Chapter 7 – Symmetry and Spectroscopy – Molecular Vibrations – p ...

Chapter 7 – Symmetry and Spectroscopy – Molecular Vibrations – p. 1 -

7. Symmetry and Spectroscopy – Molecular Vibrations

7.1 Bases for molecular vibrations

We investigate a molecule consisting of N atoms, which has 3N degrees of freedom. Taking

the translations (3) and rotations (3 for non-linear, 2 for linear molecules) into account, we

obtain 3N-6 (5) vibrational degrees of freedom for the non-linear (linear) case. In order to

describe those, we choose an arbitrary set of basis vectors in order to describe the 3N

displacements from the equilibrium position (the choice is arbritrary, but a choice having

simple transformation properties might be advantageous).

Example: BF3 or CO32-

12 degrees of freedom: 3 translations, 3 rotations, 6 vibrations

displacement basis d:

7.2 Normal modes

We are interested in the collective vibrational modes of the molecules. These collective

modes Q with identical frequency and phase are denoted as normal modes.

As a pictorial representation of the modes, we indicate the set maximal displacements, e.g.:


As we will see later, normal modes can be classified with respect to the irreps of the point

group of the molecule.

In analogy to the strategy followed in chapter 5 for the electronic structure of molecules, we

can transform the displacement basis of dimension 3N into a symmetry adapted set of

displacement coordinates D:

dαDrr

= with

=

M

r

2

1

1

1

xzyx

d

In the next step we transform the basis D into the basis of normal modes

DβQrr

= .

In analogy to chapter 5, the matrix β is blockdiagonal, i.e. the normal coordinates are linear

combinations of the SALCs of displacements belonging to one irrep of the point group of the

molecule (Wigner’s theorem).

As an example we classify the normal vibrations of CO32-, BF3

(D3h):

(7.1 Character table D3h).

Note: Q3, 4 and Q5, 6 are transformed into linear combinations

of each other by symmetry operations of the group. They

belong to 2 dimensional irreps.


7.3 Symmetry of normal modes

We would like to classify the normal modes Qi of the molecules with respect to the irreps

which they belong to. For this purpose, we analyse the 3N dimensional representation of the

displacements with respect to the irreps of the group.

Example: BF3 (C3v), 12 dimensional displacement basis

E: all diagonal elements Γii(E)=1 12=⇒ )( Eχ

σh: x, y, displacements are retained, z displacement changes sign 4=⇒ )( hσχ

C2: displacements of atoms, which change position do not contribute to Γii (off diagonal

elements) 22 −=⇒ )( Cχ


C3: z1 is invariant ( 1=χ ), for x1,y1:

°°−°°

=

2

1

2

1

120120120120

yx

yx

cossinsincos

''

( 1−=χ )

03 =⇒ )( Cχ

S3: analogous to C3, but z1’=-z1 ( 1−=χ )

23 −=⇒ )( Cχ

σv: y, z, displacements at two positions are invariant, x displacement changes sign

2=⇒ )( vσχ

D3h E 2C3 3C2 σh 2S3 3σv

χ 12 0 −2 4 −2 2

''''''' EAEAA ++++=Γ⇒ 221 23

Symmetry of translations and rotations from character table:

', ETyTx =Γ ; ''2ATz =Γ

'', ERyRx =Γ ; '2ARz =Γ

(7.2 Classify translations and rotations in D3h).

The remaining vibrational normal modes are: '''' 21 2 AEAvib ++=Γ


(7.3 Excercise: classify the vibrational modes of SF6 with respect to their symmetry, substitute one F by Cl).

7.4 Analysis of normal modes

We assume that we have determined the complete potential energy surface of an N atom

molecule V(r1,r2,...r3N) and expand V into a Taylor series at the equilibrium geometry:

∑ +

∂∂∂∂

∑ ∑ +

∂∂∂

+

∂∂

+=

=

kjikji

kjii ji

jiji

ii

rrrrrr

Vrrrr

VrrVVV

,,,...

4342434232itiesanharmonic

0

3

(harmonic)k constants force

0

2

position mequilibriuat expansionfor 0

0energy

mequilibriu

0 61

21

ij

We start from the equation of motion for the molecule

∑−=j

iijii rkrm hh

and use the ansatz tiii err ω0= which yields

∑=j

iijii rkωrm 02

0 .

Introducing mass weighted coordinates iii rmR 21 /= this equation is simplified to

∑=j

i

ij

ii

iji R

Kmm

kωR 02121

20

4342//

or RKRrr

=2ω .

As usual we solve the set of 3N linear equations by determinig the eigenvalues i2ω and

eigenvectors iΩr

. By transforming to a new basis RQrr

Ω= with ( )N321 ,...,, ΩΩΩ=Ωrrr

, i.e. the

basis of normal coordinates, the matrix of force constant 1−ΩΩ= Kκ becomes diagonal:

QQrv

κω =2 or 002

0 iiijj

iji QQQ κκω == ∑ .

Thus we obtain a set of 3N decoupled degees of motion (vibrations, rotations, translations). If

we express the total energy

∑∑ +=+=ji

jiiji

ii rrkrmVTE,

2

21

21

h

in terms of the new coordinates, we obtain


∑∑∑∑∑ +=+=+=i

iiiii

iiii

iij

jiiji

i QQQQQQQE 2222

21

21

21

21

21

κκκ hhh

In a quantum mechanical description we can describe the Hamiltonian as a simple sum over

harmonic oscillators

∑=i

iHH ˆ21ˆ with 2

2

22

21

2ˆ

iiii

i QQ

H κ+∂∂

−=h

with the well known solutions

( )2

21

2/1

2/1i

i

iii

Q

ii

ivviv eQHNQ h

h

ωωφ

−

= ; ( ) iiv vE

iωh2

1+= ; 2/1ii κω = ; ,...3,2,1=iv ;

(Hj(x): Hermite polynomials)

Consequently, the vibrational wavefunction of the molecules is

( )ii

vvib Qi∏=Φ φ with ∑=

ivvib i

EE .

(7.4 Example: normal modes of C2H2)

.

7.5 Symmetry of vibrational wavefunctions: fundamental modes

We consider a molecules with N vibrational modes. The excitation from the vibrational

ground state ( Nivi ...1;0 == ) to a state with a singly excited normal mode

( Nkkivv ik ...1,1,...1;0;1 +−=== ) is referred to as a fundamental mode of the molecule.

Symmetry of the ground state:

∑=Φ

−i

ii Q

vib Ne2

21

h

ω

Effect of symmetry operation R on

(a) non degenerate modes:

ii QQ ±→

( )

vib

Q

vibi

ii

NeR Φ=∑

=Φ±− 2

21

ˆ h

ω

(a) i = 1..n degenerate modes:


∑→j

jiji QQ α with ikj

kjij δαα =∑ (orthonormalization)

vib

QQQ

vibi

ii

jj

iji

i

NeNeNeR Φ=∑

=∑ ∑

=∑

=Φ−

−− 2

2

2

21

21

21

ˆ hhh

ωαωω

Thus, the vibrational ground state is invariant under all symmetry operations, i.e. 10, Avib =Γ .

Symmetry of the excited state:

∑=Φ

−i

ii Q

kvib eNQ2

21

h

ω

Symmetry of wavefunction is equal to symmetry of Qk, i.e. kQkvib Γ=Γ =1, .

7.6 Symmetry of vibrational wavefunctions: combination modes

The excitation from the vibrational ground state ( Nivi ...1;0 == ) to a state with several

normal modes excited by one vibrational quantum each ( 1;1 == lk vv ) is referred to as a

combination mode of the molecule.


∑=Φ

−i

ii Q

lkvib eQNQ2

21

h

ω

Symmetry of wavefunction is equal to the direct product of the representations of Qk and Qk,

i.e. lk QQlkvib Γ⊗Γ=Γ == 1,1, .

(7.5 Excercise: symmetry of the combination modes of BF3).

7.7 Symmetry of vibrational wavefunctions: overtones

The excitation from the vibrational ground state ( Nivi ...1;0 == ) to a state with several

vibrational quanta in one normal mode ( 1>kv ) is referred to as a overtone of the mode.


(a) non degenerate case


( )kkvvib QNUHk

2/1γ=Φ with ∑

=−

ii

i Q

eU2

21

h

ω

and h

kk

ωγ =

Hermite polynomials:

( ) 10 =zH

( ) zzH 21 =

( ) 14 22 −= zzH

( ) zzzH 128 33 −=

( ) 124816 244 +−= zzzH

Note: even for even vk, odd for odd vk. Therefore

( )

+=ΦΓ=Φ

=Φ12for

2for ˆnvRnv

RkvibQ

kvibvib

k

or

+=Γ=

=Γ12for

2for 1

nvnvA

kQ

kvib

k

(a) degenerate case

Similar as for the problem of several electrons occupying the same set of degenerate orbitals

(chapter 6), quantum statistics restricts the number of vibrational states if several vibrational

quanta occupy the same degenerate mode. Thus the problem becomes significantly more

complicated .

We consider the case of a twofold degenerate vibrational mode (Q1, v1; Q2, v2):

1 vibrational quantum: (v1=1; v2=0) (v1=0; v2=1)

2-fold degenerate

2 vibrational quanta: (v1=2; v2=0) (v1=1; v2=1) (v1=0; v2=2)

3-fold degenerate

3 vibrational quanta: (v1=3; v2=0) (v1=2; v2=1) (v1=1; v2=2) (v1=0; v2=3)

4-fold degenerate

n vibrational quanta: n+1 fold degenerate


(analogous: n vibrational quanta in three-fold degenerate mode: ½(v+1)(v+2) vibrational

wavefunctions; see textbooks)

As an example we consider the wavefunctions for v1+v2=3:

( )aa QQNU 2/132/30,3 128 γγ −=Φ

( )( )ba QQNU 2/121,2 224 γγ −=Φ

( )( )242 22/12,1 −=Φ ba QQNU γγ

( )bb QQNU 2/132/33,0 128 γγ −=Φ

We consider linear combinations Qa, Qb, for which the transformation matrix of a given

symmetry operation R is diagonal (this is always possible for single operation, for other

symmetry operations the transformation is of course not diagonal because the functions

belong to a two dimensional representation):

bbbb

aaaa

QQQQ

Γ=Γ=

''

As the cubic and the linear terms of the wavefunctions must transform indetically, we only

consider the cubic terms:

3330,3 ': aaaa QQ Γ=Φ

babbaaba QQQQ 2221,2 '': ΓΓ=Φ

2222,1 '': babbaaba QQQQ ΓΓ=Φ

3333,0 ': bbbb QQ Γ=Φ

Therefore, the character of the 4 dimensional representation is

( ) 32233 bbabaa RRRRRRR +++=χ

Analogous: ( ) 222 bbaa RRRRR ++=χ and ( ) 3

ba RRR +=χ

Repeated application of R: ( ) nb

na

n RRR +=χ


In general, one can show that the general recursion formula is valid (see Wilson, Decius,

Cross): ( ) ( ) ( ) ( )( )vvv RRRR χχχχ += −12

1 .

(analogous for triply degenerate normal coordinates:

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )

+−+= −−

vvvv RRRRRRR χχχχχχχ 2

221 2

1231 , see Wilson, Decius, Cross)

With this information it is possible to construct the characters of an k-fold excited vibrational

states corresponding to a twofold degenerate normal coordinate and analyse the representation

(notation (Γ)n) which they span in terms of the irreps of the group (similar formulae exist for

higher degenerate normal coordinate, see Wilson, Decius, Cross).

(7.6 Exercise: What is the symmetry of the first and second overtone of an E vibrational mode of CH4?).

The intensity of overtones and combination bands is relatively low in most cases. In some

cases these weak transitions become important. We assume that there is a singly excited state

0,1 21 ==Φ vv and a 1st overtone 2,0 21 ==Φ vv with similar vibrational energy. If both states have the

same symmetry (or contain a component with identical symmetry), a coupling between both

state is possible, i.e. matrix elements 2,00,1 2121ˆ

==== ΦΦ vvvv H can be nonzero. The eigenstates

of the coupled modes are mixed vibrational wave functions 2,020,11 2121' ==== Φ+Φ=Φ vvivvii cc .

The effect is denoted as Fermi resonance and leads to an “intensity borrowing” of the weak

combination band or overtone from the fundamental as well as to a “repulsion” between both

states (compare section 5.6).

(7.7 Example: Fermi resonance in CO2).

7.8 Symmetry of vibrational wavefunctions: arbitrary excited states


According to sections 7.5 to 7.8 the symmetry of an arbitrary vibrational state with the

vibrational quantum numbers v1, v2, v3,... corresponding to the normal modes of symmetry Γ1,

Γ 2, Γ 3,... is:

( ) ( ) ( ) ...321321

vvvvib Γ⊗Γ⊗Γ=Γ

Note: We have discussed the symmetry of vibrational wavefunctions in terms of the harmonic

approximation. However, the results are not affected, if we take anharmonicities into account,

as the different parts of the Taylor expansion of the Hamiltonian (section 7.4) must be totally

symmetric (A1) within the group of the molecule. If we thus express the anharmonic wave

function as a sum over harmonic functions ∑ Φ=Φi

iianh a , only those functions contribute,

which have the same symmetry.

7.9 Selection Rules

Electrical dipole transitions are described by matrix elements if the type

if iµ where iµ are the components of the vector of the dipole operator (with i=x,y,z).

We have seen that the integrand must be totally symmetric (A1) for the integral to be non

zero. This immediately leads to the dipole selection rule:

ivibfvib iA ,,1 Γ⊗Γ⊗Γ∈ µ

Note:

We consider the components of the dipole moment operator

ii

izi

iiyi

iix zqyqxq ∑∑∑ === µµµ ˆ;ˆ;ˆ

(1) A symmetry operation exchanges equivalent atoms with identical charge.


(2) The operator contains the sums ∑∑∑i

ii

ii

i zyx ;; only, which remain unaffected by a

change of indices.

Therefore, it is sufficient to investigate the transformation properties of the functions x, y, z,

which span the same representation as the components of the dipole operator. Typically, these

are tabulated in the character table.

Remark concerning IR reflection absorption spectroscopy (IRAS) at metallic surfaces: For

metallic surfaces the parallel component of the electric field is efficiently screened by the

conduction electron. Thus the electric field vector must be nearly perpendicular to the surface,

and only the electric field component perpendicular to the surface contributes to the field

strength at the surface. Similarly, dipole fields of the oscillating molecules are screened by the

conduction electrons. This resulting field is normally constructed by assuming an (imaginary)

image charge inside the metal. The image charge leads to the effect that perpendicular dipole

moments are amplified, whereas parallel dipoles are extinguished. Thus the relevant matrix

element for IR absorption at metallic surfaces is:

if zµ .

If we assume excitation from the ground state ( 1Ai =Γ ) and take into account that zµ or z is

symmetric with respect to any possible symmetry operation of a surface, i.e. 1Az

=Γµ , we

obtain the so called metal surface selection rule (MSSR) as:

1Af =Γ .

Raman transitions are described by matrix elements if the type

if ijα where ijα are the components of the tensor of the dielectric polarizability (with

i,j=x,y,z). Analogous to the previous case, we obtain the selection rule:


ivibfvib iA ,,1 Γ⊗Γ⊗Γ∈ α

As we will discuss in the next section, the polarizability is a symmetric tensor of rank 2

connecting the electric field and the induced dipole moment:

Eχ

vr

=

332313

232212

131211

αααααααα

µ

We will see that upon a symmetry operation described by the transformation matrix Rij this

type of tensor transforms according to

klkl

jlikij RRa α∑=' .

e.g.

yzxzxyxzxzxxxyxyxxzzxzyyxyxxxxxx RRRRRRRRRa αααααα 222' 222 +++++=

Transformation of function x2 for comparison:

( ) ( ) yzRRxzRRxyRRzzRyyRxxRzRyRxRx xzxyxzxxxyxxxzxyxxxzxyxx 222' 22222 +++++=++=

Thus, the components of the tensor show the same transformation behaviour as the quadratic

forms x2, y2, z2, xy, xz, yz.

Typically, these functions are tabulated in the character table. But how are they determined?

In principle there are two methods:

(1) Application of the symmetry operation to the basis (x2, y2, z2, xy, xz, yz),

determination of the representation matrices and of their characters, analysis in terms

of the irreps of the symmetry group.

(2) Analogous to the method applied in the symmetry analysis of overtones (section 7.7)


wavefunctions of 1st overtone of 3fold deg. mode: Qa2, Qb

2, Qc2, Qa Qb, Qa Qc, Qb Qc

quadratic functions of three coordinates: x2, y2, z2, xy, xz, yz

Example:

Symmetry of the quadratic forms in Td:

Γxyz=T2

R E 8C3 3C2 6S4 6σd

R2 E C3 E C2 E

χ(R) 3 0 -1 -1 1 (E)

χ(R2) 3 0 3 -1 3

Recursion formula for triply degenate modes (see section 7.7):

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )

( )( ) ( )[ ]22

20

222

21

212

31

RR

RRRRRRR

χχ

χχχχχχχ

+=

+−+=

χ2(R) 6 0 2 0 2 ⇒ A1 + E + T2

(7.8 Exercise: CO32-: Symmetry of normal modes, IR / Raman activity of fundamentals and 1st overtones).

Note: from the IR and Raman selection rules it is immediately evident that if the inversion i is

element of the symmetry group of the molecule, a IR active transition is Raman inactive and

vice versa (exclusion principle). Moreover all 1st overtones of centrosymmetric molecules are

IR inactive.

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