Chemistry 2A - First Year Chemistry by pptfiles

VIEWS: 10 PAGES: 41

									        Chemistry 2

         Lecture 1
Quantum Mechanics in Chemistry
                  Your lecturers




           8am                            12pm
 Assoc. Prof Timothy Schmidt    Assoc. Prof. Adam J Bridgeman

           Room 315                       Room 222
timothy.schmidt@sydney.edu.au   adam.bridgeman@sydney.edu.au
           93512781                        93512731
                Learning outcomes
•Be able to recognize a valid wavefunction in terms of its being
single valued, continuous, and differentiable (where potential is).
•Be able to recognize the Schrödinger equation. Recognize that
atomic orbitals are solutions to this equation which are exact for
Hydrogen.
•Apply the knowledge that solutions to the Schrödinger equation are
the “observable energy levels” of a molecule.
•Use the principle that the mixing between orbitals depends on the
energy difference, and the resonance integral.
•Rationalize differences in orbital energy levels of diatomic
molecules in terms of s-p mixing.
   How quantum particles behave
• Uncertainty principle: DxDp > ℏ
• Wave-particle duality




• Need a representation for a quantum
  particle: the wavefunction (ψ)
How quantum particles behave
           The Wavefunction
• The quantum particle is described by a
  function – the wavefunction – which may be
  interpreted as a indicative of probability
  density

• P(x) = Ψ(x)2

• All the information about the particle is
  contained in the wavefunction
The Wavefunction




       Must be single valued
       Must be continuous
       Should be differentiable (where potential is)
           The Schrödinger equation

 • Observable energy levels of a quantum
   particle obey this equation
                                   wavefunction




Hamiltonian operator
                          Energy
               Hydrogen atom
• You already know the solutions to the
  Schrödinger equation!
• For hydrogen, they are orbitals…

                                          3s
                       2s



       1s

                       2p                 3p
    No nodes

                      1 node                   2 nodes
           Hydrogen energy levels
E

                 2 nodes

                           3p
                                               (also 3d orbitals…)
           3s

           2s          2p
            1 node                        Rydberg constant
                                energy level
                                               Z          2
                                         En     2
            1s
                                                n Nuclear charge
No nodes
                                 Principal quantum number
     ionization potential
4s                          4p
3s                          3p

2s                          2p



                 UV (can’t see)



      1s
Solutions to Schrödinger equation
• Hamiltonian operates on wavefunction and
  gets function back, multiplied by energy.
                  ˆ
                  H  
•   More nodes, more energy (1s, 2s &c..)
•   Solutions are observable energy levels.
•   Lowest energy solution is ground state.
•   Others are excited states.
Solutions to Schrödinger equation
             ˆ
            H  
Exactly soluble for various model problems
• You know the hydrogen atom (orbitals)

In these lectures we will deal with
• Particle-in-a-box
• Harmonic Oscillator
• Particle-on-a-ring

But first, let’s look at approximate solutions.
Approximate solutions to Schrödinger
             equation

• For molecules, we solve the problem
  approximately.
• We use known solutions to the Schrödinger
  equation to guide our construction of
  approximate solutions.
• The simplest problem to solve is H2+
               Revision – H2+
• Near each nucleus, electron should behave as a
  1s electron.
• At dissociation, 1s orbital will be exact solution
  at each nucleus

 




                         r
                  Revision – H2+
• At equilibrium, we have to make the lowest energy possible
  using the 1s functions available
  




  
                              r                      ?

                              r
                Revision – H2+
                               
                                          = 1sA – 1sB
          anti-bonding
                                   1sA                   1sB
E



    1sA                  1sB
                                          = 1sA + 1sB
                               

           bonding

                                   1sA                   1sB
                 Revision – H2
                               
                                          = 1sA – 1sB
          anti-bonding
                                   1sA                   1sB
E



    1sA                  1sB
                                          = 1sA + 1sB
                               

           bonding

                                   1sA                   1sB
                Revision – He2
                               
                                          = 1sA – 1sB
          anti-bonding
                                   1sA                   1sB
E



    1sA                  1sB
                                          = 1sA + 1sB
                               

           bonding

                                   1sA                   1sB
      NOT BOUND!!
     2nd row homonuclear diatomics
• Now what do we do? So many orbitals!


2p                                       2p

2s                                       2s




1s                                       1s
                Interacting orbitals
Orbitals can interact and combine to make new approximate
solutions to the Schrödinger equation. There are two
considerations:
1. Orbitals interact inversely proportionally to their energy
    difference. Orbitals of the same energy interact completely,
    yielding completely mixed linear combinations.
2. The extent of orbital mixing is given by the integral
                                       total energy operator (Hamiltonian)

                             ˆ
                         2 H 1d
                                                    integral over all space

               orbital wavefunctions
                  Interacting orbitals
   1. Orbitals interact inversely proportionally to their energy
      difference. Orbitals of the same energy interact completely,
      yielding completely mixed linear combinations.

                                        
                                             1
                                                ( 2 sA  2 sB )
                                              2

    2p                                                              2p

     2s                                                             2s


 2 sA                                  
                                             1
                                              2
                                                ( 2 sA  2 sB )    2 sB



    1s                                                              1s
                  Interacting orbitals
1. The extent of orbital mixing is given by the integral

                ˆ
            2 H 1d  something

2p                                                                      2p

2s                 2 s                                2 p           2s


        The 2s orbital on one atom can interact with the 2p from the
         other atom, but since they have different energies this is a
         smaller interaction than the 2s-2s interaction. We will deal
                                with this later.

1s                                                                      1s
                   Interacting orbitals
1. The extent of orbital mixing is given by the integral

                           ˆ
                       2 H 1d  0
                            cancels
2p                                                                             2p

2s                  2 s                          2 p                       2s




               There is no net interaction between these orbitals.
       The positive-positive term is cancelled by the positive-negative term

1s                                                                             1s
               (First year) MO diagram
Orbitals interact most with the corresponding orbital on the other atom
    to make perfectly mixed linear combinations. (we ignore core).




    2p                                                           2p




    2s                                                           2s
Molecular Orbital Theory - Revision
•Molecular orbitals may be classified according to their symmetry
•Looking end-on at a diatomic molecule, a molecular orbital may
resemble an s-orbital, or a p-orbital.
•Those without a node in the plane containing both nuclei resemble an
s-orbital and are denoted -orbitals.
•Those with a node in the plane containing both nuclei resemble a p-
orbital and are denoted -orbitals.




                                                             
 Molecular Orbital Theory - Revision
•Molecular orbitals may be classified according to their contribution to
bonding
•Those without a node between the nuclei resemble are bonding.
•Those with a node between the nuclei resemble are anti-bonding,
denoted with an asterisk, e.g. *.




                       *                            *
 Molecular Orbital Theory - Revision
• Can predict bond strengths qualitatively
                              2 p *

                                     2 p *
                                       2 p
                              2 p

                           2s *

  diamagnetic                      N2 Bond Order = 3

                           2s
             More refined MO diagram
                 orbitals can now interact
   2 p *
2 p *
                                   
2 p
    2 p

    2s *



       2s
                                   
             More refined MO diagram
                 * orbitals can interact
                                  *
   2 p *
2 p *
                                  
2 p
    2 p

    2s *                         *



       2s
                                   
             More refined MO diagram
                 orbitals do not interact
                                  *
   2 p *
2 p *
                         *
                                  
2 p
                         
    2 p

    2s *                         *



       2s
                                   
               More refined MO diagram
                       sp mixing
                                   *
   2 p *
2 p *
                        *
                                   
2 p
                        
    2 p
                                           This new interaction energy
    2s    *
                                      *
                                           Depends on the energy
                                           spacing between the 2s
                                           and the 2p


       2s
                                   
Smallest energy gap,
                       sp mixing      Largest energy gap, and
                                      thus smallest mixing
and thus largest                      between 2s and 2p is for
mixing between 2s                     Fluorine.
and 2p is for Boron.


                             2p




                                                Z 2
                             2s    c.f.   En      2
                                                 n
                                   sp mixing
               *                      *                     *                 *

*                      *                      *                     *
                                                       
                                                                         


                   *                      *
                                                                  *             *


    weakly bound             paramagnetic            diamagnetic


                                                                              
         Be2                      B2                     C2                 N2
sp mixing in N2
            sp mixing in N2




-bonding orbital   Linear combinations of lone pairs



                    :N≡N:
                       Summary
• A valid wavefunction is single valued, continuous, and
  differentiable (where potential is).
• Solving the Schrödinger equation gives the observable energy
  levels and the wavefunctions describing a sysyem:
• Atomic orbitals are solutions to the Schrödinger equation for
  atoms and are exact for hydrogen.
• Molecular orbitals are solutions to the Schrödinger equation for
  molecules and are exact for H2+.
• Molecular orbitals are made by combining atomic orbitals and
  are labelled by their symmetry with σ and π used for diatomics
• Mixing between atomic orbitals depends on the energy
  difference, and the resonance integral.
• The orbital energy levels of diatomic molecules are affected by
  the extent of s-p mixing.
               Next lecture
• Particle in a box approximation
  – solving the Schrödinger equation.


           Week 10 tutorials
• Particle in a box approximation
  – you solve the Schrödinger equation.
                 Practice Questions
1. Which of the wavefunctions (a) – (d) is
   acceptable as a solution to the Schrödinger
   equation?
2. Why is s-p mixing more important in Li2
   than in F2?
3. The ionization energy of NO is 9.25 eV
   and corresponds to removal of an
   antibonding electron
   (a) Why does ionization of an antibonding
        electron require energy?
   (b) Predict the effect of ionization on the
        bond length and vibrational frequency
        of NO
   (c) The ionization energies of N2, NO, CO
        and O2 are 15.6, 9.25, 14.0 and 12.1
        eV respectively. Explain.

								
To top