Basis sets Be

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

At the very least it is desirable to project out the contaminant functions from the basis, even if the integrals are evaluated over Cartesians. Including the contaminants exacerbates linear dependence problems, and if their inclusion makes a big difference the basis set being used should be viewed with suspicion. 

Bonifacic and Huzinaga[3] use explicit core orbital projection operators, while orbital angular momentum projection operators are used by Goddard, Kahn and Melius[4], by Barthelat and Durand[5] and others. Explicit core orbital projection operators can, in the full basis set, be viewed as shift operators which ensure that the first root in the Fock matrix really corresponds to a valence orbital. However, in applications the basis set is always modified and the role of the core orbital projection operators thus partly changes. 

We summarize the results of this section. We start expressing hermitian operators as a Fourier-like expansion, with the basis set being provided by... 

One desirable feature of a basis set is that it should be balanced. How might a basis set be unbalanced ... 

Glukhovtsev, M. N., Should the standard basis sets be augmented with diffuse functions on hydrogens to provide a reasonable de.scription of the lowest Rydberg state of hydrogen-containing molecules , J. Mol. Struct. (Theochem) 357, 237-242 (1995). 

Since there is no simple relation between bonds and eigen-functions in a many-atom molecule, the discussion will generally use a molecular orbital rather than a valence bond approach. The molecular orbitals used to rationalise the bonding are linear combinations of atomic orbitals, this arbitrarily limited basis set being easy to visualise. We shall occasioncdly speak of altering the size and shape of these orbitals this is equivalent to using an expanded basis set. 

The STO-3G basis set is to be avoided for studies of H-bonding. If one is restricted to the use of minimal basis sets, MlNI-1 would be a better choice. Under no circumstances, however, can a minimal basis set be expected to furnish an adequate framework for correlation. Hence, components, such as dispersion, should be estimated in some manner other than application of Cl or M0ller-Plesset to a basis set of this sort. The 3-21G basis set is also not recommended among the class of double- sets. 

In order to exemplify the above procedure let us consider the simplest case of the SA = SB — 1/2 system. Let the members of the basis set be ordered as follows... 

In their seminal 1969 paper (33) on the design of Gaussian basis sets to model Slater functions, Hehre, Stewart and Pople proposed that basis sets be determined using the least-squares minimum conditions. 

Figures 3.13 and 3.14 demonstrate that the modelling of atomic radial functions using Gaussian basis sets need not involve the step that the basis. sets be proved against Slater orbitals and so connect to early work with Gaussian basis sets. This should not be...

Chapter 1 ended with a short survey of modem methodology in the application of Gaussian basis set theory to molecular orbital theory calculations. In real calculations it is standard to use a variety of larger basis sets tailored to particular applications (49) with the minimum basis sto-3g) sets widely used for general use and then these more complicated basis sets being applied to obtain better agreement with particular experimental results. 

It is required that the largest possible basis set be used (mathematics we approach fire complete set), but we may also ask if a basis set dimension may be decreased freely (economy ). 

When developing the large basis sets which are required for the reliable calculation of van der Waals interactions, it is important to ensure that the basis set be constructed and extended in a systematic fashion. In recent work, Wells and Wilson have employed systematic sequences of even-tempered basis sets of Gaussian-type functions in conjunction with many-body perturbation theory calculations to study van der Waals interaction potentials, thereby ensuring basis set superposition errors and size-inconsistency problems are controlled. They used the Boys-Bernardi procedure as a test for the magnitude of the basis set superposition error rather than as a correction. 

In the close-coupling expansion, basis functions with different states are coupled together and this can lead to an enormous basis set being required Expansion of the I J-jr operator gives... 

The EMSL (the environmental molecular sciences laboratory) [463] library sup>-plies a wide selection of atomic basis sets optimized for molecules. This library allows extraction of Gaussian basis sets, and any related effective core potentials (the effective core potentials are considered in the next sections), from the Molecular Science Research Center s Basis Set Library. A user may request the basis set be formatted appropriately for a wide variety of popular molecular electronic-structure packages. In addition to the exponents and contraction coefficients that define the basis set, the user can obtain descriptive data that include the overall philosophy behind the basis literature citations, the angular momentmn composition of the basis and many other pieces of information. 

Let the atomic basis set be well localized on atoms (or the atomic fragments under consideration) and be orthonormal. The unit operator can be expanded as... 

We will discuss in some detail a few of the approaches used to prove or disprove the validity of the FCP. It is important to note that in many studies, BSSE has been incorrectly blamed for not reproducing a value of the interaction energy close to the experimental value. As stated by Van Lenthe et al., the FCP does, in fact, eliminate the BSSE for the basis sets being used. It does not, however, correct for the inadequacy of the basis sets. Furthermore, one should not expect to obtain a particular value for the interaction energy such as the experimental value or some Hartree-Fock limit value, just because the energies are being corrected for the BSSE. The CP correction is supposed to correct for BSE effects, but it will not correct for the use of finite basis sets. 

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