Split valence zeta

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. 

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). 

The next step up in basis set size is a Triple Zeta (TZ). Such a basis contains three times as many functions as tire minimum basis, i.e. six s-functions and three p-functions for the first row elements. Some of the core orbitals may again be saved by only splitting the valence, producing a triple split valence basis set. Again the term TZ is used to cover both cases. The names Quadruple Zeta (QZ) and Quintuple Zeta (5Z, not QZ) for the next levels of basis sets are also used, but large sets are often given explicitly in terms of the number of basis functions of each type. 

For most molecules studied, modest Hartree-Fock calculations yield remarkably accurate barriers that allow confident prediction of the lowest energy conformer in the S0 and D0 states. The simplest level of theory that predicts barriers in good agreement with experiment is HF/6-31G for the closed-shell S0 state (Hartree-Fock theory) and UHF/6-31G for the open-shell D0 state (unrestricted Hartree-Fock theory). The 6-31G basis set has double-zeta quality, with split valence plus d-type polarization on heavy atoms. This is quite modest by current standards. Nevertheless, such calculations reproduce experimental barrier heights within 10%. 

Each CGTO can be considered as an approximation to a single Slater-type orbital (STO) with effective nuclear charge f (zeta). The composition of the basis set can therefore be described in terms of the number of such effective zeta values (or STOs) for each electron. A double-zeta (DZ) basis includes twice as many effective STOs per electron as a single-zeta minimal basis (MB) set, a triple-zeta (TZ) basis three times as many, and so forth. A popular choice, of so-called split-valence type, is to describe core electrons with a minimal set and valence electrons with a more flexible DZ (or higher) set. 

The keyword label of a standard Pople-style split-valence basis specifies the sp sets (groups of CGTOs of s and p symmetry) with distinct zeta values for each atomic shell, as well as the contraction length of each CGTO. The keyword contains the following syntactical components ... 

In the early calculations of IR spectra of molecules, small basis sets (e.g., STO-3G, 3-21G, or 4-31G) were used because of limitations of computational power. At present typically a basis set consists of split valence functions (double zeta) with polarization functions placed on the heavy atoms (i.e., non-hydrogens) of the molecule (the so-called DZ+P or 6-31 G basis set). Such basis sets have been... 

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. 

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. 

Double zeta and split valence 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. 

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). 

As said above it is possible to use the same Gaussian-type standard basis sets of ab initio theory for DFT calculations. Concerning the quality of the basis set which is necessary to obtain reliable results, it is advisable to use for Ge at least a split-valence basis set which should be augmented by a d-type polarization function such as 6-31G(d). Better basis sets of triple-zeta quality with more polarization functions up to 6-311G(3df) have been developed for Ge which belong to the standard basis sets in Gaussian 9831. Other basis sets for Ge are available, e.g., from the compilations of Huzinaga et al.32 and Poitier et al.33 and from the work of Aldrichs et al 34. 

Split-valence or double zeta. Orbitals represented by 3 Gaussians near the nucleus and 2 away from the nucleus. Hydrogen atoms represented by one Gaussian. This is a good basis for conformational analysis of moderate-sized molecules... 

Since most of chemistry focuses on the action of the valence electrons, Pople developed the split-valence basis sets, SZ in the core and DZ in the valence region. A double-zeta split-valence basis set for carbon has three s basis... 

Weigend, R Ahhichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn design and assessment of accuracy, Phys. Chem. Chem. Phys. 2005, 7, 3297-3305. 

As far as the basis set is concerned, increasing its quality from split valence to double zeta does not lead to any improvement of the situation a slight increase in the energy dificrence was found on going from a (14,9,6/9,5/6) set of primitives contracted to < 6,4,3/3,2/3 > for the iron atom, the first row atoms and the hydrogen atom respectively, to the (14,11,6/10,6/6) < 8,6,3/4,2/3 > basis set (14). The addition of a p polarisation function on the hydrogen atom decreased this value somewhat, down to 1.8 kcal/mol, but in every case the trans isomer remained the most stable one (14). 

One should mention however that our conclusions have been very recently questionned by Axe and Marynick (42) who carried out calculations on the reaction (3) with various basis sets ranging from split valence to double zeta quality, with and without polarization functions on C, O and H atoms. They found a marked increase in the endothermicity value on going from the unpolarized basis sets ( values ranging between 8.7 and 15.2 kcal/mol) to the polarized basis s.ets (with values between 19.5 and 25.2 kcal/mol, i.e. close to our SD-CI values). We have now carried out calculations adding to our original split valence basis set polarization functions on C, O and H. One polarized set includes the two sets of polarization functions ( = 0.920... 

---