Hydrogen atom, wave mechanical treatment

In summary, the wave mechanical treatment of the hydrogen atom assumes that the electronic motion is described by tp, obtained as solutions of the equation (in atomic units)... 

The wave mechanical treatment of the hydrogen atom does not provide more accurate values than the Bohr model did for the energy states of the hydrogen atom. It does, however, provide the basis for describing the probability of finding electrons in certain regions, which is more compatible with the Heisenberg uncertainty principle. Note that the solution of this three-dimensional wave equation resulted in the introduction of three quantum numbers (n, /, and mi). A principle of quantum mechanics predicts that there will be one quantum number for... 

Part XI. A Wave Mechanical Treatment of the Change with Distortion of the Interaction Energy of Carbon 2p,i Orbitals. R. G. Parr and B. L. Crawford, J. Chem. Phys., 16, 1049 (1948). On Certain Integrals Useful in Molecular Orbital Calculations. J. O. Hirschfelder and J. W. Linnett,. Chem. Phys., 18, 130 (1950). The Energy of Interaction between Two Hydrogen Atoms. 

The problem of the structure of the hydrogen atom is the most important problem in the field of atomic and molecular structure, not only because the theoretical treatment of this atom is simpler than that of other atoms and of molecules, but also because it forms the basis for the discussion of more complex atomic systems. The wave-mechanical treatment of polyelectronic atoms and of molecules is usually closely related in procedure to that of the hydrogen atom, often being based on the use of hydrogenlike or closely related wave functions. Moreover, almost without exception the applications of qualitative and semiquantitative wave-mechanical arguments to chemistry involve the functions which occur in the treatment of the hydrogen atom. 

The year 1926 was an exciting one. Schrddinger, Heisenberg and Dirac, all working independently, solved the hydrogen atom problem. Schrddinger s treatment, which we refer to as wave mechanics, is the version that you will be fanuliar with. The only cloud on the horizon was summarized by Dirac, in his famous statement ... 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

The discussion begins with the consideration of diatomic molecules formed by univalent elements—molecules in which there are two atoms held to one another by a single bond. The hydrogen molecule is the only molecule of this kind for which an accurate solution of the SchrSdinger wave equation has been obtained. The approximate quantum-mechanical treatment of more complex molecules has provided interesting information about their electronic structure, but work along these lines has not been sufficiently extensive to permit the... 

Now that we have examined some of the mathematical details of the quantum mechanical treatment of the hydrogen atom, we need to consider what it all means. What is a wave function, and what does it tell us about the electron to which it applies First, a warning There is always danger in taking a mathematical description of nature and using our human experiences to interpret it. Although our attempts to attach physical significance to mathematical descriptions are quite useful to us as we try to understand how nature operates, they must be viewed with caution. Simple pictorial models of a particular natural phenomenon always oversimplify the phenomenon and should not be taken too literally. With that caveat we will proceed to try to picture what the quantum mechanical atom is like. 

One of the satisfying results from this treatment of the electron using wave mechanics is that an elaborate application of the wave equation can explain satisfactorily the numerous lines that appear in the spectrum of the hydrogen atom, discussed in the first part of this book. In other words, it is possible to predict that there will be differences in energy between the different orbitals of a particular principal quantum number. 

The quantum-mechanical treatment of the hydrogen atom gives an expression for the wave function, i f, of the l.y orbital ... 

The hydrogen molecule H2 is the simplest molecule which forms an electron-pair bond. Many calculations have been made for this molecule, which is a prototype for many other chemical bonds. One of the two basic quantum-mechanical treatments of the hydrogen molecule involves constructing a molecular orbital for the bond from a linear combination of atomic orbitals (LCAO method). The other involves constructing the molecular orbital as the product of wave functions for each of the two electrons forming the bond. Both of these methods will be outlined. 

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