Quadruple zeta

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-... 

Calculations at the 6-31G and 6-31G level provide, in many cases, quantitative results considerably superior to those at the lower STO-3G and 3-21G levels. Even these basis sets, however, have deficiencies that can only be remedied by going to triple zeta (6-31IG basis sets in HyperChem) or quadruple zeta, adding more than one set of polarization functions, adding f-type functions to heavy atoms and d-type functions to hydrogen, improving the basis function descriptions of inner shell electrons, etc. As technology improves, it will be possible to use more and more accurate basis sets. 

The next step up in basis set size is a Triple Zeta (TZ). Such a basis contains three times as many functions as tire minimum basis, i.e. six s-functions and three p-functions for the first row elements. Some of the core orbitals may again be saved by only splitting the valence, producing a triple split valence basis set. Again the term TZ is used to cover both cases. The names Quadruple Zeta (QZ) and Quintuple Zeta (5Z, not QZ) for the next levels of basis sets are also used, but large sets are often given explicitly in terms of the number of basis functions of each type. 

Dunning has developed a series of correlation-consistent polarized valence n-zeta basis sets (denoted cc-pVnZ ) in which polarization functions are systematically added to all atoms with each increase in n. (Corresponding diffuse sets are also added for each n if the prefix aug- is included.) These sets are optimized for use in correlated calculations and are chosen to insure a smooth and rapid (exponential-like) convergence pattern with increasing n. For example, the keyword label aug-cc-pVDZ denotes a valence double-zeta set with polarization and diffuse functions on all atoms (approximately equivalent to the 6-311++G set), whereas aug-cc-pVQZ is the corresponding quadruple-zeta basis which includes (3d2flg,2pld) polarization sets. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

Weigend, R Ahhichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn design and assessment of accuracy, Phys. Chem. Chem. Phys. 2005, 7, 3297-3305. 

Augmented Gaussian basis sets of triple and quadruple zeta valence quality for the atoms H and from Li to Ar applications, in HF, MP2, and DFT calculations-of molecular dipole moment and dipole (h5q)er)polarizability ... 

In a study of tin dibromide, it was prudent to allow for the presence of the dimer, Sn2Br2, although in the end the refined amount of dimer was just 1.4(5)%.14 The Sn-Br distance (rg) refined to 251.5(5) pm and the BrSnBr angle to 97.9(4)°. The structure proved to be rather difficult to reproduce theoretically, requiring a very large basis set (quadruple zeta quality). 

There are other aspects of the application of the MCSCF method that have not been discussed in this review. The most notable of these probably is the lack of a discussion of orbital basis sets. Although the orbital basis set choice is very important in determining the quality of the MCSCF wavefunction, the general principles determined from other electronic structure methods also hold for the MCSCF method with very little change. For example, the description of Rydberg states requires diffuse basis functions in the MCSCF method just as any other method. The description of charge-transfer states requires a flexible description of the valence orbital space, triple or quadruple zeta quality, in the MCSCF method just as in other methods. Similarly, the efficient transformation of the two-electron integrals is crucial to the overall efficiency of the MCSCF optimization procedure. However, this is a relatively well understood problem (if not always well implemented) and has been described adequately in previous discussions of the MCSCF method and other electronic structure methods . ... 

As an example of such a study we pick a small molecule PbF. Suppose we want to compute the potentials for the lower electronic states of this molecule, with the relativistic CASSCF/CASPT2 approach, how do we proceed Well, it should not be difficult. Relativistic basis sets (ANO-RCC) are available for Pb and F [30] and we choose a reasonably extended set Pb 25s22pl6dl2f4g/9s8p6d4f3g and F 14s9p4d3f2g/ 5s4p3d2flg. It is of quadruple zeta accuracy. 

A triple-zeta (TZ) basis in which three times as many STOs or CGTOs are used as the number of core and valence AOs (and, yes, there now are quadruple-zeta (QZ) and higher-zeta basis sets appearing in the literature). 

In the case of B3LYP computations the convergence of the results, as a function of the basis set dimension, has been tested using basis sets of increasing size starting from the double zeta (6-31G ) to arrive to quadruple zeta (AUG-cc-pVQZ) quality. As it is shown in Table 2, the 6-3IG basis set does not give reliable results. The introduction of the diffusion functions (6-31 1-m-G ) improves strongly the values of PA. The deviation, with respect to the experimental data, is of only 0.2 kcal/mol. 

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. 

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