Unlinked clusters

In fact, for tightly localized electron pairs, the dominant excitation level is the value of k nearest "vO.OlN (i.e., for about 200 electrons the double excitations in aggregate are more important than the SCF configuration and for 400 electrons quadruple excitations should dominate). Even for molecules with only 40 electrons quadruple and higher excitations must be considered in order to reproduce excitation energies (30) or potential surfaces to an accuracy of 0.1 eV. Thus, configuration interaction calculations for very large molecules are hopeless unless perturbation theory can be used to correct for unlinked cluster effects. 

Table 1. Lowest order of perturbation theory (LOPT) in which various linked and unlinked clusters first appear51)...

Linked clusters LOPT Unlinked clusters LOPT... 

Table 2. Four-electron correlation and unlinked clusters in Be atom47)...

Orbitals were generated in the single-configuration SCF approximation and dynamical correlation was treated in one-reference state Contracted Cl (CCI) calculations (15). The Davidson correction (16) was finally added to the CCI energies to account for unlinked cluster contributions. 

CEPA is not a variational method, but the high accuracy achieved in the calculated properties suggests that the advantage of furnishing an upper bound to the energy is no so important. In contrast to PNO-CI, the approximate inclusion of unlinked clusters ensures the correct dependence with respect to the number of particles. As with any theoretical approach going beyond the Hartree-Fock level, the portion of the correlation energy accounted for depends on the size of the basis set. Polarization functions were found to be very important because they... 

It is the need to remove the unlinked clusters and the introduction of Feynman diagrams which make MBPT [and CC theory] appear unfamiliar to quantum... 

Correlation calculations were carried out at the 6-31G HF optimized geometries. In the MP and Cl calculations, all single (S) and double (D) excitations were included, with core-like orbitals frozen. The energies obtained by the Cl calculations were further improved with the Davidson formula to allow for unlinked cluster quadruple correction (QC). b From Reference 175b. 

As we shall see below, the main mathematical function of unlinked clusters is to cancel the normalization terms in the denominator of Eq. (37.) This cancellation of unlinked clusters - plays an important role in infinite many-body problems, because — co as N — Rayleigh-Schrodinger (R.S.) perturbation theory but it must be recognized in variational methods by augmenting the trial functions with imlinked clusters. > ... 

One can see in a very simple way how some of the unlinked clusters come in. Take the terms for example. These are corrections to the H.F. orbitals i, so suppose that each i in is replaced by (i -j- ft) and the products are expanded. Then... 

Once the are known, a better w.f. than Xs is obtained by including all their unlinked clusters. This is actually quite significant, since with x, alone we will get normalization terms in , Eq. (58), 1 -f- which are of comparable magnitude... 

The prime on denotes that unlinked clusters are included. Note that in the products of the [Pg.350]

With unlinked clusters many more diagrams arise. The complete variational energy of the many-electron theory becomes, after the substitution of x s Eq. (77), in E, Eq. (58),... 

Dfj contains all the pairs except the (ij) one. It arises - because the unlinked clusters in Eq. (77) give rise to factors multiplpng the Eq. (78b). 

TABLE la. Unlinked Clusters vs. Four-Electron Correlations in the Beryllium Atom... 

The differences between the full C.I. coefficients and the unlinked cluster coefficients are due to the true four-electron correlations, L/1234. They are very small. 

These 3e contributions were estimated from the C.I. results for Be and LiH. They came out less than 0.01 ev in LiH. In Be, the only energy effect of unlinked clusters was shown to come from DJD [compare Eqs. (78) and (82) partial cancellation of normalization], showing that the rest of R is also negligible (see Table I). 

The theory is greatly simplified, because not only the but also their many unlinked clusters [Eqs. (49)-(55)] drop out. This makes the complications that Brueckner-type generalized SCF theories would introduce unnecessary. One may still estimate the/-, but being small they do not have to be made self-consistent with the M,/s. 

In Watson s 37 configuration C.I., the ls (2/>) configuration contributed about 1.2 ev (corrected for unlinked clusters). Note that in the many-electron theory, even when C.I. is needed, it is done only on one pair at a time, requiring very few matrix elements and a very small secular equation compared with the full C.I. treatment applied to the entire f and H. 

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