Minimization Principle for Linear Response Eigenvalue Problem with Applications
Zhaojun Bai, Ren‐Cang Li · 2011
We present a minimization principle for the sum of the first few smallest positive eigenvalues and Cauchy-like interlacing inequalities for the linear response (a.k.a random phase approximation) eigenvalue problem arising from the calculation of excitation states of many-particle systems, a hot topic among computational material scientists today for materials design to advance energy science. Subsequently, we develop the best approximations of these smallest positive eigenvalues by a structurepreserving subspace projection. Based on these newly established theoretical results, we outline conjugate gradient-like algorithms for simultaneously computing the first few smallest positive eigenvalues and associated eigenvectors. Finally, we present numerical examples to illustrate essential convergence behaviors of the proposed conjugate