  Linear-scaling electronic structure methods based on the calculation of
moments of the underlying electronic Hamiltonian offer a computationally
efficient and numerically robust scheme to drive large-scale atomistic
simulations, in which the quantum-mechanical nature of the electrons is
explicitly taken into account. We compare the kernel polynomial method to the
Fermi operator expansion method and establish a formal connection between the
two approaches. We show that the convolution of the kernel polynomial method
may be understood as an effective electron temperature. The results of a number
of possible kernels are formally examined, and then applied to a representative
tight-binding model.
