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									Lecture 26: Homonuclear Diatomic Molecules-I

 The material in this lecture covers the following in Atkins.
14 Molecular structure
          Molecular Orbital Theory
          14.5 The structure of diatomic molecules
          (b) bond order
          (c) Period 2 diatomic molecules
          (d) p-orbitals
          (e) The overlap integral

Lecture on-line

          Homonuclear diatomic molecules (PowerPoint)
              Homonuclear diatomic molecules (PDF)
     Handout for this lecture
Audio-visuals on-line

  Shape of molecular orbitals in homonuclear diatomic molecules
(PowerPoint)(From the Wilson Group,***)
 Shape of molecular orbitals in homonuclear diatomic molecules
(PDF)(From the Wilson Group,***)
Composition of orbitals in homonuclear molecules
 (6 MB MBQuick-Time with music)
(A must from the Wilson Group,*****)
The Occupation of homonuclear diatomic orbitals
(PowerPoint)(From the Wilson Group,***)
The Occupation of homonuclear diatomic orbitals(PDF)
(From the Wilson Group,***)
 Molecular Orbital Theory         Diatomics
 We used the orbitals of the
 one - electron hydrogen to build up
 wavefunctions for many- electron
 atoms


We shall use the orbitals of
the one - electron H+ molecule
                    2
to describe diatomic molecules

 The molecular orbitals are
 written as linear combinations
 of atomic orbitals
  The atomic orbitals are in general those centered on
  the atoms of our molecule
 Molecular Orbital Theory     Diatomics                   
                                                         H2
For H2 we have one 1sH orbital on each
hydrogen : 1sA (1) = A(1); and 1sB (1) = B(1)
                                                      :
From these we can form two different molecular orbitals
            1                                1
   (1)         [A(1)  B(1)]    (1)          [A(1)  B(1)]
          2(1 S)                          2(1 S)
                                      J+K     e2 1
       With energies : E   E1sH         
                                    (1+ S) 4o R
                    J-K    e2 1
     ;E   E1sH        
                   (1- S) 4o R




  E1sH
Molecular Orbital Theory     Diatomics    H2




E1sH




The H2 molecule has two electrons. They
                       
will be in the bonding 1 orbital
Molecular Orbital Theory         Diatomics     He 2
                                 The ground electronic
                                  configuration of the
                                  hypothetical four-electron
                                  molecule He2 has two
                                 bonding electrons and
                                 two antibonding electrons.
                                  It has a higher energy
                                 than the separated atoms,
                                 and so is unstable.

                                   The bond order is:
The He2 molecule has four
                                      1
electrons. They will be in the     b = (n  n* )
                                      2
           
 bonding 1 orbital and in the
                               n  occupied bonding orb
anti - bonding 2 * orbital:
  2      2                   n*  occupied anti - bonding orb
1 (2*)
Molecular Orbital Theory Diatomics        Second row
In second row elements
we have both 2s and 2p    S    A Bdv
orbitals
The 2s orbitals can
form strong overlaps                   2p     2p
with each other


We also have two p                      2s      2s
orbitals pointing along
the A - B bond vector
They can overlap with And with 2s
each other

These are the  - orbitals, they do not change
sign with rotation around A- B vector
Molecular Orbital Theory         Diatomics
We finally have two
sets of p - orbitals    S    A Bdv
perpendicular to the
A - B bond
                                         2p   2p
vector
They can overlap with
each other in pairs
                                         2s       2s


  and


 These are the  - orbitals, they change
 sign with rotation of 180  around A- B vector
Molecular Orbital Theory         Diatomics

The  - and  - orbitals
do not overlap

         S0                                2p   2p



         S0
                                            2s    2s
           S0
                                                 negative
             S0           S    A Bdv
In all cases positive                            positive
and negative contributions
cancel
Molecular Orbital Theory        Diatomics
                           (a) When two orbitals are on
                            atoms that are far apart, the
                            wavefunctions are small where
                            they overlap, so S is small.

                           (b) When the atoms are closer,
                            both orbitals have significant
                            amplitudes where they overlap,
                            and S may approach 1. Note
                            that S will decrease again as
                            the two atoms approach more
                            closely than shown here,
                           because the region of negative
                            amplitude of the p orbital starts
                            to overlap the positive overlap
                            of the s orbital. When the centres
                           of the atoms coincide, S = 0.
Molecular Orbital Theory             Diatomics
Overlap is 1 when functions
                                                 Data 1
coinside               1



                     0.8



                     0.6

  Zero at infinite
                     0.4
  separation
                     0.2



                      0
                           0   0.1   0.2   0.3     0.4    0.5   0.6   0.7   0.8


                               The overlap integral, S,
                               between two H1s orbitals
                               as a function of their
                               separation R.
Molecular Orbital TheoryDiatomics                     J-K       e2 1
                                         E   o           
Consider two orbitals A                            (1- SAB ) 4o R
and A on nuclei A and B                            1
                                                        [A  B ]
of the same energy  o :                        2(SAB  1)
 They will interact to
 form a bonding
 orbital + of energy E+                   o                o
                                                                   
                                     A                               B
  And the anti- bonding
  orbital - of energy E-

  The interaction intergral K
                                                   1
  will be proportional to S .
                           AB                             [A  B ]
                                                2(SAB  1)
   K ~ SAB   A Bdv                       J+K      e2 1
                                E   o           
                                           (1+ SAB ) 4o R
 Molecular Orbital Theory         Diatomics



            2p    2p




            2s     2s




According to molecular orbital theory,  orbitals
are built from all orbitals that have the appropriate symmetry.
In homonuclear diatomic molecules of Period 2, that means
that two 2s and two 2pz orbitals should be used. From these
four orbitals, four molecular orbitals can be built.
 Molecular Orbital Theory       Diatomics

 From two 2s orbitals and two
 2p orbitals we can form 4
 molecular orbitals:
1  c1 2sA  c1 2sB
      2sA      2sB
                                            4
c1 A 2pA  c1 B 2pB
  2P          2P

2  c2 2sA  c2 2sB
      2sA      2sB                     2p        2p
                                            3
c2 A 2pA
  2P          c2 B 2pB
                2P
                                            2
3  c3 2sA
      2sA      c3 2sB
                 2sB                   2s         2s
c3 A 2pA  c3 B 2pB
  2P          2P

 4  c2sA 2sA  c4 2sB
       4
                  2sB                       1
 c4 A 2pA  c4 B 2pB
   2P          2P
Molecular Orbital Theory      Diatomics
To a first approximation
2s and 2p are separated
sufficiently in energy so that:

We can form two orbitals
made up of 2s                                      4

1  c1 2sA  c1 2sB
      2sA      2sB
                                          2p               2p
                                               
                                                       3
2    c2 2sA
        2sA      c2 2sB
                   2sB

and two orbitals                               
                                                   2
made up of 2p
3  c3 A 2pA  c3 B 2pB
      2P          2P                      2s               2s

4  c2PA 2p A  c4 B 2pB
      4
                   2P                          1
Molecular Orbital Theory
                                 Diatomics
A representation of the
composition of bonding                       4
and antibonding  orbitals
built from the overlap of p           2p             2p
orbitals. These illustrations                
                                                 3
are schematic.




 3  c3 A 2pA  c3 B 2pB
       2P          2P             4  c2PA 2p A  c4 B 2pB
                                        4
                                                     2P

   Or from symmetry                  Or from symmetry
         1                                1
3            [2pA  2p B ]   4           [2pA  2p B ]
       2(1 S)                         2(1 S)
        bonding                       anti  bonding
Molecular Orbital Theory             Diatomics
 We also have two p- orbitals perpendicular
 to the bond- vector
  They will form the  - orbitals :
                                               1
           1                         1y              [2pyA  2pyB ]
1x               [2p xA  2p xB ]         2(S   1)
        2(S   1)
                                                  1
2 x 
   *        1
                    [2p xA  2p xB ]  2 y 
                                         *
                                                         [2pyA  2pyB ]
         2(S  1)                            2(S   1)


                                                         2y
               2x
                                                   2py          2py
         2px          2px
               1x



                                                          1y
Molecular Orbital Theory      Diatomics
                      orbitals do not change sign on
                    rotation arounf A- B bond vector

                                   A schematic
                                   representation
                                    of the structure of
                                    bonding and
                                   antibonding
                                    molecular orbitals.

  orbitals change sign once on
rotation around A- B bond vector
Molecular Orbital Theory    Diatomics

 For oxygen and flourine
 where 2p and 2s are well
 separated we get
 the orbital diagram
        What you must learn from this lecture
 1. Understand the difference between bonding
 and anti- bonding orbitals in diatomic molecules
2. Understand the difference between constructive
interferrence (in bonding orbitals) and destructive
interferrence in (anti - bonding) orbitals
 3. Understand the difference between  - orbitals
 with complete rotational symmetry around bond
 vector and  - orbitals that change sign on 180
rotation.
 4. Be able to construct qualitatively the
molecular orbitals of the homonuclear
 diatomic molecules as a linear combination
 of atomic orbitals

 5. Be able to deduce the bond order for a diatomic
 molecule from its electronic configuration

								
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