Types of Basis Functions

The transition to the analytic approximation is effected by introducing a set of functions and expressing the large and small components of the one-electron 2-spinor as a linear combination of these basis functions 

If all we cared about was hydrogen-like atoms, we would not need to employ expansion methods, but could solve the differential equations numerically, as is done in existing codes for 4-component calculations on atoms [1,2]. However, the electron-electron interaction of many-electron systems gives rise to integrals of the form 

In the case of a general polyatomic molecule, the I of the integral above may be located on different atoms, in the worst case giving rise to a four-center integral. For calculations within mean-field or independent particle approximations, such as Hartree-Fock or Dirac-Hartree-Fock (DHF), the bulk of the computational effort lies in the evaluation and handling of these two-electron integrals. This has consequences for our choice of expansion functions for the analytic approximation. 

For the non-relativistic case various functions have been tried, and discussions of their respective merits may be found in the literature [3]. The simplest has been to use hydrogenic functions, or suitably modified La-guerre polynomials. This may be useful for purposes of analysis in simple atomic systems, but has had little impact on the molecular field. The reason for this is the complicated form of the integrals. A somewhat more efficient choice is the Slater type orbital (STO) of the form 

For the relativistic case there are three analogous choices of expansion functions to those discussed above. The hydrogenic functions have their analogue in the L-spinors obtained from the solution of the Dirac-Coulomb equation [4]. Again their use is mainly restricted to analytic work in atomic calculations, due to the difficulties in evaluating the integrals [5]. The analogue of the STO is the S-spinor which may be written in the form 

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This type of basis functions is frequently used in popular quantum chemishy packages. We shall discuss the way to evaluate different kinds of matrix elements in this basis set that are often used in quantum chemistt calculation. 

There are several types of basis functions listed below. Over the past several decades, most basis sets have been optimized to describe individual atoms at the EIF level of theory. These basis sets work very well, although not optimally, for other types of calculations. The atomic natural orbital, ANO, basis sets use primitive exponents from older EIF basis sets with coefficients obtained from the natural orbitals of correlated atom calculations to give a basis that is a bit better for correlated calculations. The correlation-consistent basis sets have been completely optimized for use with correlated calculations. Compared to ANO basis sets, correlation consistent sets give a comparable accuracy with significantly fewer primitives and thus require less CPU time. 

There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schrodinger equation) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals have die functional form... 

It is not easy to see why the authors believe that the success of orbital calculations should lead one to think that the most profound characterization of the properties of atoms implies such an importance to quantum numbers as they are claiming. As is well known in quantum chemistry, successful mathematical modeling may be achieved via any number of types of basis functions such as plane waves. Similarly, it would be a mistake to infer that the terms characterizing such plane wave expansions are of crucial importance in characterizing the behavior of atoms. 

First, with respect to the type of basis functions used in G, smoothness is by no means restrictive. As it is intuitively clear and proved in practice, weird nonsmooth basis functions have to be excluded from consideration but beyond that, all normal bases are able to create smooth approximations of the available data. Accuracy is not a constraint either. Given enough basis functions, arbitrary accuracy for the prediction on the data is possible. 

The conclusion is that for every particular set of basis functions and given data, there exists an appropriate size of G that can approximate both accurately and smoothly this data set. A decisive advantage would be if there existed a set of basis functions, which could probably represent any data set or function with minimal complexity (as measured by the number of basis functions for given accuracy). It is, however, straightforward to construct different examples that acquire minimal representations with respect to different types of basis functions. Each basis function for itself is the most obvious positive example. A Gaussian (or discrete points... 

In recent years some theoretical results have seemed to defeat the basic principle of induction that no mathematical proofs on the validity of the model can be derived. More specifically, the universal approximation property has been proved for different sets of basis functions (Homik et al, 1989, for sigmoids Hartman et al, 1990, for Gaussians) in order to justify the bias of NN developers to these types of basis functions. This property basically establishes that, for every function, there exists a NN model that exhibits arbitrarily small generalization error. This property, however, should not be erroneously interpreted as a guarantee for small generalization error. Even though there might exist a NN that could... 

The nature of the input transformation, type of basis functions, and optimization criteria discussed in this section provide a common framework for comparing the wide variety of techniques for input transformation and input-output modeling. This comparison framework is useful for understanding the similarities and differences between various methods it may be used to select the best method for a given task and to identify the challenges for combining the properties of various techniques (Bakshi and Utojo, 1999). 

The number and type of basis functions strongly influence the quality of the results. The use of a single basis function for each atomic orbital leads to file minimal basis set. In order to improve the results, extended basis sets should be used. These basis sets are named double-f, triple-f, etc. depending on whether each atomic orbital is described by two, three, etc. basis functions. Higher angular momentum functions, called polarization functions, are also necessary to describe the distortion of the electronic distribution due... 

In quantum chemistry it is quite common to use combinations of more familiar and easy-to-handle "basis functions" to approximate atomic orbitals. Two common types of basis functions are the Slater type orbitals (STO s) and gaussian type orbitals (GTO s). STO s have the normalized form ... 

The use of different types of basis functions in molecular calculations continues. The Os or Hulthdn-type function has been used in calculations on Hs and Ha,33 and is more effective than the Is-type function. Hj-type elliptical orbitals have also been employed in variational calculations on Hi, Ha, He 4, and H.4.34 The H orbitals were also used as basis orbitals for SCF calculations. A four-function basis set with two (non-linear) variational parameters yields 99% of the Hartree-Fock energy. 

A basis set may be employed that is of the same form throughout the space of the system, or one in which the orbitals are expanded in different types of basis functions in different parts of space. Such partitioned bases are often used in solid-state calculations in which one must describe an overall wave function that is rapidly varying near the nuclei and slowly varying and free-electron-like when far from the nuclei. Such partitioned bases will be considered further in our discussion of band-theoretical calculations and the multiple-scattering Xa molecular-orbital method. 

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis function may in principle be used expo ... 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

This notation covers both the non-relativistic and relativistic cases (scalar orbitals and 4-component spinors, respectively), the indices i and j carry information to identify the basis functions imambiguously. The integrals in Eq. (122) may involve only a single centre A = B — C [ atomic integrals ]), two centres, or three centres A, B, C all different). The difficulty of their evaluation increases with the number of centres. In addition, every type of basis functions requires its own implementation of nuclear attraction integrals. This task has been accomplished, at least to some paxt, for various potentials and Slater-type or Gauss-type basis functions. For technical reasons (ease of evaluation of multi-centre integrals) the latter type is usually preferred. 

We focus on periodic systems (ID ordered polymers, 2D slabs and ultra-thin films, 3D —> crystals). A considerable variety of DFT implementations exists in codes for such systems, depending on the choice of basis set. (Somewhat confusingly for begiimers, in solid-state physics the choice of a basis commonly has been called a method , presumably because special techniques evolved to exploit the advantages and minimize the difficulties of each choice.) With few exceptions, modem codes are based on some approximate eigenvalue problem for an effective Hamiltonian, hence they expand the eigensolutions in linear combinations of one of four types of basis functions ... 

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