Pople basis set

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. 

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

As the Pople basis sets have further expanded to include several sets of polarization functions, / functions and so on, there has been a need for a new notation. In recent years, the types of functions being added have been indicated in parentheses. An example of this notation is 6—31G(dp,p) which means that extra sets of p and d functions have been added to nonhydrogens and an extra set of p functions have been added to hydrogens. Thus, this example is synonymous with 6—31+G. ... 

Regarding current ab initio calculations it is probably fair to say that they are not really ab initio in every respect since they incorporate many empirical parameters. For example, a standard HF/6-31G calculation would generally be called "ab initio", but all the exponents and contraction coefficients in the basis set are selected by fitting to experimental data. Some say that this feature is one of the main reasons for the success of the Pople basis sets. Because they have been fit to real data these basis sets, not surprisingly, are good at reproducing real data. This is said to occur because the basis set incorporates systematical errors that to a large extent cancel the systematical errors in the Hartree-Fock approach. These features are of course not limited to the Pople sets. Any basis set with fixed exponent and/or contraction coefficients have at some point been adjusted to fit some data. Clearly it becomes rather difficult to demarcate sharply between so-called ab initio and semi-empirical methods.4... 

Although there is no strict relationship between the basis sets developed for, and used in, conventional ah initio calculations and those applicable in DFT, the basis sets employed in molecular DFT calculations are usually the same or highly similar to those. For most practical purposes, a standard valence double-zeta plus polarization basis set (e.g. the Pople basis set 6-31G(d,p) [29] and similar) provides sufficiently accurate geometries and energetics when employed in combination with one of the more accurate functionals (B3LYP, PBEO, PW91). A somewhat sweeping statement is that the accuracy usually lies mid-way between that of M P2 and that of the CCSD(T) or G2 conventional wave-function methods. 

The P3 approximation to the self-energy was applied to the atoms Li through Kr and to neutral and ionic molecular species from the G2 set [47]. For the atoms, a set of 22 representative basis sets was tested. Results for the molecular set were obtained using standard Pople basis sets as described below. 

Aside from the results for the individual atoms, some trends in basis set performance may be observed. Pople basis sets produced results that were fairly accurate, especially for alkali and alkaline earth metals. Although the results are much less accurate for the p group elements, they are certainly within acceptable error for this simple approximation. The steady decrease in errors observed in the progression from the P3/6-31G to the P3/6-311++G(3df,3pd) level for nontransition elements also attests to the sound design of these basis sets. 

One feature of the Pople basis sets is that they use a so-called segmented contraction. This implies that the primitives used for one basis function are not used for another of the same angular momentum (e.g., no common primitives between the 2s and 3s basis functions for phosphorus). Such a contraction scheme is typical of older basis sets. Other segmented split-valence basis sets include the MIDI and MAXI basis sets of Huzinaga and co-workers, which are named MIDI-1, MIDI-2, etc., MAXI-1, MAXI-2, etc. and vary in the number of primitives used for different kinds of functions. 

The Pople basis sets have seen sufficient use in the literature that certain trends have clearly emerged. While a more complete discussion of the utility of HF theory and its basis-set dependence appears at the end of this chapter, we note here that, in general, the 4-3IG basis set is inferior to the less expensive 3-21G, so there is little point in ever using it. The 6-21G basis set is obsolete. 

Recognizing the tendency to use more than one set of polarization functions in modem calculations, the standard nomenclature for the Pople basis sets now typically includes an explicit enumeration of those functions instead of die star nomenclature. Thus, 6-31G(d) is to be preferred over 6-3IG because the fonner obviously generalizes to allow names like 6-31G(3d2fg,2pd), which implies heavy atoms polarized by three sets of d functions, two sets of f functions, and a set of g functions, and hydrogen atoms by two sets of p functions and one of d (note that since this latter basis set is only valence double-C, it is somewhat unbalanced by having so many polarization functions). 

In die Pople family of basis sets, the presence of diffuse functions is indicated by a + in die basis set name. Thus, 6-31- -G(d) indicates that heavy atoms have been augmented with an additional one s and one set of p functions having small exponents. A second plus indicates the presence of diffuse s functions on H, e.g., 6-311- -- -G(3df,2pd). For the Pople basis sets, die exponents for the diffuse functions were variationally optimized on the anionic one-heavy-atom hydrides, e.g., BH2 , and are die same for 3-21G, 6-3IG, and 6-3IIG. In the general case, a rough rule of thumb is diat diffuse functions should have an exponent about a factor of four smaller than the smallest valence exponent. Diffuse sp sets have also been defined for use in conjunction widi die MIDI and MIDIY basis sets, generating MIDIX+ and MIDIY-I-, respectively (Lynch and Truhlar 2004) the former basis set appears pardcularly efficient for the computation of accurate electron affinities. 

Recognizing the tendency to use more than one set of polarization functions in modern calculations, the standard nomenclature for the Pople basis sets now typically includes an... 

Before leaving this topic, it should be mentioned that, in addition to the (Pople) basis sets discussed so far, there are others as well. The more popular ones include the Dunning-Huzinaga basis sets, correlation consistent basis sets, etc. These functions will not be described here. 

The following strategy for electronic structure calculations on DRAs was employed. Geometry optimization and harmonic frequency analysis for the cations were performed at the HF level with a standard Pople basis set.14,18 These structures were used as initial guesses in the optimization of the respective anions, where the 6-311 + +G(d,p) basis set, which includes diffuse functions, was used. By this stage, optimizations and frequency calculations could be refined using a higher level of theory therefore, MP2 and QCISD calculations were performed for all the molecular systems. The diffuse... 

A,A -Dimethylformamide [DMFH][N03] ion pairs were optimized with the 6-31G Pople basis set and the B3LYP hybrid functional [41]. Subsequently, energies with the 6-311++G basis set were obtained. The enol form of the [DMFH]+ cation was observed to form three stable conformers with the anion, while the cation of the keto form is unstable and the proton transfer occurs to form three kinds of neutral molecule pairs [41], Moreover, the neutral pairs were more stable than the ion pairs, and the ion pairs tended to tautomerize to neutral pairs without barriers, which was interpreted as decomposition of the ILs [41],... 

The zero-order regular approximation (ZORA), a two-component form of the fully-relativistic Dirac equation, is currently used for organotin computational calculations using basis sets specifically designed for ZORA. It should be noted that while all-electron calculations, whether non-relativistic or relativistic, can be used for organotin systems, the 6-3IG Pople basis set is not available for tin and therefore, most all-electron calculations involving tin employ the smaller 3-21G basis set. 

Fig. 6.3 R factors for optimized 2VBliJ structures using nine different levels of theory. Only 126 even numbered residues from the 2x2x2 superceU with unambiguous inner coordination spheres were optimized. The numbers show Rwork in per cent for the optimized structures. TZ means del2-TZVP , DZ means def2-SV(P) and PDZ stands for the Pople basis set 6-3IG. The lines are drawn to better guide the eye...

Figure 14. Variation of the MP2 BSSE-uncorrected and BSSE-corrected interaction energies of the methane-water complex with the basis set. Circles indicate results obtained using Pople basis set, while squares indicate results obtained with the aug-cc-pVxZ basis sets. The geometry is the optimum one computed at the MP2/aug-cc-pVTZ level.

Figure 1.17 The Hehre, Stewart and Pople basis sets for Slater H2s using Is Gaussians.

Two commonly used families of extended basis sets are the Pople basis sets (8) and the Dunning basis sets (27). It is worth making a few brief comments on their notation. 

Pople basis sets have names such as 6-311- -G(3df,2p). The 6-311 part... 

TABLE 2 Commonly Used Pople Basis Sets and the Number of Functions for the Second Row Atoms (Li-Ne)... 

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