Hartree-Fock method, limitations

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. 

As a final comment, it is interesting to note that this FS(K) study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances. 

The Hartree-Fock method was in any case the method of choice for the first quantitative calculations related to homogeneous catalysis. It was the method, for instance, on a study of the bonding between manganese and hydride in Mn-H, published in 1973 [28]. The first studies on single steps of catalytic cycles in the early 1980 s used the HF method [29]. And it was also the method applied in the first calculation of a full catalytic cycle, which was the hydrogenation of olefins with the Wilkinson catalyst in 1987 [30]. The limitations of the method were nevertheless soon noticed, and already in the late 1980 s, the importance of electron correlation was being recognized [31]. These approaches will be discussed in detail in the next section. 

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

The great speed and known properties of RHF calculations are not sufficient justification for a limitation to RHF methods when they are inherently inappropriate. It is worth remarking that most potential-energy surfaces describing reactions, and many describing dissociations are inappropriate for RHF methods. Restricted Hartree-Fock methods are also of limited validity in many situations where two or more surfaces are at nearly the same energy. 

It is clear that these early DFT attempts suffered from basis set limitations. At the same time, the advantages of having a post-Hartree-Fock method available... 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

Since the problem of correlated motion of the N electrons cannot be dealt with adequately by the Hartree-Fock method, even in the limit of an infinite basis set (i.e., B —> oo), it is expected that the ground-state electronic energy and estimation of the first few excited-state energy levels of even small molecules can be several electron volts away from thermodynamic or spectroscopic reality the difference Eexp—EHF is often called the correlation energy Ecorr ... 

It is interesting to observe that, if one limits oneself to study the original real and self-adjoint Hamiltonian with H+= H = H and applies the complex symmetric Hartree-Fock method to this particular case without any transformations whatsoever, one... 

Fig. 11.13(a) shows the summed momentum profiles for the states of the 6p manifold at 7.4 eV and 9.2 eV. Figs. 11.13(b) and (c) describe states that are identified by the plane-wave impulse approximation with the Dirac—Fock orbital as belonging to the 6s manifold. Since the valence states of lead are diffuse in coordinate space most of the momentum profile is within the 1 a.u. limit of validity of the plane-wave impulse approximation for the profile shape. The experiment agrees with the Dirac—Fock profile but rules out the nonrelativistic Hartree—Fock method. 

For a long time, the Hartree-Fock (HF) theory combined with the self-consistent field (SCF) procedure proved arguably to be the most useful method for computational chemists despite its well-known limitations. Within its framework, orbitals can be constmcted to reflect paired or unpaired electrons. If the molecular system has a singlet spin state, then the same orbital spatial function can be used for both the Gland (3-spin electrons in each pair. This assumption is called the restricted Hartree-Fock method (RHF). 

AEC is usually explicitly defined with respect to the Hartree-Fock method - i.e. it is the difference between the HF limiting energy and the true electronic energy. The statement that AECOII is smaller in DFT than in HF theory therefore implies that the DFT energy is lower than the HF limit... 

---