S-atomic orbitals combination

When two s-atomic orbitals combine, a molecular orbitals result. Atomic orbitals Molecular orbitals... 

An answer was provided in 1931 by Linus Pauling, who showed how an s orbital and three p orbitals on an atom can combine mathematically, or hybridize, to form four equivalent atomic orbitals with tetrahedral orientation. Shown in Figure 1.10, these tetrahedrally oriented orbitals are called sp3 hybrids. Note that the superscript 3 in the name sp3 tells how many of each type of atomic orbital combine to form the hybrid, not how many electrons occupy it. 

Figure 1.17 Molecular orbitals of H2- Combination of two hydrogen 1 s atomic orbitals leads to two H2 molecular orbitals. The lower-energy, bonding MO is filled, and the higher-energy, antibonding MO is unfilled.

Having just seen a resonance description of benzene, let s now look at the alternative molecular orbital description. We can construct -tt molecular orbitals for benzene just as we did for 1,3-butadiene in Section 14.1. If six p atomic orbitals combine in a cyclic manner, six benzene molecular orbitals result, as shown in Figure 15.3. The three low-energy molecular orbitals, denoted bonding combinations, and the three high-energy orbitals are antibonding. 

The atomic orbitals of atoms are combined to give a new set of molecular orbitals characteristic of the molecule as a whole. The number of molecular orbitals formed is equal to the number of atomic orbitals combined. When two H atoms combine to form H two s orbitals, one from each atom, yield two molecular orbitals. In another case, six p orbitals, three from each atom, give a total of six molecular orbitals. 

Figure 3.6 The molecular orbital diagram for the combination of two identical 1 s atomic orbitals...

The molecule of hydrogen fluoride, HF, belongs to the Coov point group. The hydrogen atom uses its 1 s atomic orbital to make bonding and antibonding combinations with the 2pz orbital of the fluorine atom, the z... 

For the integrations in ab initio calculations we need the actual mathematical form of the spatial functions, and the hydrogenlike expressions are Slater functions [1]. For atomic and some molecular calculations Slater functions have been used [3]. These vary with distance from where they are centered as exp(-constant.r), where r is the radius vector of the location of the electron, but for molecular calculations certain integrals with Slater functions are very time-consuming to evaluate, and so Gaussian functions, which vary as exp(-constant.r2) are almost always used a basis set is almost always a set of (usually linear combinations of) Gaussian functions [4]. Very importantly, we are under no theoretical restraints about their precise form (other than that in the exponent the electron coordinate occurs as exp(-constant.r2)). Neither are we limited to how many basis functions we can place on an atom for example, conventionally carbon has one 1 s atomic orbital, one 2s, and three 2p. But we can place on a carbon atom an inner and outer Is basis function, an inner and outer 2s etc., and we can also add d functions, and even f (and g ) functions. This freedom allows us to devise basis sets solely with a view to getting... 

Now, how can the carbon s 2s and 2p atomic orbitals combine with the four hydrogen Is atomic orbitals The carbon s 2s orbital can overlap with all four hydrogen Is orbitals at once with all the orbitals in the same phase. In more complicated systems like this, it is clearer to use a diagram of the AOs to see what the MO will be like. 

This is a reducible representation of the D point group which reduces to ug + uu. Two molecular orbitals must be generated, one with ag and the other with antibonding orbitals which can be formed from the two 1. s atomic orbitals. 

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. 

Alternatively, a Bloch sum of s atomic orbitals could combine with a Bloch sum of p atomic orbitals. The symmetric (bonding) combinations of the basis atomic orbitals for the latter case are illustrated for one CC4 subunit in Figure 3.9c. The actual COs are delocalized over aU the atoms with the space group symmetry of the diamond lattice. A LCAO-CO construction from Bloch sums is thus completely analogous to a... 

Since there are two atoms per primitive cell, or lattice point, and consideration is still on just s atomic orbitals, two separate Bloch sums are required. These combine to form two COs with energies given by the sum and difference in energy between the nearest neighbor and second-nearest neighbor interactions. The nearest neighbor... 

Orbital (Section 1.8) . A hybrid orbital derived by combination of an s atomic orbital with tw o p atomic orbitals. The three sp hybrid orbitals that result lie in a plane at angles of 120" to each other. 

Molecular orbitals are formed by combining atomic orbitals. The number of MO s formed is the same as the number of atomic orbitals combined. 

When we discussed sp hybrid orbitals in Section 1.7, we said that all four of carbon s valence-shell atomic orbitals combine to form four equivalent sp hybrids. Imagine instead that the 2s orbital combines with only two of the three 2p orbitals. Three sp hybrid orbitals result, and one 2p orbital remains unchanged. The three sp orbitals lie in a plane at angles of 120° to one another, with the remaining p orbital perpendicular to the sp plane, as shown in Figure 1.14. 

Just as bonding and antibonding a molecular orbitals result from the combination of two s atomic orbitals in H2 (Section 1.6), so bonding and antibonding tt molecular orbitals result from the combination of two p atomic orbitals in ethylene. As shown in Figure 1.17, the n bonding MO has no node between nuclei and results from combination of p orbital lobes with... 

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