Molecular Orbitals

1
Molecular Orbitals

• Chapter 5

2
Molecular Orbital Theory

• Molecular orbital theory uses the methods of
  group theory to describe bonding. Symmetry and
  relative energies largely determine how they
  interact to form molecular orbitals.
• If the total energy of the electrons in the
  molecular orbitals is less than in the atomic
  orbitals, the molecule is predicted to form and
  be stable.

3
Molecular Orbital TheoryApproximations to the MO
Theory

• Born-Oppenheimer approximation
• ?(r,R)?el(r)?(R)
• ?el(r) describes the electrons and ?(R) describes
  the nuclei.
• (R) describes the nuclear coordinates and (r)
  describes the electron coordinates.
• Nuclei are much heavier and move slowly relative
  to the electrons. The motion of the electrons
  can be separated out.
• ?el(r) will be uniquely considered

4
Molecular Orbital TheoryApproximations to the MO
Theory

• Orbital Approximation
• It is not possible to find an exact solution for
  the electronic wavefunction in a many-electron
  system. It can be expressed as a product of
  one-electron wavefunctions.
• ?el(e1,e2en)?1(e1)?2(e2),?n(en) where ?i are
  the molecular orbitals of the system.
• In this expression, electron-electron
  interactions are neglected. More complicated
  expressions treat the interactions between
  electrons for a many-electron system.

5
Molecular Orbital TheoryApproximations to the MO
Theory

• Linear combinations of atomic orbitals, LCAO
  approximation.
• where cij is the weight or
  amplitude of the atomic orbital, ?j, in the
  molecular orbital, ?i.
• Usually the atomic orbitals are known and only
  the coefficients need to be determined.
• ?1c11?1c12?2
• ?2c21?1c22?2 The coefficients can be positive
  or negative.

6
Molecular Orbital TheoryApproximations to the MO
Theory

• Linear combinations of atomic orbitals, LCAO
  approximation.
• Minimal basis set (how many AOs are used).
• Orbitals describing core electrons are ignored
  (coefficients are extremely small).
• The sum includes all valence electron orbitals.
• H, O, and Na

7
Three Conditions for Overlap/Combining of Atomic
Orbitals

• Symmetry of the orbitals must be such that
  regions with the same sign of ? overlap.
• Overlap of s atomic orbital with p atomic
  orbitals
• Energies of the overlapping orbitals must be
  similar.
• The distance between overlapping orbitals must be
  short to be effective.

8
Construction of Molecular Orbitals

• The molecular orbital for diatomic hydrogen, H2.
• ?Nca?(1sa) cb?(1sb) LCAO for bonding
• ?-Nca?(1sa) - cb?(1sb) LCAO for antibonding
• ca and cb are adjustable coefficients to reflect
  the contribution to bonding. Here, we assume
  that the coefficients are the same for the
  bonding and antibonding. Since the orbitals are
  idential, cacb, and these can be set equal to 1.
• The normalization constant, N, is introduced to
  verify that the integral, ,
  is equal to 1.
• The probability density (location of the
  electron), is obtained by squaring the
  wavefunction. The electron has to be located
  somewhere in space.

9
Construction of Molecular Orbitals

• ?(ca?(1sa) cb?(1sb))2 ca2?2(1sa)cb2?2(1sb)
  2cacb?(1sa)?(1sb)
• ca2?2(1sa) is the probability of finding the
  electron on the 1sa atomic orbital.
• 2cacb?(1sa)?(1sb) is the interaction term and
  relates to the probability of finding the
  electron between the atoms.
• The positive term indicates bonding between the
  atoms.

10
Construction of Molecular Orbitals

• ?(ca?(1sa) - cb?(1sb))2 ca2?2(1sa)cb2?2(1sb)-
  2cacb?(1sa)?(1sb)
• ca2?2(1sa) is the probability of finding the
  electron on the 1sa atomic orbital.
• In this situation, -2cacb?(1sa)?(1sb) is the
  interaction term. The electrons in this orbital
  are excluded from the region between the atoms.
  
• The negative term indicates antibonding between
  the atoms. The surface where the electron is
  excluded is called a nodal surface. Make sure
  that you understand Figure 5-1.

11
Finding the Normalization Constant

• The electron must be somewhere in space.

12
Nonbonding Orbitals

• Nonbonding orbitals occur when there is not a
  corresponding orbital of the correct symmetry.
• There is no net overlap between the wavefunctions
  or the wavefunctions are orthogonal.
• The energy of the nonbonding orbital is
  essentially equal to the atomic orbital.
• Illustrate the combining of an s orbital with the
  three p orbitals.
• Nonbonding orbitals may also result because of
  the energy differences of combining orbitals. It
  may also occur if the separation between atoms is
  too great.

13
Formation of Molecular Orbitals from p Atomic
Orbitals

• Illustrate with the different p orbitals.
• ? and ? bonding molecular orbitals.
• The symmetry properties of the orbitals must
  match (e.g. C2 rotation) for the wavefunctions to
  combine.
• The character tables reveal the symmetry of
  particular orbitals indicating if they will or
  will not combine.
• If there is no symmetry match for an orbital, the
  orbital is nonbonding.
• When overlapping regions have the same sign,
  there will be an increased probability in the
  overlap region. If opposite signs exist, the
  combination produces decreased electron
  probability in the overlap region.

14
Formation of Molecular Orbitals from d Atomic
Orbitals

• The same rules apply for the combination of d
  orbitals.
• Types of bonding with d orbitals.
• ? bonding with dz2 orbitals
• ? bonding with dyz and dxz orbitals
• ? bonding (new) with and dxy orbitals.
• This type of bonding is invoked to account for
  quadruple bonds.
• Discuss nodal surfaces and their locations.
  These can be illustrated with orbital viewer
  software.

15
Homonuclear Diatomics

• Figure 5-5 illustrates a simplified molecular
  orbital diagram for homonuclear diatomics.
• Aufbau, Hunds, and Pauli-exclusion
• Bond order ½( of bonding electrons - of
  antibonding electrons.
• g and u stand for gerade and ungerade which
  relate to the inversion symmetry operation.
  Gerade, g, indicates that the orbital is
  symmetric with respect to the inversion
  operation.
• Illustrate this with a few orbitals.

16
Orbital Mixing

• The diagram shown in Fig. 5-5 is not entirely
  correct homonuclear diatomics since other
  orbitals of appropriate symmetry may interact.
• ?ici,2sa?(2sa) ci,2sb?(2sb) ci,2pa?(2pa)
  ci,2pb?(2pb)
• How many molecular orbitals would be produced
  from this combination?
• ?1c1,2sa?(2sa)c2,2sb?(2sb) c3,2pa?(2pa)
  c4,2pb?(2pb)
• This is an example of one of the s-type orbitals.
• The result of this mixing is that the lowest
  energy orbitals move lower and the higher energy
  orbitals move higher in energy. The mixing
  inverts the order of the ?g and ?u bonding
  orbitals (Examine Fig. 5-6).
• When overlapping the AOs the valence shell
  orbitals are commonly the only orbitals
  considered. This is called the minimal basis
  set. Illustrate for N.

17
Electron Configurations for Homonuclear Diatomics

• The mixing effect decreases as a progression is
  made across the periodic table.
• The order of ?g and ?u switch back at O2.
• The electron configurations for some homonuclear
  diatomics.
• H2, C2, and O2
• The frontier orbitals (HOMO and LUMO)
• Important when considering bonding and
  reactivity.
• Paramagnetic versus diamagnetic.

18
Photoelectron Spectroscopy

• Method for determining orbital energies.
• O2(g) h?(photon)?O2(g) e-
• Ionization energy h?-KE (of expelled electron)
• Figures 5-10 and 5-11. The ionization energies
  of the low-energy orbitals are indicated.
• The fine structure is a result of the interaction
  between the electron energy and the vibration
  energy. The peaks with pronounced vibration
  structure are involved strongly in bonding.

19
Molecular Orbitals of Polar Bonds

• A greater nuclear charge shifts the atomic energy
  levels lower in energy. The atomic orbitals have
  different energies and a given MO receives
  unequal contributions from the atomic orbitals.
• Orbital energies are given in Table 5-1 and Fig.
  5-13.
• The contribution of an atomic orbital to a
  molecular orbital is directly related to the
  energy. Generally, the MO has the most character
  of the AO closest to it in energy (i.e. it will
  have the greatest coefficient value).

20
The Molecular Orbitals for CO

• The symmetry of the molecule will be reduced to
  C2v for easier discussion.
• The px and py orbitals have this symmetry.
• Combination of orbitals with the same symmetry.
• s and pz have A1 symmetry (character table).
• px and py possess B1 and B2 symmetry,
  respectively.
• Compare with C?v (px and py, together, behave
  like the E1 representation).
• Energies of the orbitals and MO orbitals were
  derived from Spartan using semi-empirical
  calculations.
• Examine the MO diagram and PES.

21
The Molecular Orbitals for CO

• Molecular orbitals derived from Spartan (semi).
• ?-40.0,2?0.82?(O2s)-0.21?(O2pz)0.41?(C2s)0.34
  ?(C2pz)
• ?-20.7,2?-0.48?(O2s)-0.66?(O2pz)0.56?(C2s)
• ?-16.2,1?-0.85?(O2px)-0.51?(C2px)
• ?-16.2,1?0.85?(O2py)0.51?(C2py)
• ?-13.0,3?-0.46?(O2pz)-0.66?(C2s)0.59?(C2pz)
• ?1.0,1?0.51?(O2px)-0.85?(C2px)
• Discuss and understand the various contributions
  to the MOs. Examine the MOs. The character of
  the MO is determined by the relative amounts of
  contribution from the combining orbitals.
• Atomic orbital contributions with small
  coefficients are ignored (lt0.13).

22
The Molecular Orbitals for CO

• The MOs of greatest interest are the frontier
  orbitals. Identify these in the diagram.
• HOMO contributes electrons in reactions and LUMO
  accepts electrons (future discussion).
• For the HOMO, the greater electron density is on
  the carbon (larger lobe).
• Actual bonding in most compounds is M-C-O. In
  fact, a higher overall electron density is on the
  carbon (show with Spartan). Why?
• Examine the fine structure in the PES. Why is
  there the fine structure originating from the 1?
  orbitals.

23
The Molecular Orbitals for LiF

• Examine the diagram for LiF.
• Small interactions due to poor energy overlap.
• In a real compound, each Li is surrounded by six
  F- ions. A modification to this simple picture
  is needed (developed later).
• Utilize Spartan to construct correct MO energy
  diagrams for HF and HI.

24
Molecular Orbitals for Larger Molecules

• Determine the point group of the molecule.
• Linear D?h ? D2h and C?v ? C2v
• Assign x, y, and z axes.
• The principal rotation axis is chosen as the
  z-axis.
• In non-linear molecules, the y axes of the outer
  atoms point toward the central atom.
• Find the characters of the representations for
  the combination of atomic orbitals on the outer
  atoms.
• Reduce each representation to its irreducible
  representations.
• This determines the groups orbitals or SALCs.

25
Molecular Orbitals for Larger Molecules

• Identify the appearance and character of the
  SALCs by using the projection technique.
• For nonlinear molecules
• Find the orbitals on the central atom with the
  same symmetries as the SALCs.
• Combine the atomic orbitals of the central atom
  with the SALCs of the same symmetry to produce
  the MOs.
• Energy similarities are also considered to
  determine amount of contribution in a particular
  MO.

26
Linear Molecule, FHF-

• The symmetry is D?h but we shall use D2h.
• This retains the symmetry of the p orbitals.
• Examine the D2h character table and identify the
  IRs that the same symmetry as the AOs.
• The z-axis is down the internuclear axis.
• Fig. 5-16 illustrates the group orbitals (i.e.
  SALCs) that form on the fluorine atoms.
• Make sure that you can identify the symmetry or
  IR.
• There does not have to be direct boding between
  the outer orbitals.

27
Linear Molecule, FHF-

• For a linear species, these representations do
  not need to be reduced to IRs (but you can do it
  his way).
• Only the 1s atomic orbital is considered for
  hydrogen, and it has an Ag type symmetry.
• Two SALCs have the correct symmetry to interact
  with the the hydrogen atom.
• The SALC from the 2s atomic orbital is too low in
  energy. The H 1s atomic orbital interacts most
  strongly with the SALC from the 2pz orbitals on
  fluorine.
• (-13.6 eV and -18.7 eV).
• Five of the SALCs do not interact with the
  central atom. These are essentially nonbonding.

28
Linear Molecule, FHF-

• The MO picture illustrates a 3-center, 2-electron
  bond(s). This is different than the Lewis
  approach which utilizes a localized description
  of bonding between 2 atoms.
• Involving a large number of atoms coordinated to
  a central atom usually decreases the bonding
  orbitals even further.

29
Bonding in the CO2 Molecule

• The molecule reduces to D2h symmetry for easier
  analysis.
• Construct the group orbitals as before and
  determine their symmetry (Fig. 5-19).
• Determine the symmetry of the C atomic orbitals
  and group interactions to determine MOs.
• Label the interactions according to symmetry
  taking into account the energy differences.
  Discuss these interactions.
• A large energy difference indicates that the
  interaction is probably insignificant.

30
Bonding in the CO2 Molecule

• Construct the MO diagram from the combinations
  determined previously.
• Notice the multitude of 3-center, 2-electron
  orbitals.
• All the bonding orbitals are occupied as well as
  two nonbonding MOs.
• Identify the ?-type and ?-type bonds. Where is
  the electron density for the nonbonding
  electrons?
• Note Use capital when describing symmetry and
  lower case letter when describing the actual
  orbitals (see diagram).
• A1g versus alg

31
H2O A Nonlinear Molecule

• The point group is C2v.
• C2 axis is determined as the z axis and as the xz
  plane of the molecule.
• Only the 1s orbital will be considered on the
  hydrogens so it is not necessary to assign axes
  to hydrogen.
• Find the representation for the group orbitals.
• The book utilizes transformation matrices to find
  the reducible representation for the SALCS. We
  will use a slightly different approach throughout
  the semester which is simpler (especially for
  larger molecules).

32
H2O A Nonlinear Molecule

• The book largely finds the IRs of ? by
  inspection. We will reduce the RR into its
  component IRs by a systematic approach.
• Transformation matrix exist which will reduce the
  matrix to one consisting of blocks along the
  diagonal. Each of these matrices belongs to an
  IR.
• Find the Irs and normalize the SALCs.

33
H2O A Nonlinear Molecule

• Determine the central atomic orbitals that can
  combine with the SALCs.
• The 2pz and 2s possess A1 symmetry and the px
  orbital possesses B2 symmetry. What about the py
  central atomic orbital?
• Combine the central atomic orbitals with the
  SALCs considering the differences in potential
  energy.
• How many MOs will form?
• How many a1orbitals will form? Why? Roughly
  determine the contribution of the SALC and the
  center pz and 2s atomic orbitals to each al MO
  (discuss this in some detail).

34
H2O A Nonlinear Molecule

• ?1-0.88?(O2s)-0.11?(O2pz)-0.33?(Ha)-0.33?(Hb)
• or c1?(O2s)c2(?(Ha)?(Hb)) (ignoring the O2pz
  contrib.)
• ?20.77?(O2px)0.45?(Ha)-0.45?(Hb)
• Or c3?(O2s)c4(?(Hb)-?(Ha))
• ?3-0.33?(O2s)0.83?(O2pz)0.31?(Ha)0.31?(Hb)
• ?41.00?(O2py)
• ?5-0.34?(O2s)-0.54?(O2pz)0.54?(Ha)0.54?(Hb)
• ?6-0.64?(O2px)0.54?(Ha)-0.54?(Hb)
• The 2py atomic orbital is nonbonding (1b2).
• Examine Table 5-3 and Fig. 5-29. All four MOs
  are different. What are the major differences
  when compared to the Lewis structure? Can also
  view the MOs with Spartan.

35
Ammonia NH3

• Point group is C3v
• The C3 axis is determined as the z-axis.
• Find the RR (only consider the 1s atomic orbitals
  on the hydrogens).
• Determine the IR components of the RR.
• There are three SALCs, one with A1 symmetry and
  two (considered together) with E symmetry. What
  does the E representation indicate? The A1 SALC
  is easy to visualize. What about the two SALCs
  with E symmetry?
• Sum of the squares of the coefficients for each
  AO must equal 1.
• Symmetry of the central atom orbitals matches the
  symmetry of the SALCs. There must be one nodal
  surface in each E SALC.

36
Ammonia NH3

• Determine the symmetry representations of the
  atomic orbitals on the central atom.
• s and pz have A1 symmetry and px and py (as a
  pair) possess E symmetry.
• Combine the central atomic orbitals with the
  SALCs of appropriate symmetry to form MOs.
• Spartan helps in understanding the SALCs and MOs.
• ?1e0.37?(2px)-0.63?(2py)0.58?(Ha)-0.28?(Hb)-0
  .27?(Hc)
• ?1e0.63?(2px)0.37?(2py)0.49?(Hb)-0.48?(Hc)
• SALCs

37
Ammonia NH3

• MO diagram in Figure 5-31
• A1 symmetry orbitals
• A bonding, nonbonding, and antibonding MO
  (roughly)
• E symmetry orbitals
• These are doubly degenerate orbitals which means
  that there is a pair at low energies and a pair
  at high energies (the same energy).
• Lone pair chemistry there is a lone pair of
  electrons largely located on the nitrogen atom.
  This can act as a Lewis base. The LUMO/HOMO
  chemistry will be discussed in detail later.
• BF3 species

38
The Pi Bonding in C4H4

• What is the point group?
• Construction of the SALCs.
• Find the RR and IRs.
• The appearance of the SALCs may not be obvious
  when there is more than a two group atoms.
  Obviously, Spartan can be used to determine the
  wavefunctions and appearance of the orbitals.
  There is another way!!! This involves the use of
  projection operators.

39
Using the Projection Technique

• The most important and frequent use for
  projection operators is to determine the proper
  way to combine atomic wave functions on
  individual atoms into Mos that correspond to the
  molecular symmetry.
• A particular atomic orbital or wavefunction will
  be projected by the symmetry operations. Lets
  perform this for C4H4. This reveals how the
  atomic group orbitals combine to form the SALCs
  of a given symmetry determined earlier.

40
Phosphorus Pentafluoride, PF5

• What is the point group?
• Find the z-axis and determine the y-axes for the
  fluorine ligands.
• Determine the RR representing the group orbitals
  from the fluorine ligands.
• The axial and equatorial have to be considered
  separately since they are not interconverted by
  symmetry.
• Determine the IRs components contained in ?.
• How many SALCs will there be?
• Website http//www.mpip-mainz.mpg.de/gelessus/g
  roup.html

41
Phosphorus Pentafluoride, PF5

• Use the projection technique to determine the
  appearance of the group orbitals.
• Determine the symmetry types of the central atom
  atomic orbitals.
• Combine the SALCs and the atomic orbitals on
  phosphorus to make the MOs.
• Draw the interaction diagram considering the
  potential energy differences.

42
Molecular Shapes

• Determining the actual shapes of molecules using
  the MO approach usually involves the use of
  molecular modeling software (Spartan).
• The overall energy at different bond distances
  and angles is calculated until the minimum is
  found. Any energy that is calculated will be
  equal to or greater than the true energy.

43
Hybrid Orbitals vs. Molecular Orbitals

• The hybrid orbitals point from a central atom
  toward surrounding atoms or lone pairs.
• Therefore, the symmetry properties of a set of
  hybrid orbitals will be identical to the
  properties of a set of vectors with origins at
  the nucleus of the central atom and pointing
  toward the surrounding atoms and lone pairs.
• Td example in the book and PtCl42-