Accelerated Density Matrix Iterations Using Eigenvalue Gaps

Abstract: Density and Fock matrices play a significant role in electronic structure calculations, and there exist several methods to compute the density matrix from a given Fock matrix. One such method isdensity matrix purification, in which the eigenspectrum of the Fock matrix is scaled into the [0,1]interval and the Fock matrix is afterwards iterated with a sequence of polynomials that push theeigenvalues towards 1 and 0. A variant of density matrix purification exists, where theeigenspectrum of the iterate matrix is first scaled beyond the [0,1] interval before applying theiterating polynomial. The eigenspectrum is said to be folded back into the [0,1] interval, and thisresults in faster convergence to the density matrix. However, large eigenvalue gaps can give rise toan empty subinterval after the folding. In this thesis, we propose a scheme in which the eigenspectrum is scaled again after the folding in a given iteration so that the empty subinterval becomes filled. We have found that this additional scaling can decrease the number of iterations toconverge to the density matrix. However, we have also found the scaling to be significant only inthe first few iterations, as the eigenvalue gaps rapidly diminish throughout the iterations. We notethat our proposed scheme requires computing additional eigenvalues during the iterations, whichincurs additional computational cost in practice. The added computational cost per iteration should therefore be investigated as part of a future work