Configuration-interaction theory size-extensivity

At that time, the best-known method for electron correlation in molecules was undoubtedly configuration interaction (Cl). This tool had developed in the hands of Slater and Condon, with early applications by Boys, Parr, Matsen, and their coworkers around 1950 (see Ref. [2] for an excellent review). Somewhat less known to the quantum chemistry community was the parallel development in the mid 1950s of the correlation problem in physics that originated with Brueckner [3] and Goldstone [4], termed many-body perturbation theory (MBPT) because it was applicable to many-electron systems. This feature, that we now call size-extensivity [5], was not shared by Cl, but was a necessity for the physics applications to nuclear matter and the electron gas. Important questions at this time included the correlation treatment of the high- and low-density electron... 

The coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

It is important to note that, at each level of coupled-cluster theory, we include through the exponential parameterization of Eq. (28) all possible determinants that can be generated within a given orbital basis, that is, all determinants that enter the FCI wave function in the same orbital basis. Thus, the improvement in the sequence CCSD, CCSDT, and so on does not occur because more determinants are included in the description but from an improved representation of their expansion coefficients. For example, in CCS theory, the doubly-excited determinants are represented by ]HF), whereas the same determinants are represented by (T2 + Tf) HF) in CCSD theory. Thus, in CCSD theory, the weight of each doubly-excited determinant is obtained as the sum of a connected doubles contribution from T2 and a disconnected singles contribution from Tf/2. This parameterization of the wave function is not only more compact than the linear parameterization of configuration-interaction (Cl) theory, but it also ensures size-extensivity of the calculated electronic energy. 

One popular modification of the standard coupled-cluster model is the quadratic configuration-interaction (QCI) model, originally introduced as a size-extensive amendment of the Cl model [33]. We here discuss the QCI singles-and-doubles (QCISD) model within the framework of similarity-transformed (linked) coupled-cluster theory, from which it is obtained by omitting certain commutators in the CCSD equations. Expanding the remaining commutators, we then go on to express the QCISD equations in a form that illustrates its historical connection to CISD theory. 

In Chapter 11, we treat configuration-interaction (Cl) theory, concentrating on the full Cl wave function and certain classes of truncated Cl wave functions. The simplicity of the Cl model allows for efficient methods of optimization, as discussed in this chapter. However, we also consider the chief shortcomings of the Cl method - namely, the lack of compactness in the description and the loss of size-extensivity that occurs upon truncation of the Cl expansion. 

---