Symmetry and molecules:Symmetry operations (e.g., permutation-inversion operations) for a molecule leavethe molecular Hamiltonian invariant.

They can be organized in symmetrygroups.

The symmetry groups have irreduciblerepresentations.

The wavefunctions for the moleculareigenstates transform according to theirreducible representations of the symmetry group.

Literature: • Per Jensen and P. R. Bunker: The Symmetry of Molecules, in:

"Encyclopedia of Chemical Physics and Physical Chemistry" (J. H.Moore and N. D. Spencer, Eds.), IOP Publishing, Bristol, 2001. pdf-file available!

• P. R. Bunker and Per Jensen: "Molecular Symmetry and Spectroscopy, 2nd Edition," NRC Research Press, Ottawa, 1998 (ISBN 0-660-17519-3).

• P. R. Bunker and Per Jensen: “Fundamentals of Molecular Symmetry," IOP Publishing, Bristol, 2004 (ISBN 0-7503-0941-5).

Symmetry and molecules:We determine the molecular symmetry group forthe molecule considered.

It consist of the feasible permutation-inversionoperations.

We consider here two examples, H2O and PH3.

Permutation-inversion symmetry of H2O

C2v(M) = {E, (12), E*, (12)* }

Permutation-inversion operationscan be multiplied...

C2v(M) = {E, (12), E*, (12)* }

Multiplication table (Rrow Rcolumn)

C2v(M) is a (CNPI) group!

12

E (12) E* (12)* E E (12) E* (12)*

(12) (12) E (12)* E* E* E* (12)* E (12)

(12)* (12)* E* (12) E

Group axioms

•All possible products RS = T belong to the group•Group contains neutral element E (which does nothing)•Each element has inverse element R−1 (R−1R =RR−1 =E) in the group•Associative law (AB)C = A(BC) holds

C2v(M)

C2v(M) = {E, (12), E*, (12)* }

E (12) E* (12)* E E (12) E* (12)*

(12) (12) E (12)* E* E* E* (12)* E (12)

(12)* (12)* E* (12) E

Symmetry example: H2 OThe rovibronic states of H2O are labeled by theirreducible representations A1, A2, B1, B2, of C2v(M)

S Ψrve = χ(S) Ψrve

A1, A2 ,B1, B2 :

CNP group for PH3

12

3

GCNP={E, (12), (13), (23), (123), (132) }

The CNP group has 6 elements – but what are thecyclic permutations (123) and (132) ?

(12) 1 2 3 2 1 3=

(123) 1 2 3 2 3 1=

(132) 1 2 3 3 1 2=

Transposition

Cyclic permutations

{E, (12), (13), (23), (123), (132)}are all possible permutations

(123)(12) 1 2 3 2 1 3= (123)

= 3 2 1 (13)= 1 2 3

(123)(12) = (13)

Examples of „multiplication“

(12)(123) 1 2 3 2 3 1= (12)

= 1 3 2 (23)= 1 2 3

(12)(123) = (23)

CNPI group for PH3

12

3

GCNPI={E, (12), (13), (23), (123), (132), E*, (12)*, (13)*,(23)*, (123)*, (132)*}

The CNPI group has 12 elements

2

3113

2

PH3

Very, veryhighpotentialbarrier

No tunneling through barrier – energies doubly degenerate

We cannot see any effectsof the tunneling through thebarrier

We do not really need the symmetry operations accompanied by this tunneling

They are unfeasible!

GCNPI={E, (12), (13), (23), (123), (132), E*, (12)*, (13)*,(23)*, (123)*, (132)*}

PH3:

GMS ={E, (123), (132), (12)*, (13)*,(23)*}

The six feasible elements are

C3v(M) =

Symmetry example: PH3

The rovibronic states of PH3 are labeled by theirreducible representations A1, A2, E of C3v(M)

S Ψrve = χ(S) Ψrve

A1, A2 :

E :

Ψrve(1)

Ψrve(2)

Ψrve(1)

Ψrve(2)S C21(S)

C11(S) C12(S)C22(S)=

χE(S) = C11(S) + C22(S)

The E states are doubly degenerate

Law of Nature, applied to molecules:The total internal wavefunction of a molecule changes sign under the interchange of two fermions (two nuclei with half-integral spin;Fermi-Dirac statistics)

The total internal wavefunction of a molecule is unchanged by the interchange of two bosons (two nuclei with integral spin;Bose-Einstein statistics)

What is the total internal wavefunction?

Ψint = Ψrve Ψnspin

Ψnspin is a nuclear spin function

Fermi-Dirac statistics for H2OProtons are fermions (I = ½), so Fermi-Dirac statistics apply

1 2

The total internalwavefunction ΦH2O has B1 orB2 symmetry

Total internal wavefunction

ΦH2O = Ψrve Ψnspin

What are the possible Ψnspin functions for H2O ?

H has the functions α = |½, ½⟩, β = |½, − ½⟩

16O has the function δ = |0,0⟩

Four functions in total: α α δ, α β δ, β α δ, β β δ

Nuclear spin functions for H2O

Nuclear spin functions for H2O

B2

A1

A1

A1

E* leaves spin functionsunchanged

Symmetrized spinfunctions

Nuclear spin statistical weights for H2O

The total internal wavefunction ΦH2O has B1 or B2 symmetry

Symmetry labels for H2OMolecules with twospin modifications:

para: lowest weightfactor (here 1)

ortho: highest weightfactor (here 3)

Selection rules:

A1 ↔ A2 (p-p)

B1 ↔ B2 (o-o)

Intensity alternation for H2O

Difficult to observe in H2O (irregular spectrum); much easier in NCCN

Spin statistical weight factorsaccording to Landau and LifshitzShorthand notation for the spin-statistical weight factors,Example H2O

Γrve = A1 + A2 + 3 B1 + 3 B2

Coefficient of irrep is gns for that irrep.

Landau and Lifshitz give a recipe for determining the characters in Γrve

sw

sw

For a permutation-inversion operation P*,

χrve [P*] = 0

For a pure permutation operation P,

where the product contains one factor for each set of na nuclei with spinIa permuted by P

sw

Example H2O P=E:Proton 1 forms a set with I1=1/2Proton 2 forms a set with I2=1/2Oxygen nucleus forms a set withI3=0

χrve = 2×2×2×1 = 8

P=(12):Protons 1 and 2 form a set with I1=1/2 and n1=2Oxygen nucleus forms a set withI2=0

χrve = 2×(-2)×1 = -4Γrve 8 -4 0 0

Γrve = A1 + A2 + 3 B1 + 3 B2

sw

sw

sw

sw

Example CH4

P operations are E, (123),and (14)(23)

Example CH4

P=E:Proton i forms a set with Ii=1/2, i = 1, 2, 3, 4,Carbon nucleus forms a set with I5=0χrve = 2×2×2×2×2×1 = 32

P=(123):Protons 1, 2, 3, form a set with I1=1/2 and n1=3Proton 4 forms a set with I2=1/2Carbon nucleus forms a set with I3=0χrve = 2×2×2×1 = 8

P=(14)(23):Protons 1, 4 form a set with I1=1/2 and n1=2Protons 2, 3 form a set with I2=1/2 and n2=2Carbon nucleus forms a set with I3=0χrve = 2×(-2)×(-2)×1 = 8

sw

sw

sw

Γrve 32 8 8 0 0

Example CH4

Example XNYCYCXN

p p*

p = (12)(34)XN has spin INYC has spin IC

P=EEach nucleus forms a set on its own

P=(12)(34):The two N form a set and the two C form at set

14N12C12C14N 3:6

15N12C12C15N 3:1

14N13C13C14N 21:15

15N12C12C14N 6:6

1 0 0 1

1/2 0 0 1/2

1 1/2 1/2 1

1/2 0 0 1

Nuclear spin statistical weights for CO2

The total internal wavefunction ΦCO2 has A1 or A2 symmetry

In 12C16O2, all nuclei have I = 0

Only one spin function of A1 symmetry

Missinglevels

B1

A1A2B2

Nuclear spin statistical weights for C60

The total internal wavefunction ΦC60 has Ag or Au symmetry

In 12C60, all carbon nuclei have I = 0

Only one spin function of Ag symmetry

gns = 1

All other levelsare missing!

Nuclear spin statistical weights for C60In 13C60, all carbon nuclei have I = 1/2

260 spin functions available

Selection rules for H2OMolecules with twospin modifications:

para: lowest weightfactor (here 1)

ortho: highest weightfactor (here 3)

Selection rules:

A1 ↔ A2 (p-p)

B1 ↔ B2 (o-o)Not th

e whole

truth!

So far: All effects of the nuclear spin on the energieshave been neglected

Nuclear spin only manifest in the spin statistical weights.

Reality: Molecular Hamiltonian contains nuclearspin terms (hyperfine terms) that have extremelysmall contributions to the energy, but can coupleortho and para levels.

Ortho-para interaction in H2O

Ortho-para interaction in H2O

Ortho-para interaction in H2O

Ortho-para interaction in H2O

Intensity stealing (intensity borrowing)Φo is a bright state

Transition (b) steals (or borrows) intensity from Transition (a)

Intensity stealing/borrowing is a common phenomenonin molecular spectra

Φp is a dark state

Symmetry and molecules – really?Current research in symmetry breakdown:

E* is not a universal symmetry operation (paritybreakdown of weak interaction force in nuclei)

Observable in molecules??? Energy differencebetween enantiomers observable spectroscopically???

Permutation symmetry universal???

It is possible to observe „missing transitions“ –for example in CO2 ???