Hartree-Fock calculations, momentum density atoms

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

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... 

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,... 

In practice, using currently available exchange and correlation potentials, this path leads to results [113] worse than those obtained with the Hartree-Fock method. This is illustrated for momentum moments in Table 19.2 which shows median absolute percent errors of (p ) for 78 molecules relative to those computed by an approximate singles and doubles coupled-cluster method often called QCISD [114,115]. The molecules are mostly polyatomic, and contain H, C, N, O, and F atoms. The correlation-consistent cc-pVTZ basis set [110] was used for these computations. Table 19.2 shows the median errors for the Hartree-Fock method, for second-order Mpller-Plesset permrbation theory (MP2), and for DFT calculations done with the B3LYP hybrid density functional... 

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