Symmetry Elements

The symmetry of a molecule can be described by 5 types of symmetry elements.

Symmetry axis: an axis around which a rotation by 360/n results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system.
Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σd). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).
Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF4) where the inversion center is at the Xe atom, and benzene (C6H6) where the inversion center is at the center of the ring.
Rotation-reflection axis: an axis around which a rotation by 360/n, followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn. Examples are present in tetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis.
Identity, abbreviated to E, from the German 'Einheit' meaning unity. This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, its consideration is necessary for the group-theoretical machinery to work properly. It is so called because it is analogous to multiplying by one (unity).

Objectives

This exercise consists of two activities.
Activity 1: To use software to draw and manipulate structures in order to identify the generators and determine their Point Groups.
Activity 2: To Use Group Theory and Spectroscopy to identify the isomers of bisglycinatocopper(II) hydrate (Cu(gly)2.H2O)

Activity 1

Using Argus Lab, draw each molecule.

Then, by studying the molecule in various orientations, identify the generators and use the flowchart to predict the point group. Consult the character tables and list the remaining symmetry operations in each group. Use diagrams to show the location of all symmetry elements.

The molecules to be studied include:

BrF₅
ferrocene (eclipsed)
S₈ (puckered)
1,5-dibromonaphthalene
Ni(cyclobutadiene)₂ (staggered)
CH₄
CHCl₃
CH₂Cl₂
acetylene
cis- and trans-[Cu(gly₂)]
cis- and trans-1,2-dichloroethylene (DCE)

Activity 2a

To identify the observed and calculated differences between the spectra of the isomers of 1,2-dichloroethylene (DCE) based on Group Theory.
You are provided with the IR and Raman spectra for DCE. Identify the major differences in the spectra between the cis- and the trans- isomers. Using Group Theory, and showing all reasoning, rationalise these differences.

Activity 2b

To identify the observed and calculated differences between the spectra of the isomers of bis-(glycinato)copper(II).hydrate (Cu(gly)₂).H₂O, based on Group Theory.
You are provided with the IR and Raman spectra for the cis- and trans- isomers of Cu(gly)₂.H₂O. Identify the major differences in the spectra between the cis- and the trans- isomers. Using Group Theory, and showing all reasoning, rationalise these differences.

At the end of this exercise you should be able to:

Use ACDLabs/Arguslab to draw chemical structures.
Find the generators and determine the point group of any molecule.
Determine which irreducible representation of a Point Group labels the symmetry of a particular molecular vibration.
Predict the number of bands expected in the IR or Raman spectra of simple molecules.
Use SGT to differentiate between cis and trans isomers.

Point group	Symmetry elements	Simple description, chiral if applicable	Illustrative species
C₁	E	no symmetry, chiral	CFClBrH, lysergic acid
C_s	E σ_h	mirror plane, no other symmetry	thionyl chloride, hypochlorous acid
C_i	E i	Inversion center	anti-1,2-dichloro-1,2-dibromoethane
C_∞v	E 2C_∞ σ_v	linear	hydrogen chloride, dicarbon monoxide
D_∞h	E 2C_∞ ∞σ_i i 2S_σ ∞C₂	linear with inversion center	dihydrogen, azide anion, carbon dioxide
C₂	E C₂	"open book geometry," chiral	hydrogen peroxide
C₃	E C₃	propeller, chiral	triphenylphosphine
C_2h	E C₂ i σ_h	planar with inversion center	trans-1,2-dichloroethylene
C_3h	E C₃ C₃² σ_h S₃ S₃⁵	propeller	Boric acid
C_2v	E C₂ σ_v(xz) σ_v'(yz)	angular (H₂O) or see-saw (SF₄)	water, sulfur tetrafluoride, sulfuryl fluoride
C_3v	E 2C₃ 3σ_v	trigonal pyramidal	ammonia, phosphorus oxychloride
C_4v	E 2C₄ C₂ 2σ_v 2σ_d	square pyramidal	xenon oxytetrafluoride
D₂	E C₂(x) C₂(y) C₂(z)	twist, chiral	cyclohexane twist conformation
D₃	E C₃(z) 3C₂	triple helix, chiral	Tris(ethylenediamine)cobalt(III) cation
D_2h	E C₂(z) C₂(y) C₂(x) i σ(xy) σ(xz) σ(yz)	planar with inversion center	ethylene, dinitrogen tetroxide, diborane
D_3h	E 2C₃ 3C₂ σ_h 2S₃ 3σ_v	trigonal planar or trigonal bipyramidal	boron trifluoride, phosphorus pentachloride
D_4h	E 2C₄ C₂ 2C₂' 2C₂ i 2S₄ σ_h 2σ_v 2σ_d	square planar	xenon tetrafluoride
D_5h	E 2C₅ 2C₅² 5C₂ σ_h 2S₅ 2S₅³ 5σ_v	pentagonal	ruthenocene, eclipsed ferrocene, C₇₀ fullerene
D_6h	E 2C₆ 2C₃ C₂ 3C₂' 3C₂ i 3S₃ 2S₆³ σ_h 3σ_d 3σ_v	hexagonal	benzene, bis(benzene)chromium
D_2d	E 2S₄ C₂ 2C₂' 2σ_d	90° twist	allene, tetrasulfur tetranitride
D_3d	E C₃ 3C₂ i 2S₆ 3σ_d	60° twist	ethane (staggered rotamer), cyclohexane chair conformation
D_4d	E 2S₈ 2C₄ 2S₈³ C₂ 4C₂' 4σ_d	45° twist	dimanganese decacarbonyl (staggered rotamer)
D_5d	E 2C₅ 2C₅² 5C₂ i 3S₁₀³ 2S₁₀ 5σ_d	36° twist	ferrocene (staggered rotamer)
T_d	E 8C₃ 3C₂ 6S₄ 6σ_d	tetrahedral	methane, phosphorus pentoxide, adamantane
O_h	E 8C₃ 6C₂ 6C₄ 3C₂ i 6S₄ 8S₆ 3σ_h 6σ_d	octahedral or cubic	cubane, sulfur hexafluoride
I_h	E 12C₅ 12C₅² 20C₃ 15C₂ i 12S₁₀ 12S₁₀³ 20S₆ 15σ	icosahedral	C₆₀, B₁₂H₁₂^2-

C_2v Point Group

Abelian, 4 irreducible representations

Subgroups of C_2v point group: C_s, C₂

Character table for C_2v point group

	E	C₂ (z)	σ_v(xz)	σ_v(yz)	linear, rotations	quadratic
A₁	1	1	1	1	z	x², y², z²
A₂	1	1	-1	-1	R_z	xy
B₁	1	-1	1	-1	x, R_y	xz
B₂	1	-1	-1	1	y, R_x	yz