Variational calculations Hartree-Fock approximation

Below is a brief review of the published calculations of yttrium ceramics based on the ECM approach. In studies by Goodman et al. [20] and Kaplan et al. [25,26], the embedded quantum clusters, representing the YBa2Cu307 x ceramics (with different x), were calculated by the discrete variation method in the local density approximation (EDA). Although in these studies many interesting results were obtained, it is necessary to keep in mind that the EDA approach has a restricted applicability to cuprate oxides, e.g. it does not describe correctly the magnetic properties [41] and gives an inadequate description of anisotropic effects [42,43]. Therefore, comparative ab initio calculations in the frame of the Hartree-Fock approximation are desirable. 

The calculation of x-ray emission spectra of molecules or solids are one of the most successful applications of the discrete variational (DV) Hartree-Fock-Slater (Xa) MO method using cluster approximation [8-10], which was originally coded by Ellis and his CO workers [11-14] based on Slater s Xa exchange potential [15]. The DV-Xa method has several advantages for the calculation of x-ray transition process as follows. 

To overcome the deficiencies of the Hartree-Fock wave function (for example, improper behavior R oo and incorrect values), one can introduce configuration interaction (Cl), thus going beyond the Hartree-Fock approximation. Recall (Section 11.3) that in a molecular Cl calculation one begins with a set of basis functions Xi, does an SCF calculation to find SCf occupied and virtual (unoccupied) MOs, uses these MOs to form configuration (state) functions writes the molecular wave function i/ as a linear combination 2/ of the configuration functions, and uses the variation method to find the ft, s. In calculations on diatomic molecules, the basis functions can be Slater-type AOs, some centered on one atom, the remainder on the second atom. 

The pure Hartree-Fock approximation works fairly well for predicting the geometries of metal complexes. Vibrational frequencies are overestimated while formation energies are underestimated. These errors are rather systematic so that, for example, it is standard practice to scale calculated frequencies some empirical factor (Pople et al. 1993) to predict experimental values. The problem is that the single-Slater-determinant approximation to the wavefunction is too restrictive it doesn t allow the electrons with opposite spin to avoid each other as much as they would like. In accordance with the variational principle, the Hartree-Fock total energy is always too... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

There are two ways to improve the accuracy in order to obtain solutions to almost any degree of accuracy. The first is via the so-called self-consistent field-Hartree-Fock (SCF-HF) method, which is a method based on the variational principle that gives the optimal one-electron wave functions of the Slater determinant. Electron correlation is, however, still neglected (due to the assumed product of one-electron wave functions). In order to obtain highly accurate results, this approximation must also be eliminated.6 This is done via the so-called configuration interaction (Cl) method. The Cl method is again a variational calculation that involves several Slater determinants. 

If the Kohn-Sham orbitals [52] of density functional theory (DFT) [53] are used instead of Hartree-Fock orbitals in the reference state [54], the RI can become essential for the realization of electron propagator calculations. Modern implementations of Kohn-Sham DFT [55] use the variational approximation of the Coulomb potential [45,46] (which is mathematically equivalent to the RI as presented above), and four-index integrals are not used at all. A very interesting example of this combination is the use of the GW approximation [56] for molecular systems [54],... 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

The correspond to different electron configurations. In configuration interaction o is the Hartree-Fock function (or an approximation to it in a truncated basis set) and the other 4>t are constructed from virtual orbitals which are the by-product of the Hartree-Fock calculation. The coefficients Ci are found by the linear-variation method. Unfortunately, the so constructed are usually an inadequate basis for the part of the wavefunction not represented by [Pg.5]

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