Basis set exponents

Table 1. vD + D basis set exponents of the additional polarization functions included in the basis set... 

For accurate calculations of TM compounds, f-type polarization functions should be added to the basis set. Exponents for f-polarization functions have been optimized by us for the Hay-Wadt ECP. No other sets of f-type functions optimized for use with pseudopotentials are known to us. However, because the valence orbitals of the pseudopotentials mimic the all-electron orbitals, the f-type functions determined for all-electron cases can also be used for pseudopotential calculations. 

Even-tempered basis sets [40] consist of GTOs in which tlie orbital exponents belonging to series of... 

The solution to this problem is to use more than one basis function of each type some of them compact and others diffuse, Linear combinations of basis Functions of the same type can then produce MOs with spatial extents between the limits set by the most compact and the most diffuse basis functions. Such basis sets arc known as double is the usual symbol for the exponent of the basis function, which determines its spatial extent) if all orbitals arc split into two components, or split ualence if only the valence orbitals arc split. A typical early split valence basis set was known as 6-31G 124], This nomenclature means that the core (non-valence) orbitals are represented by six Gaussian functions and the valence AOs by two sets of three (compact) and one (more diffuse) Gaussian functions. 

Th e con traction expon en ts and cocfTicien ts of th e d-type functions were optinii/ed using five d-primitives (the first set of d-type functions) for the STO-XG basis sets and six d-primitives (the second set of d-type functions ) for the split-valence basis sets. Thus, five d orbitals are recommended for the STO-XG basis sets and six d orhitals for the split-valence basis sets. 

Usually, con traction s arc determ in cd from atom ic SCh calcula-tioris. In these calculations one uses a relatively large basis of uncontracted Gaussiaris, optim i/.es all exponents, and determines th e SCH coefficien is of each of the derived atom ic orbitals, fh c optim i,red e.spon en ts and SCH coefficien Ls cati th en be used to derive sii itable con traction expon cn is an d con traction coefficien is fora smaller basis set to be used in subsequent rn olecu lar ca leu la-lion s. 

Sin cc til e basis set is oblairicd from atom ic calcii laliori s, it is still desirable to scale expon eti ts for the rn oleeular en viron tn eti t, Th is is accom piished by defiri in g an in ri er valen ce scale factor 1 and an outer valence scale factor C" ( doiihle zeta ) and multiplying the correspon din g in ri er an d otiler ct s by th e square of these factors. On ly the valen ce sh ells arc scaled. 

The even-tempered basis set consists of the following sequence of functions Is, 2p, 3d, 4f,. .., which correspond to increasing values oi k. The advantage of this basis set is that ii relatively easy to optimise the exponents for a large sequence of basis functions. 

Basis sets can be constructed using an optimisation procedure in which the coefficients and the exponents are varied to give the lowest atomic energies. Some complications can arise when this approach is applied to larger basis sets. For example, in an atomic calculation the diffuse functions can move towards the nucleus, especially if the core region is described... 

Even-tempered basis sets (M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3961 (1979)) consist of GTOs in which the orbital exponents ak belonging to series of orbitals consist of geometrical progressions ak = a, where a and P characterize the particular set of GTOs. 

The values of the orbital exponents ( s or as) and the GTO-to-CGTO eontraetion eoeffieients needed to implement a partieular basis of the kind deseribed above have been tabulated in several journal artieles and in eomputer data bases (in partieular, in the data base eontained in the book Handbook of Gaussian Basis Sets A. Compendium for Ab initio Moleeular Orbital Caleulations, R. Poirer, R. Kari, and I. G. Csizmadia, Elsevier Seienee Publishing Co., Ine., New York, New York (1985)). 

A second issue is the practice of using the same set of exponents for several sets of functions, such as the 2s and 2p. These are also referred to as general contraction or more often split valence basis sets and are still in widespread use. The acronyms denoting these basis sets sometimes include the letters SP to indicate the use of the same exponents for s andp orbitals. The disadvantage of this is that the basis set may suffer in the accuracy of its description of the wave function needed for high-accuracy calculations. The advantage of this scheme is that integral evaluation can be completed more quickly. This is partly responsible for the popularity of the Pople basis sets described below. 

This section gives a listing of some basis sets and some notes on when each is used. The number of primitives is listed as a simplistic measure of basis set accuracy (bigger is always slower and usually more accurate). The contraction scheme is also important since it determines the basis set flexibility. Even two basis sets with the same number of primitives and the same contraction scheme are not completely equivalent since the numerical values of the exponents and contraction coefficients determine how well the basis describes the wave function. 

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. 

Diffuse functions are those functions with small Gaussian exponents, thus describing the wave function far from the nucleus. It is common to add additional diffuse functions to a basis. The most frequent reason for doing this is to describe orbitals with a large spatial extent, such as the HOMO of an anion or Rydberg orbitals. Adding diffuse functions can also result in a greater tendency to develop basis set superposition error (BSSE), as described later in this chapter. 

The contracted basis in Figure 28.3 is called a minimal basis set because there is one contraction per occupied orbital. The valence region, and thus chemical bonding, could be described better if an additional primitive were added to each of the valence orbitals. This is almost always done using the even-tempered method. This method comes from the observation that energy-optimized exponents tend to nearly follow an exponential pattern given by... 

The Slater exponents partially listed in the table above are used for all the STO-NG basis sets. The exponents for all the atoms with atomic numbers less than and equal to 54 are available from HyperChem basis function. BAS files. 

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