The Frozen-Core Approximation

From a formal point of view, we can represent the spinors of any system of interest in terms of a complete orthonormal set of spinors located on a single center. These spinors include both bound state and continuum spinors and range over all angular momenta. In an application, the particular choice of spinors will be determined by the state of the atom whose core we wish to freeze. But for the purpose of analysis, it does not matter which spinors we choose provided the set is complete and orthonormal. 

The core spinors selected from this set are occupied in all possible configurations. The many-electron wave function is partitioned into a core and a valence part, 

we integrate the contribution due to the core electrons out of the Hamiltonian, and work with a valence Hamiltonian and a valence wave function. The valence Hamiltonian (from which we exclude the core energy) is 

The sums now range over the valence electrons. The one-electron operator has been replaced by the core Fock operator. 

In order to be able to relax the constraints of orthogonality to the core, we must now introduce projection operators. The core projection operator is 

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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. 

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 ... 

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... 

The structural parameters and vibrational frequencies of three selected examples, namely, H2O, O2F2, and B2H6, are summarized in Tables 5.6.1 to 5.6.3, respectively. Experimental results are also included for easy comparison. In each table, the structural parameters are optimized at ten theoretical levels, ranging from the fairly routine HF/6-31G(d) to the relatively sophisticated QCISD(T)/6-31G(d). In passing, it is noted that, in the last six correlation methods employed, CISD(FC), CCSD(FC),..., QCISD(T)(FC), FC denotes the frozen core approximation. In this approximation, only the correlation energy associated with the valence electrons is calculated. In other words, excitations out of the inner shell (core) orbitals of the molecule are not considered. The basis of this approximation is that the most significant chemical changes occur in the valence orbitals and the core orbitals remain essentially intact. On... 

Frozen-core orbitals are doubly occupied and a spin integration has been performed for the core electrons. The summation index k therefore runs over spatial orbitals only. Employing the frozen-core approximation considerably shortens the summation procedure in single excitation cases. Double excitation cases are left unaltered at this level of approximation. The computational effort can be substantially reduced further if one manages to get rid of all explicit two-electron terms. 

Intramolecular nucleophilic substitution to form thiiranes was studied by means of ab initio MO computations based on the 6-31G basis set <1997JCC1773>. Systems studied included the anions SCH2CH2F and CH2C(=S)CH2F which would afford thiirane and 2-methylenethiirane, respectively (Equations Z and 3). It was important to include electron correlation which was done with the frozen-core approximation at the second-order Moller-Plesset perturbation level. Optimized structures were confirmed by means of vibrational frequency calculations. The main conclusions were that electron correlation is important in lowering AG and AG°, that the displacements are enthalpy controlled, and that reaction energies are strongly dependent on reactant stabilities. 

A good example is provided by the alkali-metal atoms, which consist of one electron outside a closed-shell core in the single-configuration model. If the frozen-core approximation is valid a frozen-core calculation of the orbital occupied by one electron will give the same result as a Hartree—Fock calculation and the core orbitals will not depend on the state. 

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