Exponential cluster expansions

C(I) determined by exponential cluster expansion non-variational, size extensive. 

The exponential cluster expansion has a long history in statistical physics, where it is known as the Ursell and Mayer linked-cluster expansion for the partition function [68]. A very general argument for the exponential form of the exact wave function can be found in a paper on the origins of the CC method by Kiimmel [69]. Nonetheless,... 

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],... 

This approximation has a philosophical and mathematical resemblance to the linked-cluster expansion that has been applied successfully to the small polaron problem. The linked-cluster expansion is an exponential resummation of... 

The 1inked-cluster theorem for energy, from the above analysis, is a consequence of the connectivity of T, and the exponential structure for ft. Size-extensivity is thus seen as a consequence of cluster expansion of the wave function. Specfic realizations of the situation are provided by the Bruckner—Goldstone MBPT/25,26/, as indicated by Hubbard/27/, or in the non-perturbative CC theory as indicated by Coester/30,31/, Kummel/317, Cizek/32/, Paldus/33/, Bartlett/21(a)/ and others/30-38/. There are also the earlier approximate many-electron theories like CEPA/47/, Sinanoglu s Many Electron Theory/28/ or the Cl methods with cluster correction /467. 

The cluster expansion methods are based on an excitation operator, which transforms an approximate wave function into the exact one according to the exponential ansatz... 

In the standard SR CC approach, the exact (nonrelativistic) N-electron wave function I1 ) for the state of interest (assumed to be energetically the lowest state of a given symmetry species) is represented by the so-called cluster expansion relative to some IPM wave function 4>0). This expansion is concisely expressed via the exponential cluster ansatz... 

The exponential character of the cluster expansion warrants the size-extensivity of the resulting formalism regardless of the truncation scheme employed, as implied by a comparison with the standard linear Cl expansion of (intermediately normalized) I T),... 

The thermodynamics of a l-d Fermi system can be perfectly mapped onto the thermodynamics of a two-component classical real gas on the surface of a cylinder. The relationship between these two infrared problems (cf. Zittartz s contribution) is exploited as follows. We treat the classical plasma by a modified Mayer cluster expansion method (the lowest order term corresponding to the Debye Hiickel theory), and obtain an exponentially activated behavior of the specific heat (cf. Luther s contribution) of the original quantum gas by simply reinterpreting the meaning of thermodynamic variables. 

Obtain the 4-cluster terms in the exponential operator expansion (9.3.9) and show that if 1-cluster corrections to the reference function are neglected then considerable reduction occurs. Hence, neglecting irreducible cluster operators for more than 2 electrons, express the wavefunction in the correlated-pair form (9.3.17). What condition must the reference function satisfy in order to eliminate the 1-cluster terms [Hint The effect of the 1-cluster terms in the exponential operator is equivalent to an orbital variation in the operand. Look for a variational condition.]... 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples... 

To understand the structure of the coupled-cluster wavefunction, let us Taylor expand the exponential in Eq. (2.1). Sorting the resulting expansion according to the level of excitation, we obtain... 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

So far, we have specified the wave operator H in the BW form (15). If we adopt an exponential ansatz for the wave operator Cl, we can speak about the single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory. The simplest way how to accomplish the idea of an exponential expansion is to exploit the so-called state universal or Hilbert space exponential ansatz of Jeziorski and Monkhorst [23]... 

Mass spectra for clusters formed by the adiabatic expansion of liquid droplets of different mole fraction (Xdio) 1,4-dioxane-water mixtures have been studied. For Xdio = 0.01, the hydrogen-bonded networks of water are predominant in the water-rich region with 1,4-dioxan molecules probably being captured in the network to form clathrates, but decrease exponentially with increasing Xdio <1999JML163>. 

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