A Comparison of the Ab Initio Methods and Limitations
Hartree Fock theory is very useful for providing initial, first-level predictions for many molecular systems. However, the restricted HF theory has some basic deficiencies. It is insufficient for an accurate description of the energetics of reactions and bond dissociations, such as the dissociation of the H2 molecule:
A "dissociation catastrophe" occurs because the separated, one-electron hydrogen atoms cannot be described using doubly occupied orbitals. This means H2 tends to dissociate into H+ and H-, which can be described by a doubly occupied orbital on H- (closed shell). This problem does not occur in the UHF procedure, however, UHF does not give pure spin states.
Generally speaking, the HF method is reasonably good at calculating the structures and vibrational frequencies of stable molecules. An additional inaccuracy stems from the neglect of the electron correlation arising from the interaction of electrons with antiparallel spin. Thus in HF, all computations of main group compounds result in bond distances that are too short, since antibonding states are not considered.
A variety of theoretical methods have been developed to attempt to account for electron correlation. To be satisfactory technique, the post-HF methods should ideally have six features, as discussed in Chapter 3.1.
Mφller- Plesset perturbation theory treats the correlation part of the Hamiltonian as a perturbation of the HF part and truncates the energy expansion at some order. This method is size-consistent at any order, but not variational. Moreover, MPn results can oscillate with the order of perturbation applied. An example from literature is the ab initio study on CrF6. In this case, the total energy of both isomers (Oh and D3h) "oscillate", depending on the order n of the applied MPn (n =1, 2, 3, 4 ) perturbation series. The Oh isomer is energetically favored by 18 (MP2) and 44 (MP4) kcal mol1, whereas the D3h isomer is more stable by 16 kcal mol1, when the perturbation is truncated after the third order (MP3) (See Chapter. 14.2).[i] The principal deficiency is that MPn series sometimes converge slowly and might become unreliable, when a large amount of electron correlation has to be considered..
The CI methods do not really represent an alternative to the MP method as a full CI is not feasible for most molecules. Limited CI methods (such as CID or CISD) have been used extensively. These methods are variational, but not size-consistent, which is, for many applications, a disadvantage, since it is generally more important that they are size-consistent, than obey the variation principle.
Alternative methods are CC and QCI, as well as the DFT (especially for larger systems). The CC method satisfies the size-consistency condition, but does not have the variational property. Further, it offers a better approximation than perturbation theory, if the correlation energy correction is large. For a two-electron system, it is equivalent to CID. Comparing the MP with the CCD method in terms of mathematical construction (and thereby in terms of energies), it can be shown that the CCD expansion is equivalent in l to third order (MP3). In fourth order, the energy is equivalent to the MP fourth order energy, in the limited space of double and quadruple substitutions (MP4(DQ), rather than the full energy to fourth order, which would be the energy at MP4(SDTQ) level. The QCI methods are, in a sense, intermediate between CI and CC theory. The idea is to modify the CI equations in a simple manner, so as to restore size-consistency, but the loss of variational character. This requires additional terms, which are quadratic in the general substitution operators (See CC method) and leads to a treatment that is somewhat simpler than CC methods, at least for higher substitutions.
Within the HF-SCF procedure, the spin-orbital coefficients (cip) are optimized, whereas in a CI calculation the Slater determinant coefficients (ak) are variationally optimized, while the spin orbital coefficients from which the Slater determinants are constructed, remain unchanged. From a theoretical point of view, the following order of methods should lead to very good results:
1. HF-SCF calculation, to optimize the spin-orbitals
2. MCSCF calculation, to optimize the spin-orbital coefficients while simultaneously
applying a CI calculation to obtain suitable reference determinants (CSFs)
3. MRCISD calculation, with the MCSCF orbitals and the correct MCSCF-CSFs as
reference determinants
Unfortunately, these precise calculations are limited to very small systems and require considerable care in the selection of the basis set and especially, the active space. They should not be considered for routine use.
Although the approximate density functional theory does not strictly belong to ab initio methods, it has been employed successfully to obtain thermodynamic data, molecular structures, force fields and frequencies and is particularly useful for large systems, e.g. systems with transition metals. Here, correlation effects are included at much lower cost (CPU). DFT already contains the necessary exchange-correlation terms and for the wave function no additional improvements are needed (beside the improvement of the basis set). This is in contrast to the HF method, where the approximation lies in the inadequate description of the wave function. In all post-HF methods (CI, CC, MPn ...), the wave function is successively improved.
The complexity of the approximate Y determines the level of theory.
Finally, the HF theory is a special case of the DFT, with EX[r] given by the exchange integral and with EC[r] = 0. Moreover, hybrid methods of the DFT and the HF methods are widely used. These hybrids include a mixture of HF exchange with DFT exchange-correlation by defining the exchange functional as a linear combination of HF, local, and gradient-corrected exchange terms; this exchange functional is then combined with a local and/or gradient corrected correlation functional.