The subtraction of the last term accounts for the fact that E j contains a term due to the interaction with the electron present at site i , but after moving the electron this term should not be considered. It is easy to see from this that there exists an energy E f such that all sites with energies above it are empty, and below it are full (this is the Fermi energy, but since we are dealing with a system with interactions it is not obvious a-priori that it is still well-defined). Assume we have a finite single-particle DOS at the Fermi energy, g ( E f ) . For every possible transfer of an electron from an occupied site i to an unoccupied site j , the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., Δ Δ E >= 0 . Assuming we have a large system, consider all the sites with energies in the interval [ E f − − ϵ ϵ , E f + ϵ ϵ ] . The number of these, by assumption, is N = 2 ϵ ϵ g ( E f ) . As explained, N / 2 of these would be occupied, and the others unoccupied. Of all pairs of occupied and unoccupied sites, let us choose the one where the two are closest to each other. If we assume the sites are randomly distributed in space, we find that the distance between these two sites is of order: R ∼ ∼ ( N / V ) − − 1 / d , where d is the dimension of space. Plugging the expression for N into the previous equation, we obtain the inequality: E j − − E i − − C e 2 ( ϵ ϵ g ( E f ) / V ) 1 / d > 0 where C is a coefficient of order unity. Since E j − − E i < 2 ϵ ϵ , this inequality will necessarily be violated for small enough ϵ ϵ . Hence, assuming a finite DOS at E f led to a contradiction. Repeating the above calculation under the assumption that the DOS near E f is proportional to ( E − − E f ) α α shows that α α >= d − − 1 . This is an upper bound for the Coulomb gap. Efros considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound).