Molecular orbitals in hydrogen

Although mixing of s and p orbitals is represented in Fig. 5.24 as a sepamle step preceding the formation of molecular orbitals, the entire process can be combined into a single step. For example, the bonding molecular orbital in hydrogen chloride may be considered to be formed as... 

FIGURE 10.2 Molecular orbitals in hydrogen fluoride, plotted using two isosurfaces, at 0.02 e 1/2 Bohr 3/2. (See the color version of this figure in Color Plates section.)... 

Relative energies of atomic and molecular orbitals in hydrogen molecule are shown in figure below. 

In molecular hydrogen, where we are building LCAO-MOs from two atomic orbitals, we expect two molecular orbitals. In the second molecular orbital, the two atomic orbitals interfere destructively where they overlap. This orbital has the form... 

Molecular orbitals are generated by combining atomic orbitals. The number of molecular orbitals formed is always equal to the number of atomic orbitals that combine. So, if two atomic orbitals combine, then two molecular orbitals will be formed. This is the case when two hydrogen Is atomic orbitals combine to produce two molecular orbitals in a hydrogen molecule (H ). 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

We could simply proceed to inspect these orbitals to see which overlap with each other, and then begin to make molecular orbitals in the way described in the previous section. Unfortunately, the situation is now quite complicated. The hydrogen number 1 interacts with allfour of the carbon valence orbitals. 

In this case, it is assumed that the molecular orbitals in a hydrogen molecule can be represented in the usual way (see Section 2.3.1) and that the bonding molecular orbital is then used with the third atomic orbital to give the bonding molecular orbital for the entire species. Because H3+ contains only two electrons, the bonding molecular orbital is occupied by two electrons, but it encompasses all three atomic centers. This type of three-center bonding will also be discussed in later chapters. 

With a total of fourteen valence electrons to accommodate in molecular orbitals, ethane presents a more complicated picture, and we now meet a C—C bond. We will not go into the full picture—finding the symmetry elements and identifying which atomic orbitals mix to set up the molecular orbitals. It is easy enough to see the various combinations of the Is orbitals on the hydrogen atoms and the 2s, 2px, 2py and 2pz orbitals on the two carbon atoms giving the set of seven bonding molecular orbitals in Fig. 1.19. 

The molecular orbitals in the HF2 system in Fig. 2.17 resemble those of the allyl anion—a low-energy orbital with no nodes, and an orbital with a node at the central atom. The node at the hydrogen atom leaves it with no interactions with the two fluorine atoms, which are far enough apart to be essentially nonbonding. For this arrangement to be stabilised, i/ , and ip2 must together be lower in energy... 

FIGURE 6-6 Molecular Orbitals for Hydrogen Bonding in FHF. The Is and the 2p nonbonding orbitals of F are omitted. Figure 5-18 shows the full set of molecular orbitals. 

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