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Many-particle systems, computational

As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics - the Schrodinger equation - lies beyond the potentialities of modem mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. [Pg.109]

Although we have the theoretical tool in the time-dependent Schrodinger equation to deal with the dynamics of the nuclei and the electrons, it is still far beyond the capacities of modem computers to apply it to many-particle systems. Instead of waiting for the computers to become several orders of magnitude faster, we can investigate other possibilities to include the electrons (and other quantum particles) in current computer simulations of molecular systems by compromising between theory and practice. [Pg.98]

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. [Pg.271]

Another field where dielectric continuum models are extensively used is the statistical mechanical study of many particle systems. In the past decades, computer simulations have become the most popular statistical mechanical tool. With the increasing power of computers, simulation of full atomistic models became possible. However, creating models of full atomic detail is still problematic from many reasons (1) computer resources are still unsatisfactory to obtain simulation results for macroscopic quantities that can be related to experiments (2) unknown microscopic structures (3) uncertainties in developing intermolecular potentials (many-body correlations, quantum-corrections, potential parameter estimations). Therefore, creating continuum models, which process is sometimes called coarse graining in this field, is still necessary. [Pg.20]

So far, our discussion has focussed on stationary quantum chemical methods, which yield results for fixed atomic nuclei, i.e. for frozen molecular structures like minimum structures on the Born-Oppenheimer potential energy surface. Processes in supramolecular assemblies usually feature prominent dynamical effects, which can only be captured through explicit molecular dynamics or Monte Carlo simulations [95-98]. Molecular dynamics simulations proved to be a useful tool for studying the detailed microscopic dynamic behavior of many-particle systems as present in physics, chemistry and biology. The aim of molecular dynamics is to study a system by recreating it on the computer as close to nature as possible, i.e. by simulating the dynamics of a system in all microscopic detail over a physical length of time relevant to properties of interest. [Pg.433]

We note that, even if we start here from the same truncated basis B = B, B2,. . . , Bm as in the EOM method, the results are not necessarily the same, since (2.16) is a single-commutator secular equation whereas (1.50) is a double-commutator secular equation. It should be observed, however, that the column vectors d obtained by solving (2.15) are optimal in the sense of the variation principle, whereas this is not necessarily true for the vectors obtained by solving (1.49). In the following analysis, we will discuss the connection between these two approaches in somewhat greater detail. Since the variation principle (2.10) would provide an optimal approximation, the essential question is whether the theoretical and computational resources available today would permit the proper evaluation of the single-commutator matrix elements defined by (2.13) for a real many-particle system this remains to be seen. [Pg.303]

There is little question that the double-commutator expressions in (3.46) and (3.47) greatly simplify the algebraic and computational aspects of the calculation of the m2 matrix elements occurring in the equation system (3.42) see Ref. 9. The theoretical results as to excitations, ionizations, etc., of certain many-particle systems are in such good agreement with experimental experience that one can probably only expect that part of this agreement will be lost, if one tries to refine the theory. [Pg.327]

The history of quantum chemistry is very closely tied to the history of computation, and in order to place Carl Ballhausen s work in context, it is relevant to review the enormously rapid development of computing during the twentieth century. The fundamental equations governing the physical properties of matter, while deceptively simple to write down, are notoriously difficult to solve. Only the simplest problems, for example the harmonic oscillator and the problem of a single electron moving in the field of a fixed nucleus, can be solved exactly. However, no solutions to the wave equations for interacting many-particle systems such as atoms or molecules are known, and it is quite possible that no simple solutions exist. In 1929, P.A.M. Dirac summarized the position since the discovery of quantum theory with his famous remark ... [Pg.54]

With the development of material science, fine chemistry, molecular biology and many branches of condensed-matter physics, the problem of how to deal with the quantum mechanics of many-particle systems formed by thousands of electrons and hundreds of nuclei has attained relevance. An alternative of ab-initio methods is the density functional approach [8-10] which gives results of an accuracy comparable to ab-initio methods. The density-functional method bypasses the calculation of the n-electron wave-function by using the electron density p r). The energy of a many-electron system is a unique function of electron density. The computational work grows like instead of in HF. [Pg.94]

There used to be two realities in the world of physics Experiment and Theory. Now there are three, and the third one is The Computer. In the community of physicists outside of the computer-simulation enclave, there is a good deal of skepticism about computer simulations of many-particle systems. This skepticism is certainly justified at the present time. Nevertheless, in my opinion, computer simulation will affect the progress of physics in a profound way. The many-body problems that we worked on for decades will finally yield to computer simulation. This does not mean that many-body problems will suddenly become simple the complications of these problems will appear in a new form. The question will be, how are computer simulations to be interpreted in terms of the mathematically posed problem, or in terms of physical reality. A computer-generated many-particle process contains an enormous amount of information, and the challenge will be to extract the information we want, leaving computer artifacts behind. In other words, the computer may have the answer, but we will have to figure out the question. [Pg.521]

For additional studies of the ultrafast photodissociation of small alkali clusters, the support of detailed qualitative and quantitative predictions of the stability and photoinduced fragmentation of these many-particle systems is essential. Knowing that even for the three-body system Nas, state-of-the-art time-dependent quantum dynamical simulations are extremely costly, requiring considerable computer time, the application of other concepts is necessary. In the near future, the density matrix formalism might be an appropriate approach here. [Pg.180]

However, once this fundamental difference in the role played by probabilistic elements in quantum and clas.sir mechanics is accepted, there is essentially no operational diffcnuicc in the way in wlikJi one would compute any property of a thermal many-particle system regardless of whether the system is treated at the quantum level. Our discussion below will reveal that in any case the key quantity required is the so-called partition function, which conveys information about the probability with which a particular (quantum or classic) microstate is realized. [Pg.37]


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