1 Equipe de Chimie Théorique et Réactivité ECP -IPREM UMR CNRS 5254 MODELLING THE VIBRATIONAL SPECTRA OF MOLECULES Claude POUCHAN IPREM UMR 5254 Université

1

Equipe de Chimie Théorique et Réactivité

ECP -IPREM UMR CNRS 5254

MODELLING THE VIBRATIONAL SPECTRA MODELLING THE VIBRATIONAL SPECTRA

OF MOLECULESOF MOLECULES

Claude POUCHANIPREM UMR 5254

Université de Pau et des Pays de l’Adour

European MasterIntensive Course Madrid September 2007

2


UMR 5254

ANSWER TO AN EXPERIMENTAL PROBLEM

- EXPERIMENTAL PROBLEM NATURE

2 31- To give a theoretical explanation to an experimental interpretation.

- To find correlation between and intrinsic properties for comparable systems.

Help to identifyreaction species

(by identification oftheir spectra) When several

products appear(photolyse,interstellar

environment..)

Predict or explain bands expected (fundamentals,

harmonics, overtones,Hot bands,

resonances) in a spectral area.

(vibrational, vibronicstructure)

Access toreactionnalmechanisms

INCREASING COMPLEXITY

Compute accurately Band position and

intensity

3

I – Electronic and nuclear motions

General Equation :...HHTH O.SeN

Limitation for H

LAPLACIANT

VVTH

RN

eeeNee

.2/2

Quantum mechanic : H diagonalizationEigenvalues and eigenvectors ----- all states

In fact1 – We choose a basis of states (electronic states ) from a partial Hamiltonian (He) issued from H ( He electronic Hamiltonian) for all nuclear configurations R.

i

R,rei

EH

4

2 – Development of the total wave function on the basis of R,rei

3 – Development of the exact Schrodinger equation

RERVRT2 iii

2iiR

2

COUPLING

KINETICS

jR*ie

31ij

jR*ie

32ij

rdRT

rdRT

i

eii R,rR

Exact coupling equations

RRVRT2RT2 j

jiijR

1ij

2ij

2

jeieij HrdRVCOUPLING

POTENTIAL *3

5

ijiiij RVRVPC 0 ADIABATIC BASISDiagonalisation of He

00 21 ijij TetTKC DIABATIC BASIS

If « PC » and « KC » are neglected by approximation

Only the first member of the equation remains

Nuclear and electronic motions are deconnected and can be separated

Nuclei move along a PES defined by RVRT ii2

ii

6

If in the adiabatic approach we don’t keep the diagonal term Born Oppenheimer approximation RT 2

ii

0RERV2

R,rRVR,rH

iiiR

2

eiiieie

• Diagonalisation of with fixed nuclei• appear as the potential in the nuclear equation

RVH iie RVii

P.E.S is defined by

All solutions of the electronic Schrödinger equation for all nuclear (R) fixed position.

RVii

7

Remark: Our study will be limited to the adiabatic surfaces for wich:

• only one PE surface is considered• BO approximation applicable

4 – Separation of the nuclear motions

TranslationalSpeed

Relativespeed

Angular speed

VvR

This implies very slow internuclear variations

.

8

Z

X

Yo’x

o y

z

R

r mv

Kinetic energy:

2T

2Vm

In summary:

2

rvRm

rrmvmmRT2 22n

vrm2rmR2vmR2

9

This expression is written with the 3N qi coordinates. These 3N coordinates can be splitted in two parts :

6 coordinates 3N - 6 coordinates

Which define theGlobal position of the

Molecule in theO’XYZ lab referential

Which define theRelative positions of the

Atoms in the Oxyz referential

10

Two Eckart’s conditions

a) Oxyz is in translation with the molecule: 1st condition

0rm

0vm

0rvmdt

rdm

2Tn : the crossed terms t/r et t/v are cancelled

11

b) Oxyz is in rotation with the molecule: 2nd condition

Hypothesis :

Angular momentum is zero

0vrm

orr

Second Eckart’s condition :

0vrm o

12

x

o y

z

r2r

0M

S M

v

But

Srr 0

Then

CoriolisvSm2

In consequence : neglectedTTTT VRVRn 2222

Rotational Schrödinger equation

Vibrational Schrödinger Equation

RRR

RRR ET̂

VRV

VRVV EV̂T̂

222n rmvmmRT2

13

II – Classical resolution of the vibrational motions in the harmonic approximation

1 - Lagrangian equations

tiq

ti

ti

ti zyxz

x

y

i

12

3

O

iq

V

0q

V

q

T

dt

d

ii

T, V, qii

i

iii

i

2ii

qmq

T

dt

d

qmq

T

qm2

1T

iii qmF

14

2 – Potential energy expressionUsually V(q) is expressed by means of a Taylor serie development.

i

0i i0 q

q

VqVqV

ji

0j,i ji

2

qqqq

V

!2

1

kji

0k,j,i kji

3

qqqqqq

V

!3

1

+ ...

nji

n

q...qq

V

= n order force constants

15

Equilibrium

Harmonic hypothesis : All terms of order > 2 are neglected ( small amplitude motions). Then

j,i

ji

0ji

2

i qqqq

VqV2

j,ijiiji qqFqV2

iall

for

q

V

originqV

i

o

0

:

16

3 – Resolution of the Lagrangian equations

a) Cartesian coordinates space of displacements (X)

i

2ii XMXxmT2

j,i

xji

xij XFXxxFV2

Lagrangian Equations : 0x

V

x

T

dt

d

ii

i = 1... 3N

3N equations

N3

1jj

xijii 0xFxm

CCFM x1

17

Diagonalisation of x1 FM

Eigenvalues

Eigenvectors Cik

3N solutions among them (3N-6) or 3N-5 are not null

b) Internal coordinates space of displacements R

6N3

j,i

Rji

Rij RFRRRFV2

Problem : How to express 2T ?

Find a linear transformation to gofrom X to R.

2k

2k c4

18

XMXT2

XBRRB

XRB

XRB

BXR

11

1

1

matrixWilsonBMBGwith

RGR

RBMBR

G

1

1

11

1

Lagrangian equations : i = 1... 3N-60R

V

R

T

dt

d

ii

3N-6

19

3N-6 equations

LLFG R

Diagonalisation of GFR

Eigenvalues

Eigenvectors Lik

k

kiki QLr

63

1

1 63...10N

jj

Rijjij NiwhateverrFrg

20

c) Taking into account the symmetry

with :

d) Normal coordinates space Q

It is the space of the solutions

k

2kk

k

2k

QQQV2

QQQT2

UGUG

UFUF

SGST2

SFSV2

URS

RS

RS

1S

S

21

ConjugatedMomentum

kk

k QQ

TP

k k

2k

2k PPPQT2

III – The quantic resolution of vibrational motions in the harmonic hypothesis

6N3

1kkk

2kVV QQ

2

1VTH

QiP̂Q̂

QQ̂

classical

operators

22

Vibrational Schrödinger equation :

6N3

1k

6N3

1k

2kk22

k

2

0Q2

1E

2

Q

• Variables splitting : product of mono modes functions

6N36N322116N321 Q...QQQ...Q,Q

• Resolution of 3N-6 one-variable equations

0QQ2

E2

dQ

Qdkk

2k

kk22

k

kk2

• Variable change dimensionless normal coordinates

23

k

4/1

k

2

k qQ

solutions

2

1vhcE kkvk

2

q

kv

2/1

kv

kk

2k

k

k eqH!v2q

Where is an Hermite Polynomial of vk order for the qk coordinate with a vk parity .

kv qHk

0qqE2

dq

qdkk

2k2/1

k

k2k

kk2

24

2k

2k q

vk

vqv

kv

k3kk3

2kk2

kk1

k0

edq

de1qH

q12q8qH

2q4qH

q2qH

1qH

Recurency relations between the Hermite polynomials :

0qvH2qHq2qH k1vkvkk1v

• Degenerated vibrations :

Laguerre (2) or Legendre (3) polynomials

25

IV – The anharmonic approach of the vibrational motions

1 – The potential anharmonic function :• usually an n order polynomial function with n = 2,3,4…• dissociation case : Morse potential function• double-well case : polynomial and gaussian functions.

2 – Force field determination .• M.O calculations

F(2) structural parameters dependant CCSD(T) ; MRCI ; MPn... DFT (B3LYP) … Bases : ccpVQZ ; ccpVTZ ; DZ ou TZP+ Diffuse

• Determination of the force field analytical process : (3) et (4) : HF, MP2 analytico-numerical : analytical gradient and/orHessian numerical : linear regression E ; G; H..

26

3 - Resolution of the vibrational Schrödinger equation in the anharmonic case.

• Vibrational Hamiltonian expression

We take into account the 3 and 4 order termsof the potential function. Kinetic function usually not affected….

We express the Fijk Fijkl terms in the dimensionless normal coord. basis

1, cmijklijk

k k,j,i l,k,j,i

lkjiijklkjiijk2k

2k

2/1kV qqqq

!4

1qqq

!3

1qp

2H

27

• Vibrational equation processing :

Development basis of : They are the eigenfunctions of

VHH OVV

21OV VVH

OVHO

VV

The eigenfunctions of HV are developped on the eigenfunctions of

OVH

i

OViV C

Equation resolution operated by:

• Matricial representation of the vibrational Hamiltonian

28

• Integrals computation of the following terms :

'v/q/v

'v/q/v

'v/q/v

'v/p/v

4

3

2

2

• resolution processing from : perturbational method variational method variation-perturbation method

29

4 – Presentation of the main resolution methods ofthe vibrational equation

4.1 - Perturbational method :

• Second order example : 21O

VV VVHH

with :

1V

1OV

OV

2OV

2V

OV

1OV

1V

l,k,j,ilkjiijkl

2

k,j,ikjiijk

1

/V//V/E

0/V/E

qqqq24

1V

qqq6

1V

30

• The second order energy correction requires to know the first order vibrational wavefunction :

1V

• We develop on the eigenfunctions basis of

• Thus :

OVH

1V

V'VO

'VOV

OV

1O'V

O'V

1OVO

V2O

V2

V EE

VVVE

Then :

V'VO

'VOV

2'VV

VV2

V EE

WWE

impliesthe terms

impliesthe terms

iijjiiii ; ijkiijiii

31

• Vibrational energies from a second order correction :

2V

OV

V EEhc

E 1cmen

With ij = anharmonicity constants

For non-degenerated modes :

ij expressed in function of and

iijjiiiiji ,,, ijkiijiii ,,

ij2j

2ij

2j

2i2

iijiiiiii 4

38

16

1

16

1

6N3

1i j,ijiijii

V

2

1v

2

1v

2

1v

hc

E

32

j,ik j,ik ijk

2j

2i

2k

kijkk

kjjikkiijjij 2

1

4

1

4

1

with :

kjikjikjikjiijk

If we consider the vibration-rotation we must add :• the Coriolis terms• the centrifugal distorsionAll terms are computed as perturbative corrections

ii = identicij = corrected by

...8 2

222

cI

hAwithCBA

eB

ei

j

j

icije

bije

aije

33

Vibrational equation resolution

Diatomic molecules

...qkqkq2

V 4ssss

3sss

221

4ssss

3sss

221 qkqkq2

First order term v(1)

Second order term V(2)

Partition choice

4-1.1 Perturbational approach: details

34


ORDER 1:

vHvdqHE )1()0(v

)1(*)0(v

)1(v

vk

)0(k)0(

k)0(

v

)1()1(

vEE

vHk

4-1.1 Perturbational approach :details

diagonal term of H(1) Hamiltonian

)1(vE Given by the diagonal terms of H

35


ORDER 2:

Diagonal terms of H(2) andNon-Diagonal terms of H(1)

vk

)0(k

)0(v

2)1()2()2(

vEE

)vHk(vHvE

Vibrational energy level

vk

)0(k

)0(v

2)1()2()1()0(

vvEE

)vHk(vHvvHvEE


36


vk

)0(k

)0(v

2)1()2()1()0(

vvEE

)vHk(vHvvHvEE


...

2

''''''''''''''''','''','','''''','''','',

''''''''','','''','',

221

ssssssssssss

ssssssssss

ss

qqqqk

qqqkqV

Diagonal terms

Diagonal and non diagonal terms

37


vk kv

vv

EE

vHk

vHv

vHv

EE

)0()0(

2)1(

)2(

)1(

)0(

)(

4-1.1Perturbational approach: details

38


vk kv

vv

EE

vHk

vHv

vHv

EE

)0()0(

2)1(

)2(

)1(

)0(

)(

3sssqk

vHvdqHE )1()0(v

)1(*)0(v

)1(v = 0

)0(3)0()0(

3

3)0(1)0()0(

1

3

)0(1)0()0(

1

3)0(3)0()0(

3

3

)0()0()0(

)1()1(

31

13

vvv

vvv

vvv

vvv

sssvk

kkv

v

EE

vqv

EE

vqv

EE

vqv

EE

vqv

kEE

vHk

)vk()2

1v()

2

1k(EE )0(

k)0(

v

avec

)0(3

)0(1

2/3

)0(1

2/3)0(3

)1(

26

)2)(1)((

22

3

)1(22

3

26

)3)(2)(1(

vv

vvsss

vvvv

v

vvvv

k

Need the knowledge of the harmonic wavefunction

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

4-1.1 Perturbational approach : details

39


vk kv

vv

EE

vHk

vHv

vHv

EE

)0()0(

2)1(

)2(

)1(

)0(

)(

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

)1v2v2(k4

3vqvkvHv 2

ssss4

ssss)2(

?

vqvvqv

vqvvqv

vqvvqv

vqvvqv

vqvvqv

vqv

3

3

22

22

22

4

11

11

22

22


40


vk kv

vv

EE

vHk

vHv

vHv

EE

)0()0(

2)1(

)2(

)1(

)0(

)(

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

60

7

2

1v

8

k15

EE

)vHk( 22sss

vk)0(

k)0(

v

2)1(

60

7

2

1

8

15)122(

4

3

)(

222

)0()0(

2)1()2()2(

vk

vvk

EE

vHkvHvE

sssssss

vk kvv

tconsvk

kssss

sss tan2

1

2

3

4

1522

Anharmonic term


41


vk kv

vv

EE

vHk

vHv

vHv

EE

)0()0(

2)1(

)2(

)1(

)0(

)(

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

...2

1vhcy

2

1vhcx

2

1vhE

32

v

3N-6 modes de vibration

t; s : degenerated modes (example CO2) :

Asymetric stretching

Degenerated bending

Symetric stretching


42

4.2 – Variational method

The problem consists to diagonalize the H matrixwhich is the projection of the HV Hamiltonianin the eigenfunctions basis of

OVH

HC = E C

Eigenvalues :Vibrational energy

levels

Eigenvectors :Vibrational states

i

OViV C

i

OiiJJ C

What space to be diagonalized ? Limitations : Size of the system to be solved.

43

IC choice :

• an usual choice of configurations space• a selective generation of configurations

a) Usual ‘a priori ‘ choice The selected space must provide a correct description of the system we want to solve .

• Upper limit : quantic vibrational number v(max) fixedfor each configurations

- Excitation criteria We consider all possible excitations (S,D,T,Q..)of the studied subspace. Dependent on the potentialfunction form.

44

- Energetic criteria :

• All possible configurations are selected in a given energetic domain.

6N3

iMAXiiMIN E

2

1VE

- Usual criteria :All possible excitations (S,D,T,Q,...) of the studied subspace are selected (Direct Interaction)with a cutting imposed by the form of the potentialfunction (usually 4).

MAXi VV

i

MAXi NVand

b) Perturbational criteria

45

Vibrational processing

In summary How to choose the subspace to be diagonalized

,

V 0

0V

,,

V 00

0

Excitation criteria

44 000 VVV '

Perturbational criteria

'V 0 00

'VV

'VV

EE

Wif Thre-

shold

Energy criteria

1000 0 cmn,V,V,V '''

Usually 3 criteria

46

OOO4V'V4V

OV

Diagonalization

47

c) Main variational method• UAO-CI (Uncoupled Anharmonic Oscillator• VSCF-CI (Vibrational Self-Consistent-Field)

6N3

1I

mmi Phq,pH

'v

m'v'vvv C

mv

mi

mv

mi

mv

6N3

1i

mv

ii

i

h

6N3

1j

ovij

mv ji

C

48

Effective operator definition

4iiiii

3iiii

oi

UAOi Q

24

1Q

6

1hh

m =UAO-CI m =UAO-CI

iUAOi

VSCFi Vhh

Non orthogonality for

VSCv

VSCF/virtual-CI

• VMFCI Method (P.Cassam;J.Lievin)• P-VMWCI Method (N.Gohaud;D.Bégué;C.Pouchan)

49

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(


4-2.1. Variational method Direct IC: details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

Example: 3 modes

...32222

22

11122

12

21112

31111 qkqqkqqkqkV

000

?

We take only a cubic potential in this example

50

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(


4-2.1 Variational methodDirect IC : details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

.....32222

22

11122

12

21112

31111

qk

qqk

qqk

qkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

003

001

Notation :

Remark : normation

1 13

11311 vqv

31311 vqv

13033 vqv

51

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(



j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

22

11122

12

21112

31111

qk

qqk

qqk

qkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

012

010

Notations : 2 212

1211 vqv

21211 vqv

1222 vqv

1222 vqv

52

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(


4-2.1 Variational method Direct IC : details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

22

11122

12

21112

31111

qk

qqk

qqk

qkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

021

001

Notations : 1 122

2222 vqv

22222 vqv

1111 vqv

1111 vqv

53

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(



j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

32222

22

11122

12

21112

31111

qk

qqk

qqk

qkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

030

010

Notations : 2 23

12322 vqv

32322 vqv

54


4-2.1 Variational method Direct IC: details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

...32222

22

11122

12

21112

31111 qkqqkqqkqkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

003

001

012

010

021

030

010

22

11122

31111

qqk

qk

31111qk

12

21112 qqk

12

21112 qqk

22

11122 qqk

32222qk

32222qk

55


4-2.1Variational method Direct IC: details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

...32222

22

11122

12

21112

31111 qkqqkqqkqkV

000

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3

003

001

012

010

021

030

010

22

11122

31111

qqk

qk

31111qk

12

21112 qqk

12

21112 qqk

22

11122 qqk

32222qk

32222qk

Others coupling ?Yes

56


4-2.1 Variational method Direct IC: details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

22

11122

12

21112

31111 qkqqkqqkqkV

1vqv

1vqv

2vqv 2

vqv 2

2vqv 2

3vqv 3

1vqv 3

1vqv 3

3vqv 3 012 021What couplings for and

A quartic constant is involved :

This quartic term is not considered in a perturbative approach

1111 vqv

1222 vqv

3233 vqv

23

12

111233 qqqk

57

jN

ip

v0

11

0),,(

jN

ip

v1

11

1),,(

jN

ip

v2

11

2),,(

jnN

ip

nv1

12),,(

1<E<10%

E<1%


4-2.1Variational method Direct IC: details

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

58


4-2.2 Variational method Direct IC with parallel process

jN

ij

v0

1

0),,(

jN

ij

nv1

1

0),,(

thresholdEE

vHv

vv

v )0(

')0(

2)','(),(

),( vN

Parallel process

Parallel process

BASIS

59


P-VMWCI method4-2.2. Variationalmethod P-VMWCI

j

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

60

4.3 – Variation-Perturbation method– Interacting configurations with the studied subspace are selected from a perturbational criteria.– Selection criteria :

– Iterative improvment of the subspace :

– At iteration n :

• Initial space choice : SO

• Interacting configurations with SO

Partition following a fixed thresholdS’M’

• Subspace S1 = SO + S’• Interacting configurations with S1

PartitionS’1M’1

• Subspace Sn is diagonalised• Corrective energies are given by from a perturbative method.

1n'M

61

0

n

'V

nV

1nV

1nn

diag

Vibrational processing

Iterative method

variation – perturbation Method

- Configurations are selected iteratively by means of perturbative method- Configurations with weights greater than a given threshold are included in the subspace- At the end :• a primary subspace containing the major configurations diagonalization• a secondary subspace M interacting weakly with S perturbative correction.

V

62

Remark :Limit of a perturbation method for quasi degenerated statesor for high density of states.

Si V'V

OV

O'V W2EE

Equivalent to

0'EW

WEdet

OV'VV

'VVO

V

O'V

OV

2'VVO

'V'V

O'V

OV

2'VVO

VV

EE

WEE

EE

WEE

63

A main difficulty for resonances processing

Contribution

20

1'VV cm20W

O'V

OV EE 20 100 200

perturbation

diag

64

V- IR Intensities calculations

• A vibrational spectra is characterized by bands possessing a wavenumber and an intensity.

• IR and Raman spectroscopies give vibrational spectra but the selections rules are different

• If the wavenumbers are the same for the 2 spectroscopies for assigned the modes, the intensities are different

• Only IR intensities will be studied in paragraph V

65

What informations can be deduced from a spectra ?

1 2

2

1

2

1)()(

.1

)( 0 dI

ILn

Cld

Integrated IR intensity : defined in the validity domain of Beer-Lambert by

IR intensities

Frequency : 0

Energetic transitions allowed energetic levels Full Width at Half Maximum

Parameter very sensitive at the

intermolecular forces gaseous

Transitions Momentum final and initial wavefunctions

66

One PES)R,x( Aa

eF

IR spectrumIR spectrum

)R().R,x()R,x( AN

k,IAaeIAak,I

Born-Oppenheimer Approximation :

)R().R()R( Ark,IA

vk,IA

Nk,I

Rigid Rotator :

Bands with

vibronic structuration

UV-V spectrumUV-V spectrum

Two PES)R,x( et )R,x( Aa

eFAa

eI

IR intensities

Vibrational properties of molecules

)B(A~)A(X~ u21

g11

absorption UV – V of benzene

67

)m,F()k,I(I )m,F()k,I(E

km)m,I()k,I( hcE

vmI,

vI,km,Ik,I

m,Fk,I NN

(Nk-Nm) : Population given by Botzmann statistic

2

m,Fk,I

Vibrational Transitions Intensities

IR intensities

)().,(),( ,, AN

mFAaeFAamF RRxRx )().,(),( ,, A

NkIAa

eIAakI RRxRx

Initial state

Final state

One PES : The electronic states I and F are the same.

)N(N I mk

2vmI,

vI,kkmkm

68

)N(N )3hc(4

8I mk

2vmI,

vkI,km

0

3

km

AN

General formulation of the IR intensity

kmv

k,Iv

m,I hcEE

k

vk,I

vk,I

k kT

Eexp

kT

EexpN

...21 63

1,1

63

10

N

tstsst

N

sss QQQ

Electrical harmonicity

Electrical anharmonicity

Knowledge of :Vibrational modes and corresponding wavenumbers and derivatives of dipolar moment

IR intensities

0

ss Q

.

QQ0ts

2

st

with and

69

Mechanical and electrical harmonicities

IR intensities

s

N

ssk hcvE

k

)21

(63

1 Energy levels :

2s

63

1

Q )(

N

sss

eI aQE Mecanical harmonicity quartic form of

V:

Vibrational wavefunctionVibrational wavefunction :

h

cQQHNQ s

sss

q

ssvvsv

s

ksksks

22

2/1 4 avec )

2exp(- )()(

)(63

1, s

N

sv

vkI Q

ks

Electrical harmonicity : s

N

ss Q

63

10

gradient of dipolar moment

70

Mechanical and Electrical harmonicities

)N(N )3hc(4

N8I mk

2vmI,

vkI,km

0

A3

km

s

N

ss Q

63

10

)(

63

1, s

N

sv

vkI Q

ks

)()()()(vmI,

vkI, svssv

sttvtvs QQQQQQ

msksmtkt

0vmI,0

vkI,

mtkt vv1

2/1

2

1

msks

k

vvs

sv

Only the vibrational GS is occupied: s 0 ksv

2/1

2

2/1

vmI,

vI,0 82

1

sss c

hQ

IR intensities

71

Mechanical and Electrical harmonicities

+ + + + Only the fundamental bands Only the fundamental bands s s possess an IR possess an IR intensity Iintensity Iss

km.mol-1 e.u-1/2

Gaussian

22A

22A

3

3 N

21

4

3 N8

ss

sss cch

hcI

2

0

892.974

ss Q

I

IR intensities

72

Mechanical harmonicity - Electrical anharmonicity

IR intensities

Mechanical harmonicity

Electric anharmonicity : DM second derivatives

ts

N

tssts

N

ss QQQ

63

1,1

63

10

21

0vmI,0

vkI,

Overtones (s+ t) Harmonics :2 s

2 types terms

vmI,

vkI, tsst QQ

vmI,

2vkI, sss Q

vmI,

vkI, sQ

22A 3 N

ss cI

Fond: s

s 0 ksv

73

Mechanical harmonicity - Electrical anharmonicity

IR intensities

2,2

,2A

3

3

N8 vmIs

vkIsskm Q

hc

)()()()( 2,

2, svssv

sttvtv

vmIs

vkI QQQQQQ

msksmtkt

mtvktv

2

2/12/1

2

21

msvksvs

ksks vv

sss c

hQ 2

vmI,

2vI,0 8

22

2

s 0 ksv

2

s

2sss

A3

s 22

2h c 3

N8)2(I

Term vmI,

2vkI, sss Q

74

Mechanical Harmonicity - Electrical anharmonicity

IR intensities

2

,,2A

3

21

3

N8

vmIts

vkIstkm QQ

hc

)()()()()()(,2

, tvttvsvssv

tusu

uvuvvmIs

vkI QQQQQQQQQ

mtktmsksmuku

muvkuv

2/12/1

vmI,

vI,0 2

121

tstsQQ

s 0 ksv

1

2/1

2

1

msks

k

vvs

sv1

2/1

2

1

mtkt

k

vvt

tv

tssttsts hc

I21

21

21

3

N8)(

22A

3

Term vmI,

vkI, tsst QQ

75

Mechanical anharmonicity - Electrical harmonicity

IR intensities

ututs

stus

sse QQaaQE s,,

2s Q Q )(

Mechanical anharmonicity : n greater than 2

Example for n= 3

Perturbational processing (order 2)

Electrical harmonicity : s

N

ss Q

63

10

Only the vibrational GS is occupied : s 0 ksv

76

Mechanical anharmonicity – Electrical harmonicity

ExampleExample : two symmetric modes : Q : two symmetric modes : Q11 and Q and Q22

32222

2211222

21112

31111 QQQQQQ aaaaP

22110 QQ

Study of the transition : 0 21

s

ssnv

)Q(C sn

6N3

1s

Iv

Ikn

vk,I sn

Multiconfigurational Wavefunction :

Problem : What are the configurations concerned in the 2 states development ?

IR intensities

77


IR intensities

32222

2211222

21112

31111 QQQQQQ aaaaP

Problem : configurations concerned in the 2 states description ?

1

13

2

212 1

21 2 23

2

2

212

214 13

15

1 1

21 2

13

21 23

212

21 32

0

12

78


IR intensities

Problem : What are the non null transitions moment ?vmI,

vkI, sQ

112Q0

12 2Q0

11Q0 111 2Q 111 2Q3

12Q0 1221 2Q2

2

22

21

221122

1

11111

A3

2 )4(2a

2a

2h c 3

N8I

211

1

Mechanical anharmonicity contribution

79

Mechanical and electrical anharmonicities

IR intensities

2

22

21

221122

1

1111111

A3

2 )4(2a

2a

22 2

h c 3 N8

I2111

1

Electrical anharmonicity

Mechanical anharmonicity

Objective : comparison of the two contributions

(21) is allowed whatever the development of the initial and final wavefunctions possess a component on

1 13212

0

(21) is allowed if:

possess a component on

1 212

80

IR intensities calculations in the electrical harmonicity hypothesis

s

vm,Is

vk,Is

vm,I

vk,I Q

Electrical harmonicity

2ss 892.974I

Only fundamentals are active

Mechanical anharmonicity

Mechanical harmonicity

sr

rrnssnssr

rrnssn v)1v(Qvv

Intensity modified through the coupling with the fundamental modes :ssQ0

s rr'rns

rrrn

'n,n

I'mn

Ikns vQvCC

Activitity allowed for harmonics and overtones

r20 tr0

81

s sr

vm,Irs

vk,Isr

vm,I

2s

vk,Iss

vm,Is

vk,Is

vm,I

vk,I QQQ

21

Q

IR intensities calculations taking into account the electrical anharmonicity

Non diagonal terms

Diagonal terms

sr

rrnssn2s

srrrnssn v)2v(Qvv

sr

rrnssn2s

srrrnssn vvQvv

IR intensity contribution to harmonics

s2s 2Q0

rtst

ttnrrnssnrs

rtst

ttnrrnssn v)1v()1v(QQvvv

IR intensity contribution to overtones

)(QQ0 rsrs

IR intensities

82

Exp. km )1(kmI Vrr Vrt )2(

kmI Iexp

62 Q 852 843 4.84 8.48 0.02 24.53 11.6 ± 1.2

82 Q 2294 2233 0.11 0.32 - 0.95

42 P 2290

Q 2317

R 2330

2391 4.76 0.06 - 5.29 7.4 ± 0.8

32 2842 0.04 0.39 - 0.03

22 4393 0.76 0.59 - 2.21

Conditions : method B3LYP + basis set DZP

Application to diazomethane molecule CH2N2H

H

C N N

CH2 wagging

326.0 a.u 1810.03

CN

6279.0 u.a. 269.066

41 19.0 19.0 .a.u 123.01 .a.u 040.04

4286.0 .a.u 035.044

IR intensities

83


UMR 5254

Claude POUCHANDidier BEGUE

Philippe CARBONNIERE

Neil GOHAUD

Isabelle Baraille

Documents

1 Equipe de Chimie Théorique et Réactivité ECP -IPREM UMR CNRS 5254 MODELLING THE VIBRATIONAL SPECTRA OF MOLECULES Claude POUCHAN IPREM UMR 5254 Université