Finite basis set expansion

The error for negative ions is due to the wrong asymptotic form of the LSD potential. An LSD electron far from a neutral atom still sees itself and the extra repulsion leads to instability. Reasonable estimates of electron affinities can be obtained by preventing the electron from entering this poorly described asymptotic region, either by imposing a potential barrier or through the use of a finite basis-set expansion with no very diffuse functions . 

Finite Element Methods Applied to Many-body Perturbation Theory. - Over the past ten years, the finite element method, which is a classical tool in classical science and engineering applications, has been developed into a technique for the accurate solution of the atomic243 and molecular244,245 electronic structure problem. The piece-wise definition of the form functions employed in the finite element method prevents the computational linear dependencies which occur in the finite basis set expansion method and, moreover, leads to sparse, band structured matrices for which efficient solvers are available. 

It should be mentioned that problems in RCI calculations are not limited to finite basis set expansions of one-electron radial wave functions and can occur even if P(r) and Q r)... 

Finite basis set expansions are ubiquitous in ab initio molecular elecuonic sUucture studies and are widely recognized as one of the major sources of error in contemporary calculations [1-5]. Since the pioneering work by Hartree and his co-workers in the 1930s, finite difference methods have been used in atomic Hartree-Fock calculations. It is only in the past fifteen years or so that finite difference techniques [6-10] (and more recently, finite element methods [11-14]) have been applied to the molecular Hartree-Fock problem. By exploiting spheroidal co-ordinates, two-dimensional Haitree-Fock calculations for diatomic molecules have become possible. These calculations have provided benchmarks which, in turn, have enabled the finite basis set approach to be refined to the point where matrix Hartree-Fock calculations for diatomic molecules can yield energies which approach the p-Hartree level of accuracy [15-18]. Furthermore, these basis sets can then be employed in calculations for polyatomic molecules [19,20] which are not, at present, amenable to finite difference or finite element techniques. 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

Of these schemes, quantum many-body theories implemented within the algebraic approximation (using finite basis set expansions) have come to dominate quantum chemistry over the past 50 years underpinning a systematic approach to the theoretical study of molecular electronic structure. 

It must be based on a careful and systematic realization of the algebraic approximation (i.e., the use of finite basis set expansions), since this can often be the dominant source of error in calculations which aim to achieve high precision. Over the past 20 years a great deal of progress has been made in the systematic and accurate implementation of the algebraic approximation in many-body quantum chemical studies. This will not be reviewed here and the interested reader is referred elsewhere for detailed reviews [89,110]. 

A more recent development are DHF or MCDHF programs based on the algebraic approximation, i.e. finite basis set expansions are used instead of finite difference methods, cf e.g. Grant and Quiney (1988). The first work on closed and open shell atoms was published by Kim (1967) and Kagawa (1975), respectively. Nevertheless these algebraic... 

It must be based on a careful and systematic realization of the algebraic approximation (i.e. the use of finite basis set expansions), since this can often be the dominant source of error in calculations which aim to achieve high precision. 

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. 

In the algebraic approximation, finite basis set expansions are employed to approximate the orbital functions. The orbital cpk is then written in the form... 

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