Tighter multipole-based integral estimates and parallel implementation of linear-scaling AO-MP2 theory

Within an atomic-orbital-based (AO-based) formulation of second-order M?ller-Plesset perturbation theory (MP2), we present a novel screening procedure which allows us to preselect numerically significant two-electron integrals more efficiently, especially for large basis sets. The screening is based on our recently introduced multipole-based integral estimates (MBIE) method [J. Chem. Phys., 2005, 123, 184102], that allows to exploit the 1/R(4) or 1/R(6) coupling between electronic charge distributions in transformed integral products within AO-MP2. In this way, linear scaling is attained with fully-controlled numerical accuracy. Furthermore, a parallel implementation of our linear-scaling AO-MP2 method is described, which also allows us to perform calculations with larger basis sets. First calculations reveal that for e.g. linear alkanes the scaling of the number of required transformed integral products is almost equal for 6-31G* and cc-pVTZ basis sets. Using the improved MBIE screening, the largest parallel calculation was performed for a ribozyme fragment consisting of 497 atoms and 5697 basis functions, while our largest AO-MP2 calculation was performed for a stacked DNA system (16 base pairs) comprising 1052 atoms and 10 674 basis functions on a single processor.     

Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals

We derive multipole-based integral estimates (MBIE) as rigorous and tight upper bounds to four-center two-electron integrals in order to account for the 1/R distance decay between the charge distributions, which is missing in the Schwarz screening commonly used in ab initio methods. Our screening criteria are valid for all angular momenta and can be formulated for any order of multipoles. We have found the expansion limited to dipoles to be sufficiently tight for estimating the integrals in Hartree-Fock and density-functional theories, while the screening effort is negligible. For, e.g., a DNA fragment with 1052 atoms and 10,674 basis functions (6-31G*) the exchange part is faster by a factor of 2.1 as compared to the Schwarz screening both within our linear exchange scheme, whereas a smaller factor of 1.3 is gained for the Coulomb part within the continuous fast multipole method. Most importantly, our new MBIE screening is perfectly suited to exploit the strong distance decay of electron-correlation effects of at least 1/R4 in atomic-orbital-based formulations of correlation methods.     

Rigorous integral screening for electron correlation methods

We derive rigorous multipole-based integral estimates (MBIE) in order to account for the distance dependence occurring in atomic-orbital (AO) formulations of electron correlation theory, where our focus is on AO-MP2 theory within a Laplace scheme. We find for the exact transformed integral products an extremely early onset of a linear-scaling behavior and a very small number of significant products. To preselect the significant integral products we adapt our MBIE method as rigorous upper bound. In this way it is possible to exploit the favorable scaling behavior observed and to reduce the scaling of estimated products asymptotically to linear, without sacrificing accuracy or reliability. By separating Coulomb- and exchange-type contractions only half-transformed integrals need to be computed. Furthermore, our scheme of rigorously preselecting transformed integral products via MBIE seems to offer particularly interesting perspectives for a direct formation of half- or fully transformed integrals by using multipole expansions and auxiliary basis sets.     

Methods for parallel computation of SCF NMR chemical shifts by GIAO method: Efficient integral calculation,multi-Fock algorithm,and pseudodiagonalization

We implemented our gauge-including atomic orbital (GIAO) NMR chemical shielding program on a workstation cluster, using the parallel virtual machine (PVM) message-passing system. On a modest number of nodes, we achieved close to linear speedup. This program is characterized by several novel features. It uses the new integral program of Wolinski that calculates integrals in vectorized batches, increases efficiency, and simplifies parallelization. The self-consistent field (SCF) step includes a multi-Fock algorithm, i.e., the simultaneous calculation of several Fock matrices with the same integral set, increasing the efficiency of the direct SCF procedure. The SCF diagonalization step, which is difficult to parallelize, has been replaced by pseudodiagonalization. The latter, widely used in semiempirical programs, becomes important in ab initio type calculations above a certain size, because the ultimate scaling of the diagonalization step is steeper than that of integral computation. Examples of the calculation of the NMR shieldings in large systems at the SCF level are shown. Parallelization of the density functional code is underway. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 816–825, 1997     

Ab initio calculations on large molecules: The multiplicative integral approximation

In the multiplicative integral approximation (MIA), two-electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self-consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA-assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock-matrix allows important time savings. Results are presented for a number of test cases.     

Comment on “Analytical evaluation for two-center nuclear attraction integrals over Slater type orbitals by using Fourier transform method” by S. ?zcan and E. ?ztekin (J. Math. Chem. DOI 10.1007/s10910-008-9398-z)

S. ?zcan and E. ?ztekin, (J. Math. Chem. doi:) published formulas for evaluating the two-center nuclear attraction integrals over Slater type orbitals. It is shown that the analytical relations for these integrals through the expansion coefficients of the electron charge density for the one-center case and the overlap integrals presented in Sect. 3 of this work can easily be derived by means of a simple algebra from the formulas published in our papers (I.I. Guseinov, J Mol Struct (Theochem) 417:117, 1997; J Math Chem 42:415, 2007 and B.A. Mamedov, Chin J Chem 22:545, 2004). It should be noted that the formulas of overlap integrals presented by E. ?ztekin et al., in previous paper (E. ?ztekin, M. Yavuz, Ş. Atalay, J Mol Struct (Theochem) 544:69, 2001) for the calculation of two-center nuclear attraction integrals also are obtained from our papers (see Comment: I.I. Guseinov, J Mol Struct (Theochem) 638:235, 2003).     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. II. Two‐center expansion method

Using expansion formulas for the charge‐density over Slater‐type orbitals (STOs) obtained by the one of authors [I. I. Guseinov, J Mol Struct (Theochem) 1997, 417, 117] the multicenter molecular integrals with an arbitrary multielectron operator are expressed in terms of the overlap integrals with the same screening parameters of STOs and the basic multielectron two‐center Coulomb or hybrid integrals with the same operator. In the special case of two‐electron electron‐repulsion operator appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules the new auxiliary functions are introduced by means of which basic two‐center Coulomb and hybrid integrals are expressed. Using recurrence relations for auxiliary functions the multicenter electron‐repulsion integrals are calculated for extremely large quantum numbers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 117–125, 2001     

Efficient evaluation of short-range Hartree-Fock exchange in large molecules and periodic systems

We present an efficient algorithm for the evaluation of short-range Hartree-Fock exchange energies and geometry gradients in Gaussian basis sets. Our method uses a hierarchy of screening levels to eliminate negligible two-electron integrals whose evaluation is the fundamental computational bottleneck of the procedure. By applying our screening technique to the Heyd-Scuseria-Ernzerhof [J. Chem. Phys. 118, 8207 (2003)] short-range Coulomb hybrid density functional, we achieve a computational efficiency comparable with that of standard nonhybrid density functional calculations.     

Pharmacophore alignment search tool: influence of the third dimension on text-based similarity searching

Previously (Hähnke et al., J Comput Chem 2010, 31, 2810) we introduced the concept of nonlinear dimensionality reduction for canonization of two‐dimensional layouts of molecular graphs as foundation for text‐based similarity searching using our Pharmacophore Alignment Search Tool (PhAST), a ligand‐based virtual screening method. Here we apply these methods to three‐dimensional molecular conformations and investigate the impact of these additional degrees of freedom on virtual screening performance and assess differences in ranking behavior. Best‐performing variants of PhAST are compared with 16 state‐of‐the‐art screening methods with respect to significance estimates for differences in screening performance. We show that PhAST sorts new chemotypes on early ranks without sacrificing overall screening performance. We succeeded in combining PhAST with other virtual screening techniques by rank‐based data fusion, significantly improving screening capabilities. We also present a parameterization of double dynamic programming for the problem of small molecule comparison, which allows for the calculation of structural similarity between compounds based on one‐dimensional representations, opening the door to a holistic approach to molecule comparison based on textual representations. © 2011 Wiley Periodicals, Inc. J Comput Chem , 2011.     

Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates. II

The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. On the basis of the prolate spheroidal coordinates, the overlap and the kinetic integral are reduced to the integral over the rectangular domain. The integration over the rectangular domain is performed by the adaptive integration scheme. The developed algorithm is applied to calculate the integrals for the pairs of hydrogen and gallium eigenfunctions. It is demonstrated that high accuracy can be obtained for small number of integrand evaluations what guarantees the efficiency of the presented algorithm. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Multipole expansions of orbital products about an intermediate center

A numerical method described previously (Talman, Int J Quantum Chem, 2007, 107, 1578) is reviewed and a modified formulation that may be more computationally efficient is presented. Modifications required for the application to real harmonics are described. The method requires the numerical evaluation of a one‐dimensional integral on a finite interval, and an improved method for this is described. The application to the evaluation of electron–electron repulsion four‐center integrals is also discussed. It is indicated that μ H accuracy is obtainable with modest computational effort. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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     

Evaluation of bielectronic integrals for 1s Slater orbitals by using averages

A new approach for evaluating the four‐center bielectronic integrals (12|34), involving 1s Slater‐type orbitals, is presented. The method uses the multiplication theorem for Bessel functions. The bielectronic integral is expressed in terms of a finite sum of functions, and a scaling parameter is introduced. In the present work, the scaling parameter used is an average. The results show that the first term in the sum is always the principal contribution, and the remainder has a corrective character. The whole scheme and its numerical trend are understood on the basis of a theorem recently proved. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Three‐center Coulomb repulsion integrals with Slater functions

The translation method for products of two Slater functions is improved and combined with the short–long range separation in order to develop a robust and efficient algorithm for the evaluation of three‐center Coulomb integrals with Slater functions. Several tests are carried out showing that the algorithm reported here yields integrals with an absolute error below 10^?12 hartree and a computational cost of a few microseconds per integral. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Computation of two‐center Coulomb integrals over Slater‐type orbitals using elliptical coordinates

A general analytic formula is obtained for the two‐center Coulomb integrals over Slater‐type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well‐known auxiliary functions A_k(p) and B_k(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Communication: Linear-expansion shooting techniques for accelerating self-consistent field convergence

Based on the corrected Hohenberg-Kohn-Sham total energy density functional [Y. A. Zhang and Y. A. Wang, J. Chem. Phys. 130, 144116 (2009)], we have developed two linear-expansion shooting techniques (LIST)- direct LIST (LISTd) and indirect LIST (LISTi), to accelerate the convergence of self-consistent field (SCF) calculations. Case studies show that overall LISTi is the most robust and efficient algorithm for accelerating SCF convergence, whereas LISTd is advantageous in the early stage of an SCF process. More importantly, LISTi outperforms Pulay's direct inversion in the iterative subspace (DIIS) [P. Pulay, J. Comput. Chem. 3, 556 (1982)] and its two recent improvements, energy-DIIS [K. N. Kudin, G. E. Scuseria, and E. Cance?s, J. Chem. Phys. 116, 8255 (2002)] and augmented Roothaan-Hall energy-DIIS [X. Hu and W. Yang, J. Chem. Phys. 132, 054109 (2010)].     

Quasi‐random integration in quantum chemistry: Efficiency,stability, and application to the study of small atoms and molecules constrained in spherical boxes

This project consists of two parts. In the first part, a series of test calculations is performed to verify that the integrals involved in the determination of atomic and molecular properties by standard self‐consistent field (SCF) methods can be obtained through Halton, Korobov, or Hammersley quasi‐random integration procedures. Through these calculations, we confirm that all three methods lead to results that meet the levels of precision required for their use in the calculation of properties of small atoms or molecules at least at a Hartree–Fock level. Moreover, we have ensured that the efficiency of quasi‐random integration methods that we have tested is Halton=Korobov>Hammersley?pseudo‐random. We also find that these results are comparable to those yielded by ordinary Monte Carlo (pseudo‐random) integration, with a calculation effort of two orders of smaller magnitude. The second part, which would not have been possible without the integration method previously analyzed, contains a first study of atoms constrained in spherical boxes through SCF calculations with basis functions adapted to the features of the problem: Slater‐type orbitals (STOs) trimmed by multiplying them by a function that yields 1 for 0 < r < (R‐δ), polynomial values for (R‐δ) < r < R and null for r > R, R being the radius of the box and δ a variationally determined interval. As a result, we obtain a equation of state for electrons of small systems, valid just in the limit of low temperatures, but fairly simple. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Fast and stable algorithm for the analytical computation of two-center Coulomb and overlap integrals over Slater-type orbitals

Proceeding from analytical expressions for two-center kernel functions that we derived recently, we present new analytical formulas for the two-center Coulomb and overlap integrals over Slater-type orbitals. These formulas are of an exceptionally simple analytical structure and high numerical efficiency. An especially important point is that for the most frequently needed ranges of discrete quantum numbers, the formulas are completely stable in the cases of nearly equal scaling parameters or vanishing interatomic distances, except for one particular case of the Coulomb integral. No special asymptotic formulas are needed any more to compute the two-center integrals over Slater-type orbitals in these case. Furthermore, a largely recursive formulation makes the integral evaluation very economical and fast. In particular, we assess the numerical performance of a new kind of angular momentum recurrences that we have proposed in a previous article [W. Hierse and P.M. Oppeneer, J. Chem. Phys. 99 , 1278 (1993)]. © 1994 John Wiley & Sons, Inc.