There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Ultrafast vibrational dynamics of hydrogen bonds in the condensed phase.

This review of the role of reaction kinetics in combustion chemistry traces the historical evolution and present state of qualitative and quantitative understanding of a number of reaction systems. Starting from the H2–O2 system, in particular from the reaction between H and O2, mechanisms and key reactions for soot formation, for the appearance of NOx, and for processes of peroxy radicals in hydrocarbon oxidation are illustrated. The struggle for precise rate constants on the experimental and theoretical side is demonstrated for the example of the reaction H + O2 → OH + O. The intrinsic complexity of complex-forming bimolecular reactions, such as observed even in this reaction, also dominates most other key reactions of the systems considered and can be unravelled only with the help of quantum-chemical methods. The multi-channel character of these reactions often also requires the combination with master equation codes. Although kinetics provides an already impressive database for quantitative modelling of simple combustion systems, considerable effort is still required to quantitatively account for the complexities of more complicated fuel oxidation processes.

Unravelling combustion mechanisms through a quantitative understanding of elementary reactions

Vibrational and vibronic relaxation and related energy redistribution play an important role in many photoinduced processes of molecules. In the condensed phase, in which collision-induced relaxation is of particular relevance, these phenomena occur on a very rapid time scale between several tens of femtoseconds and 100 ps (1, 2). Spectroscopy with ultrashort laser pulses has been widely applied to elucidate the dynamics and physical mechanisms of intraand intermolecular energy redistribution (3, 4). Indeed, time resolved experiments have provided extensive information not accessible by other experimental techniques. In this article, we present a review on vibrational and vibronic relaxation of large molecules in liquids. Photoinduced redistribution processes of molecules, which consist typically of 20 or more atoms, are considered. We discuss. femtoand picosecond spectroscopy of the intramolecular redistribution and the intermolecular energy transfer from vibrationally hot molecules to their (cold) surrounding. We focus on measurements of vibrational and vibronic lifetimes, as well as on studies of energy redis­ tribution and transport. However, we will not consider dephasing of vibranic or vibrational states, relaxation processes connected to the

VIBRATIONAL AND VIBRONIC RELAlXATION OF LARGE POL-yrATOMIC MOLECULES IN LIQUIDS

In this article, we review state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches. We illustrate the power of those methods by presenting applications to molecules with more than four atoms. This demonstrates the great progress that has been made in this field in the last decade in dealing with the exponential scaling with the number of vibrational degrees of freedom. In this review we present three methods that effectively obviate this bottleneck. The first important idea is the n-mode representation of the Hamiltonian and notably the potential. The potential (and other functions) is represented as a sum of terms that depend on a subset of the coordinates. This makes it possible to compute matrix elements, form a Hamiltonian matrix, and compute its eigenvalues and eigenfunctions. Another approach takes advantage of this multimode representation and represents the terms as a sum of products. It then exploits the powerful...

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Variational quantum approaches for computing vibrational energies of polyatomic molecules

A new discrete variable representation (DVR) in generalized vibrational coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence coordinates as well as nonlinear maps of coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usually are hampered by the typically large number of (avoided) crossings in dense spectra. This problem is particularly severe for discrete variable representations. A solution is provided by the new technique of diabatic rotations leading to a systematic construction of locally diabatic channels. This allows the treatment of very dense spectra where conventional truncation techniques fail. Applying the new approach to the vibrational problem of tetratomic molecules demonstrates its flexibility and efficiency. The examples of formaldehyde, ammonia, and hydrogen peroxide cover the whole range from semirigid (CH2O) to large amplitude inversion (NH3) and torsional tunnelling dynamics (H2O2). In solving the full six-dimensional vibrational eigenvalue problems for CH2O and NH3 the Z-matrix DVR shows at least comparable if not superior numerical efficiency compared with specialized techniques. In the case of H2O2 the technique of diabatic rotations and adiabatic contraction for the first time allows the treatment of the tunneling dynamics significantly above the dissociation threshold up to the fifth OH stretch overtone. The calculated decrease of the tunneling rate by about one order of magnitude agrees well with experimental observations.

6D vibrational quantum dynamics: Generalized coordinate discrete variable representation and (a)diabatic contraction

We report calculations of the electronic ground state potential energy surface (PES) of hydrogen peroxide covering, in an almost global fashion, all six internal degrees of freedom by two different ab initio techniques. Density functional theory (DFT) calculations using the Becke 3 parameter Lee-Yang-Parr (B3LYP) hybrid functional and multiconfigurational second order perturbation theory (CASPT2) calculations, both using large basis sets, are performed for a wide range of geometries (8145 DFT and 5310 CASPT2 single-point energies). We use a combined data set of mostly DFT with additional CASPT2 ab initio points and the complete CASPT2 surface to fit a total of four different 6D analytical representations. The resulting potentials contain 70-76 freely adjusted parameters and represent the ground state PES up to 40000 cm(-1) above the equilibrium energy with a standard deviation of 100-107 cm(-1) without any important artifacts. One of the model surfaces is further empirically refined to match the bond dissociation energy D-0 for HOOH --> 2OH. The potentials are designed for energy regions accessible by vibrational fundamental and overtone spectroscopy including the dissociation channel into hydroxyl radicals. Characteristic properties of the model surfaces are investigated by means of stationary point analyses, torsional barrier heights, harmonic frequencies, low-dimensional cuts and minimum energy paths for dissociation. Overall good agreement with high-level ab initio calculations, especially for the CASPT2 based potentials, is achieved. The drastic change in geometry at intermediate O-O distances, which reflects the transition from covalent to hydrogen bonding, is reproduced quantitatively. We calculate fully 6D anharmonic zero point energies and ground state torsional splittings with the diffusion quantum Monte Carlo method in perfect agreement, within statistical error bars, with experiment for the CASPT2 based potentials. Variational vibrational calculations in the (4+2)D adiabatic approximation yield energy levels and torsional splittings from the ground state up to predissociative states, satisfactorily reproducing the experimental transition wavenumbers. (C) 1999 American Institute of Physics. [S0021-9606(99)30205-1].

A new six-dimensional analytical potential up to chemically significant energies for the electronic ground state of hydrogen peroxide

The OH stretching fundamental band system of carboxylic acid dimers is studied using acetic acid and its isotopomers as a model system. Comparing experimental jet spectra with multidimensional quantum mechanical calculations the origin of the extremely broad vibrational band structure (Δν≈800 cm−1) is found in strong anharmonic resonances involving the OH stretching vibration. Within an adiabatic picture of hydrogen bonding a new monomers-in-dimers model allows to analyze the observed vibrational band structure in terms of the anharmonic quantum dynamics of the CO2H functional group. The results are discussed in terms of the time-dependent population dynamics and its implications for the mode-specificity of the vibrational predissociation of the hydrogen bonds. On a subpicosecond time scale the intramolecular vibrational energy redistribution of the dimer remains effectively localized within the six-dimensional manifold of the internal vibrations of the carboxyl group, conserving its local CS symmetry.

A monomers-in-dimers model for carboxylic acid dimers

The FIR rotational spectrum of gaseous CH35ClF2 (chlorodifluoromethane, CFC22) is analysed improving vibrational ground state parameters for J up to 80. A new set of rotational and centrifugal distortion constants is given for the C-Cl stretching fundamental v 4. The Coriolis coupling between the two C-F stretching fundamentals v 3 and v 8 is investigated. An effective coupling constant ξ a = 0·38 ± 0·03 cm-1(ξ a 3,8 = 0·56 ± 0·04) was obtained from a rotational analysis of the high resolution FTIR room temperature spectrum (apodized experimental bandwidth: 0·005 cm-1). A ratio of 2 : 1 was estimated for the squares of the vibrational transition moments for the fundamentals v 3 and v 8. A fortran program for simulations and least squares refinements of asymmetric top spectra is described, which allows for the inclusion of any kind of rovibrational coupling between several vibrational states.

David Luckhaus

Papers

6D vibrational quantum dynamics: Generalized coordinate discrete variable representation and (a)diabatic contraction

A new six-dimensional analytical potential up to chemically significant energies for the electronic ground state of hydrogen peroxide

A monomers-in-dimers model for carboxylic acid dimers

The far infrared pure rotational spectrum and the Coriolis coupling between v 3 and v 8 in CH35ClF2

Mode selective stereomutation tunnelling in hydrogen peroxide isotopomers