Wavefunctions hydrogenlike

The normalized radial wavefunctions for hydrogenlike atoms can be expressed by... 

In treating complex atoms and molecules, these hydrogenlike wavefunctions are still too complicated in most applications. In treating low-energy (up to a few eV) problems, the behavior of the wavefunctions near the nuclei of the atoms is not important. Therefore, in the hydrogen wavefunctions, only the term with the highest power of r is significant. Based on this observation, Zener (1930) and Slater (1930) proposed a simplified form of the atomic wavefunctions ... 

The spirit of the Slater atomic wavefunction is to use the term with the highest power r" to represent the entire algebraic factor in the hydrogenlike wavefunctions. This approach is particularly suitable for treating STM-related problems. In the processes pertinent to STM, only the values of atomic wavefunctions a few Angstroms away from the nucleus of the atom are relevant, not the values near the core. Within the same spirit, we can derive all the Slater functions from a single function... 

The simpler wavefunctions for helium atom, for example (8.5), can be interpreted as representing two electrons in hydrogenlike s orbitals, designated as a configuration. Pauli s exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers, requires the two s electrons to have different spins one spin-up ora, the other spin-down or A product of an orbital with a spin function is called a spinorbital. For example, electron 1 might occupy a spinorbital which we designate... 

Since this function is nodeless, we identify it with the ground state of the hydrogenlike atom. Multipyling (7.35) by the spherical harmonic Too = l/ /4jr, we obtain the complete wavefunction (7.27), which in this case reduces to... 

The wavefunctions and radial distribution functions for the li, 2s, and hs hydrogenlike atomic orbitals are shown in Figure 1.29 as functions of r. Note that the wavefunctions for the li, 2s, and hs orbitals have 0, 1, and 2 nodes (points where the wavefunction is zero), respectively. For a general s orbital with principal quantum number n, the number of nodes is 1. This increase in the number of nodes as... 

Figure 4-5 s wavefunctions versus r and volume-weighted electron densities versus r for the hydrogenlike ion. 

This method is best illustrated by exanple, and we will now proceed with the problem of a hydrogen atom in a z-directed uniform electric field of strength F a.u. As mentioned earlier, a suitable choice of functions to mix together to approximate the accurate wavefunction is the Is and 2p, hydrogenlike functions. The choice of two basis functions leads to a 2 x 2 secular determinantal equation ... 

For what wavefunction(s) and which hydrogenlike atom(s) would the total energy of the species be... 

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