Modified hydrogen atom wave functions

CALCULATIONS WITH MODIFIED HYDROGEN ATOM WAVE FUNCTIONS... 

Although the calculated molecular parameters De = 3.15 eV, re = 1.64 a0 do not compare well with experiment the simplicity of the method is the more important consideration. Various workers have, for instance, succeeded to improve on the HL result by modifying the simple Is hydrogenic functions in various ways, and to approach the best results obtained by variational methods of the James and Coolidge type. It can therefore be concluded that the method has the correct symmetry to reproduce the experimental results if atomic wave functions of the correct form and symmetry are used. The most important consideration will be the effect of the environment on free-atom wave functions. 

Humberston and Wallace, 1972), is shown in Figure 6.4. Also shown there is the distribution function obtained using the Born approximation, in which neither the positron nor the atomic wave function is modified by the interaction. This latter curve therefore represents the momentum distribution of the electron in the undistorted hydrogen atom. The distribution function for the accurate wave function is narrower than that for the undistorted case because the positron attracts the electron towards itself and away from the nucleus, thereby enhancing the probability of low values of the momentum of the pair. 

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

In deriving these formulae it is assumed that the wave functions fa are orthogonal to one another, as for instance are Wannier functions. If they are atomic functions falling off as e ar, a formula such as (9) must be modified by a term in the denominator to take account of the non-orthogonality. Reviews of the appropriate formulae are given in textbooks see e.g. Callaway (1964) and Wohlfarth (1953), who considered a linear chain of hydrogen atoms. 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

This equation differs from that of the hydrogen atom only in that the factor Z, instead of unity, appears for the nuclear charge. Consequently, we have a set of quantum numbers n, I, m, nis for each electron in the atom. The presence of Z modifies the wave function but not the quantum numbers. For example, the Is wave function becomes... 

Once again it must be emphasized that we are here dealing not with a phenomenon predicted by quantum mechanics, but with a convenient mode of description, in terms of approximations, of matters to which those approximations ought really never to have been applied. That they have been so applied in an imperfect world is a necessity imposed by the absence of methods which are at the same time precise and manageable. A correct solution of the wave equation for a combined carbon atom (in methane) would presumably predict four symmetrically disposed axes of maximum electric density for the configuration of minimum energy, and not the existence of 2s and 2p wave functions. The latter apply to isolated atoms in any case. Interaction with other atoms modifies the density distribution, as is seen from the fact that two hydrogen atoms, each with... 

Such a calculation for non-hydrogen atoms was carried out numerically by a modified Hartree-Fock-Slater procedure [23]. The boundary condition for each wave function was introduced on defining a cutoff radius by the step function ... 

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