Basis functions semiempirical methods

Common language of electronic structure theory, validating density functional theory, standardization of Gaussian basis sets, semiempirical methods, B3LYP functional (Becke s three-parameter hybrid functional using the LYP correlation functional), well-defined procedures for QM algorithms... 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

Semiempirical methods are widely used, based on zero differential overlap (ZDO) approximations which assume that the products of two different basis functions for the same electron, related to different atoms, are equal to zero [21]. The use of semiempirical methods, like MNDO, ZINDO, etc., reduces the calculations to about integrals. This approach, however, causes certain errors that should be compensated by assigning empirical parameters to the integrals. The limited sets of parameters available, in particular for transition metals, make the semiempirical methods of limited use. Moreover, for TM systems the self-consistent field (SCF) procedures are hardly convergent because atoms with partly filled d shells have many... 

Although ASCF methods are more likely to be successful, it is critical that diffuse functions be included in the basis set so that the description of the radical anion is adequate with respect to the loosely held extra electron. In general, correlated methods are to be preferred, and DFT represents a reasonably efficient choice that seems to be robust so long as the radical anion is not subject to overdelocalization problems. Semiempirical methods do rather badly for EAs, at least in part because of their use of minimal basis sets. 

In 1951 Roothaan and Hall independently pointed out [26] that these problems can be solved by representing MO s as linear combinations of basis functions (just as in the simple Hiickel method, in Chapter 4, the % MO s are constructed from atomic p orbitals). Roothaan s paper was more general and more detailed than Hall s, which was oriented to semiempirical calculations and alkanes, and the method is sometimes called the Roothaan method. For a basis-function expansion of MO s we write... 

The electron distribution around an atom can be represented in several ways. Hydrogenlike functions based on solutions of the Schrodinger equation for the hydrogen atom, polynomial functions with adjustable parameters, Slater functions (Eq. 5.95), and Gaussian functions (Eq. 5.96) have all been used [34]. Of these, Slater and Gaussian functions are mathematically the simplest, and it is these that are currently used as the basis functions in molecular calculations. Slater functions are used in semiempirical calculations, like the extended Hiickel method (Section 4.4) and other semiempirical methods (Chapter 6). Modem molecular ab initio programs employ Gaussian functions. 

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