Hartree-Fock results

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

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. 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

In Table VIII, we present the local-scaling- transformation-energy results for lithium and beryllium and compare them with results obtained with other methods. It is worth mentioning that the Hartree-Fock results for these atoms are a first instance of atxurate energy values obtained within the context of a formalism based on density functional theory. 

In the Hartree-Fock approximation, the GS is given by HF) = + +) and Fhf = —t + U. Let us examine the Hartree-Fock results by defining the ionic and nonionic geminals, respectively ... 

Figure 2. Total energy of the alternate chain versus A-subband filling. Upper curve Hartree-Fock result, Lower curve Gutzwiller result. The 2 minima are clearly different.

If comparison with experiment is not appropriate, what should be used for reference values Clearly, the desirable thing would be to eliminate as many sources of error as possible. For instance, if we wish to establish the reliability of an SCF treatment in a given one-particle basis, we could use numerical Hartree-Fock results for diatomic molecules (that is, essentially complete basis results) as benchmarks. Any difference between the finite basis and numerical results would presumably be due to inadequacies in the former, as otherwise the same approximations are made in both treatments. It is crucial to understand that this approach gives much more informa-... 

The remaining error in the dipole moment Green8 attributes to lack of highly excited configurations. For open-shell molecules it is probable that Hartree-Fock results will be unreliable (see above) and a limited amount of Cl will be essential. Thus even for a Hartree-Fock function the calculated one-electron properties may not agree well with experiment (it should be remembered that, in the most favourable cases where the substance can be studied in a molecular-beam spectrometer and the dipole moment obtained from Stark effect measurements, the experimental error is much less than 0.001 D).28... 

The geometry adopted for the cluster has symmetry C2h (Fig-1) and the set of structures spans symmetry species Ag and Bu. All the point energy calculations we performed resulted in wave functions of symmetry species Ag for the ground state, as would be expected. The energy for selected points are listed in Table 2, where we have included Hartree-Fock results for comparison. 

While basis sets composed of a sufficient number of s- and p-type functions will yield the Hartree-Fock results for atoms up to scandium,53 such s-p saturated sets are not suitable for molecular SCF calculations. Instead, the basis sets must be augmented with polarizing functions, i.e. those functions, p for hydrogen and d and / for first- and second-row atoms, which are absent in the description of the isolated atoms, but whose presence is essential for a proper description of the distortions in the charge distribution caused by chemical bonding. As the examples in the following section illustrate, the inclusion of polarizing functions in the basis set is essential in the calculation of potential surfaces. 

The friction term has been evaluated for CO/Cu(l 00) by Hartree-Fock cluster calculations using single excitations [110]. A parametrized form of the Hartree-Fock results has been used for the molecular dynamics simulations. The interaction potential of CO/Cu(l 00) in the nuclear degrees of freedom, however, was derived empirically. 

Hobza P, Kabelac M, Sponer J, Mejzlik P, Vondrasek J (1997) Performance of empirical potentials (AMBER, CFF95, CVFF, CHARMM, OPS, POLTEV), semiemprical quantum chemical methods (AMI, MNDO/M, PM3) and ab initio Hartree-Fock method for interaction of DNA bases comparison of nonempirical beyond Hartree-Fock results, J Comp Chem, 18 1136-1150... 

We will discuss Vs(r) and Is(r) calculated for (6,0) all-carbon and C/B/N model nanotubes. The computational level, Hartree-Fock (HF) STO-5G//STO-3G, was dictated by the large sizes of the systems. It has been confirmed, however, that minimum basis set STO-5G Hartree-Fock results for both v(r)64,65 and Is(r)65 are quite satisfactory on a relative basis, for seeing trends, which has been our objective. The validity of the HF/STO-3G geometry optimizations can be seen from the good agreement between our C-C and B-N bond lengths and relevant experimental data.28-31... 

The resulting atom polarizabilities fitted to experiment are summarized in Table 3-2 together with numerical Hartree-Fock results for the free atoms [112],... 

Ground State o/N2. In Table 13, the spectroscopic constants of the N2 ground state OSJ) obtained by Gunnarsson et al. are compared with experiment104 and with the Hartree-Fock results of Cade et al.10 ... 

---