Subshells, atomic

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]. 

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. 

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... 

Transverse exchange energies ( j) for closed subshell atoms Selfconsistent ROPM, RLDA and B88-RGGA results in comparison with perturbative RHF values (Coulomb gauge for j in the case of RHF — all energies in Hartree [172]). 

Table 3. Total ground-state energies of noble gas and closed s-subshell atoms as determined within Slater theory, the Work-interpretation Pauli-correlated approximation, and Hartree-Fock theory. The negative values of the energies in atomic units are quoted...

The orbital energy s, in the Hartree-Fock equations (11.12) can be shown to be a good approximation to the negative of the energy needed to ionize a closed-subshell atom by removing an electron from spin-orbited i (Koopmans theorem Section 15.6). 

Thus a closed-shell molecular configuration has both 5 and A equal to zero and gives rise to only a 2 term. An example is the ground electronic configuration of H2. (Recall that a filled-subshell atomic configuration gives only a 5 term.) In deriving molecular terms, we need consider only electrons outside filled shells. 

Hartree-Fock expression for the total energy of the atom involves exchange integrals in addition to the Coulomb integrals that occur in the Hartree expression (11.10). See Section 14.3. [Actually, Eq. (11.12) applies only when the Hartree-Fock wave function can be written as a single Slater determinant, as it can for closed-subshell atoms and atoms with only one electron outside closed subshells. When the Hartree-Fock wave function contains more than one Slater determinant, the Hartree-Fock equations are more complicated than (11.12).]... 

One immediately realizes that only regions in space with non-vanishing density contribute to the correlation energy. Now consider two neutral closed-subshell atoms which are so far apart that there exists no overlap between the their densities, as depicted in Fig. 2.1. The density of this system is identical... 

Table 2.2. Exchange-only ground-state energies of closed-subshell atoms Self-consistent OPM results [48] versus KLl, LDA, PW91-GGA [30] and HE [49] energies (all energies in mhartree)...

Table 2.9. Correlation energies —Ec) of closed-subshell atoms EDA [29], PW91-GGA [30], CS [23], e [18] and ISl [21] results (all energies obtained by insertion of x-only densities) in comparison with MP2 [84,85] and exact [83] energies (in mhartree). The contribution (2.85) to E is also listed separately...

After establishing the basic ability of to deal with dispersion forces, the next step is a quantitative study of more conventional systems. In Table 2.9 the correlation energies obtained with this functional for closed-subshell atoms are compared with various other approximations and the exact correlation energies [83] (which have been extracted by combining variational results for two- and three-electron systems with experimental data for the ionization energies of the remaining electrons). The LDA energies show the... 

One might first hope that this divergence is a consequence of the KLI approximation. Unfortunately, this is not the case. One can explicitly verify that the divergence is present within the full 0PM [25]. For the closed-subshell atoms considered in this section the 0PM equation reduces to a radial integral equation [3],... 

For the S of a multielectron, unfilled-subshell atom, a similar relationship applies. For the simple case of two electrons, the possible values of S are given by... 

---