Enquiry about Kernel Polynomial Method for Hamiltonian

I read a paper (Alexander Weiße, Gerhard Wellein, Andreas Alvermann, and Holger Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275, 2006), which discusses how to use the Kernel Polynomial Method (KPM) to deal with the large Hamiltonian matrix.

This techniques enable people to compute the density of states without diagonalizing the Hamiltonian. It processes the Delta function through a series of Chebyshev expansion. In this way, it makes the computation faster than the diagonalization of Hamiltonian.

I have a question about this technique. Suppose there is a Hamiltonian, containing 150 orbitals. After the spin degeneracy is released, the Hamiltonian contains both spin-up and spin-down columns for each orbital so the size of this Hamiltonian matrix is 300 by 300.

Now, I want to calculate the exchange potential for such a system. To get this quantity, I subtract eigenvalues for the spin-up and spin-down columns of the same orbital; then, sum all the eigenvalues subtraction for all 150 orbitals in this system. This is how I calculate the exchange potential.

I wonder whether there is any way to calculate this exchange potential without diagonalizing the Hamiltonian? Is it possible to compute this quantity through the KPM? Would anyone please give me some suggestions? Thank you very much in advance.