Quantum many-body dynamics approximation

There are two different aspects to these approximations. One consists in the approximate treatment of the underlying many-body quantum dynamics the other, in the statistical approach to observable average quantities. An exlmistive discussion of different approaches would go beyond the scope of this introduction. Some of the most important aspects are discussed in separate chapters (see chapter A3.7. chapter A3.11. chapter A3.12. chapter A3.131. 

The difficulty in simulating the full quantum dynamics of large many-body systems has stimulated the development of mixed quantum-classical dynamical schemes. In such approaches, the quantum system of interest is partitioned into two subsystems, which we term the quantum subsystem, and quantum bath. Approximations to the full quantum dynamics are then made such that... 

The computation of the equilibrium properties of quantum systems is a challenging problem. The simulation of dynamical properties, such as transport coefficients, presents additional problems since the solution of the quantum equations of motion for many-body systems is even more difficult. This fact has prompted the development of approximate methods for dealing with such problems. 

As a consequence of the breakdown of the independent particle approximation, it then emerged that the quantisation of individual electrons was not completely reliable. This was referred to in the classic texts on the theory of atomic spectra [309] as a breakdown in the I characterisation, and it manifests itself in the appearance of extra lines, which could not be classified within the independent electron scheme. The proper solution would, of course, be to revisit the initial theory and correct its inadequacies by a proper understanding of the dynamics of the many-electron problem, including where necessary new quantum numbers to describe the behaviour of correlated groups of electrons. Unfortunately, this plan of action cannot be followed through it would require a deeper understanding of the many-body problem than exists at present (see, e.g., chapter 10 for some of the difficulties). 

It should also be noted that full symmetry unconstrained structural relaxation is essential before the theoretical determination of any physical properties. In quantum mechanical simulations, use of a Hamiltonian with only one rr-electron orbital is untenable for dynamical relaxations even for graphite. Most authors attempt to circumvent this problem by using classical many body potentials for obtaining relaxation while still making use of the ir-electron orbital approximation for conductivity calculations. Use of two completely different methods for the same system, can introduce inconsistency in the prediction of physical properties. 

The choice of the adjustable parameters used in conjunction with classical potentials can result to either effective potentials that implicitly include the nuclear quantization and can therefore be used in conjunction with classical simulations (albeit only for the conditions they were parameterized for) or transferable ones that attempt to best approximate the Born-Oppenheimer PES and should be used in nuclear quantum statistical simulations. Representative examples of effective force fields for water consist of TIP4P (Jorgensen et al. 1983), SPC/E (Berendsen et al. 1987) (pair-wise additive), and Dang-Chang (DC) (Dang and Chang 1997) (polarizable, many-body). The polarizable potentials contain - in addition to the pairwise additive term - a classical induction (polarization) term that explicitly (albeit approximately) accounts for many-body effects to infinite order. These effective potentials are fitted to reproduce bulk-phase experimental data (i.e., the enthalpy at T = 298 K and the radial distribution functions at ambient conditions) in classical molecular dynamics simulations of liquid water. Despite their simplicity, these models describe some experimental properties of liquid... 

Quantum dynamical calculations are reviewed, in different approximations and for sudden and adiabatic energy transfer. The inverse scattering problem is briefly covered, as well as the many-body approach to molecular collisions. 

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