Correlation core-valence

In addition, for thermochemical purposes we are primarily interested in the core-valence correlation, since we can reasonably expect the core-core contributions to largely cancel between the molecule and its constituent atoms. (The partitioning between core-core correlation -involving excitations only from inner-shell orbitals - and core-valence correlation - involving simultaneous excitations from valence and inner-shell orbitals - was first proposed by Bauschlicher, Langhoff, and Taylor [42]). 

Optimization of augmenting functions for the description of electron affinities, weak interactions, or core-valence correlation effects. 

In the case of core-valence correlation effects, correlating functions were optimized at the CISD level of theory using the weighted core-valence scheme (5). In this case a cc-pwCVTZ-PP set consisted of the cc-pVTZ-PP basis set with the addition of 2.y2p2(5fl/core-valence correlating functions. 

Relativistic Quantum Defect Orbital (RQDO) calculations, with and without explicit account for core-valence correlation, have been performed on several electronic transitions in halogen atoms, for which transition probability data are particularly scarce. For the atomic species iodine, we supply the only available oscillator strengths at the moment. In our calculations of /-values we have followed either the LS or I coupling schemes. 

In Tables -A, we report oscillator strengths for some fine structure transitions in neutral fluorine, chlorine, bromine and iodine, respectively. Two sets of RQDO/-values are shown, those computed with the standard dipole length operator g(r) = r, and those where core-valence correlation has been explicitly introduced, Eq. (10). As comparative data, we have included in the tables /-values taken from critical compilations [15,18], results of length and velocity /-values by Ojha and Hibbert [17], who used large configuration expansions in the atomic structure code CIVS, and absolute transition probabilities measured through a gas-driven shock tube by Bengtson et al. converted... 

It is interesting to note here that at the time of these calculations the best value for the separation was virtually the same as the [4s 3p 2d 1//3s 2p Id] result, but was obtained with a basis roughly twice the size. In order to emphasize the importance of computing all terms, I have included theoretical estimates of the zero-point vibration and core-valence correlation contribution to the splitting. It can be seen that accounting for only zero-point contributions would give a theoretical To value of 8.97 kcal/mol, in almost exact agreement with experiment. However, core-valence... 

In addition, none of these calculations involve correlation of more than ten electrons, so no correlation effects from the core electrons axe included at all. Explicit inclusion of the core electrons at the CPF level was found to increase De by about 0.7 kcal/mol in calculations by Ahlrichs and co-workers [69], while in calculations by Almlof and co-workers [68] the same increase was obtained by a completely different technique (inclusion of only core-valence correlation effects, as described in Sec. 6.2). Hence it appeared safe to assume that core correlation would increase De by less than 1 kcal/mol. However, recent calculations by Werner and Knowles [70] give a larger effect of about 1.5 kcal/mol, so this question is not yet settled. 

For the very highest accuracy, the effect of at least core-valence correlation should be explored. This must be accompanied by some serious effort to extend the basis so that core-correlating functions are included. Using valence-optimized basis sets and including core correlation is not only a waste of computer time, but a potential source of problems, as it can substantially increase BSSE. This point is not well appreciated the prevailing view appears to be that no harm can come of correlating the core when the basis set is inadequate. This is not so. 

Several empirical corrections are added to the resulting energies in the CBS methods to remove the systematic errors in the calculations (see Table 10). The CBS-Q method contains a two-electron correction term similar in spirit to the higher level correction used in G2 theory, a spin correction term to account for errors resulting from spin contamination in UHF wavefunctions for open-shell systems, and a correction to the sodium atom to account for core-valence correlation effects. The CBS-4 and CBS-q methods also contain a one-electron... 

Another series of composite computational methods, Weizmann-n (Wn), with n = 1-4, have been recently proposed by Martin and co-workers W1 and W2 in 1999 and W3 and W4 in 2004. These models are particularly accurate for thermochemical calculations and they aim at approximating the CBS limit at the CCSD(T) level of theory. In all Wn methods, the core-valence correlations, spin-orbit couplings, and relativistic effects are explicitly included. Note that in G2, for instance, the single-points are performed with the frozen core (FC) approximation, which was discussed in the previous section. In other words, there is no core-valence effect in the G2 theory. Meanwhile, in G3, the corevalence correlation is calculated at the MP2 level with a valence basis set. In the Wn methods, the core-valence correlation is done at the more advanced CCSD(T) level with a specially designed core-valence basis set. 

In our calculations we associate to each Li atom a (10s2p) atomic centered gaussian basis set contracted to [4s,2p] (Table 3). We treat only the valence electrons at the VB level and keep the inner shell electrons (LiIs) in a core obtained by HF calculations. Therefore we are neglecting the core-core and core-valence correlation effects, which are small for these small lithium clusters (Fig.5). 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

G. Igel, and H. PreulS. Int. J. Quantum Chem., 26, 725 (1984). Pseudopotential Calculations Including Core-Valence Correlation Alkali and Noble-Metal Compounds. M. Dolg, U. Wedig, H. Stoll, and H. PreulS, /. Chem. Phys., 86, 866 (1987). Energy-Adjusted Ab Initio Pseudopotentials for the First Row Transition Elements. M. Dolg, H. Stoll, A. Savin, and H. PreulS, Theor. Chim. Acta, 75,173 (1989). Energy-Adjusted Pseudopotentials for the Rare Earth Elements. 

Although the frozen-core approximation underlies all ECP schemes discussed so far, both static (polarization of the core at the Hartree-Fock level) and dynamic (core-valence correlation) polarization of the core may accurately and efficiently be accounted for by a core polarization potential (CPP). The CPP approach was originally used by Meyer and co-workers (MUller etal. 1984) for all-electron calculations and adapted by the Stuttgart group (Fuentealba et al. 1982) for PP calculations. The... 

The use of CPPs to account for core-valence correlation effects of inner shells in combination with accurate relativistic small- or medium-core ECPs (Yu and Dolg 1997) may be a useful direction for future developments, especially in view of the large computational effort for an explicit treatment of core-valence correlation in case of d and/or f shells and the significant basis-set superposition errors occurring at the correlated level (Dolg et al. 2001). 

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