Impact of Parametrizations of the One-Body Reduced Density Matrix on the Energy Landscape

TL;DR

This paper investigates how different parametrizations of the one-body reduced density matrix affect the energy landscape in electronic structure methods, revealing implications for optimization strategies and the nature of critical points.

Contribution

It compares various parametrizations of the 1-RDM, showing how they influence the energy landscape and the occurrence of critical points, especially in the context of functional theory.

Findings

Cayley and Householder parametrizations avoid extra critical points.

Degeneracies in occupation numbers can create saddle points.

Certain parametrizations like Givens rotations introduce additional critical points.

Abstract

Many electronic structure methods rely on the minimization of the energy of the system with respect to the one-body reduced density matrix (1-RDM). To formulate a minimization algorithm, the 1-RDM is often expressed in terms of its eigenvectors via an orthonormal transformation and its eigenvalues. This transformation drastically alters the energy landscape. Especially in 1-RDM functional theory this means that the convexity of the energy functional is lost. We show that degeneracies in the occupation numbers can lead to additional critical points which are classified as saddle points. Using a Cayley or Householder parametrization for the orthonormal transformation, no extra critical points arise. In case of Given's rotations or the exponential, additional critical points can arise, which are of no concern in practical minimization. These findings provide an explanation for the success of recent minimization procedures using second-order information.