Diffuse basis functions

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. 

Basis sets can be extended indefinitely. The highest MOs in anions and weakly bound lone pairs, for instance, are very diffuse maybe more so than the most diffuse basis functions in a spht valence basis set. In this case, extra diffuse functions must be added to give a diffuse augmented basis set. An early example of such a basis set is 6-31+G [26]. Basis sets may also be split more than once and have many sets of polarization functions. 

This means that diffuse basis functions will be required, just as they were for absolute acidity comparisons. Note, however, that in the case of absolute acidities, none of the models, except the local density model, gave satisfactory results even with the 6-311+G basis set. 

Numerous calculations [61] of the electronic tensors with different basis sets have shown, on the other hand, that the computed size of the couplet depends critically on the presence or absence of diffuse basis functions with valence angular momentum numbers. It is the diffuse part of the electron distribution of a molecule which is primarily affected by nonspecific interactions in the condensed phase. This suggests that the absence of a sizable couplet in the condensed phase, in substance as well as in trideuterioacetonitrile, is the result of the change of the electron distribution of (+)-(P)-1,4-dimethylenespiropentane by nonspecific interactions. 

Table 7-2. Main geometrical parameters for the ground state (GS) and theintramolecular charge transfer state (ICT) of pNA in gas phase and in the two solvents. The values in parentheses refer to calculations including diffuse basis functions on the heavy atoms...

In Table 8.6 we present the results obtained with the CCSD method and the aug-cc-pV(T- -d)Z and aug-cc-pV(T-l-d)Z-t-3 basis sets for the first 12 excited states (the CCS method is prohibitively expensive for this many states with this basis set). The aug-cc-pV(T- -d)Z- -3 basis set contains additional very diffuse basis functions necessary for the adequate description of Rydberg states. Comparison of results obtained with the aug-cc-pV(D-fd)Z-l-3 basis set and those obtained with the much larger aug-cc-pV(D-t-d)-l-7 basis set showed that H2SO4 is saturated with diffuse basis functions with the aug-cc-pV(D-fd)Z-i-3 basis set [76]. For the 4 lowest energy excited states shown in Table 8.6, we see that the difference in energy between the aug-cc-pV(T- -d)Z and aug-cc-pV(T-i-d)Z-F3 basis sets with the CCSD... 

High-accuracy calculations clearly require very flexible basis sets extended in a number of ways such as with multiple polarization sets, with diffuse basis function augmentation, and with other than atom-centered functions. Use of smaller bases goes along with less reliability however, with bases smaller than double-zeta in the valence plus one well-chosen set of polarization functions on all centers, including hydrogens, the reliability is so limited that results are not likely to be meaningful for most contemporary problems of weak interaction. 

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 . ... 

Whenever the truncated basis sets are used, matching will be important, because the profiles of charge density and energy density are better represented in one region than others in this case. The subsystem orbitals with the lowest eigenvalues are attributed most to the most localized basis functions for the subsystem. The diffuse basis functions contribute much less to these orbitals. If a mismatch between basis set and projection weight occurs, the energy obtained from eqs.(10), (13), (20) and (22) will not be the optimal one. 

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