Coupled size consistency

There is a variation on the coupled cluster method known as the symmetry adapted cluster (SAC) method. This is also a size consistent method. For excited states, a Cl out of this space, called a SAC-CI, is done. This improves the accuracy of electronic excited-state energies. 

The electron correlation problem remains a central research area for quantum chemists, as its solution would provide the exact energies for arbitrary systems. Today there exist many procedures for calculating the electron correlation energy (/), none of which, unfortunately, is both robust and computationally inexpensive. Configuration interaction (Cl) methods provide a conceptually simple route to correlation energies and a full Cl calculation will provide exact energies but only at prohibitive computational cost as it scales factorially with the number of basis functions, N. Truncated Cl methods such as CISD (A cost) are more computationally feasible but can still only be used for small systems and are neither size consistent nor size extensive. Coupled cluster... 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

A filter of realistic size consists of several thousand channels so its direct simulation via the numerical solution of a coupled discrete multichannel problem is an intractable task with the currently available computational resources. Alternatively the scale-up problem may be faced employing a continuum model of the filter honeycomb structure. 

The final topic we will discuss in this chapter is size-consistency, which has been mentioned several times already. A method is said to be size-consistent if the computed energy of the composite system A + B, with A and B at infinite distance from each other, yields the same energy as if the method is applied to A and B separately and the energies axe added, i.e. E(A+B)=E(A)+E(B). Some of the methods we have discussed are automatically size-consistent. This is true, for example, for the Hartree-Fock method and the complete Cl method, and it is also true for the methods discussed in the chapter on perturbation theory, such as the coupled cluster method. It is, however, not true for the SD-CI or the MR-CI method. We will in this section show that it is possible, by a slight modification of the formalism, to correct these Cl methods to be approximately size-consistent. The experience gathered over the past two decades on size-consistency corrections indicates that the calculated results axe much improved at the SD-CI level, whereas relative energies axe improved at the MR-CI level but the situation for geometries is less clear at this level. 

Another point worth making is that since the SD-CI method is exact within the chosen basis set for a two-electron system, it must be size-consistent in this particular case. Nevertheless, when Davidson s correction is applied to an SD-CI wave-function for a two-electron system it will give a non-zero contribution, which is thus an artefact of this correction. (The same error appears also when the functional (10.1) is used with g=0.) This artefact can be simply removed and this is done in the Averaged Coupled Pair Functional (ACPF) method. In this method the factor g is considered to be a function of the number of electrons N, g=g(N), and one considers the special case of n separated He atoms. If the denominator De in (10.1) for one He atom is... 

One way of achieving size-consistency for a dissociation process is to use an MCSCF wave function as the reference. Unfortunately, as noted above, there are as yet no general multireference perturbation theory or multireference coupled-cluster treatments that can be applied to such an MCSCF reference function. For rather few electrons, as we shall see, the MRCI approach performs acceptably. 

For the treatment of electron correlation, Cizek uses classical techniques as well as techniques based on mathematical methods of quantum field theory, namely, a coupled-cluster approach. A rapid development and deployment of these methods during the past decade was stimulated by the realization of the importance of size consistency or size extensivity in the studies of reactive chemical processes. Although truly remarkable accuracy and development have been achieved for ground states of closed-shell systems, an extension to quasidegenerate and general open-shell systems is most challenging. Cizek also works on the exploitation of these approaches to study the electronic structure of extended systems (molecular crystals, polymers107). His many interests in-... 

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 ... 

Mahapatra, U. S. Datta, B. Mukheijee, D. A size-consistent state-specific multireference coupled cluster theory Formal developments and molecular applications, J. Chem. Phys. 1999,110, 6171-6188. 

In principle, a Cl approach provides an exact solution of the many-electron problem. In practice, however, only a finite set of Slater determinants can be handled in the linear expansion. A common procedure is to retain all Slater determinants that differ from the HF determinant by one or two excitations (although one-electron excitations do not couple directly to the ground state they couple with two-electron excitations, which in turn affect the ground state indirectly). Unfortunately, such a procedure is not size consistent. For example, the energy of two highly separated monomers will not be twice that of a single monomer in such a truncated Cl calculation. Fortunately, a slightly modified approach called quadratic Cl has recently been developed (Pople et al., 1987) that is size consistent. 

This property of the coupled cluster energy is commonly known as size consistency. ... 

This size inconsistency occurs because the two open-shell electrons on the atoms must be singlet-coupled to produce the correct dissociation limit, and a supermolecule, two-determinant approach is therefore required. This difficulty also applies to coupled cluster or perturbation-based wavefunctions that use the RHF determinant as a reference these methods cannot be size consistent for a given molecular system unless the reference wavefunction is size consistent. 

---