Core correlation

The orbitals from which electrons are removed can be restricted to focus attention on the correlations among certain orbitals. For example, if the excitations from the core electrons are excluded, one computes the total energy that contains no core correlation energy. The number of CSFs included in the Cl calculation can be far in excess of the number considered in typical MCSCF calculations. Cl wavefimctions including 5000 to 50 000 CSFs are routine, and fimctions with one to several billion CSFs are within the realm of practicality [53]. 

Additional empirical corrections 1 and 2-electron higher-order corrections (size-consistent), spin contamination 2-electron higher-order correction (size-consistent), spin contamination, core correlation for sodium... 

The magnitude of the core correlation can be evaluated by including the oxygen Is-electrons and using the cc-pCVXZ basis sets the results are shown in Table 11.9. The extrapolated CCSD(T) correlation energy is —0.370 a.u. Assuming that the CCSD(T) method provides 99.7% of the full Cl value, as indicated by Table 11.7, the extrapolated correlation energy becomes —0.371 a.u., well within the error limits on the estimated experimental value. The core (and core-valence) electron correlation is thus 0.063 a.u.. 

The effect of core-electron correlation is small, as shown in Table 11.16. It should be noted that the valence and core correlation energy per electron pair is of the same magnitude, however, the core correlation is almost constant over the whole energy surface and consequently contributes very little to properties depending on relative energies, like vibrational frequencies. It should be noted that relativistic corrections for the frequencies are expected to be of the order of 1 cm" or less. ... 

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

Urban, M. and Sadie), A.J. (2000) Core correlation effects in weak interactions involving transition metal atoms. Journal of Chemical Physics, 112, 5—8. 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

In the calculations presented so far, all electrons have been correlated. However, chemical reactions involve mainly the valence electrons, leaving the core electrons nearly unaffected. It is therefore tempting to correlate only the valence electrons and to let the core orbitals remain doubly occupied. In this way, we avoid the calculation of the nearly constant core-correlation energy, concentrating on the valence correlation energy. The freezing of the core electrons simplifies the calculations as there are fewer electrons to correlate and since it enables us to use the cc-pVXZ basis sets rather than the larger cc-pCVXZ sets. 

Nevertheless, core-correlation contributions to AEs are often sizeable, with contributions of about 10 kJ/mol for some of the molecules considered here (CH4, C2H2, and C2H4). For an accuracy of 10 kJ/mol or better, it is therefore necessary to make an estimate of core correlation [9, 56]. It is, however, not necessary to calculate the core correlation at the same level of theory as the valence correlation energy. We may, for example, estimate the core-correlation energy by extrapolating the difference between all-electron and valence-electron CCSD(T) calculations in the cc-pCVDZ and cc-pCVTZ basis sets. The core-correlation energies obtained in this way reproduce the CCSD(T)/cc-pCV(Q5)Z core-correlation contributions to the AEs well, with mean absolute and maximum deviations of only 0.4 kJ/mol and 1.4 kJ/mol, respectively. By contrast, the calculation of the valence contribution to the AEs by cc-pCV(DT)Z extrapolation leads to errors as large as 30 kJ/mol. 

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]). 

It has been found repeatedly [1, 43, 45] that scalar relativistic contributions are overestimated by about 20 - 25 % in absolute value at the SCF level. Hence inclusion of electron correlation is essential we found the ACPF method (which is both variational and approximately size extensive) to be an excellent compromise between quality and cost. It is reasonable to suppose that for a property that becomes more important as one approaches the nucleus, one wants maximum flexibility of the wavefunction near the nucleus as well as correlation of all electrons thus we finally opted for ACPF/MTsmall as our approach of choice. Typically the cost of the scalar relativistic step is a fairly small fraction of that of the core correlation step, since only n2N4 scaling is involved in the ACPF calculations. 

Table 2.3 Comparison of core correlation contributions to TAEo (kcal/mol) for the W2-1 test set.

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. 

For all results in this paper, spin-orbit coupling corrections have been added to open-shell calculations from a compendium given elsewhere I0) we note that this consistent treatment sometimes differs from the original methods employed by other workers, e.g., standard G3 calculations include spin-orbit contributions only for atoms. In the SAC and MCCM calculations presented here, core correlation energy and relativistic effects are not explicitly included but are implicit in the parameters (i.e., we use parameters called versions 2s and 3s in the notation of previous papers 11,16,18)). 

As the MSFT (38) additivity model predicts only very weak core correlation and scalar relativistic contributions to the proton affinity, we have not attempted their explicit (and very expensive) calculation. 

The real power in the multi-coefficient models, however, derives from the potential for the coefficients to make up for more severe approximations in the quantities used for (/) in Eq. (7.62). At present, Truhlar and co-workers have codified some 20 different multicoefficient models, some of which they term minimal , meaning that relatively few terms enter into analogs of Eq. (7.62), and in particular the optimized coefficients absorb the spin-orbit and core-correlation terms, so they are not separately estimated. Different models can thus be chosen for an individual problem based on error tolerance, resource constraints, need to optimize TS geometries at levels beyond MP2, etc. Moreover, for some of the minimal models, analytic derivatives are available on a term-by-term basis, meaning that analytic derivatives for the composite energy can be computed simply as the sum over tenns. 

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. 

An important task for theory in the quest for experimental verification of N4 is to provide spectral characteristics that allow its detection. The early computational studies focused on the use of infrared (IR) spectroscopy for the detection process. Unfortunately, due to the high symmetry of N4(7)/) (1), the IR spectrum has only one line of weak intensity [37], Still, this single transition could be used for detection pending that isotopic labeling is employed. Lee and Martin has recently published a very accurate quartic force field of 1, which has allowed the prediction of both absolute frequencies and isotopic shifts that can directly be used for assignment of experimental spectra (see Table 1.) [16]. The force field was computed at the CCSD(T)/cc-pVQZ level with additional corrections for core-correlation effects. The IR-spectrum of N4(T>2 ) (3) consists of two lines, which both have very low intensities [37], To our knowledge, high level calculations of the vibrational frequencies have so far only been performed... 

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