Frozen-core Approximation Research Articles

Accelerating Analytic-Continuation GW Calculations with a Laplace Transform and Natural Auxiliary Functions.

We present a simple and accurate GW implementation based on a combination of a Laplace transform (LT) and other acceleration techniques used in post-self-consistent field quantum chemistry, namely, natural auxiliary functions and the frozen-core approximation. The LT-GW approach combines three major benefits: (a) a small prefactor for computational scaling, (b) easy integration into existing molecular GW implementations, and (c) significant performance improvements for a wide range of possible applications. Illustrating these advantages for systems consisting of up to 352 atoms and 7412 basis functions, we further demonstrate the benefits of this approach combined with an efficient implementation of the Bethe-Salpeter equation.

Implementation of the Time-Dependent Complete-Active-Space Self-Consistent-Field Method for Diatomic Molecules.

We present a computational approach that implements the time-dependent complete-active-space self-consistent-field method, as introduced in [Phys. Rev. A 88, 023402 (2013)]. Our implementation addresses the challenge of diatomic molecules subjected to an intense laser pulse by considering the full dimensionality of the problem using prolate spheroidal coordinates. The method incorporates the gauge-invariant frozen-core approximation, boosts the evaluation of the electron-electron interaction term using finite-element discrete-variable representation with Neumann expansion, and utilizes an exponential time differencing scheme tailored for the stable propagation of the stiff nonlinear orbital functions. We have successfully applied this methodology to study high-harmonic generation in diatomic molecules such as H2, LiH, and N2, shedding light on the impact of electron correlations in these systems.

Convergent close-coupling calculations of positron scattering from neon and argon

Positron scattering from the noble gas atoms neon and argon is calculated using the single-center convergent close-coupling (CCC) method. Unlike previous CCC calculations, we have relaxed the frozen-core approximation by fully opening the outermost s and p shells, and find better agreement with experiment for excitation energies and dipole polarizability. We have used this improved structure model to calculate elastic, elastic differential, total, momentum-transfer, electron-loss, and total bound excitation cross sections. We have applied a complex model potential calculation, scaled to our single-center CCC results, to obtain direct ionization and positronium-formation cross sections for these systems. This complex model potential was also used to calculate results between the positronium-formation and ionization thresholds, giving complete results from 10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-5}$$\\end{document} eV to 5000 eV. Overall, good agreement is observed with past theory and experiment for most transitions.Graphic abstract

Nonunitary projective transcorrelation theory inspired by the F12 ansatz.

An alternative nonunitary transcorrelation, inspired by the F12 ansatz, is investigated. In contrast to the Jastrow transcorrelation of Boys-Handy, the effective Hamiltonian of this projective transcorrelation features: 1. a series terminating formally at four-body interactions. 2. no spin-contamination within the non-relativistic framework. 3. simultaneous satisfaction of the singlet and triplet first-order cusp conditions. 4. arbitrary choices of pairs for correlation including frozen-core approximations. We discuss the connection between the projective transcorrelation and F12 theory with applications to small molecules, to show that the cusp conditions play an important role to reduce the uncertainty arising from the nonunitary transformation.

Transcorrelated selected configuration interaction in a bi-orthonormal basis and with a cheap three-body correlation factor.

In this work, we develop a mathematical framework for a selected configuration interaction (SCI) algorithm within a bi-orthogonal basis for transcorrelated (TC) calculations. The bi-orthogonal basis used here serves as the equivalent of the standard Hartree-Fock (HF) orbitals. However, within the context of TC, it leads to distinct orbitals for the left and right vectors. Our findings indicate that the use of such a bi-orthogonal basis allows for a proper definition of the frozen core approximation. In contrast, the use of HF orbitals results in bad error cancellations for ionization potentials and atomization energies (AE). Compared to HF orbitals, the optimized bi-orthogonal basis significantly reduces the positive part of the second-order energy (PT2), thereby facilitating the use of standard extrapolation techniques of hermitian SCI. While we did not observe a significant improvement in the convergence of the SCI algorithm, this is largely due to the use in this work of a simple three-body correlation factor introduced in a recent study. This correlation factor, which depends only on atomic parameters, eliminates the need for re-optimization of the correlation factor for molecular systems, making its use straightforward and user-friendly. Despite the simplicity of this correlation factor, we were able to achieve accurate results on the AE of a series of 14 molecules on a triple-zeta basis. We also successfully broke a double bond until the full dissociation limit while maintaining the size consistency property. This work thus demonstrates the potential of the BiO-TC-SCI approach in handling complex molecular systems.

An efficient implementation of analytical nuclear gradients for linear-response time-dependent density functional theory in the plane wave basis

We present an efficient implementation of the analytical nuclear gradient of linear-response time-dependent density functional theory (LR-TDDFT) with the frozen core approximation (FCA). This implementation is realized based on the Hutter’s formalism and the plane wave pseudopotential method. Numerical results demonstrate that the LR-TDDFT/FCA method using a small subset of Kohn–Sham occupied orbitals are accurate enough to reproduce the LR-TDDFT results. Here, the FCA remarkably reduces the computational cost in solving the LR-TDDFT eigenvalue equation. Another challenge in the calculations of analytical nuclear gradients for LR-TDDFT is the solution of the Z-vector equation, for which the Davidson algorithm is a popular choice. While, for large systems the standard Davidson algorithm exhibits a low convergence rate. In order to overcome this problem, we generalize the two-level Davidson algorithm to solve linear equation problems. A more stable performance is achieved with this new algorithm. Our method should encourage further studies of excited-state properties with LR-TDDFT in the plane wave basis.

Wave-packet continuum discretization approach to He2+−He collisions

The two-center wave-packet convergent close-coupling method is generalized to multiply charged ion collisions with helium and applied to the ${\mathrm{He}}^{2+}\text{\ensuremath{-}}\mathrm{He}$ scattering problem. The approach is applicable in a wide range of collision energies including low and intermediate energies, where coupling between various reaction channels and electron exchange between the fragments in the rearrangement channel are important. The target structure is treated using the configuration-interaction method within the frozen-core approximation, where one of the electrons of the target is assumed to stay in the ground state throughout the collision. This accounts for the indistinguishability and the correlations between the electrons of the target. We also use a simpler alternative method that is based on an effective one-electron target description that neglects the electron-electron correlations. In both methods, the continuum of the target atom and the hydrogenlike atom formed after electron capture by the projectile is discretized using the wave-packet approach. We present cross sections for total and state-selective electron capture, excitation, and single ionization of the target. The results are provided for the incident energies from 10 keV/u to 5 MeV/u, where one-electron processes are expected to dominate. Particular attention is focused on the intermediate-energy region where substantial deviations between various theoretical results are found. However, overall, the present results are in good agreement with experimental data and other calculations, where available. We demonstrate that both effective single-electron and two-electron methods can provide a realistic picture of all single-electron scattering processes taking place in ${\mathrm{He}}^{2+}\text{\ensuremath{-}}\mathrm{He}$ collisions in terms of the total cross sections. This creates a reliable platform for modeling differential scattering and ionization in this four-body system.

Benchmarking Gaussian Basis Sets in Quantum-Chemical Calculations of Photoabsorption Spectra of Light Atomic Clusters.

The choice of Gaussian basis functions for computing the ground-state properties of molecules and clusters, employing wave function-based electron-correlated approaches, is a well-studied subject. However, the same cannot be said when it comes to the excited-state properties of such systems, in general, and optical properties, in particular. The aim of the present study is to understand how the choice of basis functions affects the calculations of linear optical absorption in clusters, qualitatively and quantitatively. For this purpose, we have calculated linear optical absorption spectra of several small charged and neutral clusters, namely, Li2, Li3, Li4, B2 +, B3 +, Be2 +, and Be3 +, using a variety of Gaussian basis sets. The calculations were performed within the frozen-core approximation, and a rigorous account of electron correlation effects in the valence sector was taken by employing various levels of configuration interaction (CI) approach both for the ground and excited states. Our results on the peak locations in the absorption spectra of Li3 and Li4 are in very good agreement with the experiments. Our general recommendation is that for excited-state calculations, it is very important to utilize those basis sets which contain augmented functions. Relatively smaller aug-cc-pVDZ basis sets also yield high-quality results for photoabsorption spectra and are recommended for such calculations if the computational resources are limited.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation.

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.

General Analytical Nuclear Forces and Molecular Potential Energy Surface from Full Configuration Interaction Quantum Monte Carlo.

The full configuration interaction quantum Monte Carlo (FCIQMC) is a state-of-the-art stochastic electronic structure method, providing a methodology to compute FCI-level state energies of molecular systems within a quantum chemical basis. However, especially to probe dynamics at the FCIQMC level, it is necessary to devise more efficient schemes to produce nuclear forces and potential energy surfaces (PES) from FCIQMC. In this work, we derive the general formula for nuclear forces from FCIQMC, and clarify different contributions of the total force. This method to obtain FCIQMC forces eliminates previous restrictions and can be used with frozen core approximation and free selection of orbitals, making it promising for more efficient nuclear forces calculations. After some numerical checks of this procedure on the binding curve of N2 molecule, we use the FCIQMC energy and force to obtain the full-dimensional ground state PES of the water molecule via Gaussian processes regression. The new water FCIQMC PES can be used as the basis for H2O ground state nuclear dynamics, structure optimization, and rotation-vibrational spectrum calculation.

Computational Prediction of Graphdiyne-Supported Three-Atom Single-Cluster Catalysts

Computational Prediction of Graphdiyne-Supported Three-Atom Single-Cluster Catalysts

Effect of Nonequilibrium Transient Electronic Structures on Lattice Stability in Metals: Density Functional Theory Calculations

The electronic structures of metals undergo transient nonequilibrium states during the photoexcitation process caused by isochoric heating of X-ray free-electron laser, and their lattice stability is, thus, significantly affected. By going beyond frozen core approximation, we manually introduced nonequilibrium electron distribution function in finite-temperature density functional theory with the framework of Kohn–Sham–Mermin to investigate such transient states, and their effect on lattice stability in metals is demonstrated by phonon dispersion calculated using the finite displacement method. We found that the perfect lattice of a metal collapses due to the exotic electronic structure of nonequilibrium transient state created by isochoric heating of X-ray free-electron laser. Further increase of the number of holes created in the sample (i.e., an increase of laser fluence) still results in lattice instability for aluminum, while for copper, it results in phonon hardening. The potential energy surface is calculated for the extreme case of both Al and Cu with exactly one hole created in its inner shell for each one of the atoms. A double-well structure is clearly observed for Al, while the potential energy surface becomes steeper for Cu.

Double Photoionization of Atomic Carbon

Double photoionization events provide a direct evaluation of electron correlation. The recent focus on few-electron targets continues to reveal the consequences of electron correlation for targets that possess several electrons. We consider the double photoionization of the 2p2 valence electrons of atomic carbon and focus on the first energetically accessible final-state symmetries that originate from coupling the active electrons in 3P configurations, which are doubly ionized by a single photon. Comparison of this process in carbon with neon provides an analogous case for the resulting final-state symmetries within the framework where the ejected electrons are influenced by the remaining bound electrons in a frozen-core approximation. Choosing this symmetry allows for comparison with previous theoretical results for total and energy sharing cross-sections of carbon. Fully differential cross-sections for both carbon and neon are also compared.

Accurate frozen core approximation for all-electron density-functional theory.

We implement and benchmark the frozen core approximation, a technique commonly adopted in electronic structure theory to reduce the computational cost by means of mathematically fixing the chemically inactive core electron states. The accuracy and efficiency of this approach are well controlled by a single parameter, the number of frozen orbitals. Explicit corrections for the frozen core orbitals and the unfrozen valence orbitals are introduced, safeguarding against seemingly minor numerical deviations from the assumed orthonormality conditions of the basis functions. A speedup of over twofold can be achieved for the diagonalization step in all-electron density-functional theory simulations containing heavy elements, without any accuracy degradation in terms of the electron density, total energy, and atomic forces. This is demonstrated in a benchmark study covering 103 materials across the Periodic Table and a large-scale simulation of CsPbBr3 with 2560 atoms. Our study provides a rigorous benchmark of the precision of the frozen core approximation (sub-meV per atom for frozen core orbitals below -200 eV) for a wide range of test cases and for chemical elements ranging from Li to Po. The algorithms discussed here are implemented in the open-source Electronic Structure Infrastructure software package.

Dirac–Fock internal conversion coefficients at low [formula omitted]-ray energy

Presented here are internal conversion coefficients (ICCs) for nuclear transitions of low γ-ray energies from 0.1keV to 1keV in elements with 10≤Z≤118. The low-energy nuclear transitions attract significant interest since the novel experimental technique and methods make it possible to study the transitions as well as to discover new low-lying isomers. Our ICC calculations are shown to be in excellent agreement with experimental data for the fairly low-energy nuclear transitions. The calculations are based on the Dirac–Fock method. The vacancy in the atomic shell from which an electron has been emitted is taken into account in the framework of the frozen core approximation. Experimental binding energies are used for elements Z≤95. Theoretical binding energies obtained in a good approximation are used for Z≥96. Peculiarities of the ICC behavior at low energies are considered. The resonance-like structure of ICC and the drastic decrease in magnitude as the energy increases are demonstrated to be responsible for a pronounced dependence of ICC on theoretical assumptions underlying the calculations. We discuss the influence on the low-energy ICCs of the exact taking account the exchange interaction between electrons, the inclusion of the vacancy after conversion, and a credible choice of the binding energy.

Machine-learned electron correlation model based on frozen core approximation.

The machine-learned electron correlation (ML-EC) model is a regression model in the form of a density functional that reproduces the correlation energy density based on wavefunction theory. In a previous study [T. Nudejima et al., J. Chem. Phys. 151, 024104 (2019)], the ML-EC model was constructed using the correlation energy density from all-electron calculations with basis sets including core polarization functions. In this study, we applied the frozen core approximation (FCA) to the correlation energy density to reduce the computational cost of the response variable used in machine learning. The coupled cluster singles, doubles, and perturbative triples [CCSD(T)] correlation energy density obtained from a grid-based energy density analysis was analyzed within FCA and correlation-consistent basis sets without core polarization functions. The complete basis set (CBS) limit of the correlation energy density was obtained using the extrapolation and composite schemes. The CCSD(T)/CBS correlation energy densities based on these schemes showed reasonable behavior, indicating its appropriateness as a response variable. As expected, the computational time was significantly reduced, especially for systems containing elements with a large number of inner-shell electrons. Based on the density-to-density relationship, a large number of data (5 662 500 points), which were accumulated from 30 molecules, were sufficient to construct the ML-EC model. The valence-electron correlation energies and reaction energies calculated using the constructed model were in good agreement with the reference values, the latter of which were superior in accuracy to density functional calculations using 71 exchange-correlation functionals. The numerical results indicate that the FCA is useful for constructing a versatile model.

Electron capture by fast projectiles from lithium, carbon, nitrogen, oxygen and neon

The three-body boundary-corrected continuum intermediate state method is used to compute total cross sections for single charge exchange in ion-atom collisions at intermediate and high impact energies. Detailed illustrations are given for several scattering systems: H+ + Li, He2+ + Li, H+ + C, He2+ + C, H+ + N, H+ + O and H+ + Ne, He2+ + Ne. An independent particle model and the frozen-core approximation are employed with only one target electron taken as being active. The initial ground state of the active electron in a multi-electron target is described by five wave functions. These are two Roothaan-Hartree-Fock (RHF) wave functions, the single- as well as double-zeta functions and the hydrogen-like functions. Comparisons among the resulting cross sections are made to check their sensitivity to the choice of the initial target wave functions. In the case of a lithium target, the separate cross sections for electron capture from the K-shell and L-shell are reported. The present theoretical total cross sections are compared with the available experimental data and overall good agreement is found, especially when using the RHF wave functions for multi-electron targets.

Pauli Shielding and Breakdown of Spin Statistics in Multielectron Multi-Open-Shell Dynamical Atomic Systems

We report on C^{3+}(1s2ℓ2ℓ^{'} ^{2S+1}L)-resolved cross sections of electron capture in collisions of swift C^{4+}(1s2s ^{3}S) ions with helium and hydrogen. The study focuses on the formation of doubly excited triply open-shell C^{3+}(1s2s2p) ^{4}P and ^{2}P_{±} states with emphasis on the ratio R of their cross sections as a measure of spin statistics. Using zero-degree Auger projectile spectroscopy and a three-electron close-coupling semiclassical approach, we resolve a long-standing puzzle and controversy on the value of R and on the effect of cascades, to clarify the underlying physics. The present results invalidate the frozen core approximation generally used in the past when considering electron capture in multielectron multi-open-shell quantum systems. A distinctive screening effect due to the Pauli exclusion principle (Pauli shielding) is proposed to account for the value of R, consistent with our findings.

Characterization of the 2-methylvinoxy radical + O2 reaction: A focal point analysis and composite multireference study

Vinoxy radicals are involved in numerous atmospheric and combustion mechanisms. High-level theoretical methods have recently shed new light on the reaction of the unsubstituted vinoxy radical with O2. The reactions of 1-methylvinoxy radical and 2-methylvinoxy radical with molecular oxygen have experimental high pressure limiting rate constants, k∞, 5-7 times higher than that of the vinoxy plus O2 reaction. In this work, high-level ab initio quantum chemical computations are applied to the 2-methylvinoxy radical plus O2 system, namely, the formation and isomerization of the 1-oxo-2-propylperoxy radical, the immediate product of O2 addition to the 2-methylvinoxy radical. Multireference methods were applied to the entrance channel. No barrier to O2 addition could be located, and more sophisticated treatment of dynamic electron correlation shows that the principal difference between O2 addition to the vinoxy and 2-methylvinoxy radicals is a larger steric factor for 2-methylvinoxy + O2. This is attributed to the favorable interaction between the incoming O2 molecule and the methyl group of the 2-methylvinoxy radical. Via the focal point approach, energetics for this reaction were determined, in most cases, to chemical accuracy. The coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] correlation energy and Hartree-Fock energies were independently extrapolated to the complete basis set limit. A correction for the effect of higher excitations was computed at the CCSDT(Q)/6-31G level. Corrections for the frozen-core approximation, the Born-Oppenheimer approximation, the nonrelativistic approximation, and the zero-point vibrational energy were included. From the 1-oxo-2-propylperoxy radical, dissociation to reactants is competitive with the lowest energy isomerization pathway. The lowest energy isomerization pathway ultimately forms acetaldehyde, CO, and ·OH as the final products.

Wave-packet continuum-discretization approach to proton collisions with helium

We extend the two-center wave-packet convergent close-coupling method to proton collisions with helium. The target is treated as a three-body system, where correlations between the electrons are taken into account. We apply a frozen-core approximation, where one of the electrons is described by the He+1s orbital, and obtain the helium singlet wave functions as well as the energy levels using a numerical approach. The wave-packet approach is used to discretize the continuum of the target and the hydrogen atom formed after electron capture by the projectile. Convergence of the results is studied in terms of the included projectile- and target-centered states. We present electron-capture and single- and double-ionization cross sections for protons incident on He in the ground state in the energy range from 15 keV to 1 MeV. We also provide partial cross sections for electron capture and direct excitation into the n=2 shell states of hydrogen and helium, respectively. Our results are in good agreement with available experimental data and other calculations, where available.