inorganic chemistry - Hard and Soft Acid and Bases in terms of HOMO and LUMO

It has been given in a book and on Wikipedia that:

hard bases have highest-occupied molecular orbitals (HOMO) of low energy, and hard acids have lowest-unoccupied molecular orbitals (LUMO) of high energy.[7][8]

soft bases have HOMO of higher energy than hard bases, and soft acids have LUMO of lower energy than hard acids. (However the soft-base HOMO energies are still lower than the soft-acid LUMO energies.)[7][8]

Let us consider the energy difference between HOMO and LUMO is given by $\Delta E$. Thus, $$\Delta E (S-S)<\Delta E(H-S) < \Delta E (H-H)$$

Hence the hard-hard interaction should have been weaker than hard-soft, which is not the case. Please explain.

P.S.: I went through the reference (Meissler and Tarr) and this question but could not find a satisfying explanation.