Hole formalism

Hole formalism in quantum chemistry states that for many electronic properties, one may consider systems with e or (n-e), the number of unoccupied sites or “holes”, to be equivalent.
The number of microstates (N) of a system corresponds to the total number of distinct arrangements for a number “e” of electrons to be placed in a number “n” of possible orbital positions:
formula_1
In hole formalism, the commutative property of multiplication applies, meaning that in the above equation, formula_2
Example.
For a set of p orbitals, n = 6 since there are two vacant positions in each of the three orbitals (p"x", p"y", p"z"). Therefore, for P2 (e = 2 and n = 6):
formula_3
Based on the hole formalism, microstates of P4 (e = 4 and n = 6) are:
formula_4
Here it is seen that P4 (2 holes) and P2 (4 holes) give equivalent values of N. The same is true for all p, d, f, … systems such as d1/d9 or f2/f12.
This is characterised by the symmetry of "k"-combinations in general, that is:
formula_5
For all positive integers "n", "e" where "n" ≥ "e".