CORE AND VALENCE ELECTRONS

A familiar way of handling this question is offered by the notion of electronic shells. By definition, an electronic shell collects all the electrons with the same principal quantum number. The K shell, for example, consists of U electrons, the L shell collects the 2s and 2p electrons, and so on. The valence shell thus consists of the last occupied electronic shell, while the core consists of all the inner shells. This segregation into electronic shells is justified by the well-known order of the successive ionization potentials of the atoms. 

in what is called Hartree-Fock orbital. space—or simply orbital space—the total energy is partitioned from the outset into orbital energies, e,- = ejj, 2, ... Hence we can always consider a collection of electrons and deduce their total energy from the appropriate sum of their orbital energies, remembering, however, that one must also correct for the interelectronic repulsions which are doubly counted in any sum of Hartree-Fock eigenvalues. No special problem arises with 

Atomic Charges, Bond Properties, and Molecular Energies, by Sandor Fliszar Copyright 2009 John Wiley Sons, Inc. 

Incidentally, let us mention that the essence of the pseudopotential methods [52] is to replace core electrons by an appropriate operator. The point is that the core-valence partitioning involved in these methods refers to the same orbital space as the corresponding all-electron calculations. 

what if we abandon the orbital-by-orbital electron partitioning in favor of a description based on the stationary ground-state electron density p(r) Clearly, this will oblige us to redefine the coie-valence separation. In sharp contrast with what was done in orbital space, we need a partitioning in real space. Let us begin with isolated atoms. 

Recent theoretical studies and accurate experimental determinations of electron densities in molecules have confirmed that the majority of electrons do indeed form a concentrated core near the nucleus which appears very atomic like [120]. The electTOTi density is very monotonic as the radial distance from the nucleus increases. Chemists sometimes wrongly consider that the electron density in the cores of atoms has outer maxima corresponding to the shell structures. The inner electron cores are almost transferable entities and cmisequently endorse the valence-care partition proposed by Lewis and Kossel, and this property is utilised in frozen core approximations [138] 

A pseudo-potential is an effective potential which effectively replaces the atomic all-electron potential such that core states are eliminated and the valence electrons are described by pseudo-wave functions with significantly fewer nodes. Only the chemically significant valence electrons are dealt with explicitly, while the 

The differences resulting from variations in the radial distribution functions of different valence orbitals become more pronounced for atoms which have nd and nf valence orbitals, because they are significantly more contracted than the (n+l)s and (n+l)p orbitals and therefore behave in a more core like manner [140]. 

Consequently transition metal carbonyls behave like main group compounds where the stoichiometry changes to satisfy the inert gas rule, and this similarity underpins the isolobal relationships described in Sects. 3.6 and 3.7. The f orbitals of 

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Figure 8.19 X-ray photoelectron spectrum, showing core and valence electron ionization energies, of Cu, Pd, and a 60% Cu and 40% Pd alloy (face-centred cubic lattice). The binding energy is the ionization energy relative to the Fermi energy, isp, of Cu. (Reproduced, with permission, from Siegbahn, K., J. Electron Spectrosc., 5, 3, 1974)...

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

The structure of a molecule is given by the three-dimensional distribution of atomic cores and valence electrons. This structure has been elucidated for many molecules with the use of X-ray or electron diffraction data. Chemical properties of molecules are observed under conditions which permit internal motions. Such observations yield views which may differ markedly as a function of time. Thus, observable properties are determined from equilibrated ensembles of species differing in geometry and energy. 

When a sample maintained in a high vacuum is irradiated with soft X-rays, photoionization occurs, and the kinetic energy of the ejected photoelectrons is measured. Output data and information related to (he number of electrons that arc detected as a function of energy are generated. Interaction of the soft X-ray photon with sample surface results in ionization from the core and valence electron energy levels of the surface elements. 

Let us consider approximations in accounting for the Breit interaction, that we made when outer core and valence electrons are included in GRECP calculations with Coulomb two-electron interactions, but inner core electrons are absorbed into the GRECP. When both electrons belong to the inner core shells, the Breit effect is of the same order as the Coulomb interaction between them. Though Bff does not contribute to differential (valence) properties directly, it can lead to essential relaxation of both core and valence shells. This relaxation is taken into account when the Breit interaction is treated by self-consistent way in the framework of the HEDB method [33, 34]. 

A typical x-ray photoelectron spectrum consists of a plot of the intensity of photoelectrons as a function of electron EB or E A sample is shown in Figure 8 for Ag (21). In this spectrum, discrete photoelectron responses from the core and valence electron eneigy levels of the Ag atoms are observed. These electrons are superimposed on a significant background from the Bremsstrahlung radiation inherent in nonmonochromatic x-ray sources (see below) which produces an increasing number of photoelectrons as EK decreases. Also observed in the spectrum are lines due to x-ray excited Auger electrons. 

Fadley, C.S. "Core and Valence Electronic States Studied With XPS". Lawrence Radiation Laboratory Report, 1970,... 

Core and valence electrons can be divided based on a criterion of whether their distributions and populations are changed or not owing to bond formation. The electron population analysis conventionally uses spherical form factors, f, which are given by the spherically averaged Fourier transform of electron-density distributions. 

X-ray photoelectron (XPS) studies of the core and valence electrons were also carried out (see Tables III -Table V and Figures 11 and 12). The Sn core level peaks were shifted to higher energies 1.2 eV shift for Sn 3p and 0.4 eV for Sn 3d. These results could be explained by considering that there is considerable charge delocalization toward Au in these small particles, which is consistent with electronegativity... 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

The pseudopotential method relies on the separation (in both energy and space) of electrons into core and valence electrons and implies that most physical and chemical properties of materials are determined by valence electrons in the interstitial region. One can therefore combine the full ionic potential with that of the core electrons to give an effective potential (called the pseudopotential), which acts on the valence electrons only. On top of this, one can also remove the rapid oscillations of the valence wavefunctions inside the core region such that the resulting wavefunction and potential are smooth. 

The results of a preliminary study of a sample of berkelium oxide (Bk02, Bk203, or a mixture of the two) via X-ray photoelectron spectroscopy (XPS) included measured core- and valence-electron binding energies (162). The valence-band XPS spectrum, which was limited in resolution by photon broadening, was dominated by 5f-electron emission. 

If it is assumed that the orbitals representing the core and valence electrons comprise an orthonormal set, the Dirac-Fock equation for a single valence... 

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