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  • 1
    Electronic Resource
    Electronic Resource
    Quadratic steepest descent on potential energy surfaces. III. Minima seeking along steepest descent lines (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 5276-5280 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A simplified quadratic steepest descent method, based on the availability of energies and gradients, is formulated for use in minimum searching. It requires only a fraction of the computational effort needed for the previously developed accurate steepest descent procedures and, typically, involves less work than standard quasi-Newton minimum searches. At the same time, it follows the true steepest descent curves reasonably closely and reaches the closest minima. This is in contrast to quasi-Newton procedures which cannot be relied upon to do so. The performance is documented by applications to a variety of searches on the Müller–Brown surface.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465996
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  • 2
    Electronic Resource
    Electronic Resource
    Quadratic steepest descent on potential energy surfaces. I. Basic formalism and quantitative assessment (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 5257-5268 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A novel second-order algorithm is formulated for determining steepest-descent lines on potential energy surfaces. The reaction path is deduced from successive exact steepest-descent lines of local quadratic approximations to the surface. At each step, a distinction is made between three points: the center for the local quadratic Taylor expansion of the surface, the junction of the two adjacent local steepest-descent line approximations, and the predicted approximation to the true steepest-descent line. This flexibility returns a more efficient yield from the calculated information and increases the accuracy of the local quadratic approximations by almost an order of magnitude. In addition, the step size is varied with the curvature and, if desired, can be readjusted by a trust region assessment. Applications to the Gonzalez–Schlegel and the Müller–Brown surfaces show the method to compare favorably with existing methods. Several measures are given for assessing the accuracy achieved without knowledge of the exact steepest-descent line. The optimal evaluation of the predicted gradient and curvature for dynamical applications is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465994
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  • 3
    Electronic Resource
    Electronic Resource
    Quadratic steepest descent on potential energy surfaces. II. Reaction path following without analytic Hessians (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 5269-5275 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A second order method is developed for determining steepest descent lines of potential energy surfaces by following steepest curves of successive local quadratic surface approximations. The basic principle is similar to that of a previously developed method where, however, the availability of analytically calculated exact Hessians was assumed wherever needed. By contrast, only the analytically calculated exact values of the energy and its gradient are used here and this difference entails marked changes in strategy. Applications to the Gonzalez–Schlegel and the Müller–Brown surfaces show that the method compares favorably with existing methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.465995
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  • 4
    Electronic Resource
    Electronic Resource
    Second-order many-body perturbation-theory calculations in extended systems (1996)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 104 (1996), S. 8553-8565 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Explicit expressions for electron correlation at the second-order many-body perturbation-theory [MBPT(2)] level are presented and implemented for the total energy per unit cell and for the band structure of extended systems. In the latter case, a formula is presented for a direct evaluation of the band gap rather than obtaining it as a difference of two large numbers. Application is made to alternating trans-polyacetylene. We assess the convergence of MBPT(2) with the number of unit cells (N) included in the lattice summations, the number of k-points (K) taken for the integrals over k in the first Brillouin zone, and the cutoff threshold (10−C) for the two-electron integrals. The MBPT(2) correlation correction to the band structure converges very slowly with N and demands a large K while the MBPT(2) correction to the total energy per unit cell converges much faster with N and needs a much smaller K. Neither MBPT(2) correction is sensitive to the cutoff of the two-electron atomic orbital integrals, 10−C, when C≥5. For polyacetylene, the MBPT(2) band gap is much improved from the SCF result, but does not agree with previous numerical data. Analysis shows that the previous MBPT(2) results were obtained either with too few unit cells such that the convergence with N had not been reached, or that the zeroth-order Hartree–Fock results were inadequately converged. MBPT(2) with a DZP basis improves the Hartree–Fock band gap from 5.57 to 3.22 eV at the experimentally estimated geometry, compared to the measured ∼2 eV peak in the absorption spectrum of the system. We verify our results with three independent programs. We also study the band gap as a function of geometry. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.471545
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  • 5
    Electronic Resource
    Electronic Resource
    A simple prediction of approximate transition states on potential energy surfaces (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 101 (1994), S. 2168-2174 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Given the locations, the energies, and the force constants of a reactant minimum and a product minimum and assuming that no other information is available, an analytical algorithm is formulated for determining the optimal conjecture for a surmized transition state between them. It is based on a model surface obtained by combining the two quadratic basin approximations. The method is illustrated by applications to transition states on an analytical surface and on an ab initio surface.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467722
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  • 6
    Electronic Resource
    Electronic Resource
    Locating transition states by quadratic image gradient descent on potential energy surfaces (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 101 (1994), S. 2157-2167 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: An analysis is given of the so-called "image function'' approach to finding transition states. It is demonstrated that, in fact, such functions do not exist for general potential energy surfaces so that a plain minimum search is inappropriate. Nonconservative image gradient fields do exist, however, and their field lines, defined by Euler's equation, can lead to transition states as exemplified by quantitative integrations of these equations for the Müller–Brown surface. As do gradient fields, image gradient fields contain streambeds and ridges, but their global structure is considerably more complex than that of gradient fields. In particular, they contain certain singular points where the image gradients change sign without passing through zero. They are the points where the two lowest eigenvalues of the Hessian are degenerate. Some of them can act as singular attractors for the image gradient descent and any algorithm must contain safeguards for avoiding them. (Such regions are equally troublesome for quasi-Newton-type transition-state searches.) Image gradient fields appear to have considerably larger catchment basins around transition states than do quasi-Newton-type or gradient-norm-type transition-state searches. A quantitative quadratic image-gradient-following algorithm is formulated and, through applications to the Müller–Brown surface, shown to be effective in finding transition states.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467721
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  • 7
    Electronic Resource
    Electronic Resource
    Gradient fields of potential energy surfaces (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 5836-5848 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: While contour plots provide conceptual pictures of potential energy surfaces and exhibit their critical points, quantitative determinations of critical points and reaction paths as well as dynamical calculations require a knowledge of the gradient fields. The corresponding orthogonal trajectory maps are more complex than contour maps, but they provide additional insights. They are found to contain certain frequently occurring structural elements and these patterns are here examined. It is shown that many of them result from local confluences of orthogonal trajectory bundles with gradient extremals. The analysis leads to the distinction between eight different kinds of such gradient extremal channels. The most important ones are the streambeds and ridges, the former being typical conceptual prototypes of reaction channels, the latter being prototypes of reaction barriers. Gradient extremal channels emanate from second order critical points in the directions of all normal modes, but they do not necessarily follow along the entire reaction path of any one reaction. They can also exist unrelated to critical points. The conclusions are exemplified on a number of model potential energy surfaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467147
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  • 8
    Electronic Resource
    Electronic Resource
    Quadratic steepest descent on potential energy surfaces. IV. Adaptation to singular Hessians (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 6101-6101 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: It is shown that the quadratic steepest descent procedure developed in three previous papers goes smoothly and naturally into a limiting form, presenting no computational problems, when the Hessian becomes singular. This is in contrast to quasi-Newton algorithms which, in the case of singular or near-singular Hessians, yield unrealistic search directions and are kept from breaking down only by introducing arbitrary additional assumptions. The reason is that quasi-Newton algorithms give inadequate weight to the first order variation of the energy, i.e., its gradient.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467123
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  • 9
    Electronic Resource
    Electronic Resource
    Erratum: Gradient extremals and steepest descent lines on potential energy surfaces [J. Chem. Phys. 98, 9707 (1993)] (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 1779-1779 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467291
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  • 10
    Electronic Resource
    Electronic Resource
    Convergence of many-body perturbation methods with lattice summations in extended systems (1997)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 106 (1997), S. 5554-5563 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.473577
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