Single reference model spaces

Two important remarks are in order here. The first is that eq. (3) does not impose any limitation on the degree of excitation of ([) , so that can be calculated whatever the composition and symmetry of Sq. In this way, even if eqs. (1), (2) and (4) assume a single reference determinant, the SDCI space that constitutes Sq can be built as a MR-SDCI and the procedure described above can be applied. The second is that care must be taken to remove the possible redundancies that could be present if Dj( i belongs to the model space Spg. In such a case, the physical effects included in each term of eq. (3) would be included twice in the diagonalization of H + A. ... 

One of the simplest ways of extending this single-pore model is to assume that variations in the size of pore spaces can be represented by a variable-diameter assembly of such pores, referred to as a parallel bundle of pores. An example is shown in Fig. 3, for a variation of the model applied to a supported zeolite cracking catalyst. In this example [10] the zeolite pores are simply configured along the pore walls, so that the parallel bundle represents the Si/Alumina-support pore spaces. 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

A further restriction on the use of many-body perturbation techniques arises from the (quasi-) degenerate energy structure, which occurs for most open-shell atoms and molecules. In these systems, a single reference state fails to provide a good approximation for the physical states of interest. A better choice, instead, is the use of a multi-configurational reference state or model space, respectively. Such a choice, when combined with configuration interactions calculations, enables one to incorporate important correlation effects (within the model space) to all orders. The extension and application of perturbation expansions towards open-shell systems is of interest for both, the traditional order-by-order MBPT [1] as well as in the case of the CCA [17]. 

For open-shell systems, therefore, the model space is no longer simple and sometimes not even known before the computations really start. For closed-shell systems, in contrast, the model space can be formed by a single many-electron state, which is taken as the reference state and which is sufficient in order to classify the single-electron states into particle (unoccupied) and hole (occupied or core) states, respectively. In open-shell systems, instead, the valence shells are neither empty nor completely filled. To facilitate the handling of such open shells, we shall provide (and discuss) a orbital notation in Subsection 3.4 which is appropriate for the derivation of perturbation expansions. 

In the single reference Cl method, the model space (Fig. 10.8) is formed by a single Slater determinant. In the multireference Cl method, the set of determinants constitute the model space. This time, the Cl expansion is obtained by replacement of the spinorbitals participating in the model space by other virtual orbitals. We proceed further as in Cl. 

Multireference coupled cluster (MRCC) models provide a generalization of the single-reference CC approach (3.1) for applications where several reference determinants contribute with similarly large weights to the wave function of the molecular system under consideration. The MRCC models can be divided into state-specific (SS) and multi-state approaches. One of the state-specific MRCC (SSMRCC) approaches is the active space CC method proposed in the works of Adamowicz and co-workers [14—18]. This method established the foundation for the approach developed and implemented by the authors of this article [19-27]. 

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