Connections between Coupled Cluster. Configuration Interaction--------and Perturbation Theory

10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 

10 COUPLED CLUSTER, CONFIGURATION INTERACTION AND PERTURBATION THEORY 

The principal deficiency of CISD is the lack of the TI term, which is the main reason for CISD not being size extensive. Furthermore, this term becomes more and more important as the number of electrons increases, and CISD therefore recovers a smaller and smaller percentage of the correlation energy as the system increases. There are various approximate corrections for this lack of size extensivity which can be added to standard CISD. The most widely known of these is the Davidson correction, sometimes denoted CISD - - Q(Davidson), where the quadruples contribution is approximated as 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples 

Since the singly excited determinants effectively relax the orbitals in a CCSD calculation, non-canonical HF orbitals can also be used in coupled cluster methods. This allows for example the use of open-shell singlet states (which require two Slater determinants) as reference for a coupled cluster calculation. 

The general cluster operator is given by eq. (4.69), where terms have been collected according to the excitation they generate. 

If the renormalization of the wave function is also taken into account, the (1 - al) quantity is divided by ai, and the corresponding correction is called the renormalized Davidson correction. The effect of higher order excitations is thus estimated from the correlation energy obtained at the CISD level times a factor that measures how important the single-determinant reference is at the CISD level. The Davidson correction does not yield zero for two-electron systems, where CISD is equivalent to full Cl, and it is likely that it overestimates the higher order corrections for systems with few electrons. More complicated correction schemes have also been proposed, but are rarely used. 

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