Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems

The Fermi operator, i.e., the Fermi-Dirac function of the system Hamiltonian, is a fundamental quantity in the quantum mechanics of many-electron systems and is ubiquitous in condensed-matter physics. In the last decade the development of accurate and numerically efficient representations of the Fermi operator has attracted lot of attention in the quest for linear scaling electronic structure methods based on effective one-electron Hamiltonians. These approaches have numerical cost that scales linearly with N, the number of electrons, and thus hold the promise of making quantummechanical calculations of large systems feasible. Achieving linear scaling in realistic calculations is very challenging. Formulations based on the Fermi operator are appealing because this operator gives directly the single-particle density matrix without the need for Hamiltonian diagonalization. At finite temperature the density matrix can be expanded in terms of finite powers of the Hamiltonian, requiring computations that scale linearly with N owing to the sparse character of the Hamiltonian matrix. 1 These properties of the Fermi operator are valid not only for insulators but also for metals, making formulations based on the Fermi operator particularly attractive. Electronic structure algorithms using a Fermi operator expansion FOE were introduced by Baroni and Giannozzi 2 and Goedecker et al. 3,4 see also the review article 5 . These authors proposed polynomial and rational approximations of the Fermi operator. Major improvements were made recently in a series of publications by Parrinello and coauthors, 6‐11 in which a new form of Fermi operator expansion was introduced based on the grand canonical formalism.