Hydrogen orbitals angular functions

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The symbol for the normal hydrogen molecule is and means the following (/) there is no net orbital angular momentum around the axis of the molecule (/ /) the two electron spins are paired (as they have to be if the Pauli Exclusion Principle is not to be violated) (iii) the sum of the f s is an even number (zero in this case) and leads to the subscript g, which means that the wave function does not change sign by inversion through the center of symmetry (otherwise the symbol u would be used). 

Because the orbital angular momentum of the positron-hydrogen system is zero for s-wave scattering, the total wave function is spherically symmetric and depends only on the three internal coordinates which specify the shape of the three-body system. The kinetic energy operator... 

Figure 7.1 Radial eigenfunctions Pn((r) = rR fr) for the electron in the hydrogen atom (in atomic units) where n is the principal quantum number,

It turns out that there is not one specific solution to the Schrodinger equation but many. This is good news because the electron in a hydrogen atom can indeed have a number of different energies. It turns out that each wave function can be defined by three quantum numbers (there is also a fourth quantum number but this is not needed to define the wave function). We have already met the principal quantum number, n. The other two are called the orbital angular momentum quantum number (sometimes called the azimuthal quantum number), , and the magnetic quantum number, mi. 

The canonical real d-orbital wave functions are linear combinations of the complex eigen functions (solutions) of the hydrogenic Schrodinger equation and the orbital angular momentum operator. Thus, instead of a set of five degenerate orbitals that may be indexed by values of orbital... 

Table 1.1 The radial and angular components of the hydrogenic atomic orbitals with distinct normalization constants for the radial and angular functions. The parameter, p = extends the application of the functions in the table entries for non-hydrogen one-electron atomic species. Remember that the solutions to the angular equation in are exp(-f / — im0) and the real forms given are obtained by taking the sums and differences of the expansions of the complex exponentials and then applying equations 1.1 to 1.3 to these results. The column headed -I-/- indicates the particular choices of sum when relevant. ...

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