Minimal parametrization of an n-electron state

Abstract

The Hamiltonian H for an n-electron system in a finite one-electron basis of dimension m is characterized by d=O(m⁴) matrix elements. The eigenstates of this Hamiltonian-i.e., the full-CI states Ψ-depend, however, on the usually much larger set of N=O(mⁿ) parameters. One can, nevertheless, write a full-CI state as Ψ=e^SΦ with Φ a reference function and S an operator familiar from traditional coupled cluster (TCC) theory. The "exact" operator S can be expressed (though in an implicit and rather complicated way) in terms of d parameters. An alternative ansatz Ψ=e^ΤΦ with Τ depending in a very simple way on d parameters only (namely, with Τ having the same structure in Fock space as H) has been studied by Nooijen and by Nakatsuji and been called coupled-cluster with generalized single and double excitations (CCGSD). Nooijen has conjectured that the full-CI equations can be fulfilled with this ansatz. This paper is devoted to a comprehensive analysis of the Nooijen conjecture (NC). Several features make this analysis difficult and even intriguing. (a) One deals with coupled nonlinear systems of equations, for which theorems concerning the existence of their solution are hardly available. (b) There are different possible interpretations of the NC, especially as far as the choice of the reference function Φ is concerned. (c) There are solutions of the CCGSD equations, for which some elements of Τ becomes negative infinite, and e^Τ becomes a projection operator. Such solutions are undesired but difficult to eliminate. We show by direct comparison of the exact wave operator with that of CCGSD theory, for a closed-shell state with Φ a single Slater determinant, using a perturbation expansion, that CCGSD cannot be exact. This required a reformulation of the CCGSD operator e^Τ to an equivalent exponential form e^R, with R similar to the S of TCC theory, but with constraints on the cluster amplitudes, such that these depend on d parameters only. The CCGSD ansatz simulates three-particle and higher excitations, in a perturbation expansion even with diagrams of the correct topology, but with incorrect energy denominators. Differences to the exact formulation arise to second order in perturbation theory for the wave function and to fourth order in the energy, but even the contributions which are not exact, are often acceptable approximations. Essential for the proof that the CCGSD ansatz is not potentially exact is that the set of one- and two-particle basis operators, into which H can be expanded, does not span a Lie algebra. What matters is actually that the variations of the wave function describable within a CCGSD ansatz are different from those required to satisfy the two-particle contracted Schrodinger equations. The first attempt, due to Nakatsuji, to prove in this philosophy that the CCGSD ansatz is not potentially exact, was essentially correct, but some subtle yet important aspects were not sufficiently stressed. A refinement was given by Mazziotti. Papers by Ronen and Davidson addressed a somewhat academic problem-namely, the possible validity of the CCGSD ansatz for a completely arbitrary reference function-using mainly dimensionality arguments.