Orbitals for Diatomic Molecules

The atomic orbitals considered above can be used to help describe the wave functions of electrons in chemical bonds. To see this, we start with the simple problem of the H2 molecule (Fig. 1.1). 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as 

To find the optimal coefficients CA and CB one can use the variation principle, which states that any trial solution for the wave function will give a larger value for e(R) than the value obtained with the exact wave function. With this in mind, we should try to find the minimum of e(R) as a function of CA and CB. This is done by expressing e(R) as 

This leads to the well-known secular equation 

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Similarly, Slater orbitals for diatomic molecules give integrals of the form (22)... 

The actual energies of molecular orbitals for diatomic molecules are intermediate between the extremes of this diagram, approximately in the region set off by the vertical lines. Toward the right within this region, closer to the separated atoms, the energy sequence is the normal one of O2 and F2 further to the left, the order of molecular orbitals is that of B2, C2 and N2, with a-g(2p) above TT (2p). 

The same principles that we used to build up molecular orbitals for diatomic molecules can be applied to larger and, indeed, very large molecules. We shall consider a few examples and then some general points that apply to calculations on any molecule. For our first example, we pick up on the molecules that we only partially dealt with using hybrid orbitals. 

Look back at the orbitals for diatomic molecules and suggest what type of molecular orbital these 2p orbitals might form. 

In an extension to the labelling used for diatomic molecules, this orbital is referred to as a 7t orbital. We can describe the bonding in ethene as consisting of bonding orbitals in the plane of the molecule formed by overlap of the sp2 hybrid orbitals with each other and with the s orbitals on hydrogen and an out-of-plane Tt-bonding orbital. By analogy with orbitals for diatomic molecules, the orbitals formed by the sp2 hybrid orbitals are called a orbitals. 

Estimate the energies of the bonding and antibonding molecular orbitals of diatomic molecules from the secular determinant Construct simple molecular orbitals for diatomic molecules from a linear combination of atomic orbitals and describe their symmetry... 

Figure 14.68 Bonding molecular orbitals for diatomic molecules, Xj, of elements in the second period of the periodic table... 

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