Split-valence double-zeta basis set

The column (AE(MP2/G3)) contains the differences between the MP2(full) energies with a basis set of augmented-triple-zeta quality and the 6-31G(d) basis set. The size of these numbers clearly demonstrates the fact that the split-valence double-zeta basis sets are far from being complete. 

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. 

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. 

At the SCF or MCSCF level, the basis set requirements are fairly simple. We can imagine that the occupied molecular orbitals are given as a simple linear combination of atomic orbitals this corresponds to a minimal basis set. The results so obtained are fairly crude, but by admitting extra functions to represent the atomic orbitals more flexibly (split-valence, double zeta, etc) we can obtain a much better description. However, some effects require going beyond the occupied atomic orbitals ... 

G and 3-21G Split Valence and Double-Zeta Basis Sets... 

A still more malleable basis set would be one with all the basis functions, not just those of the valence AO s but the core ones too, split this is called a double zeta (double 0 basis set (perhaps from the days before Gaussians, with exp(—xr2). had almost completely displaced Slater functions with exp(— r) for molecular calculations). Double zeta basis sets are much less widely used than split valence sets, since the former are computationally more demanding and for many purposes only the contributions of the chemically active valence functions to the MO s need to be fine-tuned, and in fact double zeta is sometimes used to refer to split valence basis sets. 

The aforementioned split valence (or double zeta) basis sets can be further improved if polarization functions are added to the mix. The polarization functions have a higher angular momentum number i so they correspond to p orbitals for hydrogen and helium and d orbitals for elements lithium to neon, etc. So if we add d orbitals to the split valence 6-31G set of a non-hydrogen element, the basis now becomes 6-31G(d). If we also include p orbitals to the hydrogens of the 6-31G(d) set, it is then called 6-31G(d,p). 

Other hybridization procedures for double-zeta basis sets may be imagined, for instance by using projection techniques [63] Starting with a set of hybrids obtained from a parent minimal basis (e.g., STO 3G versus 6-31 G) we can project the latter hybrids into the space spanned either by the inner subset or the outer subset of the original split-valence basis. The extension of the Del Re algorithm for hybridization to multiple-zeta sets does not raise any major difficulty, which illustrates both the flexibility of the method and the perenniality of the concept. 

To obtain information on how the electronic and eneiigetic properties of the macromolecules depend on the choice of basis set, the band-structure calculations were performed using four different atomic basis sets. On the one hand, the minimal basis sets STO-3G and dementi s basis set ls3p (denoted by C-7s3p) were used, and, on the other, the valence-split set 4-31G (contracted to a 3s2p basis ) and dementi s double-zeta basis set 10 /5p (denoted by C-10 5p) contracted to a 4s2p basis. 

One may use two contracted GTOs per atomic orbital, as in the so-called double-zeta basis sets (Lochan et al. 2005), as this provides much greater flexibility. Split-valence basis sets partition the atomic orbitals into core and valence regions. The core AO s are assigned a minimal basis, while the valence orbitals are described at the double-zeta level. 

The chemical bonding occurs between valence orbitals. Doubling the 1 s-functions in for example carbon allows for a better description of the 1 s-electrons. However, the Is-orbital is essentially independent of the chemical environment, being very close to the atomic case. A variation of the DZ type basis only doubles the number of valence orbitals, producing a split valence basis. In actual calculations a doubling of tire core orbitals would rarely be considered, and the term DZ basis is also used for split valence basis sets (or sometimes VDZ, for valence double zeta). 

Further improvements in the flexibility with which the AOs in Eq. 4 are described mathematically can be obtained by adding a third independent basis function to a split valence basis set. In an anion, electrons are likely to be spread over a greater volume than in a neutral molecule, so adding very dijfuse basis functions to the basis set for a negatively charged molecule is usually important. A fiuther improvement in the basis set for a molecule would be to use two or three independent basis functions to describe, not only the valence AOs, but also the core AOs. Such basis sets are called, respectively, double-zeta or triple-zeta basis sets. 

If only valence orbitals are described by double zeta basis, while the inner shell (or core) orbitals retain their minimal basis character, a split valence basis set is obtained. In the early days of computational chemistry, the 3-21G basis was fairly popular. In this basis set, the core orbitals are described by three Gaussian functions. The valence electrons are also described by three Gaussians the inner part by two Gaussians and the outer part by one Gaussian. More recently, the popularity of this basis set is overtaken by the 6-31G set, where the core orbitals are a contraction of six Gaussians, the inner part of the valence orbitals is a contraction of three Gaussians, and the outer part is represented by one Gaussian. 

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