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  • 1
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 113 (2000), S. 10037-10043 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Hierarchical cubature is a new method for achieving linear scaling computation of the exchange-correlation matrix central to Density Functional Theory. Hierarchical cubature combines a k-dimensional generalization of the binary search tree with adaptive numerical integration involving an entirely Cartesian grid. Hierarchical cubature overcomes strong variations in the electron density associated with nuclear cusps through multiresolution rather than spherical-polar coordinate transformations. This unique Cartesian representation allows use of the exact integration error during grid construction, supporting O(log N) range-queries that exploit locality of the Cartesian Gaussian based electron density. Convergence is controlled by τr, which bounds the local integration error of the electron density. An early onset of linear scaling is observed for RB3LYP/6-31G * * calculations on water clusters, commencing at (H2O)30 and persisting with decreasing values of τr. Comparison with nuclear weight schemes suggests that the new method is competitive on the basis of grid points per atom. Systematic convergence of the RPBE0/6-31G* Ne2 binding curve is demonstrated with respect to τr. © 2000 American Institute of Physics.
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  • 2
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 464-472 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A new, highly optimized implementation of the Hiller–Sucher–Feinberg (HSF) identity is presented. The HSF identity, when applied to molecular wave functions calculated with Gaussian-type basis functions, not only improves the overall accuracy of the electron density by more than an order of magnitude, but also yields approximate cusps at nuclei. The three classes of molecular integrals, L, U, and V, which are encountered in the calculation of the HSF density, are derived in compact form. Efficient algorithms for the accurate evaluation of these integrals are detailed, including a novel approach to the necessary numerical quadratures and the thresholding of two-electron V integrals. Hartree–Fock (HF) electron densities calculated with both the conventional definition and from the HSF identity are compared to their respective HF limits for a variety of diatomic molecules and basis sets. The average error in the calculated HSF electron densities at non-hydrogen nuclei equals 0.17%, which constitutes a marked improvement over an error of 5.77% in the conventional densities.
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  • 3
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 111 (1999), S. 6223-6229 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A new method for the multipole evaluation of contracted Cartesian Gaussian-based electron repulsion integrals is described, and implemented in linear scaling methods for computation of the Hartree–Fock exchange matrix. The new method, which relies on a nonempirical multipole acceptability criterion [J. Chem. Phys. 109, 8764 (1998)], renders the work associated with integral evaluation independent of the basis set contraction length. Benchmark calculations on a series of three-dimensional water molecule clusters and graphitic sheets with highly contracted basis sets indicate that the new method is up to 4.6 times faster than a well optimized direct integral evaluation routine. For calculations involving lower levels of contraction a factor of 2 speedup is typically observed. Importantly, the method achieves these large gains in computational efficiency while maintaining numerical equivalence with standard direct self consistent field theory. © 1999 American Institute of Physics.
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  • 4
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 110 (1999), S. 2332-2342 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A simplified version of the Li, Nunes and Vanderbilt [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)] density matrix minimization is introduced that requires four fewer matrix multiplies per minimization step relative to previous formulations. The simplified method also exhibits superior convergence properties, such that the bulk of the work may be shifted to the quadratically convergent McWeeny purification, which brings the density matrix to idempotency. Both orthogonal and nonorthogonal versions are derived. The AINV algorithm of Benzi, Meyer, and Tu(ring)ma [SIAM J. Sci. Comp. 17, 1135 (1996)] is introduced to linear scaling electronic structure theory, and found to be essential in transformations between orthogonal and nonorthogonal representations. These methods have been developed with an atom-blocked sparse matrix algebra that achieves sustained megafloating point operations per second rates as high as 50% of theoretical, and implemented in the MondoSCF suite of linear scaling SCF programs. For the first time, linear scaling Hartree–Fock theory is demonstrated with three-dimensional systems, including water clusters and estane polymers. The nonorthogonal minimization is shown to be uncompetitive with minimization in an orthonormal representation. An early onset of linear scaling is found for both minimal and double zeta basis sets, and crossovers with a highly optimized eigensolver are achieved. Calculations with up to 6000 basis functions are reported. The scaling of errors with system size is investigated for various levels of approximation. © 1999 American Institute of Physics.
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  • 5
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 105 (1996), S. 2726-2734 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Thresholding criteria are introduced that enforce locality of exchange interactions in Cartesian Gaussian-based Hartree–Fock calculations. These criteria are obtained from an asymptotic form of the density matrix valid for insulating systems, and lead to a linear scaling algorithm for computation of the Hartree–Fock exchange matrix. Restricted Hartree–Fock/3-21G calculations on a series of water clusters and polyglycine α-helices are used to demonstrate the O(N) complexity of the algorithm, its competitiveness with standard direct self-consistent field methods, and a systematic control of error in converged total energies. © 1996 American Institute of Physics.
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  • 6
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 109 (1998), S. 8764-8769 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Accurate and computationally inexpensive estimates of multipole expansion errors are crucial to the success of several fast electronic structure methods. In this paper, a new nonempirical multipole acceptability criterion is described that is directly applicable to expansions of high order moments. Several model calculations typical of electronic structure theory are presented to demonstrate its performance. For cases involving small translation distances, accuracies are increased by up to five orders of magnitude over an empirical criterion. The new multipole acceptance criterion is on average within an order of magnitude of the exact expansion error. Use of the multipole acceptance criterion in hierarchical multipole based methods as well as in traditional electronic structure methods is discussed. © 1998 American Institute of Physics.
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  • 7
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 104 (1996), S. 4685-4698 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Fast methods based on a representation of the electron charge density in a Hermite Gaussian basis are introduced for constructing the Coulomb matrix encountered in Hartree-Fock and density functional theories. Simplifications that arise from working in a Hermite Gaussian basis are discussed, translations of such functions are shown to yield rapidly convergent expansions valid in both the near- and far-field, and the corresponding truncation errors are derived in compact form. The relationship of such translations to hierarchical multipole methods is pointed out and a quantum chemical tree code related to the Barnes-Hut method is developed. Novel methods are introduced for the independent thresholding of "bra'' and "ket'' distributions as well as for screening out insignificant multipole interactions. Recurrence relations for computing the Cartesian multipole tensor are used to efficiently calculate far-field electrostatic interactions using high-order expansions. Application of the quantum chemical tree code to assembly of the Coulomb matrix for HF/3-21G calculations on sequences of polyglycine α-helices and water clusters demonstrate scalings as favorable as N1.6, where N is the number of basis functions. Comparisons with a commercial electronic structure program indicate that our method is highly competitive. Speed is obtained without sacrificing precision, truncation errors are controlled with a single parameter, and the method performs equally well with a contracted or uncontracted LCAO basis. © 1996 American Institute of Physics.
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  • 8
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 107 (1997), S. 10131-10140 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The seminal work of Nijboer and De Wette [Physica 23, 309 (1957)] enables the calculation of lattice sums of spherical harmonics, but has long been overlooked. In this article, their central result is recast in a simplified form suitable for modern multipole algorithms that employ the solid harmonics. This formulation makes possible the imposition of periodic boundary conditions within modern versions of the fast multipole method, and other fast N-body methods. The distinction between the extrinsic values obtained with the lattice sums M of the multipole interaction tensors, and the intrinsic values associated with Taylor's expansion of the Ewald formulas, is made. The central constants, M, are computed to 32 digit accuracy using extended precision arithmetic. Timings and corresponding errors obtained with a periodic version of the fast multipole method are presented for particle numbers spanning [103,106], and a range of expansion orders. A qualitative comparison is made between the present implementation, other periodic versions of the fast multipole method, and fast Ewald methods. © 1997 American Institute of Physics.
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  • 9
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 106 (1997), S. 5526-5536 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Computation of the Fock matrix is currently the limiting factor in the application of Hartree-Fock and hybrid Hartree-Fock/density functional theories to larger systems. Computation of the Fock matrix is dominated by calculation of the Coulomb and exchange matrices. With conventional Gaussian-based methods, computation of the Fock matrix typically scales as ∼N2.7, where N is the number of basis functions. A hierarchical multipole method is developed for fast computation of the Coulomb matrix. This method, together with a recently described approach to computing the Hartree-Fock exchange matrix of insulators [J. Chem. Phys. 105, 2726 (1900)], leads to a linear scaling algorithm for calculation of the Fock matrix. Linear scaling computation the Fock matrix is demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory, and corresponding accuracies in converged total energies are shown to be comparable with those obtained from standard quantum chemistry programs. Restricted Hartree-Fock/3-21G calculations on several proteins of current interest are documented, including endothelin, charybdotoxin, and the tetramerization monomer of P53. The P53 calculation, involving 698 atoms and 3836 basis functions, may be the largest Hartree-Fock calculation to date. The electrostatic potentials of charybdotoxin and the tetramerization monomer of P53 are visualized and the results are related to molecular function. © 1997 American Institute of Physics.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Theoretical chemistry accounts 104 (2000), S. 344-349 
    ISSN: 1432-2234
    Keywords: Key words: Linear Scaling ; Exact exchange ; Electron repulsion integrals ; Gaussian basis functions ; Permutational symmetry
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract.  A direct comparison is made between two recently proposed methods for linear scaling computation of the Hartree–Fock exchange matrix to investigate the importance of exploiting two-electron integral permutational symmetry. Calculations on three-dimensional water clusters and graphitic sheets with different basis sets and levels of accuracy are presented to identify specific cases where permutational symmetry may or may not be useful. We conclude that a reduction in integrals via permutational symmetry does not necessarily translate into a reduction in computation times. For large insulating systems and weakly contracted basis sets the advantage of permutational symmetry is found to be negligible, while for noninsulating systems and highly contracted basis sets a fourfold speedup is approached.
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