IMA Seminar on Mathematics and Chemistry

October 8, 2008, 10:00am-11:00am, Lind Hall 409
Steven M. Valone (Structure-Property Relations Group, Materials Science and Technology, Division, Los Alamos National Laboratory, Los Alamos, NM, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN)

A view of outstanding problems in density functional theory^*
Slides:  pdf

Abstract: Constrained-search density functional theory (DFT) pioneered by Levy [1] poses the problem of the theory as one of searching over subsets of Hilbert space. As such it provides a hypothetical means of constructing density-based energy functionals for use in electronic structure applications. I will illustrate the constrained-search form with simple examples [2]. Early results on continuity of the energy functional [3] and the advent of "open-system" DFT [4] will be reviewed. The construction of energy functionals will discussed in the context of the Colle-Salvetti functionals [5] that played a subtle, but important, role in the 1998 Nobel Prize in Chemistry [6]. Alternative constructions based on constrained-search DFT will be discussed. Finally topics pertaining to excitations in homogeneous electron gases and from the introduction of other constraints to DFT calculations [7,8] will be entertained.

[1] M Levy, Proc Natl Acad Sci USA 76, 6062 (1979).
[2] SM Valone, "Vignette on Constrained-Search Density Functional Theory," private communication, August (2008).
[3] SM Valone, J Chem Phys 73, 1344 (1980).
[4] JP Perdew, RG Parr, M Levy, and JL Balduz, Jr, Phys Rev Lett 49, 1691 (1982).
[5] R Colle and O Salvetti, Theoret Chim Acta (Berl) 37, 329 (1975).
[6] C Lee, W Yang, and RG Parr, Phys Rev B 37, 785 (1988).
[7] X-Y Pan, V Sahni, and L Massa, Phys Rev Lett 93, 130401 (2004).
[8] Q Wu and T van Voorhis, Phys Rev A 72, 024502 (2005); J Behler, B Delley, K Reuter, and M Scheffler, Phys Rev B 75, 115409 (2007).

^*LA-UR-08-06035

October 15, 2008, 4:00pm-5:00pm, Lind Hall 409
Dexuan Xie (Department of Mathematical Sciences, University of Wisconsin-Milwaukee)
http://www.uwm.edu/~dxie/

New efficient algorithms for a general blood tissue transport-metabolism model and stiff differential equations

Abstract: Fast algorithms for simulating mathematical models of coupled blood-tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissue. This talk will introduce a general blood tissue transport-metabolism model governed by a large system of one-dimensional hyperbolic partial differential equations, and then present a new parallel algorithm for solving it. The key part of the new algorithm is to approximate the model as a group of independent ordinary differential equation (ODE) systems such that each ODE system has the same size as the model and can be integrated independently. The accuracy of the algorithm is demonstrated for solving a simple blood-tissue transport model with an analytical solution. Numerical experiments were made for a large-scale coupled blood tissue transport-metabolism model on a distributed-memory parallel computer and a shared-memory parallel computer, showing the high parallel efficiency of the algorithm.

In the second part of this talk, a well-known implicit Runge-Kutta algorithm called the Radau IIA method will be discussed, which is a favorite stiff ODE solver for the new parallel algorithm. The most time consuming part of the Radau IIA method is to solve a large scale nonlinear algebraic system of stage values. Currently, the widely-used nonlinear solver was still a simplified Newton method proposed by Liniger & Willoughby in 1970. In practice, it may suffer poor convergence problems, forcing the Radau IIA method to select too small step sizes in order to guarantee the convergence. To provide the Radau IIA method with a robust nonlinear solver, this talk will present a new simplified Newton algorithm and discuss its convergence and performance. Numerical results confirm that the new algorithm can have better convergence properties than the current one and can significantly improve the performance of the Radau IIA method.

November 12, 2008, 4:00pm-5:00pm, Lind Hall 305
Daniel M. Chipman (Radiation Laboratory, University of Notre Dame)
http://www.rad.nd.edu/faculty/chipman.htm

Fermi contact interactions evaluated from hidden relations in the Schrödinger equation

Abstract: Fermi contact interactions between electrons and nuclei govern important properties such as the hyperfine coupling constants observed in Electron Spin Resonance Spectroscopy and the spin-spin coupling constants observed in Nuclear Magnetic Resonance Spectroscopy. But approximate wavefunctions of the kinds commonly used for molecules are generally optimized through some kind of overall energy criterion, and so may have significant errors for the electron density at position of a nucleus where the Fermi contact interaction occurs. It will be shown how hidden relations that are implicit in the Schrödinger equation allow the Fermi contact interactions to be reexpressed in terms of more global properties of the electron density. For exact wavefunctions the hidden relations would give the same results as would direct pointwise evaluation of the electron density, but for approximate wavefunctions the results may differ and in fact provide improved accuracy. The relevant equations will be derived, and numerical examples will be given that demonstrate the point. An extension to higher order will also be developed for the well-known Kato cusp condition that constrains the behavior of the wavefunction at the singularity that occurs when two particles coalesce.

December 3, 2008, 1:15pm-2:15pm, Lind Hall 305 [note time change]
Tong Li (Department of Mathematics, The University of Iowa)
http://www.math.uiowa.edu/~tli/

Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion

Abstract: We establish the existence and the stability of traveling wave solutions of quasi-linear hyperbolic systems with both relaxation and diffusion. The traveling wave solutions are shown to be asymptotically stable against small perturbations provided that the diffusion coefficient is bounded by a constant multiple of the relaxation time. The result provides an important first step toward the understanding of the transition from stability to instability as the diffusion coefficient increases.

January 28, 2009, 1:00pm-2:00pm, Lind Hall 305
Maurizio Persico (Dipartimento di Chimica e Chimica Industriale Facoltà di Scienze Matematiche, Fisiche e Naturali, Università di Pisa)
http://perseo.dcci.unipi.it/

Computational strategy options in tackling a problem of molecular excited state dynamics

Abstract: We shall briefly introduce a particular (but typical) problem concerning the photophysical and photochemical behaviour of a class of compounds, namely azobenzene and its derivatives. We shall also motivate the goal of running computer simulations of such processes, mainly consisting in the interplay with very refined experiments and with proposals of new applications. We shall examine the computational tools currently available and we shall motivate our choices, namely, direct trajectory methods with semiempirical electronic energies and wavefunctions. We shall conclude by underlying strong and weak points of the computational approach, and by discussing the proposal of a Molecular Dynamics method based on a time-dependent force-field for excited states.

February 5, 2009, 11:15pm-12:15pm, Lind Hall 409
Xavier Blanc (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie - Paris 6)
http://www.ann.jussieu.fr/~blanc/

Fast rotating Bose-Einstein condensates in asymmetric harmonic traps

Abstract: A trapped rotating Bose-Einstein condensate is described by minimizing the Gross-Pitaevskii energy with an angular momentum term. In the fast rotating regime, one can restrict the minimization space to the lowest Landau level (LLL), which is the first eigenspace of the linear part of the Hamiltonian of the system. In the case of a symmetric harmonic trap, this framework allows to recover, both analytically and numerically, the lattice of vortices of experiments. In the case of an asymmetric trap, an LLL can still be defined, but the behaviour is drastically different: the condensate has no vortex. Furthermore, contrary to the symmetric case, convergence of minimizers can be proved, and a limit profile can be computed.

February 13, 2009, 2:30pm-3:00pm, Lind Hall 305 [Joint with the Department of Chemical Engineering and Materials Science, University of Minnesota.]
Alexander L. Efros (Center for Computational Material Science, Naval Research Laboratory)

Surface effect on the quantum size energy levels in semiconductor nanocrystals

Abstract: We study the effect of the surface on the electron and hole energy level structure in spherical semiconductor nanocrystals within 8 band effective mass approximation. The surface properties are modelled by the General Boundary Conditions that allow us to exclude spurious and wing contributions to the eight band envelope function. The boundary conditions contain a surface parameter that is independent of the energy of the electronic states and should be considered as additional to the set of effective mass parameters describing the bulk semiconductor. We have shown that this parameter: (i) effects strongly the size dependence of the electron and hole quantum size energy levels, (ii) changes the symmetry of the lowest energy levels in the valence band, (iii) leads to the existence of surface localized states with energies within the forbidden gap, (iv) induces the spin-orbit splitting of the conduction band states, and (v) causes additional magnetic moment of the electrons.

February 18, 2009, 11:15am-12:15pm, Lind Hall 409
Heinz Siedentop (Lehrstuhl für Analysis, Ludwig-Maximilians-Universität München (LMU))
http://www.mathematik.uni-muenchen.de/%7Ehkh/

The ground state energy of atoms: Functionals of the one-particle-reduced density matrix and their relation to the full Schrödinger equation

Abstract: To have an explicit formula for the ground state energy (lowest spectral point) E(Z) of the Schrödinger operator of (neutral) atoms of atomic number Z is an elusive goal for Z>1 since it is a matrix differential operator in 3N dimensions with 2^z components.

Shortly after the advent of quantum mechanics efforts were made to reduce the dimensions to 3 and the components to one. The first steps were taken by Thomas and Fermi and by Hartree and Fock. A modern version of this idea is due to Hohenberg and Kohn (density functional theory) and Gilbert (density matrix functional theory). The price to pay is to give up the linearity of the problem.

In this talk I will explain the general idea of density matrix functional theory and show how a particular type of density matrix functionals (Müller functional and variants thereof, see the kick-off meeting of IMA's year on mathematics and chemistry) can be used to get information on the asymptotic behavior of E(Z). Among other things, we will show, that the infimum E_M(Z) has the same asymptotic expansion

E_M(Z) = a Z^7/3 + 1/4 Z² - c Z ^5/3+ o(Z^5/3)

as the quantum case.

February 25, 2009, 1:00pm-2:00pm, Lind Hall 409
Frédéric Legoll (LAMI, Ecole Nationale des Ponts et Chaussées (ENPC))
http://cermics.enpc.fr/~legoll/home.html

Effective dynamics using conditional expectations

Abstract: We consider a system described by its position X_t, that evolves according to the overdamped Langevin equation. At equilibrium, the statistics of X are given by the Boltzmann-Gibbs measure. Suppose that we are only interested in some given low-dimensional function ξ(X) of the complete variable (the so-called reaction coordinate). The statistics of ξ are completely determined by the free energy associated to this reaction coordinate. In this work, we try and design an effective dynamics on ξ, that is a low-dimensional dynamics which is a good approximation of ξ(X_t). Using conditional expectations, we build an original dynamics, whose accuracy is supported by error estimates obtained following an entropy-based approach. Numerical simulations will illustrate the accuracy of the proposed dynamics according to various criteria.

This is joint work with T. Lelievre (ENPC and INRIA).

April 1, 2009, 11:15am-12:15pm, Lind Hall 409
James W. Evans (Ames Laboratory - US DOE / Iowa State University)
http://www.ameslab.gov/pbchem/PI%20info/evans.htm

Stochastic "interacting particle systems" models for reaction-diffusion systems: Non-linear kinetics, steady-state bifurcations (phase transitions), reaction fronts

Abstract: Traditionally, non-linear reaction kinetics and associated spatiotemporal reaction-diffusion behavior have been analyzed with mean-field rate and reaction-diffusion equations. This formulation assumes that the reactants are well-mixed, ignoring spatial correlations and fluctuations. This is akin to the mean-field Van der Waals equation of state for a fluid which has long since been surpassed by statistical mechanical treatments of phase transitions and critical phenomena. The recent USDOE Basic Science Grand Challenges report proposes an analogous sophisticated treatment of such far-from-equilibrium systems (such as chemical reactions), where the thermodynamic framework available for equilibrium systems does not apply. Here, we investigate a statistical mechanical lattice-gas or "interacting particle systems" (IPS) realization of Schloegl's 2nd model for autocatalysis. The mean-field model displays bistability between a reactive and a poisoned state. In contrast, the IPS realization exhibits a discontinuous phase transition between these states with associated metastability and nucleation phenomena. This is mostly analogous to behavior in equilibrium fluid systems. However, the IPS realization also exhibits "generic two-phase coexistence," behavior never seen in an equilibrium system.

References: Phys. Rev. Lett. 98 (2007) 050601; Phys. Rev. E 75 (2007); Physica A 387 (2008); J Stat. Phys. (2009); J. Chem. Phys. 130 (2009) 074106.

May 13, 2009, 11:15am-12:15pm, Lind Hall 409
Frédéric Legoll (LAMI, Ecole Nationale des Ponts et Chaussées (ENPC))
http://cermics.enpc.fr/~legoll/home.html

Non-ergodicity of the Nosé-Hoover dynamics

Abstract: The Nosé-Hoover dynamics is a deterministic method that is commonly used to sample the canonical Gibbs measure. This dynamics extends the physical Hamiltonian dynamics by the addition of a thermostat variable, that is coupled nonlinearly with the physical variables. The validity of the method depends on the dynamics being ergodic. It has been numerically observed for a long time that such a thermostat, applied to some model problems (including the one-dimensional harmonic oscillator), is actually not ergodic.

In this work, we first show that, for some multidimensional systems, the averaged dynamics, obtained in the limit of infinite thermostat mass, has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Next, in the case of one-dimensional Hamiltonian systems, we go further and prove non-ergodicity of the thermostat for large (but finite) thermostat masses.

Numerical experiments will illustrate the theoretical results.

May 14, 2009, 11:15am-12:15pm, Lind Hall 409
Tony Lelievre (CERMICS, Ecole Nationale des Ponts et Chaussées (ENPC))
http://cermics.enpc.fr/~lelievre/home.html

Adaptive methods for efficient sampling. Applications in molecular dynamics

Abstract: One aim of molecular dynamics simulations is to sample Boltzmann-Gibbs measures associated to some potentials in high dimensional spaces, to compute macroscopic quantities (such as chemical reaction constants, or diffusions constants) by statistical means in the canonical (NVT) ensemble. Numerical methods typically rely on ergodic limits for processes solution to well-chosen stochastic differential equations (SDEs). The main difficulty comes from existence of metastable states in which the stochastic processes remain for long time: this may slow down dramatically the convergence of the ergodic limit. We present a class of adaptive importance sampling methods which enable fast exploration of the configurational space, by modifying the potential seen by the particles (the associated SDE becomes non-homogeneous and nonlinear). These methods accelerate the longtime convergence while they allow to obtain, in the longtime limit, the quantities of practical interest (the marginal law associated to the slow variables in the system). We propose a proof of convergence for some of these methods, based on entropy techniques.

References:

- T. Lelievre, M. Rousset and G. Stoltz, Computation of free energy profiles with parallel adaptive dynamics, Journal of Chemical Physics 126, 134111 (2007)

- T. Lelievre, M. Rousset and G. Stoltz, Long-time convergence of an Adaptive Biasing Force method, Nonlinearity, 21, 1155-1181 (2008)

- T. Lelievre, A general two-scale criteria for logarithmic Sobolev inequalities, to appear in Journal of Functional Analysis

May 22, 2009, 1:00pm-2:00pm, Lind Hall 409
Gabriel Stoltz (CERMICS - ENPC)
http://www-rocq.inria.fr/MICMAC/rubrique.php3?id_rubrique=8

Adiabatic switching for degenerate ground states

Abstract: The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian H₀ into eigenstates of the modified Hamiltonian H₀ + V. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator \cP₀ V \cP₀, where \cP₀ is the projection onto the degenerate eigenspace of H₀.