Browsing by Author "Hagedorn, George A."
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- Accurate Calculations of Molecular Properties with Explicitly Correlated MethodsZhang, Jinmei (Virginia Tech, 2014-08-13)Conventional correlation methods suffer from the slow convergence of electron correlation energies with respect to the size of orbital expansions. This problem is due to the fact that orbital products alone cannot describe the behavior of the exact wave function at short inter-electronic distances. Explicitly correlated methods overcome this basis set problem by including the inter-electronic distances (rij) explicitly in wave function expansions. Here, the origin of the basis set problem of conventional wave function methods is reviewed, and a short history of explicitly correlated methods is presented. The F12 methods are the focus herein, as they are the most practical explicitly correlated methods to date. Moreover, some of the key developments in modern F12 technology, which have significantly improved the efficiency and accuracy of these methods, are also reviewed. In this work, the extension of the perturbative coupled-cluster F12 method, CCSD(T)F12, developed in our group for the treatment of high-spin open-shell molecules (J. Zhang and E. F. Valeev, J. Chem. Theory Comput., 2012, 8, 3175.), is also documented. Its performance is assessed for accurate prediction of chemical reactivity. The reference data include reaction barrier heights, electronic reaction energies, atomization energies, and enthalpies of formation from the following sources: (1) the DBH24/08 database of 22 reaction barriers (Truhlar et al., J. Chem. Theory Comput., 2007, 3, 569.), (2) the HJO12 set of isogyric reaction energies (Helgaker et al., Modern Electronic Structure Theory, Wiley, Chichester, first ed., 2000.), and (3) the HEAT set of atomization energies and heats of formation (Stanton et al., J. Chem. Phys., 2004, 121, 11599.). Two types of analyses were performed, which target the two distinct uses of explicitly correlated CCSD(T) models: as a replacement for the basis-set-extrapolated CCSD(T) in highly accurate composite methods like HEAT and as a distinct model chemistry for standalone applications. Hence, (1) the basis set error of each component of the CCSD(T)F12 contribution to the chemical energy difference in question and (2) the total error of the CCSD(T)F12 model chemistry relative to the benchmark values are analyzed in detail. Two basis set families were utilized in the calculations: the standard aug-cc-p(C)VXZ (X = D, T, Q) basis sets for the conventional correlation methods and the cc-p(C)VXZ-F12 (X = D, T, Q) basis sets of Peterson and co-workers that are specifically designed for explicitly correlated methods. The conclusion is that the performance of the two families for CCSD correlation contributions (which are the only components affected by the explicitly correlated terms in our formulation) are nearly identical with triple- and quadruple-ζ quality basis sets, with some differences at the double-ζ level. Chemical accuracy (~4.18 kJ/mol) for reaction barrier heights, electronic reaction energies, atomization energies, and enthalpies of formation is attained, on average, with the aug-cc-pVDZ, aug-cc-pVTZ, cc- pCVTZ-F12/aug-cc-pCVTZ, and cc-pCVDZ-F12 basis sets, respectively, at the CCSD(T)F12 level of theory. The corresponding mean unsigned errors are 1.72 kJ/ mol, 1.5 kJ/mol, ~ 2 kJ/mol, and 2.17 kJ/mol, and the corresponding maximum unsigned errors are 4.44 kJ/mol, 3.6 kJ/mol, ~ 5 kJ/mol, and 5.75 kJ/mol. In addition to accurate energy calculations, our studies were extended to the computation of molecular properties with the MP2-F12 method, and its performance was assessed for prediction of the electric dipole and quadrupole moments of the BH, CO, H2O, and HF molecules (J. Zhang and E. F. Valeev, in preparation for submission). First, various MP2- F12 contributions to the electric dipole and quadrupole moments were analyzed. It was found that the unrelaxed one-electron density contribution is much larger than the orbital response contribution in the CABS singles correction, while both contributions are important in the MP2 correlation contribution. In contrast, the majority of the F12 correction originates from orbital response effects. In the calculations, the two basis set families, the aug-cc-pVXZ (X = D, T, Q) and cc-pVXZ-F12 (X = D, T, Q) basis sets, were also employed. The two basis set series show noticeably different performances at the double-ζ level, though the difference is smaller at triple- and quadruple-ζ levels. In general, the F12 calculations with the aug-cc- pVXZ series give better results than those with the cc-pVXZ-F12 family. In addition, the contribution of the coupling from the MP2 and F12 corrections was investigated. Although the computational cost of the F12 calculations can be significantly reduced by neglecting the coupling terms, this does increase the errors in most cases. With the MP2-F12C/aug-cc-pVDZ calculations, dipole moments close to the basis set limits can be obtained; the errors are around 0.001 a.u. For quadrupole moments, the MP2-F12C/aug-cc-pVTZ calculations can accurately approximate the MP2 basis set limits (within 0.001 a.u.).
- Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time VersionsAmaya, Austin J. (Virginia Tech, 2012-04-26)Given a full-range simply-invariant shift-invariant subspace M of the vector-valued L2 space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function W so that M may be represented as the image of of the Hardy space H2 on the disc under multiplication by W. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces (M,MÃ ) which together form a direct-sum decomposition of L2. In the case where the pair (M,MÃ ) are finite-dimensional perturbations of the Hardy space H2 and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function W; this realization was parameterized in terms of zero-pole data computed from the pair (M,MÃ ). Later work by Ball-Raney extended this analysis to the case of nonrational functions W where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs (M,MÃ ) of the L2 spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans.
- The Born-Oppenheimer Approximation for Triatomic Molecules with Large Angular Momentum in Two DimensionsBowman, Adam Shoresworth (Virginia Tech, 2010-12-08)We study the Born-Oppenheimer approximation for a symmetric linear triatomic molecule in two space dimensions. We compute energy levels up to errors of order ε⁵, uniformly for three bounded vibrational quantum numbers n₁, n₂, and n₃; and nuclear angular momentum quantum numbers â ≤ kε-3/4 for k > 0. Here the small parameter ε is the fourth root of the ratio of the electron mass to an average nuclear mass.
- The Born-Oppenheimer approximation in scattering theoryKargol, Armin (Virginia Tech, 1994-05-05)We analyze the Schrödinger equation i𝜖 ¬2â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬-4 and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬2). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.
- Born-Oppenheimer Corrections Near a Renner-Teller CrossingHerman, Mark Steven (Virginia Tech, 2008-07-03)We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner-Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of ε, where ε4 is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter bË , which is equivalent to the parameter originally used by Renner. For 0 < bË < 1, we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near bË = 0.925. We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for 0 < bË < 0.925 and non-degenerate for 0.925 < bË < 1.
- Born-Oppenheimer Expansion for Diatomic Molecules with Large Angular MomentumHughes, Sharon Marie (Virginia Tech, 2007-10-29)Semiclassical and Born-Oppenheimer approximations are used to provide uniform error bounds for the energies of diatomic molecules for bounded vibrational quantum number n and large angular momentum quantum number l. Specifically, results are given when (l + 1) < κ𝛜⁻³/². Explicit formulas for the approximate energies are also given. Numerical comparisons for the H+₂ and HD+ molecules are presented.
- Conical Intersections and Avoided Crossings of Electronic Energy LevelsGamble, Stephanie Nicole (Virginia Tech, 2021-01-14)We study the unique phenomena which occur in certain systems characterized by the crossing or avoided crossing of two electronic eigenvalues. First, an example problem will be investigated for a given Hamiltonian resulting in a codimension 1 crossing by implementing results by Hagedorn from 1994. Then we perturb the Hamiltonian to study the system for the corresponding avoided crossing by implementing results by Hagedorn and Joye from 1998. The results from these demonstrate the behavior which occurs at a codimension 1 crossing and avoided crossing and illustrates the differences. These solutions may also be used in further studies with Herman-Kluk propagation and more. Secondly, we study codimension 2 crossings by considering a more general type of wave packet. We focus on the case of Schrödinger equation but our methods are general enough to be adapted to other systems with the geometric conditions therein. The motivation comes from the construction of surface hopping algorithms giving an approximation of the solution of a system of Schrödinger equations coupled by a potential admitting a conical intersection, in the spirit of Herman-Kluk approximation (in close relation with frozen/thawed approximations). Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase and its proof is based on a normal form approach.
- Contribution of the First Electronically Excited State of Molecular Nitrogen to Thermospheric Nitric OxideYonker, Justin David (Virginia Tech, 2013-05-13)The chemical reaction of the first excited electronic state of molecular nitrogen, N₂(A), with ground state atomic oxygen is an important contributor to thermospheric nitric oxide (NO). The importance is assessed by including this reaction in a one-dimensional photochemical model. The method is to scale the photoelectron impact ionization rate of molecular nitrogen by a Gaussian centered near 100 km. Large uncertainties remain in the temperature dependence and branching ratios of many reactions important to NO production and loss. Similarly large uncertainties are present in the solar soft x-ray irradiance, known to be the fundamental driver of the low-latitude NO. To illustrate, it is shown that the equatorial, midday NO density measured by the Student Nitric Oxide Explorer (SNOE) satellite near the Solar Cycle 23 maximum can be recovered by the model to within the 20% measurement uncertainties using two rather different but equally reasonable chemical schemes, each with their own solar soft-xray irradiance parameterizations. Including the N₂(A) changes the NO production rate by an average of 11%, but the NO density changes by a much larger 44%. This is explained by tracing the direct, indirect, and catalytic contributions of N₂(A) to NO, finding them to contribute 40%, 33%, and 27% respectively. The contribution of N₂(A) relative to the total NO production and loss is assessed by tracing both back to their origins in the primary photoabsorption and photoelectron impact processes. The photoelectron impact ionization of N₂ is shown to be the main driver of the midday NO production while the photoelectron impact dissociation of N₂ is the main NO destroyer. The net photoelectron impact excitation rate of N₂, which is responsible for the N₂(A) production, is larger than the ionization and dissociation rates and thus potentially very important. Although the conservative assumptions regarding the level-specific NO yield from the N₂(A)+O reaction results in N₂(A) being a somewhat minor contributor, N₂(A) production is found to be the most efficient producer of NO among the thermospheric energy deposition processes.
- A Discontinuous Galerkin Method for Higher-Order Differential Equations Applied to the Wave EquationTemimi, Helmi (Virginia Tech, 2008-03-17)We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions.
- Dynamics of Competition using a Bit String Model with Age Structure and MutationsAstalos, Robert Joseph (Virginia Tech, 2001-04-17)Using Monte Carlo simulations and analytic methods, we examine the dynamics of inter-species competition using the Penna bit-string model. We begin with a study of the steady state with a single species, then proceed to the dynamics of competition between two species. When the species are not evenly matched in fitness, a simple differential equation provides a satisfactory model of the behavior of the system. However, when the species are equally fit, we show that a model, originally proposed to describe population genetics [Fisher,Wright], is required. When mutations are allowed between the competing species, the dynamics becomes more interesting. The mutation rate becomes a parameter that dictates the steady state behavior. If the two species are not equally fit, the value of the mutation rate determines whether the longer-lived or faster reproducing species is favored. With two species that are equally fit, the steady state varies with mutation rate from a single peaked to a double peaked distribution. This behavior is shown to be well described by an extension to the Fisher-Wright model mentioned above. Finally, we describe the preliminary results of a few new lines of investigation, and suggest ideas for further study of the dynamics of this model.
- Eigenvalue Statistics for Random Block OperatorsSchmidt, Daniel F. (Virginia Tech, 2015-04-28)The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.
- Equilibrium states of ferromagnetic abelian lattice systemsMiekisz, Jacek (Virginia Polytechnic Institute and State University, 1984)Ferromagnetic abelian lattice systems are the topic of this paper. Namely, at each site of ZV-invariant lattice is placed a finite abelian group. The interaction is given by any real, negative definite, and translation invariant function on the space of configurations.Algebraic structure of the system is investigated. This allows a complete · description of the family of equilibrium states for given. interaction at low temperatures. At the same time it is proven that the low temperature expansion for Gibbs free energy is analytic. It is also shown that it is not necessary to consider gauge models in the case of Zm on ZV lattice.
- Exponentially Accurate Error Estimates of Quasiclassical EigenvaluesToloza, Julio Hugo (Virginia Tech, 2002-12-11)We study the behavior of truncated Rayleigh-Schröodinger series for the low-lying eigenvalues of the time-independent Schröodinger equation, when the Planck's constant is considered in the semiclassical limit. Under certain hypotheses on the potential energy, we prove that, for any given small value of the Planck's constant, there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and actual eigenvalue is smaller than an exponentially small function of the Planck's constant. We also prove the analogous results concerning the eigenfunctions.
- A foundation for translating user interaction designs into OSF/Motif-based softwareHinson, Kenneth Paul (Virginia Tech, 1994-04-05)The user interface development process occurs in a behavioral domain and in a constructional domain. The development process in the behavioral domain focuses on the "look and feel" of the user interface and its behavior in response to user actions. The development process in the constructional domain focuses on developing software to implement the user interface. Although one may attempt to design a user interface from a constructional view, it is important to concentrate design efforts in the behavioral domain to improve software usability. User Action Notation (UAN) is a useful technique for representing user interaction designs in the behavioral domain. Primary abstractions in UAN-expressed designs are user tasks. Information about interface objects is encapsulated in user task descriptions and scenarios. Primary abstractions in a GUI such as Motif™ are interface objects. Motif implements objects' behavior and appearance using system functions that are encapsulated within pre-defined object classes. Therefore, user interaction developers and software developers must communicate well to translate UAN-expressed interaction designs into Motif-based software designs. Translation is not trivial since it is a translation between two significantly different domains. This thesis contributes to understanding of the user interface development process by developing a foundation to assist translation of user interaction designs into Motif-based software designs. This thesis develops the foundation as follows: 1. Adapt UAN for use with Motif. 2. Summarize Motif concepts about objects and object relationships. 3. Develop new approaches for discussing objects and object relationships. 4. Develop a partial translation guide containing VAN descriptions of selected Motif abstractions.
- Global existence in L1 for the square-well kinetic equationLiu, Rongsheng (Virginia Tech, 1993-04-04)An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in L¹. global in time and for arbitrary initial data. is proved. A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates. Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved.
- Hardy-space Function Theory on Finitely Connected Planar DomainsGuerra Huaman, Moises Daniel (Virginia Tech, 2008-04-17)Hardy space scalar theory on the disk is now classical. Some extensions have been done, one of them is the approach done by Donald Sarason using Laurent series. We present the more complicated function theory, without the use of either power series or Laurent series, for finitely-connected planar domains.
- A High Order Correction of the Energy of a One Dimensional Model of an H2+ MoleculeHumfeld, Keith Daniel (Virginia Tech, 1998-07-31)The ground state electron wavefunction of some molecules has a non-zero angular momentum about the internuclear axis. Molecular rotational momentum can couple with this angular momentum, splitting the energy degeneracy of the two directions of motion about the internuclear axis. Performing a Born-Oppenheimer approximation of such a system will break the relevant energy degeneracy at eighth order. This degeneracy breaking is known as L-doubling.
- Higher-Degree Immersed Finite Elements for Second-Order Elliptic Interface ProblemsBen Romdhane, Mohamed (Virginia Tech, 2011-08-01)A wide range of applications involve interface problems. In most of the cases, mathematical modeling of these interface problems leads to partial differential equations with non-smooth or discontinuous inputs and solutions, especially across material interfaces. Different numerical methods have been developed to solve these kinds of problems and handle the non-smooth behavior of the input data and/or the solution across the interface. The main focus of our work is the immersed finite element method to obtain optimal numerical solutions for interface problems. In this thesis, we present piecewise quadratic immersed finite element (IFE) spaces that are used with an immersed finite element (IFE) method with interior penalty (IP) for solving two-dimensional second-order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. An analysis of the constructed IFE spaces and their dimensions is presented. Shape functions of Lagrange and hierarchical types are constructed for these spaces, and a proof for the existence is established. The interpolation errors in the proposed piecewise quadratic spaces yield optimal O(h³) and O(h²) convergence rates, respectively, in the L² and broken H¹ norms under mesh refinement. Furthermore, numerical results are presented to validate our theory and show the optimality of our quadratic IFE method. Our approach in this thesis is, first, to establish a theory for the simplified case of a linear interface. After that, we extend the framework to quadratic interfaces. We, then, describe a general procedure for handling arbitrary interfaces occurring in real physical practical applications and present computational examples showing the optimality of the proposed method. Furthermore, we investigate a general procedure for extending our quadratic IFE spaces to p-th degree and construct hierarchical shape functions for p=3.
- Identification of Transient Nonlinear Aeroelastic PhenomenaChabalko, Christopher C. (Virginia Tech, 2007-03-05)Complex nonlinear aspects of aeroelastic phenomena include unsteady nonlinear aerodynamic loads, structural nonlinearities, as well as nonlinear couplings between the flow and the structural response. Nonlinearities in aerodynamic loads originate from unsteady shocks and/or flow separation. Structural nonlinearities are geometric, or a result of free play. Nonlinear fluid structure couplings result from nonlinear resonance between the aerodynamic load and structural modes. Under different conditions, one or a combination of these aspects could yield flutter or Limit Cycle Oscillations (LCO). The overall goal of this work is to develop the capabilities to quantify the role that these different nonlinear mechanisms could play in observed flutter and LCO. The realization of such a goal would help in providing a benchmark for the detection of nonlinear aeroelastic instabilities and possibly effective means for obtaining improved performance and reduced uncertainties through operation beyond conventional boundaries that are based on linear analysis. Additionally, this effort will provide a benchmark for the validation of computational methodologies. In this thesis, wavelet-based higher order spectra are applied to identify different nonlinear aeroelastic phenomena as encountered in two experiments. First, the analysis is applied to a set of experiments involving a flexible semispan model (FSM) of a High Speed Civil Transport (HSCT) wing configuration conducted by Silva et al. (Experimental Steady and Unsteady Aerodynamic and Flutter Results for HSCT Semispan Models; AIAA/ASME/ASCE/AHS/ASC 41st Structures, Structural Dynamics, and Materials Conference, 2000). The interest is in the identification of nonlinear aeroelastic phenomena associated with a high dynamic response region which was measured over a large range of dynamic pressures around Mach number 0.98. At the top of this region is a ``hard'' flutter point that resulted in the loss of the model. The results show that ``hard'' flutter is related to intermittent nonlinear coupling between the shock motion and large amplitude structural motions. Second, the analysis is applied to identify nonlinear aspects of LCO encountered during test flights of an F-16 aircraft. The results show quadratic and cubic couplings in the acceleration signals of the under-wing launchers and high quadratic coupling levels between flaperon motions and wing oscillations. The implications of applying these techniques in the capacity of a ``flutterometer'' are also discussed.
- Kinetic theory and global existence in Lp1s for a dense square-well fluidYao, Aixiang (Virginia Tech, 1995)In this paper, we consider the kinetic equation for a dense square-well fluid and the geometric factor Y _ 1, provide the related kinetic theory, and prove a global existence theorem in L1 for the kinetic equation under rather general initial value condition. An analogue of the classical H-theorem is verified.
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