Shell Correlation Energy

In section 4, we established that the orbital truncation error represents a serious obstacle to the accurate calculation of AEs. Next, in section 5, we found that this problem may be solved in two different ways we may either employ wave functions that contain the interelec-tronic distance explicitly (in particular the R12 model), or we may try to extrapolate to the basis-set limit using energies obtained with finite basis sets. In the present section, we shall apply both methods to a set of small molecules, to establish whether or not these techniques are useful also for systems of chemical interest. 

Hartree-Fock convergence does not present an insurmountable difficulty there are clear indications that the molecular Hartree-Fock energy converges as exp (—aX) and thus rather rapidly [51-53], 

This convergence is significantly accelerated by applying the extrapolation formula (5.14), the errors being reduced to 1 - 3 kJ/mol for all extrapolations except for the cc-pV(DT)Z energies. (Here and elsewhere we shall use the notation cc-pV(X - 1,X)Z for the energy obtained by extrapolation from the cc-pV(X - 1)Z and cc-pVXZ correlation energies.) 

6 Obtained from experimental dissociation energies and estimated total atomic energies, see text. c lAi state. 

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Table 5. Valence shell correlation energy in H2O1) given by MB-RSPT treatment116) and the INO-CI calculations96) including all singly and doubly excited configurations (Cl-SD)...

The valence shell correlation energies in Tables I and II were based on a frozen ls-3p UHF core. Additional calculations with only a Is or ls-2p UHF core indicate that correlation energy differences for different d-electronic configurations can change typically by < 0.1 eV when correlation of the 3s/3p shell is included in the MP model. While these changes are rather small, they can correspond to appreciable relative changes (typically 10-20%). So as to eliminate this source of uncertainty, we have employed a frozen ls-2p core for... 

Table I. Valence Shell Correlation Energy (eV) for the 3d / D States of Cr and Co " " ...

Table II. Valence Shell Correlation Energy for Co " " and Co Ions...

MH2 j which permits the valence-shell correlation energy to be picked... 

Valence shell correlation energy E/E,) in HpO given by MB-RSPT... 

The entries in Tables 4.6 and 4,7 are only valence shell correlation energies but no restriction was imposed upon the number of used unoccupied orbitals given by the particular basis set. From Tables 4.5... 

In the case of the basis applied one can estimate the valence-shell correlation, which is in its calculation about 70-75%. [The total energy per unit cell at the MP2 level is — 77.168 a.u., at the HF level —76.893 and their difference is 7.5 eV.55 On the other hand the valence shell-correlation energy of an acetylene unit was estimated to be 10 eV56]. Extrapolation to 100% correlation has given a gap of 2.5 eV.M... 

An individual interpair (or interbond) correlation energy srs increases considerably in absolute value on going from LiH to CH4. The reason is that the bond angle decreases steadily from LiH to CH4 and so the electrons in the different bonds come increasingly closer to each other. The closer two pairs are, the stronger is their interpair correlation interaction. Since the number of interpair terms increase (compared to the number of pairs) in the same series, one finds a substantial increase of the interpair contribution to the total valence-shell correlation energy on going from LiH (0%) to CH4 (almost 50%). 

A. Pipano, R. R. Gilman, and I. Shavitt, Invariance of Inner Shell Correlation Energy with Geometry Changes in a Polyatomic Molecule, Chem. Phys. Lett. 5, 285-287 (1970). 

One would like to know, of course, the percent of the full valence-shell correlation energy included in Ei with the best spd basis used. One can obtain an approximate answer to this question if we recall that the valence-shell correlation energy of an acetylene unit was estimated to be about — 10 eV therefore our best energy ( 7.5 eV) should cover 70 to 75% of the total value. Nearly the same result was also obtained recently for an infinite atomic-hydrogen model chain (see Table 5.2). 

In this way hf/n = 392.036284 H was obtained for the infinite polymer. The valence-shell correlation energy per monomer is — 0.663012 H ( 18.5 mH per valence electron), which from previous experience indicates that about half of the estimated full correlation energy can be achieved with the given method and basis set. ... 

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