Basis sets in ab initio calculations

Any set of one-electron functions can be a basis set in the LCAO approximation. However, a well-defined basis set will predict electronic properties using fewer terms than a poorly-defined basis set. So, choosing a proper basis set in ab initio calculations is critical to the reliability and accuracy of the calculated results. 

The choice of basis set in ab initio calculations has been the subject of numerous theoretical studies. Early SCF calculations utilized mainly spht-va-lence basis sets such as 3-21G and 4-31G. The importance of inclusion of d polarization functions on sulfur atoms has been stressed by several authors. For instance, Suleimenov and Ha found that the omission of d polarization functions leads to a substantially lower barrier for the internal rotation ( 16 kj mol for the central bond of H2S4) and produces an unreahstically large S-S bond length for the most stable rotamer [4]. In general, the use of... 

A fortunate finding of test calculations was that the same basis sets of Gaussian-type functions which are used as standard basis sets in ab initio calculations can be used for DFT calculations. It was also found that the same ECPs which have been optimized for ab initio methods can be employed for DFT methods20. Users of the program package Gaussian may, e.g., simply choose DFT/6-31G(d) instead of HF/6-31G(d) or MP2/6-31G(d). The only choice which one has to make is the DFT functional. 

The calculations of Lehn and Wipff (1975) included d-functions to a very limited extent. Similarly, Gorenstein and coworkers have almost exclusively employed a minimal STO-3G basis set in ab initio calculations on P(K) species (Gorenstein et al., 1979). This appears to be a major omission. However, two observations suggest that, despite the use of a minimal basis set, calculations on the stereoelectronic effect may remain valid. Firstly, the inclusion of d-functions in Lehn and Wipflf s (1975) study shows similar effects with and without these functions. Secondly, recent studies at the STO-3G and 3-21G( ) level on H2SO4 and dimethyl sulphate indicate the presence of similar stereoelectronic effects, independent of the inclusion of d-functions on sulphur (Lowe et al., 1988). 

Using the variation condition one can derive a linear equation for Xq which, in turn, can be used to evaluate the first-order correction to the density matrix. This leads to an iterative scheme involving matrix manipulations in the local space. It is important to note that Kirtman s treatment describes not only inductive but delocalization effects, as well. A practical limitation is that one should work with orthogonal AO basis sets therefore the calculations have been done with semiempirical ZDO model Hamiltonians [91], or with explicitly Lowdin-orthogonalized basis sets in ab initio calculations. 

In the later 1970s there were, however, some papers which asserted the importance of including d orbitals in the basis set in ab initio MO calculations. For instance, calculations... 

The great advance in the capacity of modern computers has contributed to a major expansion in the detail of theoretical investigations of small molecules and also an expansion of the size of systems suitable for study. Pteridine chemistry has aspects of both. In small molecule work, it is important to establish the appropriateness of a number of possible basis sets for ab initio calculations to yield useful results. For vibrational spectra of pteridines and related tetraazanaphthalenes, the B3LYP/6-31G basis set has been reported to give the best correlation between theory and experiment but the smaller 4-31G basis set also gave acceptable results <2000PCA7386>. Calculations relevant to... 

The method of McConnell [13] assumes the presence of a set of virtual orbitals all with higher energy than the electron to be transferred, in the form of Hiickel combinations of C 3d orbitals. This somewhat odd assumption leads to exponential decrease. Incorrectly one gets the impression that only unoccupied MO s are useful for ET. The virtual orbitals of the kind used by McConnell, are nowadays standard in any reasonable basis set for ab initio calculations, but it is fair to say that the electronic factor is not much dependent on whether they are included or not. Ordinary valence MO s of the bridge, occupied or unoccupied are of a much greater importance for ET. The McConnell method [13] is therefore mainly of historical interest. 

Although quantum-mechanical calculations have challenged the possible involvement of 2>d orbitals in sulfoxides and sulfones [5-8] this bonding model involving participation of M orbitals at sulfur is reviving again as the result of new basis sets for ab initio calculations [3,9]. It is noted, however, that the role of d functions in hypervalent molecules seems to be to provide additional flexibility for the orbitals to increase their overlap and thereby increase the stabilization of the system (d-orbitals polarization effect) [10], rather than to participate directly in bonding, as depicted in Scheme 3.1. 

To this point, the basic approximation is that the total wave function is a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ab initio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ab initio calculation. However, there are two main th ings to be con sidered in the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222), will require the fewest possible terms for an accurate representation of a molecular orbital. The second one is the speed of two-electron integral calculation. 

Feller, D. and Davidson, E. R. 1990. Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions in Reviews in Computational Chemistry, Vol. 1. Lipkowitz. K. B. and Boyd, D. B. Eds., VCH New York, 1. 

Probably the most important contribution so far has been a comparison of the relative merits of utilizing sp and spd basis sets for selenium in ab initio calculations on selenophene. The inclusion of the five 4 -orbitals of the selenium atom was found to have only a marginal effect upon the total energy, and their main role appears to be that of polarization functions,... 

Feller D, Davidson ER. Basis sets for ab initio molecular orbital calculations and intermolecular interactions. In Lipkowitz KB, Boyd DB, eds. Reviews in Computational Chemistry. Vol. I. New York VCH, 1990 1 4-3. 

Fig. 9. Torsional potentials for HS-SH molecule calculated in 6-31G basis set (a) ab initio SCF results, lab initio SCF (b) CMMM estimates (c = 6) up to quadrupole-quadrupole term, + CMM (Q-Q) (c) PCM results (c = 6, s = 9, R = 0.1 au), PCM (R = 0.1 au) (Reproduced from [90] copyright-Springer-Verlag)...

We need at least enough spatial MO s i// to accommodate all the electrons in the molecule, i.e. we need at least n ij/ s for the 2n electrons (recall that we are dealing with closed-shell molecules). This is ensured because even the smallest basis sets used in ab initio calculations have for each atom at least one basis function corresponding to each orbital conventionally used to describe the chemistry of the atom, and the number of basis functions

initio calculation on CH4, the smallest basis set would specify for C ... 

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