Basis Functions Gauss-Type

For the evaluation of Eq. (100), we expand the one-electron functions into a sum over M atom-centered basis functions ) of Slater or Gauss type,... 

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. 

Since the introduction of the Gauss-t3q>e finite nucleus model (see Sect. 4.4) into relativistic quantum chemistry by Visser et al. [96], the combination of this model with Gauss-type basis functions has been most widely distributed, see, e.g., the programs developed by Dyall et al. [97], by Viss-cher et al. [98,99] (MOLFDIR), and by Saue et al. [100] (DIRAC). 

Nuclear attraction integrals for the combination of the homogeneous finite nucleus model (see Sect. 4.3) with Gauss-type basis functions were implemented by Ishikawa et al. [101], by Matsuoka et al. [102,103], and by dementi et al. [104-106]. 

The spatial part of each ip], is expanded in terms of the fundamental set of basis functions (Slater- or Gauss-type functions) gm... 

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

Bishop and Leclerc [35] investigated several unconventional basis functions including EGTOs, generalized Slater-Gauss-type, non-integer n Slater, rational, Hulthdn, and Bessel functions and found that the non-integer n Slater basis performed best. However, the scope of their study was fairly limited only one system, the H2 molecule, was probed. 

BSE = basis set expansion FDA = finite difference approximation raA = finite element approximation GTF = gauss-ian-type function ODE = ordinary differential equation nD = n-dimensional PDE = partial differential equation PW = partial-wave STF = Slater-type function. 

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