Symmetry atomic orbitaLs

Symmetry enters the approximate solution of the electronic Schrd-dinger equation in two ways. In the first place, the exact MOs are eigenfunctions of an operator which commutes with all Om of the point group concerned, they therefore generate irreducible representations of that point group (see Chapter 8) and can be classified accordingly. The same is true for the approximate MOs and consequently one constructs them from combinations of atomic orbitals (symmetry orbitals) which generate irreducible representations. 

Fig. 3.25 Central atom orbital symmetries and degeneracies Tor Oh AB , ABS, and Cltl AB4 species.

Fig. 3.25 Ceniral atom orbital symmetries and degeneracies for 0 AB, ABj, and Cj AB.I species.

Fig. 33a-c. d-valence electron distribution at the (111) surface, a. The out-of-plane lobes of the degenerate d,y, dy, and d atomic orbitals [37]. b. The linear combinations of the plane lobes of the djy, dy, and d, atomic orbitals symmetry adapted to the (111) surface, c. Scheme of surface d-electron density of states... 

King RB (2000) Atomic orbitals, symmetry, and coordination polyhedra. Coord Chem Rev 197 141... 

From the reducible representation we must identify the irreducible representations for the H(ls) orbitals so that those that match with the B atom orbital symmetries listed above can be identified. The reduction is laid out in Table 7.5. 

Since 0a nd 0 normally identical atomic orbitals, symmetry requires that Cg and be of equal magnitude. Thus the molecular orbital may be written... 

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

These one-dimensional matriees ean be shown to multiply together just like the symmetry operations of the C3V group. They form an irredueible representation of the group (beeause it is one-dimensional, it ean not be further redueed). Note that this one-dimensional representation is not identieal to that found above for the Is N-atom orbital, or the Ti funetion. 

The Huckel method and is one of the earliest and simplest semiempirical methods. A Huckel calculation models only the 7t valence electrons in a planar conjugated hydrocarbon. A parameter is used to describe the interaction between bonded atoms. There are no second atom affects. Huckel calculations do reflect orbital symmetry and qualitatively predict orbital coefficients. Huckel calculations can give crude quantitative information or qualitative insight into conjugated compounds, but are seldom used today. The primary use of Huckel calculations now is as a class exercise because it is a calculation that can be done by hand. 

The PES of 2-aminothiazole and derivatives have been interpreted the first peak (8, 45 eV) is associated with an MO of tr symmetry mainly located on C-5 and N-3 the weiehts" of the atomic orbital in this MO... 

The researchers established that the potential energy surface is dependent on the basis set (the description of individual atomic orbitals). Using an ab initio method (6-3IG ), they found eight Cg stationary points for the conformational potential energy surface, including four minima. They also found four minima of Cg symmetry. Both the AMI and PM3 semi-empirical methods found three minima. Only one of these minima corresponded to the 6-3IG conformational potential energy surface. 

The atomic orbital contributions for each atom in the molecule are given for each molecular orbital, numbered in order of increasing energy (the MO s energy is given in the row labeled EIGENVALUES preceding the orbital coefficients). The symmetry of the orbital and whether it is an occupied orbital or a virtual (unoccupied) orbital appears immediately under the orbital number. 

Boron is unique among the elements in the structural complexity of its allotropic modifications this reflects the variety of ways in which boron seeks to solve the problem of having fewer electrons than atomic orbitals available for bonding. Elements in this situation usually adopt metallic bonding, but the small size and high ionization energies of B (p. 222) result in covalent rather than metallic bonding. The structural unit which dominates the various allotropes of B is the B 2 icosahedron (Fig. 6.1), and this also occurs in several metal boride structures and in certain boron hydride derivatives. Because of the fivefold rotation symmetry at the individual B atoms, the B)2 icosahedra pack rather inefficiently and there... 

For the minute, imagine an HF-LCAO treatment of just the jr-electrons in ethene where each carbon atom contributes just one electron and one atomic orbital of the correct symmetry to the conjugated system. Without any particular justification except chemical intuition, we make the following assumptions. 

Looking back, 1 seem to have made two contradictory statements about the basis fiinctions Xt used in the PPP model. On the one hand, I appealed to your chemical Intuition and prior knowledge by suggesting that the basis functions should be j garded as ordinary atomic orbitals of the correct symmetry (i.e. 2p orbitals). On the other hand, 1 told you that the basis functions used in such calculations are taken to be orthonormal and so... 

Carbon atoms in free space have spherical symmetry, but a carbon atom in a molecule is a quite different entity because its charge density may well distort from spherical symmetry. To take account of the finer points of this distortion, we very often need to include d, f,. .. atomic orbitals in the basis set. Such atomic orbitals are referred to as polarization functions because their inclusion would allow a free atom to take account of the polarization induced by an external electric field or by molecule formation. 1 mentioned polarization functions briefly in Section 9.3.1. 

Because a [1,5] sigmatropic rearrangement involves three electron pairs (two ir bonds and one cr bond), the orbital-symmetry rules in Table 30.3 predict a suprafacial reaction. In fact, the 1,5] suprafacial shift of a hydrogen atom across... 

The valence atomic orbitals which are available to form the orbitals of a CC single bond, directed along the x axis, are the 2s and 2px atomic orbitals on each carbon atom. Their admixture—in proportions which depend on the number of neighbors at each carbon and on the subsequent hybridization—creates two (s, p ) hybrids on each atom. One of these hybrids points away from the other atom and can be used for bonding to additional atoms. The pair of hybrids which point at each other overlap and interact in the conventional fashion [we symbolize the non-interacting orbitals by an interruption of the bond axis (Fig. 1)]. The two bond orbitals which are formed in this manner both have [Pg.3]

If the pair of carbon atoms shown above each have only two neighbors so that they are doubly-bonded in the conventional sense, there is an extra p orbital available on each atom. These p orbitals point along the (z) direction, perpendicular to the plane of the molecular fragment. The interaction of these two atomic orbitals via overlap creates a new pair of bond orbitals with local tt symmetry (Fig. 2, where again we have symbolized the non-interacting or-... 

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