Infinite mass approximation

Exact values of critical exponents are more difficult to obtain, because variational bounds do not give estimations of the exponents. Then the result presented by M. Hoffmann-Ostenhof et al. [64] for the two-electron atom in the infinite mass approximation is the only result we know for /V-body problems with N > 1. They proved that there exists a minimum (critical) charge where the ground state degenerates with the continuum, there is a normalized wave function at the critical charge, and the critical exponent of the energy is a = 1. 

Drawbaugh and G. Sadowsky, An IBM 704 Program for Calculating Effective Absorption Cross Sections by the Narrow Resonance or Narrow Resonance Infinite Mass Approximation, Combustion Engineering report CEND-94. 

Since the water mass getting into contact with the suspended sediment particles is probably very large compared to the mass of solids that will spill and settle through the water body, the infinite bath approximation can be used (y = 0). Thus from Eq. 19-74 ... 

Let us consider now a relativistic hydrogenlike atom. In the infinite nucleus mass approximation a hydrogenlike atom is described by the Dirac equation (h = c = l)... 

It is possible to write eqn.(ll) in a form that makes scaling and perturbation corrections to the infinite nuclear mass approximation more obvious [13]. Since for the cases being considered p = p = p2, coordinate scaling pi = (p/me) rt gives a Hamiltonian,... 

Schrodinger equation is then solved exactly. Note that exact is this context is not the same as the experimental value, as the nuclei are assumed to have infinite masses (Bom-Oppenheimer approximation) and relativistic effects are neglected. Methods which include eleclroti correlation are thus two-dimensional, the larger the one electron-expansion (basis set size) and the larger the many-electron expansion (number of determinants), the better are the results. This is illustrated in Figure 4.2. 

In this section we plan to review the analytical properties of the eigenvalues of the Hamiltonian for two-electron atoms as a function of the nuclear charge. This system, in the infinite-mass nucleus approximation, is the simplest few-body problem that does not admit an exact solution, but has well-studied ground-state properties. The Hamiltonian in the scaled variables [96] has the form... 

For each of the ionomer systems studies as well as for cations in zeolites, the cation-motion bands are easy to identify, because they vary in frequency approximately as M"1, where M is the mass of the cation. However, M-i 1 clearly is not the correct complete reduced mass for the vibration, since that would require the site to be of infinite mass. A better approximation to the proper reduced mass, p, can be obtained by considering at least the atoms immediately surrounding the cation. Typically, p is calculated from models in which the cation has either an octahedral surrounding (for the T mode) or a tetrahedral surrounding (for the T2 mode). With it and the band frequency, the force field element, (force constant), for the vibration can be calculated. 

Here, r denotes the scalar distance of the electron measured from the centre of mass (assumed to be at rest) of the complete system in the infinite proton mass approximation this corresponds to the electron-proton distance. The resulting reduced Schrodinger equation for the relative electronic motion is found to be soluble in several different orthogonal coordinate systems. In particular, in spherical polar coordinates (r, 0, ) referred to the centre of mass, the natural unconfined ranges of these variables are... 

Within the framework of the effective mass approximation, improvement can be made by considering the effect of finite well depth [23,27], The approach recognizes that the matrix cannot be represented by an infinite... 

In the rest of this paper, we will use this infinite proton mass approximation. The inaccuracies it produces are smaller than those resulting from other sources of error in the computations we have performed so far, and if desired can be corrected by either a first order perturbation theory approach or a repetition of the calculations without using the approximation. The last two expressions explicitly display the symmetry of the system, and lead to interesting insights, which justify the slight error they produce. 

Which of these surface rearrangements takes place upon cleavage is determined by the surface electronic structure. This follows within the adiabatic approximation which is applicable for a wide range of molecular systems and is based on the fact that the ratio me I Mis small. Here, me and M are the electron and the mean nuclei masses, respectively. In the zeroth-order approximation, which corresponds to infinite masses of nuclei, one can neglect their kinetic energies and consider the states of the electronic subsystem at fixed nuclei positions specified by a multidimensional vector R. Then the electronic energy E(R) plays the role of a potential in which the nuclei move. The equilibrium positions of the nuclei, Rq, are found at the minimum... 

---