Single-zeta function

The examples to be considered are the ground states of He and the related two-electron ions from H to Nes+. In all cases, the single-zeta function of Kellner will be the approximate wave function used [16]. This function is T° =Nz ""r. The local chemical potential is given by... 

An older, but still used, notation specihes how many contractions are present. For example, the acronym TZV stands for triple-zeta valence, meaning that there are three valence contractions, such as in a 6—311G basis. The acronyms SZ and DZ stand for single zeta and double zeta, respectively. A P in this notation indicates the use of polarization functions. Since this notation has been used for describing a number of basis sets, the name of the set creator is usually included in the basis set name (i.e., Ahlrichs VDZ). If the author s name is not included, either the Dunning-Hay set is implied or the set that came with the software package being used is implied. 

Table 7.1 Atomic orbital parameters used in extended Huckel calculations. Single zeta STO functions are used for B and C and double zeta STO functions are used for the transition metals...

With the addition of polarization functions and/or diffuse functions to the basis sets, the Pople notation can become rather cumbersome. For example, the 6-311++G(3df,2pd) set has a single zeta core and triple zeta valence shell, diffuse functions for all the atoms. Regarding polarized functions, there are three sets of d functions and one set of f functions on the non-hydrogens and two sets of p functions and one set of d orbitals on the hydrogens. 

A more sequential approach to the analysis of the systematic error of ab initio methods has been proposed in [14]. The same set of molecules as in [10] has been analyzed there. For this set the series of calculations using the basis sets aug-cc-pVxZ containing both polarization and diffuse functions with the number of exponents x in their respective radial parts up to x = 6 (single zeta x = 1, double zeta - DZ -x = 2, triple zeta - TZ - x = 3, etc.) and with the account of correlation effects in the range of methods from MP2 up to CCSD(T) had been performed and then fitted to the formulae [15-18] ... 

A number of factors define the basis set for a quantum chemical computation. First, how many basis functions should be used The minimum basis set has one basis function for every formally occupied or partially occupied orbital in the atom. So, for example, the minimum basis set for carbon, with electron occupation ls 2s 2p, has two s-type functions and p, p, and Pj functions, for a total of five basis functions. This minimum basis set is referred to as a single zeta (SZ) basis set. The use of the term zeta here reflects that each basis function mimics a single STO, which is defined by its exponent, C-... 

The (HF) radial 3d function of a first transition series atom is compact and not accurately representable by a single exponential (r e 0- This is strikingly illustrated in Table III, which compares 3d orbital energies from single-zeta... 

Triple-zeta STF basis sets were used for the valence shells, extended by single-zeta STF to accurately represent the nodal structure of the core. For the transition metals, (n-l)d and ns were considered as valence shells and one np polarization function was added. For C and O, 2s and 2p were the valence shells and a 3d polarization function was added. For H one 2p polarizaition function was added. Core shells were frozen. 

The exponents were often denoted by the Greek letter zeta. Thus, a single-zeta" basis set would have a single exponential function representing each atomic orbital. A double-zeta basis would have two exponential functions for each AO. The terminology has carried over into Gaussians. 

Widely used are the GTO basis sets of the 6-31G and 6-31G types. They correspond to extended STO basis sets which include polarization function. One asterisk denotes addition of the polarization d-GTO to each p function, while two asterisks mean that, besides that orbital, a p-GTO is added to the Is orbitals of the hydrogen atoms There are cases when the 6-31 G -type basis sets do not satisfy accuracy requirements in the calculation of physical characteristics of molecules. Pople and his co-workers [14] have suggested in this connection the basis sets of the types 6-311G (single zeta core, triple zeta valence and polarization functions on all atoms) and 6-311 + -hG (3d/, 3pd) which differ from the previous ones in further additions of the polarization... 

In Figure 7.6, we have plotted the error in this single-zeta Cl energy as more and more terms from (7.3.5) are included in the wave function, truncating the expansion according to the principal quantum number N ... 

A more flexible approach is to optimize the exponents and coefficients simultaneously, obtaining in some sense the best segmented basis set for a fixed number of primitive functions and a fixed number of contracted functions. Several such sets have been designed. Particularly popular are the split-valence 3-21G [12] and 6-31G [13] basis sets of Pople and coworkers. (The term split-valence indicates that there is a single-zeta representation of the core shell and a double-zeta representation of the valence shell.) In the 6-3IG basis, for example, the 1j core orbital is described by a single... 

Barone also introduces two new basis sets, EPR-Il and EPR-llI. These are optimized for the calculation of hyperfine coupling constants by density functional methods. EPR-Il is a double zeta basis set with a single set of polarization functions and an enhanced s part. EPR-III is a triple zeta set including diffuse functions, double d polarization functions and a single set off functions. 

---