Helium Hartree-Fock

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

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... 

Fischer C.F. Average-Energy of Configuration Hartree-Fock Results for the Atoms Helium to Radon./M tomic Data.-1972. -No 4. -p. 301-399. 

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). 

In a few cases, the wave-function F of a monatomic entity can be used for calculating a, e.g. 4.5 bohr3 for the hydrogen atom, or 0.205 A3 for the helium atom in agreement with the experimental value. Gaseous H does not have a Hartree-Fock function stable relative to spontaneous loss of an electron, and it is necessary to introduce correlation effects in order to calculate a which is said to be 31 A3. The value 1.8 A3 for H(-I) in Table 2 derives from NaCl-type LiH, NaH and KH. The anion B2Hg2 has a = 6.3 A3 to be compared with the isoelectronic C2H6 4.47 A3. Since CH4 has a =... 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Figure 7.6 Radial wavefunctions Pls(r) = rRls(r) of helium. HYDR is the hydrogenic wavefunction with Z = 2 HF is the Hartree-Fock wavefunction. From [BJ066].

The simplest kind of ab initio calculation is a Hartree-Fock (HF) calculation. Modem molecular HF calculations grew out of calculations first performed on atoms by Hartree1 in 1928 [3]. The problem that Hartree addressed arises from the fact that for any atom (or molecule) with more than one electron an exact analytic solution of the Schrodinger equation (Section 4.3.2) is not possible, because of the electron-electron repulsion term(s). Thus for the helium atom the Schrodinger equation (cf. Section 4.3.4, Eqs. 4.36 and 4.37) is, in SI units... 

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. 

A symmetry-adapted perturbation theory approach for the calculation of the Hartree-Fock interaction energies has been proposed by Jeziorska et al.105 for the helium dimer, and generalized to the many-electron case in Ref. (106). The authors of Refs. (105-106) developed a basis-set independent perturbation scheme to solve the Hartree-Fock equations for the dimer, and analyzed the Hartree-Fock interaction energy in terms of contributions related to many-electron SAPT reviewed in Section 7. Specifically, they proposed to replace the Hartree-Fock equations for the... 

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