Inner-shell correlation

The contribution of inner-shell correlation is taken as the difference between the CCSD(T)/MTsmall TAE with and without constraining the inner-shell orbitals to be doubly occupied. 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

We should note that inner polarization is strictly an SCF-level effect while, for instance, switching from an A VDZ to an A,VDZ+2d basis set affects the computed atomization energy of SO3 by as much as 40 kcal/mol ( ), almost all of this effect is seen in the SCF component of the TAE [28], In fact, we have recently found [29] that the effect persists if the (1, v, 2s, 2p) orbitals on the second-row atom are all replaced by a pseudopotential. What is really getting polarized here is the inner part of the valence orbitals, which requires polarizations functions that are much tighter (higher-exponent) than those required for the outer part of the valence orbital. The fact that these inner polarization functions are in the same exponent range as the d and / functions required for correlation out of the (2s, 2p) orbitals is merely coincidental the inner polarization effect has nothing to do with correlation, let alone with inner-shell correlation. 

Inner-shell correlation is a substantial part of the absolute correlation energy even for late first-row systems for second-row systems, it in fact rivals the absolute valence correlation energy in importance. However, its relative contribution to molecular TAEs is fairly small in benzene, for instance, it amounts to less than 0.7 % of the TAE. Even so, at 7 kcal/mol, its contribution is important by any reasonable thermochemical standard. By the same token, a 1 % relative error in a 7 kcal/mol contribution is tolerable even by benchmark thermochemistry standards, while the same relative error in a 300 kcal/mol contribution would be unacceptable even by the chemical accuracy standards. 

For these reasons, we feel justified in treating the inner-shell correlation contribution to TAE as a separate contribution, rather than together with the valence correlation. There are substantial cost advantages to this rather than having to carry out very elaborate all-electrons-correlated CCSD(T) calculations in basis sets near saturation for both valence and inner-shell correlation, we can limit these costly calculations to a basis set that is primarily saturated for inner-shell correlation. 

Inner-shell correlation contributions are found to be somewhat more important for ionization potentials than for electron affinities, which is understandable in terms of the creation of a valence hole by ionization... 

A Bond-Equivalent Model for Inner-Shell Correlation... 

In a pilot Wlh calculation on benzene [1], it was found that 85 % of the CPU time was spent on the inner-shell correlation step. Given that this contribution is about 0.5 % of the TAE of benzene, the CPU time proportion appears to be lopsided to say the least. On the other hand, a contribution of 7 kcal/mol clearly cannot be neglected by any reasonable standard. However, inner-shell correlation is by its very nature a much more local phenomenon than valence correlation, and a relative error of a few percent in such a small contribution is more tolerable than a corresponding error in the major contributions, Martin, Sundermann, Fast and Truhlar (MSFT) [43] investigated the applicability of a bond equivalent model. 

We started by generating a data base of inner-shell correlation contributions for some 130 molecules that cover the first two rows of the periodic table. In order to reduce the number of parameters in the model to be fitted, we introduced a Mulliken-type approximation for the parameters Dab (Da+Db)/2. Furthermore we did retain different parameters for single and multiple bonds, but assumed Da=b (3/2)Da=b-... 

It was recently suggested by Nicklass and Peterson [60] that the use of core polarization potentials (CPPs) [61] could be an inexpensive and effective way to account, for the effects of inner shell correlation. The great potential advantage of this indeed rather inexpensive method over the MSFT bond-equivalent model is that it does not depend on... 

Here we propose a new reduced-cost variant of W1 theory which we shall denote Wlc (for cheap ), with Wlch theory being derived analogously from Wlh theory. Specifically, the core correlation and scalar relativistic steps are replaced by the approximations outlined in the previous two sections, i.e. the MSFT bond additivity model for inner-shell correlation and scaled B3LYP/cc-pVTZuc+l Darwin and mass-velocity corrections. Representative results (for the W2-1 set) can be seen in Table 2.1 complete data for the molecules in the G2-1 and G2-2 sets are available through the World Wide Web as supplementary material [63] to the present paper. 

Inner-shell correlation, at 7 kcal/mol, is of quite nontrivial importance, but even scalar relativistic effects (at 1 kcal/mol) cannot be ig-... 

In systems where a large number of inner-shell electrons makes the inner-shell correlation (and, to a lesser extent, scalar relativistic) steps in W1 and W2 theory unfeasible, the use of a bond equivalent model for the inner-shell correlation and scaled B3LYP/cc-pVTZuc+l scalar relativistic corrections offers an alternative under the name of Wlc and Wlch theories. 

The Wlc total atomization energy at 0 K of aniline, 1468.7 kcal/mol, is in satisfying agreement with the value obtained from heats of formation in the NIST WebBook 39), 1467.7 0.7 kcal/mol. (Most of the uncertainty derives from the heat of vaporization of graphite.) The various contributions to this result are (in kcal/mol) SCF limit 1144.4, valence CCSD correlation energy limit 359.0, connected triple excitations 31.7, inner shell correlation 7.6, scalar relativistic effects -1.2, atomic spin-orbit coupling -0.5 kcal/mol. Extrapolations account for 0.6, 12.1, and 2.5 kcal/mol, respectively, out of the three first contributions. 

Comparison with experimental values are meaningful only for boron and aluminum. The MRCI values for AI (0.45 eV) and the MCDF results for B (0.26 eV) and A1 (0.43 eV) are in good agreement with the Hotop and Lineberger values (0.28 and 0.44 eV, respectively). The MRCI and MCDF EAs for the other atoms agree with each other (0.29 and 0.30 eV for Ga, 0.38 and 0.39 eV for In, 0.27 and 0.29 eV for Tl). The RCC EA of TI is much higher at 0.40(5) eV. A major difference between the RCC and the other two methods lies in the number of electrons correlated. While [52] and [53] correlate valence electrons only, three for the neutral atom and four for the anion, we correlated 35 electrons in Tl and 36 in TE. A RCC study of all five elements was undertaken [54], with the aim of determining all five EAs and, in particular, the effect of inner-shell correlation and virtual space used on the calculated values. 

The first seven configurations remain almost exclusively perfect-paired. As for the eighth configuration, it too turns out to be almost exclusively perfect-paired the coefficient of the perfectly-paired YK spin function is in fact 0.982. In any case, the fact that its first two orbitals are identical rules out nine YK spin functions out of fourteen, and symmetry requirements further mandate three linear constraints on the coefficients of the five allowed spin functions, so that only two of them are truly independent. Anyway, one can legitimately conclude that the seven-configuration wavefunction is qualitatively robust with respect to the inclusion of this kind of inner shell correlation. 

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). 

For the other elements in the second row, there have been few direct calculations on the importance of 2s2p correlation. Possibly the best indication of the effect of inner-shell correlation comes from benchmark calculations determining the valence (complete) basis set limit. In Table 7 we show the intrinsic error computed at the CASSCF/ICMRCI level reported by Woon and Dunning for a series of second row molecules. The intrinsic error is the difference in the CBS limit at a specified level of correlation and experiment. It is the composite of a number of small effects including the error in the valence electron correlation treatment (n-particle error), any error in the CBS extrapolation, the relativistic correction, and the effect of inner-shell correlation as well as any error in... 

By including the most important left-right, angular, and in-out pair correlation effects the Li2 bond dissociation energy (De = 1.05 eV) is reproduced with a small error of 6% while at the HF level (De = 0.17 eV) the error is larger than 80%. Mostly it is sufficient to consider just the correlation effects of valence electron pairs. Inner-shell correlation effects can be ignored (frozen core description). However, for larger atoms the core is polarized and inner-shell correlation becomes important. 

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