One-electron Integrals Research Articles

GBasis: A Python library for evaluating functions, functionals, and integrals expressed with Gaussian basis functions.

GBasis is a free and open-source Python library for molecular property computations based on Gaussian basis functions in quantum chemistry. Specifically, GBasis allows one to evaluate functions expanded in Gaussian basis functions (including molecular orbitals, electron density, and reduced density matrices) and to compute functionals of Gaussian basis functions (overlap integrals, one-electron integrals, and two-electron integrals). Unique features of GBasis include supporting evaluation and analytical integration of arbitrary-order derivatives of the density (matrices), computation of a broad range of (screened) Coulomb interactions, and evaluation of overlap integrals of arbitrary numbers of Gaussians in arbitrarily high dimensions. For circumstances where the flexibility of GBasis is less important than high performance, a seamless Python interface to the Libcint C package is provided. GBasis is designed to be easy to use, maintain, and extend following many standards of sustainable software development, including code-quality assurance through continuous integration protocols, extensive testing, comprehensive documentation, up-to-date package management, and continuous delivery. This article marks the official release of the GBasis library, outlining its features, examples, and development.

PyBEST: Improved functionality and enhanced performance

PyBEST: Improved functionality and enhanced performance

Quantum Mechanics/Molecular Mechanics Simulations on NVIDIA and AMD Graphics Processing Units.

We have ported and optimized the graphics processing unit (GPU)-accelerated QUICK and AMBER-based ab initio quantum mechanics/molecular mechanics (QM/MM) implementation on AMD GPUs. This encompasses the entire Fock matrix build and force calculation in QUICK including one-electron integrals, two-electron repulsion integrals, exchange-correlation quadrature, and linear algebra operations. General performance improvements to the QUICK GPU code are also presented. Benchmarks carried out on NVIDIA V100 and AMD MI100 cards display similar performance on both hardware for standalone HF/DFT calculations with QUICK and QM/MM molecular dynamics simulations with QUICK/AMBER. Furthermore, with respect to the QUICK/AMBER release version 21, significant speedups are observed for QM/MM molecular dynamics simulations. This significantly increases the range of scientific problems that can be addressed with open-source QM/MM software on state-of-the-art computer hardware.

LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions

Generalized relativistic pseudopotentials (GRPP) of atomic cores implying the use of different potentials for atomic electronic shells with different principal quantum numbers give rise to accurate and reliable relativistic electronic structure models of atoms, molecules, clusters, and solids. These models readily incorporate the effects of Breit electron–electron interactions and one-loop quantum electrodynamics effects. Here, we report the computational procedure for evaluating one-electron integrals of GRPP over contracted Gaussian functions. This procedure was implemented in a library of routines named LIBGRPP, which can be integrated into existing quantum chemistry software, thus enabling the application of various methods to solve the many-electron problem with GRPPs. Pilot applications to electronic transitions in the ThO and UO2 molecules using the new library and intermediate-Hamiltonian Fock space relativistic coupled cluster method are presented. Deviations of excitation energies obtained within the GRPP approach from their all-electron Dirac–Coulomb–Gaunt counterparts do not exceed 50 cm−1 for the 31 lowest-energy states of ThO and 110 cm−1 for the 79 states of UO2. The results clearly demonstrate that rather economical tiny-core GRPP models can exceed in accuracy relativistic all-electron models defined by Dirac–Coulomb and Dirac–Coulomb–Gaunt Hamiltonians.

Periodic plane-wave electronic structure calculations on quantum computers

A procedure for defining virtual spaces, and the periodic one-electron and two-electron integrals, for plane-wave second quantized Hamiltonians has been developed, and it was validated using full configuration interaction (FCI) calculations, as well as executions of variational quantum eigensolver (VQE) circuits on Quantinuum’s ion trap quantum computers accessed through Microsoft’s Azure Quantum service. This work is an extension to periodic systems of a new class of algorithms in which the virtual spaces were generated by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. In this extension, the integration of the first Brillouin zone is automatically incorporated into the two-electron integrals. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, that were able to capture a significant amount of correlation. The focus in this manuscript is on comparing the simulations of small molecules calculated with plane-wave basis sets with large periodic unit cells at the Gamma-point, including images, to results for plane-wave basis sets with aperiodic unit cells. The results for this approach were promising, as we were able to obtain good agreement between periodic and aperiodic results for an LiH molecule. Calculations performed on the Quantinuum H1-1 quantum computer produced surprisingly good energies, in which the error mitigation played a small role in the quantum hardware calculations and the (noisy) quantum simulator results. Using a modest number of circuit runs (500 shots), we reproduced the FCI values for the 1 COVO Hamiltonian with an error of 11 milliHartree, which is expected to improve with a larger number of circuit runs.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C5Me5)2Y(μ-S)2Mo(μ-S)2Y(C5Me5)2]-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C6Cl5)4]- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

Wavefunction-Based Electrostatic-Embedding QM/MM Using CFOUR through MiMiC.

We present an interface of the wavefunction-based quantum chemical software CFOUR to the multiscale modeling framework MiMiC. Electrostatic embedding of the quantum mechanical (QM) part is achieved by analytic evaluation of one-electron integrals in CFOUR, while the rest of the QM/molecular mechanical (MM) operations are treated according to the previous MiMiC-based QM/MM implementation. Long-range electrostatic interactions are treated by a multipole expansion of the potential from the QM electron density to reduce the computational cost without loss of accuracy. Testing on model water/water systems, we verified that the CFOUR interface to MiMiC is robust, guaranteeing fast convergence of the self-consistent field cycles and optimal conservation of the energy during the integration of the equations of motion. Finally, we verified that the CFOUR interface to MiMiC is compatible with the use of a QM/QM multiple time-step algorithm, which effectively reduces the cost of ab initio MD (AIMD) or QM/MM-MD simulations using higher level wavefunction-based approaches compared to cheaper density functional theory-based ones. The new wavefunction-based AIMD and QM/MM-MD implementations were tested and validated for a large number of wavefunction approaches, including Hartree-Fock and post-Hartree-Fock methods like Møller-Plesset, coupled-cluster, and complete active space self-consistent field.

Faster Self-Consistent Field (SCF) Calculations on GPU Clusters.

A novel implementation of the self-consistent field (SCF) procedure specifically designed for high-performance execution on multiple graphics processing units (GPUs) is presented. The algorithm offloads to GPUs the three major computational stages of the SCF, namely, the calculation of one-electron integrals, the calculation and digestion of electron repulsion integrals, and the diagonalization of the Fock matrix, including SCF acceleration via DIIS. Performance results for a variety of test molecules and basis sets show remarkable speedups with respect to the state-of-the-art parallel GAMESS CPU code and relative to other widely used GPU codes for both single and multi-GPU execution. The new code outperforms all existing multi-GPU implementations when using eight V100 GPUs, with speedups relative to Terachem ranging from 1.2× to 3.3× and speedups of up to 28× over QUICK on one GPU and 15× using eight GPUs. Strong scaling calculations show nearly ideal scalability up to 8 GPUs while retaining high parallel efficiency for up to 18 GPUs.

Optimizing two-electron repulsion integral calculations with McMurchie-Davidson method on graphic processing unit.

In this article, several optimization methods of two-electron repulsion integral calculations on a graphic processing unit (GPU) are presented. These methods are based on the investigations of the method presented by McMurchie and Davidson (MD). A new Boys function evaluation method for the GPU calculation is introduced. The series summation, the error function, and the finite sum formula method are combined; thus, good performance on the GPU can be achieved. By taking some theoretical study of the McMurchie-Davidson recurrence relations, three major optimization approaches are derived from the deduction of the general term formula for the Hermite expansion coefficient. The three approaches contain a new form of the Hermite expansion coefficients with corresponding recurrence relations, which is more efficient for one-electron integrals and [ss|∗∗] or [∗∗|ss] type two-electron integrals. In addition, a simple yet efficient new recurrence formula for the coefficient evaluation is derived, which is more efficient both in float operations and memory operations than its original one. In average, the new recurrence relation can save 26% float operations and 37% memory operations. Finally, a common sub-expression elimination (CSE) method is implemented. This CSE method is directly generated from some equalities we discovered from the general term formula other than by computer algebra system software. This optimized method achieved up to 3.09 speedups compared to the original MD method on the GPU and up to 92.75 speedups compared to the GAMESS calculation on the central processing unit.

Computing Four-Center Two-Electron Coulomb Integrals Using Exponential Transformations and Trapezoidal Rule

The numerical evaluations of the four-center two-electron Coulomb integrals are among the most time-consuming computations involved in molecular electronic structure calculations. In the present paper we extend the double exponential (DE) transform method, previously developed for the numerical evaluation of threecenter one-electron molecular integrals [J. Lovrod, H. Safouhi, Molecular Physics (2019) DOI:10.1030/0026867.2019.1619854], to four-center two-electron integrals. The fast convergence properties analyzed in the numerical section illustrate the advantages of the new approach.

Ab initio quantum mechanics/molecular mechanics method with periodic boundaries employing Ewald summation technique to electron-charge interaction: Treatment of the surface-dipole term.

We have developed a combined quantum mechanics/molecular mechanics (QM/MM) method with periodic boundary condition (PBC) treatment of explicit electron-charge interactions in a theoretically rigorous manner, for an accurate description of electronic structures for molecules in the condensed phase. The Ewald summation technique is employed for the calculation of the one-electron Hamiltonian in an ab initio framework. We decompose the Coulomb interactions into two components: those within the same cell and those between different cells. The former is calculated in the same way as the conventional QM/MM calculation for isolated systems; this article focuses on our novel method for calculating the latter type of Coulomb interactions. The detailed formulation of the Hamiltonian of this new QM/MM-PBC method, as well as the necessary one-electron integrals and their gradients, is given. The novel method is assessed by applying it to the dilute water system and a system with a coumarin molecule in water solvent; it successfully reproduces the electronic energies, frontier orbital energies, and Mulliken population charge of the real-space limit calculated by QM/MM using large isolated systems. We investigated the contribution from each term of the Hamiltonian and found that the surface-dipole term in the Ewald summation technique is indispensable for QM/MM-PBC calculations. The newly developed QM/MM-PBC method is promising for tackling chemical reactions and excited states of molecules in the condensed phase.

Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals.

We introduce tight upper bounds for a variety of integrals appearing in electronic structure theories. These include electronic interaction integrals involving any number of electrons and various integral kernels such as the ubiquitous electron repulsion integrals and the three- and four-electron integrals found in explicitly correlated methods. Our bounds are also applicable to the one-electron potential integrals that appear in great number in quantum mechanical (QM), mixed quantum and molecular mechanical (QM/MM), and semi-numerical methods. The bounds are based on a partitioning of the integration space into balls centered around electronic distributions and their complements. Such a partitioning leads directly to equations for rigorous extents, which we solve for shell pair distributions containing shells of Gaussian basis functions of arbitrary angular momentum. The extents are the first general rigorous formulation we are aware of, as previous definitions are based on the inverse distance operator 1/r12 and typically only rigorous for simple spherical Gaussians. We test our bounds for six different integral kernels found throughout quantum chemistry, including exponential, Gaussian, and complementary error function based forms. We compare to previously developed estimates on the basis of significant integral counts and their usage in both explicitly correlated second-order Møller-Plesset theory (MP2-F12) and density functional theory calculations employing screened Hartree-Fock exchange.

On the development of the exact two-component relativistic method for calculating indirect NMR spin-spin coupling constants

Recently, to obtain the proper basis functions of small component spinors from the basis functions of large component spinors, the restricted magnetic balance (RMB) has been used in relativistic calculations of molecular magnetic properties, instead of the restricted kinetic balance (RKB). In this study, we attempt to develop the RMB-based exact two-component relativistic method, i.e., RMB-based normalized elimination of the small component (RMB-NESC) method for calculating indirect NMR spin-spin coupling constants (SSCCs), as well as the RKB-NESC method. The RMB-NESC SSCC formulas require complicated one-electron integrals, unlike the RKB-NESC SSCC formulas. However, the standard integral evaluation method for the complicated integrals has not yet been established, while we calculate them by using the finite-size nuclear model and partial integration. By comparison with the RKB-NESC and experimental SSCC values, we discuss whether the RMB-NESC results are valid. Finally, we conclude that our calculation for the complicated integrals must have some problems.

Efficient Method for Calculating Effective Core Potential Integrals.

Effective core potential (ECP) integrals are among the most difficult one-electron integrals to calculate due to the projection operators. The radial part of these operators may include r0, r-1, and r-2 terms. For the r0 terms, we exploit a simple analytic expression for the fundamental projected integral to derive new recurrence relations and upper bounds for ECP integrals. For the r-1 and r-2 terms, we present a reconstruction method that replaces these terms by a sum of r0 terms and show that the resulting errors are chemically insignificant for a range of molecular properties. The new algorithm is available in Q-Chem 5.0 and is significantly faster than the ECP implementations in Q-Chem 4.4, GAMESS (US) and Dalton 2016.

Hartree-Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms.

An implementation of the Hartree-Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be ZHFC=1.031 177 528.This article is part of the theme issue 'Modern theoretical chemistry'.

Potential Energy Curves for the Low-Lying Electronic States of K2+ from ab Initio Calculations with All Electrons Correlated

The electron affinity (EA) calculations based on the equation-of-motion coupled cluster method proved to be an efficient scheme in the treatment of potential energy curves (PECs) for alkali molecular ions, Me2+. The EA approach provides description of states obtained by an attachment of one electron to the reference, which for the Me2+ is a doubly ionized Me2+2 system. The latter has a very concrete advantage in the calculations of the PECs, since it dissociates into the closed-shell fragments (Me2+2 → Me+ + Me+); hence, the restricted Hartree-Fock reference can be used in the whole range of interatomic distances. In this work accurate PECs and spectroscopic constants are obtained for the six lowest-lying electronic states of the K2+ ion. The relativistic effects are included by adding appropriate terms of the Douglas-Kroll Hamiltonian to the one-electron integrals.

Determination of consistent semiempirical one-centre integrals based on coupled-cluster theory

ABSTRACTWe examine the one-centre integrals used in semiempirical molecular orbital theory, for the elements H–Ne. The currently used parameters do not provide good estimates for the relative energies of ionised states of atoms. Directly calculating the one-electron integrals Uss and Upp with coupled-cluster theory and fitting the two-electron repulsion integrals Gss and Gpp to accurate coupled-cluster ionisation curves improves this behaviour. Since all the remaining parameters can be derived from these, the number of fitted variables is reduced from seven to two. The two-parameter model provides qualitative agreement with coupled-cluster theory for all ionisation potentials (IPs) and the principal electron affinity (EA). To obtain quantitative agreement for the principal IP and EA, Uss and Upp are included as variables in a four-parameter model. We discuss the new parameters and implications for the development of new, consistent semiempirical Hamiltonians.

Solution of the Dirac equation using the Rayleigh-Ritz method: Flexible basis coupling large and small components. Results for one-electron systems.

An algebraic solution of the Dirac equation is reinvestigated. Slater-type spinor orbitals and their corresponding system of differential equations are defined in two- and four-component formalism. They describe the radial function in components of the wave function of the Dirac equation solution to high accuracy. They constitute the matrix elements arising in a generalized eigenvalue equation. These terms are evaluated through prolate spheroidal coordinates. The corresponding integrals are calculated by the numerical global-adaptive method taking into account the Gauss-Kronrod numerical integration extension. Sample calculations are performed using flexible basis sets generated with both signs of the relativistic angular momentum quantum number κ. Applications to one-electron atoms and diatomics are detailed. Variationally optimum values for orbital parameters are obtained at given nuclear separation. Methods discussed in this work are capable of yielding highly accurate relativistic two-center integrals for all ranges of orbital parameters. This work provides an efficient way to overcome the problems that arise in relativistic calculations.

Calculations on rovibrational spectra of two lowest electronic states in sulfur monoxide molecule by explicitly correlated approach

Accurate spectroscopic constants and rovibrational level information are of great importance in molecular physics and astrophysical field. In this work, a new computational scheme is presented to further improve the accuracy of spectroscopic constants, and the two lowest electronic states of sulfur monoxide molecule are investigated as a typical case study. High-level ab initio calculations are carried out to compute the potential energy curves (PECs) of the lowest bound states, the ground triplet states X3Σg-, and the first excited singlet states a1Δg of SO molecule. The explicitly correlated multi-reference configuration interaction method (MRCI-F12) and cc-pCVQZ-F12 basis set are adopted in the electronic structure computations. The Davidson correction is taken into account to eliminate the size-consistency error. The core-valence (CV) electron correlations of the n=2 shell of S atom and n=1 shell of O atom are estimated by the MRCI-F12 method, whereas only the 1 s orbital of S atom is excluded in the CV calculations. Moreover, we introduce the scalar relativistic effect into our study by utilizing the second-order Douglas-Kroll and Hess (DKH) one-electron integrals by the MRCI method through combining with the uncontracted cc-pCVQZ-F12 basis set. On the basis of the PECs of the SO dimer, the spectroscopic constants (Te, Re, ωe, ωeχe, Be, αe and De) of the two electronic states are determined by solving the one-dimensional nuclear rovibrational Schrödinger equations. Our spectroscopic constants are found to be in excellent agreement with previous experimental and theoretical values. Furthermore, the detailed information about the vibrational energy levels and rotation spectroscopic constants (Bν, Dν) of the two states is also presented with a deviation of 0.1% order of magnitude from the available experimental results. Our present computational work is valuable for future experimental studies on the rovibrational energy levels for the SO molecule and also indicates that the MRCI-F12 approach is cheap and accurate and expected to have wide applications in the PECs of other small molecular systems.

Effect of generalized Breit interaction in dielectronicrecombination for highly charged ions

Dielectronic recombination (DR) for highly charged Fe, I, Ba, Bi and U ions are calculated with our developed codes. The calculations are based on isolate resonance approximation, and the free electrons are described with a fully relativistic distorted wave. The Auger decay rates are calculated in Coulomb plus generalized Breit interaction (GBI) and Møller interaction, respectively. The two-electron integrals in the imaginary parts of GBI and Møller interactions are derived to be the product of two one-electron integrals. By comparing our calculations with published theoretical and experimental results, residual relativistic and GBI effects are discussed. DR transitions through optical forbidden decays are investigated.