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The determination of point groups of molecules

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					The determination of point groups of
            molecules
only one rotational   two σv but no σh mirror planes means
axis = C2                  point group is C2v




      The point group of the water molecule is C2v
             Naming point groups:
   The name of the point group has information about the
   symmetry elements present. The letter is the rotational group
   and the subscript number after the letter indicates the order
   of the principal rotational axis (e.g. 3-fold or 4 fold etc.):

                                         A ‘D’ indicates an n-fold
                                         principal rotation axis
     A ‘C’ indicates only                plus n 2-fold axes at
     one rotational axis                 right angles to it



   C3                 C3v             D4d                D4h
3-fold rotational has σv but          4-fold      d = no       ‘h’ indicates
     axis          no σh mirror       principal   σh mirror     a σh mirror
                planes in a C group      axis      plane          plane
          Naming point groups (contd.):

    A subscript ‘h’ means that there is a σh mirror plane at
    right angles to the n-fold principal axis:

                   C4   principal axis        C3   principal axis

                                                          only one
                                                          of the three
                                                          σv planes
                                                          is shown

                                     σh                     σv

                                D4h                              D3d
A subscript ‘d’ (or v for C groups) means there is no σh mirror
plane, but only n σv mirror planes containing the principal Cn axis.
     Naming platonic solids:

     Platonic solids:
     T = tetrahedral = 4 three-fold axes
     O = octahedral = 3 four-fold axes
      I = icosahedral = 6 five-fold axes




                                               C60
                                            ‘bucky-ball’
                                           or ‘Fullerene’

Td         Oh                Ih
Flow chart for determining point groups.
The point group of the carbon dioxide
              molecule
              i      We start at the top of the
                         flow-chart, and can see that
                   C∞    the CO2 molecule is linear,
                         and has a center of inversion
                         (i) so it is D∞h. Note the C∞
                D∞h       principal rotation axis.
             Other linear molecules:
 The top row of linear molecules all have a center of
 inversion (i) and so are D∞h.

               i                D∞h              i


       N2              O2               F2              H2




            HC≡N                HI                   C≡O
The bottom row have no           C∞v         All have a C∞
i and so are C∞v                                  axis
     The Platonic solids:
tetrahedron   octahedron   icosahedron




    Td         Oh          Ih         C60
                                    ‘buckyball’
     The Cs point group:
             I             σ
Cs
             C
                 Cl        chloro-difluoro-iodo-
     F                           methane
         F
Most land animals have bilateral symmetry,
    and belong to the Cs point group:




  Cs        Mirror planes (σ)      Cs
               The C1 point group:

Molecules that have no symmetry elements at all except
the trivial one where they are rotated through 360º and
remain unchanged, belong to the C1 point group. In
other words, they have an axis of 360º/360º = 1-fold, so
have a C1 axis. Examples are:
                I
                                 I        N
                          Cl
                C                                H
        Br
                                 Cl
   C1                     F                          C1
     Bromo-chloro-fluoro-iodo-        chloro-iodo-amine
         methane
The division into Cn and Dn point groups:

After we have
decided that
there is a
principal rotat-
ional axis, we
come to the
red box. If there
are n C2 axes
at right angles
to the principal                       Dn
axis, we have a
Dn point group,
If not, it is a Cn                     Cn
point group.
          The Cn point groups:
The Cn point groups all have only a single rotational
axis, which can theoretically be very high e.g. C5 in
the complex [IF6O]- below. They are further divided
                         into Cn, Cnv, and Cnh point
          C5                   groups. The Cn point
groups        O                        have no other
symmetry
  iodine                                     elements,
the Cnv point groups                               have
also n mirror planes
F containing the Cn rotational
           axis, while the Cnh point
           groups also have a σh mirror
        F         plane at right angles to the
               [IF6O]- principal rotational axis.
The point group of the water molecule

               We start at the top of the
               flow-chart, and can see that
               the water molecule is not
               linear, and is not tetrahedral (Td),
               octahedral (Oh), or icosahedral,
               (Ih) so we proceed down the chart
C2   Yes, there is a principal Cn axis,
     so we proceed down the chart, but
     in answer to the next question, there
     are no further C2 axes at right angles
     to the principal axis, which is the only
     axis, so we proceed down the chart
                            C2         there is no σh plane
C2           C2
                                       at right angles to
                      σv          σv   the C2 axis, but
                                       there are two σv
                                       planes containing
                                       the C2 axis.




     The point group of the water molecule is C2v
        Other Cnv molecules:

water                                     ammonia
                    σv

                    C2v
                              C3v
                                            σv


          C4v      V
                                σv



           Vanadyl tetrafluoride (VOF4)
        Some more C2v molecules:

           C2    σv             C2       σv            C2   σv


       P                    S
                                                   C




                 σv                      σv                 σv
Phosphorus iodo-         sulfur tetra-           carbonyl
tetrafluoride (PF4I)   fluoride (SF4)         chloride (COCl2)
                         The Cn point groups:
    These have a Cn axis as their only symmetry element. They
    generally resemble propellers which have the front and back
    different. Important examples are (hydrogens omitted for clarity):


   triphenyl                                         C3        Cobalt(III)
 phosphine                                                  tris-glycinate
viewed down                                                 viewed down
  C3 axis           C3                                         C3 axis


               C3                                         C3
                           C3                  C3
   triphenyl                                                   Cobalt(III)
 phosphine                                                  tris-glycinate
viewed from                                                 viewed from
  the side                                                     the side
            The Dnh point groups:




  C4           four C2
                                    mirror plane
principal      axes at
               rt. angles           at rt. angles
  axis         to C4 axis           to C4 axis

                  C2
                 C2                         σh
                       C2   C2

                                     D4h
Examples of molecules belonging to Dnh point
                 groups:
     C2                               C3         C3
                      C3




           D2h             D3h             D3h        D3h
C4
                                 C5
                 C4                              C5




          D4h          D4h                 D5h        D5h
Benzene, an example of the D6h point group:
    C6                C2
principal axis                               C2




                           C2               C2
     C6          σh
                                σv                 σv

                                                        D6h

                                                      C6
                                                  principal axis
                               C6
                           principal axis
              The Dn point groups:

                                               these have a principal
                                               n-fold axis, and n
                                               2-fold axes at right
                                               angles to it, but no
                                               mirror planes.

                      C2 principal axis
                                          C
                  N
         C2                                        [Cu(en)2]2+ complex
                  Cu
     N                                               with H-atoms
                                                   omitted for clarity.
                                                    (en = ethylene
                                                      diamine)
C2                                            D2
     Some further views of the symmetry elements
            of [Cu(en)2]2+, point group D2 :
                                C2 principal axis
                                                    [Cu(en)2]2+ complex
                      C2                              with H-atoms
                D2                                  omitted for clarity.
                                                     (en = ethylene
                                                       diamine)
 C2 principal    C2
      axis
                C2                   C2 principal         C2 principal
                                         axis                 axis
C2                         C2                                        C2
                                C2                  C2
          Some views of the symmetry elements of
                [Co(en)3]3+, point group D3.

                   C2
                                                      C3 principal axis
C2




     C2

     C3                             D3   C2
principal axis                           axis

           view down the C3 axis             view down one of the
           of [Co(en)3]3+ showing         three C2 axes of [Co(en)3]3+
             the three C2 axes.               at right angles to C3
          Other examples of the D3 point group

                                   C3                    C2
     C2                C2     principal axis

                                        C2
                                                                    C2

C2



                             D3                                      D3
          [Co(oxalate)3]3-                     [Co(bipyridyl)3]3+
  Some cobalt(III) complexes belonging to
           the D3 point group:
                                                       C2
    C2                         C2                               CH 3
                      3+                          3+
          NH 2 H                                   H3C           O          CH 3
                                       N
                 2
   H2N         N                                            O          O
                                  N         N
         Co                           Co                        Co
                                                            O          O
   H2N           N                N         N
                 H2                                H3C           O          CH 3
          NH 2                         N

                                                                CH 3



tris(ethylenediamine)      tris(2,2’-bipyridyl)   tris(acetylacetonato)
cobalt(III) cation          cobalt(III) cation    cobalt(III)

                                                                       D3
  Comparison of C3 and D3 tris(chelates)
                               no C2 axis at
                               this point

    C2
                  D3                      O               C3
                          3+
                                               O     H2
            NH2 H
                 2                  H2N              N
     H2N        N
           Co                                 Co
                                      O              O
     H2N          N                                            O
                  H2                           NH2
            NH2                 O



tris(ethylenediamine)cobalt(III) tris(glycinato)cobalt(III)
Molecules belonging to the Dnd point groups

 These have mirror planes parallel to the principal
 axis, but not at right angles to it.
                                       C5 axis
    C3 axis          σv planes
                     contain the
                     principal
                     axis



                   D3d                                D5d
 Staggered form
 of ethane                 Staggered form of ferrocene
                      The D4d point group:
                                   C4 principal axis             σv
       C2        C2                                    σv                σv
C2                    C2
                                             C2                               σv



            C4                                              C4
                                 [ZrF8 ]4-             principal axis
     principal axis
                            Square antiprism                            D4d
 As predicted by VSEPR, the [ZrF8]4- anion has a square anti-prismatic
 structure. At left is seen the C4 principal axis. It has four C2 axes at
 right angles to it, so it has D4 symmetry. One C2 axis is shown side-on
 (center). There are four σv mirror planes (right), but no mirror plane at
 right angles to C4, so the point group does not rate an h, and is D4d.
        [K(18-crown-6)]+, an example of a D3d
                   point group:
                           C3
                      principal axis        C3 principal axis       σv
   C2            C2
                                 K+
                                                          σv
                         C2
                          C2                         C2
                                   C2                     σv
                                              D3d
The complex cation [K(18-crown-6)]+ above is an important structure that
has D3d symmetry. It has a C3 principal axis with 3 C2 axes at right
angles to it, as well as three σv mirror planes that contain the C3 axis,
but no σh mirror plane (because it’s not flat, as seen at center), so is D3d.
Some Point groups

				
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posted:11/29/2011
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