Atomic basis set

Representation of each molecular orbital as a linear combination of atomic orbitals (atomic basis sets). Atomic basis sets are usually represented as Slater type orbitals or as combinations of Gaussian functions. The latter is very popular, due to a very fast algorithm for the computation of bielectronic integrals. 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

An alternative analysis of sigmatropic reactions involves drawing the basis set atomic orbitals and classifying the resulting system as Htickel or Mobius in character. When this classification has been done, the electrons involved in the process are counted to determine if the TS is aromatic or antiaromatic. The conclusions reached are the same as for the frontier orbital approach. The suprafacial 1,3-shift of hydrogen is forbidden but the suprafacial 1,5-shift is allowed. Analysis of a 1,7-shift of hydrogen shows that the antarafacial shift is allowed. This analysis is illustrated in Figure 10.31. These conclusions based on orbital symmetry considerations are supported by HF/6-31G calculations, which conclude that 1,5-shifts should be suprafacial, whereas... 

Bader analysis, 667 balance, kinetic, 132 band, conduction, 533-534 band gap, 537 band structure, 523, 527 band, valence, 537, 610 bandwidth, 532 barrier as shell opening, 948 barrier of dissociation, 801 barriers of reaction, 948 basis, biorthogonal, 513 basis set, atomic, 428, 431,el37... 

An alternative analysis involves drawing the basis set atomic orbitals and, classifying the resulting system as Hiickel or Mobius in character. Once this classification is complete and the electrons involved in the process are counted, the transition state can be recognized as aromatic or antiaromatic. This analysis is illustrated in Fig. 10.7. 

According to the ab initio molecular orbital theory methodology, atomic orbitals (set of functions, also called basis sets) combine in a way to form molecnlar orbitals that snrronnd the molecule. The molecular orbital theory considers the molecnlar wave function as an antisymmetiized product of orthonormal spatial molecular orbitals. Then they are constructed as a Slater determinant [56], Essentially, the calculations initially use a basis set, atomic wave functions [57, 58], to constract the molecular orbitals. The first and basic ab initio molecular orbital theory approach to solve the Schrodinger equation is the Hartree-Fock (HF) method [59, 60], Almost all the ab initio methodologies have the same basic numerical approach but they differ in mathematical approximations. As it is clear that finding the exact solution for the Schrodinger equation, for a molecular system, is not possible, various approaches and approximations are used to find the reliable to close-to-accurate solutions [61-68]. 

The molecular basis set atomic-hke orbitals can be considered as a starting point to generate atomic basis sets to be used in crystalline compounds. Therefore, we begin with the molecular basis-sets description. 

Cpv ore tbe LCAO coefficients for the basis set atomic orbitals p and v. The i and j are indices of molecular ofbitals. are overly integrals. By expressions (4.80) and (4.81) a partitioning of Ae non-classical d le term arising from orbitals centered on different atoms is accmtt didied so that all dipole crmtributions are associated widi atomic sites. The last term in Eq. (4.79) comprises the dipole contributions from die second and third terms as defined in Eq. (3.30). On the odier hand, the first and second terms in Eq. (4.79) arc associated widi the same dipole contributions as defined in Eqs. (3.34) and (3.35), respectively. 

Table 8.15 The Hartree-Fock energy and the CISD valence correlation energy for the ground state of the carbon atom and for the 5 ground state of the carbon anion calculated using the cc-pVXZ and aug-cc-pVXZ basis sets (atomic units). The Hartree-Fock basis-set limits for the neutral and anionic atoms are —37.688619 and —37.708844 Eh, respectively. The estimated CISD basis-set limits are —99.7 0.4 mEh and —120.7 0.4 mEh, respectively...

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