General recurrence-relation generation scheme for molecular integral evaluation.

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

This paper presents a new framework for hierarchical neural networks. It is based on the interaction between bottom-up and topdown procedure to realize human-like cognitive processes. The proposed system consists of 3 layers, input layer, feature extraction layer and symbol layer. Input layer is a kind of associative memory and associate by itself. When the association fails, feature extraction through bottom-up procedure and attention to the parts of input pattern through top-down procedure make it possible to associate target pattern. Some simulation results show the feasibility of proposed system.

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

A method is presented for the evaluation of molecular multi-centre spin-orbit integrals of Coulomb, hybrid and exchange types using cartesian gaussian functions. It is shown that if gaussian bases are employed the spin-same-orbit integrals reduce to one-electron field integrals and the spin-other-orbit integrals to two-electron field integrals. The final formulae for these integrals are found to be convenient for computer programming. Some relationships among the nuclear attraction, the field and the field-gradient integrals as well as among the electronic repulsion, the two-electron field, and the zero-field splitting integrals are discussed in connection with the spin-orbit integrals. Some calculations are carried out for the radicals CH2 and NH.

The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation scheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries.

General recurrence formulas for evaluating molecular integrals over contracted Cartesian Gaussian functions are derived by introducing auxiliary contracted hyper-Gaussian (ACH) functions. By using a contracted Gaussian function, this ACH represents an extension of the Gaussian function named derivative of Fourier-kernel multiplied Gaussian [J. Chem. Phys. 94, 3790 (1991)]. The ACH is reducible to contracted Cartesian Gaussian functions, contracted modified Hermite Gaussian functions, and to contracted Gaussian functions multiplied by phase factors, or the so-called GIAO, and is also reducible to various spatial operators necessary for ab initio molecular orbital calculations. In our formulation, all molecular integrals are expressed in terms of ACH. Therefore, the formulations have wide applicability for calculating various kinds of molecular integrals in ab initio calculations. Recursive calculations based on our formulation do not depend on the number of contraction terms, because the contraction step is completed at the evaluation of the initial integrals. Therefore, we expect that more efficient recursive calculations will be accomplished by using our formulas for evaluating molecular integrals over contracted Gaussian functions.

Multi-centre molecular integrals over spherical Gaussian-type orbitals modified with plane-wave phase factors, which arise in the close-coupling treatment of ion - atom collisions as well as in some molecular structure calculations, are evaluated analytically in closed form by means of the Fourier transform and the addition theorem of harmonic polynomials. The resulting integral formulae for spherical Gaussian orbitals, consisting of vector-coupling coefficients and well known functions, are more compact than those for their counterparts, Cartesian Gaussian orbitals. Practical techniques are suggested for the efficient implementation of the present formulae.

This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bagci and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

Traditional database query languages such as datalog and SQL allow the user to specify only mandatory requirements on the data to be retrieved from a database. In many applications, it may be natural to express not only mandatory requirements but also preferences on the data to be retrieved. Lacroix and Lavency10) extended SQL with a notion of preference and showed how the resulting query language could still be translated into the domain relational calculus. We explore the use of preference in databases in the setting of datalog. We introduce the formalism of preference datalog programs (PDPs) as preference logic programs without uninterpreted function symbols for this purpose. PDPs extend datalog not only with constructs to specify which predicate is to be optimized and the criterion for optimization but also with constructs to specify which predicate to be relaxed and the criterion to be used for relaxation. We can show that all of the soft requirements in Reference10) can be directly encoded in PDP. We first develop anaively-pruned bottom-up evaluation procedure that is sound and complete for computing answers to normal and relaxation queries when the PDPs are stratified, we then show how the evaluation scheme can be extended to the case when the programs are not necessarily stratified, and finally we develop an extension of themagic templates method for datalog14) that constructs an equivalent but more efficient program for bottom-up evaluation.

Recurrence relations between auxiliary functions to calculate the electron-repulsion integrals (ERIs) over Cartesian Gaussian-type orbitals (GTOs) can be derived. With the use of the Rys polynomials, this serves as a general formula method for evaluating ERIs and the ERI analytical derivatives. The computer code of the present method is quite general (i.e., it has no limitation for the shell type of GTOs), very compact (i.e., it has no coding difficulty), and extremely rapid (e.g., 2.1 μs per one ERI, 0.51 μs per one ERI first derivative, and 0.40 μs per one ERI second derivative for the [II‖II] shell block). It is found that the present method is especially efficient for evaluating ERIs including g-type or higher GTOs, the first derivatives including f-type or higher GTOs, and the second derivatives including d-type or higher GTOs.

The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further.

The reduced multiplication scheme of the Rys quadrature is presented. The method is based on new ways in which the Rys quadrature can be developed if it is implemented together with the transfer equation applied to the contracted integrals. In parallel to the new scheme of the Rys quadrature improvements are suggested to the auxiliary function based algorithms. The two new methods have very favorable theoretical floating point operation (FLOP) counts as compared to other methods. It is noted that the only significant difference in performance of the two new methods is due to the vectorizability of the presented algorithms. In order to exhibit this, both methods were implemented in the integral program seward. Timings are presented for comparisons with other implemenations. Finally, it is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two-electron integrals of such basis functions. It is demonstrated both theoretically and with actual performance that the use of spherical harmonic Gaussians offers a clear advantage over the traditional evaluation of the two-electron integrals in the Cartesian Gaussian basis.

Software development relies heavily on manual processes for transforming requirements into software artifacts such as models, source code, or test cases. Requirements are the starting point for these transformations, and they are typically written in natural language. However, hardly any automated tools exist that translate natural language texts into software artifacts. We propose to adapt recent advances in natural language processing and semantic technologies to generate UML models, test cases, and perhaps even source code, from natural language input. Though earlier efforts in automatic programming had limited success, we are now in a situation where extracting meaning from text has made substantial progress. For example, encouraging results for generating UML models from textual requirements have been achieved. It might even be possible to generate executable test cases. An intermediate step would be to generate tests from API documentation (which would also be a useful capability in itself). An even greater advance would be to automate rote coding. In all of these cases, entirely new classes of software tools would be needed to extract and process the semantics inherent in natural language texts, augment them with tacit knowledge from ontologies and domain models, and, where necessary, ask humans to clarify ambiguities. Though speculative, a determined, long-term effort in translating text to software could automate and accelerate software development to an unprecedented degree.

A vector efficient implementation of the McMurchie and Davidson algorithm for the calculation of one‐ and two‐electron molecular integrals is presented, as available in the Cray version of the ASTERIX program system. The implementation and performance of a vector‐oriented strategy for the generation and processing of the P supermatrix is also discussed. This program system has been applied to the ab initio SCF computation of the ground‐state wave function for the [V10O28]6− ion, with a basis set of triple‐zeta quality for the valence shell of oxygen generating 1404 GTOS and 574 CGTOS for the complete system. The performance and the bottlenecks of the integral calculation are discussed as a function of the integral classes. Two‐dimensional maps of the electrostatic potential are presented for this molecule and compared to experimental information about proton fixation.

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.