Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Strategies for an efficient implementation of the Gauss-Bessel quadrature for the evaluation of multicenter integral over STFs

In a previous work, a new Gauss quadrature was introduced with a view to evaluate multicenter integrals over Slater-type functions efficiently. The complexity analysis of the new approach, carried out using the three-center nuclear integral as a case study, has shown that for low-order polynomials its efficiency is comparable to the SD. The latter was developed in connection with multi-center integrals evaluated by means of the Fourier transform of B functions. In this work we investigate the numerical properties of the Gauss-Bessel quadrature and devise strategies for an efficient implementation of the numerical algorithms for the evaluation of multi-center integrals in the framework of the Gaussian transform/Gauss-Bessel approach. The success of these strategies are essential to elaborate a fast and reliable algorithm for the evaluation of multi-center integrals over STFs.     

Use of analytical relations in evaluation of exponential integral functions

Analytical formulas through the initial values suitable for numerical computation are developed for the exponential integral functions E_n (x). The relationships obtained are numerically stable for all values of n and for x < 1. Numerical results are also given.PACS No: 31.15.+q, 31.20.Ej AMS subject classification:81-V55, 81V45     

An accurate numerical multicenter integration for molecular orbital theory

A powerful and accurate numerical three‐dimensional integration scheme was developed especially for molecular orbital calculations. A multicenter integral is decomposed into the sum of single‐center integrals using nuclear weight functions and calculated using Gaussian quadrature rules. The decomposed single‐center integrands show strong anisotropy. With a careful selection of the Gaussian quadrature rule according to the anisotropy, it is possible to obtain an accuracy of 13 digits with a small number of integration points for the overlap integrals, normalization integrals, and molecular integrals for the hydrogen molecule. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 509–523, 1999     

Components for integral evaluation in quantum chemistry

Sharing low-level functionality between software packages enables more rapid development of new capabilities and reduces the duplication of work among development groups. Using the component approach advocated by the Common Component Architecture Forum, we have designed a flexible interface for sharing integrals between quantum chemistry codes. Implementation of these interfaces has been undertaken within the Massively Parallel Quantum Chemistry package, exposing both the IntV3 and Cints/Libint integrals packages to component applications. Benchmark timings for Hartree-Fock calculations demonstrate that the overhead due to the added interface code varies significantly, from less than 1% for small molecules with large basis sets to nearly 10% for larger molecules with smaller basis sets. Correlated calculations and density functional approaches encounter less severe performance overheads of less than 5%. While these overheads are acceptable, additional performance losses occur when arbitrary implementation details, such as integral ordering within buffers, must be handled. Integral reordering is observed to add an additional overhead as large as 12%; hence, a common standard for such implementation details is desired for optimal performance.     

Integral algorithm and density matrix integration scheme for ab initio band structure calculations on polymeric systems

A new program for band structure calculations of periodic one-dimensional systems has been constructed. It is distinguishable from other codes by the efficient two-electron integral evaluation and the integration schemes of the density matrix in the first Brillouin zone. The computation of polymeric two-electron integrals is based on the McMurchie Davidson algorithm and builds batches of the different cell indices included in the polymeric system. Consequently it presents efficient scaling with respect to the number of unit cells taken into account. Our algorithm takes into account fully the polymeric symmetry rather than the molecular symmetry. A semidirect procedure where only exchange integrals are computed at each SCF cycle is proposed in order to maintain balance between computation time and disk space. In addition, the integration of the density matrix over a large number of cell indices can be performed by different methods, such as Gauss-Legendre, Clenshaw-Curtis, Filon, and Alaylioglu-Evans-Hyslop. This last scheme is able to obtain an accuracy of 10(-13) a.u. on each individual density matrix element for all cell indices with only 48 k-points.     

Exact Gaussian expansions of Slater-type atomic orbitals

Three exact Slater-type function (STO) integral transforms are presented. The STO-NG basis set can then be developed using either only 1s Gaussian functions, the same Gaussian exponents for each shell, or using the first Gaussian of each symmetry. The use of any of these three alternatives depends only on appropriate numerical integration techniques.     

Recurrence formulas for molecular integrals over Laguerre Gaussian-type functions

Recurrence formulas for overlap, nuclear attraction, and electron-repulsion integrals over Laguerre Gaussian-type functions are presented. They have been derived using compact recurrence relations for homogeneous solid spherical harmonic operators but are rather lengthy as compared to those over Cartesian Gaussian-type functions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 273–279, 1998     

Derivative of electron repulsion integral using accompanying coordinate expansion and transferred recurrence relation method for long contraction and high angular momentum

In this study, an early‐working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy‐element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE‐TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE‐TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.     

Evaluation of molecular integral of Cartesian Gaussian type basis function with complex‐valued center coordinates and exponent via the McMurchie–Davidson recursion formula,and its application to electron dynamics

McMurchie–Davidson recursion formula is extended to derive the ab initio molecular integrals with higher angular quantum number complex Gaussian type basis function which has complex‐valued center coordinates and a complex‐valued exponent. Using the analytical recursion formulae, some calculations of electronic dynamics after beta decay of tritium hydride molecular ion HT⁺ are performed by a quantum wave packet method with thawed Gaussian basis functions of s‐ and p‐type. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Recursion formula for electron repulsion integrals over Hermite polynomials

A method for computing electron repulsion integrals over contracted Gaussian functions is described in which intermediate integrals over Hermite polynomials are generated by a “pre‐Hermite” recursion (PHR) step before the conversion to regular integrals. This greatly reduces the floating‐point operation counts inside the contraction loops, where only simple “scaling”‐type operations are required, making the method efficient for contracted Gaussians, particularly of high angular momentum. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Automatic code generation for quantum chemistry applications

A unified, computer algebra system‐based scheme of code‐generation for computational quantum‐chemistry programs is presented. Generation of electron‐repulsion integrals and their derivatives as well as exchange‐correlation potential and its derivatives is discussed. Application to general‐purpose computing on graphics processing units is considered.     

Evaluation of expansion coefficients for translation of Slater‐type orbitals using complete orthonormal sets of exponential‐type functions

By the use of exponential‐type functions (ETFs) the simpler formulas for the expansion of Slater‐type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001     

A toolchain for the automatic generation of computer codes for correlated wavefunction calculations

In this work, the automated generator environment for ORCA (ORCA‐AGE) is described. It is a powerful toolchain for the automatic implementation of wavefunction‐based quantum chemical methods. ORCA‐AGE consists of three main modules: (1) generation of “raw” equations from a second quantized Ansatz for the wavefunction, (2) factorization and optimization of equations, and (3) generation of actual computer code. We generate code for the ORCA package, making use of the powerful functionality for wavefunction‐based correlation calculations that is already present in the code. The equation generation makes use of the most elementary commutation relations and hence is extremely general. Consequently, code can be generated for single reference as well as multireference approaches and spin‐independent as well as spin‐dependent operators. The performance of the generated code is demonstrated through comparison with efficient hand‐optimized code for some well‐understood standard configuration interaction and coupled cluster methods. In general, the speed of the generated code is no more than 30% slower than the hand‐optimized code, thus allowing for routine application of canonical ab initio methods to molecules with about 500–1000 basis functions. Using the toolchain, complicated methods, especially those surpassing human ability for handling complexity, can be efficiently and reliably implemented in very short times. This enables the developer to shift the attention from debugging code to the physical content of the chosen wavefunction Ansatz. Automatic code generation also has the desirable property that any improvement in the toolchain immediately applies to all generated code. © 2017 Wiley Periodicals, Inc.     

GPGPU for orbital function evaluation with a new updating scheme

We have accelerated an ab initio quantum Monte Carlo electronic structure calculation using general purpose computing on graphical processing units (GPGPU). The part of the code causing the bottleneck for extended systems is replaced by Compute Unified Device Architecture‐GPGPU subroutine kernels which build up spline basis set expansions of electronic orbital functions at each Monte Carlo step. We have achieved a speedup of a factor of 30 for the bottleneck for a simulation of solid TiO₂ with 1536 electrons. To improve the performance with GPGPU we propose a new updating scheme for Monte Carlo sampling, quasi‐simultaneous updating, which is intermediate between configuration‐by‐configuration updating and the widely used particle‐by‐particle updating. The error in the energy due to by the single precision treatment and the new updating scheme is found to be within the required accuracy of ～10^?3 hartree per primitive cell. © 2012 Wiley Periodicals, Inc.     

Extension of accompanying coordinate expansion and recurrence relation method for general‐contraction basis sets

An algorithm of the accompanying coordinate expansion and recurrence relation (ACE‐RR), which is used for the rapid evaluation of the electron repulsion integral (ERI), has been extended to the general‐contraction (GC) scheme. The present algorithm, denoted by GC‐ACE‐RR, is designed for molecular calculations including heavy elements, whose orbitals consist of many primitive functions with and without higher angular momentum such as d‐ and f‐orbitals. The performance of GC‐ACE‐RR was assessed for ‐, ‐, ‐, and ‐type ERIs in terms of contraction length and the number of GC orbitals. The present algorithm was found to reduce the central processing unit time compared with the ACE‐RR algorithm, especially for higher angular momentum and highly contracted orbitals. Compared with HONDOPLUS and GAMESS program packages, GC‐ACE‐RR computations for ERIs of three‐dimensional gold clusters Au_n (n = 1, 2, …, 10, 15, 20, and 25) are more than 10 times faster. © 2014 Wiley Periodicals, Inc.     

Wavelet algorithm for solving integral equations of molecular liquids. A test for the reference interaction site model

A new efficient method is developed for solving integral equations based on the reference interaction site model (RISM) of molecular liquids. The method proposes the expansion of site-site correlation functions into the wavelet series and further calculations of the approximating coefficients. To solve the integral equations we have applied the hybrid scheme in which the coarse part of the solution is calculated by wavelets with the use of the Newton-Raphson procedure, while the fine part is evaluated by the direct iterations. The Coifman 2 basis set is employed for the wavelet treatment of the coarse solution. This wavelet basis set provides compact and accurate approximation of site-site correlation functions so that the number of basis functions and the amplitude of the fine part of solution decrease sufficiently with respect to those obtained by the conventional scheme. The efficiency of the method is tested by calculations of SPC/E model of water. The results indicated that the total CPU time to obtain solution by the proposed procedure reduces to 20% of that required for the conventional hybrid method.     

Fourier transform general formula for systematic potentials

For calculating molecular integrals of systematic potentials, a three‐dimensional (3D) Fourier transform general formula can be derived, by the use of the Abel summation method. The present general formula contains all 3D Fourier transform formulas which are well known as Bethe–Salpeter formulas (Bethe and Salpeter, Handbuch der Physik, Bd. XXXV, 1957) as special cases. It is shown that, in several of the Bethe–Salpeter formulas, the integral does not converge in the meaning of the Riemann integral but converges in the meaning of a hyper function as the Schwartz distribution. For showing an effectiveness of the present general formula, the convergence condition of molecular integrals is derived generally for all of the present potentials. It is found that molecular integrals can be converged in the meaning of the Riemann integral for the present potentials, except for those for extra super singular potentials. It is also found that the convergence condition of molecular integrals over the Slater‐type orbitals is exactly the same as that of the corresponding integrals over the Gaussian‐type orbitals for the present systematic potentials. For showing more effectiveness, the molecular integral over the gauge‐including atomic orbitals is derived for the magnetic dipole‐same‐dipole interaction. © 2012 Wiley Periodicals, Inc.     

生物炭模板构筑Fe-Ni复合载氧体及其化学链制氢反应性能研究

采用松木热解生物炭为模板构筑Fe-Ni复合载氧体（Fe-Ni/BC）,并与溶胶-凝胶法制备的NiFe₂O₄载氧体（NiFe₂O₄/SG）对比,采用SEM、XRD、XPS、BET、H₂-TPR、TG-redox循环等表征方法考察载氧体的物理、化学性质,并在固定床上进行化学链制氢循环实验。结果表明,Fe-Ni/BC载氧体为Ni_0.6Fe_2.4O₄与Fe₂O₃的混合晶体,保留了生物炭骨架并形成了大孔结构。与NiFe₂O₄/SG相比,Fe-Ni/BC平均粒径更小,比表面积更大,吸附氧含量更高,更有利于氧的释放。在固定床实验过程中,Fe-Ni/BC表现出更强的化学链制氢与抗积炭性能,其最大产氢速率是NiFe₂O₄/SG的1.58倍,制取H₂的相对浓度可达到99.5%以上。     

Use of semiempirical molecular orbital calculations for the evaluation of force fields

Various methods, employing molecular orbital calculations of varying approximations, for evaluation of force fields of polyatomic molecules have been reviewed. Applications ofcndo/force method for the force field calculations are specially dealt with in detail because of its ease of operation and being economically more viable in terms of computer time. The calculated C=O stretching force constants for a series of organic molecules are shown to have linear relationship with substituent constants.