MOLECULAR SYMMETRYAND SPECTROSCOPY
P. R. Bunker and Per Jensen:
Molecular Symmetry and Spectroscopy,
2nd Edition, 3rd Printing,
NRC Research Press, Ottawa, 2012.
P. R. Bunker and Per Jensen:
Fundamentals of Molecular Symmetry,
Taylor and Francis, 2004.
Learning about molecular symmetry from books
C$ 23.95
P. R. Bunker and Per Jensen:
Molecular Symmetry and Spectroscopy,
2nd Edition, 3rd Printing,
NRC Research Press, Ottawa, 2012.
P. R. Bunker and Per Jensen:
Fundamentals of Molecular Symmetry,
Taylor and Francis, 2004.
Learning about molecular symmetry from books
C$ 23.95
Chapter 1 in here
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν
frequency
Transmittance = Itr(ν)/I0(ν)
(number of waves per cm)frequency
Transmittance = Itr(ν)/I0(ν)
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν
f
iM
hνif = Ef – Ei = ΔEif
νif
Absorption can only occur at resonance
E = Einternal
= Erve-spin
EACH “LINE’’ SIGNALS A RESONANCE;
Molecule undergoes a “transition.’’
Transmittance = Itr(ν)/I0(ν)
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν
f
iM
hνif = Ef – Ei = ΔEif
νif
Absorption can only occur at resonance
E = Einternal
= Erve-spin
EACH “LINE’’ SIGNALS A RESONANCE;
Molecule undergoes a “transition.’’
[“Weak radiation”: Transmittance ≠ f(I0)]
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν, ν or λ ~
~ ~
ν = 1/ λ ~ = ν /c
frequency wavenumber wavelength
cm-1
Transmittance = Itr(ν)/I0(ν) ~ ~
(number of waves per cm)
Transmittance = Itr(ν)/I0(ν)
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν, ν or λ ~
f
iM
hνif = Ef – Ei = ΔEif
νif
Absorption can only occur at resonance
~ ~
~ ~
E = Einternal
= Erve-spin
hνif = Ef – Ei = ΔEif ν = 1/ λ ~ = ν /ccm-1
Transmittance = Itr(ν)/I0(ν)
I0(ν) molecules Itr(ν)
Absorption spectrum: a plot of transmittance vs. ν, ν or λ ~
f
iM
hνif = Ef – Ei = ΔEif
νif
Absorption can only occur at resonance
~ ~
~ ~
E = Einternal
= Erve-spin
hνif = Ef – Ei = ΔEif ν = 1/ λ ~ = ν /ccm-1
νif /c = Ef /hc - Ei /hc = ΔEif /hc
Divide by hc
wavenumber
Term values
Term value difference
cm-1
cm-1
cm-1
I0(ν) moleculesconc = c* Itr(ν) ~ ~
Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]~ ~ ~
Absorption coefficient
hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f
iM
νif
Pierre_Bouguer
Beer-Lambert Law (weak radiation):
ℓ
I0(ν) moleculesconc = c* Itr(ν) ~ ~
Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]~ ~ ~
Absorption coefficient
hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f
iM
νif
Pierre_Bouguer
Beer-Lambert Law (weak radiation):
ℓ
http://www.google.com/
(youtube feynman names)
I0(ν) moleculesconc = c* Itr(ν) ~ ~
Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]~ ~ ~
Absorption coefficient
hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f
iM
νif
Pierre_Bouguer
The exponential attenuation law (weak radn):
ℓ
I0(ν) moleculesconc = c* Itr(ν) ~ ~
Transmittance = Itr(ν)/I0(ν) = exp[-ℓ c* ε(ν)]~ ~ ~
Absorption coefficient
I(f ← i) = ∫8π3 Na______
(4πε0)3hcF(Ei ) S(f ← i) = νif
~line
ε(ν)dν~ ~
Integrated absorption coefficient (i.e. intensity) for a line is:
hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f
iM
νif
Pierre_Bouguer
ℓ
Frequency Factor[1 – exp(-hυif /kT)]
Boltzmann Factor Line Strength
Stimulated Emission Factor
Rstim(f→i)
| ∫ Φf* μA Φi dτ |2 ∑A=X,Y,Z
gi exp [-Ei /kT ]
∑j gj exp [-Ej /kT ]
The exponential attenuation law (weak radn):
This depends on the wavefunctions that describe eachof the levels involved in the transition. They occur in theelectric dipole transition moment integral.
μA is the component of the molecular electric dipolemoment along the space fixed A (= X, Y or Z) axis:
μA = Σ Cre Arr
Charge onparticle r
A coordinateof particle r
S(f ← i) = | ∫ Φf* μA Φi dτ |2 ∑A=X,Y,Z
The Line Strength
Oscillating E and B fields contribute to the energy of emradiation. Above we have given the intensity of resonantlyabsorbed electric field energy in terms of the electric dipoletransition moment.
Molecules also absorb magnetic field energy; there is amagnetic dipole transition moment. But typically mdtm ≈ 10-5 edtm. Usually ignored, just as we ignore the electric quadrupole tm.
However sometimes we need the mdtm or the eqtm.ESR and NMR are magnetic dipole transitions. Electric field energy absorbed by low pressure H2 gas is elec. quadrupole.
AN ASIDE
Here ends the summary of Ch.1 15
Proportional to S(f ← i)
hνif = Ef – Ei = ΔEif POSITION: At a resonancef
iM
νif
INTENSITY: Line strength
μA = Σ Cre ArrS(f ← i) = | ∫ Φf* μA Φi dτ |2
~
∑A=X,Y,Z
CAN SIMULATE A SPECTRUM BY
CALC OF E and Φ
Quantum mechanics
Ch.2
Quantum mechanics
Schrödinger equation
For discussion of full Hincluding spin terms see pages 126-131
For spin-free H see pages 26-29 and 32-33
Q.M. Wave
Quantum mechanics
For discussion of full Hincluding spin terms see pages 126-131
For spin-free H see pages 26-28 and 32-33
Spin-free H is Hrve (rovibronic H)
The spin-free (rovibronic) Hamiltonian
Vee + VNN + VNe
THE GLUE
In a world of infinitely powerful computers we could solve the Sch. equation numerically and that would be that. However, we usually have to start by making approximations. We thenselectively correct for the approximations made.
(after separating translation)
Off-diagonal matrix elements (ODME)
An pxp matrix A has elements Amn
A11 A12 A13 … A1p
A21 A22 A23 … A2p
A31 …………… A3p
A41
.
.Ap1 …………… App
Diagonalelements
Off-diagonalelements
A matrix is ‘diagonal’ if all off-diagonal elements are zero
Quantum mechanics and off-diagonal matrix elements
(ODME) Schrödinger equation
Eigenvalues and eigenfunctions are found by diagonalization of a matrix with elements
Eigenfunctions of an approximate H are “basis” functions
Exact Ej and ψj are obtained from ψn0 by
p 21-26
ODME has m ≠ n; perturbation Problem 2.5on page 41
Ch 2
0 0 0 0
ψm0
ψn0
Em0
En0
APPROXIMATE OR ZEROTHORDER SITUATION
Δmn0 = Em
0 - En0
Effect of nonvanishing ODME
ψm0
ψn0
Δmn0 = Em
0 - En0
Em
En
- - - - -
- - - - -
+δ
-δ
δ ~ Hmn2 / Δmn
0
,ψm
,ψn
ψm ~ ψm0 + [Hmn /Δmn
0 ] ψn0
Em0
En0
Using PerturbationTheory
ψelec ψvib ψrot ψns
Born-Oppapprox
NeglectCent. DistCoriolis
NeglectOrtho-para
mixing
MolecularOrbitals
Harmonicoscillator
Rigidrotor
Uncoupledspins
Ψe-v-r-ns0
(espin and Slater determinants)
ODME of H
μfi = ∫ (Ψf )* μA Ψi dτ
ODME of μA
To simulate a spectrum we need energies and line strengths i.e. weneed to calculate:
The Role of ApproximationsTo solve the Sch. Eq. we make approximations and then correct for them. In so doing we introduce concepts thatenable us to UNDERSTAND molecules.Key approximationis the BORN-OPPENHEIMER APPROXIMATION
Concepts that come about because we make approximations:
Electronic state, molecular orbital, electronic configuration,potential energy surface, molecular structure, force constants,electronic angular momentum, vibrational state, vibrational angular momentum, rotational state, Coriolis coupling, centrifugal distortion,…
We could call the approach that aims to solve the Sch. Eq. numerically without approximations using a gigantic computer “The few-concepts big-computer” approach.
BO Approx. Pages 43-47
When I am grumpy and sarcastic I like to call it “The small-brain big-computer approach.”
I(f ← i) = ∫8π3 Na______
(4πε0)3hcF(Ei ) S(f ← i) Rstim(f→i) = νif
~line
ε(ν)dν~ ~
Integrated absorption coefficient (i.e. intensity) for a line is:
hνif = Ef – Ei = ΔEif Absorption can only occur at resonance f
iM
νif
ODME of H
μfi = ∫ (Ψf )* μA Ψi dτODME of μA
Use Q. Mech. to calculate:
Here ends the summary of Chapters 1 and 2
P. R. Bunker and Per Jensen:
Fundamentals of Molecular Symmetry,
Taylor and Francis, 2004.
P. R. Bunker and Per Jensen:
Molecular Symmetry and Spectroscopy,
2nd Edition, 3rd Printing,
NRC Research Press, Ottawa, 2012.
To buy it go to:
http://www.crcpress.comDownload pdf file from
www.chem.uni-wuppertal.de/prb
Chapter 1 (Spectroscopy)Chapter 2 (Quantum Mechanics) andSection 3.1 (The breakdown of the BO Approx.)
The first 47 pages:
28
Now for Molecular Symmetry
http://www.google.com/
Wikipedia molecular symmetry
Now for Molecular Symmetry
http://www.google.com/
Wikipedia molecular symmetry
No discussion of the fundamentals of how or why the geometrical symmetry of the equilibrium structureof a moleule in a particular electronic state allows oneto do the things it says you can do. No discussion ofwhat the “rotations” and “reflection” symmetryoperations do to molecular coordinates.
Molecular Symmetry(as it should be explained)
Knowing the symmetry labels it is easy to determine which ODME of H
and have to be zero.
An important use of symmetry is to put “symmetry labels” on the zeroth
order energy levels.
μA
Molecular Symmetry(as it should be explained)
GROUP THEORYFERMI:“Just a bunch of
definitions”
andPOINT GROUPS
(uses geometrical symmetry)
The Point Group of H2O
z
x
(+y)
The point group of H2O consists of the four symmetry operations
E, C2x, xz, and xy
It is called the C2v point group
The identityoperation
Examples of point group symmetry
H2O
CH3F
C60
C3H4
C2v
C3v
D2d
Ih120 symmetry operations
z
x
(+y)σxz
C2x
z
x
(+y)
z
x
(+y)
+
+
-
σxy
Successive application = “multiplication”
σxy C2xσxz=
+ -
+
z
x
(+y)
+
Successive application = “multiplication”
σxy C2xσxz=
+
C2v multiplication table
E C2x σxz σxy
E E C2x σxz σxy
C2x C2x E σxy σxz
σxz σxz σxy E C2x
σxy σxy σxz C2x EA “Group” contains all
products of its membersand it contains E
z
x
(+y)
+
Successive application = “multiplication”
σxy C2xσxz=
+
E C2x σxz σxy
E E C2x σxz σxy
C2x C2x E σxy σxz
σxz σxz σxy E C2x
σxy σxy σxz C2x EIS NOT A GROUP
A “Group” contains allproducts of its members
and it contains E E C2x σxz{ }
z
x
(+y)
+
Successive application = “multiplication”
E C2x C2x=
+
C2 multiplication table
E C2x σxz σxy
E E C2x σxz σxy
C2x C2x E σxy σxz
σxz σxz σxy E C2x
σxy σxy σxz C2x ETHE C2 GROUP – A SUBGROUP OF C2v
A “Group” contains allproducts of its members
and it contains E
PH3 at equilibrium: The C3v point group
Symmetry operations:C3v = {E, C3, C3
2, 1, 2, 3 }
3
12
Multiplication table for C3v
C32 = σ2σ1
C3 = σ1σ2
Multiplication is not necessarily commutative
A matrix group
1 0
0 1
21
23
23
21
21
23
23
21
1 0
0 1
21
23
23
21
21
23
23
21
´M1 = M4 =
´ M5 =
´ M6 =
´M2 =
´M3 =
M1
M2
M3
M4
M5
M6
M1
M1
M2
M3
M4
M5
M6
M2
M2
M3
M1
M6
M4
M5
M3
M3
M1
M2
M5
M6
M4
M4
M4
M5
M6
M1
M2
M3
M5
M5
M6
M4
M3
M1
M2
M6
M6
M4
M5
M2
M3
M1
Multiplication table for the matrix group
M3 = M5 M4
E C3 C32 σ1 σ2 σ3
This matrix group forms a “representation” of the C3v group
Multiplication tableshave the ‘same shape’
M1 M2 M3 M4 M5 M6
C32 = σ2σ1
M3 = M5 M4
Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations
E C3 C32 σ1 σ2 σ3
This matrix group forms a “representation” of the C3v group
Multiplication tableshave the ‘same shape’
M1 M2 M3 M4 M5 M6
C32 = σ2σ1
M3 = M5 M4
Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations
E C3 C32 σ1 σ2 σ3
This matrix group forms a “representation” of the C3v group
Multiplication tableshave the ‘same shape’
M1 M2 M3 M4 M5 M6
C32 = σ2σ1
M3 = M5 M4
Also (1,1,1,1,1,1} and {1,1,1,-1,-1,-1} are representations
E C3 C32 σ1 σ2 σ3
This matrix group forms an “irreducible representation” of the C3v group
Multiplication tableshave the ‘same shape’
M1 M2 M3 M4 M5 M6
C32 = σ2σ1
M3 = M5 M4
The characters of this irreducible rep.
1 0
0 1
21
23
23
21
21
23
23
21
1 0
0 1
21
23
23
21
21
23
23
21
´M1 = E M4 = 1
´ M5 = 2
´ M6 = 3
´M2 = C3
´M3 = C32
C3v C3v
2
-1
-1
0
0
0
Characters of the 2d matrix irrep of C3v
E (123) (12)
1 2 3
A1 1 1 1
A2 1 1 1
E 2 1 0
The 2D representation M = {M1, M2, M3, ....., M6}of C3v is the irreducible representation E. In thistable we give the characters of the matrices.
E C3 σ1
C32 σ2
σ3
Elements in the same class have the same characters
Two 1Dirreduciblerepresentationsof the C3v group
3 classes and 3 irreducible representations
Character table for the point group C3v
E (123) (12)
1 2 3
A1 1 1 1
A2 1 1 1
E 2 1 0
The 2D representation M = {M1, M2, M3, ....., M6}of C3v is the irreducible representation E. In thistable we give the characters of the matrices.
E C3 σ1
C32 σ2
σ3
Elements in the same class have the same characters
Two 1Dirreduciblerepresentationsof the C3v group
3 classes and 3 irreducible representations
The matrices of the A1 + E reducible rep.
21
23
23
21
21
23
23
21
1 0
0 1
21
23
23
21
21
23
23
21
´M1 = E M4 = 1
´ M5 = 2
´ M6 = 3
´M2 = C3
´M3 = C32
A1 + E C3v A1 + E C3v
1 0 00 1 00 0 1
1 0 00
0
1 0 00
0
1 0 00
0
1 0 000
1 0 00
0
‘
‘
‘
‘
‘
‘
A similarity transformation would make the ‘origin’ of each of these reducible reps lessobvious…..but the characters would give theirreduction away!
A-1Mi’A = Ni
(A-1Mi’A)(A-1Mj’A) = A-1Mi’(A A-1)Mj’A
= A-1Mi’Mj’A
NiNj =
The matrices N form an equivalent representationand the character of Ni is the same as that of Mi’Both representations reduce to A1 + E.
‘
The matrices of the A2 + E reducible rep.
21
23
23
21
21
23
23
21
1 0
0 1
21
23
23
21
21
23
23
21
´M1 = E M4 = 1
´ M5 = 2
´ M6 = 3
´M2 = C3
´M3 = C32
A2 + E C3v A2 + E C3v
1 0 00 1 00 0 1
-1 0 00
0
1 0 00
0
1 0 00
0
-1 0 000
-1 0 00
0
‘
‘
‘
‘
‘
‘
‘
‘
‘
‘
‘
‘
Irreducible representationsThe elements of irrep matrices satisfy the„Grand Orthogonality Theorem“ (GOT).
We do not discuss the GOT here, but we list threeconsequences of it: • Number of irreps = Number of classes in the group.
• Dimensions of the irreps, l1, l2, l3 … satisfy
l12 + l2
2 + l32 + … = h,
where h is the number of elements in the group.
• Orthogonality relation
Character table for the point group C2v
E (12) E* (12)*
A1 1 1 1 1
A2 1 1 1 1
B1 1 1 1 1
B2 1 1 1 1
E C2 σyz σxy
4 classes and 4 irreducible representations
z
x
(+y)
“Group”
A set of operations that is closed wrt “multiplication”
“Point Group”
All rotation, reflection and rotation-reflection operationsthat leave the molecule (in its equilibrium configuration)“looking” the same.
“Matrix group”
A set of matrices that forms a group.
“Irreducible representation”
An irreducible matrix group that maps onto the group.
Definitions a la Fermi:
As we shall see later, these are used as ‘‘symmetry labels‘‘ on energy levels.
We can use these symmetry labels to determine which ODME of H and μA
are zero.
IRREDUCIBLE REPRESENTATIONS
The “why” and the “how” will be explained
That’s it for Point Groups
Ch.6(15 pages)
This is where chemistry courses describing the Fundamentals of molecular symmetry usually end
That’s it for Point Groups
Ch.6(15 pages)
This is where chemistry courses describing the Fundamentals of molecular symmetry usually end
59
PAUSEGroups, Point Groups, Matrix Groups and Irreducible Representations.
That’s it for Point Groups
Ch.6(15 pages)
This is where chemistry courses describing the Fundamentals of molecular symmetry usually end
THERE CAN BE PROBLEMS IF WE TRY TO USE POINT
GROUP SYMMETRY.
BUT
How do we use point group symmetry when a molecule rotates and distorts?
H3+
D3h
How do we use point group symmetry when a molecule rotates and distorts?
H3+
D3h C2v
How do we use point group symmetry when a molecule rotates and distorts?
H3+
D3h C2vCH4 also
2
3
113
2
How do we use point group symmetry when the molecule tunnels?
NH3
C3vD3h
What are the symmetriesof B(CH3)3, CH3.CC.CH3, (CO)2, (NH3)2,(C6H6)2,…?
Nonrigid molecules (i.e. moleculesthat tunnel) are a problemif we try to use a point group.
What should we do if we study transitions (or interactions) between electronic states that have different point group symmetries at equilibrium?
Also
Further
People claim to be able to use thepoint group (or its rotational subgroup) to determine nuclear spin statistical weights. Thisis not really possible since point group operations do not just permute nuclei.
Point groups used for classifying:
The electronic states for any moleculeat a fixed nuclear geometry and
The vibrational states for molecules,called “rigid” molecules, undergoing infinitesimal vibrations about a unique equilibrium structure.
Rotations andreflections
Permutationsand the inversion
J.T.Hougen, JCP 37, 1422 (1962); ibid, 39, 358 (1963)H.C.Longuet-Higgins, Mol. Phys., 6, 445 (1963)P.R.B. and Per Jensen, JMS 228, 640 (2004) [historical introduction]
To understand how we use symmetrylabels and where the point groupgoes wrong we must studywhat we mean by “symmetry”
70
Symmetry not from geometry since molecules are dynamic
• Centrifugal distortion
eg. H3+ or CH4 dipole
moment
• Nonrigid molecules: eg. ethane, ammonia, (H2O)2, (CO)2,…
• Breakdown of BOA: eg. HCCH* - H2CC
Also symmetry appliesto atoms, nuclei and fundamental particles. Geometrical point group symmetry is not possible for them.
We need a more general definition of symmetry
We define symmetry operationsas being those operations that leave the
energy of the molecule unchanged.
Doing this allows us to treat molecular symmetryon the same footing as it is used elsewhere in Physics.It makes it easy to see how energy levels are labelled.
And finally it allows us to understand what Point Group symmetry operations really do to a molecule
and how they too also involve energy invariance.
72
• Uniform Space-----------Translation• Isotropic Space----------Rotation • Identical electrons------Permute electrons• Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q,s,I) (-p,-q,s,I)
Symmetry Operations (energy invariance)
We might say something about Charge Conjugation and Time Reversal (p,q,s,I) (-p,q,-s,-I)at the end
• Uniform Space-----------Translation• Isotropic Space----------Rotation • Identical electrons-------Permute electrons• Identical nuclei-----------Permute identical nuclei • Parity conservation-----Inversion (p,q) (-p,-q) E*
Symmetry Operations (energy invariance)
CNPI group = Complete Nuclear Permutation Inversion Group
EXAMPLE:The CNPI group for H2O is {E, (12),E*,(12)*}
Permutation of protons inversion (12)* = (12)E* = E*(12)
The Complete Nuclear Permutation Inversion (CNPI) group
for the water molecule is {E,(12)} x {E,E*} = {E, (12), E*, (12)*}
H1 H2
O e+
H2 H1
O e+ H1 H2
Oe-(12) E*
(12)*
Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinates
E* Inverts coordinates of nuclei and electrons.Does not change spins.
The CNPI Group for the Water Molecule
The Complete Nuclear Permutation Inversion (CNPI) group
for the water molecule is {E,(12)} x {E,E*} = {E, (12), E*, (12)*}
H1 H2
O e+
H2 H1
O e+ H1 H2
Oe-(12) E*
(12)*
Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinates
E* Inverts coordinates of nuclei and electrons.Does not change spins.
The CNPI Group for the Water Molecule
CNPI group of OCO?
The Complete Nuclear Permutation Inversion (CNPI) group
for the water molecule is {E,(12)} x {E,E*} = {E, (12), E*, (12)*}
H1 H2
O e+
H2 H1
O e+ H1 H2
Oe-(12) E*
(12)*
Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinates
E* Inverts coordinates of nuclei and electrons.Does not change spins.
The CNPI Group for the Water Molecule
CNPI group of H2?
The Complete Nuclear Permutation Inversion (CNPI) group
for the water molecule is {E,(12)} x {E,E*} = {E, (12), E*, (12)*}
H1 H2
O e+
H2 H1
O e+ H1 H2
Oe-(12) E*
(12)*
Nuclear permutations permute nuclei (coordinates and spins).Do not change electron coordinates
E* Inverts coordinates of nuclei and electrons.Does not change spins.
The CNPI Group for the Water Molecule
CNPI group of CH2D13CFClBr?
The CNPI group of PH3
1
2
3
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
N! ways of permuting N identical nuclei
GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*}
Number of elements = 3! x 2 = 6 x 2 = 12
1 is replaced by 22 is replaced by 33 is replaced by 1
The CNPI group of O3
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
N! ways of permuting N identical nuclei
GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*}
Number of elements = 3! x 2 = 6 x 2 = 12
The ozonemolecule
O OO
1 3
2
The CNPI group of HD12C=13CH2
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
N! ways of permuting N identical nuclei
GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*}
Number of elements = 3! x 2 = 6 x 2 = 12
H H12C 13C
D H
1 2
3
I
F
H
D
SubstitutedAllene molecule C1
C2
C3
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
GCNPI = {E, (12), (13), (23), (123), (132)} x {E, E*}
Number of elements = 3! x 2 = 6 x 2 = 12
The CNPI Group of a Substituted Allene Molecule
The same CNPI group as for PH3, O3, H3+ and HD12C=13CH2
GCNPI = {E, (12), (13), (23), (123), (132)} x{E, (45)} x {E, E*}
(45), (12)(45), (13)(45), (23)(45), (123)(45), (132)(45),
E*, (12)*, (13)*, (23)*, (123)*, (132)*,
(45)*, (12)(45)*, (13)(45)*, (23)(45)*, (123)(45)*, (132)(45)*}
Number of elements = 3! x 2! x 2 = 6 x 2 x 2 = 24
The CNPI Group of a slightly less substituted Allene Molecule
I
H5
D
C1
C2
C3
H4
= {E, (12), (13), (23), (123), (132),
How many elements in the CNPI groups of SF6, C2H6, C6H6, C6H5CH3, C60
(use ! In answers)
I
H5
D
C1
C2
C3
H4
Number of elements in CNPI group = 3! x 2! x 2
CH3FNumber of elements in CNPI group = 3! x 2
C3H2ID
The number of elements in a group is called the “order” of the groupand is denoted by the letter “h”
CNPI GROUPDONE
Ch.7
An important number
Molecule PG h(PG) h(CNPIG) h(CNPIG)/h(PG)
H2O C2v 4 2!x2=4 1
PH3 C3v 6 3!x2=12 2
Allene D2d 8 4!x3!x2=288 36 C3H4
Benzene D6h 24 6!x6!x2=1036800 43200C6H6
86This number means something!