Closed subshell

The numerical value of the quantity 25 + 1 (total multiplicity) is used as a left superscript with the corresponding term code to define the total atomic term symbol, e.g. for L = 2 and 5 = 1, the term symbol is ZD. Closed subshells make no contribution to L and 5 and may be ignored. 

Ec = E c - Ex have been employed. On the one hand, LDA and GGA type correlation functionals have been used [14], However, the success of the LDA (and, to a lesser extent, also the GGA) partially depends on an error cancellation between the exchange and correlation contributions, which is lost as soon as the exact Ex is used. On the other hand, the semiempirical orbital-dependent Colle-Salvetti functional [22] has been investigated [15]. Although the corresponding atomic correlation energies compare well [15] with the exact data extracted from experiment [23], the Colle-Salvetti correlation potential deviates substantially from the exact t)c = 8Ecl5n [24] in the case of closed subshell atoms [25]. 

Table 1 X-only Coulomb ground state energies Selfconsistent ROPM [16], RHF [60], RLDA and PW91-GGA [8] results for neutral atoms with closed subshells. Also given is E up,Eq. (4.8) (all energies in mhartree).

The very first guess might be that, outside of the two Is closed subshells, a single a bond is formed from the two p orbitals in the a orientation. A singlet state is expected. 

The zeroth-order antisymmetric wave functions of closed-subshell states, states that have only one electron outside a closed-subshell configuration, and states that are one electron short of having a closed-subshell configuration can be expressed as a single Slater determinant [e.g. (1.259)]. However, for open-subshell states in general, one has to take an appropriate linear combination of a few Slater determinants to get a state that is an eigenfunction of L2 and S2. 

In the Hartree-Fock method, the molecular (or atomic) electronic wave function is approximated by an antisymmetrized product (Slater determinant) of spin-orbitals each spin-orbital is the product of a spatial orbital and a spin function (a or ft). Solution of the Hartree-Fock equations (given below) yields the orbitals that minimize the variational integral. Thus the Hartree-Fock wave function is the best possible electronic wave function in which each electron is assigned to a spatial orbital. For a closed-subshell state of an -electron molecule, minimization... 

The direct product is also useful for determining the symmetry of ip9l from the symmetry of the occupied MOs. For two nonequivalent electrons outside closed subshells, the symmetry species of el is given by the direct product of the species of the orbitals of the two electrons. For example, the first excited electronic configuration of H20 is... 

This configuration is found in copper(ll) compounds but is otherwise unimportant. It has neither the closed subshell stability of Jw nor the LFSE possible for [Pg.305]

From these formulas and the definitions of fk (20.29) and (20.30) there follow directly simple expressions for the coefficients considered for one closed subshell, namely,... 

Evidence, based on recent (n,n y) reaction experiments and previous particle transfer studies is presented in support of a coexisting four-particle, four-hole band built on the 1581-keV first excited 0+ state in the doubly closed subshell nucleus 9 >Zr. An alternative explanation for this band in terms of alpha-clustering appears reasonable. 

It is important to understand what gives rise to the established coexisting intruder band in doubly closed subshell 9(>Zr. The slight predominance of the intruder deformed configuration in the ground state wavefunction of lO Mo shown in Fig. 1(b) cannot explain the tremendous difference between the alpha-pickup strengths to the two 0+ states of 9 >Zr clearly demonstrated by Fig. 1(a). This is not unexpected, since the simultaneous occurrence of proton and neutron subshell closures should produce a marked difference with respect to other nuclei near the middle of the 50 to 82 neutron shell where two-proton, two-hole excitations can account entirely for the observed shape coexistence phenomena. 

Fig. 4. Partial level scheme for doubly closed subshell (>Zr showing the 4p-4h intruder band.

Because of the contraction and stabilization of the 6s orbital, the outermost, or valence, shell of Au is formed by both the 5d and 6s orbitals. Indeed, electronically, Au is halogen-like, with one electron missing from the pseudo noble gas (closed subshell) configuration. Hence, similar to the existence of halogen X2 molecule, gold also forms the covalent Au2 molecule. In addition, gold also forms ionic compounds such as RbAu and CsAu, in which the Au- anion has the pseudo noble gas electronic configuration. 

The coefficients have been determined by a least squares fit to the exact relativistic x-only energies of a number of closed subshell atoms keeping the form of < ( ) fixed. For the PW91 GGA this procedure leads to the parameters listed in Table 1 [28]. As has been demonstrated in [28] the resulting RGGA produces much more accurate atomic results than both the RLDA and the corresponding nonrelativistic GGA. 

Table 3.1. Longitudinal ground state energies ( - / ,) and highest occupied eigenvalues ( — iu) for closed subshell atoms from nonrelativistic OPM (NROPM [59]), relativistic OPM (ROPM [36]) and relativistic HF (RHF [58] ) calculations [69] (all energies are in hartree).

In this section we summarise the properties of the approximations to tc[M] discussed in Section 4 in applications to atoms. All results presented in the following [36] are based on the direct numerical solution of Eqs. (3.25-3.29) using a nuclear potential which corresponds to a homogeneously charged sphere [69]. Only spherical, i.e. closed subshell, atoms and ions are considered. Whenever suitable we use Hg as a prototype of all high-Z atoms. 

Configuration neither the closed subshell stability of d nor the LFSE possible for d. Cop-... 

Binding in clusters with closed-subshell atoms... 

The study of the binding in clusters with closed subshell atoms is performed. The study is based on the accurate calculations of the Be ,... 

Binding in Clusters with Closed-Subshell Atoms... 

Atoms with closed subshells have no multipole moments and their electrostatic and induction interactions have a pure overlap origin from which follows their short-range character. The main contribution to Ef F gives the exchange interaction ch- Between atoms with closed subshells, it is repulsive (as in the noble-gas atom systems). This determines the unstability of the alkaline earth dimers at the SCF approximation. They are stabilized by the attractive electron correlation forces. 

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