Quantum Mechanical Description of electrons in orbitals

Electrons distribute themselves in atoms and molecules according to Quantum Mechanics. The Schrodinger equation is the starting point for determining the distribution of electrons;

H^{(op)}ψ = Eψ (Schrodinger equation)

Here H^{(op)} is the Hamiltonian operator, E the energy of the system and Ψ the wavefunction.

The solution to the Schrodinger equation yields the wavefunctions (orbitals) and a set of 4 quantum numbers for each electron in the atom. Quantum numbers describe the state of the electrons and each electron in an atom or a molecule has a unique set of these four quantum numbers. The quantum numbers are n, l, m_{l}and m_{s}.

The principal quantum number (n) indicates the energy and size of the orbital and corresponds to rows on the periodic table. For example, a hydrogen atom has n=1 since it is in the first row.

The angular momentum quantum (l) (or azimuthal) specifies the shape of the orbital. An s orbital corresponds to l = 0, while a p orbital has l = 1 and so on. Thus an electron in a p orbital of boron atom would have n = 2 (2nd row) and l=1.

Magnetic quantum number m_{l}is related to the orientation of the orbitals in space. Possible values for this quantum number depends on l. There are 2l+1 possible values of m_{l}. For example an s orbital has l = 0 there are 2(0)+1 = 1 possible orientations in space. Not very surprising considering an s orbital is spherically symmetric. On the other hand, a p orbital has l=1 and therefore 2(1)+1 = 3 possible values of m_{l}. These correspond to the degenerate p_{x}, p_{y,} and p_{z} orbitals and have values of -1, 0, +1. Thus an electron in the 2p_{x} orbital of a boron atoms has n = 2, l = 1 and m_{l }=-1

The spin quantum number m_{s} can have two possible values +1/2 and -1/2. this corresponds to the two possible spin states (spin up or down) an electron can have. Thus an electron in the 2py orbital of carbon would have n = 2, l = 1, m_{l} = 0, m_{s }= +1/2. The following table summarizes the possible values of the quantum numbers for the first three rows.