Frozen core approximation

HyperChem supports MP2 (second order Mdllcr-l Icsset) correlation energy calcu latiou s u sin g any available basis set. lu order to save main memory and disk space, the HyperChem MP2 electron correlation calculation normally uses a so called frozen-core approximation, i.e. th e in n er sh el I (core) orbitals are omitted. A sett in g in CHHM.IX I allows excitation s from th e core orbitals to be include if necessary (melted core). Only the single poin t calcula-tion is available for this option. 

Most of the scale factors in this table are from the recent paper of Wong. The HF/6-31G(d) and MP2(Full) scale factors are the traditional ones computed by Pople and coworkers and cited by Wong. Note that the MP2 scale factor used in this book is the one for MP2(Full) even though our jobs are run using the (defriultj frozen core approximation. Scott and Radom computed the MP2(FC) and HF/3-21G entries in the table, but this work came to our attention only just as this book was going to press. 

It is usual to make the frozen core approximation in calculations of this type. This means that the seven inner shells are left frozen and not included in the Cl calculation. 

The HF-LCAO calculation follows the usual lines (Figure 11.10) and the frozen core approximation is invoked by default for the CISD calculation. CISD is iterative, and eventually we arrive at the improved ground-state energy and normalization coefficient (as given by equation 11.7) — Figure 11.11. 

The MP2 and CCSD(T) values in Tables 11.2 and 11.3 are for correlation of the valence electrons only, i.e. the frozen core approximation. In order to asses the effect of core-electron correlation, the basis set needs to be augmented with tight polarization functions. The corresponding MP2 results are shown in Table 11.4, where the A values refer to the change relative to the valence only MP2 with the same basis set. Essentially identical changes are found at the CCSD(T) level. 

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

If the electron density were known at high resolution, the antishielding effects would be represented in the experimental distribution, and the correction in Eq. (10.31a) would be superfluous. However, the experimental resolution is limited, and the frozen-core approximation is used in the X-ray analysis. Thus, for consistency, the Rcore shielding factor should be applied in the conversion of the... 

The suitability of light-atom crystals for charge density analysis can be understood in terms of the relative importance of core electron scattering. As the perturbation of the core electrons by the chemical environment is beyond the reach of practically all experimental studies, the frozen-core approximation is routinely used. It assumes the intensity of the core electron scattering to be invariable, while the valence scattering is affected by the chemical environment, as discussed in chapter... 

The interaction energy and its many-body partition for Bejv and Lii r N = 2 to 4) were calculated in by the SCF method and by the M/ller-Plesset perturbation theory up to the fourth order (MP4), in the frozen core approximation. The calculations were carried out using the triply split valence basis set [6-311+G(3df)]. 

The Self-Consistent (SfC) (G)RECP version [23, 19, 24, 27] allows one to minimize errors for energies of transitions with the change of the occupation numbers for the OuterMost Core (OMC) shells without extension of space of explicitly treated electrons. It allows one to take account of relaxation of those core shells, which are explicitly excluded from the GRECP calculations, thus going beyond the frozen core approximation. This method is most optimal for studying compounds of transition metals, lanthanides, and actinides. Features of constructing the self-consistent GRECP are ... 

Such wave functions are orthogonal if we use the so-called frozen core approximation, in which the core wave functions are fixed and only those of excited electrons are recalculated. Such calculations are much simpler and the functions obtained in this way are very close to those obtained with full self-consistency. 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

It follows from Table V that spin-orbit effects are relevant for the heavy metal shieldings and, since the spin-orbit contribution does not always have the same sign, for the relative chemical shifts. In this connection, it is interesting to note that the ZORA spin-orbit numbers are shifted as compared to their Pauli spin-orbit counterparts. This effect can be attributed, at least partly, to core contributions at the metal while scalar contributions of the core orbitals are approximately accounted for by the frozen core approximation (6,7), spin-orbit contributions of the core orbitals are neglected. Hence, the more positive (diamagnetic) shieldings from the ZORA method are due to spin-orbit/Fermi contact contributions of the s orbitals in the uranium core. 

Thus, F(oj) has a complicated codependence. The latter will be mirrored in the photoionization cross section of the encaged atom. Correspondingly, the photoionization cross section of the encaged atom in the dynamical-cage approximation might differ greatly from that in the frozen-core approximation, both quantitatively and qualitatively. 

Note that in the frozen core approximation the orthonomality of the basis functions leads for k = k to vanishing matrix elements < d H single-particle eigenvalues are given by... 

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