A remark on the disconnected nature of Lagrange equations in the context of a linear-scaling implementation of the coupled-cluster energy gradients

Abstract

It is known that the Ã-tensor (an array of Lagrange multipliers), necessary for evaluating analytic energy gradients in the coupled-cluster theory, is diagrammatically disconnected in general. This means that the number of non-negligible elements in the Ã-tensor grows faster than linearly with the number of calculated particles. At a formal level, when evaluating the gradients of the coupled-cluster energy, this could prevent obtaining a linear scaling of the operational cost with respect to the number of correlated particles. It is shown that in ground/ excited-state coupled-cluster calculations, based on localized orbitals, the disconnected part of the Ã-tensor, as well as the disconnected part of the left-hand excited-state eigenvector, can be ignored, thus justifying the use of standard screening techniques employed in linear-scaling schemes.

Coupled-cluster Green's function: Analysis of properties originating in the exponential parametrization of the ground-state wave function

“…The Λ operator has been extensively discussed in the literature. [77] In particular, the Λ operator is defined by linked diagrams (in the present context, linked diagrams refer to the open diagrams which do not contain disconnected closed part). In the limit of the exact theory discussed in this paper, the Φ|(1+Λ) is equivalent to the exponential ansatz based on the de-excitation cluster operator S, i.e.,…”

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

“…Alternatively, the local spatial interaction metric can naturally induce a fragmentation of the underlying (chemical) system into dense domains that significantly interact only with their close neighbors (at best), a fact used in closely related fragmentation‐based methods (this nomenclature is rather fuzzy as no rigorous distinction can be made in general). Here, however, we would like to stress that the techniques based on local tensor screening can be efficient only for the so‐called connected/linked correlated many‐body methods , as it can be shown that the number of non‐negligible elements in a connected tensor grows at most linearly with the number of simulated particles . Moreover, in many connected many‐body methods for excited states (e.g., the equation‐of‐motion [EOM] coupled‐cluster [CC] method, the Fock‐space multireference CC theory, etc.…”