Single-orbital cluster operators

Using these so-called second-quantized operators, we may define the single-orbital cluster operator... 

If one continues this process to include all cluster functions for up to N orbitals (four in the case discussed here), as well as single-orbital cluster functions that account for adjustment of the one-electron basis as other cluster functions are added, we could obtain the exact wavefunction within the space spanned by the (j)p. On the other hand, we might assume that clusters larger than pairs are less important to an adequate description of the system—an assumption supported by the fact that the electronic Hamiltonian contains operators describing pairwise electronic interactions at most. We therefore write a four-electron wavefunction that includes all clusters of only one and two orbitals as ° ... 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

Following the customary terminology, we will call inactive holes the inactive occupied orbitals, doubly filled in every model CSF. The inactive particles will refer to aU the orbitals unoccupied in every CSF. Orbitals which are occupied in some (singly or doubly) but unoccupied in others are the active orbitals. In our spin-free form, the labels are for orbitals only, and not for spin orbitals. From the mode of definition, no active orbital can be doubly occupied in every model CSF. We want to express the cluster operator T, inducing excitations to the virtual functions, in terms of excitations of minimum excitation rank, and at the same time wish to represent them in a manifestly spin-free form. To accomplish this, we take as the vacuum—for excitations out of 4> — the largest closed-shell portion of it, For each such vacuum, we redefine the holes and particles, respectively, as ones which are doubly occupied and unoccupied in < 0 a-The holes are denoted by the labels. .., etc. and the particle orbitals are denoted as a, etc. The particle orbitals are totally unoccupied in any or are necessarily... 

In the Ansatz above, the full cluster operator is split into two parts T and T. The cluster operator T, taken to be independent of p, is restricted to purely inactive excitations of the type singles [Tp (inactive h inactive p)] and doubles [T2- (inactive 2h —> inactive 2p)] and the associated projector is denoted by the symbol Q. The excitations due to the other cluster operator involve at least inactive (3h — 3p) type or at least one active orbital line. The corresponding projector is labeled by Qi. In our formalism, the excitations due to the cluster operators and T are described by the symbol exi and ex2, respectively. The virtual space is spanned by the sets of exi and ex2 type of functions and hence... 

The most common approximation in coupled-cluster theory is to truncate the cluster operator at the doubles level, yielding the coupled-cluster singles-and-doubles (CCSD) model [5]. In this model, the T2 operator describes the important electron-pair interactions and T carries out the orbital relaxations induced by the field set up by the pair interactions. The CCSD wave function contains contributions from all determinants of the FCl wave function, although the highly excited determinants, generated by disconnected clusters, are in general less accurately described than those that also contain connected contributions. However, the disconnected contributions may... 

For the singles and doubles, the singlet parametrization of the cluster operator was discussed in Section 13.7.1. We shall not carry out a similar, rigorous derivation of the singlet operato for the triples (14.4.6) but note that the given parametrization of the triples space is redundant. Thus, for a set of three different virtual orbitals a, b and c and three different occupied orbitals i, j and k, there are six amplitudes t, and not related by the permutational symmetries... 

Results. Calculations were carried out at two internuclear separations, the equilibrium Re = 2.0844 A as in Ref. [89], and 2.1 A, for comparison with Ref. [127]. The relativistic coupled cluster (RCC) method [130, 131] with only single (RCC-S) or with single and double (RCC-SD) cluster amplitudes is used (for review of different coupled cluster approaches see also [132, 133] and references). The RCC-S calculations with the spin-dependent GRECP operator take into account effects of the spin-orbit interaction at the level of the one-configurational SCF-type method. The RCC-SD calculations include, in addition, the most important electron correlation effects. 

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