the states corresponding to isolated particles overlap and we obtain two new energy eigenstates one having [*lower*]{} energy than the ground state energy of a single atom (the so-called bonding state for which the wavefunction is symmetric) and the other one having higher energy (the anti-bonding state for which the wavefunction is antisymmetric). In a hydrogen-like molecule, therefore, it makes sense for both the electrons to anti-align and occupy the bonding energy (symmetric) state to lower the energy of the whole system. Therefore we see that by bringing two atoms close to each other can in principle lower the energy of the whole system by the formation of bonding and anti-bonding states. Note that here we have ignored the Coulomb interaction between two charged electrons.

Now instead of just two atoms, if we take many such atoms to form a solid, there are several of these bonding and anti-bonding states. In case of a real macroscopic solid where the number of atoms is of the order of $10^{23}$, these states have to be so closely spaced that the discreteness of their energies can be ignored and instead we will have bands where such states exist. In a more complex solid, 