Energy Levels

1. Quantum number
Since an electron moves in three-dimensions, three kinds of quantum numbers :principal quantum number, azimuthal quantum number and magnetic quantum number, are required to express one wave function. The 4th quantum number, spin magnetic quantum number is associated with the quantization of angular momentum of an electron caused by its spinning. The axial component of angular momentum is allowed to have only two values, +1/2 (h/2?) and - 1/2(h/2?). The spin magnetic quantum number is related to these values (ms = +1/2 or -1/2). Only this quantum number is not an integer.
principal n 1, 2, 3,...
azimuthal l 0, 1, 2, 3, ...n – 1
magnetic m(ml) 0, ±1, ±2,...±l
spin magnetic ms +1/2, -1/2

2. Orbital
A wave function for an electron is called an orbital. If the principal quantum number is n = 1, there is only one value for l, i.e., 0. In this case there is only one orbital, and the set of quantum numbers for this orbital is (n = 1, l = 0). When n = 2, two values, 0 and 1, are allowed for l. In this
case there are four orbitals which are defined by the following set of quantum numbers, (n = 2, l = 0), (n = 2, l = 1, m = -1), (n = 2, l = 1, m = 0), (n = 2, l = 1, m = +1).





3. Many electrons
The Hamiltonian for an atom containing many electrons:

....(34)
The symbol ? i>j means we sum over all values of j = 1 to N (for N electrons) and values of i from j + 1 to N (thus avoiding counting pairs twice).
The numerical problem of solving Schr?dinger s equation with more than a few electrons is extremely difficult. It is necessary to simplify the task, and the self-consistent field, or Hartree method is the best way to do this. The idea is to reason that each electron moves in the total electric field produced by the nucleus and all the other electrons, and this total electric field is close to spherically symmetric (though with a complicated dependence on r).
The list of quantum numbers is called the configuration of the atom. In a given configuration, the filled shells, also called `closed shells, form the core and the electrons lying outside the core are called valence electrons.
One way to get a self-consistent (self-consistent method to calculate appropriate potential energy function) argument is to take the net charge distribution of all Z electrons and multiply it by (Z - 1)/Z to obtain the charge distribution whose field is experienced by any one electron.