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Carlos F Bunge - One of the best experts on this subject based on the ideXlab platform.
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symmetric dissociation of the water molecule with truncation Energy Error a benchmark study
Physical Chemistry Chemical Physics, 2019Co-Authors: Cesar X Almoradiaz, A Ramirezsolis, Carlos F BungeAbstract:We use selected configuration interaction with truncation Energy Error (SCI-TEE) and CI by parts (CIBP) to study the symmetric dissociation of the water molecule with Roos' triple-ζ double polarization basis set and with the Dunning cc-pVTZ basis. The calculations comprise CISDTQ (CI-4x) through CI-8x for H2O at its equilibrium geometry (Req) and up to fifteen times Req. With the Dunning basis our SCI-TEE-8x energies differ from full CI by less than 0.01 mHartree (0.006 kcal mol−1) at all O–H distances, representing the best upper bounds for this system outside Req. We compare our results with those of other relevant ab initio methods finding good agreement with recent DMRG calculations. The non-parallelity Error (NPE) for SCI-TEE-6x remains stable below 0.1 mHartree when moving from the Roos to the Dunning orbitals. For the present system, CBS Energy Errors at the experimental equilibrium geometry and at dissociation can accurately be evaluated as the difference between non-relativistic total electronic energies taken from the literature, and our SCI-TEE-8x energies obtained with Dunning's or Roos' orbitals. In both cases, the difference between CBS Energy Errors at the equilibrium geometry and dissociation is not smaller than 10 mH, showing that chemically accurate NPE values do not guarantee a chemically accurate potential Energy surface.
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Present Status of Selected Configuration Interaction With Truncation Energy Error
Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems, 2018Co-Authors: Carlos F BungeAbstract:Abstract Configuration interaction (CI) starts from a matrix-eigenvalue equation involving an atomic or molecular electronic Hamiltonian represented by a complete set of Slater determinants made up of a given orbital basis. Full CI scales unfavorably with number of orbitals and number of electrons relative to all other orbital methods. Recent work on 10-electron systems (Ne and H 2 O ground states, the latter at many internuclear distances), and using large orbital bases, shows that up to sextuply excited configurations can be selected a priori, quantitatively and very efficiently by means of Brown's formula, leading to unsurpassed accuracy and understanding. Selected CI (SCI) suggests an array of promising and unexplored models and calls for new vistas demanding new algorithms. Here I review SCI with truncation Energy Error in the light of new software suitable for considerably larger systems.
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Recent Progress in the Variational Orbital Approach to Atomic and Molecular Electronic Structure
Concepts of Mathematical Physics in Chemistry: A Tribute to Frank E. Harris - Part B, 2016Co-Authors: César X. Almora-díaz, Herzain I. Rivera-arrieta, Carlos F BungeAbstract:Abstract Recent progress in selected configuration interaction (CI) with truncation Energy Error (SCI-TEE) is discussed together with applications. In molecular CI, we take up (i) preselection of huge numbers of configurations and sensitivity analyses, (ii) highlights of SCI-TEE applied to H 2 O ground state, and (iii) symmetric dissociation of H 2 O ground state. We describe automatic optimization of atomic orbital bases to within a prescribed complete basis set Energy Error and an application of it to Ne ground state. New perspectives on the use of optimized orbital bases are briefly outlined. We discuss opportunities for new theory and new predictions in connection with a genuine variational theorem for the Breit–Dirac Hamiltonian. We explain the meaning of positive- and negative-Energy orbitals in contrast with positive-Energy and unphysical N-electron states. We conclude with an overview of current and planned work.
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A priori selected configuration interaction with truncation Energy Error, general sensitivity analysis and application to the Ne atom
Molecular Physics, 2010Co-Authors: Carlos F BungeAbstract:The a priori selected configuration interaction (SCI) method [J. Chem. Phys. 125, 014107 (2006)] seeks to approximate full CI results by performing CI on quantitatively selected spaces while estimating truncation Energy Errors for the discarded spaces. New selection methods are here introduced increasing efficiency therefore improving accuracy for a fixed amount of computer time. SCI is explained within a historical context, from my first steps as a graduate student until present. An application to the ground state of Ne with an Energy-optimized basis of 205 radial functions up to l=20 yields an Energy upper bound E = −128.937477 au(Ne) thus recovering more than 99.97% of the estimated correlation Energy. The sources of the remaining 0.03% of the correlation Energy are mentioned and challenges to be met for the incorporation of the missing correlation Energy into the wave function are outlined.
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selected configuration interaction with truncation Energy Error and application to the ne atom
Journal of Chemical Physics, 2006Co-Authors: Carlos F BungeAbstract:Selected configuration interaction (SCI) for atomic and molecular electronic structure calculations is reformulated in a general framework encompassing all CI methods. The linked cluster expansion is used as an intermediate device to approximate CI coefficients BK of disconnected configurations (those that can be expressed as products of combinations of singly and doubly excited ones) in terms of CI coefficients of lower-excited configurations where each K is a linear combination of configuration-state-functions (CSFs) over all degenerate elements of K. Disconnected configurations up to sextuply excited ones are selected by Brown’s Energy formula, ΔEK=(E−HKK)BK2∕(1−BK2), with BK determined from coefficients of singly and doubly excited configurations. The truncation Energy Error from disconnected configurations, ΔEdis, is approximated by the sum of ΔEKs of all discarded Ks. The remaining (connected) configurations are selected by thresholds based on natural orbital concepts. Given a model CI space M, a us...
David J. Tozer - One of the best experts on this subject based on the ideXlab platform.
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Relationship between long-range charge-transfer excitation Energy Error and integer discontinuity in Kohn–Sham theory
The Journal of Chemical Physics, 2003Co-Authors: David J. TozerAbstract:Charge-transfer (CT) electronic excitation energies are known to be very poorly predicted by time-dependent density functional theory (TDDFT) using local exchange-correlation functionals. Insight into this observation is provided by a simple analysis of intermolecular CT excitations at infinite separation. It is argued that the first TDDFT CT excitation Energy approximately underestimates the experimental excitation by the average of the integer discontinuities of the donor and acceptor molecules; Errors are of the order of several electron volts.
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relationship between long range charge transfer excitation Energy Error and integer discontinuity in kohn sham theory
Journal of Chemical Physics, 2003Co-Authors: David J. TozerAbstract:Charge-transfer (CT) electronic excitation energies are known to be very poorly predicted by time-dependent density functional theory (TDDFT) using local exchange-correlation functionals. Insight into this observation is provided by a simple analysis of intermolecular CT excitations at infinite separation. It is argued that the first TDDFT CT excitation Energy approximately underestimates the experimental excitation by the average of the integer discontinuities of the donor and acceptor molecules; Errors are of the order of several electron volts.
Daniel Tamayo - One of the best experts on this subject based on the ideXlab platform.
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On the accuracy of symplectic integrators for secularly evolving planetary systems
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, Garett Brown, Daniel TamayoAbstract:Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators. We find that the specific choice of metric for determining a simulation's accuracy is important. Only looking at metrics related to integrals of motions such as the Energy Error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the accuracy of secular frequencies compared to the standard Wisdom-Holman method without symplectic correctors, despite the fact that the Energy Error is three orders of magnitudes smaller. We present a framework to trace the origin of this apparent paradox to one term in the shadow Hamiltonian. Specifically, we find a term that leads to negligible contributions to the Energy Error but introduces non-oscillatory Errors that result in artificial periastron precession. This term is the dominant Error when determining secular frequencies of the system. We show that higher order symplectic methods such as the Wisdom-Holman method with a modified kernel or the SABAC family of integrators perform significantly better in secularly evolving systems because they remove this specific term.
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WHFAST: a fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long-term gravitational simulations
Monthly Notices of the Royal Astronomical Society, 2015Co-Authors: Hanno Rein, Daniel TamayoAbstract:We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves Energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the Energy Error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the Energy Error is dominated by an unbiased random walk due to floating-point round-off Errors. We implement symplectic correctors up to order eleven that significantly reduce the Energy Error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.
Martin Vohralík - One of the best experts on this subject based on the ideXlab platform.
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A posteriori estimates distinguishing the Error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Jad Dabaghi, Vincent Martin, Martin VohralíkAbstract:We consider in this paper a model parabolic variational inequality. This problem is discretized with conforming Lagrange finite elements of order $p ≥ 1$ in space and with the backward Euler scheme in time. The nonlinearity coming from the complementarity constraints is treated with any semismooth Newton algorithm and we take into account in our analysis an arbitrary iterative algebraic solver. In the case $p = 1$, when the system of nonlinear algebraic equations is solved exactly, we derive an a posteriori Error estimate on both the Energy Error norm and a norm approximating the time derivative Error. When $p ≥ 1$, we provide a fully computable and guaranteed a posteriori estimate in the Energy Error norm which is valid at each step of the linearization and algebraic solvers. Our estimate, based on equilibrated flux reconstructions, also distinguishes the discretization, linearization, and algebraic Error components. We build an adaptive inexact semismooth Newton algorithm based on stopping the iterations of both solvers when the estimators of the corresponding Error components do not affect significantly the overall estimate. Numerical experiments are performed with the semismooth Newton-min algorithm and the semismooth Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative algebraic solver to illustrate the strengths of our approach.
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Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework
Numerische Mathematik, 2018Co-Authors: Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin VohralíkAbstract:This paper develops a general framework for a posteriori Error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the Energy Error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i -th eigenvalue lies between the exact $$(i-1)$$ ( i - 1 ) -th and $$(i+1)$$ ( i + 1 ) -th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the accuracy of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredients are equivalences between the i -th eigenvalue Error, the associated eigenvector Energy Error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.
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Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework
Numerische Mathematik, 2018Co-Authors: Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin VohralíkAbstract:This paper develops a general framework for a posteriori Error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the Energy Error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i-th eigenvalue lies between the exact (i−1)-th and (i+1)-th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the precision of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredient are equivalences between the i-th eigenvalue Error, the associated eigenvector Energy Error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.
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unified primal formulation based a priori and a posteriori Error analysis of mixed finite element methods
Mathematics of Computation, 2010Co-Authors: Martin VohralíkAbstract:We derive in this paper a unified framework for a priori and a posteriori Error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori Energy Error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the Energy Error, represent Error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall Error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori Error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.
Hanno Rein - One of the best experts on this subject based on the ideXlab platform.
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On the accuracy of symplectic integrators for secularly evolving planetary systems
Monthly Notices of the Royal Astronomical Society, 2019Co-Authors: Hanno Rein, Garett Brown, Daniel TamayoAbstract:Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators. We find that the specific choice of metric for determining a simulation's accuracy is important. Only looking at metrics related to integrals of motions such as the Energy Error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the accuracy of secular frequencies compared to the standard Wisdom-Holman method without symplectic correctors, despite the fact that the Energy Error is three orders of magnitudes smaller. We present a framework to trace the origin of this apparent paradox to one term in the shadow Hamiltonian. Specifically, we find a term that leads to negligible contributions to the Energy Error but introduces non-oscillatory Errors that result in artificial periastron precession. This term is the dominant Error when determining secular frequencies of the system. We show that higher order symplectic methods such as the Wisdom-Holman method with a modified kernel or the SABAC family of integrators perform significantly better in secularly evolving systems because they remove this specific term.
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WHFAST: a fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long-term gravitational simulations
Monthly Notices of the Royal Astronomical Society, 2015Co-Authors: Hanno Rein, Daniel TamayoAbstract:We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves Energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the Energy Error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the Energy Error is dominated by an unbiased random walk due to floating-point round-off Errors. We implement symplectic correctors up to order eleven that significantly reduce the Energy Error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.