Fock expansion properties

A very compact and highly accurate wave function for the ground state of the He atom has already been constructed by Hylleraas long ago [21]. He expressed this in terms of the coordinates ri,r2 and ru with ri and T2 the distances of the first and second electron from the nucleus, and ri2 the distance between the electrons. Thus the cusp conditions [20] could be satisfied. Essentially in the same philosophy Pekeris performed a calculation on the He-ground state [22], that remained an undisputed landmark for quite some time. A progress beyond this was possible when analytic properties of the exact wave function of a three-particle system (one nucleus and two electrons) were talren into account, which were ignored in earlier formulations. The keyword to this is Fock expansion and it requires terms that are logarithmic in the coordinates [23, 24, 25]. 

Considering the different calculated values for an individual complex in Table 11, it seems appropriate to comment on the accuracy achievable within the Hartree-Fock approximation, with respect to both the limitations inherent in the theory itself and also to the expense one is willing to invest into basis sets. Clearly the Hartree-Fock-Roothaan expectation values have a uniquely defined meaning only as long as a complete set of basis functions is used. In practice, however, one is forced to truncate the expansion of the wave function at a point demanded by the computing facilities available. Some sources of error introduced thereby, namely ghost effects and the inaccurate description of ligand properties, have already been discussed in Chapter II. Here we concentrate on the... 

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

A landmark in atomic theory was provided by the work of Layzer,9 who pointed out that regularities in the properties of atomic ions,10 which were hard to relate via numerical Hartree-Fock (HF) studies, could be understood via the so-called 1/Z expansion. Layzer9 showed that the total non-relativistic energy of an atomic ion could be expanded as... 

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

We can also formulate this in a different manner and say that the self-consistent field procedure plays a crucial role in 4-component theory because it serves to define the spinors that isolate the n-electron subspaces from the rest of the Fock space. In this manner it determines in effect the precise form of the electron-electron interaction used in the calculations. Both aspects are a consequence of the renormalization procedure that was followed when fixing the energy scale and interpretation of the vacuum. The experience with different realizations of the no-pair procedure has learned that the differences in calculated chemical properties (that depend on energy differences and not on absolute energies) are usually small and that other sources of errors (truncation errors in the basis set expansion, approximations in the evaluation of the integrals) prevail in actual calculations. 

The truncation procedure explored in the present smdy is described in detail in section 2. An analysis of the orbital expansion coefficients for the ground state of the BF molecule is presented in section 3, where the truncated basis sets employed in the present study are defined. The results of both matrix Hartree-Fock calculations and second-order many-body perturbation theory studies are given in section 4 together with a discussion of the properties of the truncated basis sets. The final section, section 5, contains a discussion of the results and conclusions are given. 

This corresponds to describing the function/in an M-dimensional space of the basis functions x- For a fixed basis set size M, only the components of/that lie within this space can be described, and/is therefore approximated. As the size of the basis set M is increased, the approximation becomes better since more and more components of / can be described. If the basis set has the property of being complete, the function / can be described to any desired accuracy, provided that a sufficient number of functions are included. The expansion coefficients C are often determined either by variational or perturbational methods. For the expansion of the molecular orbitals in a Flartree-Fock wave function, for example, the coefficients are determined by requiring the total energy to be a minimum. 

The convergence properties of Fock s expansion have been studied mathematically by Macek [10], Leray [11], and me [12], and it now is essentially certain that Fock s expansion does provide a convergent representation of many-electron wavefunctions, although a few mathematical details remain to be wrapped up. 

---