Post-HF calculations

As can be seen from Table I, the C-C bond distance as described by LDF is closer to experiment than the corresponding HF value obtained with a 6-3IG basis. Including correlation via second and third order Moller-Plesset perturbation theory and via Cl leads to very close agreement with experiment. The C-H bond length is significantly overestimated in the LDF calculations by almost 2%. The HCH bond angle is reasonably well described and lies close to all the HF and post-HF calculations. Still, all the theoretical values are too small by more than one degree compared with experiment the deviation from experiment is particularly pronounced for the semi-empirical MNDO calculation. 

The scheme described above, reconforted by the post-HF calculations [57] where the coordinate representing the distance between the nuclei in the diatomic molecule (or any bond in polyatomic molecules), lead to the pervading picture of a diatom connected adiabatically with two non-interacting atoms at infinite distance. From a compuational point of view, this picture is quite useful and widely employed. 

Regarding the height of the insertion barrier, the situation is much more controversial, since pure density functionals and some MP2 calculations suggest that this a barrierless reaction, or it occurs with a negligible barrier. HF, hybrid density functionals and several post-HF calculations, instead, suggest a barrier in the range of 5-10 kcal/mol, roughly. 

In order to use wave-function-based methods to converge to the true solution of the Schrodinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both FIF and post-HF calculations. Even with these improvements, however you can appreciate the problem with... 

Of course, with HF wave functions in hand, it is possible to carry out post-HF calculations to partially correct for electron correlation effects. The poor quality of the HF wave functions, however, militate against any treatment much less sophisticated than coupled-cluster. At the CCSD(T)/cc-pVDZ level, the predicted energy of the lowest closed-shell singlet is in fair agreement with experiment (other data in the table suggest that use of a triple- basis set would improve the CCSD(T) estimate). The energy of the second closed-shell singlet state... 

Ab initio and DFT calculations of the BO surfaces, which are used to describe hydrogen-bonded systems, are computationally demanding. Computational practice has shown that a flexible basis set is required. The Hartree-Fock (HF) level is typically insufficient, and electron correlation must be included. DFT is an attractive alternative to the post HF calculations. The hypersurface is obtained in such a way pointwise. One can fit it to a computationally efficient form that allows for an inexpensive evaluation needed in thermal averaging or calculation of matrix elements when performing vibrational analysis. 

At the HF-level, Mn04 is unstable with respect to dissociation into one electron and the neutral species. Introducing post-HF calculations makes Mn04 stable with respect to such a dissociation. The problems encountered for HF in 3d metal complexes are not present in systems involving nd (n = 4,5) elements since the corresponding ns, np orbitals are much more contracted than nd for n = 4,5. Also, the HF error seems to be more severe for late 3d elements than early 3d elements. 

We will focus the discussion on the reliability of DFT methods since these are most often used in zeolite modeling. It is much more difficult to assess the reliability of various exchange-correlation functionals since these cannot be systematically improved and the form of the exact functional is not known. Many benchmark calculations on small molecules were performed and results were compared with the reliable post-HF calculations and experimental... 

Tsipis [73] gives extensive tabulations of M-L bond energies. The vast majority of post-HF calculations are restricted to simple ML or ML2 systems where L is monatomic (H, O or halide) or a small ligand such as CO, CH3, NH3, PH3 or C2H4. These molecules are small enough to facilitate the necessary correlation treatment. Multi Reference Cl and Complete Active Space SCF are the most popular correlated methods and appear capable of good accuracy (i.e. to within 5-10 kcal/mole). 

Electron correlation is the phenomenon of the motion of pairs of electrons in atoms or molecules being connected ( correlated ) [56]. The purpose of post-HF calculations is to treat such correlated motion better than does the HF method. In the HF treatment, electron-electron repulsion is handled by having each electron move in a smeared-out, average electrostatic field due to all the other electrons (sections 5.2.3.2 and 5.2.3.6b), and the probability that an electron will have a particular set of spatial coordinates at some moment is independent of the coordinates of the other electrons at that moment. In reality, however, each electron at any moment moves under the influence of the repulsion, not of an average electron cloud, but rather of individual electrons (in fact current physics regards electrons as point particles - with wave properties of course). The consequence of this is that the motion of an electron in a real atom or molecule is more complicated than that for an electron moving in a smeared-out field [57] and the electrons are thus better able to avoid one another. Because of this enhanced (compared to the HF treatment) standoffishness, electron-electron repulsion is really smaller than... 

Valence-only methods are computationally far more efficient than allelectron methods, especially up to the integral transformation step necessary for most post-HF calculations. 

If the method is in principle exact we do not need to do perturbation theory or Cl or any other post HF calculation, so the basis-function... 

Table 5. Comparison between DFT calculated FC, PSO, SD, and DSO terms of /(A,B) using the PW exchange-correlation functional and the corresponding HF and post-HF calculations (in Hz).

In post-HF calculations, the core electrons (e.g., Is for C, ls 2s 2p for Si) are usually left at the HF level, i.e., they are not correlated. This is called the frozen-core approximation. When desired, the contribution from core correlation is generally computed directly, although core-polarization potentials are effective [71,72] and parameterized estimation schemes are available [73]. 

Unless new and better XC functionals are introduced and DFT results are systematically improved, conventional wavefunction-based approaches should have a renaissance and remain competitive. Ultimately, when computing power evolves to the point where the cost of correlated post-HF calculations on large biomolecular systems is no longer prohibitive, DFT in its present form will have outlived its usefulness. All is not gloom and doom, however. A great deal of ongoing research is concentrated on the development new XC functionals that hopefully will push DFT beyond MP2 and other post-HF methods. The form of these XC functionals and the increased computational effort they will introduce are not yet known. 

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