We are carrying out research in several areas of dynamics and electronic structure, with a special emphasis on applying quantum mechanics to the treatment of large and complex systems. Dynamical calculations are being carried out for combustion (with a special emphasis on biofuel mechanisms) and atmospheric reactions in the gas phase and catalytic reactions in the condensed phase. Both thermal and photochemical reactions are under consideration. New orbital-dependent density functionals are being developed to provide an efficient route to the potential energy surfaces for these studies. New methods are also being developed for representing the potentials and for combined quantum mechanical and molecular mechanical methods, with a special emphasis in the latter case on improving the electrostatics. New techniques for modeling vibrational anharmonicity and for Feynman path integral calculations are also under development.
Much of the work is carried out at Minnesota Supercomputing Institute, and the group also has a computational grand challenge grant from the Environmental Molecular Sciences Laboratory and an allocation on the INCITE resources of the Argonne Leadership Computing Facility (the second largest computer in the world).
Several collaborations with other Chemistry faculty are also in progress: studies of solvatochromic shifts, computational electrochemistry, and drug design with Professor Cramer, development of explicit polarization models for aqueous protein chemistry with Professor Gao, studies of atmospheric nucleation with Professor Siepmann, and studies of actinoid chemistry with Professor Gagliardi. Collaborations with faculty outside of chemistry include studies of the collision processes occurring in hypersonic flow around re-entry of space shuttles with Professor Graham Candler and Tom Schwartzentruber of Aerospace Engineering, studies of the thermochemistry and dynamics of metal nanoparticles with Professor Chris Hogan of Mechanical Engineering, modeling of zinc biocenters with Professor Elizabeth Amin of Medicinal chemistry, and modeling of plasma chemistry with Professor Steven Girshick of Mechanical Engineering.
This work is supported in part by the NSF, DOE, NIH, AFOSR, and ARO.
Density functional design
We are employing two paths to improving density functional theory. On the one hand we are designing new exchange-correclation functionals with improved performance in Kohn-Sham theory. We have summarized the quest for a universal exchange-correlation functional in three overview articles. See
- "Density Functionals with Broad Applicability in Chemistry," Y. Zhao and D. G. Truhlar, Accounts of Chemical Research 41, 157-167 (2008). dx.doi.org/10.1021/ar700111a
- "Applications and Validations of the Minnesota Density Functionals," Y. Zhao and D. G. Truhlar, Chemical Physics Letters 502, 1-13 (2011). dx.doi.org/10.1016/j.cplett.2010.11.060
- "The Quest for a Universal Density Functional: The Accuracy of Density Functionals Across a Broad Spectrum of Databases in Chemistry and Physics," R. Peverati and D. G. Truhlar, Philosophical Transactions of the Royal Society A 372, 20120476/1-51 (2014).dx.doi.org/10.1098/rsta.2012.0476
The newest Minnesota functionals are xe-PBE0, GAM, MN15-L, and MN15.
In a second initiative, in collaboration with Professor Laura Gagliardi, we are designing new on-top density functionals for multi-configuration pair-density functional theory. This theory goes beyond Kohn-Sham theory to use the diagonal element of the two-particle density matrix, whereas Kohn-Sham theory is restricted to the singe-particule density. Initial results are extremely encouraging. This theory has now been applied successfully to a broad database, to charge transfer, to a dirhenium complex, to Rydberg states, to bond energies of diatomic and triatomic molecules, and to a database with both valence and Rydberg excitations.
Inorganometallic catalyst design
The Inorganometallic Catalyst Design Center (ICDC) is developing structure-function relationships that will guide further catalyst discovery. Our catalysts will consist of precisely synthesized clusters dispersed on supports. We are pushing into new territory in efficient computational materials design by making highly uniform materials selected on the basis of modeling reinforced by feedback from results of catalyst characterization, including rapid-throughput performance testing. Our Center is developing catalytic materials and catalytic conversion strategies that are inaccessible with current methodologies. The key to our approach is the development of well isolated cluster catalysts that are highly uniform in structure yet present in high-density arrays accessible by reactants, while relatively amenable to structural characterization and theoretical modeling. Our major goals are to introduce an entirely new class of cluster catalysts, to discover and explain fundamental relationships between atomic-scale structure and catalytic performance, and to apply methods to discover new catalysts for energy-relevant conversions. We are also developing new methodologies for computational design by addressing the challenge of three-dimensinal modeling that includes the effect of supports and catalyst environments that enhance reactivity at active catalytic centers. We are computationally predicting, with atomic-scale precision, cluster catalysts for conversion of light hydrocarbons in natural gas and investigating their reactivities with state-of-the-art quantum chemical and classical simulation techniques.
Quantum mechanical dynamics
A critical focus area in our research group is incorporation of quantum mechanical effects into dynamics simulations. For small-molecule systems, we have developed an efficient linear algebraic variational method for calculating converged quantum mechanical transition probabilities for reactive collisions, both electronically adiabatic and electronically nonadiabatic. Our present focus, however, is on incorporating quantum mechanical effects into simulations on large and complex systems. For this we use several semiclassical tools, including variational transition state theory with quantized vibrations, multidimensional semiclassical tunneling calculations, and semiclassical treatments of coupled electronic states, including coherence, decoherence, and nonadiabatic transitions.
An exciting area of special interest is the application of these methods to enzyme reactions, and we have made good progress at incorporating quantum mechanical effects such as zero point energy, tunneling, and substrate polarization into the quantitative modeling of proton and hydride transfer reactions catalyzed by enzymes. An interesting area for future study is the incorporation of quantal effects in enzyme reactions that proceed by radical mechanisms. An area of future work is a better treatment of open-shell effects in metalloenzymes.
The development of new methods for studying photochemistry and other non-Born-Oppenheimer processes is our current focus area for dynamics research. The convenient generation of potential energy surfaces in the dynamically preferred electronic representation and the incorporation of decoherence into non-Born-Oppenheimer trajectories are two areas of the utmost importance where progress has been made. Two examples of recent progress that will enable photochemical studies of large and complex systems are the development of the anchor points reactive potential and the development of a way to include multidimensional tunneling into multi-surface molecular dynamics calculations:
- "Anchor Points Reactive Potential for Bond-Breaking Reactions," K. R. Yang, X. Xu, and D. G. Truhlar, Journal of Chemical Theory and Computation 10, 924-933 (2014).dx.doi.org/10.1021/ct401074s
- "Including Tunneling in Non-Born-Oppenheimer Simulations," J. Zheng, R. Meana-Pañeda, and D. G. Truhlar, Journal of Physical Chemistry Letters 5, 2039-2043 (2014).dx.doi.org/10.1021/jz500653m
In related work we are developing improved Monte Carlo methods for the calculation of Feynman path integrals for quantum statistical mechanics, and we are developing a vibrational configuration interaction method for calculating thermal rate constants in terms of quantum mechanical flux correlation functions.
Kinetics
Variational transition state theory (VTST) is being further developed and used for the calculation of the rates of electronically adiabatic chemical reactions. Electronically adiabatic reactions are those that take place entirely on a single potential energy surface, usually that corresponding to the ground electronic state; this includes the field of traditional chemical kinetics, namely thermally activated reactions. We have developed a powerful technique for studying such reactions, namely, variational transition-state theory with multidimensional semiclassical tunneling contributions (VTST/MT). VTST/MT involves finding the free energy bottleneck for over barrier processes and the optimal tunneling paths for through-barrier processes. This theory, including three important generalizations for complex reactions
- ensemble-averaged VTST
- multi-structural VTST
- multi-path VTST
has been developed for reactions in the gas phase and at gas-solid interfaces based on potential energy surfaces and in liquid solution and at enzyme active sites based on free energy surfaces. We have developed practical methods for including multidimensional tunneling effects on canonical and microcanonical rate constants, kinetic isotope effects, and state-selective chemistry. We have also developed methods for treating nonequilibrium solvation effects on reactions in the condensed phase. In the gas phase, we distinguish two kinds of reactions, those proceeding over a barrier and those without a barrier; the former are studying using a potential expanded around the minimum-energy reaction path, in some cases augmented to include a wider reaction swath and large-amplitude vibrations, such as torsions; the latter class of reactions (barrierless ones) are studied using a variable reaction coordinate with multifaceted dividing surfaces. Application areas include combustion, atmospheric chemistry, environmental chemistry, clusters (from microhydrated species to nanoparticles), and catalysis (heterogeneous, organometallic, and biological).
We have also developed new methods for calculating potential energy surfaces, including the development of new density functionals that give more accurate barrier heights for complex processes.
Our work on adding quantum effects to simulations of complex chemical systems and on density functionals is funded in part by by the National Science Foundation, and our work on fundamental gas-phase kinetics is funded in part by the U.S. Department of Energy. Our work on combustion of biofuels is funded in part by the Combustion Energy Frontier Research Center. Our work on orbital-dependent density functionals for catalysis is funded by the Air Force Office of Scientific Research.
We have recently extended variational transition state theory to calculate pressure-dependent rate constants.
Photochemistry
Electronically nonadiabatic reactions, that is, reactions involving coupled potential energy surfaces, are also a major area of study. Such reactions are traditionally called photochemistry. We developed accurate semiclassical methods for multisurface trajectories and validated them against accurate quantum mechanical results obtained with the unique capabilities of our quantum mechanical electronically nonadiabatic rearrangement chemistry scattering programs, and we are developing new methods. Two types of semiclassical (in particular mixed quantum-classical) methods have been developed: improved trajectory surface hopping (also called molecular dynamics with quantum transitions and time uncertainty) and self-consistent potential methods (also called time-dependent self-consistent-field methods). We have recently developed a new method that combines the best features of both of these approaches into a single formalism. This new method is called decay of mixing with coherent switches, and it is more accurate than previously available methods for the whole range of problems encountered in photochemistry. Furthermore, it is practical to apply this method, at least when state-specific zero-point energy (ZPE) effects are not expected to be too large, to both simple and complex photochemical reactions, and we are currently carrying out (in various stages) or have recently carried out such calculations for ammonia, chlorobromomethane, OH...HH, bromoacetyl chloride, phenol, and thioanisole. New studies of electronically nonadiabatic processes of oxygen atoms and oxygen molecules have also been begun. When ZPE effects play a major role, traditional mixed quantum-classical methods may need to be improved and we have recently made progress on including the effect of these adjustments on simulation results.
An important aspect of studying both electronically nonadiabatic reactions is the calculation of potential energy surfaces and their multi-state couplings, and we are currently carrying this out by the fourfold way direct diabatization procedure based on multi-configuration quasidegenerate perturbation theory and complete-active-space self-consistent-field diabatic molecular orbitals.
Additional work under study in this area includes state-specific non-equilibrium and equilibrium continuum solvation effects for the computation of excited-state wave functions, algorithms for the treatment of electronically nonadiabatic and ultrafast dynamics in both the gas-phase and solution, electrostatically embedded multiconfiguration molecular mechanics and molecular mechanics (EE-MCMM/MM) schemes for coupled potential energy surfaces, multiscale approaches for the treatment of explicit local solvation environments with embedding to include longer-range solvent effects, and Monte Carlo strategies for efficient conformational sampling of large and flexible chromophores employing direct dynamics.
Most photochemical simulations are limited to a few dozen trajectories of less than 1 picosecond without tunneling on potential energy surfaces at the CASSCF level or some other level without full dynamical correlation. We recently reported a study of the photodissociation of phenol with 177,097 multi-state 10 ps trajectories with nonadiabatic tunneling on a surface with full dynamical correlation.
Our work on charge transfer and charge transport on photoactivated systems is funded in part as a SciDAC partnership by the U.S. Department of Energy.
Force fields, potential energy surfaces, direct dynamics, and computational thermochemistry
In addition to dynamics calculations, a large effort is being devoted to the development of new potential energy functions and force fields, using new techniques of ab initio and semiempirical electronic structure theory, as well as molecular modeling techniques. Knowledge of the potential energy surface or force field is a prerequisite for either dynamics or thermochemistry calculations.
One area of active work is the extension of molecular mechanics force fields to be able to treat reactive systems that involve bond breaking. An approach called multi-configuration molecular mechanics (MCMM) has been developed for this purpose, and it is very promising. Recent progress on this method is mentioned in the Kinetics and Photochemistry section. Other general strategies for treating reactions in complex systems are combined quantum mechanical and molecular mechanical (QM/MM) methods, electrostatically embedded many-body (EE-MB) expansions, and the explicit polarization (X-Pol) method. The most widely used strategy for calculating electronic energies and potential energy surfaces for large molecular systems such as biopolymers or large clusters is partitioning the system into subsystems or fragments, as in the QM/MM method, the EE-MB method, electrostatically embedded multiconfiguration Shepard interpolation (EE-MCSI), and the explicit polarization (X-pol) method, as well as many others. Two important problems that arise in essentially all such methods are (i) including the electrostatic potential of one subsystem in the Hamiltonian of another and (ii), when the fragments are covalently bonded, treating the boundary between fragments in calculations on the partitioned system. For problem ii, we developed a tuned pseudoatom scheme with a balanced redistributed charge algorithm that has much better performance than conventional link atom methods, especially for boundaries through a polar bond. For problem i, we developed and parameterized a general method for including charge penetration effects in a distributed monopole scheme for generating electrostatic potentials.
One area of special concentration is in the interface of electronic structure theory and dynamics. We are developing a variety of single-level and dual-level methods for direct dynamics calculations, where direct dynamics denotes the calculation of rate constants or other dynamical quantities directly from electronic structure calculations without the intermediacy of fitting a potential energy function. In such a case the potential energy surface is implicit but is never actually constructed.
A very exciting recent development is the parameterization of multi-coefficient methods for scaling components of the correlation energy and extrapolating electronic structure calculations to an infinite basis set. These methods allow one to calculate accurate gas-phase heats of formation, atomization energies, and potential energy surfaces for large systems at affordable cost. These methods have better scaling properties than pure ab initio calculations, and they often yield more accurate results with far less computer time. We have now shown how these methods can be improved by adding static correlation with density function theory for even great performance-to-cost ratios.
The direct calculation of free energies from potential energy surfaces, without first calculating the energy spectrum, is also of great interest, and we are developing improved Monte Carlo sampling methods for doing this by the Feynman path integral method.
Solvation effects
Solvation effects are important for several of the above mentioned projects, and our group has a special focus on solvation effects; much of this work involves a close collaboration with Professor Chris Cramer and his research group.
The general goal of our work on solvation has been to allow the treatment of energetics and dynamics in the condensed phase to be as accurate as their treatment for gas-phase species and processes. To this end, we are working on theoretical problems involving solvation effects on organic, biochemical, and environmental processes in aqueous and nonaqueous solutions. The role of the solvent in polarizing the solute is especially interesting. Solvation models for both aqueous and organic solvents have been developed and are under development, and a variety of applications to structure and reactivity in solution are underway.
One important aspect of our solvation work is the incorporation of the new solvation models into dynamics calculations, including both equilibrium and nonequilibrium solvation, and spectroscopy. The dynamics calculations employ VTST methods discussed above, and spectroscopic applications involve a new vertical excitation model. We are also applying solvation models to nonhomogeneous media (e.g., cell membrane boundaries), to supercritical fluids, and to environmentally green ionic liquids.
Further information about solvation models and solvation software is available on another page:http://comp.chem.umn.edu/solvation .
Our work on solvation is funded in part by by the National Science Foundation and in part by the Army Research Office.
Scientific Discovery Through Advanced Computing
Another project in our group, in collaboration with several other groups, funded by the U.S. Department of Energy through the Scientific Discovery Through Advanced Computing (SciDAC) program, is the development of new electron-correlated methods for excited-state structure and dynamics for incorporation in generally available software, especially in the NWChem software suite.
Biochemistry
Many enzymatic reactions involve proton and hydride transfer, but until recently techniques for simulating the dynamics of these processes were usually based entirely in classical mechanics. We are working on a number of initiatives for including quantum effects in biological simulations. This includes tunneling, zero point effects, and the effect of quantization on thermally averaged quantities. We have shown that proton transfers catalyzed by enolase and hydride transfer catalyzed by liver alcohol dehydrogenase are dominatedby quantum mechanical events, and that these can be well modeled by semiclassical dynamics methods developed in our group. Further development are applications are underway.
An important application of solvation modeling is the calculation of the partitioning of organic and biological molecules between aqueous and cell membranes. This has an important effect on bioavailability of drugs. We are developing and applying new molecular modeling methods that make such calculations more accurate.
Much of our work on enzyme kinetics involves collaboration with Professor Jiali Gao. This work is funded by the in part by by the National Science Foundation and in part by the National Institutes of Health.
Nanomaterials
Nanotechnology is the art of manipulating materials on a scale of the order of a nanometer, to build molecular scale devices or to take advantage of the unique chemical, physical, and material properties of nanostructured materials. Our research in this area focuses on computational studies of nanoparticle growth and dynamics. We are concerned with the development and implementation of new methods for modeling and simulation of nanoparticles and their elementary processes, including nucleation, deposition, melting, and surface reactions. Nanoscale systems present a challenge to computation because they display properties that are not well modeled by methods developed for use in bulk simulations and because they are expensive to treat using methods developed for molecular systems. We have therefore taken a bootstrap approach with the goal of developing a set of accurate methods for predicting the energetics and structures of Al particles from Al dimer to the bulk, including nanoparticles with 40 to 200 Al atoms. Critical to the success of this project is the development of new electronic structure methods for aluminum nanoparticles. To study larger aluminum nanoparticles, we are currently also developing a novel tight binding-configuration interaction (TBCI) method that incorporates charges non-iteratively into tight binding by applying a configuration interaction-type treatment to the tight binding wavefunction. We are especially concerned with multi-scale modeling, i.e., the development of new techniques for extending the time and length scales of simulations and their application to problems involving semiconductor nanoparticles and metal nanoparticles. To study of the importance of quantum effects in nanoparticle reactivity, for example, the reaction of metal particles with hydrocarbons and hydrocarbon fragments, we are developing multilevel methods, such as QM/MM methods, that combine quantum mechanics (QM) and molecular mechanics (MM). The efficiency of these methods potentially allows one to perform accurate calculations for large reactive systems over long time scales. For the simulation of systems with non-localized active areas, it is necessary to adaptively redefine the region to be treated by quantum mechanics. For such systems, we are developing new methods for combining multilevel methods with modern sampling schemes, such as our molecular dynamics code, ANT, or Monte Carlo codes.
Our most recent work on nanoparticles is a collaboration with Chris Hogan and Steven Girshick. This work is funded by the in part by the National Science Foundation.
Earth and Planetary Materials Research
Research is being carried out on the incorporation of quantum mechanics into the simulations of materials, especially under conditions relevant to earth and planetary sciences. This includes work on density functional theory and quantum dynamics. For further details see the VLab home page.
Atmospheric Nucleation Processes
Work is underway on understanding how particles nucleate in a multi-component gas mixture. This has important implications not only for climate and weather but also wide-ranging technological applications including gas separations, pollution control, and nanotechnology. Atmospheric nucleation involves multi-scale processes ranging from proton transfer to molecular condensation and evaporation events and culminating in the rare formation of the critical nucleus. Pre-critical clusters are characterized by emergent behavior (e.g., formation of zwitterionic particles due to proton transfer) and self-organization due to differences in micro-solubility. Many fundamental questions about atmospheric nucleation pathways remain unanswered. We are working on developing computational algorithms and analysis tools for efficient investigations of multi-component gas-to-particle nucleation processes and elucidating atmospherically relevant nucleation processes.
This research was initiatiated in an NSF-funded collaborative project with Professor J. Ilja Siepmann for cyber-enabled chemistry.
Fundamental Processes in High-Temperature Hypersonic Flows
Hypersonic vehicles that are in the atmosphere for extended period of times are exposed to high-temperature interactions between gases and the vehicle surfaces, resulting in the need for new materials for advanced thermal protection systems. We are developing theoretical and computational methods that describe the potentials and dynamics of molecule-molecule and molecule-surface collisions in order to enable more realistic hypersonic flow simulations of shock layers around the vehicles in terms of molecular-scale interactions between gases and vehicle surfaces for computer-aided design of these new materials. This involves quantum mechanical electronic structure theory, fitting of potential energy surfaces and their couplings, and electronically adiabatic and nonadiabatic dynamics of energy transfer and reactive collisions. We have found that complete active space perturbation theory is particularly well suited for this problem. Primarily we will apply recently developed, highly successful density functionals and wave function theory to develop interaction potentials. Further information on this project is available at https://www.sites.google.com/site/muriafosr/
Integrated Tools for Computational Chemical Dynamics
This was a joint project between the Chemistry Department of the University of Minnesota and the Supercomputing Institute of the University of Minnesota (PI: D. G. Truhlar) and Pacific Northwest National Laboratory (co-PI: Bruce C. Garrett). The University of Minnesota faculty investigators participaating in this initiative, in addition to DGT, are: Chris Cramer, Jiali Gao, Ilja Siepmann, and Darrin York. The goal of this project is to develop more powerful simulation methods and incorporate them into a user-friendly high-throughput integrated software suite for chemical dynamics. Recent advances in computer power and algorithms have made possible accurate calculations of many chemical properties for both equilibria and kinetics. Nonetheless, applications to complex chemical systems, such as reactive processes in the condensed phase, remain problematic due to the lack of a seamless integration of computational methods that allow modern quantum electronic structure calculations to be performed with state-of-the-art methods for electronic structure, chemical thermodynamics, and reactive dynamics. These problems are often exacerbated by invalidated methods, non-modular and non-portable computer codes, and inadequate documentation that drastically limit software reliability, throughput, and ease of use. The goal of the Integrtated Tools consortium is to develop an integrated software suite that combines electronic structure packages with dynamics codes and efficient sampling algorithms for the following kinds of condensed-phase modeling problems:
thermochemical kinetics and rate constants
photochemistry and spectroscopy
chemical and phase equilibria
computational electrochemistry
heterogeneous catalysis
Photochemical creation of excited states offers a means to control chemical transformations because different wavelengths of light can be used to create different vibronic states, thereby directing chemical reactions along different pathways. It is crucial to understand how energy deposited into the system is used; this is particularly complicated in condensed phase systems where many channels lead to dissipation of excess energy. Similar opportunities and challenges present themselves in the areas of electrochemistry and catalysis. This work is supported in part by the Office of Naval Research. For further information, please see the Integrated Tools home page and Grand Challenge Project: Computational Chemical Dynamics of Complex Systems.
Grand Challenge: Computational Chemical Dynamics of Complex Systems
"Computational Chemical Dynamics of Complex Systems" was a Computational Grand Challenge project of The William R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a U.S. Department of Energy national scientific user facility located at Pacific Northwest National Laboratory (PNNL) in Richland, Washington. Resources are provided by EMSL's Molecular Science Computing Facility (MSCF). The project is a collaborative effort involving scientists in the Department of Chemistry at the University of Minnesota and scientists at PNNL. Our consortium is focusing on a variety of condensed-phase modeling problems including thermochemical kinetics and rate constants, photochemistry and spectroscopy, chemical and phase equilibria, electrochemistry, and heterogeneous catalysis.
A second DOE grant for high-performance computing has been obtained under the INCITE program. This grant, entitled "Potential energy surfaces for simulating complex chemical processes" supports computing on the IBM Blue Gene/P of Argonne National Laboratory. The computer-intensive part of our INCITE research consists of electronic structure calculations required for structural characterization and rate constant and dynamics calculations. The main software packages for this project are the GAMESS to obtain accurate energies and stationary points for systems whose electronic structure has high multireference character (in molecular electronics, medicinal chemistry, and combustion energy research), POLYRATE to obtain rate constants, and GPAW for density functional calculations applied to catalytic reactions at gas-solid and gas-nanoparticle-solid interfaces and to charge transfer at material interfaces. Computational parallelism is exploited both in the electronic structure steps and in the dynamics steps.
Our high-perfrmance computing is is also supported by the Department of Defense (for computing at Maui High-Perfromance Computing Center and other DoD computers) for research on "Orbital-Dependent Density Functionals for Chemical Catalysis."
Finally our work is supported by Minnesota Supercomputing Institute for a broad program of research on "Computational Chemical Dynamics." The latter grant supports research on the structure, dynamics, and thermodynamics of few-body systems; the reaction dynamics of organic, metal-organic, neurochemical, and enzymatic systems; photochemistry; combustion kinetics and atmospheric kinetics; the structure of mixed aqueous clusters; hydrogen and hydrocarbon diffusion in solid oxides; the structure and reactivity of nanoparticles; electrochemistry; catalysis; new methods for electronic structure calculations; and the influence of solvation on structure and dynamics in water, organic liquids, and environmental media and their application to relevant experimental phenomena. This research utilizes quantum mechanical, quantum statistical, semiclassical, and classical mechanical methods. We utilize commercially available electronic structure and molecular modeling packages, appropriately modified for our work. At the same time, we are developing our own integrated software tool suite plus stand-alone software packages.
Disclaimer
Opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, U. S. Department of Energy, the Air Force Office of Scientific Research, or other sponsors.