Random-matrix theories in quantum physics : common concepts

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

The fact that a new text book introducing the essentials of computational chemistry contains more than 500 pages shows impressively the grown and still growing size and importance of this field of chemistry. The author’s objectives of the book, using his own words, are “to provide a survey of computational chemistry its underpinnings, its jargon, its strengths and weaknesses that will be accessible to both the experimental and theoretical communities”. This design as a general introduction into computational chemistry makes it an alternative to Andrew R. Leach’s well-established “Molecular Modeling” (Prentice Hall) and Frank Jensen’s “Introduction to Computational Chemistry” (Wiley), although the latter focuses on the theory of electronic structure methods. Cramer’s “Essentials” covers force field and molecular orbital theory, Monte Carlo and Molecular Dynamics simulations, thermodynamic and electronic (spectroscopic) property calculation, condensed phase treatment and a few more topics. Moreover, the book contains thirteen selected case studies sexamples taken from the literature sto illustrate the application of the just presented theoretical and computational models. This especially makes the text book well suited for both classroom discussion and self-study. Each chapter of “Essentials” covers a main topic of computational chemistry and will be briefly described here; all chapters are ended by a bibliography and suggested additional readings as well as the literature references cited in the text. In chapter 1 the author defines basic terms such as “theory”, “model”, and “computation”, introduces the concept of the potential energy surface and provides some general considerations about hardware and software. Interestingly, the first equation occurring in the text is not Schro ̈dinger’s equation, as is the case for most computational chemistry introductions, but the famous Einstein relation. The second chapter deals with molecular mechanics. It explains the different potential energy contributions, introduces the field of structure optimization, and provides an overview of the variety of modern force fields. Chapter 3 covers the simulation of molecular ensembles. It defines phase space and trajectories and shows the formalism of, and problems and difference between, Monte Carlo and molecular dynamics. In chapter 4 the author introduces the foundations of molecular orbital theory. Basic concepts such as Hamilton operator, LCAO basis set approach, many-electron wave functions, etc. are explained. To illuminate the LCAO variational process, the Hu ̈ckel theory is presented with an example. Chapter 5 deals with semiempirical molecular orbital (MO) theory. Besides the classical approaches (extended Hu ̈ckel, CNDO, INDO, NDDO) and methods (e.g., MNDO, AM1, PM3) and their performance, examples are provided from the ongoing development in that still fascinating area. Ab initio MO theory is presented in chapter 6; the basis set concept is discussed in detail, and, after some considerations from an user’s point of view, the general performance of ab initio methods is explicated. The next chapter covers the problem of electron correlation and gives the most prominent solutions for its treatment: configuration interaction, theory of the multiconfiguration self-consistent field, perturbation, and coupled cluster. Practical issues are also discussed. Chapter 8’s topic is density functional theory (DFT). Its theoretical foundation, methodology, and some functionals as well as its pros and cons compared to MO theory are presented together with a general performance overview. The next two chapters deal with charge distribution, derived and spectroscopic properties (e.g., atomic charges, polarizability, rotational, vibrational, and NMR spectra), and thermodynamic properties (e.g., zero-point vibrational energy, free energy of formation, and reaction). The modeling of condensed phases is addressed in chapters 11 (implicit models) and 12 (explicit models), which closes with a comparison between the two approaches. Chapter 13 familiarizes the reader with hybrid quantum mechanical/molecular mechanical (QM/MM) models. Polarization as well as the problematic implications of unsaturated QM and MM components are discussed, and empirical valence bond methods are also presented. The treatment of excited states is the topic of chapter 14; besides CI and MCSCF as computational methods, transition probabilities and solvatochromism are discussed. The last chapter deals with reaction dynamics, mostly adiabaticskinetics, rate constants, reaction paths, and transition state theory are section topics here sbut also nonadiabatic, introducing curve crossing and Marcus theory in brief. The appendix is divided into four parts: an acronym glossary (which is very helpful), an overview of symmetry and group theory, an introduction to spin algebra, and finally a section about orbital localization. A rather detailed index ends the book. The “Essentials” writing style fits the fascinating topic: one reads on and on andssurprise! sanother chapter has been absorbed. The text is complemented by a large number of black and white figures and clear tables, mostly self-explanatory with descriptive captions. The use of equations and mathematical formulas in general is well-balanced, and the level of math should be understandable for every natural scientist with some basic knowledge of physics. There are only a few minor shortcomings: for example, a literature reference cited in the text (“Beck et al.”, p 142) is missing in the bibliography; “Kronecker” is mistyped with o ̈; and the author completely forgot to reference Leach’s text book when referring to other computational chemistry introductions. However, the author has established a specific errata web page (http://pollux.chem.umn.edu/ ∼cramer/Errors.html) with all known errors. These will be corrected in the next printing or next revised edition, respectively. With its emphasis, on one hand, on the basic concepts and applications rather than pure theory and mathematics, and on the other hand, coverage of quantum mechanical and classical mechanical models including examples from inorganic, organic, and biological chemistry, “Essentials” is a useful tool not only for teaching and learning but also as a quick reference, and thus will most probably become one of the standard text books for computational chemistry.

Essentials Of Computational Chemistry Theories And Models

A pedagogical overview of intramolecular vibrational redistribution (IVR) phenomena in vibrationally excited molecules is presented. In the interest of focus and simplicity, the topics covered deal primarily with IVR in the ground electronic state, relying on examples from the literature to illustrate key points. The experimental topics discussed attempt to sample systematically three different energy regimes on the full potential surface corresponding to (i) “low”, e.g., moderate- to high-resolution vibrational spectroscopies, (ii) “intermediate”, e.g., stimulated emission pumping and high overtone spectroscopies, and (iii) “high”, e.g., photofragment/predissociation dynamical spectroscopies. The interplay between experiment and theory is highlighted here because it has facilitated enormous advances in the field over the past decade.

Vibrational Energy Flow in Highly Excited Molecules: Role of Intramolecular Vibrational Redistribution

Multiconfiguration Self-Consistent Field and Multireference Configuration Interaction Methods and Applications

Spectra of S0 D2CO rovibrational eigenstates with 28 300 cm−1 of vibrational excitation are measured by Stark level‐crossing spectroscopy. In this new method, the lifetime of a single J, K, M‐resolved S1 state is monitored as a function of electric field. Enhanced nonradiative decay causes the S1 lifetime to decrease as S0 states are Stark tuned into resonance. Analysis of the resulting resonance lineshapes yields complete distributions of S0 decay rates (linewidths) and S1‐S0 coupling matrix elements. The S0 decay rates represent the first measurements of unimolecular dissociation rates of a polyatomic molecule at the eigenstate‐resolved level. S0 decay widths vary from 6.4×10−5 to 3.8×10−3 cm−1 and S1‐S0 coupling matrix elements vary from 3.5×10−7 to 4.7×10−5 cm−1, demonstrating that chemical properties of neighboring eigenstates fluctuate by over two orders of magnitude. The observed density of S0 vibrational states is ∼400 per cm−1, six times greater than an estimate including first‐order anharmonic c...

Stark level‐crossing spectroscopy of S0 formaldehyde eigenstates at the dissociation threshold

Static light scattering from aqueous poly(ethylene oxide) solutions in the temperature range 20-90°C

Transition States and Rate Constants for Unimolecular Reactions

The Stark level‐crossing spectra of S0 D2CO are analyzed for evidence of energy level correlations and quantum ergodicity. The analysis for short and long range level correlations gives an apparent energy level spacing distribution which lies between the Poisson and GOE limits. However, the Stark level‐crossing method diminishes any existing correlations. The true level spacing distribution must be closer to the Gaussian orthogonal ensemble (GOE) than to the Poisson limit. Complete distributions of S1−S0 coupling matrix elements and S0 dissociation rates are reported and subjected to statistical tests for ergodicity. The distribution of S1−S0 coupling matrix elements indicates that the dynamics of intramolecular vibrational redistribution of energy (IVR) is very nearly quantum ergodic. There is strong coupling and free flow of energy among vibrational degrees of freedom in this molecule above its dissociation threshold. The average S0 decay rate can be accounted for by RRKM theory with tunneling correctio...

Eigenstate‐resolved unimolecular reaction dynamics: Ergodic character of S0 formaldehyde at the dissociation threshold

A statistical distribution of state‐specific unimolecular decay rates is derived (within the framework of random matrix theory) that is determined completely by the transition state properties of the potential energy surface It includes the standard χ‐square distributions as a special case Model calculations are presented to show the extent to which it can differ from the χ‐square distribution, and specific application is made to the state‐specific unimolecular decay rate data for D2CO→D2+CO

William F. Polik

Papers

Stark level‐crossing spectroscopy of S0 formaldehyde eigenstates at the dissociation threshold

Static light scattering from aqueous poly(ethylene oxide) solutions in the temperature range 20-90°C

Transition States and Rate Constants for Unimolecular Reactions

Eigenstate‐resolved unimolecular reaction dynamics: Ergodic character of S0 formaldehyde at the dissociation threshold

A transition state theory‐based statistical distribution of unimolecular decay rates with application to unimolecular decomposition of formaldehyde