Molecular Elements of Computers - Molecular Computing: Origins and Promises - page 53

Fig. 3.3 Energy levels of the hydrogen atom

Fig. 3.4 Angular parts of the wave functions of the hydrogen atom

Electronic energy levels of the hydrogen atom (the energy levels of the electron
in the Coulomb potential) are shown in Fig. 3.3 . The form of the basic angular
functions for the “ s ” and “ p ” states of the hydrogen atom is shown in Fig. 3.4 .
The hydrogen atom problem forms the basis for describing molecular systems.
Schr¨dinger equations for multi-electron atoms and molecules (even for the sim-
plest two-electron helium atom) do not have an exact solution. In this case, strictly
speaking, the symmetry of the atoms with more than one electron does not satisfy
the spherical symmetry group, since the Schr¨dinger equation contains a term
corresponding to the interaction of electrons. Nevertheless, it is assumed that at
least for the first half of the atoms of the periodic system, the spherical symmetry is
approximately preserved. Therefore, as in the case of the hydrogen atom, each
electron is assigned four quantum numbers n , l , m , and a s -spin quantum number,
which takes only two values ½

. The overall structure of energy levels is also
preserved. In addition, the state filling principle is introduced, according to which in
the ground electronic state, the electrons occupy successive levels, starting from the