Hartree-Fock calculations beryllium

The use of a systematic sequence of basis sets can, of course, be usefully combined with the use of a universal basis set(52). In Figure 8, we display the results of Hartree-Fock calculations on the radial beryllium-like ions Li", B+, C2+, N +, Ol++, F5+, Ne6+, using the basis set given by Schmidt and Ruedenberg(56) specifi-ically for the beryllium atom. It can be seen that for the positive ions the basis sets give a uniform convergence rate. In... 

Numerical Hartree-Fock calculations, free from basis set artifacts, have been used to establish that the ground state momentum densities of all the atoms and their ions can be classihed into three types [84,85]. Type I and III momentum densities are found almost exclusively in metal atoms He, N, all atoms from groups 1-14 except Ge and Pd, and all the lanthanides and actinides. These momentum densities all have a global maximum at p = 0 and resemble the momentum density shown in Fig. 19.3 for the beryllium atom. The maximum atp = 0 comes mainly from the outermost s-subshell, 2s in this case. Type I and III densities dilfer in that the latter have a secondary maximum that is so small as to be invisible on a diagram such as Fig. 19.3. Type II densities are the norm for non-metallic atoms and are found in Ge, Pd and all atoms from groups 15-18 except He and... 

We describe in this Subsection the application of local-scaling transformations to the calculation of the energy for the lithium and beryllium atoms at the Hartree-Fock level [113]. (For other reformulations of the Hartree-Fock problem see [114] and referenres therein.) The procedure described here involves three parts. The first part is orbital transformation already discussed in Sect. 2.5. The second is intra-orbit optimization described in Sect. 4.3 and the third is inter-orbit optimization discussed in Sect. 4.6. 

A set of calculations on the beryllium atom and its isoelectronic series have been carried out [48]. The starting basis set used was dementi s double-zeta [82]. This basis was then transformed into the Hartree-Fock one, and the initial RDMs corresponded to Slater determinants built with this basis. Note that in... 

Although for brevity s sake the 3-MCSE has not been considered here, it may be convenient to mention it in these final comments. This equation, which depends on the 1-CSE, does not have a unique solution. Indeed, this equation is satisfied not only by the FCI 3-RDM but also by the Hartree-Fock one. Alcoba [48] performed a series of calculations with the 3-MCSE for the beryllium iso-electronic series. Alcoba took as initial data a set of RDMs that corresponded to a state that had already some correlation and whose energy was below the Hartree-Fock s one. The results of these calculations showed that there was a smooth although very slow convergence toward the exact solution. For larger systems the situation will probably be similar to the 4-MCSE one and a strict... 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Calculation of the energy and wavefunction for the beryllium atom at the Hartree-Fock level by in the context of local-scaling transformations... 

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

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