Semi-classical simulations

Computer simulations are methods addressed to perform computer experimentation . The importance of computer simulations rests on the fact that they provide quasi-experimental data on well-defined models. As there is no uncertainty about the form of the interaction potential, theoretical results can be tested in a way that is usually impossible with results obtained by experiments on real liquids. In addition, it is possible to get information on quantities of no direct access to experimental measures. 

There are basically two ways of simulating a many-body system through a stochastic process, sueh as the Monte Carlo (MC) simulation, or through a deterministic process, such as a Molecular Dynamics (MD) simulation. Numerical simulations are also performed in a hybridized form, like the Langevin dynamics which is similar to MD except for the presence of a random dissipative force, or the Brownian dynamics, which is based on the condition that the acceleration is balanced out by drifting and random dissipative forces. 

Both the MC and the MD methodologies are used to obtain information on the system via a classical statistical analysis but, whereas MC is limited to the treatment of static properties, MD is more general and can be used to take into account the time dependence of the system states, allowing one to calculate time fluctuations and dynamic properties. 

In the following, we shall briefly describe the main features of MD and MC methodologies, focusing the attention to their use in the treatment of liquid systems. 

Molecular Dynamics is the term used to refer to a technique based on the solution of the classical equation of motion for a classical many-body system described by a many-body Hamiltonian H. 

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Semi-classical simulations are not aimed at providing an exhaustive and definitive understanding of photoreactivity. They should be improved on different fronts to become more predictive. Gaussian-based quantum dynamics could also provide a more accurate description of the quantum effects involved in such processes. However, semi-classical simulations are useful to generate ideas for designing new experiments, which in turn can be designed to validate specific aspects of the... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

Each of the semi-classical trajectory surface hopping and quantum wave packet dynamics simulations has its pros and cons. For the semi-classical trajectory surface hopping, the lack of coherence and phase of the nuclei, and total time per trajectory are cons whereas inclusion of all nuclear degrees of freedom, the use of potentials direct from electronic structure theory, the ease of increasing accuracy by running more trajectories, and the ease of visualization of results are pros. For the quantum wave packet dynamics, the complexity of setting up an appropriate model Hamiltonian, use of approximate fitted potentials, and the... 

We have therefore shown that adiabatic surfaces can be said to cross off the real coordinate axes, and indeed if the classical equations of motion are solved in complex coordinate space then it is possible to simulate non-adiabatic processes. This can be considered as the basis of the Stuckelberg semi-classical approach to non-adiabatic transitions in atom-atom collisions (64) and it has been recently extended to more degrees of freedom (65). Moreover the actual form of potential surfaces in the complex plane has been obtained by direct calculation (66). 

A theoretical understanding of the diffusion of hydrocarbons through the porous catalyst layer (see Fig. 2.45) may be obtained by simulations using semi-classical molecular dynamics (as in Fig. 2.3). Such calculations have been performed for the penetration of various hydrocarbons through AljOj catalysts with and without Pt insertions (Szczygiel and Szyja, 2004). As indicated in Fig. 2.46, it is found that fuel transport depends on both cavity structure and the adsorption on internal catalyst walls. 

In their simulations, L6vesque et used a standard Leimard-Jones interaction potential between hydrogen molecules, and included the effect of quadrupolar interactions by adding a Coulomb interaction term in which each hydrogen molecule is represented by three effective charges q (q = 0.4829e at the position of the protons and q = -e at the centre of mass of the molecule). The adsorbate-adsorbent interaction was modeled with a standard Lennard-Jones potential. In order to partially account for quantum effects at 77 K, a semi-classical approach based on the Feynman-Hibbs effective potential was used ... 

Stilbene and Related compounds. As a result of calculations on the isomerization of ethene, it has been suggested that there is a need for the reconsideration and refinement of the photoisomerization mechanism of stilbene . Both a semi-classical approachand simulations have been employed to study the cisprans-isomQnzdiiion of stilbene. Simulations have examined the photoisomerization of stilbene, and ab initio quantum chemistry has also been utilized to study this system. 

A group of theoretical methods exists where the electronic wavefuntion is computed, and the atomic nuclei are propagated (using classical equations of motion). The Car-Parrinello MD method is one of this type [22-24]. These methods he between the extremes of the classical and ab initio methods, as they include some (quantum) electronic information and some (classical) dynamics information. These methods are called ah initio or first principles MD if you come from the classical community and semi-classical MD if you come firom the quantum community [9], Ah initio MD methods are far more expensive and cannot simulate as many molecules for as long as the classical simulations, but they are more flexible in that structures are not predetermined and information on the electronic structure is retained. Semi-classical MD can be carried out under periodic boundary conditions and thus the local liquid environment, and any extended bonding network, vyill be present. These methods hold a great deal of promise for the future study of ionic liquid systems, the first such calciilations on ionic liquids were reported in 2005 [21,25]. 

Here T represents the configurational integral of the associated potential, the square brackets indicate an independent simulation, and the superscript SCL denotes a QFH semi-classical approximation (Eq. 18) to m or Since we are interested... 

We have implemented the PIMC method with two approaches based on semi-classical beads (SCB-QFH, SCB-TI) and MSMC to compute more precise quantum virial coefihdents for helium. The SCB results agree well with CB results as they are within statistical uncertainties of each other. The decomposition algorithm of Shaul et al. [18] was implemented to achieve better efficiency of quantum virial coefficient calculations. We observed similar trends in decompositions of simulations in our SCB based approaches as was the case for the CB approach. For lower temperatures, the approximation to u for finite P is... 

Parallel to the developments achieved in methodology and hardware, the conventional methods and some of the new approaches have been employed to study several types of photoinduced processes which are relevant mainly in biology and nanotechnology. In particular, important contributions have been made related to the topics of photodissociations, photostability, photodimerizations, photoisomerizations, proton/hydrogen transfer, photodecarboxylations, charge transport, bioexcimers, chemiluminescence and bioluminescence. In contrast to earlier studies in the field of computational photochemistry, recent works include in many cases analyses in solution or in the natural environment (protein or DNA) of the mechanisms found in the isolated chromophores. In addition, semi-classical non-adiabatic molecular dynamics simulations have been performed in some studies to obtain dynamical attributes of the photoreactions. These latter calculations are however still not able to provide quantitative accuracy, since either the level of theory is too low or too few trajectories are generated. Within this context, theory and hardware developments aimed to decrease the time for accurate calculations of the PESs will certainly guide future achievements in the field of photodynamics. 

Using empirical potentials, solute-solvent and solvent-solvent interactions can be modeled, allowing many aspects of solvent clusters to be studied semi-quantitatively by classical simulation techniques (Monte Carlo, molecular dynamics). As a specific example we discuss two cluster/sub-strate-type "phase" transitions of the carbazole Ar cluster. This is the smallest system that exhibits all types of two-dimensional transitions observed so far in MC simulations, and, at the same time, allows a direct and intuitive interpretation of the associated structural changes [9-14]. 

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