1 equipe de chimie théorique et réactivité ecp -iprem umr cnrs 5254 modelling the vibrational...

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

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

I – Electronic and nuclear motions

General Equation :...HHTH O.SeN

Limitation for H

LAPLACIANT

eeeNee

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.

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

3 – Development of the exact Schrodinger equation

RERVRT2 iii

COUPLING

KINETICS

eii R,rR

Exact coupling equations

RRVRT2RT2 j

jeieij HrdRVCOUPLING

POTENTIAL *3

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

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

0RERV2

R,rRVR,rH

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.

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

This implies very slow internuclear variations

Yo’x

Kinetic energy:

In summary:

rrmvmmRT2 22n

vrm2rmR2vmR2

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

Two Eckart’s conditions

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

0rvmdt

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

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

Hypothesis :

Angular momentum is zero

Second Eckart’s condition :

0vrm o

CoriolisvSm2

In consequence : neglectedTTTT VRVRn 2222

Rotational Schrödinger equation

Vibrational Schrödinger Equation

RRR ET̂

VRVV EV̂T̂

222n rmvmmRT2

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

1 - Lagrangian equations

ti zyxz

T, V, qii

iii qmF

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

0i i0 q

0j,i ji

0k,j,i kji

qqqqqq

q...qq

= n order force constants

Equilibrium

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

i qqqq

j,ijiiji qqFqV2

originqV

3 – Resolution of the Lagrangian equations

a) Cartesian coordinates space of displacements (X)

2ii XMXxmT2

xij XFXxxFV2

Lagrangian Equations : 0x

i = 1... 3N

3N equations

xijii 0xFxm

CCFM x1

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

Rij RFRRRFV2

Problem : How to express 2T ?

Find a linear transformation to gofrom X to R.

matrixWilsonBMBGwith

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

3N-6 equations

LLFG R

Diagonalisation of GFR

Eigenvalues

Eigenvectors Lik

kiki QLr

1 63...10N

Rijjij NiwhateverrFrg

c) Taking into account the symmetry

with :

d) Normal coordinates space Q

It is the space of the solutions

ConjugatedMomentum

2k PPPQT2

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

2kVV QQ

QiP̂Q̂

classical

operators

Vibrational Schrödinger equation :

• Variables splitting : product of mono modes functions

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

• Resolution of 3N-6 one-variable equations

• Variable change dimensionless normal coordinates

solutions

1vhcE kkvk

k eqH!v2q

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

kv qHk

q12q8qH

Recurency relations between the Hermite polynomials :

0qvH2qHq2qH k1vkvkk1v

• Degenerated vibrations :

Laguerre (2) or Legendre (3) polynomials

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..

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

2/1kV qqqq

• Vibrational equation processing :

Development basis of : They are the eigenfunctions of

VHH OVV

21OV VVH

The eigenfunctions of HV are developped on the eigenfunctions of

OViV C

Equation resolution operated by:

• Matricial representation of the vibrational Hamiltonian

• Integrals computation of the following terms :

'v/q/v

'v/p/v

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

4 – Presentation of the main resolution methods ofthe vibrational equation

4.1 - Perturbational method :

• Second order example : 21O

VV VVHH

with :

l,k,j,ilkjiijkl

k,j,ikjiijk

/V//V/E

qqqq24

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

• We develop on the eigenfunctions basis of

• Thus :

Then :

impliesthe terms

iijjiiii ; ijkiijiii

• Vibrational energies from a second order correction :

V EEhc

E 1cmen

With ij = anharmonicity constants

For non-degenerated modes :

ij expressed in function of and

iijjiiiiji ,,, ijkiijiii ,,

iijiiiiii 4

1i j,ijiijii

j,ik j,ik ijk

kjjikkiijjij 2

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

hAwithCBA

Vibrational equation resolution

Diatomic molecules

...qkqkq2

V 4ssss

221 qkqkq2

First order term v(1)

Second order term V(2)

Partition choice

4-1.1 Perturbational approach: details

ORDER 1:

vHvdqHE )1()0(v

)1(*)0(v

)0(k)0(

)1()1(

4-1.1 Perturbational approach :details

diagonal term of H(1) Hamiltonian

)1(vE Given by the diagonal terms of H

ORDER 2:

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

2)1()2()2(

)vHk(vHvE

Vibrational energy level

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

)vHk(vHvvHvEE

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

)vHk(vHvvHvEE

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

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

ssssssssssss

ssssssssss

qqqkqV

Diagonal terms

Diagonal and non diagonal terms

)0()0(

4-1.1Perturbational approach: details

)0()0(

3sssqk

vHvdqHE )1()0(v

)1(*)0(v

)1(v = 0

)0(3)0()0(

3)0(1)0()0(

)0(1)0()0(

3)0(3)0()0(

)0()0()0(

)1()1(

)vk()2

1k(EE )0(

2/3)0(3

)2)(1)((

)3)(2)(1(

Need the knowledge of the harmonic wavefunction

2vqv 2

3vqv 3

1vqv 3

3vqv 3

4-1.1 Perturbational approach : details

)0()0(

2vqv 2

3vqv 3

1vqv 3

3vqv 3

)1v2v2(k4

3vqvkvHv 2

ssss)2(

vqvvqv

)0()0(

2vqv 2

3vqv 3

1vqv 3

3vqv 3

)vHk( 22sss

15)122(

)0()0(

2)1()2()2(

vHkvHvE

sssssss

vk kvv

tconsvk

sss tan2

Anharmonic term

)0()0(

2vqv 2

3vqv 3

1vqv 3

3vqv 3

3N-6 modes de vibration

t; s : degenerated modes (example CO2) :

Asymetric stretching

Degenerated bending

Symetric stretching

4.2 – Variational method

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

HC = E C

Eigenvalues :Vibrational energy

levels

Eigenvectors :Vibrational states

OViV C

OiiJJ C

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

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.

- Energetic criteria :

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

iMAXiiMIN E

- 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

MAXi NVand

b) Perturbational criteria

Vibrational processing

In summary How to choose the subspace to be diagonalized

Excitation criteria

44 000 VVV '

Perturbational criteria

'V 0 00

Wif Thre-

Energy criteria

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

Usually 3 criteria

OOO4V'V4V

Diagonalization

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

mmi Phq,pH

m'v'vvv C

Effective operator definition

4iiiii

UAOi Q

m =UAO-CI m =UAO-CI

VSCFi Vhh

Non orthogonality for

VSCF/virtual-CI

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

4-2.1. Variational method Direct IC: details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

Example: 3 modes

...32222

31111 qkqqkqqkqkV

We take only a cubic potential in this example

4-2.1 Variational methodDirect IC : details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

.....32222

2vqv 2

3vqv 3

1vqv 3

3vqv 3

Notation :

Remark : normation

11311 vqv

31311 vqv

13033 vqv

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

2vqv 2

3vqv 3

1vqv 3

3vqv 3

Notations : 2 212

1211 vqv

21211 vqv

1222 vqv

4-2.1 Variational method Direct IC : details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

2vqv 2

3vqv 3

1vqv 3

3vqv 3

Notations : 1 122

2222 vqv

22222 vqv

1111 vqv

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

2vqv 2

3vqv 3

1vqv 3

3vqv 3

Notations : 2 23

12322 vqv

32322 vqv

4-2.1 Variational method Direct IC: details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

...32222

31111 qkqqkqqkqkV

2vqv 2

3vqv 3

1vqv 3

3vqv 3

31111qk

21112 qqk

11122 qqk

32222qk

4-2.1Variational method Direct IC: details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

...32222

31111 qkqqkqqkqkV

2vqv 2

3vqv 3

1vqv 3

3vqv 3

31111qk

21112 qqk

11122 qqk

32222qk

Others coupling ?Yes

4-2.1 Variational method Direct IC: details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

....32222

31111 qkqqkqqkqkV

2vqv 2

3vqv 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

111233 qqqk

12),,(

1<E<10%

4-2.1Variational method Direct IC: details

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

4-2.2 Variational method Direct IC with parallel process

thresholdEE

2)','(),(

),( vN

Parallel process

P-VMWCI method4-2.2. Variationalmethod P-VMWCI

vjjinviCvvvv )0(

,21 ,...,,

,, ,, vEvH vv

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.

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.

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

Si V'V

O'V W2EE

Equivalent to

A main difficulty for resonances processing

Contribution

1'VV cm20W

OV EE 20 100 200

perturbation

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

What informations can be deduced from a spectra ?

)( 0 dI

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

One PES)R,x( Aa

IR spectrumIR spectrum

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

k,IAaeIAak,I

Born-Oppenheimer Approximation :

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

Rigid Rotator :

Bands with

vibronic structuration

UV-V spectrumUV-V spectrum

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

IR intensities

Vibrational properties of molecules

)B(A~)A(X~ u21

absorption UV – V of benzene

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

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

vI,km,Ik,I

m,Fk,I NN

(Nk-Nm) : Population given by Botzmann statistic

m,Fk,I

Vibrational Transitions Intensities

IR intensities

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

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

eIAakI RRxRx

Initial state

Final state

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

)N(N I mk

vI,kkmkm

)N(N )3hc(4

vkI,km

General formulation of the IR intensity

m,I hcEE

...21 63

tstsst

sss QQQ

Electrical harmonicity

Electrical anharmonicity

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

IR intensities

with and

Mechanical and electrical harmonicities

IR intensities

ssk hcvE

1 Energy levels :

eI aQE Mecanical harmonicity quartic form of

Vibrational wavefunctionVibrational wavefunction :

cQQHNQ s

ssvvsv

ksksks

2/1 4 avec )

2exp(- )()(

Electrical harmonicity : s

gradient of dipolar moment

Mechanical and Electrical harmonicities

)N(N )3hc(4

N8I mk

vkI,km

)()()()(vmI,

vkI, svssv

sttvtvs QQQQQQ

msksmtkt

0vmI,0

mtkt vv1

Only the vibrational GS is occupied: s 0 ksv

vI,0 82

IR intensities

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

sss cch

892.974

IR intensities

Mechanical harmonicity - Electrical anharmonicity

IR intensities

Mechanical harmonicity

Electric anharmonicity : DM second derivatives

ss QQQ

0vmI,0

Overtones (s+ t) Harmonics :2 s

2 types terms

vkI, tsst QQ

2vkI, sss Q

vkI, sQ

22A 3 N

Fond: s

s 0 ksv

Mechanical harmonicity - Electrical anharmonicity

IR intensities

N8 vmIs

vkIsskm Q

)()()()( 2,

2, svssv

sttvtv

vkI QQQQQQ

msksmtkt

mtvktv

2/12/1

msvksvs

ksks vv

2vI,0 8

s 0 ksv

2h c 3

N8)2(I

Term vmI,

2vkI, sss Q

Mechanical Harmonicity - Electrical anharmonicity

IR intensities

vkIstkm QQ

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

, tvttvsvssv

uvuvvmIs

vkI QQQQQQQQQ

mtktmsksmuku

muvkuv

2/12/1

vI,0 2

tstsQQ

s 0 ksv

tssttsts hc

Term vmI,

vkI, tsst QQ

Mechanical anharmonicity - Electrical harmonicity

IR intensities

sse QQaaQE s,,

2s Q Q )(

Mechanical anharmonicity : n greater than 2

Example for n= 3

Perturbational processing (order 2)

Electrical harmonicity : s

Only the vibrational GS is occupied : s 0 ksv

Mechanical anharmonicity – Electrical harmonicity

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

2211222

31111 QQQQQQ aaaaP

22110 QQ

Study of the transition : 0 21

)Q(C sn

vk,I sn

Multiconfigurational Wavefunction :

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

IR intensities

2211222

31111 QQQQQQ aaaaP

Problem : configurations concerned in the 2 states description ?

21 2 23

214 13

IR intensities

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

vkI, sQ

12 2Q0

11Q0 111 2Q 111 2Q3

12Q0 1221 2Q2

221122

2 )4(2a

2h c 3

Mechanical anharmonicity contribution

Mechanical and electrical anharmonicities

IR intensities

221122

1111111

2 )4(2a

h c 3 N8

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

(21) is allowed if:

possess a component on

IR intensities calculations in the electrical harmonicity hypothesis

vk,I Q

Electrical harmonicity

2ss 892.974I

Only fundamentals are active

Mechanical anharmonicity

Mechanical harmonicity

rrnssnssr

rrnssn v)1v(Qvv

Intensity modified through the coupling with the fundamental modes :ssQ0

s rr'rns

Ikns vQvCC

Activitity allowed for harmonics and overtones

r20 tr0

vm,Irs

vk,Isr

vk,Iss

vk,I QQQ

IR intensities calculations taking into account the electrical anharmonicity

Non diagonal terms

Diagonal terms

rrnssn2s

srrrnssn v)2v(Qvv

rrnssn2s

srrrnssn vvQvv

IR intensity contribution to harmonics

s2s 2Q0

ttnrrnssnrs

ttnrrnssn v)1v()1v(QQvvv

IR intensity contribution to overtones

)(QQ0 rsrs

IR intensities

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

CH2 wagging

326.0 a.u 1810.03

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

UMR 5254

Claude POUCHANDidier BEGUE

Philippe CARBONNIERE

Neil GOHAUD

Isabelle Baraille

1 equipe de chimie théorique et réactivité ecp -iprem umr cnrs 5254 modelling the vibrational...

q slide

intensity slide

oxyz referential slide

electronic schrdinger

nuclear equation p

electronic motions

vibrational motions

z x y o x o y z kinetic

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