Solutions to the many-body problem

A central issue in statistical thermodynamic modelling is to solve the best model possible for a system with many interacting molecules. If it is essential to include all excluded-volume correlations, i.e. to account for all the possible ways that the molecules in the system instantaneously interact with each other, it is necessary to do computer simulations as discussed above, because there are no exact (analytical) solutions to the many-body problems. The only analytical models that can be solved are of the mean-field type. 

In 1965 Kohn and Sham used the variational principle of Hohenberg and Kohn to derive a system of one-electron equations which, like the Hartree approach, can be self-consistently solved. For this case, however the electron densities obtained from the orbitals (called Kohn-Sham orbitals) are an exact solution to the many-body problem (for a complete basis) given the density functional. Hence the task of determining the electronic energy is changed from calculating the full many-body wavefunction to determining the best approximation to the density functional. 

Ah, the crux of the problem, is it not Up until now, we ve just assumed we have some set of molecular orbitals i or Vu which we can manipulate at will. But how does one come up with even approximate solutions to the many body Schrodinger equation without having to solve it Start with the celebrated linear combination of atomic orbitals to get molecular orbitals (LCAO-MO) approximation. This allows us to use some set of (approximate) atomic orbitals, the basis functions which we know and love, to expand the MOs in. In the most general terms,... 

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

Such fundamental knowledge about the microscopic processes on the gas-solid interface is also necessary for optimization of many catalytic processes. A statistical mechanical approach, which enables the solution of the many-body problem constituted by the adsorbate layer on the catalytic surface, is essential in the case when lateral interactions between adatoms and molecules are significant. In such cases, non-ideal surface adlayer mixing is often important and the adsorbates form islands on the surface. Hence, microscopic simulations of catalytic processes are necessary to develop an ab-initio approach to kinetics in catalysis. 

One of the aims of this chapter, then, is to discuss the problem of calculating a property of a many-electron atom with suflicient precision so that the new physics of radiative corrections can be studied. The challenge to many-body theory is quite specific. As will be discussed below, properties of cesium, the atom in which the most accurate PNC measurement has been made [5] must be calculated to the fraction of a percent level to accurately study PNC and radiative corrections to it can this level in fact be reached by modern many-body methods While great progress has been made, the particular nature of this problem, in which relativity has to be incorporated from the start, and a transition between two open-shell states calculated in the presence of a parity-nonconserving interaction, has not permitted solution of the many-body problem to the desired level. It may well be that a reader of this chapter has developed techniques for some other many-electron problem that are of sufficient power to resolve this issue this chapter is meant to clearly lay out the nature of the calculation so that the reader can apply those techniques to what is, after all, a relatively simple system by the standards of quantum chemistry, an isolated cesium atom. 

A technique of the solid simulation starting from the first principles ah initio theory) is the subject of Chapter 8. We study milestones in solution of the many-body problem. We describe the density functional theory as an essence of the technique. The Kohn-Sham approach, pseudopotential method, iterative technique of calculations are described here. These methods enable one to determine and calculate the equilibrium structure of a solid quantitatively and self-consistently. 

One method for finding a numerical solution to a many-body problem is the Hartree-Fock method [3], devised in the late 1920s by Hartree [4] and refined by Vladimir Fock (though it wasn t until later that the equations were refined and implemented in computational code). Hartree-Fock methods are computationally very expensive, and inherent in the equations are a number of approximations that introduce artefacts in the computational model. These must be borne in mind when interpreting the results. Not least of them is the Bom-Oppenheimer approximation, which separates the nuclear and electronic wavefiinctions. ... 

These equations have been considered by Zwanzig [21]. He reduced the solution of the many-body problem to one equation... 

The solution of the many-body problem as defined in the previous section on classical mechanics, is numerically demanding. Considering, furthermore, that the primary effect of the surface atoms is assumed to be that of a heat bath maintained at a certain temperature, it is naturally to think that some approximate description would suffice. 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

Liquids are difficult to model because, on the one hand, many-body interactions are complicated on the other hand, liquids lack the symmetry of crystals which makes many-body systems tractable [364, 376, 94]. No rigorous solutions currently exist for the many-body problem of the liquid state. Yet the molecular properties of liquids are important for example, most chemistry involves solutions of one kind or another. Significant advances have recently been made through the use of spectroscopy (i.e., infrared, Raman, neutron scattering, nuclear magnetic resonance, dielectric relaxation, etc.) and associated time correlation functions of molecular properties. 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

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). 

Much progress has been made in other areas of quantum physics earlier in the absence of complete solutions of the many-body Schrodinger equation. In our view, until recently the lack of direct experimental information and direct unambiguous computational results have conspired with the high dimensionality and richness of mechanism in the IVR problem to inhibit the development of simple pictures of quantum intramolecular vibrational energy flow. 

If an approximate separation is not possible, the many-body problem can often be transformed into a pseudo one-particle system by taking the average interaction into account. For quantum mechanics, this corresponds to the Hartree-Fock approximation, where the average electron-electron repulsion is incorporated. Such pseudo one-particle solutions often form the conceptual understanding of the system, and provide the basis for more refined computational methods. 

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