Core correlation, calculations

Since such core-correlation calculations are extremely demanding in terms of computer time, both because of the extra basis functions required and because of the extra electrons in the correlation treatment, the use of additivity corrections may be a more appealing alternative. At least for first-row compounds, this appears to work very well. ... 

In this article we briefly look at the computational requirements for determining the core correlation contribution to atomic and molecular properties. The focus in the next two sections will be on the differences in the computational requirements between valence-only and core-correlated calculations. Specifically, the importance of using a size-extensive method and the importance of augmenting valence optimized basis sets will be stressed. In the next section, the core polarization potential will be introduced. This has application with... 

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]. 

HypcrChcrn sup )orLs MP2 (second order Mollcr-Plessct) correlation cn crgy calculation s tisin g ah initio rn cth ods with an y ava liable basis set. In order lo save mam memory and disk space, the Hyper-Chern MP2 electron correlation calculation normally uses a so called frozen -core" appro.xiniatioii, i.e. the in n er sh ell (core) orbitals are om it ted,. A sett in g in CHKM. INI allows excitation s from the core orbitals lo be included if necessary (melted core). Only the single poin t calcii lation is available for this option. ... 

Rappe, Smedley and Goddard (1981) Stevens, Basch and Krauss (1984) Used for ECP (effective core potentitil) calculations Dunning s correlation consistent basis sets (double, triple, quadmple, quintuple and sextuple zeta respectively). Used for correlation ctilculations Woon and Dunning (1993)... 

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. 

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. 

In general, in the correlation calculations referred to in this chapter, the core electrons are frozen. In the few cases where they are included in the correlation space, the designation fu (standing for full) is used. 

As in the recent QCCD study by Head-Gordon et al. (28, 128), we tested the ECCSD, LECCSD, and QECCSD methods, based on eqs (52)-(59), using the minimum basis set STO-3G (145) model of N2. In all correlated calculations, the lowest two core orbitals were kept frozen. As in the earlier section, our discussion of the results focuses on the bond breaking region, where the standard CCSD approach displays, using a phrase borrowed from ref 128, a colossal failure (see Table II and Figure 2). 

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. 

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. 

The splitting factor of the d-polarization functions for the 3df basis set extension is 3 rather than the factor of 4 used for first- and second-row atoms. The 3d core orbitals and Is virtual orbitals are frozen in the single-point correlation calculations. 

Example 4. Calculation of CBS-Q Energy for CH4 The geometry is first optimized at the HF/6-31G(d ) level and the HF/6-31G(d ) vibrational frequencies are calculated. The 6-31G(d ) basis set combines the sp functions of 6-31G with the polarization exponents of 6-311G(d,p). A scale factor of 0.91844 is applied to the vibrational frequencies that are used to calculate the zero-point energies and the thermal correction to 298 K. Next the MP2(FC)/6-31G(d ) optimization is performed and this geometry is used in all subsequent single-point energy calculations. In a frozen-core (FC) calculation, only valence electrons are correlated. 

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... 

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... 

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