Basis Set Size Research Articles

A multi-layer multi-configurational time-dependent Hartree approach to lattice models beyond one dimension.

The multi-layer multi-configurational time-dependent Hartree (MCTDH) approach is an efficient method to study quantum dynamics in real and imaginary time. The present work explores its potential to describe quantum fluids. The multi-layer MCTDH approach in second quantization representation is used to study lattice models beyond one dimension at finite temperatures. A scheme to map the lattice sites onto the MCTDH tree representation for multi-dimensional lattice models is proposed. A statistical sampling scheme previously used in MCTDH calculations is adapted to facilitate an efficient description of the thermal ensemble. As example, a two-dimensional hard-core Bose-Hubbard model is studied considering up to 64 × 64 lattice sites. The single particle function basis set size required to obtain converged results is found to not increase with the lattice size. The numerical results properly simulate the finite temperature Berezinskii-Kosterlitz-Thouless phase transition.

Partial density of states representation for accurate deep neural network predictions of X-ray spectra.

The performance of a machine learning (ML) algorithm for chemistry is highly contingent upon the architect's choice of input representation. This work introduces the partial density of states (p-DOS) descriptor: a novel, quantum-inspired structural representation which encodes relevant electronic information for machine learning models seeking to simulate X-ray spectroscopy. p-DOS uses a minimal basis set in conjunction with a guess (non-optimised) electronic configuration to extract and then discretise the density of states (DOS) of the absorbing atom to form the input vector. We demonstrate that while the electronically-focused p-DOS performs well in isolation, optimal performance is achieved when supplemented with nuclear structural information imparted via a geometric representation. p-DOS provides a description of the key electronic properties of a system which is not only concise and computationally efficient, but also independent of molecular size or choice of basis set. It can be rapidly generated, facilitating its application with large training sets. Its performance is demonstrated using a wide variety of examples at the sulphur K-edge, including the prediction of ultrafast X-ray spectroscopic signal associated with photoexcited 2(5H)-thiophenone. These results highlight the potential for ML models developed using p-DOS to contribute to the interpretation and prediction of experimental results e.g. in operando measurements of batteries and/or catalysts and femtosecond time-resolved studies, especially those made possible by emergent cutting-edge technologies, especially X-ray free electron lasers.

Numerical analysis of the complete active-space extended Koopmans's theorem.

We investigate the numerical accuracy of the extended Koopmans's theorem (EKT) in reproducing the full configuration interaction (FCI) and complete active-space configuration interaction (CAS-CI) ionization energies (IEs) of atomic and molecular systems calculated as the difference between the energies of N and (N - 1) electron states. In particular, we study the convergence of the EKT IEs to their exact values as the basis set and the active space sizes vary. We find that the first FCI EKT IEs approach their exact counterparts as the basis set size increases. However, increasing the basis set or the active space sizes does not always lead to more accurate CAS-CI EKT IEs. Our investigation supports the observation of Davidson et al. [J. Chem. Phys. 155, 051102 (2021)] that the FCI EKT IEs can be systematically improved with arbitrary numerical accuracy by supplementing the basis set with diffuse functions of appropriate symmetry, which allow the detached electron to travel far away from the reference system. By changing the exponent and the center of the diffuse functions, our results delineate a complex pattern for the CAS-CI EKT IE of LiH, which can be important for the spectroscopic studies of small molecules.

Improving the precision of work-function calculations within plane-wave density functional theory

Abstract Work function is a fundamental property of metals and is related to many surface-related phenomena of metals. Theoretically, it can be calculated with a metal slab supercell in density functional theory (DFT) calculations. In this paper, we discuss how the commensurability of atomic structure with the underlying fast Fourier transform (FFT) grid affects the accuracy of work function obtained from plane-wave pseudopotential DFT calculations. We show that the macroscopic average potential, which is an important property in work function calculations under the ‘bulk reference’ method, is more numerically stable when it is calculated with commensurate FFT grids than with incommensurate FFT grids. Due to the stability of the macroscopic average potential, work function calculated with commensurate FFT grids shows better convergence with respect to basis set size, vacuum length and slab thickness of a slab supercell. After we control the FFT grid commensurability issue in our work function calculations, we obtain well-converged work functions for Al, Pd, Au and Pt of (100), (110) and (111) surface orientations. For all the metals considered, the ordering of our calculated work functions of the three surface orientations agrees with experiment. Our findings reveal the importance of the FFT grid commensurability issue, which is usually neglected in practice, in obtaining accurate metal work functions, and are also meaningful to other DFT calculations which can be affected by the FFT grid commensurability issue.

Compactification of determinant expansions via transcorrelation.

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects.

Relativistic Coupled Cluster with Completely Renormalized and Perturbative Triples Corrections.

We have implemented noniterative triples corrections to the energy from coupled-cluster with single and double excitations (CCSD) within the 1-electron exact two-component (1eX2C) relativistic framework. The effectiveness of both the CCSD(T) and the completely renormalized (CR) CC(2,3) approaches are demonstrated by performing all-electron computations of the potential energy curves and spectroscopic constants of copper, silver, and gold dimers in their ground electronic states. Spin-orbit coupling effects captured via the 1eX2C framework are shown to be crucial for recovering the correct shape of the potential energy curves, and the correlation effects due to triples in these systems change the dissociation energies by about 0.1-0.2 eV or about 4-7%. We also demonstrate that relativistic effects and basis set size and contraction scheme are significantly more important in Au2 than in Ag2 or Cu2.

Analytic Gradients for Density Fitting MP2 Using Natural Auxiliary Functions.

The natural auxiliary function (NAF) approach is an approximation to decrease the size of the auxiliary basis set required for quantum chemical calculations utilizing the density fitting technique. It has been proven efficient to speed up various correlation models, such as second-order Møller-Plesset (MP2) theory and coupled-cluster methods. Here, for the first time, we discuss the theory of analytic derivatives for correlation methods employing the NAF approximation on the example of MP2. A detailed algorithm for the gradient calculation with the NAF approximation is proposed in the framework of the method of Lagrange multipliers. To assess the effect of the NAF approximation on gradients and optimized geometric parameters, a series of benchmark calculations on small and medium-sized systems was performed. Our results demonstrate that, for MP2, sufficiently accurate gradients and geometries can be achieved with a moderate time reduction of 15-20% for both small and medium-sized molecules.

Electric Field Gradient Calculations for Ice VIII and IX Using Polarizable Embedding: A Comparative Study on Classical Computers and Quantum Simulators.

We test the performance of the polarizable embedding variational quantum eigensolver self-consistent field (PE-VQE-SCF) model for computing electric field gradients with comparisons to conventional complete active space self-consistent-field (CASSCF) calculations and experimental results. We compute quadrupole coupling constants for ice VIII and ice IX. We find close agreement of the quantum-computing PE-VQE-SCF results with the results from the classical PE-CASSCF calculations and with experiment. Furthermore, we observe that the inclusion of the environment is crucial for obtaining results that match the experimental data. The calculations for ice VIII are within the experimental uncertainty for both CASSCF and VQE-SCF for oxygen and lie close to the experimental value for ice IX as well. With the VQE-SCF, which is based on an adaptive derivative-assembled problem-tailored (ADAPT) ansatz, we find that the inclusion of the environment and the size of the different basis sets do not directly affect the gate counts. However, by including an explicit environment, the wavefunction and therefore the optimization problem become more complicated, which usually results in the need to include more operators from the operator pool, thereby increasing the depth of the circuit.

Fixed-node diffusion Monte Carlo shows promise for modeling reaction thermochemistry of hydrocarbon-based radicals.

Reliable thermodynamic and kinetic properties of free radical polymerization reactions are essential for synthesizing both primary polymeric materials and specialty polymers. The computational generation of these data from quantum chemistry requires a time-efficient method capable of capturing the essential physics. One such method, fixed-node diffusion Monte Carlo (FN-DMC) (using single Slater-Jastrow trial wavefunctions), has demonstrated the capability to recover 90%-95% of missing dynamic correlation energy for typical systems. In this study, methyl radical addition to ethylene serves as a simple model to test FN-DMC's ability to calculate enthalpies of reaction and activation energies with different time steps, antisymmetric trial wavefunctions, basis set sizes, and effective core potentials. The FN-DMC computational protocol thus defined for methyl radical addition to ethylene is subsequently benchmarked against Weizmann-1 and experimental reaction enthalpies from Lin et al.'s test set of 21 radical addition and 28 hydrogen abstraction enthalpies. Our findings reveal that FN-DMC consistently generates reaction enthalpies with chemical accuracy, exhibiting mean absolute deviation of 3.5(7) and 1.4(8) kJ/mol from the Weizmann-1 reference for radical addition and hydrogen abstraction reactions, respectively. Given its favorable computational scaling and high degree of parallelizability, we, therefore, recommend more comprehensive testing of FN-DMC with effective core potentials to address more extensive and intricate polymerization reactions and reactions with other radicals.

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

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

QUANTUM CHEMICAL STUDY OF THE ELECTRONIC STRUCTURE OF YTTERBIUM HALIDES

The study of ytterbium halide crystals using the compound-tunable embedding potential (CTEP) method is carried out in the framework of the density functional theory. For subsequent calculations using the coupled-cluster methods, the optimization of atomic bases is carried out, and for this purpose stochiometric molecular systems were studied. The chemical shift of the lines of the X-ray emission spectrum, Kα1 and Kα2, in YbHal3 relative to YbHal2 was chosen as a criterion for verifying the computational accuracy of the properties localized on the nucleus of a heavy atom, Yb, since this method is a unique tool for analyzing partial electron densities near a heavy nucleus speci cally for compounds of d- and f-elements. In the study, ve main versions for the halogen basis set sizes were considered. The stability of the results was obtained using the CCSD and CCSD(T) coupled cluster methods for molecular systems YbF2, YbF3, YbCl2, YbCl3.

Molecular NMR shieldings, J-couplings, and magnetizabilities from numeric atom-centered orbital based density-functional calculations

This paper reports and benchmarks a new implementation of nuclear magnetic resonance shieldings, magnetizabilities, and J-couplings for molecules within semilocal density functional theory, based on numeric atom-centered orbital (NAO) basis sets. NAO basis sets are attractive for the calculation of these nuclear magnetic resonance (NMR) parameters because NAOs provide accurate atomic orbital representations especially near the nucleus, enabling high-quality results at modest computational cost. Moreover, NAOs are readily adaptable for linear scaling methods, enabling efficient calculations of large systems. The paper has five main parts: (1) It reviews the formalism of density functional calculations of NMR parameters in one comprehensive text to make the mathematical background available in a self-contained way. (2) The paper quantifies the attainable precision of NAO basis sets for shieldings in comparison to specialized Gaussian basis sets, showing similar performance for similar basis set size. (3) The paper quantifies the precision of calculated magnetizabilities, where the NAO basis sets appear to outperform several established Gaussian basis sets of similar size. (4) The paper quantifies the precision of computed J-couplings, for which a group of customized NAO basis sets achieves precision of ∼Hz for smaller basis set sizes than some established Gaussian basis sets. (5) The paper demonstrates that the implementation is applicable to systems beyond 1000 atoms in size.

Toward Reliable Dipole Moments without Single Excitations: The Role of Orbital Rotations and Dynamical Correlation.

The dipole moment is a crucial molecular property linked to a molecular system's bond polarity and overall electronic structure. To that end, the electronic dipole moment, which results from the electron density of a system, is often used to assess the accuracy and reliability of new electronic structure methods. This work analyses electronic dipole moments computed with the pair coupled cluster doubles (pCCD) ansätze and its linearized coupled cluster (pCCD-LCC) corrections using the canonical Hartree-Fock and pCCD-optimized (localized) orbital bases. The accuracy of pCCD-based dipole moments is assessed against experimental and CCSD(T) reference values using relaxed and unrelaxed density matrices and different basis set sizes. Our test set comprises molecules of various bonding patterns and electronic structures, exposing pCCD-based methods to a wide range of electron correlation effects. Additionally, we investigate the performance of pCCD-in-DFT dipole moments of some model complexes. Finally, our work indicates the importance of orbital relaxation in the pCCD model and shows the limitations of the linearized couple cluster corrections in predicting electronic dipole moments of multiple-bonded systems. Most importantly, pCCD with a linearized CCD correction can reproduce the dipole moment surfaces in singly bonded molecules, which are comparable to the multireference ones.

Improved treatment of the T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} molecular final-states uncertainties for the KATRIN neutrino-mass measurement

The KArlsruhe TRItium Neutrino experiment (KATRIN) aims to determine the effective mass of the electron antineutrino via a high-precision measurement of the tritium β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta $$\\end{document}-decay spectrum in its end-point region. The target neutrino-mass sensitivity of 0.2eV/c2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${0.2}~\ ext {eV}/\ ext {c}^2$$\\end{document} at 90% CL can only be achieved in the case of high statistics and good control of the systematic uncertainties. One key systematic effect originates from the calculation of the molecular final states of T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta $$\\end{document} decay. In the first neutrino-mass analyses of KATRIN the contribution of the uncertainty of the molecular final-states distribution (FSD) was estimated via a conservative phenomenological approach to be 2×10-2eV2/c4.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \ imes 10^{-2}~\ ext {eV}^{2}/\ ext {c}^{4}.$$\\end{document} In this work a new procedure is presented for estimating the FSD-related uncertainties by considering the details of the final-states calculation, i.e. the uncertainties of constants, parameters, and functions used in the calculation as well as its convergence itself as a function of the basis-set size used in expanding the molecular wave functions. The calculated uncertainties are directly propagated into the experimental observable, the squared neutrino mass mν2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_\ u ^2,$$\\end{document} and thus have to be determined individually for each experimental configuration. For the experimental conditions of the first KATRIN measurement campaign the new procedure is presented in detail, allowing for the application of this procedure to other experiments. This specific calculation leads to a constraint of the FSD-related uncertainty of 1.3×10-3eV2/c4,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.3 \ imes 10^{-3}~\ ext {eV}^{2}/\ ext {c}^{4},$$\\end{document} well below the design limit of 7.5×10-3eV2/c4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$7.5 \ imes 10^{-3}~\ ext {eV}^{2}/\ ext {c}^{4}$$\\end{document} for any individual systematic contribution.

Range-separated density functional theory using multiresolution analysis and quantum computing.

Quantum computers are expected to outperform classical computers for specific problems in quantum chemistry. Such calculations remain expensive, but costs can be lowered through the partition of the molecular system. In the present study, partition was achieved with range-separated density functional theory (RS-DFT). The use of RS-DFT reduces both the basis set size and the active space size dependence of the ground state energy in comparison with the use of wave function theory (WFT) alone. The utilization of pair natural orbitals (PNOs) in place of canonical molecular orbitals (MOs) results in more compact qubit Hamiltonians. To test this strategy, a basis-set independent framework, known as multiresolution analysis (MRA), was employed to generate PNOs. Tests were conducted with the variational quantum eigensolver for a number of molecules. The results show that the proposed approach reduces the number of qubits needed to reach a target energy accuracy.

Building a New Platform for Significantly Improving Performance of Hartree-Fock and CCSD(T) Correlation Energy Based on Two-Point Complete Basis Set Extrapolation Schemes.

The leading cause of high expense in gold standard coupled cluster theory is that calculations of electronic energies converge exceedingly slowly with an increased basis set size. Extrapolation principally allows for achieving higher-quality outcomes at reduced costs. Numerous extrapolation formulas have been developed, with attempts to predict energies up to the complete basis set limit. Unfortunately, since the intricate shape of the function hinges on the molecular properties with the highest angular momentum of the basis set, the accuracy of the extrapolated energies highly depends on the fitted empirical parameters, which rely on the quality of the data sets for fitting. In this work, to overcome the extrapolation deficiency caused by the very limited data sets and smaller basis sets in the early stages, we constructed a new benchmark platform that includes a broader data set of 183 species (containing open-shell, closed-shell, ionic, and neutral species) and a larger basis set up to aug-cc-pV6Z. The newly optimized parameters can significantly improve the energy-predictive abilities of ten published formulas. Notably, all ten formulas perform quite similarly under the new platform with the reoptimized parameters. Finally, we built an online calculator for researchers to use for these extrapolation schemes. Our work would reignite the interest and applications of the underestimated formulas.

Extending the definition of atomic basis sets to atoms with fractional nuclear charge.

Alchemical transformations showed that perturbation theory can be applied also to changes in the atomic nuclear charges of a molecule. The alchemical path that connects two different chemical species involves the conceptualization of a non-physical system in which an atom possess a non-integer nuclear charge. A correct quantum mechanical treatment of these systems is limited by the fact that finite size atomic basis sets do not define exponents and contraction coefficients for fractional charge atoms. This paper proposes a solution to this problem and shows that a smooth interpolation of the atomic orbital coefficients and exponents across the periodic table is a convenient way to produce accurate alchemical predictions, even using small size basis sets.

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals.

σ-Functionals belong to the class of Kohn-Sham (KS) correlation functionals based on the adiabatic-connection fluctuation-dissipation theorem and are technically closely related to the random phase approximation (RPA). They have the same computational demand as the latter, with the computational effort of an energy evaluation for both methods being lower than that of a preceding hybrid DFT calculation for typical systems but yield much higher accuracy, reaching chemical accuracy of 1 kcal/mol for quantities such as reactions and transition energies in main group chemistry. In previous work on σ-functionals, rather large Gaussian basis sets have been used. Here, we investigate the actual basis set requirements of σ-functionals and present three setups that employ smaller Gaussian basis sets ranging from quadruple-ζ (QZ) to triple-ζ (TZ) quality and represent a good compromise between accuracy and computational efficiency. Furthermore, we introduce an implementation of σ-functionals based on Slater-type basis sets and present two setups of QZ and TZ quality for this implementation. We test the accuracy of these setups on a large database of various physical properties and types of reactions, as well as equilibrium geometries and vibrational frequencies. As expected, the accuracy of σ-functional calculations becomes somewhat lower with a decreasing basis set size. However, for all setups considered here, calculations with σ-functionals are clearly more accurate than those within the RPA and even more so than those of the conventional KS methods. For the smallest setup using Gaussian-type basis functions and Slater-type basis functions, we introduce a reparametrization that reduces the loss in accuracy due to the basis set error to some extent. A comparison with the range-separated hybrid ωB97X-V and the double hybrid DSD-BLYP-D3 shows that σ functionals outperform in accuracy both of these accurate and, for their class, representative functionals.

Correlation consistent basis sets designed for density functional theory: Third-row atoms (Ga-Br).

The correlation consistent basis sets (cc-pVnZ with n = D, T, Q, 5) for the Ga-Br elements have been redesigned, tuning the sets for use for density functional approximations. Steps to redesign these basis sets for an improved correlation energy recovery and efficiency include truncation of higher angular momentum functions, recontraction of basis set coefficients, and reoptimization of basis set exponents. These redesigned basis sets are compared with conventional cc-pVnZ basis sets and other basis sets, which are, in principle, designed to achieve systematic improvement with respect to increasing basis set size. The convergence of atomic energies, bond lengths, bond dissociation energies, and enthalpies of formation to the Kohn-Sham limit is improved relative to other basis sets where convergence to the Kohn-Sham limit is typically not observed.

A failure of double hybrid density functional method: Out‐of‐plane bending vibrations for the carbon‐carbon double‐bonded molecules

AbstractIn this work, we report anomalous descriptions of bending modes of the carbon–carbon double bonded molecules with double hybrid density functional theories. The harmonic frequencies of the out‐of‐plane modes are found to be extremely low, and the anharmonicities are found to be very large. The unphysical large anharmonities of these modes increase significantly with a basis set having diffuse function. Increasing basis set size does not necessarily reduce or eliminate the problem. We observe that the problem with the double hybrid functional is in‐built due to the presence of the part of the second‐order perturbation terms in the functional.