Contracted basis sets, notation

There is one other detail about the contracted basis set notation. For hydrogen, there is only the valence Is orbital and this is true too for helium. But for lithium, and any other atom in the Periodic Table, we divide the electron configuration into component core and valence orbitals and leave the core representation uncontracted. Thus, for lithium the 4-31 basis is the one linear combination of four primitive Gaussians of Table 1.8 scaled by the Slater exponent for Is and then the two linear combinations of equations 1.21 and 1.22 scaled by the Slater exponent for lithium 2s and then the two linear combinations defined... 

Most calculations today are done by choosing an existing segmented GTO basis set. These basis sets are identihed by one of a number of notation schemes. These abbreviations are often used as the designator for the basis set in the input to ah initio computational chemistry programs. The following is a look at the notation for identifying some commonly available contracted GTO basis sets. 

The smallest basis sets are called minimal basis sets. The most popular minimal basis set is the STO—3G set. This notation indicates that the basis set approximates the shape of a STO orbital by using a single contraction of three GTO orbitals. One such contraction would then be used for each orbital, which is the dehnition of a minimal basis. Minimal basis sets are used for very large molecules, qualitative results, and in certain cases quantitative results. There are STO—nG basis sets for n — 2—6. Another popular minimal basis set is the MINI set 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. 

In order to describe the number of primitives and contractions more directly, the notation (6s,5p) (ls,3p) or (6s,5p)/(ls,3p) is sometimes used. This example indicates that six s primitives and hve p primitives are contracted into one s contraction and three p contractions. Thus, this might be a description of the 6—311G basis set. However, this notation is not precise enough to tell whether the three p contractions consist of three, one, and one primitives or two, two, and one primitives. The notation (6,311) or (6,221) is used to distinguish these cases. Some authors use round parentheses ( ) to denote the number of primitives and square brackets [ ] to denote the number of contractions. 

An older, but still used, notation specihes how many contractions are present. For example, the acronym TZV stands for triple-zeta valence, meaning that there are three valence contractions, such as in a 6—311G basis. The acronyms SZ and DZ stand for single zeta and double zeta, respectively. A P in this notation indicates the use of polarization functions. Since this notation has been used for describing a number of basis sets, the name of the set creator is usually included in the basis set name (i.e., Ahlrichs VDZ). If the author s name is not included, either the Dunning-Hay set is implied or the set that came with the software package being used is implied. 

Take note of Dunning s notation. He writes the primitives (10s6p) and the contracted basis functions in square brackets [5s3p]. To give a detailed example, consider the oxygen atom set in Table 9.7. 

The following short notation for basis sets will be applied throughout in this contribution (a, /S, y/A, //) indicates that a s type, / p-ty-pe, and y d-type orbitals are applied for second-row atoms and A s-type and / p-type function for H-atoms. Contractions are given in square brackets (a, ft, y j/-, fi ). Basis sets C, D, and E are almost identical to or exactly the same as the ones used in Ref. 109-111) or 88> 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... 

An alternative to the double zeta basis approach is to double the number of functions used to describe the valence electrons but to keep a single fxmction for the iimer shells. The rationale for this approach is that the core orbitals, unlike the valence orbitals, do not affect chemical properties very much and vary only slightly from one molecule to another. The notation used for such split valence double zeta basis sets is exemplified by 3-21G. In this basis set three Gaussian functions are used to describe the core orbitals. The valence electrons are also represented by three Gaussians the contracted part by two Gaussians and the diffuse part by one Gaussian. The most commonly used split valence basis sets are 3-21G, 4-31G and 6-31G. 

Since the basis set can be applied to other atoms using a suitable f, the notation has a more general meaning 4-31 is accepted to mean a 4-term contraction of primitives for the core orbitals and then a split into a single term and a 3-term contraction for the valence orbitals. For the minimal basis set of Is, 2s and three 2p orbitals this is then a single 4-term linear combination for Is and four pairs of the split basis set for the valence s and p orbitals. 

Our purpose in this section is to explicitly define the STO-3G, 4-31G, 6-31G , and 6-31G basis sets that we will be using in this and subsequent chapters. In the process, however, we will describe attributes that are characteristic of most of the basis sets that are in current use, and we will introduce some of the notation and some of the mechanics of defining and choosing a basis set. In particular, we first present a general treatment of contraction. 

Five different basis sets were used for the calculation at both the RHF and correlated levels, as shown in Tables 5.1 and 5.2, where the notation msins means that on each hydrogen atom m 5-type Gaussians are centered and are divided into n groups (contractions). The exponents and contraction coefficients of the first two sets were taken from the literature, and the exponents of the six 5-type Gaussians from another work.< > In the fourth basis set these functions were supplemented by two 5-type bond functions (2 ) centered in the middle between the hydrogen atoms, while in the fifth basis a set of three p-type atomic polarization functions p, Py, Pz) was used. The contraction coefficients and the exponents of the bond and polarization functions were optimized previously. 

For modeling the chemical bonds in saturated hydrocarbons, first, we considered methane [43] and optimized the positions and exponents of 5-type BFs for the C-H bonds together with the exponents and contraction coefficients of the AO basis functions centered on the carbon and hydrogen atoms. For consistency and fair comparison, the conventional 6-31G basis set was also reoptimized the r6-31G notation will be used hereafter for the basis set obtained in this way. The results are compiled in Table 1. 

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