Hartree calculation

Spin is not included explicitly in the Hartree calculation and the wave functions are therefore not antisymmetric as required. If anti-symmetrized orbitals are used a set of differential equations... 

Figure 1 This figure shows the ground-state energies of the 6-electron iso-electronic series of atoms and ions, C, iV, 0 +, etc., as a function of the atomic number, Z. The energies in Hartrees, calculated in the crudest approximation, with only one 6-electron Sturmian basis function (as in Table 1), are represented by the smooth curve, while dementi s Hartree-Fock values [10] are indicated by dots. 

Figure I. Partitioned total energy values (in hartree) calculated for three different water hexamer using basis set 6-31G/d.

Table 2-6. LDA and GGA(BLYP or PW91) atomic ionization potentials (in Hartree) calculated as energy differences (ASCF) or as —sH0M0 (aLDA results from Vydrov and Scuseria98 CLDA results from Perdew et al.97, bLDA results from van Leeuwen and Baerends80 GGA results from Grabo and Gross90).

D. R. Hartree, Calculation of atomic structures, John Wiley and Sons, Inc., New York, 1957. 

A.O.Williams Jr. noted in his Hartree-calculations on the closed shell atom Cu as early as 1940 (Phys. Rev. 58, 723) The charge density of each single electron turns out to resemble that for the nonrelativistic case, but with the maxima "pulled in " and raised.. .. The size of the relativistic corrections appear to be just too small to produce important corrections in atomic form factors or other secondary characteristics of the whole atom.. .. However, it must be noticed that copper is a relatively light ion, and the corrections for such an ion as mercury would be enormously greater. S.Cohen in 1955 and... 

The isodesmic reaction series shown below has been used to compare the stabilities of the cyclopolyene ketenes. The total energies (HF/6-31G ) are given in hartrees. Calculate the stabilization found for each cyclopolyene ketene for n= 1-3. Account for the differences in stabilization and compare the results of these exocyclic ketenes to the corresponding cyclopolyenones (Problem 8.7). 

Figure 5.3 The best result from Figure 5.2 as a comparison of the numerical radial function from the Herman-Skillman program and the Hartree calculation. All the graphs are of the function rR(r) against the radial distance r in atomic units. Use the figure to check the consistency of the scaling, by recovering the hydrogen result.

One should bear in mind that originally the RMF model was formulated within the Hartree and no-sea approximations. Implementing the Dirac sea may require serious revision of the model and inclusion of additional terms. Hartree calculations including the Dirac sea and Hartree-Fock calculations including exchange terms lead to smaller nucleon potentials in normal nuclei. Shallower potentials will produce smaller attraction for antinucleons, but the qualitative effect that the presence of antiprotons reduces repulsion and enhances attraction for nucleons will remain valid. We expect that the additional binding and compression of the nucleus will appear even for an antinucleon potential as low as 200 MeV. 

The starting point in this chain of equations is the Hartree (H) approximation, in which case 2 =0. An approximate "Hartree" calculation is shown in Fig.4 for Si. This calculation is "approximate" in that it is not a self-consistent H calculation, but in-stead -S based on a self-consistent pseudopotential-LDA calculation. After achieving selfconsistency the LDA exchange-correlation V was dropped in the extraction of the eigenvalues and... 

This LDA "Hartree" calculation was provided by C. Kune independently very similar results were found by D. Glotzel. 

Hammer T, Manthe U (2011) Intramolecular proton transferin malonaldehyde Accurate multilayer multi-configurational time-dependent Hartree calculations. J Chem Phys 134 224305... 

One may argue that the Dirac formula describes the trends in relativistic effects only for one-electron atoms. For neutral atoms the Pauli exclusion principle pushes valence density out of the inner region therefore, relativistic effects may be small. In 1940 A. O. Williams noted in analyzing his Hartree calculations on the closed shell atom Cu ... 

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