Hartree-Fock limit helium atom

Owing to their simplicity, the helium atom and the dihydrogen molecnle have been the object of experiments (Ref. 31 for a and 7 of He Ref. 32 for a of H2) and calcnlations, some of them near the Hartree-Fock limit (Ref. 33 for He and Ref. 34-36 forH2 ). In order to test our polarization fnnctions, we have taken the zeroth... 

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). 

Bader, Novaro, and Beltran-Lopez135 have calculated the potential surface for various geometries of three and four helium atoms near to the Hartree-Fock limit to determine the deviation of the Hartree-Fock energies of interaction from pair-wise additivity. 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

Hartree-Fock Version. As with the analogous mean field approximation in statistical mechanics, the error in the Hartree-Fock approximation is expected to diminish as D increases. This is because fluctuations decrease in proportion to D, as illustrated in Fig. 2 for the hydrogenic atom. However, whereas the mean field approximation for critical exponents of phase transitions becomes exact [86] for sufficiently large D, for the Hartree-Fock approximation the correlation energy remains nonzero and relatively large even for D oo. As a function of the total energy, AEd for the helium atom varies from 2.3% at the D — 1 limit to 1.5% for D = 3 to 0.99% at the D oo limit. 

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