Two-body cluster expansions

M. Nooijen, Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion Phys. Rev. Lett. 84, 2108 (2000). 

P. Piecuch, K. Kowalski, P.-D. Fan, and K. Jedziniak, Exactness of two-body cluster expansions in many-body quantum theory. Phys. Rev. Lett. 90, 113001 (2003). 

In Section II, we summarize the ideas and the results of Bogolubov,3 Choh and Uhlenbeck,6 and Cohen.8 Bogolubov and Choh and Uhlenbeck solved the hierarchy equations and derived two- and three-body generalized Boltzmann operators Cohen used a cluster expansion method and obtained two-, three-, and four-body explicit results which he was able to extend to arbitrary concentrations. 

T. V. Voorhis and M. Head-Gordon, Two-body coupled cluster expansions. J. Chem. Phys. 115, 5033 (2001). 

The second strategy we mention in this rapid survey replaces the QM description of the solvent-solvent and solute-solvent with a semiclassical description. There is a large variety of semiclassical descriptions for the interactions involving solvent molecules, but we limit ourselves to recall the (1,6,12) site formulation, the most diffuse. The interaction is composed of three terms defined in the formula by the inverse power of the corresponding interaction term (1 stays for coulombic interaction, 6 for dispersion and 12 for repulsion). Interactions are allowed for sites belonging to different molecules and are all of two-body character (in other words all the three- and many-body interactions appearing in the cluster expansion of the Hss and HMS terms of the Hamiltonian (1.1)... 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

It is obvious that 0o cannot serve as a vacuum in the strict sense of the traditional hole-particle formalism, since the valence orbitals in are partially occupied. A straightforward cluster expansion in the occupation number representation from tpo would thus entail two problems (a) there is no natural choice of vacuum to effect a cluster expansion, and (b) the occupation number representation of cluster operators would refer to orbital excitations with respect to the entire oi thus necessitating the considerations of virtual functions which are by themselves combination of functions. If we want to formulate a many-body theory using if>o as the reference function, we need constructs where these cause no problems. 

From the expansion (27) it can be shown that the CCA includes also those three-,. .., n-body excitations which factorize into one- and two-particle cluster amplitudes. However, despite of the benefit of CCA to incorporate selected n-body contributions to all order into the computations, setting up the equations for open-shell systems is a highly nontrivial task, in particular when compared with the usual MBPT computations. For simple shell structures, there are a number of coupled-cluster codes available today, both in the nonrelativistic as well as the relativistic framework [31-34]. 

The Q coefficients (and derived Cg) in the D3 method have been computed using a modified form of this relation, where the a(ia>) are computed nonempirically by TDDFT and A and B are reference molecules from which atomic values are derived [42]. Because the reference system can also be a molecular cluster modeling a solid environment, special coefficients for atoms in the bulk can be derived [68]. The final form for the DFT-D3 two-body part of the dispersion energy employs the so-called Becke-Johnson (BJ) damping [61, 69] and truncates the expansion at Cg... 

The convergence of this expansion is relatively fast for clusters composed by neutral molecules, less fast when there are charged species. In any case, it is not possible to interrupt the expansion to the two-body terms. This contribution gives the additive terms of the interaction energy the other terms describe non-additive effects that in principle cannot be neglected. 

The simple theory given above is valid only at rather high concentration or at small excluded volume, i.e., near the 0 condition. At both these limits there are additional difficulties. At very high concentration the precise form of the potential matters. Also, near the 0 conditions the precise details of the interaction, in the sense of a cluster expansion passed to the two-body term, can also matter. In both of these limits it is possible to make an appropriate improvement, and the results have been found to be in good agreement with experiments. ... 

ES contributions are strictly additive there are no three- or many body terms for ES. The term many-body correction to ES introduced in some reviews actually regards two other effects. The first is the electron correlation effects which come out when the starting point is the HE description of the monomer. We have already considered this topic that does not belong, strictly speaking, to the many-body effects related to the cluster expansion [8.9]. The second regards a screening effect that we shall discuss later. 

The exactness of exponential cluster expansions employing two-body operators... 

It has recently been suggested that it may be possible to represent the exact ground-state wave function of an arbitrary many-fermion pairwise interacting system, defined by the Hamiltonian iJ, Eq. (22), by an exponential cluster expansion involving a general two-body operator [92-98]. If these statements were true, completely new ways of performing ab initio quantum calculations for many-fermion (e.g., many-electron) systems might... 

We have recently provided a strong evidence that the exact ground state of a many-fermion system, described by the Hamiltonian containing one- and two-body terms, may indeed be represented by the exponential cluster expansion employing a general two-body operator by connecting the problem with the Horn-Weinstein formula for the exact energy [152],... 

---