Core Approximations

When we think of chemical bonding, we usually think only of the valence orbitals. It is these orbitals that form bonds, and the core orbitals are not involved in the chemistry. To be sure, this is only a qualitative picture, but it raises the question of whether we really need to consider the core orbitals in our calculations. 

Two solutions to this problem are in common use. One is to create a pseudopotential (PP) or effective core potential (ECP) that incorporates the core orthogonality terms along with the potential terms. The result is usually expressed as a combination of 

We will use the term pseudopotential for this kind of potential. In the second, the core orbitals are removed to a very high energy by a level-shifting procedure. As a result, any tendency of the valence orbitals to gain core character is energetically unfavorable. This second method was developed under the name ab initio model potentials (AIMP), though in some quarters this method is also referred to as an effective eore potential or pseudopotential method. We will refer to these potentials as model potentials. The form of the potential is 

Where do we draw the line between the eore and the valenee orbitals If we take the valence orbitals to be those with the highest prineipal quantum number and assign the rest to the core, the further down the periodic table the element is, the more polarizable the core is. Thus, for those elements where the use of a core approximation is most desirable, the core orbitals are also most easily influenced by changes in valenee shell electronic structure. Again, there are two main options include more orbitals in the valence space or introduce a potential to describe the core polarization. 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

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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. 

This case is particularly interesting since the surface segregation energy can be directly compared to surface core level binding energy shifts (SCLS) measurements. Indeed, if we assume that the excited atom (i. e., with a core hole) is fully screened and can be considered as a (Z + 1) impurity (equivalent core approximation), then the SCLS is equal to the surface segregation energy of a (Z + 1) atom in a Z matrixi. in this approximation the SCLS is the same for all the core states of an atom. 

Table 3. Some carbon Is binding energy shifts calculated using the equivalent cores approximation...

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 ... 

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