CHM664A : Modern Physical Methods in Chemistry
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CHM664A : Modern Physical Methods in Chemistry Manabendra Chandra Department of Chemistry Indian Institute of Technology Kanpur Lecture - 11
Transcript of CHM664A : Modern Physical Methods in Chemistry
CHM664A : Modern Physical Methods in Chemistry
Manabendra ChandraDepartment of Chemistry
Indian Institute of Technology Kanpur
Lecture - 11
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
A quick recap of the previous lecture
o We gave many examples, including some tricky ones, to illustrate our approach of determining point group.
In this lecture
• We will learn about stereographic projection and how to use this technique
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Stereographic projection of molecular point groups
o The Schönflies notation of a point group indicates only the minimum symmetry elements that point group.
o Full list of symmetry operations can be found using Stereographic projection
o Stereographic projection or stereogram is basically the projection of points of 3-D space on a given plane
o It also helps in constructing the symmetry multiplication table of a point group without considering any particular molecule or object that belongs to that point group
o To explain the method of construction of such diagrams let’s first see how points on the surface of a sphere can be represented on a plane
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Stereographic projection of molecular point groups
Projection of two points P and P’ on the equatorial plane
Different symbol of the projection of two points (one on each hemisphere) on the equatorial plane
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Stereographic projection of molecular point groups
How to draw stereographic projection?
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Construction of stereogram of a given point group
Step - I
When there is NO σh plane When there σh plane EXISTS
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Construction of stereogram of a given point group
Step - II
v Principal C2 axis : at the center of the equatorial planev Principal C3 axis : at the center of the equatorial planev Principal C4 axis : at the center of the equatorial planev Principal C5 axis : at the center of the equatorial planev Principal C6 axis : at the center of the equatorial planev C2 axis ⟂ the principal axis : a broken diametric line with a at each endv Vertical mirror plane : a solid diametric linev S4 axis :v S6 axis :
Draw the symmetry elements of the point group on the plane of the equatorial circle using the following symbols
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Construction of stereogram of a given point groupStep - III
The Schönflies notation of the point group indicates some fundamental symmetry operations. For example, C4h means that this point group has !C4, !C42= !C2, !C43, !C44 =#E, and %σh as the fundamental operators.
To generate the other operators of the group, we need to
(1) Take an arbitrary point P on the surface of the upper half of the auxiliary sphere and project it at P1 on the equatorial plane.
(2) Carry out all the fundamental symmetry operations indicated by the Schönfliesnotation and project the resultant images (Note: If #RP = P’, then P’ is called the image of P under #R) on the equatorial plane.
(3) Carry out the same symmetry operations on each of the image points generated in the previous step and project the resultant images on the equatorial plane.
(4) Stop only when no further new points are generated. This gives us the desired stereogram
Total number of image points (including P1) in the stereogram should be equal to the order of the group
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Illustration : Stereographic projection of C4 point group
!C4 on P1 !C4 on P2
i.e., !C42 on P1
!C 4on
P 3
i.e.,!C 4
3 on P 1
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
No further points can be generated
Illustration : Stereographic projection of C3v point group
!C3 on P1 and its images #σv(1),#σv(1),#σv(1)
onP1
!C3(P1) = P2, !C3(P2) = P3
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Complete stereogram of C3v point group
Total number of image points = 6.
Note, h = 6
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
!C3 on P1 and its images !C2(1), !C2(2), !C2(3) on
P1, P2 and P3 and their images
#σd(1),#σ d
(1),#σ d(1)
onthep
ointssofar
generat
ed
Illustration : Stereographic projection of D3d point group
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Complete stereogram of D3d point group
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
!σ hon
all th
e
gene
rated
point
s
"C3 on P1 and its images"C2(1), "C2(2), "C2(3) on
P1, P2 and P3 and their images
Illustration : Stereographic projection of D3h point group
CHM664A : Modern Physical Methods in Chemistry © Dr. M. Chandra, IIT Kanpur
Complete stereogram of D3h point group
