Atomic shell structure

M. PeUarin, B. Baguenard, J. L. Vialle, J. Lerme, M. Broyer, J. Miller and A. Perez, Evidence for icosahedral atomic shell structure in nickel and cobalt clusters. Comparison with iron clusters , Chem. Phys. Lett. 217 349 (1994). 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

The importance of orbital dependent functionals for a correct representation of the atomic shell structure, the correct properties of v for heteronuclear molecules, and the related particle number dependent properties will be discussed in Sect. 5.5. 

We will discuss in detail in the next section this potential, which incorporates the main features of the atomic shell structure. The screening potential is just the potential of the coupling constant integrated exchange-correlation hole. Due to the fact that this hole integrates to one electron, the screening potential has... 

The exchange-correlation potential v c reflects the atomic shell structure (see below). The shell structure of v arises from that of the pair-correlation function g. The shell structure of the latter, in its turn, originates from the antisymmetry property of the wavefunction of the system and is mainly an exchange effect. We will therefore consider first the exchange-only case. The influence of correlation effects on the atomic shell structure will be discussed at the end of this section. 

Let us now discuss the correlation effects on the atomic shell structure. We plot in Fig. 7 some of the described potentials for the case of the beryllium atom. The exact exchange-correlation potential v c is calculated from an accurate Cl (Configuration Interaction) density using the procedure described in [20]. The potentials Vx, and u" , are calculated within the optimized potential model (OPM) [21,40,41] and are probably very close to their exact values which can be obtained from the solution for of the OPM integral equation [21,40,41] by insertion of the exact Kohn-Sham orbitals instead of the OPM... 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

In addition to its role with regard to chemical reactivity, 7(r) is also linked to atomic shell structure and electronegativity, local temperature (or kinetic energy), radical characterization, bond strain and local polarizability. For a recent overall review, see Politzer and Murray. ... 

A fair amount of effort has been devoted to the manifestations of atomic shell structure in the momentum density and related functions [210-215]. The number of maxima observed in I p) varies from one to four, with 35 and 48 atoms exhibiting two and three local maxima [215]. No maxima in I p) corresponding to the most corelike shells are found in heavy atoms. Correlations have been found between 1 jpmax, where p ax is the location of the innermost maximum of I p), and the relative size of the atom [215]. 

It is also fascinating to discuss the origin of the atomic shell structure within the WDA and the SADA. Recall from Section V.2, shell structure appears for heavier atomic species (Z > 30) within the exchange-only WDA treatment. 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

It is also fascinating to discuss the origin of the atomic shell structure within the WDA and the SADA. Recall from Section V.2, shell structure appears for heavier atomic species (Z > 30) within the exchange-only WDA treatment.136140 The SADA without the proper symmetrization of the AWF in Eq. (143) behaves very much the same101 (even only with the LDA XCEDF). This implies that the WDA effectively captures most of the overall shape of the correct LR function even without any explicit enforcement, and that enforcing the correct LR behavior for the OF-KEDF alone, the SADA is able to remedy the defects of the LDA XCEDF. This is certainly encouraging for both the WDA and the SADA. On the other hand, the SADA with a proper symmetrization of the AWF in Eq. (143) is able to produce shell structure for all atomic species,101 because the kinetic-energy potential is properly symmetrized this time. This further emphasizes the importance of the symmetrization on the potential level. 

Another well-known drawback of the LDA and GGA potentials is their asymptotic behavior. They decay faster than the Coulombic asymptotic behavior vxc(ri)—>1/1 1, Ir —>oo required for the accurate xc potential. In the bulk region the LDA and GGA potentials lack the pronounced atomic shell structure of the accurate potential. The improved potentials should possess these features as well as the proper depth in the bulk region, so they should be shifted downward compared to the LDA/GGA potentials. 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

A somewhat different and promising approach has been proposed recently by Gritsenko, van Leeuwen and Baerends [95]. These authors decomposed the potential into the Slater potential vs and the response to density variations vTCtp. They showed that the latter exhibits the peaks which reflect the atomic shell structure which is poorly described by usual GGAs potentials. The potential they proposed possesses correct asymptotic and scaling properties. Although there is still room for improvements, it would be worthwhile to see how molecular properties... 

The drawback of this definition was, that it had been restricted to the atomic ground states and that it ignored the atomic shell structure, i.e. that large jumps in E(N) are expected, when atomic shells identified by their (n, l) quantum numbers are transgressed. 

The average local ionization energy 7(r) has many interesting and significant aspects and applications. It is related to local temperature and atomic shell structure, it is linked to electronegativity and shows promise as a measure of local polarizability. It permits the characterization of bonds and radical sites, and - in conjunction with volume -the prediction of molecular and group polarizabilities. Finally, it is an effective guide to reactivity towards electrophiles, especially when complemented by the electrostatic potential. All of these areas continue to be studied. 

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. 

Figure 1 Isoelectronic surface of benzene. Note the general lack of features for example, no it cloud and no atomic shell structure.

A full description is beyond the scope of this review, but it is noted that the topological method identifies other chemical features in the electron density. The union of all bond paths gives a bond path network that is normally in a 1 1 correspondence with the chemical bond network drawn by chemists. The bond paths for bonds in strained rings are curved, reflecting their bent nature. In Figure 6, we show the gradient paths in the molecular plane of cyclopropane. The C—C bond paths are distinctly bent outward. The value of the Laplacian at the bond critical point discriminates between ionic and covalent bonding." Maps of the Laplacian field reveal atomic shell structure, lone pairs, and sites of electrophilic and nucleophilic attack. The ellipticity of a bond measures the buildup of density in one direction perpendicular to the... 

---