Number correlation energy contributions

In these two equations, xp/r) are the molecular orbitals (MOs), p(r) is the electronic density, v, (r) is the external potential felt by electrons, is the set point for the constraint, and w(r) is a weight function that defines the constraint property. The Coulomb and exchange-correlation energy contributions are denoted by T[p] and Exclp]- The cDFT energy equation can be applied, for example, to constrain a given number of electrons to occupy a specific volume Q. Alternatively, if the... 

This reasoning is rather naive. Significant correlation energy contribution can result from a small energy-level paration between filled and empty MOs (rather than from merely the number of electrons), but production of a tion should normally increase this gap and lead to reduced correlation. 

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... 

The operation of Eq. (3.3) is illustrated by the results given in Table 2 out of 48 molecules of the cc-pVTZ set. They are listed in order of increasing correlation energy. The first column of the table lists the molecule. The next 6 columns show how many orbitals and orbital pairs of the various types are in each molecule, i.e. the numbers Nl, Nb, Nu, Nlb etc. The seventh column lists the CCSD(T)/triple-zeta correlation energy and the eight column lists the difference between the latter and the prediction by Eq. (3.3). The mean absolute deviation over the entire set of cc-pVTZ data set is 3.14 kcal/mol. For the 18 molecules of the CBS-limit data set it is found to be 1.57 kcal/mol. The maximum absolute deviations for the two data sets are 11.29 kcal/mol and 4.64 kcal/mol, respectively. Since the errors do not increase with the size of the molecule, the errors in the estimates of the individual contributions must fluctuate randomly within any one molecule, i. e. there does not seem to exist a systematic error. The relative accuracy of the predictions increases thus with the size of the system. It should be kept in mind that CCSD(T) results can in fact deviate from full Cl results by amounts comparable to the mean absolute deviation associated with Eq. (3.3). 

Figure 4. Contributions of correlating functions, as well as s, p, and d functions (inset), to the CISD correlation energy of the 5 d state of mercury. The absolute values of the incremental correlation energy lowerings, AEcon are plotted in mEh against the number offunctions in the expansions for spdf... functions. The solid lines are exponential fits.

E. R. Lippincott The proposed model is certainly empirical. However, the internuclear potential function used for the terms V1 and F2 may be derived from a quantum mechanical model which lends support to their use in such a treat-ment of hydrogen bond systems. Professor Pauling is quite right in suggesting that the terms Vx and F2 may include some electrostatic contribution, since it is known that the internuclear potential function used correlates properties fairly well for partial polar bonds. Nevertheless the fact that additional terms of the electrostatic type are not needed to describe a number of the important properties of hydrogen bond systems, suggests that the covalent, repulsion and dispersions energy contributions are more important than the electrostatic contribution. 

Finally, one should not overlook the possible role of correlation effects in atom-metal differences. In atoms the dominant contributions to the correlation energy arise from interaction between electrons of the same principal quantum number n, since these have the greatest overlap. As far as purely intraatomic electrons are concerned, the correlation terms differ very little between atom and metal correlation does, of course, affect interatomic screening of the final state vacancy in metals. 

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