector resolution and ion-beam monochromaticity are considered, and the results show that by using a usual bent-crystal spectrometer and a group of commercial silicon drift x-ray detectors one will be able to obtain sufficiently precise line profile and angular distribution to distinguish different theoretical descriptions of target atoms. The enhanced sensitivity to different theoretical approaches at low energies suggests that the proposed experiment will provide insights into the electromagnetic transition of an electron between a pair of relatively moving nuclei.

Half‐Projected Hartree–Fock method: History and application to excited states of the same symmetry as the ground state

AbstractSpin projected wave functions are generalizations of the Hartree–Fock wave function. Among them, the Half‐Projected Hartree–Fock (HPHF) wave function is a nearly pure wave function of spin and recovers a small part of the spin correlation energy. This paper reviews the history of the HPHF theory, not only from the conceptual point of view but also providing a compilation of the publications of this method over the years until now. In addition, the extension of the HPHF method to the calculation of excited states with the same symmetry as the ground state will be discussed. The possible variational collapse during the calculation of a singlet excited state of the same symmetry as the ground state is avoided by orthogonalizing a pair of corresponding orbitals, one occupied and one virtual orbital, at every step of the variational process. As an example, the potential energy surfaces of the S0 ground and 1S1(n, π*) first excited state of the formic acid HCOOH are calculated.

Environmental modifications of atomic properties: The ground and 1s2p excited states of compressed helium.

Atoms remaining as recognizably distinct constituents of bulk condensed phases can have properties modified from those of the isolated species. Dense helium bubbles at high pressures are a common form of radiation damage degrading the mechanical and electrical properties of host materials. Detailed knowledge is critical for predicting their long term performance. Modifications of the ground and first singlet excited states of confined compressed helium are investigated using an entirely non-empirical theory based on the results of ab initio self-consistent field calculations with corrections for the effects of electron correlation. For finite sized portions representing bulk condensed fcc and bcc phases of helium atoms, Hartree-Fock wavefunctions, energies, and charge distributions were computed as a function of different atomic densities using two models. The first model for the first excited state localizes the excitation on the central atom; in the second model, this is partially delocalized over the closest atomic neighbors. Total energies for the finite size portions are derived by adding the inter-atomic dispersive attractions and a density functional description of the short-range inter-atomic correlation energy. The experimental energy of the first allowed electronic transition increases with density being larger than in an isolated atom. The intra-atomic correlation energy does not contribute to this energy shift. The calculated energy shifts agree well with experiment for both bulk solid and liquid helium. The 2p orbital is increasingly compressed by density enhancement, thus generating the energy shifts. Consequently, calculations of the inelastic electron scattering cross sections are substantially incorrect if the compression of the final 1s2p state is not included. The character of the excitations is examined, and it is argued that these are of Frenkel rather than the Wannier type.

Density Functional Calculations Based on the Exponential Ansatz.

This work explores the application of the singles-based exponential ansatz to density functional calculations. In contrast to the standard approach where Kohn-Sham (KS) orbitals are determined prior to computing molecular quantities of interest, we consider the single-reference Hartree-Fock wave function as a starting point. Applying the exponential ansatz to this single reference gives an auxiliary wave function that is employed to calculate the electronic properties of the system. This wave function is determined self-consistently through the standard KS Hamiltonian but evaluated over the Hartree-Fock molecular orbital basis. By using spin-symmetry breaking, we recover size-consistent results free of unphysical fractional charges in the dissociation limit. Our method shows consistency with standard KS density functional calculations when the system geometry is similar to the equilibrium one or in repulsive configurations. For moderately long distances between atoms, not at dissociation, because of self-interaction the exponential ansatz may give instabilities in the form of large cluster amplitudes. To avoid these, this work introduces a relatively simple regularization method that preserves size-consistency and penalizes high amplitudes of the cluster operator, whereas the results remain physically meaningful. We also present the time-dependent extension of our theory and show that it can feature quantum states where multiple electrons are excited.