Size consistency of the CC method

The size consistency of the CC method can be proved on the basis of eq. (10.43) and (10.44). Let us assume that the system dissoeiates into two non-interacting subsystems A and B (i.e. at infinite distanee). Then the orbitals can be also divided into two separable (mutually orthogonal) subsets. We will show that the cluster amplitudes, having mixed indiees (from the first and second groups of orbitals), are equal to 0. 

Let us note first that, for an infinite distanee, the Hamiltonian H = Ha + Hb-In sueh a situation the wave operator ean be expressed as 

As we see we would have trouble if (Cm + — a — e ) is close to 0 (quasidegeneracy of the vacuum state with some other state), because then oo. 

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Size-Consistency of the CC Method Example CC with Double Excitations... 

Above we referred to the development of the CC method by Cizek and Paldus [49-51], The CC method may be viewed as a consistent summation to infinite order of certain type of linked correlation (MBPT, MP) diagrams. Thus, there is a clear relationship between many-body permrbation theory [based on the MP operator of Eq. (Al) in Appendix 37A] and coupled cluster theory. Both are supermolecule methods that give size-extensive energies. 

The coupled-cluster (CC) method is an attempt to find such an expansion of the wave function in terms of the Slater determinants, which would preserve size consistency. In this method, the wave function for the electronic ground state is obtained as a result of the operation of the wave operator exp (T) on the Hartree-Fock function (this ensures size consistency). The wave operator exp (T) contains the cluster operator T, which is defined as the sum of the operators for the Z-tuple excitations, Ti up to a certain maximum I = Zmax. Each 2) operator is the sum of the operators each responsible for a particular l-Uiple excitation multiplied by its amplitude t. The aim of the CC method is to find the t values since they determine the wave function and energy. The method... 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

Another popular post-HF technique is the CC method [4—6]. The CC method solves the size consistency problem of Cl by forming a wave function where the excitation operators are exponentiated ... 

Closely related to the term size consistency is the term size extensi vity . The meaning of both is similar and the terms are often used interchangeably. However, it should be noted that the term size extensive is based on the Linked-cluster theorem and means that there are no unlinked contributions in the energy expression (compare equation 9, for a detailed discussion, see Ref. 50). It can be stated that size-extensive methods, i.e., all CC methods, are size consistent, while the opposite is not necessarily the case. 

An important advantage of MP2 and higher-order perturbation methods is their size-consistency at every order. This is in contrast to many variational Cl methods, for which the calculated energy of two identical non-interacting systems might not be equal to twice that of an individual system. Size-consistent scaling is also characteristic of QCI and CC methods, which are therefore preferable to standard Cl-type variational methods for many applications. 

It is well known that the CC and MBPT energies are size-extensive, while this is not the case for truncated Cl methods. Indeed, it is not difficult to verify (see, e.g.. Ref. [8]) that for a system consisting of n subsystem [e.g., (He) ], the CISD energy is proportional to rather than to n. This is easy to comprehend when we realize that by restricting the excitations to, say, doubles, we ignore simultaneous double excitations on the subsystems A and B when considering the supersystem (A+B), which now represent quadruples, while such excitations are taken into account when handling the systems A and B individually. 

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