Module 2 : Molecular Structure
Lecture 7 : Homonuclear Diatomic Molecules
 

In equation (7.1) it is simplest to treat the nuclei as fixed and so, the kinetic energy terms of the nuclei are not considered. In eq (7.2) the kinetic energy terms for the nuclei are not shown. This is known as the Born_Oppenheimer approximation and helps greatly in simplifying the problem. Equations (7.1) and (7.2) do not have any exact solution. We need to develop strategies to solve these as accurately as possible. In the present chapter we will develop qualitative ideas of the nature of the solutions. A few public domain (free) softwares are available for quantitative solutions. The Schrodinger equation for (a one electron system) has an exact solution, but that is in the bipolar coordinate system. We will instead consider numerical solutions here.

 

The reason why exact solutions can not be found is that the equation (7.1) and (7.2) can not be split into n independent equations for n electrons. This is because the terms can not be separated into n terms containing the variables ri alone. What can be done is to get n effective one particle equations for the n orbitals, just as we did in the case of atomic orbitals in lecture 6.

 

An important  change  that  you  will  notice  in eq (7.1) and (7.2) is that CGS units are used here and the factor of 40 used in earlier chapters is not there. In these units, the charge on an electron is 4.8 * 10 -10 esu (electrostatic unit of charge). The repulsive force between two charges of 1 esu each separated by 1 cm is 1 erg. For two electrons separated by 1, the interaction energy (e 2 /r = e2 / 1) is 23.04 x 10 -12 erg.


In lecture 6, we have seen that an atomic orbital is a function of the coordinates of one electron. A molecular orbital is also a function of the coordinates of one (a single) electron. The only difference is that since several atoms (more than one) are present in a molecule, the electron will have different distances from the nuclei at each point in space and the value of the molecular orbital (MO) will depend on all these distances, eg : r 1A , r 1B , r 1C … for electron 1, r2A , r2B , r2C , …. for electron 2 and so on. Here r2C, for example, is the distance of electron 2 from nucleus C.
In , there is only electron and we will omit the subscript 1 and write the MO for as a linear combination of atomic orbitals on A and B.
 

.

(7.4)
This is a linear combination because the functions occur in the first power. A general linear combination will be
 
(7.5)
 

where is an MO, are atomic orbitals and ci are coefficients. We only wish to comment that the method is extensively used and is iterative in nature (ie, guess an initial solution to equations (7.1) or (7.2) and iteratively improve the solution till a converged final solution is obtained). This is analogous to finding roots (iteratively) of algebraic equations of higher than orders.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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