The continuous Coupled Cluster formulation for the electronic Schrödinger equation^∗

Sekretariat MA 5-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
rohwedde@math.tu-berlin.de

Received: 13 April 2011
Revised: 3 July 2012

Abstract

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precision method for the solution of the main equation of electronic structure calculation, the stationary electronic Schrödinger equation. Traditionally, the equations of CC are formulated as a nonlinear approximation of a Galerkin solution of the electronic Schrödinger equation, i.e. within a given discrete subspace. Unfortunately, this concept prohibits the direct application of concepts of nonlinear numerical analysis to obtain e.g. existence and uniqueness results or estimates on the convergence of discrete solutions to the full solution. Here, this shortcoming is approached by showing that based on the choice of an N-dimensional reference subspace R of H¹(ℝ³ × {± 1/2}), the original, continuous electronic Schrödinger equation can be reformulated equivalently as a root equation for an infinite-dimensional nonlinear Coupled Cluster operator. The canonical projected CC equations may then be understood as discretizations of this operator. As the main step, continuity properties of the cluster operator S and its adjoint S^† as mappings on the antisymmetric energy space H¹ are established.