One-body problem

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

Next, the effect of z on A IT through the transition matrix element Hoj is considered as follows for rigorous determination of IToi, all electrons in the system should be treated. However, for the sake of simplicity, we devote our attention only to the transferring electron the other electrons would be regarded as forming the effective potential (x) for the transferring electron (x the coordinate of the electron given from the ion center). This enables us to reduce the many-body problem to a one-body problem ... 

At this stage the assumption that the wave function ip can be factorized into com and relative-motion (rm) components, by defining E = Ecom + Erm, is commonly made. In terms of ip = body problem is decoupled into two one-body problems ... 

Instead of considering how the incorporation of a dopant ion perturbs the electronic structure of the crystal, we will face the problem of understanding the optical features of a center by considering the energy levels of the dopant free ion (i.e., out of the crystal) and its local environment. In particular, we shall start by considering the energy levels of the dopant free ion and how these levels are affected by the presence of the next nearest neighbors in the lattice (the environment). In such a way, we can practically reduce our system to a one-body problem. 

Corrections which depend on the mass ratio m/M of the light and heavy particles reflect a deviation from the theory with an infinitely heavy nucleus. Corrections to the energy levels which depend on m/M and Za are called recoil corrections. They describe contributions to the energy levels which cannot be taken into account with the help of the reduced mass factor. The presence of these corrections signals that we are dealing with a truly two-body problem, rather than with a one-body problem. 

By referring the motion to the centre of mass, the two-body problem has been reduced to a one-body problem of the vibrational motion of a particle of mass p against a fixed point, under the restraining influence of a spring of length R with a force constant k. 

It is easily seen from classical mechanics that the binary collision problem is mathematically equivalent to a one-body problem in which a body with the reduced mass... 

It may be shown quite generally, that such a two-body problem may be reduced to a one-body problem. We choose the centre of gravity of the two particles as the origin of co-ordinates 0 and determine the direction of the line joining m2 and m1 by the polar coordinates 9, tf). If then jq and r2 are the distances of the particles from 0, their polar co-ordinates will be rlt 9, and r2, tt—9, ir+ and further, r1+r2=r. The Hamiltonian function becomes... 

So far as the calculation is concerned it is immaterial whether we consider our problem as a one-body or as a two-body problem. In the first case we have a fixed centre of force, and the potential of the field of force is a function U(r) of the distance from the centre. In the second case we have two masses, whose mutual potential energy U(r) depends only on their distance apart they move about the common centre of gravity. As wc have shown generally in 20, the Hamiltonian function in polar co-ordinates is precisely the same for the two cases, if, in the one-body problem, the mass /x of the moving... 

Thus the electronic part of the present three-body problem is reduced to a one-body problem. But the function U(R) for the hydrogen-molecule ion still cannot be expressed in closed form in terms of well-known tabulated functions its values must be obtained by numerical integration of a differential equation. More insight into the three contributions to U(R) comes from examining simplified models. 

Reduction of the two-body electron-nucleus Schrodinger equation to center of mass coordinates leads to an equivalent one-body problem where the electron mass is replaced by the reduced mass fi — mM/(M -f m). The effect of this replacement is to scale the infinite-mass Rydberg constant by fi/m. The corresponding shift of energy from the infinite-mass value E i is... 

The long range attractive potential for a diatomic dissociation is given by V r) = r ", where n is the interaction parameter. Typically = 6 for a neutral molecule (recall the Lennard-Jones 6-12 potential). On the other hand, for an ionic dissociation into an ion and a polarizable neutral, n = 4. The Hamiltonian for such a two-body central force system can be expressed in the center of mass as a one-body problem with reduced mass p. and relative velocity v. In polar coordinates, this takes the form [(Eq. 2.15)]... 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

The gravitational field is described in general relativity by the set of equations (4.11). The right hand side depends on the description of matter in the system of interest and the corresponding solution consists of finding that form of the fundamental tensor that satisfies (4.11). The first successful solution of cosmological interest, obtained by Schwarzschild, is text-book material, described in detail by Adler et al. (1965). The time-independent spherically symmetric line element is of particular importance as a model of the basic one-body problem of classical astronomy. This element, of the form ... 

We use it every day, although we do not call it a mean field approach. Indeed, if we say 1 niU visit my aunt at noon, because it is easier to travel out of rush hours," or / avoid driving through the center oftoivn, because of the traffic jams, in practice we are using the mean field method. We average the motions of aU citizens (including ourselves ) and we get a map (temporal or spatial), which allows us optimize our own motion. The motion of our fellow-citizens disappears, and we obtain a one-body problem. 

The central idea of all approximate methods is the reduction of the many-body problem into a one-body problem. The one electron under investigation is assumed to move in an effective potential due to the nuclei and all the other electrons. Naturally the success or failure of the approximation hinges on the construction of the potential. This problem will receive increasing attention when more refined calculations are desired. 

The idea of the mean-field theory is to focus on one particular particle in the system. The method singles out only those fluctuations that occur within the cell and neglects the effects of fluctuations beyond the cell. It reduces the many-body statistical mechanics problem into a one-body problem. 

The self-consistent field (SCF) which ignores electron correlation effects is usually the starting point of the ab initio methods. The method treats a many-body problem (n-electron problem) as several one-body problems, in an effective field, created by rest of the - 1 bodies (electrons). The resulting equation, the Hartree-Fock equation, is usually solved iteratively to self-consistency. 

We should mention that the main idea behind MFT is to replace all interactions to any one body with an average or effective interaction. This reduces any multibody problem, which is generally very difficult to solve exactly, into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a relatively low cost even at the expense of loosing some accuracy. 

Assumption (7.5.8) permits to replace the iV-body problem involved in the calculation of the partition function of the system by a one-body problem. It corresponds to a procedure often used in the solution of quantum mecbanical many-body problems. As a consequence of (7.5.8), correlations in the motions of neighbouring molecules are neglected. 

Even witti the simplification on the nuclear parts, the solution of an N-electron Hamiltonian remains too complicated due to the correlating interactions between electrons. In this respect, the Hartree-Fock method was introduced, which assumes an approximated interaction between electrons each electron interacts with the potential field formed by the other electrons. Within the Hartree-Fock method, the system can be described by one-electron Schrodinger equations that can be solved with the variational method. As a result, the complicated N-body problem can be replaced by simple one-body problems with the Hartree-Fock method. 

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