![The Dn pointgroups:
C2
C2
C2 principal axis
D2
these have a principal
n-fold axis, and n
2-fold axes at right
angles to it, but no
mirror planes.
[Cu(en) ]2+ complex2
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
N
N
Cu
C](https://image.slidesharecdn.com/mrpresenta-thermodynamics-copy-200502110732/85/Group-theory-Part-1-43-320.jpg)
![Some further views of the symmetry elements
of [Cu(en)2]2+, point group D2 :
C2
[Cu(en)2]2+ complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C2
C2
C2
C2
C2 principal
axis
C2 principal axis
C2 principal
axis
C2 principal
axis
C2
C2
C2
D2](https://image.slidesharecdn.com/mrpresenta-thermodynamics-copy-200502110732/85/Group-theory-Part-1-44-320.jpg)
![C2
C2
C2
C3 principal axis
Some views of the symmetry elements of
[Co(en)3]3+, point group D3.
C3
principal axis
C2
axis
view down the C3 axis
of [Co(en)3]3+ showing
the three C2 axes.
D3
view down one of the
three C2 axes of [Co(en)3]3+
at right angles to C3](https://image.slidesharecdn.com/mrpresenta-thermodynamics-copy-200502110732/85/Group-theory-Part-1-45-320.jpg)
![The D4d point group:
C2 C2
C2 C2
C4
principal axis
C4 principal axis
σv
C2
σv
σv
σv
C4
principal axis[ZrF8]4-
Square antiprism
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 isD4d.
D4d](https://image.slidesharecdn.com/mrpresenta-thermodynamics-copy-200502110732/85/Group-theory-Part-1-47-320.jpg)
![[K(18-crown-6)]+, an example of a D3d
point group:
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.
D3d
σv
σv
C3
principal axis
K+
3C principal axis
σv
C2 C2
C2
C2
C2
C2](https://image.slidesharecdn.com/mrpresenta-thermodynamics-copy-200502110732/85/Group-theory-Part-1-48-320.jpg)
Group theory - Part -1
- 1. 1 Group Theory Part - I Mr.M.RAGU Assistant Professor of Chemistry Vivekananda College Tiruvedakam West –
- 2. Introduction Symmetry Symmetry element and Symmetry operation Multiplication Table Point group Class and order Examples. OutlineOutline
- 3. 3 History Mathematical study of symmetry is called group theory
- 4. 4 Symmetry
- 5. 5 Symmetry element A geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations can be carried out
- 6. 6 Symmetry operation A movement of a body such that the appearance after the operation is indistinguishable from the original appearance (if you can tell the difference, it wasn’t a symmetry operation)
- 7. 7 Symmetry Element and Symmetry operation Symmetry Element Symmetry Operation 1.Identity (E) E itself “do nothing” 2.Proper Axisis of Rotation (Cn) Axis of rotation (principal and non-principal) 3. Plane of Symmetry(σ) Plane of reflection (σh, σv, σd) 4.Centre of symmetry (i) Center of inversion 5.Improper Axisis of Rotation(Sn) Rotation followed by reflection
- 8. 8 Identity (E) The identity operator leaves a molecule unchanged. It is applied for all molecule with any degree of symmetry or asymmetry. All objects can be operated upon by the identity Eg: NH3
- 9. 9 Proper Axis of Rotation (Cn) A molecule is said to possess a proper axis of rotation of order n, Cn .If rotation about the axis by an angle ɵ=2∏/n leaves the molecule in a configuration which is indistinguishable from the original one. Proper axis of rotation => Cn where n = 2, 180o rotation n = 3, 120o rotation n = 4, 90o rotation n = 6, 60o rotation n = (1/ )o rotation principal axis of rotation, Cn
- 10. 10 Proper Axis of Rotation (Con) There are two types of rotation axis 1.Principle axis 2.Subsidary axis
- 11. 11 Proper Axis of Rotation (Con) n-fold Rotation Water has a 2-fold axis of rotation. When rotated by 180o, the hydrogen atoms trade places, but the molecule will look exactly the same.
- 12. n-fold Axis of Rotation Ammonia has a C3 axis. Note that there are two operations associated with the C3 axis. Rotation by 120o in a clockwise or a counterclockwise direction provide two different orientations of the molecule.
- 13. Mirror Planes The reflection of the water molecule in either of its two mirror planes results in a molecule that looks unchanged.
- 14. Mirror Planes The benzene ring has a C6 axis as its principal axis of rotation. The molecular plane is perpendicular to the C6 axis, and is designated as a horizontal plane, σh. C6 .
- 15. • If reflection of all parts of a molecule through a plane produces an indistinguishable configuration, the plane is a plane of symmetry 15 Plane of Symmetry (σ)
- 16. 16 There are three types of mirror planes sv => vertical mirror plane which contains the principle axis. sh => horizontal mirror plane which is perpendicular to the principle axis. sd => mirror plane bisects dihedral angle made by the principal axis of Plane of Symmetry (σ) Con
- 17. • The inversion operation projects each atom through the center of inversion, and across to the other side of the molecule. 17 Centre of Symmetry (i)
- 18. Centre of Symmetry (i) Con
- 19. Rotation about an axis, followed by reflection through a plane perpendicular to this axis If rotation through a bout an axis, followed by reflection through a plane perpendicular to that axis, yields an indistinguishable configuration, the axis is an n-fold rotation– reflection axis, also called an n-fold improper rotation axis. It is denoted by the symbol Sn. 19 19 Improper axis of Rotation (Sn)
- 20. This operator applies a clockwise rotation on the molecule followed by a reflection in a plane perpendicular to that axis of rotation. Sn = σhCn Example: Methane
- 21. 21 A collection of symmetry elements are called group. There are 1. Finite group 2. Infinite group 3. Abelian group 4. Non – Abelian group 5. Order Centre of Symmetry (i) Con Groups
- 22. Finite Group: A group containing a finite or limited number of elements is called a finite group. Ex: C2v group order is 4 C3v group order is 6 Infinite Group: A group containing a infinite or unlimited number of elements is called a infinite group. Ex: Linear molecules belong to this group. Abelian Group: A group in which all the elements commute is called an Abelian group. Two elements A and B commute if AB = BA. The set of numbers. The set of symmetry operations of water molecule represents an Abelian group since they obey the commutative law. 22
- 23. Non - Abelian Group: A group is said to be non-Abelian if all the elements do not commute with one another. The symmetry operations of the ammonia molecule provide an example for the non – Abelian group. E, C’3.δV δV’ and δV’’ do not obey the commute law. 23
- 24. Cyclic group: A group is said to be cyclic if all the elements of a group can be generated from one element. Ex : H2O2 Order: The total number of elements are called order. 24
- 25. • Definition: All the symmetry operations present in a molecule form a group is called Point group. 25 Point Group
- 26. 1. E 2. Cn 3. σn 4.i 5. Sn 26 Multiplication Table C2v water molecule
- 27. 27 Multiplication Table C3v Ammonia molecule
- 28. Point Group = the set of symmetry operations for a molecule Group Theory = mathematical treatment of the properties of the groupwhich can be used to find properties of the molecule Assigning the Point Group of a Molecule 1. Determine if the molecule is of high or low symmetry by inspection A. Low Symmetry Groups Point Groups
- 29. B. High Symmetry Groups
- 30. 2. If not, find the principle axis 3. If there are C2 axes perpendicular to Cn the molecule is in D If not, the molecule will be in C or S 4. If sh perpendicular to Cn then Dnh or Cnh If not, go to the next step 5. If s contains Cn then Cnv or Dnd If not, Dn or Cn or S2n 6. If S2n along Cn then S2n 7. If not Cn
- 31. Flow chart for determining point groups.
- 32. The determination of point groups of molecules axis = C2 only one rotational two σv but no σh mirror planes means point group is C2v The point group of the water molecule is C2v
- 33. The point group of the carbon dioxide molecule We start at the top of the flow-chart, and can see that the CO2 molecule is linear, and has a center of inversion (i) so it is D∞h. Note the C∞ principal rotation axis. i C∞ D∞h
- 34. Other linear molecules: HI C≡O All have a C∞ axis N2 O2 F2 H2 D∞h C∞v i i HC≡N The bottom row have no i and so are C∞v The top row of linear molecules all have a center of inversion (i) and so are D∞h.
- 36. Most land animals have bilateral symmetry, and belong to the Cs pointgroup: Mirror planes (σ)Cs
- 37. The C1 point group: chloro-iodo-amineBromo-chloro-fluoro-iodo- methane 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 Br F Cl C I Cl H N C1C1
- 38. Other Cnv molecules: C2v C3v C4v ammoniawater σv σv σv Vanadyl tetrafluoride (VOF4) V
- 39. These have a Cn axis as their only symmetry element. Important examples are (hydrogens omitted for clarity): The Cn point groups: C3C3 C3 C3 C3 C3triphenyl phosphine viewed down C3 axis Cobalt(III) tris-glycinate viewed down C3 axis triphenyl phosphine viewed from the side Cobalt(III) tris-glycinate viewed from the side
- 40. The Dnh pointgroups: σh four C2 axes at rt. angles to C4 axis C2 C2 C2 C2 C4 principal axis mirror plane at rt. angles to C4 axis D4h
- 41. Examples of molecules belonging to Dnh point groups: D2h D3h D3h D3h D4h D4h D5h D5h C2 C3 C3 C3 C4 C4 C5 C5
- 42. C6 principal axis C2 C2 C2 C6 C2 σv σv Benzene, an example of the D6h point group: σh C6 principal axis C6 principal axis D6h
- 43. The Dn pointgroups: C2 C2 C2 principal axis D2 these have a principal n-fold axis, and n 2-fold axes at right angles to it, but no mirror planes. [Cu(en) ]2+ complex2 with H-atoms omitted for clarity. (en = ethylene diamine) N N Cu C
- 44. Some further views of the symmetry elements of [Cu(en)2]2+, point group D2 : C2 [Cu(en)2]2+ complex with H-atoms omitted for clarity. (en = ethylene diamine) C2 C2 C2 C2 C2 principal axis C2 principal axis C2 principal axis C2 principal axis C2 C2 C2 D2
- 45. C2 C2 C2 C3 principal axis Some views of the symmetry elements of [Co(en)3]3+, point group D3. C3 principal axis C2 axis view down the C3 axis of [Co(en)3]3+ showing the three C2 axes. D3 view down one of the three C2 axes of [Co(en)3]3+ at right angles to C3
- 46. Molecules belonging to the Dnd point groups C3 axis These have mirror planes parallel to the principal axis, but not at right angles to it. C5 axis Staggered form of ethane Staggered form of ferrocene σv planes contain the principal axis D3d D5d
- 47. The D4d point group: C2 C2 C2 C2 C4 principal axis C4 principal axis σv C2 σv σv σv C4 principal axis[ZrF8]4- Square antiprism 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 isD4d. D4d
- 48. [K(18-crown-6)]+, an example of a D3d point group: 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. D3d σv σv C3 principal axis K+ 3C principal axis σv C2 C2 C2 C2 C2 C2
- 49. 49 References
- 50. 50 Thank you Mr.M.RAGU Assistant Professor of Chemistry Vivekananda College Tiruvedakam West –