Module 2 : Molecular Structure
Lecture 7 : Homonuclear Diatomic Molecules
 

The Schrödinger equation is written as H = E, where (1,2,….n) is the molecular wave function and H is the Hamiltonian operator. H ( for the molecular system in fig. 7.1) is given by

 

 
  (7.1)


Here is the kinetic energy operator for nucleus X, is the kinetic energy operator for electron i, riX is the distance between electron i and nucleus X and rij is the distance between electron i and electron j. Since the nuclei are held fixed, the nuclear kinetic energy operator terms will be set to zero.

 

Molecular orbitals of

The simplest example of a molecule is H2, which has two nuclei and two electrons. The Hamiltonian for this species is given in the following equation (7.2) and the coordinates of all the particles are shown in the following figure (fig 7.2)

 
(7.2a)
 
(7.2b)
 
The hydrogen molecular ion has two nuclei and one electron. The Hamiltonian for this species is given in the following equation (7.3). Since there is only one electron, the subscript 1 in r 1A can be dropped for convence.
 
(7.3)
 

                                   
 

Figure 7.2 The electron coordinates in H2. In the molecular ion there is only one electron.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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