Efficient linear algebra routines for symmetric matrices stored in packed form

Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.     

A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations

A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in Hartree-Fock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.     

Attractive electron–electron interactions within robust local fitting approximations

An analysis of Dunlap's robust fitting approach reveals that the resulting two‐electron integral matrix is not manifestly positive semidefinite when local fitting domains or non‐Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four‐center two‐electron integrals based on the resolution‐of‐the‐identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair‐atomic resolution‐of‐the‐identity (PARI) approach, atomic‐orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree–Fock and Kohn–Sham calculations, the indefinite integral matrix causes nonconvergence in the self‐consistent‐field iterations. In these cases, the two‐electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb‐metric RI method. The speed‐up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple‐zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky‐decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations. © 2013 Wiley Periodicals, Inc.     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.     

Cholesky decomposition of the two-electron integral matrix in electronic structure calculations

A standard Cholesky decomposition of the two-electron integral matrix leads to integral tables which have a huge number of very small elements. By neglecting these small elements, it is demonstrated that the recursive part of the Cholesky algorithm is no longer a bottleneck in the procedure. It is shown that a very efficient algorithm can be constructed when family type basis sets are adopted. For subsequent calculations, it is argued that two-electron integrals represented by Cholesky integral tables have the same potential for simplifications as density fitting. Compared to density fitting, a Cholesky decomposition of the two-electron matrix is not subjected to the problem of defining an auxiliary basis for obtaining a fixed accuracy in a calculation since the accuracy simply derives from the choice of a threshold for the decomposition procedure. A particularly robust algorithm for solving the restricted Hartree-Fock (RHF) equations can be speeded up if one has access to an ordered set of integral tables. In a test calculation on a linear chain of beryllium atoms, the advocated RHF algorithm nicely converged, but where the standard direct inversion in iterative space method converged very slowly to an excited state.     

An efficient linear scaling procedure for constructing localized orbitals of large molecules based on the one-particle density matrix

We have developed a linear-scaling algorithm for obtaining the Boys localized molecular orbitals from the one-particle density matrix. The algorithm is made up of two steps: the Cholesky decomposition of the density matrix to obtain Cholesky molecular orbitals and the subsequent Boys localization process. Linear-scaling algorithms have been proposed to achieve linear-scaling calculations of these two steps, based on the sparse matrix technique and the locality of the Cholesky molecular orbitals. The present algorithm has been applied to compute the Boys localized orbitals in a number of systems including α-helix peptides, water clusters, and protein molecules. Illustrative calculations demonstrate that the computational time of obtaining Boys localized orbitals with the present algorithm is asymptotically linear with increasing the system size.     

Density fitting with auxiliary basis sets from Cholesky decompositions

Recent progress in the use of Cholesky decomposition techniques within the density fitting approximation of two-electron integrals is reviewed with emphasis on the theoretical background. Special attention is paid to the fact that errors due to the density fitting approximation can be controlled by constructing auxiliary basis sets by means of Cholesky decomposition of either the entire or certain subblocks of the molecular two-electron integral matrix. Finally, the prospects of trivial linear-scaling calculation of fitting coefficients in the Cholesky decomposition-based density fitting scheme are outlined.     

Fast noniterative orbital localization for large molecules

We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ("Cholesky molecular orbitals") demonstrates their localized character inherited from the sparsity of the density matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.     

Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals

The authors propose a new algorithm, "local K" (LK), for fast evaluation of the exchange Fock matrix in case the Cholesky decomposition of the electron repulsion integrals is used. The novelty lies in the fact that rigorous upper bounds to the contribution from each occupied orbital to the exchange Fock matrix are employed. By formulating these inequalities in terms of localized orbitals, the scaling of computing the exchange Fock matrix is reduced from quartic to quadratic with only negligible prescreening overhead and strict error control. Compared to the unscreened Cholesky algorithm, the computational saving is substantial for systems of medium and large sizes. By virtue of its general formulation, the LK algorithm can be used also within the class of methods that employ auxiliary basis set expansions for representing the electron repulsion integrals.     

Systematic sparse matrix error control for linear scaling electronic structure calculations

Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.     

Semiclassical bound-continuum Franck-Condon factors uniformly valid at 4 coinciding critical points: 2 Crossings and 2 turning points

A uniform semiclassical approximation to bound-continuum Franck-Condon factors is derived by applying differential topological mapping techniques on a suitable three dimensional integralrepresentation. This approach even holds uniformly if two potentialcurve intersections (real or complex conjugate) and two turning points come close or coincide. The resulting Franck-Condon matrix element is expressed in terms of two generic swallowtail (A₄) integrals whose unfolding parameters are obtained from a single algebraic equation amenable to fast standard routines. Transitional approximations to this result are shown to cover all previously known approaches and to lead to a generalization of a formula of K. Sando and F. H. Mies [1]. A simple and fast trapezoidal method to evaluate the generic swallowtail integrals and other generic integrals of odd determinacy is presented, which even permits the derivation of numerical error bounds.     

Unbiased auxiliary basis sets for accurate two-electron integral approximations

We propose Cholesky decomposition (CD) of the atomic two-electron integral matrix as a robust and general technique for generating auxiliary basis sets for the density fitting approximation. The atomic CD (aCD) auxiliary basis set is calculated on the fly and is not biased toward a particular quantum chemical method. Moreover, the accuracy of the aCD basis set can be controlled with a single parameter.     

Accurate ab initio density fitting for multiconfigurational self-consistent field methods

Using Cholesky decomposition and density fitting to approximate the electron repulsion integrals, an implementation of the complete active space self-consistent field (CASSCF) method suitable for large-scale applications is presented. Sample calculations on benzene, diaquo-tetra-mu-acetato-dicopper(II), and diuraniumendofullerene demonstrate that the Cholesky and density fitting approximations allow larger basis sets and larger systems to be treated at the CASSCF level of theory with controllable accuracy. While strict error control is an inherent property of the Cholesky approximation, errors arising from the density fitting approach are managed by using a recently proposed class of auxiliary basis sets constructed from Cholesky decomposition of the atomic electron repulsion integrals.     

A pitfall of using 2‐[(2E)‐3‐(4‐tert‐butylphenyl)‐2‐methylprop‐2‐enylidene]malononitrile as a matrix in MALDI TOF MS: chemical adduction of matrix to analyte amino groups

2‐[(2E)‐3‐(4‐tert‐Butylphenyl)‐2‐methylprop‐2‐enylidene]malononitrile (DCTB) has been considered as an excellent matrix for matrix‐assisted laser desorption/ionization (MALDI) of many types of synthetic compounds. However, it might provide troublesome results for compounds containing aliphatic primary or secondary amino groups. For these compounds, strong extra ion peaks with a mass difference of 184.1 Da were usually observed, which might falsely indicate the presence of some unknown impurities that were not detected by other matrices. On the basis of the possible mechanisms proposed, these extra ions are the products of nucleophilic reactions between analyte amino groups and DCTB molecules or radical cations. In these reactions, an amino group replaces the dicyanomethylene group of DCTB forming a matrix adduct via a ? C?N‐bond. An aliphatic primary amine could react easily with DCTB and the reaction could start once they are mixed in a MALDI solution. For an aliphatic secondary amine, on the other hand, the reaction most likely occurs in the gas phase. Protonation of amino groups by adding acid seems to be a useful way to stop DCTB adduction for compounds with one single amino group, but not for compounds with multiple amino groups. Unlike aliphatic primary or secondary amines, aliphatic tertiary amines and aromatic amines do not yield DCTB adducts. This is because tertiary amines do not have the required transferrable H‐(N) atom to form an extra ? C?N‐bond, while aromatic amines are not sufficiently nucleophilic to attack DCTB. In view of the possible matrix adduction, care should be taken in MALDI time‐of‐flight mass spectrometry (TOF MS) when DCTB is used as the matrix for compounds containing amino group(s). Copyright © 2010 John Wiley & Sons, Ltd.     

Spectral representation of Hartree‐Fock exchange operator

We report a new mathematical result: it is possible to construct a spectral representation of the two particles Coulomb potential in the form of | r ? r ^′|^?1 = ∑_λλg( r ) g_λ( r ^′). We call this formula λ‐decomposition. Two special nontrivial cases of λ‐decomposition are reported together with the numerical analysis of the convergence for one of them. It is shown how λ‐decomposition allows to construct a new fast algorithm for Hartree‐Fock exchange operator calculation, in which the calculation of electron repulsion integrals (ERIs) is completely avoided. The connection between the new method and the resolution of identity and Cholesky decomposition based approaches has been established. Finally, the accuracy of ERIs evaluation within the new approach has been studied numerically. The results demonstrate that it is possible to achieve the accuracy of 10^?10 for the ERIs in wide range of their orbital exponents with relatively small number of terms in λ‐decomposition. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Truncation of small matrix elements based on the Euclidean norm for blocked data structures

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.     

化学计量学辅助紫外分光光度法同时测定洛美沙星和司帕沙星

在pH：3．0的B-R缓冲溶液中,洛美沙星（LMX）和司帕沙星（SPFX）的最大吸收波长分别为286nm和296nm,两者吸收光谱严重重叠．本文用多元线性回归（MLR）、K-矩阵（AKC）、P-矩阵（CPA）等计算方法辅助紫外分光光度法,同时测定了模拟样中LMX和SPFX的含量。发现CPA法明显优于MLR和AKC,LMX和SPFX的平均回收率分别达99．5％和99．9％,相对标准偏差RSD分别为0．59％和0．85％.     

Improved spectra for MALDI MSI of peptides using ammonium phosphate monobasic in MALDI matrix

MALDI mass spectrometry imaging (MSI) enables analysis of peptides along with histology. However, there are several critical steps in MALDI MSI of peptides, 1 of which is spectral quality. Suppression of MALDI matrix clusters by the aid of ammonium salts in MALDI experiments is well known. It is asserted that addition of ammonium salts dissociates potential matrix adducts and thereafter decreases matrix cluster formation. Consequently, MALDI MS sensitivity and mass accuracy increase. Up to our knowledge, a limited number of MALDI MSI studies used ammonium salts as matrix additives to suppress matrix clusters and enhance peptide signals. In this work, we investigated the effect of ammonium phosphate monobasic (AmP) as alpha‐cyano‐4‐hydroxycinnamic acid (α‐CHCA) matrix additive in MALDI MSI of peptides. Prior to MALDI MSI, the effect of varying concentrations of AmP in α‐CHCA was assessed in bovine serum albumin tryptic digests and compared with the control (α‐CHCA without AmP). Based on our data, the addition of AmP as matrix additive decreased matrix cluster formation regardless of its concentration, and specifically, 8 mM AmP and 10 mM AmP increased bovine serum albumin peptide signal intensities. In MALDI MSI of peptides, both 8 and 10 mM AmP in α‐CHCA improved peptide signals especially in the mass range of m/z 2000 to 3000. In particular, 9 peptide signals were found to have differential intensities within the tissues deposited with AmP in α‐CHCA (AUC > 0.60). To the best of our knowledge, this is the first MALDI MSI of peptides work investigating different concentrations of AmP as α‐CHCA matrix additive to enhance peptide signals in formalin‐fixed paraffin‐embedded (FFPE) tissues. Further, AmP as part of α‐CHCA matrix could enhance protein identifications and support MALDI MSI‐based proteomic approaches.     

The improved performance of MALDI‐TOF MS on the analysis of herbal saponins by using DHB‐GO composite matrix

Matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry (MALDI‐TOF MS) is an excellent analytical technique for rapid analysis of a variety of molecules with straightforward sample pretreatment. The performance of MALDI‐TOF MS is largely dependent on matrix type, and the development of novel MALDI matrices has aroused wide interest. Herein, we devoted to seek more robust MALDI matrix for herbal saponins than previous reported, and ginsenoside Rb1, Re, and notoginsenoside R1 were used as model saponins. At the beginning of the present study, 2,5‐dihydroxybenzoic acid (DHB) was found to provide the highest intensity for saponins in four conventional MALDI matrices, yet the heterogeneous cocrystallization of DHB with analytes made signal acquisition somewhat “hit and miss.” Then, graphene oxide (GO) was proposed as an auxiliary matrix to improve the uniformity of DHB crystallization due to its monolayer structure and good dispersion, which could result in much better shot‐to‐shot and spot‐to‐spot reproducibility of saponin analysis. The satisfactory precision further demonstrated that minute quantities of GO (0.1 μg/spot) could greatly reduce the risk of instrument contamination caused by GO detachment from the MALDI target plate under vacuum. More importantly, the sensitivity and linearity of the standard curve for saponins were improved markedly by DHB‐GO composite matrix. Finally, the application of detecting the Rb1 in complex biological sample was exploited in rat plasma and proved it applicable for pharmacokinetic study quickly. This work not only opens a new field for applications of DHB‐GO in herbal saponin analysis but also offers new ideas for the development of composite matrices to improve MALDI MS performance.