Hartree-Fock theory limit

The first cell in the last tow of the table represents the Hartree-Fock limit the best approximation that can be achieved without taking electron correlation into account. Its location on the chart is rather far from the exact solution. Although in some cases, quite good results can be achieved with Hartree-Fock theory alone, in many others, its performance ranges from orfly fair to quite poor. We ll look at some these cases in Chapters 5 and 6. 

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. 

Density functional theory (DFT),32 also a semi-empirical method, is capable of handling medium-sized systems of biological interest, and it is not limited to the second row of the periodic table. DFT has been used in the study of some small protein and peptide surfaces. Nevertheless, it is still limited by computer speed and memory. DFT offers a quantum mechanically based approach from a fundamentally different perspective, using electron density with an accuracy equivalent to post Hartree-Fock theory. The ideas have been around for many years,33 34 but only in the last ten years have numerous studies been published. DFT, compared to ab initio... 

Hartree-Fock theory as constructed using the Roothaan approach is quite beautiful in the abstract. This is not to say, however, that it does not suffer from certain chemical and practical limitations. Its chief chemical limitation is the one-electron nature of the Fock operators. Other than exchange, all election correlation is ignored. It is, of course, an interesting question to ask just how important such correlation is for various molecular properties, and we will examine that in some detail in following chapters. 

In the last chapter, the full formalism of Hartree-Fock theory was developed. While this theory is impressive as a physical and mathematical construct, it has several limitations in a practical sense. Particularly during the early days of computational chemistry, when computational power was minimal, carrying out HF calculations without any further approximations, even for small systems with small basis sets, was a challenging task. 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

Along the ordinate, a sequence of correlation-consistent cc-pVXZ basis sets with X > 2 is depicted. Along the abscissa, the FCI limit is approached—beginning with Hartree-Fock theory and followed by the first correlated level, at which the single and double excitations are described by MP2 perturbation theory. The same excitations are subsequently treated by coupled-cluster theory at the CCSD level, which is then further improved upon by a perturbation treatment of the triple excitations at the CCSD(T) level. At the CCSDT level, the triple excitations are fully treated by coupled-cluster theory, and so on. In this manner, the hierarchy Hartree-Fock -> MP2 — CCSD -> CCSD(T) > CCSDT —---------------> FCI is established. 

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