Many-body systems, application Schrodinger equation

Unlike the AT-particle picture, which in principle leads naturally to the exact solution of the many-particle Schrodinger equation, the single-particle picture has led to the development of a number of different approximation schemes designed to address particular issues in the physics of interacting quantum systems. A particularly pointed example of this state of affairs is the strict dichotomy that has set in between so-called single-particle theories and canonical many-body theory[31, 32]. Each of the two methodologies can claim a number of successful applications, which tends to reinforce the perceived formal gap between them. 

With the development of material science, fine chemistry, molecular biology and many branches of condensed-matter physics the question of how to deal with the quantum mechanics of many-particle systems formed by thousands of electrons and hundreds of nuclei has attained unusual relevance. The basic difficulty is that an exact solution to this problem by means of a straight-forward application of the Schrodinger equation, either in its numerical, variational or perturbation-theory versions is nowadays out of the reach of even the most advanced supercomputers. It is for this reason that alternative ways for handling the quantum-mechanical many-body problem have been vigorously pursued during the last few years by both quantum chemists and condensed matter physicists. As a consequence of... 

In the previous sections we showed the results of several TDDFT calculations, most of them agreeing quite well with experiment. Clearly no physical theory works for all systems and situations, and TDDFT is not an exception. It is the purpose of this section to show some examples where the theory does not work. However, before proceeding with our task, we should specify what we mean by failures of TDDFT . TDDFT is an exact reformulation of the time-dependent many-body Schrodinger equation - it can only fail in situations where quantum-mechanics also fails. The key approximation made in practical applications is the approximation for the xc potential. Errors in the calculations should therefore be imputed to the functional used. As a large majority of TDDFT calculations use the ALDA or the adiabatic GGA, we will be mainly interested in the errors caused by these approximate functionals. Furthermore, and as we already mentioned in the previous section, there... 

Unfortunately, the determination of exact solutions of the SchrOdinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree-Fock model, which is the best independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. 

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