Basis set orbitals

The indices J. and v refer to the basis set orbitals, ( ), and and are elements of the density and Fock matrices, respectively. 

The individual ab initio calculations are further identified by abbreviations for the basis set orbitals that are used. These abbreviations include, for example, STO-3G, 4-31G," and 6-31G. In general, the ab initio calculations make fewer assumptions than semiempirical methods, and therefore the computations are more demanding. 

For the butadiene-cyclobutene interconversion, the transition states for conrotatory and disrotatory interconversion are shown below. The array of orbitals represents the basis set orbitals, i.e., the total set of 2p orbitals involved in the reaction process, not the individual MOs. Each of the orbitals is tc in character, and the phase difference is represented by shading. The tilt at C-1 and C-4 as the butadiene system rotates toward the transition state is different for the disrotatory and conrotatory modes. The dashed line represents the a bond that is being broken (or formed). 

Analysis of the hexatriene-cyclohexadiene system leads to the conclusion that the disrotatory process will be favored. The basis set orbitals for the conrotatory and disrotatory transition states are shown below. 

Fig. 11.6. Classification of sigmatropic hydrogen shifts with respect to basis set orbitals.

Table 5.2 Contributions to the electron density at the iron nucleus in Fep4 unrestricted B3LYP calculations with the CP(PPP) basis set. Orbitals that electron density at the iron nucleus are printed in boldface...

The concerted mechanism shown above is allowed by the Woodward-Hoffmann rules. The TS involves the tt electrons of the alkene and enophile and the cr electrons of the allylic C-H bond. The reaction is classified as a [tt2 + tt2 + cr2] and either an FMO or basis set orbital array indicates an allowed concerted process. 

FMO orbitals for ene reactions Basis set orbital array for ene reactions... 

The CNDO procedure proposed and developed by Pople and his co-workers 5 61), as well as a number of its variants, are characterized by the assumption of vanishing overlap between basis set orbitals (CNDO = Complete Neglect of Differential Overlap), which reduces the number of non-zero two electron integrals drastically. Various other approximations and modifications had to be introduced... 

Suppose that, instead of writing the secular determinant from an n x n array of atomic orbitals, we use an n x n array of n orthonormal, linear combinations of the basis set orbitals. Suppose, furthermore—and this is the key—we require these linear combinations to be SALCs, that is, each one is required to be a function which forms a basis for an irreducible representation of the point group of the molecule. Then, as shown in Chapter 5, all integrals of the types... 

Figure 14 Basis set orbitals for the allyl system in Hiickel calculations...

Before the ready availability of high-speed computers, theoreticians realized that for them to perform MO calculations on anything but the very smallest molecules, they would need to reduce the mathematical intensity of the computation. They did this not only by making approximations that greatly reduced the size of the basis set (typically only the outer atomic orbitals are included in the calculation), but they also ignored the interaction of some of these basis set orbitals, which often turned out to be small and thus could be neglected. To make the computation yield results that were close to experimental values, numerical parameters were added that were derived empirically by experiment, hence the term semi-empirical. ... 

Errors inherent in the LSD approximation Orbital basis-set incompleteness Auxiliary basis-set incompleteness Near-linear dependencies in auxiliary basis sets Orbital basis-set superposition error Auxiliary basis-set superposition error Exchange-correlation sampling grid Model potential for core electrons (if employed)... 

The arguments are supported by a number of results on prototypical ground and excited states of atoms and molecules. Most of these are compared with results from conventional methods of quantum chemistry, where single basis sets, orbital- or Flylleraas-type, are used. 

The SOC operator contains the product of a term that acts on the spin part of a wavefunction, converting a triplet function to a singlet function and a term that acts on the space part of the wavefunction, changing one electronic configuration to another. Each component of the SOC vector, < So /z so T >, u = x, y or z, is a sum over all atoms and each atom contributes a sum over pairs of all basis set orbitals in the molecule. The terms in the sum are multiplied by numerical coefficients obtained from the Cl expansion and from the coefficients of the valence (primarily p-) orbitals in the two singly occupied MOs ofTj. The dominant contributors to the sum are those in which both basis set orbitals are located on the same atom ( one-centre terms ). Each term reflects the degree to which a 90° rotation around the axis u through the atom converts one member of the orbital pair into the other. If the space part of the wavefunctions differs only by the occupation number of two MOs ij/i and ij/j, we need to consider only the three matrix elements <... 

The output of an MO calculation includes atomic positions and the fractional contribution of each basis set orbital to each MO, that is, the values of Qp. The energy of each MO is calculated, and the total binding energy of the molecule is the sum of the binding energies of the fdled MOs ... 

In MO calculations, the total electron density is represented as the sum of all populated MOs. The electron density at any atom can be obtained by summing the electron density associated with the basis set orbitals for each atom. Electron density shared by two or more atoms, as indicated by the overlap integral, is partitioned equally among them. This is called a Mulliken population analysis ... 

Basis set orbitals for supra,supra [jt2 + Jt4] cycloaddition. Six electrons, zero nodes aromatic... 

Consideration of the HOMO-LUMO interactions also indicates that the [2Tr+2Tr] addition is allowed photochemically. The HOMO in this case is the excited alkene tt orbital. The LUMO is the tt of the ground state alkene, and a bonding interaction is present between both pairs of carbons where new bonds must be formed. Similarly, the concept of aromatic transition states shows that the reaction has an antiaromatic 4tt combination of basis set orbitals, which predicts an allowed photochemical reaction. Thus, orbital symmetry considerations indicate that photochemical [2tt- -2tt] cycloaddition of alkenes is feasible. 

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