Contracted basis sets general contraction

In the remainder of this section, we shall consider only segmented basis sets. Generally contracted basis sets have been developed primarily for the accurate calculation of electron correlation and will be discussed in Section 8.3. 

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. 

All of the above basis sets are of the segmented contraction type. Modem contracted basis sets aimed at producing very accurate wave functions often employ a general contraction scheme. The ANO and cc basis sets below are of die general contraction type. 

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... 

I Meanwhile others object to the suggestion that the optimization of basis sets are carried out by reference to experimental data. While accepting that the exponents and contraction coefficients are generally optimized in atomic calculations, they insist that these optimizations are in themselves ab initio. 

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

While the acronym STO-3G is designed to be informative about the contraction scheme, it is appropriate to mention an older and more general notation that appears in much of the earlier literature, although it has mostly fallen out of use today. In that notation, the STO-3G H basis set would be denoted (3s)/[Is]. The material in parentheses indicates the number and type of primitive functions employed, and the material in brackets indicates the number and type of contracted functions. If first-row atoms are specified too, the notation for STO-3G would be (6s3p/3s)/[2slp/ls]. Thus, for instance, lithium would require 3 each (since it is STO-3G) of Is primitives, 2s primitives, and 2p primitives, so the total primitives are 6s3p, and the contraction schemes creates a single Is, 2s, and 2p set, so the contracted functions are... 

Pople-type basis sets are generally characterized by heavier contraction (that is, fewer contracted functions for a given primitive set) and consequently somewhat lower accuracy. The earliest set was a minimal basis contraction (STO-3G) of a... 

For heavier elements primitive basis sets are available, but their contraction is a real art, at least if segmented contractions are used. The use of general contractions is far preferable and is much easier to carry out. 

The correlation-consistent basis sets of Dunning axe only beginning to appear, but they show great promise as a way to achieve accuracy similar to that of ANOs, at less computational expense in the integral evaluation (they axe generally the same size, in terms of contracted functions, as the ANO sets). It is interesting to contrast... 

In most calculations, improvement of the basis set involves an increase in both the primitive set and the contracted set. It is not generally obvious how one basis set is related to another (other than that larger basis sets are expected to perform better than smaller sets), and consequently it is not always clear how to extrapolate from a set of calculations to the basis set limit. It is preferable to use sequences of basis sets that form a well-defined chain of subspaces. 

The use of Effective Core Potential operators reduces the computational problem in three ways the primitive basis set can be reduced, the contracted basis set can be reduced and the occupied orbital space can be reduced. The reduction of the occupied orbital space is almost inconsequential in molecular calculations, since it neither affects the number of integrals nor the size of the matrices which has to be diagonalized. The reduction of the primitive basis set is of course more important, but since the integral evaluation time is in general not the bottleneck in molecular calculations, this reduction is still of limited importance. There are some cases where the size of the primitive basis set indeed is important, e.g. in direct SCF procedures. The size of the contracted basis set is very important, however. The bottleneck in normal SCF or Cl calculations is the disc storage and/or the iteration time. Both the disc storage and the iteration time depend strongly on the number of contracted functions. 

For heavy atoms the ECP s quickly become essential. The basis set size at the ECP level does not change if we go to the second or third row TM, while a generally contracted basis set for a second row TM would comprise 45 basis functions. The difference increases even more drastically for the third row TM s where in addition the 4f orbital can be included in the core in the ECP case. 

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