Double, Triple, Quadruple Zeta

Electron correlation studies demand basis sets that are capable of very high accuracy, and the 6-31IG set I used for the examples above is not truly adequate. A number of basis sets have been carefully designed for correlation studies, for example the correlation consistent basis sets of Dunning. These go by the acronyms cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z and cc-pV6Z (double, triple, quadruple, quintuple and sextuple-zeta respectively). They include polarization functions by definition, and (for example) the cc-pV6Z set consists of 8. 6p, 4d, 3f, 2g and Ih basis functions. 

Examining the results given in these two tables, it is seen that, for this small molecule, very advanced calculations can be carried out. In the tables, all the methods employed have been introduced in the previous sections. For the basis sets, aug-cc-pVnZ stands for augmented correlation consistent polarized valence n zeta, with n = 2-5 referring to double, triple, quadruple, and quintuple, respectively. Clearly, these basis functions are specially designed for... 

Rappe, Smedley and Goddard (1981) Stevens, Basch and Krauss (1984) Used for ECP (effective core potential) calculations Dunning s correlation consistent basis sets (double, triple, quadruple, quintuple and sextuple zeta respectively). Used for correlation calculations Woon and Dunning (1993)... 

The next step in iin proving a basis set could be to go to triple zeta, quadruple zeta, etc. Ifone goes in this direction rather than adding functions of higher angular quantum number, the basis set would not be well balanced. With a large number of s and p functions only, one finds, for example, that the equilibrium geometry of am monia actually becomes planar. The next step beyond double z.ela n sit ally in voices addin g polarization fn n ciion s, i.e.. addin g d-... 

DyaU, K.G. Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 5d elements Hf-Hg, Theor. Ghem. Acc. 2004,112,403. 

Several different basis sets, including pseudopotential LANL2DZ [100] as well as two Aldrich s valence split basis sets (triple zeta (TZVP) and quadruple zeta (QZVP) valence quality) [101] were used to calculate modeled reactions involving small FeS clusters. Employing an electron core potential (ECP) basis set such as LANL2DZ (Los Alamos National Laboratory 2 double for transition metals) has become popular in computations on a transition-metal-containing systems. Due to large size of calculated models in the case of the adsorption of nucleic acid bases on the clay mineral surfaces the 6-31G(d) basis set [102] was appUed. 

Double zeta, triple zeta, and quadruple zeta basis sets include additional sets of each basis function (2, 3, or 4 sets respectively), with each additional set having a different size. By varying the relative contribution of the alternative sizes to the overall molecular orbital, one effectively introduces a single function with a variable size (see Fig. 2). 

Split-valence basis sets are a simplification to the double, triple, and quadruple zeta basis sets described above. Since the inner shell orbitals are not usually involved in bonding, and their energies are reasonably independent of their molecular environment, it is usually only necessary to include the extra basis functions for the valence orbitals. Basis sets that include different numbers of basis fimctions for the inner shell and valence electrons are known as split-valence basis sets. 

Dunning basis sets have names such as cc-pVnZ. This notation stands for correlation-consistent polarized valence -zeta. For a double zeta basis set, n is replaced by a D, for a triple zeta basis set, n is replaced by a T, for a quadruple zeta basis set, n is replaced by a Q, for a quintuple basis set we use a 5, and for a sextuple basis set we use a 6. When diffuse functions are included, an aug prefix is included in the name, as in aug-cc-pVTZ. The cc-pVTZ basis set generally has a performance similar to 6-311G(2df,p). A special feature of the Dimning basis sets is that they have been designed so the series DZ, TZ, QZ, 5Z, 6Z... systematically converges on the infinite basis set limit (27). This feature has been exploited in the... 

Early comparisons of this sort were based on work by Handy and coworkers [37], who developed an efficient direct FCI approach in terms of determinants. Using a double-zeta (DZ) basis set, they considered stretching the 0-H bond lengths in the H2O molecule to 1.5 and 2.0 times their equilibrium values. The FCI results showed that even the restricted Hartree-Fock (RHF) based fourth-order many-body perturbation theory (MBPT) approach [38], which includes the effect of single, double, triple and quadruple excitations, did not accurately describe the stretching of the bond the error increased from 0.6 kcal/mole at to 10.3 kcal/mole... 

The counterpoise correction typically overestimates the BSSE since the monomer basis set is enhanced not only by empty orbitals of the other fragment, but also by orbitals occupied by electrons of the other monomer molecule which are excluded by the Pauli principle. Thus, if CP-corrected and uncorrected interaction energies are plotted as function of basis set size, they approach from above and below, respectively, the true interaction energy at the complete basis set (CBS) limit. CP corrections are mandatory for all double-zeta calculations and with MP2 or CCSD(T) also for triple-zeta basis treatments. In triple-zeta basis set (e.g., cc-pVTZ or TZVPP) DPT calculations, the BSSE is typically less than 5-10% of the interaction energy which makes the laborious CP correction unnecessary. If sets of valence quadruple-zeta are used, it seems as if the error of the CP procedure is often similar to the (uncorrected) BSSE, but this is system-dependent and more definite conclusions about this issue requires further work. 

AMBER = assisted model building with energy refinement force field CHARMM = chemistry at Harvard macromolecu-lar mechanics force field MP4SDQ = Mpller-Plesset fourth-order perturbation theory with corrections for single, double, and quadruple excitations OPLS = optimized potentials for liquid simulation force field TZP = triple-zeta -f polarization. 

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