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.     

Sparse matrix multiplications for linear scaling electronic structure calculations in an atom-centered basis set using multiatom blocks

A sparse matrix multiplication scheme with multiatom blocks is reported, a tool that can be very useful for developing linear-scaling methods with atom-centered basis functions. Compared to conventional element-by-element sparse matrix multiplication schemes, efficiency is gained by the use of the highly optimized basic linear algebra subroutines (BLAS). However, some sparsity is lost in the multiatom blocking scheme because these matrix blocks will in general contain negligible elements. As a result, an optimal block size that minimizes the CPU time by balancing these two effects is recovered. In calculations on linear alkanes, polyglycines, estane polymers, and water clusters the optimal block size is found to be between 40 and 100 basis functions, where about 55-75% of the machine peak performance was achieved on an IBM RS6000 workstation. In these calculations, the blocked sparse matrix multiplications can be 10 times faster than a standard element-by-element sparse matrix package.     

A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme

To exploit the exponential decay found in numerical studies for the density matrix and its derivative with respect to nuclear displacements, we reformulate the coupled perturbed self-consistent field (CPSCF) equations and a quadratically convergent SCF (QCSCF) method for Hartree-Fock and density functional theory within a local density matrix-based scheme. Our D-CPSCF (density matrix-based CPSCF) and D-QCSCF schemes open the way for exploiting sparsity and to achieve asymptotically linear scaling of computational complexity with molecular size (M), in case of D-CPSCF for all (M) derivative densities. Furthermore, these methods are even for small molecules strongly competitive to conventional algorithms.     

对含重元素体系的接合二分量-标量相对论密度泛函计算方法

在相对论密度泛函ZORA方法的基础上,提出一种用于含重元素体系的接合二分量-标量相对论密度泛函计算方法.对于只含少数几个重元素的较大体系,仅对其中旋轨耦合作用强的重元素作二分量相对论计算,而对体系的其余部分则作标量相对论计算,通过对动能矩阵元的近似处理实现两种计算的接合.对一系列含6p区重元素分子进行计算的结果表明,当非重元素是第三周期以前的元素时,此方法与二分量ZO-RA方法的计算结果吻合得很好.当非重元素为第四周期元素时,计算结果有一定偏差,表明在后一种情况下旋轨耦合作用已比较显著,但误差仍在目前近似密度泛函计算的精度范围内.此方法可以有效地节省计算量,而且避免了Dyall方法的缺点.     

Evaluation of the field-adapted ADMA approach: absolute and relative energies of crambin and derivatives

A large number of conformations and chemically modified variants of the protein crambin were used to extensively test the field-adapted adjustable density matrix assembler (FA-ADMA) method developed for ab initio quality quantum chemistry computations of proteins and other macromolecules, introduced in an earlier publication. In this method, the fuzzy density matrix fragmentation scheme of the original adjustable density matrix assembler (ADMA) method has been made more efficient by combining it with an approach of using point charges to approximate the effects of additional, distant parts of a given macromolecule in the quantum chemical calculation of each fragment. In this way, smaller parent molecules can be used for fragment generation, while achieving accuracy that can be obtained only with large parent molecules in the original ADMA method. Whereas in both methods the error relative to the Hartree-Fock result can be reduced below any threshold by choosing large enough parent molecules, this can be done more efficiently with the new method. In order to obtain reliable test results for the accuracy obtainable by the new method when compared to conventional Hartree-Fock calculations, we performed a large number of energy calculations for the protein crambin using various conformations available in the Protein Data Bank, various protonation states, and side chain mutations. Additionally, in order to test the performance of the method for protein-solvent interaction studies, the energy changes due to the formation of complexes with ethanol and single and multiple water molecules were investigated.     

Nonpolar dihydrogen bonds--on a gliding scale from weak dihydrogen interaction to covalent H-H in symmetric radical cations [HnE-H-H-EHn]+

The electronic structures and bonding patterns for a new class of radical cations, [HnE-H-H-EHn]+ (EHn=element hydride, E=element of Groups 15-18), have been investigated by applying quantum-chemical methods. All structures investigated give rise to symmetric potential energy minimum structures. We envisage clear periodic trends. The H--H bond length is shorter for elements toward the bottom of the periodic table of elements, and a short H--H bond corresponds to accumulation of electron density in the central H--H region. All [HnE-H-H-EHn]+ of Groups 15-17 are thermodynamically unstable towards loss of either H2 or H. The barriers for these dissociations are rather low. The Group 18 congeners, except E=Xe, appear to be global minima of the respective potential energy surfaces. The findings are discussed in terms of H2 bond activation, and a general mechanistic scheme for the standard reduction process 2H+ + 2e(-) --> H2 is given. Finally, it is proposed that some of the symmetric radical cations are likely to be observed in mass spectrometric or matrix isolation experiments.     

Density matrix purification with rigorous error control

Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification.     

液态RbCl的态密度

给出了在分子动力学模拟基础上Fumi-Tosi势离子液体的正则模式分析方法,用Fumi-Tosi势(包括长程势)代替Lennard-Jones势,并且用等效Coulomb势处理长程Coulomb作用.讨论了Hessian矩阵元的计算方法和Hessian矩阵特征值的计算方法.计算实践表明，取用余误差函数形式的等效库仑势,可以合理地得到Hessian矩阵和态密度.液态RbCl中构型平均态密度的数值结果表明，液态RbCl的态密度表现出与Lennard-Jones液体的态密度相仿的特点.     

Potentials of an alternative scheme for ab initio polymer band structure calculations. Illustration on the chain of hydrogen atoms

An alternative scheme for ab initio polymer band structure calculations based on a Filon-type quadrature is proposed. This scheme avoids the explicit calculation and the storage of the “troublesome” Fourier transforms of the LCAO density matrix elements and is a first step towards a better control of the convergence of the different lattice sums appearing in the configuration space LCAO-SCF-CO method. The potential of the proposed technique is illustrated by a minimal basis set calculations on an infinite chain of H atoms.     

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.     

Frequency dispersed transient absorption spectra of dissolved perylene: A case study using the density matrix version of the MCTDH method

A theoretical scheme is presented to calculate non-linear optical spectra of molecules in solution. Starting with electronic structure calculations of the ground and excited state, a subset of vibrational coordinates exhibiting the largest Huang–Rhys factors is assigned. It is used to set up a model Hamiltonian for density matrix multi configurational time-dependent Hartree (MCTDH) calculations. The expression derived for the dissipative part of the equation of motion goes beyond the earlier used Lindblad-form. In order to calculate the non-linear response the electric field strength is introduced into the density matrix equations used to directly determine the polarization. The whole scheme is applied to perylene as a reference case.     

Kohn-Sham Density Matrix and the Kernel Energy MethodSCI北大核心CSCD

The kernel energy method(KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries.KEM breaks a molecule into smaller subsets,called kernels,for the purposes of calculation.The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy.A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals.In this study,the kernel expansion for the density matrix is examined in the context of density functional theory(DFT) Kohn-Sham(KS) calculations.A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined,and is then converted into a normalizedprojector by using the Clinton algorithm.Such normalized projectors are factorizable into linear combination of atomic orbitals(LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis.Both straightforward KEM energies and energies from a normalized,idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion.Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules.In the case of the proof-of-concept system,calculations were performed using the STO-3 G and6-31 G(d,p) bases over a range of atomic separations,some very far from equilibrium.The water cluster was calculated in the 6-31 G(d,p) basis at an equilibrium geometry.The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases.In the case of the water cluster,the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result.The KS density matrices of this study are applicable to quantum crystallography.     

A new localization scheme for the elongation method

A different localization scheme for the elongation method is developed based on regional molecular orbitals. This scheme is more efficient and more accurate than the previous one especially for covalently bonded systems with strongly delocalized pi electrons. Ab initio test calculations have been performed on three model systems: water chains, polyglycine, and cationic cyanine chains. The dependence on the size of the starting clusters and the effect of the basis set are investigated. Our results are compared with conventional ab initio calculations and it is found in all cases that the error per added unit levels off to a satisfactorily small value as long as the starting cluster is sufficiently large.     

Simultaneous correction for drift and non-spectroscopic matrix effect in the measurement of geological samples by inductively coupled plasma-mass spectrometry using common analyte internal standardization chemometric technique

The common analyte internal standardization (CAIS) chemometric technique is extended to correct for drift in signal intensity in inductively coupled plasma-mass spectrometry. The CAIS technique is used because, unlike the conventional internal reference method, it allows for the analyte to behave differently from the internal reference under the influence of drift. Thus, only one internal reference element is sufficient to correct for drift effect for all the analytes irrespective of the difference of their mass and ionization potential from that of the internal reference element. Experimental test with 15 analytes, as a representative of the whole mass range, in different geological matrices, using only indium as internal reference element, demonstrates that the developed drift correction method is efficient in correcting up to 25% drift error and is easy to use. Further, a CAIS scheme was developed to correct for both drift and non-spectroscopic matrix effects simultaneously in a single run, using standard containing no matrix. This scheme is based on the use of two internal reference elements for simultaneous monitoring and correction of drift and matrix effects for all the analytes irrespective of their difference in mass and ionization potential from the two internal reference elements. The developed scheme was validated using river water samples containing appreciable matrix concentration and was measured under the influence of drift. The results of the experimental validation indicate that the proposed scheme is capable of removing error as large as 43% that arises from drift and non-spectroscopic matrix effects.     

A method for the rapid calculation of matrix elements with highly oscillatory JWKB radial wavefunctions

A method for the rapid computation of matrix elements with JWKB radial wavefunctions is discussed. The method consists of dividing the range of integration into segments determined by the nodes of the semiclassical wavefunction. The desired matrix elements are calculated by summing the contributions from each segment which are evaluated by integrating between nodes with a Gauss-Mehler quadrature formula. The results are compared with exact quantum mechanical calculations and were found to agree within 1–2%. The calculations with the present method were generally five to ten times faster than the quantum mechanical calculations.     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

Comparison study of several numerical integration schemes used in calculations of density functional theory

Several numerical integration schemes for the evaluation of matrix elements in density functional theory calculations have been studied and compared by computational practice. The best scheme was found to be the combination of the atomic partition function proposed by Becke with the scaled generalized Gauss-Laguerre quadrature formula for radial integration suggested by Yang, which achieve the highest convergence rate to the numerical integration. With the same number of integration points, the accuracy of the calculated results by this scheme is higher by 1 to 2 orders of magnitudes than that by other schemes. The reason for achieving higher accuracy by this scheme has been proposed preliminarily.     

Photostability of the laser dye DCM in various inorganic-organic host matrices

The laser dye, 4-dicyanomethylene-2-methyl-6-p-dimethylaminostyryl-4H-pyran (DCM) was incorporated in various inorganic-organic host matrices by the sol-gel technique. The photostability of the DCM-doped materials under CW argon laser irradiation was investigated. The absorption of DCM monomer ar 480 nm decreased with increasing irradiation time and was accompanied by an increase in the absorption at 352 nm by inactive photoproducts. Photostability was estimated by measuring the decay rates of DCM fluorescences in different host matrices. The rates exhibited two components; a rapid decay within the first several tens of seconds followed by a slower luminescence decay. The rapid decay depended upon the DCM content in the matrix while the slow decay component was related to the mobility of the DCM in the different matrices. The silica host matrix containing phenyl and a small amount of epoxy groups exhibited the best photostability of the materials examined.