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Vibrational (Infrared) SpectroscopyVibrational (Infrared) Spectroscopyvibrational modes

← C O →≣≣equilibrium bond distance re can be changed by

applying energypotential well for modified (Morse) potential classical vibrator well for a diatomic molecule

         

1. quantized – only certain energy levels may exist

E = hω( + 1/2) 

: vibrational quantum number: vibrational frequency

1 kω = ―― ―― 2 m1 × m2 = ――――― m1 + m2 

   2. too close – repulsion between nuclei and electrons

too far apart – dissociation

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ex. HCl (HCl) = 2990 cm-1

DCl (DCl) = 2145 cm-1

ex. (NO) bond order

NO+ 2273 cm-1 3 NO 1880 cm-1 2.5 NO- 1365 cm-1 2 NO2- 886 cm-1 1.5

number of vibrational modesa molecule consists of N atoms, there are 3N

degrees of freedom translation rotation vibration

nonlinear 3 3 3N – 6linear 3 2 3N – 5type of vibrational modes

stretching mode bending mode IR active absorption Raman active absorption

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frequencies for some commonly encountered groups, fragments,and linkages in inorganic and organic molecules

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ex. W(CO)6 Mn(CO)5Br

        

compound (CO) (cm-1) [Ti(CO)6]

2- 1740

[V(CO)6]- 1860

Cr(CO)6 2000

[Mn(CO)6]+ 2095

 stretching modes of CO and IR frequencies(a) terminal (b) doubly bridging (c) triply bridging

     

   ex.     

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some ligands capable of forming linkage isomers            IR spectrum for nujol         

  salt plates 

NaCl 625 cm-1

KBr 400 cm-1

CsI 200 cm-1

2925 2855

721

1377

1462

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symmetry of normal vibrationsex. CO3

2- 6 vibrational modes C3(3a) = -1/23a + 1/2 3b

C3(3b) = -3/23a - 1/2 3b

               determine the symmetry type of normal modes  E    

= 12

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C3

                  = 0  C2

 

 

= -2

 

h = 4

S3 = -2 ??v = 2

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= A1’ + A2’ + 3E’ + 2A2” + E”

3 translatory modes: E’, A2”

3 rotational modes: A2’, E”

genuine vibrational modes: g = A1’ +2E’ + A2”

IR active: E’, A2” (3 bands)

Raman active: A1’, E’ (3 bands)

particular internal coordinates to normal modesC—O bonds

E C3 C2 h S3 v

3 0 1 3 0 1 CO = A1’ + E’ in-plane stretching

OCO = A1’ + E’ in-plane bending

A2” out-of-plane bending

 ex. determine the number of IR active CO stretching

bands for the following metal carbonyl compounds : M(CO)6 M(CO)5L cis-M(CO)4L2

trans-M(CO)4L2 fac-M(CO)3L3 mer-M(CO)3L3

M(CO)5 M(CO)4L M(CO)3L2 M(CO)4

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(i) trans-M(CO)4L2

D4h E C4 C2 C2’ C2” i S4 h v d L 4 0 0 2 0 0 0 4 2 0

OC CO

OC CO ==> A1g + B1g + Eu

L IR-active: Eu

(ii) cis-M(CO)4L2

CO C2v E C2 ’ OC L 4 0 2 2 OC L ==> 2A1 + B1 + B2

CO IR-active: 2A1, B1, B2 (iii) mer-M(CO)3L3

CO C2v E C2 ’ L L 3 1 1 2 OC L ==> 2A1 + B1

OC IR-active: 2A1, B1  

(iv) M(CO)5

D3h E C3 C2 h S3 v 5 2 1 3 0 3 ==> 2A1’ + A2” + E’ IR-active: A2”, E’

 

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(v) M(CO)4LL D3v3v E C3 v

4 1 2 ==> 2A1 + EIR-active: 2A1, E

 D2v E C3 v v‘

L 4 0 2 2 ==> 2A1 + B1 + B2

IR-active: 2A1, B1, B2  

(vi) M(CO)3L2

L D3h E C3 C2 h S3 v

3 0 1 3 0 1 ==> A1‘ + E’

L IR-active: E’

L Cs E h 3 1

L ==> 2A‘ + A”IR-active: 2A’, A”

 (vii) M(CO)4

Td E C3 C2 S3 d 4 1 0 0 2==> A1 + T2

IR-active: T2

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number of CO stretching bands in IR spetcrum for metal carbonyl compounds

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number of IR bands of some common geometric arrangements

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calculation of force constantsfor diatomic molecule AB harmonic oscillator

f • -1 – = 0 for polyatomic moleculeWilson’s method “The F and G matrix method”

|FG – E| = 0

F: matrix of force constant (potential energy)G: matrix of masses and spatial relationship of

atoms (kinetic energy)E: unit matrix 

e.g. H2O g = 2A1 + B1

2 O-H distance d1, d2 A1 + B1

∠HOH θA1

using projection operator to obtain complete setof symmetry coordinates for vibrations

A1 : S1 = θS2 = 1/√2(d1 + d2)

B1 : S3 = 1/√2(d1 - d2)

F matrix 2V = fik si sk

si, sk: change in internal coordinates

for d1 d2 θd1 fd fdd fdθ

d2 fdd fd fdθ

θ fdθ fdθ fθ

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2V = fd(d1)2 + fd(d2)

2 + fθ(θ)2 + 2 fdd(d1 d2)

+ 2 fdθ(d1 θ) + 2 fdθ(d2 θ)

= [d1 d2 θ] fd fdd fdθ d1

fdd fd fdθ d2

fdθ fdθ fθ θ = s’f s

 relationship between the internal coordinates and the symmetry coordinates

S = U sU matrix 0 0 1 d1

d1 + d2 = 1/√2 1/√2 0 d2

d1 - d2 1/√2 -1/√2 0

S = U s s = U’ S s’ = (U’ S)’ = S’Us’fs = S’FS(S’U)f(U’S) = S’FSS’(UfU’)S = S’FS ==> F = UfU’

  0 0 1 fd fdd fdθ 0 1/√2 1/√2

F = 1/√2 1/√2 0 fdd fd fdθ 0 1/√2 -1/√2

1/√2 -1/√2 0 fdθ fdθ fθ 1 0 0

fθ √2 fdθ 0

= √2 fdθ fd + fdd 0

0 0 fd - fdd

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G matrix G = UgU’ 0 0 1 gd gdd gdθ 0 1/√2 1/√2

G = 1/√2 1/√2 0 gdd gd gdθ 0 1/√2 -1/√2

1/√2 -1/√2 0 gdθ gdθ gθ 1 0 0

g33 √2 g13 0

= √2 g13 g11 + g12 0

0 0 g11 - g12

g11 = H + O

g12 = O cosθ

g13 = -(O/r) sinθ

g33 = 2(H + O -O cosθ)/r2

: reciprocal of the mass

2(H + O -O cosθ)/r2 -(√2O/r) sinθ 0

G = -(√2O/r) sinθ H + O (1+ cosθ) 0

0 0H + O (1 - cosθ)

 for H2O θ= 104.3o31’ r = 0.9580 Å

2.332 -0.0893 0G = -0.0893 1.0390 0 0 0 1.0702

  fθ √2 fdθ 2.332 -0.0893 0

A1: – = 0 √2 fdθ fd + fdd -0.0893 1.0390 0

 B1: 1.0702(fd - fdd) =

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elements of the g matrix

i: reciprocal mass of the ith atom

ij: reciprocal of the distance between ith and

jth

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Raman spectroscopylight of energy less than that required to promote a molecule into an excited electronic state is absorbed by a molecule, a virtual excited state is created

virtual state is very short lifetime, the majority of the light is re-emitted over 360oC, this is called Rayleigh scattering

C. V. Raman found that the energy of a small proportion of re-emitted light differs from the incident radiation by energy gaps that correspond to some of the vibrational modes

Stokes lines

anti-Stokes line

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schematic representation of Raman spectrometer

selection rules for vibrational transitions • a fundamental will be infrared active if the

normal mode which is excited belongs to the same representation as any one or several of the Cartesian coordinates

• a fundamental will be Raman active if the normal mode involved belongs to the same representation as one or more of the components of the polarizability tensor of the molecule

the exclusion rule – in centrosymmetric molecules,no Raman-active vibration is also IR-active and no IR-active vibration is also Raman-active only fundamentals of modes belonging to g representations can be Raman active and only fundamentals of modes belonging to u representations can be IR active

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ex. Na2MoO4 dissolved in HCl exhibits Raman peaks at 964, 925, 392, 311, 246, 219 cm-1

925, 311 cm-1 being polarizedwhat can be deduced from the spectrum?

no (Mo—H) and (O—H) bandsonly M—Cl and M=O likely exist

964, 925 cm-1 Mo=O stretching bands392 cm-1 Mo=O bending mode311, 246, 219 cm-1

Mo—Cl stretching modespossible product:

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normal vibrational modes for common structures

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