Basis Sets—Modeling Atomic Orbitals

Common Basis Sets—Modeling Atomic Orbitals 

Shape of STOs. B. Shape of GTOs. C. Addition of three GTOs resembles an STO. 

Generally, basis sets are developed by optimizing the appropriate coefficients to minimize the energy or to fit experimental data. The latter approach would constitute a violation of our definition of ah initio given at the beginning of Section 14.2.1. Nevertheless, there may be some empirical parameters hidden in the basis sets used to describe the atomic orbitals. 

A common basis set used in early ab initio calculations is STO-3G, an STO mimic created by a linear combination of three GTOs (Eq. 14.34). Here the a s are coefficients that simply reflect the extent to which each G is added to create the atomic orbital j . These a s are not changed during the SCF calculation, but rather they define the basis set. The a s are therefore completely different than the C/ s used in the LCAO-MO method (Eq. 14.33), which are optimized during the SCF method. The a s were optimized to fit experimental data by computational chemist Pople, who won the 1998 Nobel Prize in Chemistry for the development of many of the essential components of modem ofc initio theory. 

The STO-3G basis set is a minimal basis set, which lacks the flexibility required for high quality calculations. As computer power grows, STO-3G sees less and less use. 

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Common Basis Sets—Modeling Atomic Orbitals... 

Construct a bonding model of molecular orbitals from a basis set of atomic orbitals. 

Until quite recently the role of a—n correlation effects was ignored in the. theoretical treatment of electronic transitions. Even now, nearly all ab initio calculations of excitation phenomena are based on independent-particle models using a minimal basis set of atomic orbitals, or involve a configuration interaction limited to the sr-electron system. In order to go far enough beyond the o—n separation, two improvements have to be simultaneously considered ... 

While semiempirical models which can be applied to molecules the size of 1 and 2 are necessarily only approximate, we were searching for trends rather than absolute values. In concept, the design of semiempirical quantum mechanical models of molecular electronic structure requires the definition of the electronic wavefunction space by a basis set of atomic orbitals representing the valence shells of the atoms which constitute the molecule. A specification of quantum mechanical operators in this function space is provided by means of parameterized matrices. Specification of the number of electrons in the system completes the information necessary for a calculation of electronic energies and wavefunctions if the molecular geometry is known. The selection of the appropriate functional forms for the parameterization of matrices is based on physical intuition and analogy to exact quantum mechanics. The numerical values of the parameters are obtained by fitting to selected experimental results, typically atomic properties. 

The advantage of Huzinaga s approach is that the model structure of the valence orbitals is preserved but the price one has to pay is the use of large basis sets of atomic orbitals which are not drastically reduced with respect to those used in standard all-electron calculations. Another approach to the... 

It is now practical to carry out these calculations for a wide variety of molecules, and here the use of density functional theory (DFT) and particularly the hybrid functionals such as B3LYP has proven to be particularly useful. A basis set (equivalent to a set of atomic orbitals) such as 6-311+G appears to be generally satisfactory. Greater accuracy may be obtained using more advanced methods such as the G2 or CBSQ model chemistries. The calculations also provide estimates of the vibrational frequencies from which the zero-point energies may be obtained. 

A Complete Basis Set Model Chemistry. IV. An Improved Atomic Pair Natural Orbital Method. 

The calculations of molecular models of siloxane sites of clay minerals with adsorbed TNB at the SCF, DFT (B3LYP) and MP2 levels of theory have been recently carried out [197]. A series of basis sets was used from the smallest 6-31G(d) up to the 6-311+G(d) set of atomic orbitals. 

The popular semiempirical methods, MNDO (Dewar and Thiel, 1977), Austin Model 1 AMI Dewar et al., 1985), Parameterized Model 3 (PM3 Stewart 1989a 1989b), and Parameterized Model 5 (PM5 Stewart, 2002), are all confined to treating only valence electrons explicitly, and employ a minimum basis set (one 5 orbital for hydrogen, and one 5 and three p orbitals for all heavy atoms). Most importantly, they are based on the NDDO approximation (Stewart, 1990a, 1990b Thiel, 1988, 1996 Zemer, 1991) ... 

In principle any function can be perfectly expressed by plane waves because the set of plane waves is complete. However, the position-independent property of plane waves makes the expressions for some sharp peak-like functions very difficult. In these cases an enormous number of plane-waves is needed to obtain near-convergent results. For this reason, the rapidly oscillating core orbitals of an atom are difficult to treat using plane wave basis sets. Because valence orbitals are orthogonal to the core orbitals, even the valence orbitals are difficult to be expressed when core orbitals are included in the calculations. In order to avoid the difficulties in expressing the core orbitals with plane wave basis set, it is necessary to freeze the core electrons and to model the integral effects of core electrons and nuclei with pseudopotentials. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

Montgomery, J.A., Ochterski, J.W., Petersson, G.A. A complete basis-set model chemistry. 4. An improved atomic pair natural orbital method. J. Chem. Phys. 1994,101(7), 5900-09. 

There are several theoretical studies on LaO and related lanthanide oxides. We have already mentioned the ligand-field theory model calculations of Field (1982) as well as Carette and Hocquet (1988). More recently, Kotzian et al. (1991a,b) have applied the INDO technique (Pople et al. (1967) extended to include spin-orbit coupling [see also Kotzian et al. (1989a, b)] to lanthanide oxides (LaO, CeO, GdO and LuO). The authors call it INDO/S-CI method. The INDO parameters were derived from atomic spectra, model Dirac Fock calculations on lanthanide atoms and ions to derive ionization potentials, Slater-Condon factors and basis sets. The spin-orbit parameter is derived from atomic spectra in this method. 

J. A. Montgomery, J. W. Ochterski, and G. A. Petersson,/. Chem. Phys., 101, 5900-5909 (1994). A Complete Basis Set Model Chemistry. IV. An Improved Atomic Pair Natural Orbital Method. 

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