INTRODUCTORY LECTURE on ROTATIONAL SPECTROSCOPY

CODECS Summer School 2013 CODECS Summer School 2013 THEORETICAL SPECTROSCOPY THEORETICAL SPECTROSCOPY

INTRODUCTORY LECTUREINTRODUCTORY LECTUREonon

ROTATIONAL SPECTROSCOPYROTATIONAL SPECTROSCOPY

Cristina PuzzariniCristina PuzzariniDip. Chimica Dip. Chimica ““Giacomo CiamicianGiacomo Ciamician””

UniversitUniversitàà di Bolognadi Bologna

ELECTRONICELECTRONIC VIBRATIONALVIBRATIONAL ROTATIONALROTATIONAL

Eel Evib ErotEE

FREQUENCY REGIONFREQUENCY REGION

mm/submm waves

Rotational Spectroscopy

Electronics Photonics

“Building” the ROTATIONAL SPECTRUM“Building” the ROTATIONAL SPECTRUM

(1) Rotational energy levels(1) Rotational energy levels

(2) Selection rules: transitions (2) Selection rules: transitions allowedallowed

(3) Intensity (transitions)(3) Intensity (transitions)

0 20 40 60 80 100frequency (cm-1)

Coordinate Coordinate SSyystemstemss

Molecule-fixed coordinate system Lab-fixed coordinate system

ROTATING RIGID BODYROTATING RIGID BODY

IωωT

= = angolar velocityangolar velocityI = inertia tensorI = inertia tensor

(CLASSIC VIEW)(CLASSIC VIEW)

Angolar Angolar VelocitVelocityy

ii rωv RIGID BODY:RIGID BODY:

zzzyzx

yzyyyx

xzxyxx

iiiixy

iiiixx

zymxrmI

Inertia TensorInertia Tensor

zzzyzx

yzyyyx

xzxyxx

PrincipalPrincipalinertiainertiasystemsystem

By cBy convenonventiontion: : IIcc IIbb IIaa

INERTIA TENSOR IINERTIA TENSOR I

RIGIDRIGID BBOODYDYJJ e e defindefineded in thein the rotating coordinate system (CM system)rotating coordinate system (CM system)

Angular MomentAngular Moment

KINETIC ENERGY of a ROTANTING RIGID BODYKINETIC ENERGY of a ROTANTING RIGID BODY

IJIωω

= angular velocy= angular velocyI = inertia tensorI = inertia tensor

(CLASSIC VIEW)(CLASSIC VIEW)

IJIωω

•• Potential energy?Potential energy?•• From classic mechanics to quantum mechanicsFrom classic mechanics to quantum mechanics

xrotrot I

KINETIC ENERGY of a ROTANTING RIGID BODYKINETIC ENERGY of a ROTANTING RIGID BODY

•• Potential energy?Potential energy?•• From classic mechanics to quantum mechanicsFrom classic mechanics to quantum mechanics

Classic view: conservation of angular momentum Classic view: conservation of angular momentum Quantum mechanics: commutation of operatorsQuantum mechanics: commutation of operators

2222222ZYXzyx JJJJJJJ

x,y,z molecule-fixed coordinate systemX,Y,Z space-fixed coordinate system

2222222ZYXzyx JJJJJJJ ˆˆˆˆˆˆˆ

The spaceThe space-- and moleculeand molecule--fixed components of fixed components of ĴĴ commute!commute!

SS = matrix that relates the coordinates of the atoms in the = matrix that relates the coordinates of the atoms in the moleculemolecule--fixed system to those in the spacefixed system to those in the space--fixed systemsfixed systems

Ĵ=SF ĴF where =x,y,z and F=X,Y,Z

ĴF= F ĴThen:[ĴF,Ĵ] = ĴFSF’ĴF’ – SF’ĴF’ĴF

= [ĴF,SF’]ĴF’ + SF’(ĴFĴF’ – ĴF’ĴF)= ieFF’F”(SF”ĴF’ + SF’ĴF”) = 0 !!

where eFF’F”=permutation symbol[ĴX,SX] = 0 [ĴX,SY] = iSZ [ĴX,SZ] = –iSY [ĴX,ĴY] = –ieXYZĴZ

EIGENVALUES of EIGENVALUES of ĴĴ22, , ĴĴZZ, , ĴĴzz

22 1)(

KMKJJMKJ

MMKJJMKJ

JJMKJJMKJ

,,ˆ,,

M=J,J-1 … -J

K=J,J-1 … -J

J=0,1,2,3, …

022 zJJ ˆ,ˆ 022 ZJJ ˆ,ˆ 022 ZJJ ˆ,ˆz

ROTATIONAL ENERGY LEVELSROTATIONAL ENERGY LEVELS

rotrotrotrot EH ˆ

xrot I

21 ˆˆˆˆ

ROTATIONAL HAMILTONIANROTATIONAL HAMILTONIAN

ClassificationClassification

CO COCO CO22

CHCH4 4 SFSF66

NHNH33

HH22OO

Examples

By cBy convenonventiontion: : IIcc IIbb IIaa

Let’s consider the simplest case

DIATOMIC/LINEAR MOLECULE:DIATOMIC/LINEAR MOLECULE:RIGID ROTOR (approx)RIGID ROTOR (approx)

iirmI 2

2RI 21

reduced mass

zIx = Iy = IIz = 0

2231231

1 rrmmrmmrmmM

1 32rr1212 rr2323

Ix = Iy = IIz = 0

ˆˆˆˆ

Ix = Iy = I

ROTATIONAL ENERGY LEVELS:ROTATIONAL ENERGY LEVELS:Diatomic and Linear moleculesDiatomic and Linear molecules

BB = rotational constant= rotational constant

)1()1(2

JBJJJI

JJ = 0,1,2,3,…. = 0,1,2,3,….

Iz = 0making use of the eigenvalues of making use of the eigenvalues of ĴĴ 22

)1( JBJErot

Erot=0

Erot=2B

Erot=6B

Erot=12B

Erot=0

Erot=2B

Erot=6B

Erot=12B

)1(2)1( JBJJE

x,y,z molecule-fixed coordinate systemX,Y,Z space-fixed coordinate system

022 zJJ ˆ,ˆ 022 ZJJ ˆ,ˆ

22 1)(

MMKJJMKJ

JJMKJJMKJ

,,ˆ,,

M=J,J-1 … -J

Rotational energy levels: Rotational energy levels: (2(2JJ+1) fold degenerate in +1) fold degenerate in MM

0 20 40 60 80 100frequency (cm-1)

SELECTION RULESSELECTION RULES

Transition moment: Transition moment: 00

Approx BO: tot=rotvibele

elevibrotele

elef ddd

dipole moment in the space-fixed coordinate system

cos where=x,y,z (molecule-fixed)F=X,Y,Z (space-fixed)

Z=direction cosines

elevibelei

elefrot

rotf ddd

molecular dipole moment components

where =x,y,zF=X,Y,Z

F=direction cosines

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

(1)(1) Selection rulesSelection rules

(2)(2) NonNon--vanishing permanent dipole momentvanishing permanent dipole moment

ifR FF

(1)(1) JJ = = 11

SELECTION RULESSELECTION RULES“Rotational” transition moment Rij:

where:

The direction-cosine matrix elements are known:

'''''''' MJJMKJJKJJMKJJKM FFFF

(1)(1) (2)(2) (3)(3)

(2)(2) KK = 0= 0(3)(3) MM = 0, = 0, 11

0 20 40 60 80 100frequency (cm-1)

Rotational energy levelsRotational energy levels++

Selection rulesSelection rules

Rotational transition frequenciesRotational transition frequencies(rotational spectrum: (rotational spectrum: xx axis) axis)

)1( JBJErot

Erot=0

Erot=2B

Erot=6B

Erot=12B

)1(2)1( JBJJE

)1(2 JBh rot

frequency

nsit y

2B/h 4B/h 6B/h 8B/h

2B/h 2B/h 2B/h 2B/h 2B/h

JJ=1=1--00 JJ=2=2--11 JJ=3=3--22 JJ=4=4--33 JJ=5=5--44

(B in energy units)

(1) (1) BoltzmannBoltzmann distributiondistribution

LINE STRENGTHSLINE STRENGTHS

(2) (2) degeneracydegeneracy

gf/g0=2J+1 exp(-Erot/kT) Nf/N0=(2J+1)exp(-Erot/kT)

221 mnNT

I mni μ

Intensity of Rotational TransitionsIntensity of Rotational Transitions

0 1 2 3 4 5 6 7 8 9 10

Intensity Population

0 20 40 60 80 100frequency (cm-1)

0 20 40 60 80 100

Rotational spectrum of CO

wavenumbers (cm-1)

( u. a

MJ ePY cos,, ||

Linear Rotor: EIGENFUNCTIONSLinear Rotor: EIGENFUNCTIONS

SPHERICAL HARMONICSSPHERICAL HARMONICS

Eigenvalues of J2: ħ2J(J+1) with J = 0, 1, 2, …

Eigenvalues of Jz : ħM with -J ≤ M ≤ J

SPHERICAL HARMONICSSPHERICAL HARMONICS

Vector Vector RappresentaRappresentation of tion of Angular MomentumAngular Momentum

Costant length (J) - 5 orientations (M)

JJ = 2 = 2 5 values for 5 values for MM

One step further …..One step further …..

Molecules are NOT rigid:Molecules are NOT rigid:centrifugal distortioncentrifugal distortion

SEMISEMI--RIGID ROTOR with RIGID ROTOR with CENTRIFUGAL DISTORTIONCENTRIFUGAL DISTORTION

'ˆˆˆdistrotrot HHH 0

perturbation theory

rigidrigid--rotorrotor

44 JDH J

distˆˆ '

22 )1( JJDE Jdist'

BD > 0 !!> 0 !!

Erot=0

Erot=2hB

Erot=6hB

Erot/h=12hB

centrifugal distortioncentrifugal distortion

22 )1()1( JJDJBJhE Jrot /

frequency

2B 4B 6B 8B

2B 2B 2B 2B 2B

3)1(4)1(2 JDJB Jrot

JJ=1=1--00 JJ=2=2--11 JJ=3=3--22 JJ=4=4--33 JJ=5=5--44

[B, DJ in frequency units]

Another step further …..Another step further …..

Other types of rotorOther types of rotor

ClassificationClassification

CO COCO CO22

CHCH4 4 SFSF66

NHNH33

HH22OO

Examples

By cBy convenonventiontion: : IIcc IIbb IIaa thus C thus C B B AA

Ia = Ib = Ic = I

Erot = B J(J+1)

Each level: (2J + 1)2 fold degenerate (K,M)

= 0 !!!

CH4, SF6 , …

SPHERICAL TOPSSPHERICAL TOPS

SYMMETRIC TOPSSYMMETRIC TOPS

ˆˆˆ//

// (z = symmetry axis)

22 11)1(

IIIJJErot

K=J,J-1 … -J

22 11)1(

IIIJJErot

Prolate: Prolate: EErotrot = = BJBJ((JJ+1)+(+1)+(AA––BB))KK22 wherewhere AA>>B=CB=COblate: Oblate: EErotrot = = BJBJ((JJ+1)+(+1)+(CC––BB))KK22 wherewhere AA==BB>>CC

BClBCl33

CHCH33FF

II < < IIoblateoblate II > > II

prolateprolate

II//// = = IIaaII//// = = IIcc>0

SYMMETRIC TOP: rotational energy levelsSYMMETRIC TOP: rotational energy levels

PROLATE PROLATE OBLATEOBLATEAA > > BB = = CC AA < < BB = = CC

J J = 6, = 6, K K = = 44

In addition to In addition to JJ = = 11::

KK = 0= 0

)1(2 JBh rotRIGID ROTOR:RIGID ROTOR:

Rotational spectrum of a symmetric-top rotor

KK structure for each structure for each JJ value (value (JJ+1 +1 JJ))

KK = 0= 0

)1(2 JBh rotRIGID ROTOR:RIGID ROTOR:

23 )1(2)1(4)1(2 KJDJDJBh JKJrotincluding CENTRIFUGAL DISTORTION:including CENTRIFUGAL DISTORTION:

Rotational spectrum of CH3CN: a small portion

1 1 1 7 2 0 0 1 1 1 7 8 0 0 1 1 1 8 4 0 0

K = 6 K = 3

F r e q u e n c y ( M H z )

C H3C N : J = 6 1 - 6 0 K = 0

KK structurestructure

5 5 4 4 0 0 5 5 4 5 0 0 5 5 4 6 0 0 5 5 4 7 0 0

K = 1 8

K = 1 5K = 1 2K = 9

F r e q u e n c y ( M H z )

1 4 N F 3 : J = 2 6 - 2 5

Rotational spectrum of NF3: a small portion

Eigenfunctions: SPHERICAL HARMONICSSPHERICAL HARMONICS

Eigenvalues of J2: ħ2J(J+1) with J = 0, 1, 2, …

Eigenvalues of JZ: ħM with -J ≤ M ≤ J

Eigenvalues of Jz: ħK with -J ≤ K ≤ J

SymmetricSymmetric--top Rotor: EIGENFUNCTIONStop Rotor: EIGENFUNCTIONS

ASYMMETRIC ROTORASYMMETRIC ROTOR

xrot I

21 ˆˆˆˆ

No longer possible to rearrange the Hamiltonian sothat it is comprised soley of and one componentof

It is not possible to describe the rotational motionIt is not possible to describe the rotational motionin terms of a in terms of a conserved motionconserved motion about a particular about a particular axis of the molecule.axis of the molecule.

xrot I

21 ˆˆˆˆ

Diagonalization: EDiagonalization: Erotrot, , For the sake of convenience:For the sake of convenience:

correlation to symmetric topcorrelation to symmetric top

Pseudo quantum numbers:Pseudo quantum numbers:KKaa limiting prolate symmetric rotorlimiting prolate symmetric rotorKKcc limiting oblate symmetric rotorlimiting oblate symmetric rotor

-101 1

PROLATEPROLATE OBLATEOBLATE

J Ka JKc(-J +J)

near oblatenear oblatenear prolatenear prolate

2B A C

Asymmetric parameter Asymmetric parameter

= -1 = +1

2 notation scheme: JKa,Kc or J

In addition to In addition to JJ = 0,= 0, 11::

KKa a ,, KKcc = 0, = 0, 11

Symmetric Rotor

Asymmetric Rotor

5 2 4 0 0 0 5 2 4 1 0 0 5 2 4 2 0 0 5 2 4 3 0 0 5 2 4 4 0 0 5 2 4 5 0 0 5 2 4 6 0 0 5 2 4 7 0 0

F re q u e n c y (M H z)

transtrans--CHCH3535Cl=CHFCl=CHF

ASYMMETRIC ROTOR: ASYMMETRIC ROTOR: small portion of rotational spectrumsmall portion of rotational spectrum

Rotational HamiltonianRotational Hamiltonian

Rotational constantsRotational constants

222CBA CBA JJJ RIGID ROTORRIGID ROTOR

++CENTRIFUGAL DISTORTIONCENTRIFUGAL DISTORTION

Rotational HamiltonianRotational Hamiltonian

Another step further …..Another step further …..

Hyperfine InteractionsHyperfine Interactions

Hyperfine structureHyperfine structureRotational HamiltonianRotational Hamiltonian

Nuclear quadrupole Nuclear quadrupole couplingcoupling

JJIIqeQ 222

)12()12(221 JIJIJI

222CBA CBA JJJ

frequency

F=-1F=+1

unperturbed

nuclear quadrupole coupling

unperturbed

F = 1/2

F = 5/2

F = 3/2

LINEAR MOLECULELINEAR MOLECULEF = J+I, J+I-1, …, |J-I |

[[IIKK 1]1] IIKK=3/2; =3/2; eQq eQq 00

Selection Rules:Selection Rules:coupling coupling II + + JJ = = F F

frequency

F=-1F=+1

unperturbed

nuclear quadrupole coupling

unperturbed

F = 1/2

F = 5/2

F = 3/2

frequency

F=-1F=+1

unperturbed

[[IIKK 1]1] IIKK=3/2; =3/2; eQq eQq 00

LINEAR MOLECULELINEAR MOLECULEF = J+I, J+I-1, …, |J-I |

hyperfine structurehyperfine structure

JJIIqeQ 222

)12()12(221 JIJIJI

SpinSpin--rotation interactionsrotation interactions

KK JCI222CBA CBA JJJ

frequency

= 2 - 1

F = +1 (F=5/2-3/2)

F = +1 (F=3/2-1/2)

F = 0 (F=3/2-3/2)

unperturbed

spin-rotation interaction

unperturbed

F = 3/2

F = 5/2

F = 1/2

F = 3/2J = 2 - 1

F = +1 (F=5/2-3/2)

F = +1 (F=3/2-1/2)

F = 0 (F=3/2-3/2)

unperturbed

frequency

[[IIKK 1/2]1/2] IIKK=1/2; =1/2; C C 00

LINEAR MOLECULELINEAR MOLECULE

hyperfine structurehyperfine structure

SpinSpin--spin (direct)spin (direct)interactionsinteractions

LKLK IDI

222CBA CBA JJJ

JJIIqeQ 222

)12()12(221 JIJIJI

SpinSpin--rotation interactionsrotation interactions

KK JCI

Selection Rules:Selection Rules:coupling Icoupling IK,LK,L + J = F+ J = FK,LK,L

1;0 LKF ,

fre que ncy

= 0, +1 1-0,1 -1 )

F = 0 ,+1 (F=1-0,1 -1,2 -1 )

F = -1 (F=0-1)

unp erturbed

direct spin-spin interaction

unperturbed

F =F'+I2

F' =J+I1

frequency

F = 0,+1 (F=1-0,1-1)

F = 0,+1 (F=1-0,1-1,2-1)

F = -1 (F=0-1)

unperturbed

IIKK=1/2 =1/2 andand IILL=1/2=1/2

LINEAR MOLECULELINEAR MOLECULE

Stark effectStark effect

x,y,z molecule-fixed coordinate systemX,Y,Z space-fixed coordinate system 022 zJJ ˆ,ˆ 022 ZJJ ˆ,ˆ

22 1)(

MMKJJMKJ

JJMKJJMKJ

,,ˆ,,

Rotational energy levels: Rotational energy levels: (2(2JJ+1) fold degenerate in +1) fold degenerate in MMM=J,J-1 … -J

Degeneracy removed by applying electric field:Degeneracy removed by applying electric field:STARK EFFECTSTARK EFFECT

ZasseE0

MJ = 0

MJ = ±1

MJ = 0

Energy0JM

STARK EFFECTSTARK EFFECT

εμ HInteraction between the applied electric field and dipole moment: perturbation theoryperturbation theoryĤ = perturbation Hamiltonian applied along Zlet’s consider a symmetric-top rotor ( along z):

By applying perturbation theory:ZzμεH ˆ

)32)(12()1()1][()1[(

)12)(12())((

222222(2)

JJJMJKJ

STARK EFFECT: the SYMMETRIC TOP caseSTARK EFFECT: the SYMMETRIC TOP case

NO FIELDNO FIELD

|000JKM

|100|10-1

|110|11-1

|111|1-10|1-1-1

1st ORDER1st ORDER

|101 |100|10-1

|111 |1-1-1|110 |1-10|11-1 |1-11

AA--BB

2nd ORDER2nd ORDER

|111 |1-1-1|110 |1-10|11-1 |1-11

|101|100

STARK EFFECT: the SYMMETRIC TOP caseSTARK EFFECT: the SYMMETRIC TOP case

NO FIELDNO FIELD

|100|10-1

|110|11-1

|111|1-10|1-1-1

1st ORDER1st ORDER

|101 |100|10-1

|111 |1-1-1|110 |1-10|11-1 |1-11

AA--BB

2nd ORDER2nd ORDER

|111 |1-1-1|110 |1-10|11-1 |1-11

|101|100

shift Stark:shift Stark: ==’’--

((’’ > > ))

’’

|000JKM

ROTATIONAL SPECTROSCOPY:ROTATIONAL SPECTROSCOPY:Computational RequirementsComputational Requirements

AccuracyAccuracy

SpectroscopicSpectroscopic pparameterarameterss::

RotationalRotational constantsconstants

CentrifugalCentrifugal--distortion constantsdistortion constants

HyperfineHyperfine parametersparametersNuclearNuclear quadrupolequadrupole coupling constantscoupling constants

SpinSpin –– rotation rotation constantsconstants

SpinSpin –– spinspin constantsconstants

Rotational SpectroscopyRotational Spectroscopy

LaboratoryLaboratory of of MillimetreMillimetre--wavewave

SpectroscopySpectroscopy of Bolognaof Bologna1071310 1071312 1071314 1071316

FREQUENCY (MHz)

HH22S: S: JJ = 8= 86,36,3 –– 885,45,4

> 4 MHz> 4 MHz

Frequency accuracy: 1 part in 107-108

SpectroscopySpectroscopy of Bolognaof Bologna1071310 1071312 1071314 1071316

FREQUENCY (MHz)

HH22S: S: JJ = 8= 86,36,3 –– 885,45,4

> 4 MHz> 4 MHz

SpectroscopySpectroscopy of Bolognaof Bologna1071313.2 1071313.4 1071313.6 1071313.8 1071314.0

FREQUENCY (MHz)

HH22S: S: JJ = 8= 86,36,3 –– 885,45,4

~100 kHz~100 kHz

Frequency accuracy: 1 kHzFrequency accuracy: 1 kHz

Frequency accuracy: better than 1 part in 109

3 8 0 1 9 7 .3 0 3 8 0 1 9 7 .3 5 3 8 0 1 9 7 .4 0 3 8 0 1 9 7 .4 5

F R E Q U E N C Y (M H z)

17 kH z 46 kH z

16O J = 4 1 4

- 3 2 1

F ' - F '' 5 - 4 4 - 3 3 - 2

QUANTUMQUANTUM--CHEMICAL CHEMICAL CALCULATIONS of CALCULATIONS of

ROTATIONAL PARAMETERS:ROTATIONAL PARAMETERS:Methodology & AccuracyMethodology & Accuracy

ROTATIONAL ROTATIONAL CONSTANTSCONSTANTS

QuantumQuantum--Chemical Calculation of Chemical Calculation of Spectroscopic ParametersSpectroscopic Parameters

• Rotational (equilibrium) constantsRotational (equilibrium) constants

requires equilibrium geometry: geometry optimization (nuclear forequires equilibrium geometry: geometry optimization (nuclear forces)rces)

INERTIA TENSOR

Accurate Accurate equilibrium structure equilibrium structure !!!!

1) 1) Principal error sourcesPrincipal error sources in in ab initio calculationsab initio calculations::

-- wf wf model model truncation truncation (N(N--ee-- errorerror))-- basisbasis--set set truncation truncation (1(1--ee-- errorerror))

2) 2) “Minor” “Minor” error sourceserror sources in in ab initio calculationsab initio calculations::

-- corecore--valencevalence (CV) (CV) correlationcorrelation

-- …………-- scalar scalar relativityrelativity (SR)(SR)

COMPOSITE APPROACHCOMPOSITE APPROACH

-- Coupled cluster method with Coupled cluster method with singlesingleand and double excitations with double excitations with aapertubativepertubative treatment of treatment of connectedconnectedtriplestriples: : CCSD(T)CCSD(T)

-- HigherHigher excitationsexcitations: : fullfull--T, Q, … …T, Q, … …

-- wf wf model model truncation truncation (N(N--ee-- errorerror))

exponential ansatz for wavefunctionexponential ansatz for wavefunction

with cluster operatorwith cluster operator

(excitations)

CoupledCoupled--Cluster TheoryCluster Theory

...! ,..,, ,...,,

...... jibat

kji cba

abcijkm

211)exp( TTTT

HFCC T )exp(

.... 321 TTTT

energyenergy

amplitudesamplitudes

coupledcoupled--cluster equationscluster equations

very efficient treatment of electronvery efficient treatment of electron--correlation effectscorrelation effects

HFHFCC TETHH )exp()exp(ˆˆ

Schrödinger equationSchrödinger equation

•• CoupledCoupled--Cluster Singles and DoublesCluster Singles and Doublesrestrict T to single and double excitations restrict T to single and double excitations (T=T(T=T11+T+T22))

CCSDCCSD

•• CoupledCoupled--Cluster Singles, Doubles, and TriplesCluster Singles, Doubles, and Triplesrestrict T to S, D, triple excitations restrict T to S, D, triple excitations (T=T(T=T1 1 +T+T2 2 +T+T33))

CCSDTCCSDT

•• approximate treatment of triple excitationsapproximate treatment of triple excitationsadd perturbative triples correctionadd perturbative triples correction CCSD(T)CCSD(T)

•• CoupledCoupled--Cluster Singles, Doubles, Triples, QuadruplesCluster Singles, Doubles, Triples, Quadruplesrestrict T to S, D, T, quadruple excitations restrict T to S, D, T, quadruple excitations (T=T(T=T1 1 +T+T2 2 +T+T3 3 +T+T44))

CCSDTQCCSDTQ

CCSD(T) T=T1 + T2 + (T) N6 + N7 (no iter)

....)(

dE CCSDTQCCSDTTCCSDtot

large basis set:large basis set:cccc--pV5Z/ccpV5Z/cc--pV6ZpV6Z

smallsmall--medium basis set:medium basis set:cccc--pVTZpVTZ small basis set:small basis set:

cccc--pVDZpVDZ

Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005) Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005)

-- HirarchicalHirarchical seriesseries of of basesbases: : cccc--pVpVnnZZ, , augaug--cccc--pVpVnnZZ, , cccc--pVpVnnZZ--PPPP

nn=D,T,Q,5,6=D,T,Q,5,6

-- Extrapolation toExtrapolation to the CBS the CBS limitlimit::

E(E(nn))[[SCFSCF]] = E= ECBSCBS[[SCFSCF]] + A+ Aexpexp((--BBnn))+ + E(E(nn))[CORR][CORR] = E= ECBSCBS[CORR][CORR] + + CCnn--33

-- basisbasis--set set truncation truncation (1(1--ee-- errorerror))

1) at ENERGY level:1) at ENERGY level:

>> >> E(E(nn))[[SCFSCF]] = E= ECBSCBS[[SCFSCF]] + A+ Aexpexp((--BBnn))+ + E(E(nn))[CORR][CORR] = E= ECBSCBS[CORR][CORR] + + CCnn--33

>> >> E(E(nn) = E) = ECBSCBS + + BeBe--((nn--1)1) + Ce+ Ce--((nn--1)1)

>> ………>> ………

Extrapolation to CBS limitExtrapolation to CBS limit

Feller, JCP Feller, JCP 9898, 7059 (1993) , 7059 (1993)

Helgaker et al., JCP Helgaker et al., JCP 106106, 9639 (1997) , 9639 (1997)

Peterson et al., JCP Peterson et al., JCP 100100, 7410 (1994) , 7410 (1994)

33--pt extrapol:pt extrapol:cccc--pVnZ, n=QpVnZ, n=Q--66

22--pt extrapol:pt extrapol:cccc--pVnZ, n=5,6pVnZ, n=5,6

Heckert, Kallay, Tew, Klopper, Gauss, JCP 125, 044108 (2006) Heckert, Kallay, Tew, Klopper, Gauss, JCP 125, 044108 (2006)

dxTCCSDEd

dxSCFHFdE

dxdEtot ))(()(

… … CV CORRELATION:… … CV CORRELATION:

-- SuitableSuitable basisbasis setssets: : cccc--pCVpCVnnZZ, , cccc--pwCVpwCVnnZZ, , cccc--pwCVpwCVnnZZ--PPPP

nn=T,Q,5=T,Q,5

-- AdditivityAdditivity approximationapproximation::EECV CV = = E E ((allall) ) –– E E ((fcfc))

2) “2) “Minor” Minor” error sourceserror sources in in ab initio calculationsab initio calculations::

dxcoreEd

dxTCCSDEd

dxSCFHFdE

dxdEtot )())(()(

mediummedium--large basis set:large basis set:cccc--p(w)CVQZ, ccp(w)CVQZ, cc--p(w)CV5Zp(w)CV5Z

Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005) Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005)

… … SCALAR RELATIVITY:… … SCALAR RELATIVITY:

-- SuitableSuitable basisbasis setssets and/or and/or approachapproach: : smallsmall--core core relativistic PPsrelativistic PPs

cccc--pVnZpVnZ--PP, PP, augaug--cccc--pVnZpVnZ--PP, PP, cccc--pwCVnZpwCVnZ--PPPP DK DK hamiltonian hamiltonian

cccc--pVnZpVnZ--DK, DK, ……. . 2nd 2nd order order direct PT direct PT

cccc--pVnZpVnZ, , cccc--pCVnZpCVnZ, , ……..

-- AdditivityAdditivity approximationapproximation

2) “2) “Minor” Minor” error sourceserror sources in in ab initio calculationsab initio calculations::

dxrelEd

dxTCCSDEd

dxSCFHFdE

dxdEtot )())(()(

DPT2:DPT2:uncontracteduncontracted--cccc--p(w)CVQZp(w)CVQZ

Michauk and Gauss, JCP 127, 044106 (2007) Michauk and Gauss, JCP 127, 044106 (2007) Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005) Heckert, Kallay, Gauss, Mol. Phys. 103, 2109 (2005)

Accuracy of Accuracy of Theoretical Rotational ConstantsTheoretical Rotational Constants

STATISTICAL ANALYSISSTATISTICAL ANALYSIS for for

•• 1616 molecules (molecules (9797 isotopologues)isotopologues)

•• 180180 rotational constantsrotational constants

Reference values: BReference values: Bee ,, BB00 from experiment from experiment

HF, N2, CO, F2, HCN, HNC, O=C=O, H2O, NH3,

CH4, HCCH, HOF, HNO, NH=NH, CH2=CH2, H2C=O

C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)

Normal Distribution of Relative ErrorsNormal Distribution of Relative Errors

Nc = normalization constant

Mean error: refi

calcii

Standard deviation:

iistd n 1

21exp)(

Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics

-3 -2 -1 0 1 2 3

CCSD(T)/TZ

normal distributions of relative errorsnormal distributions of relative errors

BBeecalccalc vs Bvs Bee

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/TZCCSD(T)/QZ

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/TZCCSD(T)/QZCCSD(T)/5Z

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6Z

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6ZCCSD(T)/6Z+ core

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/6Z+ core

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/6Z+ core

expexp

-3 -2 -1 0 1 2 3

CCSD(T)/6Z+ core

CCSD(T)/Z+ core

expexp

COMPOSITE APPROACH: COMPOSITE APPROACH: the “geometry scheme”the “geometry scheme”

2) at “PARAMETERS” level:2) at “PARAMETERS” level:

>> >> rr ((nn))[[SCFSCF]] = = rr CBSCBS[[SCFSCF]] + A+ Aexpexp((--BBnn))+ + rr ((nn))[CORR][CORR] = = rr CBSCBS[CORR][CORR] + + CCnn--33

>> >> rr ((nn) = ) = rr CBSCBS + + BeBe--((nn--1)1) + Ce+ Ce--((nn--1)1)

>> ………>> ………

Extrapolation to CBS limitExtrapolation to CBS limit

Feller, JCP Feller, JCP 9898, 7059 (1993) , 7059 (1993)

Helgaker et al., JCP Helgaker et al., JCP 106106, 9639 (1997) , 9639 (1997)

Peterson et al., JCP Peterson et al., JCP 100100, 7410 (1994) , 7410 (1994)

rr = = rrCBSCBS + + rrCVCV

wherewhererrCV CV = = r r ((((ww))CVnZCVnZ, , all all ee--) ) –– r r (((w)(w)CVnZCVnZ, , fcfc))

all electrons correlatedall electrons correlatedonly valence electrons correlatedonly valence electrons correlated

Additivity of CV effectsAdditivity of CV effects

rr = = rrtottot + + rrSRSR

wherewhere

rrSRSR = = rr ((relrel) ) –– rr ((nonnon--relrel))

relativistic optgrelativistic optgnonnon--relativistic optgrelativistic optg

Additivity of SR effectsAdditivity of SR effects

Validation: GEOM. vs GRAD.Validation: GEOM. vs GRAD.MoleculeMolecule ParameterParameter CBS/Geom. schemeCBS/Geom. scheme CBS/Grad. schemeCBS/Grad. scheme

HH22OO OO--HH 0.958390.95839 0.958360.95836OOHOOH 104.484104.484 104.478104.478

NHNH33 NN--HH 1.012101.01210 1.012061.01206HNHHNH 106.631106.631 106.641106.641

PHPH33 PP--HH 1.414351.41435 1.414641.41464HPHHPH 93.55593.555 93.55393.553

NHNH22 NN--HH 1.024761.02476 1.024741.02474HNHHNH 103.071103.071 103.060103.060

PHPH22 PP--HH 1.418251.41825 1.418461.41846HPHHPH 91.88291.882 91.87791.877

ClSiPClSiP ClCl--SiSi 2.014392.01439 2.014402.01440SiSi--PP 1.963541.96354 1.963401.96340

HCSHCS++ HH--CC 1.082001.08200 1.082141.08214CC--SS 1.478951.47895 1.479071.47907

Differences:Differences: 0.001 Å for distances 0.001 Å for distances 0.01 deg. for angles0.01 deg. for angles

VALIDATED!!VALIDATED!!Puzzarini , JPC A 113, 14530 (2009) Puzzarini , JPC A 113, 14530 (2009)

PuzzariniPuzzarini, , CazzoliCazzoli, Gauss , Gauss JMS 262, 37 (2010) JMS 262, 37 (2010)

CV correctionsCV corrections

SiHSiH33FF SiSi--F / F / ÅÅ SiSi--H / H / ÅÅ FSiH / deg.FSiH / deg.

gradient gradient schemescheme

--0.00520.0052 --0.00450.0045 0.000.00

geometry geometry schemescheme

--0.00530.0053 --0.00450.0045 0.000.00

basis = ccbasis = cc--pwCV5ZpwCV5Z

fullfull--T correctionsT corrections

ClPO: in progress ClPO: in progress SiHSiH33F: F: PuzzariniPuzzarini, , CazzoliCazzoli, Gauss , Gauss JMS 262, 37 (2010) JMS 262, 37 (2010)

gradient schemegradient scheme +0.0001+0.0001 +0.0002+0.0002 +0.00+0.00

geom. schemegeom. scheme +0.0002+0.0002 +0.0002+0.0002 +0.00+0.00

ClPOClPO ClCl--P / P / ÅÅ PP--O / O / ÅÅ ClPO/ deg.ClPO/ deg.

gradient schemegradient scheme +0.0009+0.0009 --0.00020.0002 --0.020.02

geom. schemegeom. scheme +0.0009+0.0009 --0.00020.0002 --0.020.02

basis = ccbasis = cc--pVTZpVTZ

fullfull--Q correctionsQ corrections

ClPO: in progress ClPO: in progress SiHSiH33F: F: PuzzariniPuzzarini, , CazzoliCazzoli, Gauss , Gauss JMS 262, 37 (2010) JMS 262, 37 (2010)

geom. schemegeom. scheme +0.0004+0.0004 --0.00000.0000 +0.01+0.01

ClPOClPO ClCl--P / P / ÅÅ PP--O / O / ÅÅ ClPO/ deg.ClPO/ deg.

geom. schemegeom. scheme +0.0018+0.0018 +0.0017+0.0017 +0.04+0.04

basis = ccbasis = cc--pVDZpVDZ

Which level for Which level for ““BIOMOLECULESBIOMOLECULES””??

COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule

(T)(diff)(CV)(CBS)T)diffCV(CBS rrrrr

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

RELIABLE? ACCURATE?RELIABLE? ACCURATE?

The challenge of the conformational The challenge of the conformational equilibrium in glycineequilibrium in glycine: :

can composite schemes shed light on the can composite schemes shed light on the observation of elusive conformers?observation of elusive conformers?

V. Barone, M. Biczysko, J. Bloino, C. Puzzarini, PCCP 15, 1358 (2013) V. Barone, M. Biczysko, J. Bloino, C. Puzzarini, JCTC 9, 1533 (2013) V. Barone, M. Biczysko, J. Bloino, C. Puzzarini, PCCP, in press (2013)

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

1) 1) “cheap” geom scheme“cheap” geom scheme

2) 2) “accurate” grad scheme“accurate” grad scheme

dxTCCSDEd

dxSCFHFdE

dxdEtot V)())(()(

cc-pV(T,Q)Zcc-pV(T,Q,5)Z cc-pCVTZ

The two most stable conformers ……The two most stable conformers ……

The following four stable conformers ……The following four stable conformers ……

“cheap” best “cheap” best vsvs “accurate” best: perfect match“accurate” best: perfect match

COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

RELIABLE? ACCURATE?RELIABLE? ACCURATE?

Which level for Which level for ““BIOMOLECULESBIOMOLECULES””??

Rotational constant Rotational constant

Inertia tensor Inertia tensor

More unknown parameters than data More unknown parameters than data ????

More unknown parameters than data More unknown parameters than data ????ISOTOPIC SUBSTITUTIONISOTOPIC SUBSTITUTION

16O 12C 32S

17O, 18O 33S, 34S13C

- NATURAL ABUNDANCE- ISOTOPICALLY ENRICHED

Equilibrium structureEquilibrium structure::need of need of BBee for various isotopic speciesfor various isotopic species

Bre BB

Rotational constant ofRotational constant ofvibrational ground statevibrational ground state Vibrational correctionVibrational correction

EXPERIMENTEXPERIMENT THEORYTHEORYP. Pulay, W. Meyer, J.E. Boggs, P. Pulay, W. Meyer, J.E. Boggs, J. Chem. Phys.J. Chem. Phys. 6868, 5077 (1978)., 5077 (1978).

“Semi“Semi--exp.” equilibrium structure exp.” equilibrium structure

Bre BB

fromfrom EEXXPERIMENTPERIMENT((various isotopic speciesvarious isotopic species)) from from TTHHEOREORY Y

((cubic force fieldcubic force field))

Accuracy: experimental qualityAccuracy: experimental qualityPawłowskiPawłowski, , JørgensenJørgensen, , OlsenOlsen, , HegelundHegelund, , HelgakerHelgaker, Gauss, , Gauss, BakBak, , StantonStanton JCPJCP 116116 6482 (2002)6482 (2002)

FITFIT

N3O7O8

H11H12

SemiSemi--exp equilibrium structure of large moleculeexp equilibrium structure of large molecule

URACIL: 21 independent geometrical parametersURACIL: 21 independent geometrical parameters

Isotopic substitution:Isotopic substitution:-- 1616O O 1818OO-- 1414N N 1515NN-- 1212C C 1313CC

10 isotopic species10 isotopic species

20 rotational constants20 rotational constants

Puzzarini & Barone, PCCP 13, 7158 (2011)

Vaquero, Sanz, López, Alonso, J. Phys. Chem. Lett. 111A, 3443 (2007).

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

1) 1) “cheap” geom scheme“cheap” geom scheme

2) 2) “accurate” grad scheme“accurate” grad scheme

dxTCCSDEd

dxSCFHFdE

dxdEtot V)())(()(

cc-pV(T,Q)Zcc-pV(T,Q,5)Z cc-pCVTZ

Best est. rea Semi-exp. re

b Exp. rsc

Fit 1 Fit 2 Fit 3DistancesN1-C2 1.3785 1.38175(53) 1.38163(65) 1.38161(51) 1.386(5)C2-N3 1.3756 1.3763 1.3763 1.3762N3-C4 1.3974 1.39793(40) 1.39823(47) 1.39835(45) 1.38(2)C4-C5 1.4539 1.45500(57) 1.45485(99) 1.45481(57) 1.451(4)C5-C6 1.3433 1.34496(59) 1.34576(107) 1.34473(58) 1.379(4)C6-N1 1.3723 1.37196(55) 1.37160(100) 1.37258(66) 1.352(14)C2-O7 1.2112 1.21025(21) 1.21015(26) 1.21015(21) 1.219(4)C4-O8 1.2138 1.21278(24) 1.21268(34) 1.21269(24) 1.22(2)N1-H9 1.0046 1.0004(70) N3-H10 1.0090 1.0110(96) C5-H11 1.0766 1.0695(52)C6-H12 1.0793 1.0856(32)

AnglesC2-N1-C6 123.38 123.374(19) 123.394(35) 123.370(21) 123.0(11)N1-C6-C5 121.91 121.924(10) 121.920(10) 121.9237(97) 122.3(6)C6-C5-C4 119.49 119.516(16) 119.501(20) 119.523(16) 118.8(12)C5-C4-N3 113.97 113.860(22) 113.859(33) 113.858(22) 115.4(16)C4-N3-C2 127.75 127.942 127.947 127.945N3-C2-N1 113.51 113.383 113.379 113.380N1-C2-O7 123.62 123.883(44) 123.878(54) 123.874(42) 122.3(8)C5-C4-O8 125.83 125.768(48) 125.765(75) 125.767(45) 118.8(7)C2-N1-H9 115.22 C2-N3-H10 115.70 115.52(40) C6-C5-H11 122.11 N1-C6-H12 115.34

Non-determinable Parameters: fixed at the corresponding theo values

COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule Equilibrium Rotational ConstantsEquilibrium Rotational Constants

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

Vibrational Corrections to Rotational ConstantsVibrational Corrections to Rotational Constants

Vibrational corrections to rotational constants:Vibrational corrections to rotational constants:

How to get vibrational corrections to How to get vibrational corrections to BB??SecondSecond--order vibrational perturbation theoryorder vibrational perturbation theory

(VPT2)(VPT2)

wheredimensionless normal coordinate

vibrational angular momentum

inverse inertia tensor

potential

WATSON WATSON HamiltonianHamiltonian

, = (x,y,z) r {normal coord}

unperturbed Hamiltonian:unperturbed Hamiltonian:

perturbations:perturbations:

Coriolis couplingCoriolis coupling

anharmonic anharmonic correctionscorrections

Harmonic ffHarmonic ff

How to get vibrational corrections to How to get vibrational corrections to BB??SecondSecond--order vibrational perturbation theoryorder vibrational perturbation theory

(VPT2)(VPT2)

vibrationvibration--rotation interaction constants:rotation interaction constants:

vibrational corrections to rotational constants:vibrational corrections to rotational constants:

Beyond the RigidBeyond the Rigid--Rotator Rotator ApproximationApproximation

Computation of Cubic and Quartic Force Computation of Cubic and Quartic Force FieldsFields

•• cubic force fields:cubic force fields:

single numerical differentiation along qr

•• quartic force fields:quartic force fields:

double numerical differentiation along qr

Schneider Schneider && Thiel, Thiel, Chem. Phys. LettChem. Phys. Lett.. 157157, 367 (1989), 367 (1989)Stanton et al., Stanton et al., J. Chem. PhysJ. Chem. Phys. . 108108, 7190 (1998), 7190 (1998)

Accurate force Accurate force fieldfield

>>>> >>>> Main requirementsMain requirements::-- ““correlatedcorrelated”” methodmethod-- cc basis cc basis setset

-- harmonic ffharmonic ff: : analytic analytic 2nd 2nd derivderiv. of E. of E

Schneider Schneider && Thiel, Thiel, Chem. Phys. LettChem. Phys. Lett.. 157157, 367 (1989), 367 (1989)Stanton et al., Stanton et al., J. Chem. PhysJ. Chem. Phys. . 108108, 7190 (1998), 7190 (1998)

-- harmonic ffharmonic ff: : analytic analytic 2nd 2nd derivderiv. of E. of E

-- anharmonic partanharmonic part: : numerical differnumerical differ..

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

Vibrational Corrections to Rotational ConstantsVibrational Corrections to Rotational ConstantsB3LYP/N07D MP2/cc-pVTZ

CalculatedCalculated Experiment Experiment AA00 MHzMHz 3885.4753885.475 3883.873021(60)3883.873021(60)

BB00 MHzMHz 2027.7632027.763 2023.732581(45)2023.732581(45)

CC00 MHzMHz 1332.7611332.761 1330.928108(33)1330.928108(33)

DDJ J kHzkHz 0.0610.061 0.06336(44)0.06336(44)

DDJKJK kHzkHz 0.1070.107 0.1055(23)0.1055(23)

DDKK kHzkHz 0.4470.447 0.4530(32)0.4530(32)

dd11 kHzkHz --0.0260.026 --0.02623(18)0.02623(18)

dd22 kHzkHz --0.0060.006 --0.00680(13)0.00680(13)

aaaa MHzMHz 1.7391.739 1.7600 (25)1.7600 (25)

bbbb MHzMHz 1.9521.952 1.9811(29)1.9811(29)

aaaa MHzMHz 1.8711.871 1.9255(24)1.9255(24)

bbbb MHzMHz 1.4911.491 1.5273(32)1.5273(32)

URACILURACIL

<0.2<0.2%%

Accuracy of Accuracy of Theoretical Rotational ConstantsTheoretical Rotational Constants

STATISTICAL ANALYSISSTATISTICAL ANALYSIS for for

•• 1616 molecules (molecules (9797 isotopologues)isotopologues)

•• 180180 rotational constantsrotational constants

Reference values: BReference values: Bee ,, BB00 from experiment from experiment

HF, N2, CO, F2, HCN, HNC, O=C=O, H2O, NH3,

CH4, HCCH, HOF, HNO, NH=NH, CH2=CH2, H2C=O

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZBBcalccalc vs vs BB00expexp

normal distributions of relative errorsC. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZBBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/V5ZCCSD(T)/V5ZBBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZBBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6Z + CVCCSD(T)/V6Z + CVBBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6Z + CVCCSD(T)/V6Z + CVCCSD(T)/V6Z + CV + fTCCSD(T)/V6Z + CV + fTBBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VTZCCSD(T)/VTZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6Z + CVCCSD(T)/V6Z + CVCCSD(T)/V6Z + CV + fTCCSD(T)/V6Z + CV + fTCCSD(T)/V6Z + CV + fTCCSD(T)/V6Z + CV + fT+ + fQfQ

BBcalccalc vs vs BB00expexp

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQfQ

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQ fQ + + vibvib

-4 -3 -2 -1 0 1 2 3 4

CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQ fQ + + vibvib

CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQ fQ + + vib vib + + eleele

Electronic contribution to Electronic contribution to BBelvibe BBBB 0

eel Bg

g = rotational g tensorme = mass of the electronmp = mass of the proton

=x,y,zprinc. inertia system

CCSD(T) calc: Gauss, Ruud, Kallay, JCP 127, 074101 (2007)CCSD(T) calc: Gauss, Ruud, Kallay, JCP 127, 074101 (2007)

FREQUENCY (MHz)

1 12 00 11 25 0 113 00 1 135 0 11 40 0 114 5051 50 5 200 77 50 780 0 7 850

Predictio n using Be

Experiment

1515NN22--URACIL: rotational spectrum in the 5URACIL: rotational spectrum in the 5--12 GHz range 12 GHz range

Experimental data from: Experimental data from: V. Vaquero, M. E.V. Vaquero, M. E. SanzSanz, J. C. Lo, J. C. Lopezpez and J. L.and J. L. AlonsoAlonso, , JJPCPCAA 111, 3443111, 3443 (2007)(2007)..Simulation from: Puzzarini, Simulation from: Puzzarini, PCCPPCCP 15, 6595 (2013) 15, 6595 (2013)

FREQUENCY (MHz)

5150 5200 7750 7800 7850

Prediction using Be

Prediction using B0

Exp erime nt

1120 0 1125 0 113 00 11 350 1 1400 11450

1515NN22--URACIL: rotational spectrum in the 5URACIL: rotational spectrum in the 5--12 GHz range 12 GHz range

Experimental data from: Experimental data from: V. Vaquero, M. E.V. Vaquero, M. E. SanzSanz, J. C. Lo, J. C. Lopezpez and J. L.and J. L. AlonsoAlonso, , JJPCPCAA 111, 3443111, 3443 (2007)(2007)..Simulation from: Puzzarini, Simulation from: Puzzarini, PCCPPCCP 15, 6595 (2013) 15, 6595 (2013)

CENTRIFUGALCENTRIFUGAL--DISTORTIONDISTORTIONCONSTANTSCONSTANTS

HarmonicHarmonic force force fieldfield: : quartic quartic centrifugalcentrifugal--distortion constantsdistortion constants

CubicCubic force force fieldfield: : sexticsextic centrifugalcentrifugal--distortiondistortion constantsconstants

CentrifugalCentrifugal--distortion constants distortion constants requires requires force force field calculationsfield calculations

… … … … … …

Quartic centrifugalQuartic centrifugal--distortion constants:distortion constants:combinations of combinations of ’’ss

Linear MoleculesLinear Molecules

Quartic centrifugalQuartic centrifugal--distortion constants:distortion constants:effect on rotational spectrumeffect on rotational spectrum

Sextic centrifugalSextic centrifugal--distortion constants:distortion constants:combinations of combinations of ’’ss

Aliev Aliev && Watson, J. Mol. Spectrosc. 61, 29 (1976) Watson, J. Mol. Spectrosc. 61, 29 (1976)

Sextic centrifugalSextic centrifugal--distortion constants:distortion constants:combinations ofcombinations of ’’ss

56.737 56.737 --1.673 1.673 0.854 0.854

359.120 359.120 --42.871 42.871 --10.734 10.734 1.728 1.728 4.807 4.807 3.467 3.467

231.368 231.368 --40.959 40.959 8.818 8.818

CCSD(T)/CCSD(T)/aaugugCV5ZCV5Z

95.31(73) 95.31(73) 56.253 56.253 55.113 55.113 55.733 55.733 60.687 60.687 KK / kHz/ kHz--4.05(13) 4.05(13) --1.681 1.681 --1.699 1.699 --1.631 1.631 --1.233 1.233 JKJK / kHz/ kHz0.9938(40) 0.9938(40) 0.856 0.856 0.864 0.864 0.845 0.845 0.715 0.715 JJ / kHz/ kHz

535.07(44) 535.07(44) 354.604 354.604 338.834 338.834 347.717 347.717 414.678 414.678 KK / kHz/ kHz--71.02(38) 71.02(38) --42.413 42.413 --39.905 39.905 --40.950 40.950 --47.771 47.771 KJKJ / kHz/ kHz--10.89(30) 10.89(30) --10.740 10.740 --10.880 10.880 --10.633 10.633 --9.069 9.069 JKJK / kHz/ kHz2.445(53) 2.445(53) 1.731 1.731 1.747 1.747 1.708 1.708 1.451 1.451 JJ / kHz/ kHz9.9004(51) 9.9004(51) 4.781 4.781 4.722 4.722 4.820 4.820 5.486 5.486 KK / MHz/ MHz3.69282(21)3.69282(21)3.472 3.472 3.470 3.470 3.431 3.431 3.155 3.155 JJ / MHz/ MHz

271.0554(57) 271.0554(57) 229.674 229.674 224.317 224.317 227.951 227.951 255.112 255.112 KK / MHz/ MHz--45.2241(51) 45.2241(51) --40.872 40.872 --40.303 40.303 --40.374 40.374 --40.525 40.525 JKJK / MHz/ MHz9.2889(14) 9.2889(14) 8.826 8.826 8.818 8.818 8.730 8.730 8.127 8.127 JJ / MHz/ MHz

ExperimentExperimentCCSD(T)/CCSD(T)/aaugugCVQZCVQZ

CCSD(T)/CCSD(T)/aaugugCVTZCVTZ

CCSD/CCSD/aaugugCVTZCVTZ

HFHF--SCF/SCF/aaugugCVTZ CVTZ DD22

1717OO

Quartic & sextic centrifugalQuartic & sextic centrifugal--distortion constantsdistortion constants

Puzzarini, Cazzoli, Gauss, Puzzarini, Cazzoli, Gauss, J. Chem. Phys.J. Chem. Phys. 137137, 154311 (2012), 154311 (2012)

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

CentrifugalCentrifugal--Distortion ConstantsDistortion Constants(MP2/VTZ)(MP2/aVTZ)fc)(MP2/CVTZ,all)(MP2/CVTZ,Z)(CCSD(T)/T(best) DDDDDD

CV diffuse

BB00 MHzMHz 2027.7632027.763 2023.732581(45)2023.732581(45)

CC00 MHzMHz 1332.7611332.761 1330.928108(33)1330.928108(33)

DDJ J kHzkHz 0.0610.061 0.06336(44)0.06336(44)

DDJKJK kHzkHz 0.1070.107 0.1055(23)0.1055(23)

DDKK kHzkHz 0.4470.447 0.4530(32)0.4530(32)

dd11 kHzkHz --0.0260.026 --0.02623(18)0.02623(18)

dd22 kHzkHz --0.0060.006 --0.00680(13)0.00680(13)

aaaa MHzMHz 1.7391.739 1.7600 (25)1.7600 (25)

bbbb MHzMHz 1.9521.952 1.9811(29)1.9811(29)

aaaa MHzMHz 1.8711.871 1.9255(24)1.9255(24)

bbbb MHzMHz 1.4911.491 1.5273(32)1.5273(32)

~1~1%%

URACILURACIL

HYPERFINE HYPERFINE STRUCTURESTRUCTURE

Accurate Accurate hyperfine parametershyperfine parameters

>>>> >>>> MainMain requirementsrequirements::

-- accurate accurate method method [CCSD(T)][CCSD(T)]-- cc basiscc basis set [nset [nQ]Q]

-- CV CV correction correction [[additivityadditivity]]

-- vibrational correction vibrational correction [[ffff:: correlcorrel methmeth..]]

QuantumQuantum--Chemical CalculationChemical Calculationof Hyperfine Parametersof Hyperfine Parameters

• Nuclear quadrupole couplingNuclear quadrupole coupling

firstfirst--order propertyorder property

ELECTRIC FIELD GRADIENT

RK = position of the K-th nucleus r = position of the electron

-- first derivative of first derivative of EE wrt wrt QQKK computed at computed at QQ=0=0-- expectation value of the corresponding operatorexpectation value of the corresponding operator

Nuclear quadrupoleNuclear quadrupole--coupling constants:coupling constants:from electric field gradientsfrom electric field gradients

KijKij qeQ

ij-th element of the nuclear quadrupole-coupling tensor of the K-th nucleus:

-eQK = quadrupole momentqij = ij-th element of the electric field-gradient tensorIK 1

(known!!)

462980 462990 463000 463010

trans-CH35Cl=CHF: portion of the J=0; K-1=+1; K-1 = 4 band

CALC.(without Cl quadrupole coupling)

FREQUENCY (MHz)462980 462990 463000 463010

trans-CH35Cl=CHF: portion of the J=0; K-1=+1; K-1 = 4 band

CALC.(without Cl quadrupole coupling)

FREQUENCY (MHz)

(~2 GHz freq. Shift!)(~2 GHz freq. Shift!)

Nuclear quadrupoleNuclear quadrupole--coupling constants:coupling constants:effect on rotational spectrumeffect on rotational spectrum

MP2/cc-pV(T,Q)Z

MP2/cc-pCVTZ

MP2/aug-cc-pVTZ

CCSD(T)/cc-pVTZ

Nitrogen qudrupoleNitrogen qudrupole--coupling Constantscoupling Constants

(diff)QZ)(TZVTZ)(CCSD(T)/Cest)( b

MP2/cc-pCV(T,Q)Z

MP2/aug-cc-pVTZ

BB00 MHzMHz 2027.7632027.763 2023.732581(45)2023.732581(45)

CC00 MHzMHz 1332.7611332.761 1330.928108(33)1330.928108(33)

DDJ J kHzkHz 0.0610.061 0.06336(44)0.06336(44)

DDJKJK kHzkHz 0.1070.107 0.1055(23)0.1055(23)

DDKK kHzkHz 0.4470.447 0.4530(32)0.4530(32)

dd11 kHzkHz --0.0260.026 --0.02623(18)0.02623(18)

dd22 kHzkHz --0.0060.006 --0.00680(13)0.00680(13)

aaaa MHzMHz 1.7391.739 1.7600 (25)1.7600 (25)

bbbb MHzMHz 1.9521.952 1.9811(29)1.9811(29)

aaaa MHzMHz 1.8711.871 1.9255(24)1.9255(24)

bbbb MHzMHz 1.4911.491 1.5273(32)1.5273(32)

11--22%%

URACILURACIL

QuantumQuantum--Chemical CalculationChemical Calculationof Hyperfine Parametersof Hyperfine Parameters

• SpinSpin--rotation interactionrotation interaction

secondsecond--order propertyorder property

Nuclear spinNuclear spin--rotation tensorrotation tensor

Electronic contributionElectronic contribution Nuclear contributionNuclear contribution

K = gyromagnetic ratio of the K-th nucleus

lK = electronic angular momentum defined wrt RK

l = electronic angular momentumJ = rotational angular momentumI = nuclear spin angular momentum

requires equilibrium geometry: no „electronic property“requires equilibrium geometry: no „electronic property“

addditional contribution due to: addditional contribution due to:

vibrational correctionsvibrational corrections (anharmonic force field)(anharmonic force field)

QuantumQuantum--Chemical CalculationChemical Calculationof Spectroscopic Parametersof Spectroscopic Parameters

• SpinSpin--spin couplingspin couplingDIPOLAR SPIN-SPIN COUPLING TENSOR

VIBRATIONAL VIBRATIONAL CORRECTIONCORRECTION

eqavevib PPP

Difference between vibrationally averaged Difference between vibrationally averaged value and equilibrium values (same level: i.e., value and equilibrium values (same level: i.e., same method same method and and same basis setsame basis set))

VIBRATIONAL AVERAGINGVIBRATIONAL AVERAGING

sreqsr

eq QQQQPQ

wherewhere

rrssrQQ

ExpansionExpansion of the of the expectation valueexpectation value over the over the vib wf vib wf aroundaround the the equil wrt normalequil wrt normal--coordinate coordinate displacementsdisplacements

A.A.A.A. AuerAuer, J. Gauss , J. Gauss && J.F.J.F. StantonStanton, JCP , JCP 118118, 10407 (2003), 10407 (2003)

Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011)

188308.35 188308.40 188308.45 188308.50

F1 = 28,26F1 = 27

F1 = 0 J = 273,24 - 273,25

transtrans--HCOOD: hyperfine structure due to DHCOOD: hyperfine structure due to D

188308.35 188308.40 188308.45 188308.50

F1,F = 28,57/2 26,53/2F1,F = 28,55/2

26,51/2

F1,F = 27,55/2F1,F = 27,53/2

F1, F = 0 J = 273,24 - 273,25

transtrans--HCOOD: hyperfine structure due to D and HHCOOD: hyperfine structure due to D and H

188308.35 188308.40 188308.45 188308.50

F1,F = 28,57/2 26,53/2F1,F = 28,55/2

26,51/2

F1,F = 27,55/2F1,F = 27,53/2

F1, F = 0 J = 273,24 - 273,25

transtrans--HCOOD: hyperfine structure due to D and HHCOOD: hyperfine structure due to D and H

ROTATIONAL SPECTROSCOPY:ROTATIONAL SPECTROSCOPY:Interplay Interplay ofof

Experiment Experiment andand TheoryTheory

PREDICTING PREDICTING ROTATIONAL SPECTRAROTATIONAL SPECTRA

Puzzarini , Biczysko, Barone, Pena, Cabezas,Puzzarini , Biczysko, Barone, Pena, Cabezas, Alonso, Alonso, PCCPPCCP accepted accepted

Observation of the rotational spectrum Observation of the rotational spectrum of thiouracilof thiouracil: :

Can composite schemes provide the spectroscopic Can composite schemes provide the spectroscopic parameters with the proper accuracy?parameters with the proper accuracy?

1) 1) rree (B(Bee), D’s, q’s), D’s, q’s: : “cheap” geom scheme“cheap” geom scheme

2) 2) alphasalphas: : DFTDFT

Best = CBS(MP2/TZBest = CBS(MP2/TZ--QZ) QZ) + CV(MP2/CVTZ)+ CV(MP2/CVTZ)+ diff(MP2/AVTZ)+ diff(MP2/AVTZ)+ pertT(CCSD(T)/VTZ)+ pertT(CCSD(T)/VTZ)

0 DFT = B3LYP/SNSDDFT = B3LYP/SNSD

Parameter

A0 [MHz] 3555.18805(64) 3555.458 3545.6594(11) 3545.945B0 [MHz] 1314.86002(27) 1315.287 1276.1741(51) 1276.569C0 [MHz] 960.03086(16) 960.200 938.57117(54) 938.732

14N(1)

χaa [MHz] 1.634(10) 1.609 1.616(13) 1.614χbb [MHz] 1.777(12) 1.813 1.755(17) 1.807χcc [MHz] -3.411(12) -3.422 -3.371(17) -3.422χab [MHz] - 0.314 - 0.316

14N(3)

χaa [MHz] 1.726(10) 1.739 1.732(13) 1.733χbb [MHz] 1.399(13) 1.384 1.429(19) 1.390χcc [MHz] -3.125(13) -3.123 -3.161(19) -3.123χab [MHz] - -0.336 - -0.339

Main 34SExp Theo Exp Theo

0.1%0.1%

6000 7000 8000 9000 10000 11000 12000

FREQUENCY (MHz)

EXPERIMENT THEORY

9500 9600 9700 9800 9900 10000 10100

52,4 - 51,5 41,4 - 30,3

50,5 - 41,4

FREQUENCY (MHz)

EXPERIMENT THEORY

9500 9600 9700 9800 9900 10000 10100

52,4 - 51,5 41,4 - 30,3

50,5 - 41,4

FREQUENCY (MHz)

EXPERIMENT THEORY (only B's) THEORY (B's + D's)

9601.0 9601.5 9602.0 9602.5 9603.0 9603.5 9604.0 9604.5

50,5 - 41,4

FREQUENCY (MHz)

EXPERIMENT THEORY (only B's) THEORY (B's + D's)

~3 MHz~3 MHz

Parameter

A0 [MHz] 3555.18805(64) 3555.458 3545.6594(11) 3545.945B0 [MHz] 1314.86002(27) 1315.287 1276.1741(51) 1276.569C0 [MHz] 960.03086(16) 960.200 938.57117(54) 938.732

14N(1)

χaa [MHz] 1.634(10) 1.609 1.616(13) 1.614χbb [MHz] 1.777(12) 1.813 1.755(17) 1.807χcc [MHz] -3.411(12) -3.422 -3.371(17) -3.421χab [MHz] - 0.314 - 0.316

14N(3)

χaa [MHz] 1.726(10) 1.739 1.732(13) 1.733χbb [MHz] 1.399(13) 1.384 1.429(19) 1.390χcc [MHz] -3.125(13) -3.123 -3.161(19) -3.123χab [MHz] - -0.336 - -0.339

Main 34SExp Theo Exp Theo

~~1%1%

4514.5 4515.0 4515.5 4516.0

FREQUENCY (MHz)

theory experiment

~0.4 MHz2,1 0,02,1 1,12,1 2,2

1,1 0,01,1 1,11,1 2,2

2,3 2,2

1,2 1,11,2 2,2

0,1 0,00,1 1,10,1 2,2

2,2 1,12,2 2,2

1,0 1,1

Cazzoli, Puzzarini, Stopkowicz, Gauss, Cazzoli, Puzzarini, Stopkowicz, Gauss, A A &&AA 520520, A64 (2010), A64 (2010)

SpectroscopySpectroscopy of Bolognaof Bologna107638 .20 107638 .25 107638 .30 107638 .35

H CO O H : J = 182 ,16

- 182 ,17

R F da ta : on ly S R

E xpe rim en t: Lam b-d ip

J.-C. Chardon, C. Genty, D. Guichon, & J.-G. Theobald, J. Chem. Phys. 64, 1434 (1976) “rf spectrum and hyperfine structure of formic acid”

H CO O H : J = 182 ,16

- 182 ,17

E xpe rim en t: Lam b-d ip

J.-C. Chardon, C. Genty, D. Guichon, & J.-G. Theobald, J. Chem. Phys. 64, 1434 (1976)

H CO O H : J = 182 ,16

- 182 ,17

T heo ry:on ly S R

T heo ry: S R and S S

E xpe rim en t

H CO O H : J = 182 ,16

- 182 ,17

T heo ry:on ly S R

T heo ry: S R and S S

E xpe rim en t

ExperimentExperiment TheoryTheory RFRF resultsresults

CCaaaa [H(C)][H(C)] --6.835(46)6.835(46) --7.027.02 --7.50(20)7.50(20)

CCbbbb [H(C)][H(C)] 1.0371.037 1.041.04

CCcccc [H(C)][H(C)] --0.8014(96)0.8014(96) --0.820.82

CCaaaa [H(O)][H(O)] --6.868(45)6.868(45) --6.946.94 --6.55(20)6.55(20)

CCbbbb [H(O)][H(O)] 0.781(20)0.781(20) 0.770.77

CCcccc [H(O)][H(O)] --1.290(15)1.290(15) --1.321.32

1.51.5DDaaaa 4.49(12)4.49(12) 4.624.62 ----

((DDbbbb –– DDcccc)/4)/4 --3.53(35)3.53(35) --3.473.47 ----Equil: CCSD(T)/CV5Z +Equil: CCSD(T)/CV5Z +

Vib. Corr: CCSD(T)/CVTZVib. Corr: CCSD(T)/CVTZ

Hyperfine parameters of transHyperfine parameters of trans--HCOOHHCOOH

ExperimentExperiment TheoryTheory RFRF resultsresults

CCaaaa [H(C)][H(C)] --6.835(46)6.835(46) --7.027.02 --7.50(20)7.50(20)

CCbbbb [H(C)][H(C)] 1.0371.037 1.041.04 --7.2(40)7.2(40)

CCcccc [H(C)][H(C)] --0.8014(96)0.8014(96) --0.820.82 7.5(40)7.5(40)

CCaaaa [H(O)][H(O)] --6.868(45)6.868(45) --6.946.94 --6.55(20)6.55(20)

CCbbbb [H(O)][H(O)] 0.781(20)0.781(20) 0.770.77 8.2(40)8.2(40)

CCcccc [H(O)][H(O)] --1.290(15)1.290(15) --1.321.32 --8.6(40)8.6(40)

1.51.5DDaaaa 4.49(12)4.49(12) 4.624.62 ----

((DDbbbb –– DDcccc)/4)/4 --3.53(35)3.53(35) --3.473.47 ----

Hyperfine parameters of transHyperfine parameters of trans--HCOOHHCOOH

Cazzoli, Puzzarini, Stopkowicz, Gauss, A &A 520, A64 (2010)

MOLECULAR PROPERTIESMOLECULAR PROPERTIES

-- ELECTRIC:ELECTRIC:

-- MAGNETIC:MAGNETIC:

• Dipole moment

ElectricElectric and and magneticmagnetic properties properties fromfrom RotationalRotational SpectroscopySpectroscopy

• Nuclear quadrupole coupling

• Spin-rotation interaction• Spin-spin interaction

ELECTRIC PROPERTIESELECTRIC PROPERTIES

Electric dipole momentElectric dipole moment

Analysis of the spectra completed:Analysis of the spectra completed:1. Transitions assigned 1. Transitions assigned (transition frequencies retrieved)(transition frequencies retrieved)

2. Frequecies fitted 2. Frequecies fitted (with the proper Hamiltonian)(with the proper Hamiltonian)

3. Spectroscopic parameters:3. Spectroscopic parameters:

-- rotational constants rotational constants BB-- centrifugalcentrifugal--distortion constants distortion constants D, H, D, H, ……-- hyperfine parameters hyperfine parameters (if the case)(if the case)

-- dipole moment dipole moment (if Stark spectroscopy)(if Stark spectroscopy)

1 0 6 7 8 0 1 0 6 7 9 0 1 0 6 8 0 0 1 0 6 8 1 0

F = 1 1 - 1 1

F re q u e n c y (M H z )

6 4 .8 V 7 4 .0 V 8 2 .2 V 9 1 .1 V 1 0 6 .6 V

J = 52 ,3

- 51 ,4

F = 9 - 9

CHCH22FBrFBr

UnknownUnknown molecularmolecular dipoledipole moment …moment …ExperimentExperiment: : StarkStark spectroscopyspectroscopy ……UnknownUnknown molecularmolecular dipoledipole momentmoment ……

1 0 6 7 8 0 1 0 6 7 9 0 1 0 6 8 0 0 1 0 6 8 1 0

F = 1 1 - 1 1

6 4 .8 V 7 4 .0 V 8 2 .2 V 9 1 .1 V 1 0 6 .6 V

J = 52 ,3

- 51 ,4

F = 9 - 9

CHCH22FBrFBr

UnknownUnknown molecularmolecular dipoledipole moment …moment …ExperimentExperiment: : StarkStark spectroscopyspectroscopy ……UnknownUnknown molecularmolecular dipoledipole momentmoment ……

-- POSITIVE PEAKS: unperturbed transitionsPOSITIVE PEAKS: unperturbed transitions-- NEGATIVE PEAKS: Stark componentsNEGATIVE PEAKS: Stark components

1 0 6 7 8 0 1 0 6 7 9 0 1 0 6 8 0 0 1 0 6 8 1 0

F = 1 1 - 1 1

6 4 .8 V 7 4 .0 V 8 2 .2 V 9 1 .1 V 1 0 6 .6 V

J = 52 ,3

- 51 ,4

F = 9 - 9

CHCH22FBrFBr

aVQZaVQZ -0.341 -1.696

aV5ZaV5Z -0.346 -1.700

CBSCBS -0.350 -1.702

CBS+CVCBS+CV -0.355 -1.710

Expt.Expt. -0.3466(11) -1.704(26)

CBS+CV+ZPVCBS+CV+ZPV --0.339 -1.701

CazzoliCazzoli, , PuzzariniPuzzarini, , BaldacciBaldacci && BaldanBaldan JMSJMS 241241 115 (2007)115 (2007)

ValuesValues in in debyedebye

DIPOLE MOMENT of CHDIPOLE MOMENT of CH22FIFI

2nd-order Direct Perturbation Theory

spin-free Dirac Coulomb approach

importance of relativistic effects importance of relativistic effects for heavy elementsfor heavy elements

-0.022

Values in debye

Analysis of the spectra completed:Analysis of the spectra completed:1. Transitions assigned 1. Transitions assigned (transition frequencies retrieved)(transition frequencies retrieved)

2. Frequecies fitted 2. Frequecies fitted (with the proper Hamiltonian)(with the proper Hamiltonian)

3. Spectroscopic parameters:3. Spectroscopic parameters:

-- rotational constants rotational constants BB-- centrifugalcentrifugal--distortion constants distortion constants D, H, D, H, ……-- hyperfine parameters hyperfine parameters (if the case)(if the case)

-- dipole moment dipole moment (if Stark spectroscopy)(if Stark spectroscopy)

Nuclear QuadrupoleNuclear QuadrupoleCouplingCoupling

DETERMINATIONDETERMINATIONof the of the

NUCLEAR QUADRUPOLENUCLEAR QUADRUPOLEMOMENTMOMENT

Bromine Nuclear Quadrupole MomentBromine Nuclear Quadrupole Moment

yearyear eQ eQ Lederer, ShirleyLederer, Shirley 1978 293TaqquTaqqu 1978 331(4)Kellö, SadlejKellö, Sadlej 1990 304.5Kellö, SadlejKellö, Sadlej 1996 298.9Hass, PetrilliHass, Petrilli 2000 305(5); 308.7Van Lenthe, BaerendsVan Lenthe, Baerends 2000 300(10)Bieron et al.Bieron et al. 2001 313(3)

values in mbarn for values in mbarn for 7979BrBr

Revision of the Revision of the 7979Br Quadrupole MomentBr Quadrupole Moment

nuclear quadrupole moment computed

electric field gradient

experimentalquadrupole coupling

Stopkowicz, Cheng, Harding, Puzzarini, Gauss, Mol. Phys. Stopkowicz, Cheng, Harding, Puzzarini, Gauss, Mol. Phys. 111111, 1382 (2013) , 1382 (2013)

HBrHBr

Bromine Quadrupole Coupling in CH2FBr

good agreement between theory and experimentgood agreement between theory and experiment

Theory:

χijexp χij

rel+vib Δ/%

χaa 443.431(8) 441.4 0.45

χbb-χcc 153.556(26) 154.1 0.35

χab -278.56(19) -278.4 0.06

including relativistic effects including relativistic effects && using new Q:using new Q:

Stopkowicz, Cheng, Harding, Puzzarini, Gauss, Mol. Phys. Stopkowicz, Cheng, Harding, Puzzarini, Gauss, Mol. Phys. 111111, 1382 (2013) , 1382 (2013)

MAGNETIC PROPERTIESMAGNETIC PROPERTIES

NMRNMR MWMW

Bryce Bryce & & WasylishenWasylishen, , AccAcc. . ChemChem. Res.. Res. 3636, 327 (2003), 327 (2003)

connectionconnection

nuclear magneticnuclear magneticshieldingshielding

absolute shieldingabsolute shieldingscalesscales

RamseyRamsey--FlygareFlygareequationsequations

formform of of HamiltoniansHamiltonians::coupling mechanismcoupling mechanism

vsvstensor ranktensor rank

nuclear quadrupole nuclear quadrupole couplingcoupling

nuclear quadrupole nuclear quadrupole couplingcoupling CCQQ

nuclear nuclear spinspin--rotationrotation

chemical chemical shiftshift

tensor spintensor spin--spinspincouplingcoupling ((rank rank 2)2)

scalar scalar spinspin--spinspincouplingcoupling ((rank rank 0)0)

direct direct dipolardipolarcouplingcoupling

indirect spinindirect spin--spinspincouplingcoupling

nuclear magnetic shielding nuclear magnetic shielding

DIATOMIC or LINEAR MOLECULESDIATOMIC or LINEAR MOLECULES

= = dd ++ pp

DIAMAGNETIC PARTDIAMAGNETIC PART PARAMAGNETIC PARTPARAMAGNETIC PART

DIATOMIC or LINEAR MOLECULESDIATOMIC or LINEAR MOLECULES

= = dd ++ pp

ASYMMETRICASYMMETRIC--TOPTOP MOLECULESMOLECULES

= = dd ++ pp

1717 & & 221717

Puzzarini, Cazzoli, Harding, Vázquez, Gauss, Puzzarini, Cazzoli, Harding, Vázquez, Gauss, work in progress work in progress …………

Absolute NMR shielding scaleAbsolute NMR shielding scale

SpectroscopySpectroscopy of Bolognaof Bologna

The beginning of the story ….The beginning of the story ….

385784 385786 385788 385790

J = 41,4

- 32,1

Experiment

Real+Ghost

FREQUENCY (MHz)

PuzzariniPuzzarini, , Cazzoli, Harding , Cazzoli, Harding , Vázquez Vázquez && GausGauss, s, JCP JCP 131131, 234304 , 234304 (2009) (2009)

HH221717OO::

1717OO ExperimentExperiment TheoryTheory

CCaaaa --28.477(88)28.477(88) --28.1828.18--28.6128.61

CCbbbb --28.504(71)28.504(71) --27.9427.94--27.9927.99

CCcccc --18.382(47)18.382(47) --18.4618.46--18.4918.49

results in kHzresults in kHz

Results ……. SR of Results ……. SR of 1717O O

Method:Method:CCSD(T)CCSD(T)

Equil.Equil.(exp r(exp ree))

Vib. Vib. Corr.Corr.(VPT2)(VPT2)

Vib. Vib. Corr.Corr.(DVR)(DVR)

TotalTotal(Eq+Vib)(Eq+Vib)

basisbasis augCV6ZaugCV6Z augCV5ZaugCV5Z augCV5ZaugCV5Z

CCaaaa --22.25122.251 --5.9335.933 --6.3616.361 --28.18428.184--28.61228.612

CCbbbb --25.19625.196 --2.7412.741 --2.7942.794 --27.93727.937--27.99027.990

CCcccc --17.47617.476 --0.9880.988 --1.0151.015 --18.46418.464--18.49118.491

Absolute Absolute 1717O NMR scaleO NMR scale[ppm][ppm] isotropicisotropic

(dia) (dia) calculatedcalculated

416.4416.4

(para) (para) from expfrom exp

--78.578.5

(equil)(equil)

(vib)(vib)

(T)(T)

338.1(3)338.1(3)

--11.711.7

--0.40.4

(300K)(300K) 326.2(3)326.2(3)

Best theoretical estimate Best theoretical estimate 325.6325.6 ppmppm

In search of confirmation ….In search of confirmation ….

Determination of the Determination of the 1717O spinO spin--rotation constants rotation constants

for for DD221717OO and and HDHD1717OO

221717

--1.611.61----SSaaaa (D(D--D) / kHzD) / kHz2.42.4442.11(65)2.11(65)SSaaaa (D(D--1717O) / kHzO) / kHz--2.612.61----CCcc cc (D) / kHz(D) / kHz--2.42.411----CCbb bb (D) / kHz(D) / kHz--2.92.944----CCaa aa (D) / kHz(D) / kHz--0.10.188--0.189(11)0.189(11)eQqeQqcccc (D) / MHz(D) / MHz0.020.020.041(11)0.041(11)eQqeQqbbbb (D) / MHz(D) / MHz0.150.150.1479(26)0.1479(26)eQqeQqaaaa (D) / MHz(D) / MHz--9.49.411--9.669.66((2828))CCcc cc ((1717O) / kHzO) / kHz--13.6113.61--113.343.34(2(255))CCbb bb ((1717O) / kHzO) / kHz--14.6714.67--114.574.57(2(211))CCaa aa ((1717O) / kHzO) / kHz10.010.04410.1433(68)10.1433(68)eQqeQqcccc ((1717O) / MHzO) / MHz--1.21.233--1.2716(68)1.2716(68)eQqeQqbbbb ((1717O) / MHz O) / MHz --8.88.811--8.8717(28)8.8717(28)eQqeQqaaaa ((1717O) / MHzO) / MHz

THEORYTHEORYEXPERIMENTEXPERIMENT

--14.8014.80--13.13.6060--9.419.41

Equilibrium: CCSD(T)/augCV6ZEquilibrium: CCSD(T)/augCV6ZVibrat. Corr.: CCSD(T)/augCV5ZVibrat. Corr.: CCSD(T)/augCV5Z

VPT2VPT2 DVRDVR

Absolute Absolute 1717O NMR scaleO NMR scale[ppm][ppm] HH22

1717OO

(dia) (dia) calculatedcalculated

416.4 416.4

(para) (para) from expfrom exp

--79.0(3)79.0(3)

(equil)(equil)

(vib)(vib)

(T)(T)

337.4(3)337.4(3)

--11.711.7

--0.40.4

(300K)(300K) 325.3(3)325.3(3)

DD221717OO

416.4416.4

--78.6(9)78.6(9)

337.8(9)337.8(9)

--8.48.4

--0.40.4

329.0(9)329.0(9)

MOLECULAR STRUCTUREMOLECULAR STRUCTUREDETERMINATIONDETERMINATION

Rotational constant Rotational constant

Inertia tensor Inertia tensor

Isotopic substitution Isotopic substitution

TYPES of MOLECULAR STRUCTURETYPES of MOLECULAR STRUCTURE

EFFECTIVE STRUCTURE: EFFECTIVE STRUCTURE: rr00

SUBSTITUTION STRUCTURE: SUBSTITUTION STRUCTURE: rrss

MASSMASS--DEPENDENCE STRUCTURE: DEPENDENCE STRUCTURE: rrmm

EQUILIBRIUM STRUCTURE: EQUILIBRIUM STRUCTURE: rree

EFFECTIVE STRUCTURE EFFECTIVE STRUCTURE rr00

Structure calculated directly from Structure calculated directly from BB00::leastleast--squares fit squares fit of theof the molecular structural molecular structural parameters toparameters to the the momentsmoments ofof inertia inertia II00

calcicalc

III exp

i runs over inertial moments (isotopic substitution)j runs over structural parameters

rr00 > r> ree

Accuracy: limitedAccuracy: limitedApproximation = zero-point vibrational effects are the same for different isotopic species

SO2: r(S–O)rree = 1.4308 Årr00 = 1.4336 Å

Morino et al. J. Mol. Spectrosc. 13, 95 (1964)

FCPr(F–C) r(C–P)

rree (Å) 1.27547 1.54476rr00 (Å) 1.28456 1.54097

Bizzocchi, Degli Esposti, Puzzarini Mol. Phys. 104, 2627 (2006)

SUBSTITUTION STRUCTURE SUBSTITUTION STRUCTURE rrss

Make use of isotopic substitution for deriving the Make use of isotopic substitution for deriving the position (coordinates) of the substitued atom:position (coordinates) of the substitued atom:

Kraitchman’s equationsKraitchman’s equations

[C.C. Costain, [C.C. Costain, J. Chem. Phys.J. Chem. Phys. 2929, 864 (1958)], 864 (1958)]

1) Accuracy: 1) Accuracy: rree rrss rr00Approximation = zero-point vibrational effects tend to cancel using Kraitchman’s equation2) Each non2) Each non--equivalent atoms be equivalent atoms be substitutedsubstitutedWhen not feasible:

firstfirst--moment equationsmoment equationsm

zmz sii

ClCl BB SS

1.604923(90)1.680567(89)rree

1.6040(10)1.6815(10)rrss

1.6063(22)1.6819(22)rr00

B=SCl–B(Å)

Bizzocchi, Degli Esposti, Puzzarini J. Mol. Spectrosc. 216, 177 (2002)

xPPP xxxcalc

- similar equations for Py and Pz- (x0 ,y0 ,z0) coordinate of the atom in the parent molecule

leastleast--squares treatment to obtain squares treatment to obtain rrss structures:structures:Planar moment of inertiaPlanar moment of inertia

iiiiyz

iiiixz

iiiixy

zymPzxmPyxmP

zmPymPxmP

[Mostly used for asymmetric[Mostly used for asymmetric--top molecules]top molecules]

MASSMASS--DEPENDENCE STRUCTURE DEPENDENCE STRUCTURE rrmm

Extension of the substitution method:Extension of the substitution method:to firstto first--order, the mass dependence of the order, the mass dependence of the vibrational contributions are determinedvibrational contributions are determined

Linear molecule case

mass-dependence moment of inertiambI

first-order approx

AccuracyAccuracyValidity of the first-order approximation

Major problems:Major problems:- light atoms (as H)

- missing isotopic substitution (as F)

ImprovementsImprovementsL)2(L)1()2()1(

mmmm rrrr

cbaIcII mm ,,/

rrmm(1)(1) modelmodel

It can be used for molecules that contains atoms such as F

cbaMmmdIcII NN

mm ,,/ //

)22(11

rrmm(2)(2) modelmodel

Suitable correction function based on appropriate reduced masses

Molecular structure of OCSMolecular structure of OCS

1.562021(17)1.155386(21)re

1.56120(5) 1.15619(12)rm(2)

1.56045(116)1.15764(66)rm(1)

1.56150(93)1.15842(76)rs

1.56488(92)1.15638(113)r0

r(C–S)r(C–O)OCS

Watson et al. J. Mol. Spectrosc. 196, 102 (1999)Foord et al. Mol. Phys. 29, 1685 (1975)

(XH)(XH)/

m mMmMrr

rrmm(1L)(1L) andand rrmm

(2L)(2L) modelsmodels

Laurie-type correction: introduced by using an effective bond length

To solve anomalies due to light atoms …

1.15310(24)1.06531(92)1.15404(15)1.06163(24)rm(2)

1.15324(2)1.06501(8)re

1.15338(11)1.06423(33)1.15392(20)1.06220(4)rm(1)

r(C-N)r(H-C)r(C-N)r(H-C)with corr.without corr.HCNHCN Watson et al. J. Mol. Spectrosc. 196, 102 (1999)

Comparison & AccuracyEXAMPLESEXAMPLES

1.04571.04651.0651

1.21931.21651.2075

r0rsre

H C C C C F

1.05731.05581.0614

1.20791.20781.2080

1.35251.37131.3731

1.22211.20311.2013

1.28541.27291.2735

The failure of the The failure of the rrss structurestructure

M. Bogey, C. Demuynck, and J. L. Destombes, Mol. Phys. 66, 955 (1989).P. Botschwina and C. Puzzarini, J. Mol. Spectrosc. 208, 292 (2001).

L. Dore, L. Cludi, A. Mazzavillani, G. Cazzoli, and C. Puzzarini, Phys. Chem. Chem. Phys. 9, 2275 (1999).

1.35757(13)1.3641(19)1.3674(15)

1.92854(12)1.9274(10)1.9286(8)

1.08302(8)1.0854(4)1.0699(37)

110.151(32)110.36(16)110.24(20)

107.233(8)107.36(5)107.19(5)

109.552(10)109.13(7)109.28(7)

The failure of the The failure of the rrmm structuresstructures

re in blackrm

(1) in redrm

(1L) in blue

C. Puzzarini, G. Cazzoli, A. Baldacci, A. Baldan, C. Michauk, and J. Gauss, J. Chem. Phys. 127, 164302 (2007)

ciscis--11

--chlor

loro--2

2--flu

1.0764

1.0776(4)

1.0787

1.0802(6)

1.3317(3) 1.3310

1.3249

1.3240(14)

7128(6)7107 123.10

123.07(1)

122.53

122.61(6)

120.74(9) 120.43

123.50(2)123.43

rree (emp) rrmm(2)(2) rrss rr00

C1–Cl 1.7128(6) 1.715(4) 1.721(5) 1.729(2)

C1–H 1.0776(4) 1.077(6) 1.108(5) 1.110(2)

C1–C2 1.3240(14) 1.330(7) 1.323(4) 1.314(2)

C2–F 1.3317(3) 1.327(8) 1.330(5) 1.345(3)

C2–H 1.0802(6) 1.081(5) 1.088(6) 1.083(2)

ClC1C2 123.07(1) 123.1(2) 122.9(6) 123.2(2)

HC1C2 120.74(9) 121.9(8) 126.4(6) 126.7(3)

FC2C1 122.61(6) 122.8(4) 122.8(5) 122.1(2)

HC2C1 123.50(2) 123.8(2) 124.0(6) 124.6(3)

C. Puzzarini, G. Cazzoli, L. Dore, A. Gambi PCCP 3, 4189 (2001) // C. Puzzarini, G. Cazzoli, A. Gambi, J. Gauss, JCP 125, 054307 (2006)

EQUILIBRIUM STRUCTURE EQUILIBRIUM STRUCTURE rree

-- Structure calculated from Structure calculated from BBee::leastleast--squares fit squares fit of theof the molecular structural molecular structural parameters toparameters to the the momentsmoments ofof inertia inertia IIee-- Clear physical meaning:Clear physical meaning:minimum of the Bornminimum of the Born--Oppenheimer PES, Oppenheimer PES, truly isotopic independenttruly isotopic independent

cbadBBr

rre ,,

r runs over vibrational normal modes

Main limitation:Main limitation:AAvv, B, Bvv, C, Cvv for each vibrational state v

Investigation of either pureInvestigation of either pure--rotational or vibrorotational or vibro--rotational spectra of each fundamental moderotational spectra of each fundamental mode

Approach limited to small (2Approach limited to small (2--4 atoms) molecules4 atoms) molecules

IMPOSSIBILITY OF GETTING ALL IMPOSSIBILITY OF GETTING ALL VIBRATIONVIBRATION--ROTATION INTERACTION ROTATION INTERACTION

CONSTANTS NEEDED:CONSTANTS NEEDED:HOW TO SOLVE THE PROBLEM?HOW TO SOLVE THE PROBLEM?

THE SEMITHE SEMI--EXPERIMENTAL APPROACHEXPERIMENTAL APPROACHP. Pulay, W. Meyer, J.E. Boggs, P. Pulay, W. Meyer, J.E. Boggs, J. Chem. Phys.J. Chem. Phys. 68, 5077 (1978)68, 5077 (1978)

Equilibrium structureEquilibrium structure::need of need of BBee for various isotopic speciesfor various isotopic species

Bre BB

Rotational constant ofRotational constant ofvibrational ground statevibrational ground state Vibrational correctionVibrational correction

EXPERIMENTEXPERIMENT THEORYTHEORYP. Pulay, W. Meyer, J.E. Boggs, J. Chem. Phys. 68, 5077 (1978).

BB0 0 fromfrom EXPERIMENTEXPERIMENT((variousvarious isotopicisotopic speciesspecies))

Vibrational Corrections fVibrational Corrections from rom THEORYTHEORY((cubiccubic force force fieldfield))

Actual FIT:Actual FIT:moments of inertiamoments of inertia

Requirements for accurate structure:Requirements for accurate structure: computed from computed from force field obtained with force field obtained with correlated methodcorrelated method and, at least, and, at least, tripletriple--zeta basis setzeta basis set

TypicalTypical accuracyaccuracy: : betterbetter thanthan 0.001 Å0.001 Å

10101111

1.37851.3785

1.38175(53)1.38175(53)

1.386(5)1.386(5)

1.39741.39741.39793(40)

1.39793(40)1.38(2)1.38(2)

125.83125.83125.768(48)125.768(48)118.8(7)118.8(7)

121.91121.91

121.924(10)

122.3(6)122.3(6)

CCNN NN

HH11 HH11

HH22 HH22

b) b) ExpExp data: data: spinspin--spinspin constantsconstants

a) a) ExpExp data: data: rotationalrotational constantsconstants

EquilibriumEquilibrium structurestructure determinationdetermination: : reviewreview

1) 1) ExperimentallyExperimentally: r: r00, , rrss, , rrmm, … r, … ree(?)(?)

2) 2) ComputationallyComputationally: r: ree

3) 3) MixedMixed expexp--calccalc: r: ree ((empiricalempirical))

DIRECTDIRECT spinspin--spinspin interaction interaction constantconstant::DDHHSSSS = + I= + ILL DD IIKK

INDIRECTINDIRECT spinspin--spinspin interaction interaction constantconstant::JJHHSSSS = + I= + ILL JJ IIKK

SPINSPIN--SPIN INTERACTIONSPIN INTERACTION

2KLijjKLiKLN

KLKLij

1)1) SubtractionSubtraction of the of the computedcomputed vibrationalvibrational correctioncorrection in in orderorder to getto get equilibriumequilibrium DDKLKL::

PROCEDUREPROCEDURE

KLKLeq DDD exp

2) 2) DeterminationDetermination of the of the molecularmolecular structurestructure byby invertinginverting

2KLijjKLiKLN

KLKLij

PuzzariniPuzzarini, , Metzroth Metzroth && Gauss Gauss unpublished unpublished

EquilEquil. . structurestructure fromfrom onlyonly 1 1 isotopologueisotopologue

1414NHNH33

rree [DC][DC] rree [semi[semi--exp exp BB]]

r(N-H) HNH r(N-H) HNH

Dzz (N-H)

1.0121(11) 107.05(9) 1.01139(60) 107.17(18)[Dxx-Dyy] (N-H)

Dzz (H-H)

PuzzariniPuzzarini, , Metzroth Metzroth && Gauss Gauss unpublished unpublished semisemi--exp exp BB: Pawlowski et al. : Pawlowski et al. JCPJCP 116116, 6482 (2002) , 6482 (2002)

abiabi=(=(allall)CCSD(T)/)CCSD(T)/cccc--pwCVQZpwCVQZ

Partial equilibrium structurePartial equilibrium structure

PuzzariniPuzzarini, , Metzroth Metzroth && Gauss Gauss unpublished unpublished

rree[DC][DC] rree[exp][exp] rree[abi][abi]

HH1313CNCN

HH--CC 1.064(52)1.064(52) 1.06501(8)1.06501(8) 1.06551.0655

XBO XBO (X=F,Cl)(X=F,Cl)

FF--BB 1.252(14)1.252(14) 1.2833(7)1.2833(7) 1.28091.2809

ClCl--BB 1.678(127)1.678(127) 1.68274(19)1.68274(19) 1.68361.6836

FBSFBS

FF--BB 1.282(2)1.282(2) 1.2762(2)1.2762(2) 1.27701.2770

INTRODUCTORY LECTURE on ROTATIONAL SPECTROSCOPY

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