Hartree-Fock limit method

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

Several theoretical investigations described the interaction of a d° metal center with a peroxide in order to understand oxygen transfer from a TM center to an alkene [2, 39-42] or a sulfide [43], These studies, using semiempirical methods or the ab initio Hartree-Fock SCI method, were limited to an orbital analysis ofthe ground state of metal peroxides. 

Almost all contemporary ab initio molecular electronic structure calculations employ basis sets of Gaussian-type functions in a pragmatic approach in which no error bounds are determined but the accuracy of a calculation is assessed by comparison with quantities derived from experiment[l] [2]. In this quasi-empirical[3] approach each basis set is calibrated [4] for the treatment of a particular range of atoms, for a particular range of properties, and for a particular range of methods. Molecular basis sets are almost invariably constructed from atomic basis sets. In 1960, Nesbet[5] pointed out that molecular basis sets containing only basis sets necessary to reach to atomic Hartree-Fock limit, the isotropic basis set, cannot possibly account for polarization in molecular interactions. Two approaches to the problem of constructing molecular basis sets can be identified ... 

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. 

Ab initio LCAO-MO Methods.—The so called ab initio MO method is defined by the discussion so far. The two approximations made are the MO form of the wave-function and the necessary truncation of the basis set. It is important to realize that all the ab initio calculations on large molecules are far from the Hartree-Fock limit and that even this limit may be inadequate to describe a particular feature of the molecule. An excellent introduction to ab initio calculations is the recent book by... 

MNDO, Xa, HAM/3, AMI ab initio procedures such as SCF-CI within and beyond the Hartree-Fock limit Green s function propagator methods and density functionals). 

It is apparent from our description of the HFR procedure (and has been well established numerically) that the time required for a HFR calculation increases as somewhere between the third and fourth power of the size of the basis set. Similarly, the time required for going beyond HFR by configuration interaction increases as about the sixth power of the basis-set size for conventional Cl calculations. These important results explain why dramatic increases in computer speed lead only to modest increases in the size of systems treatable by such methods. For example, an increase of 1000 in computer speed increases the size of molecules tractable by Cl by slightly more than a factor of three, and those accessible to HFR procedures by a factor of about six. Thus, it appears that Cl techniques are directly applicable to only the simplest models of the species occurring in solid minerals. Even an approach to the Hartree-Fock limit wave... 

In this chapter we attempt to survey the present state of the art. In Section 4.B. a distinction is made between the problems, where the role of the correlation energy is small and those where it is of crucial importance. As regards the calculations beyond the Hartree-Fock limit there is a large variety of computational methods. We selected those that, in our opinion, are the most suitable for practical purposes or that yield perspectives for further development. The theoretical back-round of the methods is intentionally suppressed. Instead, emphasis is laid on the fundamental idea on which the theory is based, the numerical feasibility, cost, the portion of the correlation energy recovered and reliability in chemical applications. The whole chapter is oriented to ground states. 

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