Gaussian-type Basis Functions Research Articles

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.

Pre–Born-Oppenheimer Dirac-Coulomb-Breit computations for two-body systems

The sixteen-component, no-pair Dirac--Coulomb--Breit equation, derived from the Bethe--Salpeter equation, is solved in a variational procedure using Gaussian-type basis functions for the example of positronium, muonium, hydrogen atom, and muonic hydrogen. The $\alpha$ fine-structure-constant dependence of the variational energies, through fitting a function of $\alpha^n$ and $\alpha^n\text{ln}\alpha$ terms, shows excellent agreement with the relevant energy expressions of the (perturbative) non-relativistic QED framework, and thereby, establishes a solid reference for the development of a computational relativistic QED approach.

Computing integrals for electron molecule scattering on heterogeneous accelerator systems

SummaryUsing heterogeneous accelerators to obtain high performance for mathematical kernels remains an active research frontier in computational science. The accelerators have compute architectures that are different from the CPUs and in addition have memory spaces independent of the CPU systems to which they are connected. It follows that accelerators require a different approach to writing optimal code than that needed on a multi‐CPU system. Taken together these issues have represented a significant barrier to widespread adoption of accelerators for execution with large legacy code bases. OpenCL has emerged as a common programming language with which to implement code that runs across a range of parallel architectures, including multi‐core CPUs. This article is a case study on how the instruction‐level parallelism offered by field programmable gate arrays (FPGAs) and GPUs through OpenCL can be exploited in molecular physics. The algorithm which we study is the evaluation of tail integrals between Gaussian type basis functions for the R‐matrix method, a task that arises in the study of scattering of low energy electrons by molecular targets. The results of our productivity study, which is the first application of OpenCL in this problem domain, show that significant performance can be obtained from both FPGA and graphics processing unit (GPU) accelerators for this application. We discuss suitable transformations unique to each accelerator architecture for the integrals studied and present performance results comparing the FPGA and GPU with execution on Intel multi‐core systems.

Calculation of Quantum Chemical Two-Electron Integrals by Applying Compiler Technology on GPU.

In this article, we present an effective approach to calculate quantum chemical two-electron integrals over basis sets consisting of Gaussian-type basis functions on graphical processing unit (GPU). Our framework generates several different variants called routes to the same integral problem with different integral algorithms (McMurchie-Davidson, Head-Gordon-Pople, and Rys) and precision. Each route is benchmarked on more GPU architectures, and with this data, a model is fitted to select the best available route for an integral task given a GPU architecture. Moreover, this approach supports the computation of high angular momentum orbitals up to g effectively on GPU, tested up to cc-pVQZ-sized basis sets. Rigorous analysis is shown regarding the effectiveness of our method. Molecule simulations with several basis sets are measured using NVIDIA GTX 1080 Ti, NVIDIA P100, and NVIDIA V100 cards.

Numerical calculation of the Spherical Bessel Transform from Gaussian quadrature in the complex-plane

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

English

The paper focuses on a low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization techniques due the dimensionality of the problem. However, it can be effectively performed using the tensor representation of basis functions. Furthermore, this approach can take advantage of parallel computing.

An open-source framework for analyzing N-electron dynamics. II. Hybrid density functional theory/configuration interaction methodology.

In this contribution, we extend our framework for analyzing and visualizing correlated many-electron dynamics to non-variational, highly scalable electronic structure method. Specifically, an explicitly time-dependent electronic wave packet is written as a linear combination of N-electron wave functions at the configuration interaction singles (CIS) level, which are obtained from a reference time-dependent density functional theory (TDDFT) calculation. The procedure is implemented in the open-source Python program detCI@ORBKIT, which extends the capabilities of our recently published post-processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). From the output of standard quantum chemistry packages using atom-centered Gaussian-type basis functions, the framework exploits the multideterminental structure of the hybrid TDDFT/CIS wave packet to compute fundamental one-electron quantities such as difference electronic densities, transient electronic flux densities, and transition dipole moments. The hybrid scheme is benchmarked against wave function data for the laser-driven state selective excitation in LiH. It is shown that all features of the electron dynamics are in good quantitative agreement with the higher-level method provided a judicious choice of functional is made. Broadband excitation of a medium-sized organic chromophore further demonstrates the scalability of the method. In addition, the time-dependent flux densities unravel the mechanistic details of the simulated charge migration process at a glance. © 2017 Wiley Periodicals, Inc.

An open-source framework for analyzing N-electron dynamics. I. Multideterminantal wave functions.

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many-electron dynamics of molecular systems, where an explicitly time-dependent electronic wave packet is represented as a linear combination of N-electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one-electron operator by reducing the multideterminantal, many-electron wave packet using the Slater-Condon rules. Here, we introduce a general tool for post-processing multideterminant configuration-interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom-centered Gaussian-type basis functions. The procedure is implemented in the open-source Python program detCI@ORBKIT, which shares and builds on the modular design of our recently published post-processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N-electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.

Random Phase Approximation for Periodic Systems Employing Direct Coulomb Lattice Summation.

A method to compute ground state correlation energies from the random phase approximation (RPA) is presented for molecular and periodic systems on an equal footing. The supermatrix representation of the Hartree kernel in canonical orbitals is translation-symmetry adapted and factorized by the resolution of the identity (RI) approximation. Orbital expansion and RI factorization employ atom-centered Gaussian-type basis functions. Long ranging Coulomb lattice sums are evaluated in direct space with a revised recursive multipole method that works also for irreducible representations different from Γ. The computational cost of this RI-RPA method scales as [Formula: see text](N4) with the system size in direct space, N, and as [Formula: see text](Nk2) with the number of sampled k-points in reciprocal space, Nk. For chain and film models, the exploration of translation symmetry with 10 k-points along each periodic direction reduces the computational cost by a factor of around 10-100 compared to equivalent Γ-point supercell calculations.

Reflections on the MOLECULE integral program (and its developer)

ABSTRACTIn this essay, I review some of the advances made by Jan Almlöf in the calculation of molecular integrals over Gaussian-type basis functions, and in the use of molecular symmetry to both produce integrals over symmetry-adapted combinations of basis functions and to ensure that these latter were constructed from the minimum possible number of distinct integrals over atom-centred functions. The survey is, as already noted, an essay, historical in nature, and I crave the reader's indulgence that it is structured as an essay and not as a conventional scientific manuscript.

Range-separated double-hybrid density-functional theory applied to periodic systems.

Quantum chemistry methods exploiting density-functional approximations for short-range electron-electron interactions and second-order Møller-Plesset (MP2) perturbation theory for long-range electron-electron interactions have been implemented for periodic systems using Gaussian-type basis functions and the local correlation framework. The performance of these range-separated double hybrids has been benchmarked on a significant set of systems including rare-gas, molecular, ionic, and covalent crystals. The use of spin-component-scaled MP2 for the long-range part has been tested as well. The results show that the value of μ = 0.5 bohr(-1) for the range-separation parameter usually used for molecular systems is also a reasonable choice for solids. Overall, these range-separated double hybrids provide a good accuracy for binding energies using basis sets of moderate sizes such as cc-pVDZ and aug-cc-pVDZ.

First-Principles Calculations of the Pressure Stability and Elasticity of Dense TiO2Phases Using the B3LYP Hybrid Functional

The crystal structures, pressure–volume equations of state, and pressure stability up to 100 GPa of the experimentally observed and theoretically proposed dense TiO2 phases (rutile, columbite, baddeleyite, OI, cotunnite, fluorite, pyrite, and Pca21) were calculated using the hybrid B3LYP exchange-correlation functional and two independent Gaussian-type basis functions. Overall, the B3LYP results are in good agreement with the experimental data on the structures and elastic properties. The B3LYP functional also shows superior performance relative to projector-augmented wave/pseudopotential-based planewave density functionals in predicting the elastic behaviors for most of the TiO2 phases with the exception of fluorite-TiO2. The latter structure shows significant sensitivity to the choice of basis functions. The order of phase stability with increasing pressure predicted by B3LYP is rutile → columbite → baddeleyite → OI → cotunnite, in agreement with available experimental results. In the pressure range of 20–50 GPa, the B3LYP total energy for the recently proposed Pca21 structure is very close to that for one or more of the accepted stable phases (columbite, baddeleyite, OI, and cotunnite), suggesting potential stabilization of the Pca21 structure at high temperatures. The B3LYP electron density-of-states projections suggest large band gaps for the high-pressure phases OI and Pca21.

Flexible nuclear screening approximation to the two-electron spin–orbit coupling based on ab initio parameterization

The derivation, implementation, and validation of a new approximation to the two-electron spin-orbit coupling (SOC) terms is reported. The approximation, referred to as flexible nuclear screening spin-orbit, is based on the effective one-electron spin-orbit operator and accounts for two-electron SOC effects by screening nuclear charges. A highly flexible scheme for the nuclear screening is developed, mainly using parameterization based on ab initio atomic SOC calculations. Tabulated screening parameters are provided for contracted and primitive Gaussian-type basis functions of the ANO-RCC basis set for elements from H to Cm. The strategy for their adaptation to any other Gaussian basis set is presented and validated. A model to correct for the effect of splitting of transition metal d orbitals on their SOC matrix elements is introduced. The method is applied to a representative set of molecules, and compared to exact treatment and other approximative approaches at the same level of relativistic theory. The calculated SOC matrix elements are in very good agreement with their "exact" values; deviation below 1% is observed on average. The presented approximation is considered to be generally applicable, simple to implement, highly efficient, and accurate.

Implementation of Two-Component Time-Dependent Density Functional Theory in TURBOMOLE.

We report the efficient implementation of a two-component time-dependent density functional theory proposed by Wang et al. (Wang, F.; Ziegler, T.; van Lenthe, E.; van Gisbergen, S.; Baerends, E. J. J. Chem. Phys. 2005, 122, 204103) that accounts for spin-orbit effects on excitations of closed-shell systems by employing a noncollinear exchange-correlation kernel. In contrast to the aforementioned implementation, our method is based on two-component effective core potentials as well as Gaussian-type basis functions. It is implemented in the TURBOMOLE program suite for functionals of the local density approximation and the generalized gradient approximation. Accuracy is assessed by comparison of two-component vertical excitation energies of heavy atoms and ions (Cd, Hg, Au(+)) and small molecules (I2, TlH) to other two- and four-component approaches. Efficiency is demonstrated by calculating the electronic spectrum of Au20.

An algorithm for the efficient evaluation of two-electron repulsion integrals over contracted Gaussian-type basis functions

A new algorithm for the evaluation of two-electron repulsion integrals optimized for high contraction degrees is derived. Both the segmented and general contraction versions of the algorithm show significant theoretical performance gains over the asymptotically fastest algorithms published in the literature so far. A preliminary implementation of the algorithm shows good agreement with the theoretical results and demonstrates substantial average speedups in the evaluation of two-electron repulsion integrals over commonly used basis sets with varying degrees of contraction with respect to a mature, highly optimized quantum chemical code.

Theoretical Study of Adsorption of Ag Clusters on the Anatase TiO2(100) Surface

The adsorption of silver atoms, dimers, tetramers, and octamers on the perfect anatase (100) surface has been studied theoretically at density functional theory level. Two complementary approaches based on linear combinations of atom-centered Gaussian-type basis functions and plane waves were applied. The most preferable adsorption positions were found to be hollow sites between two-coordinated oxygen atoms (O(2c)). Adsorption on the Ti–O planes is energetically less favorable. The binding mechanism was interpreted in terms of the interaction between cluster orbitals and orbitals of O(2c) atoms. The charge transfer leads to a deformation of the cluster shape compared with the stable gas-phase structures. In the Ti–O plane sites, an additional binding mechanism exists through the overlap between the silver cluster’s HOMO and unoccupied orbitals of five-coordinated titanium atoms. The same bonding mechanisms have been found previously for the adsorption of Ag particles on the rutile (110) surface1. The grow...

Calculated rotational and vibrational g factors of LiH X 1Σ+ and evaluation of parameters in radial functions from rotational and vibration‐rotational spectra

AbstractThe vibrational g factor, that is, the nonadiabatic correction to the vibrational reduced mass, of LiH has been calculated for internuclear distances over a wide range. Based on multiconfigurational wave functions with a large complete active space and an extended set of gaussian type basis functions, these calculations yielded also the rotational g factor, the electric dipolar moment, and its gradient with internuclear distance for LiH in its electronic ground state X 1Σ+. The vibrational g factor gv exhibits a pronounced minimum near internuclear distance R = 3.65 × 10−10 m; the derivative of electric dipolar moment and the nonadiabatic matrix element coupling the electronic ground state to the first electronically excited state exhibit extrema near the same location that is also near the avoided crossing of the curves for potential energy for the electronic ground state and excited state A 1Σ+. The irreducible contribution g(R) to the rotational g factor increases monotonically over the calculated domain, whereas the irreducible contribution g(R) to the vibrational g factor has a minimum at the same location as that of gv itself. From these calculated radial functions, we derived values of the rotational g factor and electric dipolar moment for LiH in vibrational states v = 0 and 1, and the corresponding rotational dependences, in satisfactory agreement with experimental values. These calculated data of rotational g factor served as constraints in new fits of 1000 vibration‐rotational spectral data of LiH in four isotopic variants, which yield estimates of adiabatic corrections for comparison with published data and of the vibrational g factor for comparison with our calculated results. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

A Linear-Scaling MP2 Method for Large Molecules by Rigorous Integral-Screening Criteria

Abstract A brief review of our linear-scaling method for atomic-orbital (AO) second-order Møller-Plesset perturbation theory (MP2) is given. The key feature of our method is the rigorous preselection of numerically significant four-center two-electron integrals based on multipole-based integral estimates (MBIE) that do not only account for the exponential coupling between the Gaussian-type basis functions forming charge distributions, but also for the 1/R coupling between the charge distributions. This coupling turns for the required integral products in AO-MP2 into at least a 1/R4 or even a 1/R6 decay behavior. Using MBIE we attain linear scaling, which is illustrated for DNA fragments with up to 1052 atoms and 10 674 basis functions as computed on a single processor. The largest molecule calculated in our present work at the scaled-opposite spin (SOS-) AO-MP2 level is an RNA system comprising 1664 atoms and 19 182 basis functions. Furthermore, we present results for the use of Cholesky-decomposed pseudo-density matrices in Laplace-based MP2, that offers the advantage of exploiting occupied/virtual blocking both with and without auxiliary basis sets.

Relativistic contracted Gaussian-type basis functions for atoms K through Xe

For 36 atoms from K to Xe, compact relativistic Gaussian-type function basis sets are developed in a segmented contraction form: ( 843 2 / 74 / 7 ) for the third row atoms and ( 843 3 / 743 / 74 ) for the fourth row atoms. Spin-free relativistic effects are considered through the third-order Douglas–Kroll approximation. The present basis sets are tested for all-electron calculations of four homo- and 12 heteronuclear diatomics in their ground states. The calculated spectroscopic constants are in good agreement with the experimental values.

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post-processing analysis of quantum-chemical electron densities is described. The code can work with Slater- and Gaussian-type basis functions of arbitrary angular momentum. It has been applied to explore the basis-set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple-zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis-set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple-zeta/higher level with Gaussian/Slater functions.