Exponentiated cluster operators

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

The coupled-cluster method (CCM) is based on the ansatz that an exact many-particle wave function can be written as an exponential cluster operator acting on an independent-particle function,... 

We wish to generate a singlet CCSD state by the application of the exponential cluster operator to a closed-shell Hartree-Fock state ... 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis,... 

Using a cluster operator, T, and an exponential ansatz [60,61], the coupled cluster wave function is written as... 

X h is an eigenstate of H with eigenvalue n o>. 4 il assumed to be of the " CC-form. In the presence of interaction, the eigenstate of the composite system will not have a definite number of photons, and the extent of correlation as a result of coupling will also be modified. These changes can be induced by the action of a second cluster operator of the exponential form exp(S). The operator S destroys/creates zero, one, two,..., photons and simultaneously induce various nh-mp excitations out of 3 . The nature of the electronic part of the cluster operator in S is dictated by the nature of the energy difference we are interested in. For IP/EA calculations, V in eq. (5.3.1) will destroy/create an electron from should involve nh-(n+l)p excitations. Similarly, for EE, V will conserve the number of electrons, and S should involve nh-np excitations. For computing the linear response, it suffices to retain only the terms linear in C/CT ... 

The various Fock space methods that we shall describe principally differ in their choice of the actual form of the cluster operator. Following the earlier analysis of Primas/29/, the form of O will be of exponential type, symbolically represented by eq.(7.1.1), but it may not strictly be exponential. There have been several choices for Ci, each having its own advantages. 

In this section we examine some of the critical ideas that contribute to most wavefunction-based models of electron correlation, including coupled cluster, configuration interaction, and many-body perturbation theory. We begin with the concept of the cluster function which may be used to include the effects of electron correlation in the wavefunction. Using a formalism in which the cluster functions are constructed by cluster operators acting on a reference determinant, we justify the use of the exponential ansatz of coupled cluster theory. ... 

For example, the amplitudes if-- or d-- , in which orbitals (j), and (j) are localized on fragment X and orbitals (j)y and are localized on fragment Y, will be zero. Thus, the total coupled cluster exponential operator may be written as a product of independent coupled cluster operators for each fragment, namely, ... 

In the event that the Cl cluster operator, C, is not truncated, however, it is possible to write the resulting full Cl wavefunction as a product of wavefunctions for each separated fragment, since the linear operator may be transformed into an exponential using a generalized form of Eq. [34]. 

The projective techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes that define the coupled cluster wavefunction, e o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, T, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression ... 

The similarity transformed EOM, STEOM-CC [183], approaches the problem somewhat differently, but it also provides an exponential ansatz for excited states, namely = exp(5) exp(7)l0), where exp(5) has a different meaning than before. The method decouples the contributions of higher cluster operators from the lower ones, by using the results for the (1,0) and (0,1) results to define the second similarity transformation, 5, leaving the excited states to be obtained now from a problem of the dimension of a Cl singles (CIS) calculation. This method is a kind of exact CIS for the excited states of molecules, at least those dominated by single excitations. It is very attractive for large-scale application as in our work for free-base porphine [184,185]. Extensions by Nooijen and Lotrich have been made for doubly excited states [186]. 

Diagrammatically, these cluster operators S,- represent the connected /-body terms, i.e. those diagrams which cannot be separated into topologically unconnected parts. Typically, the wave and cluster operators are related to each other due to the exponential ansatz... 

The CC wavefunction had been considered by Coester and Kummel [10] as the exponential S ansatz at about the same time in the nuclear physics literature however, none of these authors took the next step to develop explicit equations for the cluster amplitudes which appear in the cluster operators. 

In literature we use the argument that the wave operator ensures the size consistency of the CC. According to this reasoning, for an infinite distance between molecules A and B, both and 4>o functions can be expressed in the form of the product of the wave functions for A and B. When the cluster operator is assumed to be of the form (obvious for infinitely separated systems) T=Ta + Tb, then the exponential form of the wave operator expfT - - Tb) ensures a desired form of the product of the wave function fexpCr -l- 7b)]Oo = exp Ta exp Tb q. If we took a finite Cl expansion (Ta + then we would not get the product but the sum which... 

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... 

The above representation of the exact ground-state wave function is reminiscent of the exponential ansatz of the single-reference CC theory, Eq. (3). There is, however, a fundamental difference between Eqs. (124) and (3). The cluster operator T entering Eq. (3) is defined in terms of the excitation operators Eq. (6), where ii,..., in (ai,..., an) are the... 

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],... 

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