Basis atomic orbital contributions

Crystal orbitals are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums - one for each of the s- and p-atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. 

The total number of bands shown in a band-structure diagram is equal to the number of atomic orbitals contributed by the chemical point group, which constitutes a lattice point. As the full crystal structure is generated by the repetition of the lattice point in space, it is also referred to as the basis of the stmcture. 

Usually, the. spatial function ilt is constructed from the summation of one-electron spatial orbitals (atomic orbitals) 4>. known as the basis set. u.sed to construct a MO. This approach is known as the LCAO method (/inear combination of utomic orbitals). It is an approximation of the accurate many-elcetron wave function (Eq. 28-54). The atomic orbital contributions are weighted by coefficients c,. The summation is truncated, so the ip function is not complete, which has consequences when. solving for E. 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals approximation (LCAO). The combination of atomic orbitals chosen is called the basis set. 

In the PPP model, each first-row atom such as carbon and nitrogen contributes a single basis functiqn to the n system. Just as in Huckel theory, the orbitals x, m e not rigorously defined but we can visualize them as 2p j atomic orbitals. Each first-row atom contributes a certain number of ar-electrons—in the pyridine case, one electron per atom just as in Huckel 7r-electron theory. 

In general, all 17 s-primitives contribute to each s-derived molecular orbital. Obviously, the tighter Gaussians will contribute more strongly to the inner-shell molecular orbitals and the more diffuse Gaussians to the valence i-orbitals. Nevertheless, it is impossible (and also not desired) to make a connection between basis functions and atomic orbitals. 

Figure 4.105 A schematic perturbative-analysis diagram for occupied valence MOs (center) of PtH42 (B3LYP/LANL2DZ level), showing tie-lines for analysis in terms of AO basis functions (left) versus NAOs (right). (A tie-line is shown when the AO or NAO contributes at least 5% to the connected MO.) Note the much smaller number of contributing orbitals, the sparser tie-line patterns, and the more realistic physical range of atomic orbital energies for NAOs than for standard Gaussian-basis AOs.

The use of the Lbwdin orthogonalisation technique (or any other method of or-thogonalisation) means inevitably that the final basis of orthogonalised hybrid atomic orbitals (OHAOs) does contain many-centre orbitals in the sense that each OHAO is mainly its HAO parent but necessarily contains (minimal) contributions from overlapping HAOs. 

The right-hand side can be evaluated with mere 0(nWRl) arithmetic operations (nocc is the number of occupied spin orbitals) using just computationally tractable two-electron integrals. For atoms, nonzero contributions to this sum occur only from the RI basis functions with angular quantum numbers up to 3Locc, where Locc is the maximum angular quantum number of occupied spin orbitals. 

Molecular Orbital. A one-electron function made of contributions of Basis Functions on individual atoms (Atomic Orbitals) and delocalized throughout the entire molecule. 

However, because the atomic basis orbitals are attached to the centers, and because these centers are displaced in forming V, it is no longer true that (a%v/aX)o = 0 the variation in the wavefunction caused by movement of the basis functions now contributes to the first-order energy response. As a result, one obtains... 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

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. 

This localization scheme permits tire assignment of hybridization both to the atomic lone pairs and to each atom s contributions to its bond orbitals. Hybridization is a widely employed and generally useful chemical concept even though it has no formal basis in the absence of high-syrnmetry constraints. Witli NBO analysis, tire percent s and p character (and d, f, etc.) is immediately evident from tire coefficients of tire AO basis functions from which the NAO or NBO is formed. In addition, population analysis can be carried out using the NBOs to derive partial atomic charges (NPA, see Section 9.1.3.2). 

Attempts have been made to account for the rate enhancements in intramolecular catalysis on the basis of an effective concentration of 55 M combined with the requirement of very precise alignment of the electronic orbitals of the reacting atoms orbital steering. Although this treatment does have the merit of emphasizing the importance of correct orientation in the enzyme-substrate complex, it overestimates this importance, because, as we now know, the value of 55 M is an extreme underestimate of the contribution of translational entropy to effective concentration. The consensus is that although there are requirements for the satisfactory overlap of orbitals in the transition state, these amount to an accuracy of only 10° or so.23-24 The distortion of even a fully formed carbon-carbon bond... 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

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