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hνif = Ef – Ei = ΔEif
S(f ← i) = ∑A | ∫ Φf* μA Φi dτ |2
ODME of H and μA μfi =
Spectroscopy
Quantum Mechanics
f
iMMMM
M
∫ Φf* μA Φi dτ
Group Theory and Point Groups
C2v = {E,C2x,σxz,σxy}z
x 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 C2x σxz σxy
The Complete Nuclear Permutation Inversion Group
CNPI Group of C3H4 has 3! x 4! x 2 = 288 elementsallene
+y
(12) E* 1 1
1 -1 -1 -1 -1 1
E 1111
(12)* 1 -1 1-1
A1
A2
B1
B2
R = ±
A2 x B1 = B2, B1 x B2 = A2, B1 x A2 x B2 = A1
∫ΨaHΨbdτ = 0 if symmetries of Ψa and Ψb are different.
∫ΨaμΨbdτ = 0 if symmetry of product is not A1
ψ is a wavefunction of A2 symmetry ψ generates the A2 representation
Eψ=+1ψ (12)ψ =+1ψ E*ψ=-1ψ (12)*ψ=-1ψ
RH = HR
For example:
EIGENFUNCTIONS TRANSFORM IRREDUCIBLY
Labelling Energy Levels Using CNPIG Irreps
R2 = E
(12) E* 1 1
1 -1 -1 -1 -1 1
E 1111
(12)* 1 -1 1-1
A1
A2
B1
B2
Symmetry of μA
∫Ψa*μAΨbdτ = 0 if symmetry of ψa*μAψb is not A1,
i.e., if the symmetry of the product ΨaΨb is not = symm of μA
Symmetry of H
Using symmetry labels and the vanishingintegral theorem we deduce that:
∫Ψa*HΨbdτ = 0 if symmetry of Ψa*HΨb is not A1,i.e., if the symmetry of Ψa is not the same as Ψb
The Vanishing Integral Theorem∫f(τ)dτ = 0 if symmetry of f(τ) is not A1
ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8
Ψ1
Ψ2
Ψ3
Ψ4
Ψ5
Ψ6
Ψ7
ψ8
A1 A2 B1
A1
A2
B1
0 0
000 0
0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
. . .
. . .
. . .. . .. . .. . .
. .
. .
ψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8
Ψ1
Ψ2
Ψ3
Ψ4
Ψ5
Ψ6
Ψ7
ψ8
A1 A2 B1
A1
A2
B1
0 0
000 0
0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
. . .
. . .
. . .. . .. . .. . .
. .
. .
AllowedTransitions
A1 ↔ A2
B1 ↔ B2
A1
A2
B1Connectedby A2
Like using a big computer to calculate E from exact H, we can, in principle, use
the CNPI Group to label energy levels and to determine which ODME vanish
for any molecule
Using the CNPI Groupfor any molecule
BUT there is one big problem: Superfluous symmetry labels
Character Table of CNPI group of CH3F
12 elements 6 classes
6 irred. reps
GCNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)*
GCNPI
Using the CNPI Group approach for CH3F
μA A1”H A1’
A1’
A1’’
A2’
A2’’
E’
E’’
E’ + E’’
A1’’ + A2’
A1’ + A2’’
The CNPI Group approachCH3F
Allowedtransitionsconnectedby A1”Block diagonal H-matrix
E’ + E’’
A1’
A1’’
A2’
A2’’
E’
E’’
A1’’ + A2’
A1’ + A2’’
CH3F
Allows for tunnelingbetween two
VERSIONS
The CNPI Group approach
E’ + E’’
E’ + E’’
The number of versions of the minimum is given by: (order of CNPI group)/(order of point group)
For CH3F this is 12/6 = 2
For benzene C6H6 this is 1036800/24 = 43200,and using the CNPI group each energy level wouldget as symmetry label the sum of 43200 irreps.
Clearly using the CNPI group givesvery unwieldy symmetry labels.
3! x 2 C3v group has 6 elements
2
3113
2
Very, veryhighpotentialbarrier
No observed tunneling through barrier
Only NPI OPERATIONS FROM IN HERE
CH3F
(12) superfluousE* superfluous(123),(12)* useful
F F
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}For CH3F:
GMS ={E, (123), (132), (12)*, (13)*,(23)*}
The six feasible elements are
superfluous
superfluous
useful
useful
If we cannot see any effectsof the tunneling through thebarrier then we only needNPI operations for one version. Omit NPI elementsthat connect versions since they are not useful; they are
superfluous.
Character table of the MS group of CH3F
E (123) (12)
1 2 3
A1 1 1 1
A2 1 1 1
E 2 1 0
E (123) (12)* (132) (13)* (23)*
μA A2
H A1
(12),(13),(23)E*,(123)*,(132)*are unfeasibleelements of theCNPI Group
Use this group to block-diagonalize Hand to determine which transitions areforbidden
Using CNPIG versus MSG for PH3
E
A2
A1
Can use either to determine if an ODME vanishes.But clearly it is easier to use the MSG.
E’ + E’’
A1’’ + A2’
A1’ + A2’’
CNPIG MSG
E’ + E’’ E
The subgroup of feasible elements forms a group called
THE MOLECULAR SYMMETRY GROUP(MS GROUP)
Unfeasible elements of the CNPI groupinterconvert versions that are separated
by an insuperable energy barrier
Superfluous
useful
----------- End of Lecture Two -------
16
We are first going to set up the Molecular Symmetry Group for several non-tunneling (or “rigid”)molecules. We will notice a strange“resemblance” to the Point Groupsof these molecules. We will examinethis “resemblance” and show how ithelps us to understand Point Groups.
E (12) E* (12)*
A1 1 1 1 1
A2 1 1 1 1
B1 1 1 1 1
B2 1 1 1 1
C2v E C2x σxz σxy E (12) E* (12)*
A1 1 1 1 1
A2 1 1 1 1
B1 1 1 1 1
B2 1 1 1 1
C2v(M)
z
x
(+y)
1 2
H2O
The C2v and C2v(M) character tables
E (123) (12)
1 2 3
A1 1 1 1
A2 1 1 1
E 2 1 0
E (123) (12)
1 2 3
A1 1 1 1
A2 1 1 1
E 2 1 0
E (123) (12)* (132) (13)* (23)*
E C3 σ1
C32 σ2
σ3
C3v(M) C3v
1
2
3
CH3F
The C3v and C3v(M) character tablesF
H N1 N2 N3
MSG is {E,E*}
E E*A’ 1 1A” 1 -1
All P and P* are unfeasible (superfluous)
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
HN3 has6 versions12/2 = 6
3!x2 Cs
H N1 N2 N3
MSG is {E,E*}
E E*A’ 1 1A” 1 -1
All P and P* are unfeasible (superfluous)
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
HN3 has6 versions12/2 = 6
3!x2 Cs
PG is {E,σ}
E σA’ 1 1A” 1 -1
H3+
Point group also has 12 elements: D3h
3
1
2
GCNPI={E, (12), (13), (23), (123), (132),
E*, (12)*, (13)*,(23)*, (123)*, (132)*}
Therefore only 1 version and MSG = CNPIG
+
Character Table of MS group H3+
12 elements 6 classes
6 irred. reps
GCNPI E (123) (23) E* (123)* (23)* (132) (31) (132)* (31)* (23) (23)*
GCNPI
Character table for D3h point group
E 2C3 3C'2 σh 2S3 3σvlinear,
rotationsquadratic
A'1 1 1 1 1 1 1 x2+y2, z2
A'2 1 1 -1 1 1 -1 Rz
E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1
A''2 1 1 -1 -1 -1 1 z
E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
Number of elements in CNPIG = 3! x 4! x 2 = 288
H5
C1
C2
C3
H4
H7
H6
The MSG of allene is:
{E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*,
(4657)(13)*, (4756)(13)*}
The allene molecule C3H4
Point group is D2d has 8 elements
Thus there are 288/8 = 36 versions
Number of elements in CNPIG = 3! x 4! x 2 = 288
H5
C1
C2
C3
H4
H7
H6
The MSG of allene and PG are: E C2 C2’ C2’ σd σd
{E, (45)(67), (13)(46)(57), (13)(47)(56), (45)*, (67)*, S4 S4
(4657)(13)*, (4756)(13)*}
The allene molecule C3H4
Point group is D2d has 8 elements
Thus there are 288/8 = 36 versions
Character table for D2d point group
E 2S4 C2 (z) 2C'2 2σdlinear,rotations
quadratic
A1 1 1 1 1 1 x2+y2, z2
A2 1 1 1 -1 -1 Rz
B1 1 -1 1 1 -1 x2-y2
B2 1 -1 1 -1 1 z xy
E 2 0 -2 0 0 (x, y) (Rx, Ry) (xz, yz)29
USE OF GROUP THEORY ANDSYMMETRY IN SPECTROSCOPY
Point Group
CNPI GroupMS Group
Spoilt by rotation and tunneling
Superfluoussymmetry ifmany versions
RH=HR, thereforecan symmetry labelenergy levels
Use moleculargeometry
Use energyinvariance RH = HR
USE OF GROUP THEORY ANDSYMMETRY IN SPECTROSCOPY
Point Group
CNPI GroupMS Group
Spoilt by rotation and tunneling
Superfluoussymmetry ifmany versions
RH=HR, thereforecan symmetry labelenergy levels
Use moleculargeometry
Use energyinvariance RH = HR
USE OF GROUP THEORY ANDSYMMETRY IN SPECTROSCOPY
Point Group
CNPI GroupMS Group
Spoilt by rotation and tunneling
Superfluoussymmetry ifmany versions
RH=HR, thereforecan symmetry labelenergy levels
Use moleculargeometry
Use energyinvariance RH = HRFor rigid (nontunneling)
molecules MSG and PGare isomorphic. Leads toan understanding of PGsand how they can label energy levels.
Black are instantaneouspositions in space. White are equilibrium positions.N.B. +z is 1→2.
Then do (12). Note that axes have moved.Rotational coordinates aretransformed by MS group.
Black are instantaneouspositions in space. White are equilibrium positions.N.B. +z is 1→2.
Then do (12). Note that axes have moved.Rotational coordinates aretransformed by MS group.
MS Group and Point Group of H2O
E = p0 R0 E
(12) = p12 Rx C2x
E* = p0 Ry xz
(12)* = p12 Rz xy
MS Group Point GroupC2v(M)C2v(M) C2vC2v
MS Group and Point Group of H2O
E = p0 R0 E
(12) = p12 Rx C2x
E* = p0 Ry xz
(12)* = p12 Rz xy
MS Group Point GroupC2v(M)C2v(M) C2vC2v
Jon Hougenand what theMSG is calledby somepeople
The point group can be used to symmetrylabel the vibrational and electronic states of non-tunneling molecules.
RPG Hev = Hev RPG
Point group operations rotate/reflectelectronic coords and vib displacementsin a non-tunneling (rigid) molecule.
47
The rotational and nuclear spincoordinates are not transformedby the elements of a point group
The MSG can be used to classifyThe nspin, rotational, vibrational
and electronic states of any molecule, including those
molecules that exhibit tunneling splittings (“nonrigid” molecules).
WE USE THE ETHYL RADICAL AS AN EXAMPLE OF A NONRIGID MOLECULE
The ethyl radical
Number of elements in CNPIG = 5! x 2! x 2 = 480
The MSG of internally rotating ethyl is thefollowing group of order 12:{E, (123),(132),(12)*,(23)*,(31)*,(45),(123)(45),(132)(45),(12)(45)*,(23)(45)*,(31)(45)*}
H4
Cb
H5
a
The MSG of rigid (nontunneling) ethyl is {E,(23)*}
There are still plenty of unfeasible elementssuch as (14), (23), (145), E*, etc.
MS group can change with resolution and temperature
Recent theoretical calcs bySousa-Silva, Polyanski, Yurchenko and Tennyson
From UCL, UK.Yield the tunneling splittings given on the next slide
Schwerdtfeger, Laakkonen and Pekka Pyykkö,J. Chem. Phys., 96, 6807 (1992)
Barrier height = 12300 cm-1
THE PH3 MOLECULE AS AN EXAMPLE
MS group can change with resolution
and temperature
For NH3 MSG is usually D3h(M)
For PH3 MSG is usually C3v(M)
The subgroup of feasible elements form a group calledTHE MOLECULAR SYMMETRY GROUP
Unfeasible elements of the CNPI groupinterconvert versions that are separated
by an insuperable energy barrier
MS group is used to symmetry label the rotational, ro-vibrational, rovibronic andnuclear spin states of any molecule.
MS group can change with resolution and T
SUMMARY 54
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