Quantum Dynamical Computations

Kulander, K.C. (Ed.) (1991). Time-dependent methods for quantum dynamics, Computer Phys. Comm. 63, 1-577. 

A more advanced and, in principle, more accurate computation of dispersion interactions between mesoscopic bodies is based on quantum electrodynamics of continuous media, as presented in the Lifshitz theory. However, the simpler treatment, given above, in which the interaction energy is obtained by pairwise summation of the energies between all atoms of the interacting particles does not deviate more than, say, 30% from the absolute value. Because the materials in the system are often not sufficiently well defined to allow for accurate quantum-dynamic computations, it is reasonable to utilize the more simple theory. 

K.C. Kulander, ed., Time-Dependent Methods for Quantum Dynamics, Comput. Phys. Commun. 63 (1991) Nos. 1-3. 

As seen in second part of these thesis, the development of efficient quantum dynamical computational methods also opened the door to the study of the laser control of polyatomic molecular systems. In a first application, we used a simplified model Hamiltonian describing the excited state dynamics of pyrazine to investigate the laser control of its ultrafast radiationless decay by a strong non-resonant laser... 

Tuckerman M E and Hughes A 1998 Path integral molecular dynamics a computational approach to quantum statistical mechanics Classical and Quantum Dynamics In Condensed Phase Simulations ed B J Berne, G Ciccotti and D F Coker (Singapore World Scientific) pp 311-57... 

Another topic that received increasing attention is the incorporation of quantum methods into dynamic simulations. True quantum dynamics for hundreds of particles is beyond any foreseeable computational capability, and only approximations are viable. We should distinguish ... 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

The only feasible procedure at the moment is molecular dynamics computer simulation, which can be used since most systems are currently essentially controlled by classical dynamics even though the intermolecular potentials are often quantum mechanical in origin. There are indeed many intermolecular potentials available which are remarkably reliable for most liquids, and even for liquid mixtures, of scientific and technical importance. However potentials for the design of membranes and of the interaction of fluid molecules with membranes on the atomic scale are less well developed. 

Warshel A, Chu ZT (2001) Nature of the surface crossing process in bacteriorhodopsin computer simulations of the quantum dynamics of the primary photochemical event. J Phys Chem B 105 9857... 

Considering any of these paradigms, a minimal goal for toy models would be to manipulate the quantum dynamics of a small number of spin levels , and that requires a known and controlled composition of the wavefunction, sufficient isolation and a method for coherent manipulation. As illustrated in Figure 2.13, the first few magnetic states of the system are labelled and thus assigned qubit values. The rest of the spectrum is outside of the computational basis, so one needs to ensure that these levels are not populated during the coherent manipulation. 

The results of the previous section have already established that classical chaos and quantum mechanics are not incompatible in the macroscopic limit. The question then naturally arises whether observed quantum mechanical systems can be chaotic far from the classical limit This question is particularly significant as closed quantum mechanical systems are not chaotic, at least in the conventional sense of dynamical systems theory (R. Kosloff et.al., 1981 1989). In the case of observed systems it has recently been shown, by defining and computing a maximal Lyapunov exponent applicable to quantum trajectories, that the answer is in the affirmative (S. Habib et.al., 1998). Thus, realistic quantum dynamical systems are chaotic in the conventional sense and there is no fundamental conflict between quantum mechanics and the existence of dynamical chaos. 

The site http //www.nyu.edu/pages/mathmol/modules/water/info water.html hosts a nice discussion of water, including two short video clips (1) the quantum-mechanically computed movement of two water molecules united by means of a single hydrogen bond, at http //www.nyu.edu/pages/mathmol/modules/water/dimer.mpg-, (2) a short film of several hundred water molecules dancing within a cube at http //www. nyu. edu/pages/mathmol/modules/water/water dynamics.mpg. 

The two avenues above recalled, namely ab-initio computations on clusters and Molecular Dynamics on one hand and continuum model on the other, are somewhat bridged by those techniques where the solvent is included in the hamiltonian at the electrostatic level with a discrete representation [13,17], It is important to stress that quantum-mechanical computations imply a temperature of zero K, whereas Molecular Dynamics computations do include temperature. As it is well known, this inclusion is of paramount importance and allows also the consideration of entropic effects and thus free-energy, essential parameters in any reaction. 

Concerning quantum chemical computations, we have used the MOLE-COLE program [18a], for HF and MP2 type computations. The Molecular Dynamics simulations with analytical force fields have been performed with the DINAMICA program [18b], The MOLECOLE-DFT program [18c] has been used for both the DFT energy minimization and for the DFT-Molecular Dynamics. 

Another major, future advance in the quantum chemical computation of potential energy surfaces for reaction dynamics will be the ability to routinely compute the energies of molecular systems on the fly . The tedious and time-consuming process of fitting computed quantum chemical values to functional forms could be avoided if it were possible to compute the PES as needed during a classical trajectory or quantum dynamics calculation. For many chemical reactions, it should be practical in the near future to prudently select a sufficiently rapid and accurate electronic structure method to facilitate dynamics computations on the fly. 

Table 6.2 Tests of Variational Transition State Theory by Comparing with Exact Quantum Calculations (Extracted from Allison, T. C. and Truhlar, D. G. Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunneling, in Thompson, D. L., Ed. Modem methods for multidimensional dynamics computations in chemistry, World Scientific, Singapore 1998. pp 618-712. This reference quotes results on many more reactions and BO surfaces over broad temperature ranges.)The numbers in the table are ratios of the results of the approximate calculation to the quantum calculation, all at 300 K...

S. C. Althorpe, P. Soldn, and G. G. Balint-Kurti, eds., Time-Dependent Quantum Dynamics CCP6 Collaborative Computational Project on Heavy Particle Dynamics (Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, U.K., 2001). 

These experimental and numerical developments have posed a challenge to the theorist. Given the complexity of the phenomena involved, is it still possible to present a theory which provides the necessary concepts and insight needed for understanding rate processes in condensed phases Although classical molecular dynamics computations are almost routine, real time quantum molecular dynamics are still largely computationally inaccessible. Are there alternatives Do we understand quantum effects in rate theory These are the topics of this review article. 

In Section 9.4, we present an alternative approach to the TOR for multidimensional quantum dynamical investigations, in which the PT still plays a central role however, now the PT is combined with the concept of pseudomodes, which has found use in quantum optics [47 9], as well as in chemical physics [50-52]. This approach results to the EMDE method, which is a computationally powerful tool,... 

In this section, we present the application of the PT to problems in the time domain, when fused with the theory of pseudomodes [47,49,50], The result of this approach is the EMDE method [40 3] it is a computationally powerful tool for studying multidimensional quantum dynamics, including the effects of external electromagnetic fields of arbitrary coupling strength on the system under study. Below, we review the EMDE methodology and present its application to the IC process in 24-mode pyrazine, as well as to the quantum dynamics of the three-dimensional strong-held dissociation of the //+ molecular ion. 

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