Module 2 : Molecular Structure
Lecture 7 : Homonuclear Diatomic Molecules
 

When r (r1A or r2A ) is constant, the s orbital is a circle. On different circles, the values of the orbitals are different and they are marked as C1 , C2 ….. C8 . For both the AOs, the normalization constant is , where a0 is the Bohr radius, 0.529 . By estimating the value of r from the plot, you can easily estimate the values C1….C8 .

In fig 7.3 (b) the contours of the MO are shown. Eight contours, C1….C8 are shown. We shall illustrate the calculation of the value of MO on the contour C6 at two points, P2 and P1 . On the x axis we have x1A / a0 and x 1B /a0 and on the y axis, we have y1A /a0 and y1B /a0 . The coordinates of the nuclei are (-0.7a0, 0) and (0.7a0, 0) or in terms of , they are (-0.37 , 0) for nucleus A and (0.37 , 0) for nucleus B. The coordinates of point P2 are (-1.4a0 , .6a0 ).

 
We need the distance of this point from the nuclei A and B, ie we need r1A and r1B . The values of x1A and y 1A are –0.7a0 and 0.6a0 and the values of x1B and y1B are –2.1a0 and 0.6a0 respectively. For the point P2 , the values of r1A and r1B are

 
r1A = (0.72 + 0.6 2 ) 1/ 2 a0 = 0.922a0


r1B = ((-2.1) 2 +0.6 2 ) 1/ 2 a0 = 2.184a0 (7.5)
And the value of  is
*= (e -r1A /a0 + e-r1B /a0 )
    = (3.14 x 0.529 3 ) -1/ 2-3/2 ( e -0.9 + e –2.3 )  
    = 0.75 -3/2

(7.6)

So,Let us verify that the value of on the same contour at the point P1 is also 0.75 -3/2

In the contours it is seen that the MO looks like the corresponding AOs near the nuclei and far away from the nuclei, it looks the atomic orbitals of the “whole molecule”.

Another way to plot the MO is to sketch the values of , and = as functions of r1A and r1B. This sketch is shown in fig 7.4

          

r1A and r1B in units of a0

Fig 7.4 (a) A comparison of (red thick line) with and . (green lines). The values of (stars) are also shown.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1