How about making it really simple?  Let us assume that we want to know something only about the π system of the molecule.  OK, let's make it even simpler!   We will take for granted that only carbon atoms participate in our π system, and that all these atoms are equivalent in their bonding abilities.  Each carbon in the π network will contribute one p atomic orbital to the π system. We will build (and calculate!) MOs of our molecule from these p orbitals, and then throw in all the p electrons we want to have in the system.   Good news is that it has been done already, first by Hückel himself, and it works very well.  Well, sort of...  The method, called a Simple Hückel Molecular Orbitals (SHMO), gives us a good idea about relative energies, symmetries, and (with caution!) even the sizes of orbital lobes on the different carbons for all π molecular orbitals.

The energy of molecular orbitals is calculated in α and β units. The value of α (called the Coulomb integral) represents the energy of an electron in a p orbital on an sp²-hybridized carbon.  β (named the resonance integral) corresponds to the energy of an electron that is shared by two neighboring  sp²-hybridized carbons.  The energy of an electron in the π molecular orbital is expressed as E = α + x∙β where x is calculated by SHMO.  If an electron in a molecular orbital is more stable than an electron in a p atomic orbital, the molecular orbital is bonding. The energy of the π electrons in the molecule is obtained by adding all π electron energies together.

The wavefunction describing a molecular orbital is expressed as a linear combination of p atomic orbitals on carbons, Ψ = a₁p₁ + a₂p₂ + ... + a_ip_i, where a₁, a₂ ... a_i are coefficients (fractions) representing the different p orbitals in our molecular orbital.  In this example, we have i carbons participating in the π system, and will obtain, therefore, i wavefunctions (Ψ¹, Ψ², ... Ψⁱ) each having the form shown above. The coefficients for all these wavefunctions are calculated by SHMO, and they correspond to the lobe size on different atoms.

Enough explanations, let's calculate! Below you have an applet that can do it for you. Start by clicking "Add" and draw the structure of the π system you want to explore. Each gray circle represents a carbon atom participating in the π network, and each black line connecting them represents π bonding (we neglect σ bonds!).  If you want "smooth", symmetrical structures click on "Minimize". The calculator plots the energy of the molecular orbitals on the right and fills them with electrons, whose number can be adjusted by changing the net charge of the system (left-bottom).  By clicking on the line representing orbitals you may view their energies and coefficients (to see the numerical values chose "verbose", or look into the "Data Table").  Start with 1,3-butadiene (four dots in a row connected by single lines), or benzene (six dots in a hexagon connected by single lines) to get comfortable. Compare the results with the pictures on the π-system page.  Explore and have fun!