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

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 the 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.

Figure 1

Energy of the Müller functional for H₂ in the minimal basis, w.r.t. the 1-RDM γ (left panel) and the variables prametrizing γ, x₁₂ and n₁ as in eq 5, using an exponential (central panel) and the Cayley parametrization (right panel) for the orthonormal matrix. Note that in the present case, of only two orbitals, the Givens and exponential as well as the Cayley and Householder parametrization are identical up to a sign and so that we show only the two distinct ones. The minima are indicated by cyan dots and the saddle point by a red dot. Note that E[γ] is convex while E[x₁₂, n₁] is not.

Figure 2

Tangent vectors of the exponential parametrization U = exp(X) (with X^T = −X) for N = 3. The vectors align on the sphere x₁₂² + x₁₃² + x₂₃² = 4π² since the eigenvalues of X are and 0.