The Algebraic Approximation

In the first part of this section, the relationship between the solution of the Schrodinger equation and the hamiltonian in the space generated by a given basis set is discussed in some detail. Since basis set limitations appear to be one of the largest sources of error in most present day molecular calculations, the concept of a universal even-tempered basis set is discussed in the second part of this section. This concept represents an attempt to overcome the incomplete basis set problem, at least for diatomic molecules. Further aspects of the basis set truncation problem are discussed in the final part of this section. 

The algebraic approximation results in the restriction of the domain of the operator to a finite dimensional subspace, Sf, of the Hilbert space The algebraic approximation may be implemented by defining a suitable orthonormal basis set of M ( N) one-electron spin orbitals and constructing all unique iV-electron determinants /t using the M one-electron functions. The 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

Once the algebraic approximation has been invoked there is essentially no difference between the atomic problem and the molecular problem, except that the multicentre integrals which arise in the latter case are more difficult to evaluate. 

The algebraic approximation, i.e. the use of approximations based on finite basis set expansions, is ubiquitous in practical quanmm chemistry. Gaussian basis sets, in particular, are almost universally employed in contemporary calculations of molecular electronic structure because of the ease and accuracy with which the associated molecular integrals can be evaluated. 

We have seen in eq. (1.1) that the (cartesian) Gaussian basis functions have the form 

Within the algebraic approximation, the single-particle state functions or orbitals, of the independent particle model are approximated as 

The matrix Hartree-Fock approximation is realized within a basis set of Gaussian-type functions in terms of the following molecular integrals  

Implementation of the algebraic approximation requires the determination of the orbital expansion coefficients, in eq. (3.140). This is achieved by solving the matrix Hartree-Fock equations which may be written 

4 The Algebraic Approximation. - For atoms the use of spherical polar coordinates ( , , ) facilitates the factorization of the Hartree-Fock equations and reduces the problem to one involving a single radial coordinate r and an angular part which can be treated analytically. For diatomic molecules prolate spheroidal coordinates ( , , ) separate the non-relativistic Hartree-Fock equations into a two-dimensional part which can be solved numerical and a -dependent part which can be treated analytically. For arbitrary molecular systems there is no suitable coordinate system in which the problem can be formulated and hence it is usual to resort to the algebraic approxima- 

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Display Accuracy) presents a list of typical and extreme absolute and relative errors incurred when using the approximation note that the listed errors are in part due to the algebraic approximation as such, and in part to the finite number of digits of the tabulated values. 

Modern many-body methods have become sufficiently refined that the major source of error in most ab initio calculations of molecular properties is today associated with truncation of one-particle basis sets e.g. [1]- [4]) that is, with the accuracy with which the algebraic approximation is implemented. The importance of generating systematic sequences of basis sets capable of controlling basis set truncation error has been emphasized repeatedly in the literature (see [4] and references therein). The study of the convergence of atomic and molecular structure calculations with respect to basis set extension is highly desirable. It allows examination of the convergence of calculations with respect to basis set size and the estimation of the results that would be obtained from complete basis set calculations. 

In practical applications, we invariably invoke the algebraic approximation by parametrizing the orbitals in a finite basis set. This approximation may be written... 

This article is divided into seven parts. The many-body perturbation theory is discussed in the next section. The algebraic approximation is discussed in some detail in section 3 since this approximation is fundamental to most molecular applications. In the fourth section, the truncation of the many-body perturbation series is discussed, and, since other approaches to the many-electron correlation problem may be regarded as different ways of truncating the many-body perturbation expansion, we briefly discuss the relation to other approaches. Computational aspects of many-body perturbative calculations are considered in section 5. In section 6, some typical applications to molecules are given. In the final section, some other aspects of the many-body perturbation theory of molecules are briefly discussed and possible directions for future investigations are outlined. 

This completes a literate program for evaluating third-order ring energies in the many-body perturbation theory for closed-shell systems within the algebraic approximation. 

The Hamiltonian is, in the algebraic approximation defined by the finite MO basis, given as... 

S. Wilson and D. Moncrieff, On the accuracy of the algebraic approximation in molecular electronic structure calculations. VI. Matrix Hartree-Fock and Many-Body Perturbation Theory Calculations for the Ground State of the Water Molecule, preprint... 

H. M. Quiney, The Dirac equation in the algebraic approximation, in S. Wilson (Ed.), Handbook of Molecular Physics and Quantum Chemistry, John Wiley and Sons, Chichester, 2002. 

QED provides a framework for describing the role of the negative energy states in the Dirac theory and the divergences which arise in studies of electrodynamic interactions.83-85 In Section 2.3 it will be shown how the use of the algebraic approximation to generate a discrete representation of the Dirac spectrum has opened the way for the transcription of the rules of QED into practical algorithms for the study of many-electron systems. 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

Figure 22 Schematic representation of the Dirac spectrum and of the Schrodinger spectrum generated in the algebraic approximation...

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