Contracted functions

The double zeta basis sets, such as the Dunning-Huzinaga basis set (D95), form all molecular orbitals from linear combinations of two sizes of functions for each atomic orbital. Similarly, triple split valence basis sets, like 6-3IIG, use three sizes of contracted functions for each orbital-type. 

Typically, the contracted functions themselves are also normalized. This has two consequences ... 

In a segmented contraction each primitive as a rule is only used in one contracted function. In some cases it may be necessary to duplicate one or two PGTOs in two adjacent CGTOs. The contraction coefficients can be determined by a variational optimization, for example from an atomic HF calculation. 

In a general contraction primitives (on a given atom) and of a given angular momentum enter all the contracted functions having that angular momentum, but with different contraction coefficients. 

For both the DZ and TZ sets a contracted function was included for the 6p orbital, but this was deleted in the QZ and 5Z sets due to near linear dependence. The contraction was also deleted from the 5Z set for the same reason. Figure 7 plots the correlation energies for both nonrelativistic and DK-relativistic CISD calculations. The CBS limits using a extrapolation of the QZ and 5Z correlation energies are -391.8 and -418.0 m /, for NR and DK, respectively. 

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... 

Notice that we want to use 2 d-type functions in the basis. This is because for correlated wave functions the d-type functions serve two purposes, which cannot easily be accomplished by one basis function First they serve as polarization functions, polarizing the electron density in the bonding region, a feature that is especially important in 7t-bonded systems. Secondly, they give important contributions to the angular correlation effects, which is very important in linear molecules. Studies of the ANO s shows that the two properties of the d-type functions cannot be incorporated into one function. Hie polarization d is rather diffuse, while the correlating d is a much more contracted function. We therefore prefer to include two d-type functions in the basis set... 

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... 

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... 

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. 

In order to have a better basis set, we can replace each STO of a minimal basis set by two STOs with different orbital exponent f (zeta). This is known as a double zeta basis set. In this basis, there is a linear combination of a contracted function (with a larger zeta) and a diffuse function (with a smaller zeta) and the coefficients of these combinations are optimized by the SCF procedure. Using H2O as an example, a double zeta set has two Is STOs on each H, two Is STOs, two 2s STOs, two 2px, two 2pv and two 2pz STOs on oxygen, making a total of 14 basis functions. 

Three basis sets were used in the CCSD calculations. The first, designated ANOl, is an atomic natural orbital set consisting of 4s3pld contracted functions on C, 2slp on H, and a set of spd diffuse functions positioned at the center of the molecule. The... 

The above basis set corresponds to a total of 45 contracted functions for the Co atom and 135 contracted functions for the Co(NH3)5 complexes. 

Contracted functions are constructed from primitives that are of the same symmetry type (i.e. s-type GTF s, p-type GTF s, etc.) and that are centered on the same nucleus. Contractions of a more general type are conceivable but they are not used. 

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