The study and understanding of motion of the atoms in a molecule, i.e. rotation, vibrations, dissociation, reaction, etc., is a central problem in the molecular sciences : not only in chemistry but also in astro- and atmospheric-physics and in biology. Interpretation of experiments probing the fine details of molecular behaviour often requires computer simulations. Typical examples are found in the expanding field of femtochemistry which usesultra short laser pluses to "watch" the atoms move. The computation of the motion, called the solution of the dynamical problem, links observables such as cross sections, reaction rates, spectra, and information on internal energy transfer to the underlying molecular properties.
The correct theory to describe the dynamics is quantum dynamics.Due to its complexity, however, a full quantum-mechanical treatment is possible only for rather small molecules. For larger molecules, classical mechanics can be used, although it ignores important features such as interferences and tunneling, and suffers from the so called zero-point problem. Much effort has been invested over the last decades to improve the situation and make quantum dynamical simulations possible for larger molecules. This requires not only bigger computers, but, more, importantly, better algorithms.
MCTDH ’Multi-Configuration Time-Dependent Hartree’
A method that shows much promise is the multi-configuration time-dependent Hartree (MCTDH) method. MCTDH is an efficient method for propagating wave packets and has been developed by members of the
To demonstrate the applicability of MCTDH to large systems, it should be mentioned that converged calculations on a system (the pyrazin molecule) with two coupled electronic states and 24 nuclear degrees of freedom have been performed.
A full description can be found in the MCTDH Homepage
I have organized the second workshop "High dimensional quantum dynamics : challenges and opportunities"
In continuation of the first workshop "High dimensional quantum dynamics : challenges and opportunities" which took place in Leiden in September 2005, we have brought together scientists who are working on MCTDH, also with scientists who are exploring the use of other high-dimensional quantum dynamics methods, to discuss challenging new applications and to identify new ways of improving and extending the quantum dynamical toolbox.
Derivation of general and simple exact or constrained kinetic energy operators .
The convergence of the MCTDH method is determined by the correlation between coordinates. A set of coordinates which minimizes the correlation will improve the convergence of MCTDH, while a coordinate set which- introduces strong artificial correlations, i. e. due entirely to the unsuitable choice of coordinates, will slow down convergence. Let us consider an example. For small amplitude motions around a well defined equilibrium geometry, the vibrations are often rather harmonic. The well known normal-mode rectilinear coordinates then make the Hamiltonian operator almost separable — it is exactly separable for infinitesimal delocalizations — and the use of these coordinates will be optimal in a low energy domain. However, for more floppy systems exhibiting two or several minima, or at higher excitation energies, the vibrational amplitudes become larger and the rectilinear normal coordinates cease to describe the motion in a natural way. This introduces strong artificial correlations. In such situations, in particular when studying scattering, dissociation or isomerisation problems, the use of appropriate internal coordinates becomes important. In general curvilinear coordinates, involving angles, are the natural choice, as they usually lead to a more separable and hence less artificially correlated Hamiltonian operator.
Unfortunately, the use of curvilinear coordinates often leads to very involved expressions of the kinetic energy operator (KEO) which need to be derived for a particular system and are not easy to generalize. This is in contrast to the rectilinear coordinates which simplify the mathematical formulation of the same operator in a systematic way. The problem is not primarily to derive a formula for the KEO. An algorithmic program such as MATHEMATICA can be used to analytically evaluate the operators. A numerical computation of the action of the kinetic operator is also feasible and several contributions have been made in this direction. For efficient MCTCH calculations, the crucial point is to find a form which is (i) as compact as possible and (ii) of the required product form. The polyspherical approach, developed in a series of papers (see my list of publications) accomplishes both of these requirements. It is a general formulation of the exact KEO of an N-atom system. The approach received its name, polyspherical, because the operators are often eventually expressed in terms of spherical coordinates. This approach, however, also possesses properties which can be (and have been) exploited for other kinds of coordinates than polyspherical ones.
The polyspherical approach is characterized by the following properties : (i) It explicitly provides rather compact and general expressions of the exact kinetic energy operator including rotation and Coriolis coupling and avoids the use of differential calculus when deriving these operators. (ii) Within this approach it is very easy to find a spectral basis set (e.g. a basis set of spherical harmonics) which discards all singularities which may occur in the KEO. (iii) General expressions for the KEO are explicitly provided in two different forms. There is a large freedom in choosing the underlying set of vectors, they may be of Jacobi, Radau, Valence, satellite type, or a combination of of these. (iv) When polyspherical coordinates are used, the KEO is always separable : i.e. it can be written as a sum of products of monomodal operators. This property is, of course, very important for MCTDH and the resulting operators have been used in several applications with MCTDH.
Intramolecular vibrational-energy redistribution and infrared spectroscopy
We have developed a new field of applications of MCTDH : Intramolecular vibrational-energy redistribution and infrared spectroscopy.
Knowledge about the competition and the time scales of intramolecular vibrational energy redistribution (IVR) of selectively vibrationally excited molecules on a molecular level is essential for the understanding of rates, pathways, and efficiencies of chemical transformations. IVR studies aim at answering questions such as : starting from a well defined initial excitation of a molecule, where does the energy go ? How long does it take ? In statistical theories of Rice and Marcus, IVR is assumed to be rapid and complete. It is indeed sometimes observed that the IVR time scale is linked to the state density according to the Fermi golden rule and that the energy will be rapidly redistributed in a statistical way through all vibrational modes.
However, our understanding of IVR has since then been substantially refined by the success of numerous experimental investigations. New tools (among which, lasers, molecular beams and double resonance techniques) have indeed allowed overcoming the difficulties that plagued early experiments, most notably the impossibility to prepare and study a well defined initial state. Short pulse lasers can directly follow this flow of energy by exciting a specific vibrational motion and monitoring the transient response of the molecule. In the light of the many experimental findings for various molecules in the gas phase, it appears that neither the fast nor the statistical character of the IVR process should be taken for granted. It can even happen that IVR proceeds via very specific pathways. If these specific channels lead to a important processes, it then opens the way to the induction of chemical reactions by means of lasers. Indeed, one dream of chemists is to employ laser sources to drive selective chemical reactions in a way that only the desired products occur and it appears to be coming closer to reality. This field also offers new possibilities for the understanding of biological phenomena such as the elementary steps of vision and photosynthesis. Indeed, life seems to exploit extremely efficient and selective pathways to induce the conversion of light into chemical energy.
Much theoretical effort must thus be directed toward the development of more refined models to study systems with less drastic approximations than before. Such general methods of molecular quantum dynamics coupled with quantum chemistry calculations could predict the vibrational states leading to a desired reaction path. Pioneering works have been carried out in the 90’s and are technically based on the determination of an active space in which the dynamics is studied. The active space is extracted from the primitive space by the wave operator method based on the Bloch formalism. This approach has been applied by R. Wyatt, C. Iung and o-workers to perform full 9- and 30-dimensional studies of the spectroscopy and the Intramolecular Vibrational Energy Redistribution (IVR) of HCF3 and benzene respectively.
More recently, we have demonstrated that the Heidelberg package of the Multi-Configuration Time Dependent Hartree (MCTDH) algorithm is an efficient tool to investigate the IVR in several molecules. Here, we confine ourselves to IVR studies in molecules in their ground electronic state. In order to begin a systematic study of the IVR process, there are important reasons for focusing on the electronic ground state. First, more spectroscopic data for an individual molecule are, in general, available than for electronic excited states and these data are very important to correctly interpret the IVR process. Second, in electronic excited states, there are others several competitive processes in the overall energy redistribution (involving the internal conversion to other states) which drastically complicate the interpretation of the IVR simulations. Moreover, in the ground state, certain vibrations are related to specific chemical bonds (in particular the stretching modes of vibration) in contrast to the vibrational modes in excited states which involve the motion of many atoms and so already start out spatially delocalized. For all these reasons, it is more reasonable to start a systematic prospect of laser-enhanced, mode-selective chemistry with molecules in their electronic ground state.
As aforementioned, high resolution vibrational spectroscopy is essential for understanding flow in molecules. Consequently, IVR studies are directly linked to the simulations of infrared-spectra. The natural way to generate eigenvalues from the knowledge of a propagated wave packet is to Fourier-transform the autocorrelation function. However, this approach is useful when a high resolution of individual lines is not desired. When individual lines are to be resolved, the energy levels may be converged with MCTDH by analysing the autocorrelation function with the aid of the filter-diagonalization method. Recently, a very important progress has been made with the development of the improved relaxation method. The latter allows the convergence with a high accuracy of the eigenvalues and the eigenstates for rather large systems. For instance a comprehensive calculation of an IR spectrum by improved relaxation has been recently achieved for 6D problems, namely H2CS, and HONO. Several 15D eigenstates have also been converged for the Zundel cation H5O2+.
We have studied IVR in relatively large systems such as a 9 dimensional model of Toluene, Fluoroform, HFCO, DFCO, HONO, and H2CS in their full dimensionality (see my list of publications). Consequently, the Heidelberg package of MCTDH could offer a precious framework for a synergy between experimenters and theoreticians in the field of IVR and infrared spectroscopy.