INTRODUCTORYINTRODUCTORY LECTURELECTURE onon ROTATIONALROTATIONAL SPECTROSCOPYSPECTROSCOPY

CristinaCristina PuzzariniPuzzarini Dip.Dip. ChimicaChimica ““GiacomoGiacomo CiamicianCiamician”” UniversitUniversitàà didi BolognaBologna

CODECSCODECS SummerSummer SchoolSchool 20132013 THEORETICALTHEORETICAL SPECTROSCOPYSPECTROSCOPY EE Eel  Evib  Erot

ELECTRONICELECTRONIC VIBRATIONALVIBRATIONAL ROTATIONALROTATIONAL FREQUENCYFREQUENCY REGIONREGION mm/submm waves Rotational Spectroscopy

Electronics Photonics “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.)

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

(1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed

(3)(3) IntensityIntensity (transitions)(transitions) CoordinateCoordinate SSyystemstemss

Molecule-fixed coordinate system Lab-fixed coordinate system ROTATINGROTATING RIGIDRIGID BODYBODY (CLASSIC(CLASSIC VIEW)VIEW) 1 T  ωTIω 2

 == angolarangolar velocityvelocity II == inertiainertia tensortensor AngolarAngolar VelocitVelocityy

vi r i    x   ω   y    RIGIDRIGID BODY:BODY: v  ωr i i z  InertiaInertia TensorTensor

I xx I xy I xz  n n 2 2 2 2   I xx   mi ri  xi  mi yi  zi  i1 i1 I  I yx I yy I yz  n I xy   mi xi yi   i1 I zx I zy I zz  ByBy cconvenonventiontion:: IIc  IIb  IIa

INERTIAINERTIA TENSORTENSOR II Principal   Principal I xx I xy I xz I x 0 0    inertiainertia   I  I yx I yy I yz  systemsystem I  0 I y 0      I zx I zy I zz  0 0 I z  AngularAngular MomentMoment

RIGID BODY J e  defined in the rotating coordinate system (CM system) KINETICKINETIC ENERGYENERGY ofof aa ROTANTINGROTANTING RIGIDRIGID BODYBODY (CLASSIC(CLASSIC VIEW)VIEW) 1 1 J 2 T  ωTIω  2 2 I  == angularangular velocyvelocy II == inertiainertia tensortensor KINETICKINETIC ENERGYENERGY ofof aa ROTANTINGROTANTING RIGIDRIGID BODYBODY 1 1 J 2 T  ωTIω  2 2 I 1  J 2 J 2 J 2  E  T   x  y  z  rot rot   2  I x I y I z 

•• PotentialPotential energy?energy? •• FromFrom classicclassic mechanicsmechanics toto quantumquantum mechanicsmechanics ClassicClassic view:view: conservationconservation ofof angularangular momentummomentum QuantumQuantum mechanics:mechanics: commutationcommutation ofof operatorsoperators

2 2 2 2 2 2 2 J  J x  J y  J z  J X  J Y  J Z x,y,z molecule-fixed coordinate system X,Y,Z space-fixed coordinate system

ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 J  J x  J y  J z  J X  JY  J Z TheThe spacespace-- andand moleculemolecule--fixedfixed componentscomponents ofof ĴĴ commute!commute!

SS == matrixmatrix thatthat relatesrelates thethe coordinatescoordinates ofof thethe atomsatoms inin thethe moleculemolecule--fixedfixed systemsystem toto thosethose inin thethe spacespace--fixedfixed systemssystems

Ĵ=SF ĴF where =x,y,z and F=X,Y,Z ĴF= F Ĵ Then:

[ĴF,Ĵ] = ĴFSF’ĴF’ –SF’ĴF’ĴF = [ĴF,SF’]ĴF’ + SF’(ĴFĴF’ – ĴF’ĴF) = ieFF’F”(SF”ĴF’ + SF’ĴF”) = 0 !! where

eFF’F”=permutation symbol

[ĴX,SX] = 0 [ĴX,SY] = iSZ [ĴX,SZ] = –iSY [ĴX,ĴY] = –ieXYZĴZ 22 EIGENVALUESEIGENVALUES ofof ĴĴ ,, ĴĴZZ,, ĴĴzz

ˆ 2 ˆ 2 ˆ 2 ˆ 2 Jˆ 2 , Jˆ 2  0 J , J z  0 J , J Z  0  z Z 

ˆ 2 2 J ,K ,M J J ,K ,M   J (J 1) J=0,1,2,3, …

ˆ 2 2 2 M=J,J-1 … -J J ,K ,M J Z J ,K ,M   M

ˆ 2 2 2 J ,K ,M J z J ,K ,M   K K=J,J-1 … -J ROTATIONALROTATIONAL HAMILTONIANHAMILTONIAN

1  Jˆ 2 Jˆ 2 Jˆ 2  Hˆ   x  y  z  rot 2  I I I   x y z  ˆ H rotrot  Erot rot

ROTATIONALROTATIONAL ENERGYENERGY LEVELSLEVELS ClassificationClassification Examples

CO CO2

CH4 SF6

NH3

H2O

ByBy cconvenonventiontion:: IIc  IIb  IIa Let’s consider the simplest case

m1 m2 R DIATOMIC/LINEARDIATOMIC/LINEAR MOLECULE:MOLECULE: RIGIDRIGID ROTORROTOR (approx)(approx) R z m1 CM m2

r1 r2

2 I = 0 I = I = I z x y I   miri i m m 2   1 2 where I  R m1  m2 reduced mass rr1212 rr2323 1 2 3

1 I  m m r 2  m m r 2  m m r  r 2 M 1 2 12 2 3 23 1 3 12 23

Iz = 0 Ix = Iy = I ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules

ˆ 1 ˆ 2 1 ˆ 2 1 ˆ 2 H rot  J x J y  J 2I x 2I y 2I

Iz = 0 Ix = Iy = I makingmaking useuse ofof thethe eigenvalueseigenvalues ofof ĴĴ2

2 BB == rotationalrotational constantconstant  Erot  J (J 1)  BJ (J 1) 2I JJ == 0,1,2,3,….0,1,2,3,…. ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules

E =12B J=3 rot

Erot  BJ (J 1)

E =6B J=2 rot

E =2B J=1 rot E =0 J=0 rot ROTATIONALROTATIONAL ENERGYENERGY LEVELS:LEVELS: DiatomicDiatomic andand LinearLinear moleculesmolecules

E =12B J=3 rot

E(J 1 J )  2B(J 1)

E =6B J=2 rot

E =2B J=1 rot E =0 J=0 rot ROTATIONALROTATIONAL ENERGYENERGY LEVELSLEVELS

ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 J  J x  J y  J z  J X  JY  J Z x,y,z molecule-fixed coordinate system X,Y,Z space-fixed coordinate system

ˆ 2 ˆ 2 ˆ 2 ˆ 2 J , J z  0 J , J Z  0

ˆ 2 2 J ,K ,M J J ,K ,M   J (J 1)

ˆ 2 2 2 J ,K ,M J Z J ,K ,M   M M=J,J-1 … -J RotationalRotational energyenergy levels:levels: (2(2JJ+1)+1) foldfold degeneratedegenerate inin MM “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.)

0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions allowedallowed

(3)(3) IntensityIntensity (transitions)(transitions) SELECTIONSELECTION RULESRULES

 TransitionTransition moment:moment:  00   Approx BO:   =   tot rot vib ele    ele vib rot rot vib eled d d  f f f i i i rot vib ele

dipole moment in the space-fixed coordinate system  X Xy

Xx

Xz

  =direction cosines  Z  cos   F   F  F where       =x,y,z (molecule-fixed)     F   F=X,Y,Z (space-fixed)  F    SELECTIONSELECTION RULESRULES     where =x,y,z  F F=X,Y,Z F     =direction cosines F    rot  rot d  ele vib  vib eled d f F i rot f f i i vib ele F   (1)(1) (2)(2) molecular dipole moment components (1)(1) SelectionSelection rulesrules  (2)(2) NonNon--vanishingvanishing permanentpermanent dipoledipole momentmoment SELECTIONSELECTION RULESRULES   “Rotational” transition moment Rij:     Rij   f F i  where: F 

JKM F J' K' M'  J F J'  JK F J' K'  JM F J' M' (1)(1) (2)(2) (3)(3) The direction-cosine matrix elements are known: (1)(1) JJ == 11 (2)(2) KK == 00 (3)(3) MM == 0,0, 11 “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.)

0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions  allowedallowed

(3)(3) IntensityIntensity (transitions)(transitions) RotationalRotational energyenergy levelslevels ++ SelectionSelection rulesrules

RotationalRotational transitiontransition frequenciesfrequencies (rotational(rotational spectrum:spectrum: xx axis)axis) E =12B J=3 rot

Erot  BJ (J 1)

E =6B ++ J=2 rot

J  1 E =2B J=1 rot

E =0 J=0 rot E(J 1 J )  2B(J 1) (B in energy units) h rot  2B(J 1)

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

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

2B/h 4B/h 6B/h 8B/h frequency LINELINE STRENGTHSSTRENGTHS (1)(1) BoltzmannBoltzmann distributiondistribution E N g  rot J  J e kT N0 g0 (2)(2) degeneracydegeneracy 2J+1

E N  rot J  (2J 1)e kT N0 g /g =2J+1 f 0 exp(-E /kT) rot N /N =(2J+1)exp(-E /kT) f 0 rot

J max 0 / N f N

J E N  rot J  (2J 1)e kT N0 IntensityIntensity ofof RotationalRotational TransitionsTransitions

1 2 I  N  2 n μm T i mn

Intensity Population abs I

0 /N J N

012345678910 J “Building”“Building” thethe ROTATIONALROTATIONAL SPECTRUMSPECTRUM intensity (a.u.)

0 20 40 60 80 100 frequency (cm-1) (1)(1) RotationalRotational energyenergy levelslevels (2)(2) SelectionSelection rules:rules: transitionstransitions  allowedallowed (3)(3) IntensityIntensity (transitions)(transitions) Rotational spectrum of CO

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I 0 20 40 60 80 100 wavenumbers (cm-1)  LinearLinear Rotor:Rotor: EIGENFUNCTIONSEIGENFUNCTIONS  

M |M | iM  ,  YJ  ,  PJ cos e

SPHERICALSPHERICAL HARMONICSHARMONICS ^ Eigenvalues of J2: ħ2J(J+1) with J = 0, 1, 2, … ^ Eigenvalues of Jz : ħM with -J ≤ M ≤ J SPHERICALSPHERICAL HARMONICSHARMONICS J

M VectorVector RappresentaRappresentationtion ofof AngularAngular MomentumMomentum

JJ == 22  55 valuesvalues forfor MM

Costant length (J) - 5 orientations (M) OneOne stepstep furtherfurther …..…..

MoleculesMolecules areare NOTNOT rigid:rigid: centrifugalcentrifugal distortiondistortion SEMISEMI--RIGIDRIGID ROTORROTOR withwith CENTRIFUGALCENTRIFUGAL DISTORTIONDISTORTION

perturbation theory ˆ ˆ 0 ˆ ' H rot  H rot  H dist rigidrigid--rotorrotor

ˆ '  DJ  ˆ 4 H dist   4 J 3    4B e >> 00 !!!! DJ  2 ' 2 2  Edist  DJ J (J 1) 2 2 Erot / h  BJ (J 1)  DJ J (J 1)

E =12hB J=3 rot/h J=3 centrifugalcentrifugal distortiondistortion

E =6hB J=2 rot J=2

Erot=2hB J=1 J=1 E =0 J=0 rot J=0 3  rot  2B(J 1)  4DJ (J 1)

[B, DJ in frequency units]

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

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2B 2B 2B 2B 2B

2B 4B 6B 8B frequency AnotherAnother stepstep furtherfurther …..…..

OtherOther typestypes ofof rotorrotor ClassificationClassification Examples

CO CO2

CH4 SF6

NH3

H2O

ByBy cconvenonventiontion:: IIc  IIb  IIa thusthus CC  BB  AA SPHERICALSPHERICAL TOPSTOPS

CH4, SF6 , …

Ia = Ib = Ic = I

Erot = B J(J+1)

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

 = 0 !!! SYMMETRICSYMMETRIC TOPSTOPS

I x  I y  I

I z  I // (z = symmetry axis) 1  Jˆ 2  1 1   ˆ   ˆ 2 H rot      J z  2  I  I // I   SYMMETRICSYMMETRIC TOPSTOPS 2  J (J 1)  1 1      2 Erot      K  2  I  I // I   K=J,J-1 … -J SYMMETRICSYMMETRIC TOPSTOPS 2  J (J 1)  1 1      2 Erot      K  2  I  I // I  

CH F 3 prolateprolate oblateoblate II >> II BCl3   II << II II// == IIa II// == IIc >0 2 Prolate:Prolate: EErot == BJBJ((JJ+1)+(+1)+(AA––BB))KK wherewhere AA>>B=CB=C 2 Oblate:Oblate: EErot == BJBJ((JJ+1)+(+1)+(CC––BB))KK wherewhere AA==BB>>CC <0 SYMMETRICSYMMETRIC TOP:TOP: rotationalrotational energyenergy levelslevels

PROLATEPROLATE OBLATEOBLATE AA >> BB == CC AA << BB == CC SYMMETRICSYMMETRIC TOP:TOP: rotationalrotational energyenergy levelslevels

JJ == 6,6, KK == 44

PROLATEPROLATE OBLATEOBLATE AA >> BB == CC AA << BB == CC SELECTIONSELECTION RULESRULES

InIn additionaddition toto JJ ==  11:: KK == 00 SYMMETRICSYMMETRIC TOP:TOP: rotationalrotational energyenergy levelslevels

PROLATEPROLATE OBLATEOBLATE AA >> BB == CC AA << BB == CC SELECTIONSELECTION RULESRULES

InIn additionaddition toto JJ ==  11:: KK == 00

RIGIDRIGID ROTOR:ROTOR: h rot  2B(J 1) Rotational spectrum of a symmetric-top rotor

KK structurestructure forfor eacheach JJ valuevalue ((JJ+1+1  JJ)) SELECTIONSELECTION RULESRULES

InIn additionaddition toto JJ ==  11:: KK == 00

RIGIDRIGID ROTOR:ROTOR: h rot  2B(J 1) includingincluding CENTRIFUGALCENTRIFUGAL DISTORTION:DISTORTION: 3 2 h rot  2B(J 1)  4DJ (J 1)  2DJK (J 1)K  Rotational spectrum of CH3CN: a small portion

KK structurestructure K =6 K =3 CH CN: J = 61 - 60 K =0 3

K =9

1117200 1117800 1118400 Frequency (MHz) Rotational spectrum of NF3: a small portion 14NF : J = 26 - 25 3 K=6 K=9 K=12 K=15 K=3 K=18

554400 554500 554600 554700 Frequency (MHz) SymmetricSymmetric--toptop Rotor:Rotor: EIGENFUNCTIONSEIGENFUNCTIONS

  , ,  JKM     

Eigenfunctions: SPHERICALSPHERICAL HARMONICSHARMONICS ^ 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 ASYMMETRICASYMMETRIC ROTORROTOR

1  Jˆ 2 Jˆ 2 Jˆ 2  Hˆ   x  y  z  rot 2  I I I   x y z 

No longer possible to rearrange the Hamiltonian so that it is comprised soley of J ˆ 2 and one component ofJˆ

ItIt isis notnot possiblepossible toto describedescribe thethe rotationalrotational motionmotion inin termsterms ofof aa conservedconserved motionmotion aboutabout aa particularparticular axisaxis ofof thethe molecule.molecule. ASYMMETRICASYMMETRIC ROTORROTOR

1  Jˆ 2 Jˆ 2 Jˆ 2  Hˆ   x  y  z  rot 2  I I I   x y z 

Diagonalization:Diagonalization: EErotrot,,  ForFor thethe sakesake ofof convenience:convenience: correlationcorrelation toto symmetricsymmetric toptop

PseudoPseudo quantumquantum numbers:numbers:

KKa  limitinglimiting prolateprolate symmetricsymmetric rotorrotor KKc  limitinglimiting oblateoblate symmetricsymmetric rotorrotor ASYMMETRICASYMMETRIC ROTORROTOR ASYMMETRICASYMMETRIC ROTORROTOR

2 notation scheme: JKa,Kc or J  = 0 PROLATEPROLATE nearnear prolateprolate OBLATEOBLATE  = -1  = +1 nearnear oblateoblate 1 +1 0 1 0 0 1 -1 1 JKa  Kc J (-J    +J) AsymmetricAsymmetric parameterparameter  2B  A  C     K  K A  C a c SELECTIONSELECTION RULESRULES

InIn additionaddition toto JJ == 0,0, 11::

KKaa ,, KKcc == 0,0,  11

Ka Kc ASYMMETRICASYMMETRIC ROTORROTOR

Symmetric Rotor

Asymmetric Rotor ASYMMETRICASYMMETRIC ROTOR:ROTOR: smallsmall portionportion ofof rotationalrotational spectrumspectrum

transtrans--CHCH35Cl=CHFCl=CHF

524000 524100 524200 524300 524400 524500 524600 524700 Frequency (MHz) RotationalRotational HamiltonianHamiltonian

RotationalRotational HamiltonianHamiltonian

2 2 2 AJ A  BJ B  CJC RIGIDRIGID ROTORROTOR Rotational constants ++ CENTRIFUGALCENTRIFUGAL DISTORTIONDISTORTION AnotherAnother stepstep furtherfurther …..…..

HyperfineHyperfine InteractionsInteractions HyperfineHyperfine structurestructure RotationalRotational HamiltonianHamiltonian

2 2 2 AJ A  BJ B  CJC Rotational constants

K 1 eQK qJ  2 3 2 2   3IJ  IJ I J  2 K 2IK (2IK 1)J(2J 1)  2  Nuclear quadrupole coupling LINEARLINEAR MOLECULEMOLECULE

F = J+I, J+I-1, …, |J-I| F = 1/2

J = 1 F = 5/2 SelectionSelection Rules:Rules: F = 3/2 couplingcouplingF=0 II ++ F=+1JJ == FF F=-1 F  0;1

J=1-0 unpert urbed J = 0 F = 3/2 unperturbed nuclear quadrupole frequency coupling

[[IIKK  1]1] IIKK=3/2;=3/2; eQqeQq  00 LINEARLINEAR MOLECULEMOLECULE

F = J+I, J+I-1, …, |J-I| F = 1/2 hyperfinehyperfine structurestructure J = 1 F = 5/2

F = 3/2 F=0 F=0 F=+1F=+1 FF=-1=-1

J=1-0 J=1-0 unperturbed unpert urbed J = 0 F = 3/2 unperturbed nuclear frequency quadrupole frequency coupling

[[IIKK  1]1] IIKK=3/2;=3/2; eQqeQq  00 HyperfineHyperfine structurestructure RotationalRotational HamiltonianHamiltonian

2 2 2 I C J AJ A  BJ B  CJC  K K K Rotational constants Spin-rotation interactions

K 1 eQK qJ  2 3 2 2   3IJ  IJ I J  2 K 2IK (2IK 1)J(2J 1)  2  Nuclear quadrupole coupling LINEARLINEAR MOLECULEMOLECULE

F = +1 (F=5/2-3/2) F = +1 (F=5/2-3/2) hyperfine structure F = 3/2 hyperfine structure J = 2 F = +1 (F=3/2-1/2) F = +1 (F=3/2-1/2) F = 5/2 F = 0 (F=3/2-3/2) F = 0 (F=3/2-3/2)

F = 1/2 J = 1

F = 3/2 = 2 - 1 unperturbed J = 2 - 1 unperturbed unperturbed spin-rotation interaction frequency frequency

[[IIKK  1/2]1/2] IIKK=1/2;=1/2; CC  00 HyperfineHyperfine structurestructure SelectionSelection Rules:Rules:

RotationalRotational HamiltonianHamiltonian couplingcoupling IIK,L ++ JJ == FFK,L

FK ,L  0;1 2 2 2 I C J AJ A  BJ B  CJC  K K K Rotational constants Spin-rotation interactions

 I K D KL I L K  L K 1 eQ q  2 3  K J 3IJ  IJ I2J2 Spin-spin (direct)    interactions 2 K 2IK (2IK 1)J(2J 1)  2  Nuclear quadrupole coupling LINEARLINEAR MOLECULEMOLECULE

J F' =J+I F =F'+I2 F = 0 ,+1 1 (F= 1- 0,1 F -1,2= 0,+1 -1 ) (F=1-0,1-1,2-1) 0 = 0, +1 1 3/2 2 1-0,1-1) F = 0,+1 F = -1 (F=0-1) (F=1-0,1-1) 1/2 1 F = -1 (F=0-1) 1

1/2 J=1-0 unp erturbed 0 1 unperturbed 0

frequency fre que ncy unperturbed direct spin-spin interaction

IIKK=1/2=1/2 andand IILL=1/2=1/2 StarkStark effecteffect ROTATIONALROTATIONAL ENERGYENERGY LEVELSLEVELS

ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 ˆ 2 J  J x  J y  J z  J X  JY  J Z x,y,z molecule-fixed coordinate system ˆ 2 ˆ 2 ˆ 2 ˆ 2 X,Y,ZJ space-fixed, J z  0 coordinateJ , J Zsystem 0

ˆ 2 2 J ,K ,M J J ,K ,M   J (J 1)

ˆ 2 2 2 J ,K ,M J Z J ,K ,M   M M=J,J-1 … -J RotationalRotational energyenergy levels:levels: (2(2JJ+1)+1) foldfold degeneratedegenerate inin MM

DegeneracyDegeneracy removedremoved byby applyingapplying electricelectric field:field: STARKSTARK EFFECTEFFECT J  0

Energy M J  0 MJ = 0

J = 1

MJ = ±1 J 1

J = 0 MJ = 0 M  1 J E 0 0

M J  0  E0 asse Z STARKSTARK EFFECTEFFECT ˆ H  μ ε Interaction between the applied electric field and dipole moment: perturbationperturbation theorytheory Ĥ = perturbation Hamiltonian   applied along Z let’s consider a symmetric-top rotor ( along z): ˆ H  μεZz By applying perturbation theory: KM E (1)  Stark J (J 1)  2 2 2 2 2 2 2 2 2 2 (2)  (J  K )(J  M ) [(J 1)  K ][(J 1)  M  EStark   3  3  2hB  J (2J 1)(2J 1) (J 1) (2J 1)(2J  3)  STARKSTARK EFFECT:EFFECT: thethe SYMMETRICSYMMETRIC TOPTOP casecase

NO FIELD 1st ORDER 2nd ORDER |111 |1-11 |11-1 |1-11 |11-1 |1-11 |110 |1-10 |110|1-10 |111 |1-1-1 |110|1-10 |11-1|1-1-1 |111 |1-1-1 AA--BB |101 |100 |101 |100 |10-1  |100     |101 |10-1 |10-1

 2B2B

|000 |000 JKM |000 STARKSTARK EFFECT:EFFECT: thethe SYMMETRICSYMMETRIC TOPTOP casecase

NO FIELD 1st ORDER 2nd ORDER |111 |1-11 |11-1 |1-11 |11-1 |1-11 |110 |1-10 |110|1-10 |111 |1-1-1 |110|1-10 |11-1|1-1-1 |111 |1-1-1 AA--BB |101 |100 |101 |100 |10-1  |100     |101 |10-1 |10-1  shiftshift Stark:Stark:  2B2B ’’  ==’’-- ((’’ >> )) |000 |000 JKM |000 ROTATIONALROTATIONAL SPECTROSCOPY:SPECTROSCOPY: ComputationalComputational RequirementsRequirements && AccuracyAccuracy

CristinaCristina PuzzariniPuzzarini Dip.Dip. ChimicaChimica ““GiacomoGiacomo CiamicianCiamician”” UniversitUniversitàà didi BolognaBologna

CODECSCODECS SummerSummer SchoolSchool 20132013 THEORETICALTHEORETICAL SPECTROSCOPYSPECTROSCOPY SpectroscopicSpectroscopic pparameterarameterss:: RotationalRotational SpectroscopySpectroscopy

RotationalRotational

CentrifugalCentrifugal constantsconstants

HyperfineHyperfine--distortiondistortion constantsconstants NuclearNuclear quadrupolequadrupole couplingcoupling constantsconstants parametersparameters SpinSpin –– rotationrotation constantsconstants SpinSpin –– spinspin constantsconstants >> 44 MHzMHz LaboratoryLaboratory ofof MillimetreMillimetre--wavewave

HH2S:S: JJ == 886,3 –– 885,4

SpectroscopySpectroscopy1071310 1071312 1071314ofof BolognaBologna 10713167 8 Frequency accuracy:FREQUENCY 1 part (MHz) in 10 -10 >> 44 MHzMHz LaboratoryLaboratory ofof MillimetreMillimetre--wavewave

HH2S:S: JJ == 886,3 –– 885,4

SpectroscopySpectroscopy1071310 1071312 1071314ofof BolognaBologna 1071316 FREQUENCY (MHz) ~100~100 kHzkHz LaboratoryLaboratory ofof MillimetreMillimetre--wavewave

HH2S:S: JJ == 886,3 –– 885,4 FrequencyFrequency accuracy:accuracy: 11 kHzkHz

Frequency accuracy: better than 1 part in 109

SpectroscopySpectroscopy1071313.2 1071313.4 10713ofof13.6 BolognaBologna 1071313.8 1071314.0 FREQUENCY (MHz) 16 H O J = 4 - 3 2 1 4 2 1

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

17 kHz 46 kHz

380197.30 380197.35 380197.40 380197.45 FREQUENCY (MHz) QUANTUMQUANTUM--CHEMICALCHEMICAL CALCULATIONSCALCULATIONS ofof ROTATIONALROTATIONAL PARAMETERS:PARAMETERS: MethodologyMethodology && AccuracyAccuracy ROTATIONALROTATIONAL CONSTANTSCONSTANTS QuantumQuantum--ChemicalChemical CalculationCalculation ofof SpectroscopicSpectroscopic ParametersParameters • RotationalRotational (equilibrium)(equilibrium) constantsconstants INERTIA TENSOR

requires equilibrium geometry: geometry optimization (nuclear forces)

AccurateAccurate equilibriumequilibrium structurestructure !!!! COMPOSITECOMPOSITE APPROACHAPPROACH

1)1) PrincipalPrincipal errorerror sourcessources inin abab initioinitio calculationscalculations:: -- wfwf modelmodel truncationtruncation (N(N--ee-- errorerror)) -- basisbasis--setset truncationtruncation (1(1--ee-- errorerror))

2)2) “Minor”“Minor” errorerror sourcessources inin abab initioinitio calculationscalculations:: -- corecore--valencevalence (CV)(CV) correlationcorrelation -- scalarscalar relativityrelativity (SR)(SR) -- ………… 1)1) PrincipalPrincipal errorerror sourcessources inin abab initioinitio calculationscalculations:: -- wfwf modelmodel truncationtruncation (N(N--ee-- errorerror))

-- CoupledCoupled clustercluster methodmethod withwith singlesingle andand doubledouble excitationsexcitations withwith aa pertubativepertubative treatmenttreatment ofof connectedconnected triplestriples:: CCSD(T)CCSD(T)

-- HigherHigher excitationsexcitations:: fullfull--T,T, Q,Q, …… …… CoupledCoupled--ClusterCluster TheoryTheory exponentialexponential ansatzansatz forfor wavefunctionwavefunction

CC  exp(T ) HF

withwith clustercluster operatoroperator

T  T1 T2 T3  .... (excitations)

1  T t abc...a ib j... m  2 ijk ... (m!) ic,j ,k ,.. a ,b , ,...

1 1 exp(T ) 1T  T 2  T 3  ... 2! 3! CoupledCoupled--ClusterCluster TheoryTheory

SchrödingerSchrödinger equationequation ˆ ˆ H CC  Hexp(T ) HF  Eexp(T) HF

coupledcoupled--clustercluster equationsequations

 energyenergy

 amplitudesamplitudes veryvery efficientefficient treatmenttreatment ofof electronelectron--correlationcorrelation effectseffects CoupledCoupled--ClusterCluster TheoryTheory

•• CoupledCoupled--ClusterCluster SinglesSingles andand DoublesDoubles restrict T to single and double excitations  CCSDCCSD (T=T1+T2) •• CoupledCoupled--ClusterCluster Singles,Singles, Doubles,Doubles, andand TriplesTriples restrict T to S, D, triple excitations  CCSDTCCSDT (T=T1 +T2 +T3)

•• CoupledCoupled--ClusterCluster Singles,Singles, Doubles,Doubles, Triples,Triples, QuadruplesQuadruples restrict T to S, D, T, quadruple excitations  CCSDTQCCSDTQ (T=T1 +T2 +T3 +T4)

•• approximateapproximate treatmenttreatment ofof tripletriple excitationsexcitations add perturbative triples correction  CCSD(T)CCSD(T) CoupledCoupled--ClusterCluster TheoryTheory

6 7 CCSD(T) T=T1 + T2 + (T) N + N (no iter) dE dE dE dE tot  CCSD(T )  CCSDT  CCSDTQ .... dx dx dx dx

largelarge basisbasis set:set: cccc--pV5Z/ccpV5Z/cc--pV6ZpV6Z smallsmall--mediummedium basisbasis set:set: cccc--pVTZpVTZ smallsmall basisbasis set:set: cccc--pVDZpVDZ

Heckert,Heckert, Kallay,Kallay, Gauss,Gauss, Mol.Mol. Phys.Phys. 103,103, 21092109 (2005)(2005) 1)1) PrincipalPrincipal errorerror sourcessources inin abab initioinitio calculationscalculations:: -- basisbasis--setset truncationtruncation (1(1--ee-- errorerror))

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

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

-- ExtrapolationExtrapolation toto thethe CBSCBS limitlimit::

[SCF] [SCF] E(E(nn)) == EECBSCBS ++ AAexpexp((--BBnn)) [CORR] [CORR] --33 ++ E(E(nn)) == EECBSCBS ++ CCnn ExtrapolationExtrapolation toto CBSCBS limitlimit

1)1) atat ENERGYENERGY level:level: Feller, JCP 98, 7059 (1993) [SCF] [SCF] >>>> E(E(nn)) == EECBSCBS ++ AAexpexp((--BBnn)) [CORR] [CORR] --33 ++ E(E(nn)) == EECBSCBS ++ CCnn Helgaker et al., JCP 106, 9639 (1997)

2 >>>> E(E(nn)) == EE ++ BeBe--((nn--1)1) ++ CeCe--((nn--1)1)2 CBSCBS Peterson et al., JCP 100, 7410 (1994)

>>>> ……………… dE dE (HF  SCF) dE  (CCSD(T)) tot   dx dx dx

33--ptpt extrapol:extrapol: cccc--pVnZ,pVnZ, n=Qn=Q--66 22--ptpt extrapol:extrapol: cccc--pVnZ,pVnZ, n=5,6n=5,6

Heckert,Heckert, Kallay,Kallay, Tew,Tew, Klopper,Klopper, Gauss,Gauss, JCPJCP 125,125, 044108044108 (2006)(2006) COMPOSITECOMPOSITE APPROACHAPPROACH

1)1) PrincipalPrincipal errorerror sourcessources inin abab initioinitio calculationscalculations:: -- wfwf modelmodel truncationtruncation (N(N--ee-- errorerror)) -- basisbasis--setset truncationtruncation (1(1--ee-- errorerror))

2)2) “Minor”“Minor” errorerror sourcessources inin abab initioinitio calculationscalculations:: -- corecore--valencevalence (CV)(CV) correlationcorrelation -- scalarscalar relativityrelativity (SR)(SR) -- ………… 2)2) ““Minor”Minor” errorerror sourcessources inin abab initioinitio calculationscalculations:: …… …… CVCV CORRELATION:CORRELATION:

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

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

-- AdditivityAdditivity approximationapproximation::

EECVCV == EE ((allall)) –– EE ((fcfc)) dE dE  (HF  SCF) dE  (CCSD(T)) dE(core) tot    dx dx dx dx

mediummedium--largelarge basisbasis set:set: cccc--p(w)CVQZ,p(w)CVQZ, cccc--p(w)CV5Zp(w)CV5Z

Heckert,Heckert, Kallay,Kallay, Gauss,Gauss, Mol.Mol. Phys.Phys. 103,103, 21092109 (2005)(2005) 2)2) ““Minor”Minor” errorerror sourcessources inin abab initioinitio calculationscalculations:: …… …… SCALARSCALAR RELATIVITY:RELATIVITY:

-- SuitableSuitable basisbasis setssets and/orand/or approachapproach::  smallsmall--corecore relativisticrelativistic PPsPPs cccc--pVnZpVnZ--PP,PP, augaug--cccc--pVnZpVnZ--PP,PP, cccc--pwCVnZpwCVnZ--PPPP  DKDK hamiltonianhamiltonian cccc--pVnZpVnZ--DK,DK, ……..  2nd2nd orderorder directdirect PTPT cccc--pVnZpVnZ,, cccc--pCVnZpCVnZ,, ……..

-- AdditivityAdditivity approximationapproximation dE dE  (HF  SCF ) dE  (CCSD(T )) dE(rel) tot    dx dx dx dx

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

MichaukMichauk andand Gauss,Gauss, JCPJCP 127,127, 044106044106 (2007)(2007) Heckert,Heckert, Kallay,Kallay, Gauss,Gauss, Mol.Mol. Phys.Phys. 103,103, 21092109 (2005)(2005) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational ConstantsConstants

STATISTICALSTATISTICAL ANALYSISANALYSIS forfor •• 1616 moleculesmolecules ((9797 isotopologues)isotopologues) •• 180180 rotationalrotational constantsconstants

ReferenceReference values:values: BBee ,, BB00 fromfrom experimentexperiment

HF, N2, CO, F2, HCN, HNC, O=C=O, H2O, NH3,

CH4, HCCH, HOF, HNO, NH=NH, CH2=CH2, H2C=O

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) NormalNormal DistributionDistribution ofof RelativeRelative ErrorsErrors 2  1       ()  N exp    Nc = normalization constant c 2      std   n 1 calc ref Mean error:   i i  Bi  Bi n i1

n 1 2 Standard deviation: std  i  n1 i1 AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

CCSD(T)/TZ

-3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

CCSD(T)/QZCCSD(T)/TZ

-3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

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

-3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

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

-3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

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

-3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

CCSD(T)/QZCCSD(T)/TZCCSD(T)/5ZCCSD(T)/6Z + core +T -3 -2 -1 0 1 2 3 normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

CCSD(T)/QZCCSD(T)/TZCCSD(T)/5ZCCSD(T)/6Z + core +T -3 -2 -1 0 1 2 3 +Q normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational Constants:Constants: StatisticsStatistics

calccalc expexp BBee vsvs BBee

CCSD(T)/QZCCSD(T)/TZCCSD(T)/5ZCCSD(T)/CCSD(T)/6ZZ ++ corecore +T+T -3 -2 -1 0 1 2 3 +Q+Q normalnormal distributionsdistributions[MHz] ofof relativerelative errorserrors

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) COMPOSITECOMPOSITE APPROACH:APPROACH: thethe “geometry“geometry scheme”scheme”

1)1) PrincipalPrincipal errorerror sourcessources inin abab initioinitio calculationscalculations:: -- wfwf modelmodel truncationtruncation (N(N--ee-- errorerror)) -- basisbasis--setset truncationtruncation (1(1--ee-- errorerror))

2)2) “Minor”“Minor” errorerror sourcessources inin abab initioinitio calculationscalculations:: -- corecore--valencevalence (CV)(CV) correlationcorrelation -- scalarscalar relativityrelativity (SR)(SR) -- ………… ExtrapolationExtrapolation toto CBSCBS limitlimit

2)2) atat “PARAMETERS”“PARAMETERS” level:level: Feller, JCP 98, 7059 (1993) [SCF] [SCF] >>>> rr((nn)) == rr CBSCBS ++ AAexpexp((--BBnn)) [CORR] [CORR] --33 ++ rr((nn)) == rr CBSCBS ++ CCnn Helgaker et al., JCP 106, 9639 (1997)

--((nn--1)1) --((nn--1)1)2 >>>> rr((nn)) == rr CBS ++ BeBe ++ CeCe CBS Peterson et al., JCP 100, 7410 (1994) >>>> ……………… AdditivityAdditivity ofof CVCV effectseffects

2)2) atat “PARAMETERS”“PARAMETERS” level:level:

rr == rrCBSCBS ++ rrCVCV wherewhere

rrCV == rr ((((ww))CVnZCVnZ,, allall ee--)) –– rr (((w)(w)CVnZCVnZ,, fcfc)) allall electronselectrons correlatedcorrelated onlyonly valencevalence electronselectrons correlatedcorrelated AdditivityAdditivity ofof SRSR effectseffects

2)2) atat “PARAMETERS”“PARAMETERS” level:level:

rr == rrtottot ++ rrSRSR wherewhere

rrSR == rr ((relrel)) –– rr ((nonnon--relrel)) relativisticrelativistic optgoptg nonnon--relativisticrelativistic optgoptg Validation:Validation: GEOM.GEOM. vsvs GRAD.GRAD. Molecule Parameter CBS/Geom. scheme CBS/Grad. scheme

H2O O-H 0.95839 0.95836 OOH 104.484 104.478

NH3 N-H 1.01210 1.01206 HNHDifferences:Differences:106.631 106.641

PH3 P-H 1.41435 1.41464  0.0010.001HPH ÅÅ for93.555for distancesdistances93.553 NH2 N-H 1.02476 1.02474  0.010.01HNH deg.deg. 103.071forfor anglesangles103.060 PH2 P-H 1.41825 1.41846 HPH 91.882 91.877 ClSiP Cl-SiVALIDATED!!VALIDATED!!2.01439 2.01440 Si-P 1.96354 1.96340 HCS+ H-C 1.08200 1.08214 PuzzariniPuzzariniC-S ,, JPCJPC AA 1.47895 113,113, 1453014530 (2009)(20091.47907) CVCV correctionscorrections basisbasis == cccc--pwCV5ZpwCV5Z

SiSi--FF // ÅÅ SiSi--HH // ÅÅ FSiHFSiH // deg.deg. SiHSiH33FF gradientgradient --0.00520.0052 --0.00450.0045 0.000.00 schemescheme geometrygeometry --0.00530.0053 --0.00450.0045 0.000.00 schemescheme

PuzzariniPuzzarini,, CazzoliCazzoli,, GaussGauss JMSJMS 262,262, 3737 (2010)(2010) fullfull--TT correctionscorrections basisbasis == cccc--pVTZpVTZ SiSi--FF // ÅÅ SiSi--HH // ÅÅ FSiH / deg. SiHSiH3FF gradientgradient schemescheme +0.0001+0.0001 +0.0002+0.0002 +0.00+0.00 geom.geom. schemescheme +0.0002+0.0002 +0.0002+0.0002 +0.00+0.00

ClPOClPO ClCl--PP // ÅÅ PP--OO // ÅÅ ClPO/ClPO/ deg.deg. gradientgradient schemescheme +0.0009+0.0009 --0.00020.0002 --0.020.02 geom.geom. schemescheme +0.0009+0.0009 --0.00020.0002 --0.020.02 ClPO:ClPO: inin progressprogress

SiHSiH3F:F: PuzzariniPuzzarini,, CazzoliCazzoli,, GaussGauss JMSJMS 262,262, 3737 (2010)(2010) fullfull--QQ correctionscorrections basisbasis == cccc--pVDZpVDZ SiSi--FF // ÅÅ SiSi--HH // ÅÅ FSiH / deg. SiHSiH3FF gradientgradient schemescheme +0.0004+0.0004 +0.0001+0.0001 +0.00+0.00 geom.geom. schemescheme +0.0004+0.0004 --0.00000.0000 +0.01+0.01

ClPOClPO ClCl--PP // ÅÅ PP--OO // ÅÅ ClPO/ClPO/ deg.deg. gradientgradient schemescheme +0.0014+0.0014 +0.0013+0.0013 +0.03+0.03 geom.geom. schemescheme +0.0018+0.0018 +0.0017+0.0017 +0.04+0.04 ClPO:ClPO: inin progressprogress

SiHSiH3F:F: PuzzariniPuzzarini,, CazzoliCazzoli,, GaussGauss JMSJMS 262,262, 3737 (2010)(2010) WhichWhich levellevel forfor ““BIOMOLECULESBIOMOLECULES””??

COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ

RELIABLE?RELIABLE? ACCURATE?ACCURATE? TheThe challengechallenge ofof thethe conformationalconformational equilibriumequilibrium inin glycineglycine:: cancan compositecomposite schemesschemes shedshed lightlight onon thethe observationobservation ofof elusiveelusive conformers?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) COMPOSITECOMPOSITE APPROACHAPPROACH

1)1) “cheap”“cheap” geomgeom schemescheme r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  2)2) “accurate”“accurate” gradgrad schemescheme dE dE (HF  SCF) d E  (CCSD(T)) dE(CV) tot    dx dx dx dx

cc-pV(T,Q,5)Z cc-pV(T,Q)Z cc-pCVTZ TheThe twotwo mostmost stablestable conformersconformers ………… TheThe twotwo mostmost stablestable conformersconformers ………… TheThe followingfollowing fourfour stablestable conformersconformers …………

“cheap”“cheap” bestbest vsvs “accurate”“accurate” best:best: perfectperfect matchmatch WhichWhich levellevel forfor ““BIOMOLECULESBIOMOLECULES””??

COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ

RELIABLE?RELIABLE? ACCURATE?ACCURATE?  RotationalRotational constantconstant

 InertiaInertia tensortensor

 MoreMore unknownunknown parametersparameters thanthan datadata ????  MoreMore unknownunknown parametersparameters thanthan datadata ???? ISOTOPICISOTOPIC SUBSTITUTIONSUBSTITUTION

16O 12C 32S

17O, 18O 13C 33S, 34S

- NATURAL ABUNDANCE - ISOTOPICALLY ENRICHED EquilibriumEquilibrium structurestructure::

needneed ofof BBee forfor variousvarious isotopicisotopic speciesspecies

1 B Be  B0   r 2 r

RotationalRotational constantconstant ofof vibrationalvibrational groundground statestate VibrationalVibrational correctioncorrection EXPERIMENTEXPERIMENT THEORYTHEORY P. Pulay, W. Meyer, J.E. Boggs, J. Chem. Phys. 68, 5077 (1978). 1 B FITFIT Be  B0   r 2 r fromfrom EEXXPERIMENTPERIMENT ((variousvarious isotopicisotopic speciesspecies)) fromfrom TTHHEOREORYY ((cubiccubic forceforce fieldfield)) “Semi“Semi--exp.”exp.” equilibriumequilibrium structurestructure

Accuracy:Accuracy: experimentalexperimental qualityquality Pawłowski, Jørgensen, Olsen, Hegelund, Helgaker, Gauss, Bak, Stanton JCP 116 6482 (2002) SemiSemi--expexp equilibriumequilibrium structurestructure ofof largelarge moleculemolecule

IsotopicIsotopic substitution:substitution:bb -- 16OO  18OO -- 14NN  15H10NN O7 -- 12CC  13CC N3 C2 O8 C4 H9 aa 1010 isotopicisotopicN1 speciesspecies C6 C5 Vaquero, Sanz, López, Alonso, J. Phys. Chem. Lett. 111A, 3443 (2007). H12 H11 2020 rotationalrotational constantsconstants

URACIL:URACIL: 2121 independentindependent geometricalgeometrical parametersparameters Puzzarini & Barone, PCCP 13, 7158 (2011) COMPOSITECOMPOSITE APPROACHAPPROACH

1)1) “cheap”“cheap” geomgeom schemescheme r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  2)2) “accurate”“accurate” gradgrad schemescheme dE dE (HF  SCF) d E  (CCSD(T)) dE(CV) tot    dx dx dx dx

cc-pV(T,Q,5)Z cc-pV(T,Q)Z cc-pCVTZ a b c Best est. re Semi-exp. re Exp. rs Fit 1 Fit 2 Fit 3 Distances N1-C2 1.3785 1.38175(53) 1.38163(65) 1.38161(51) 1.386(5) C2-N3 1.3756 1.3763 1.3763 1.3762 N3-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) Angles C2-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 127.75 127.942 127.947 127.945 N3-C2-N1 113.51 113.383 113.379 113.380 N1-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 Non-determinable Parameters: fixed at the corresponding  theo values N1-C6-H12 115.34    COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule  EquilibriumEquilibrium RotationalRotational ConstantsConstants r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  VibrationalVibrational CorrectionsCorrections toto RotationalRotational ConstantsConstants  VibrationalVibrational correctionscorrections toto rotationalrotational constants:constants:

1 B B0  Be  r 2 r HowHow toto getget vibrationalvibrational correctionscorrections toto BB?? SecondSecond--orderorder vibrationalvibrational perturbationperturbation theorytheory (VPT2)(VPT2)

WATSONWATSON HamiltonianHamiltonian , = (x,y,z) r  {normal coord} where dimensionless vibrational normal angular coordinate momentum

inverse inertia tensor

potential HowHow toto getget vibrationalvibrational correctionscorrections toto BB?? SecondSecond--orderorder vibrationalvibrational perturbationperturbation theorytheory (VPT2)(VPT2)

unperturbedunperturbed Hamiltonian:Hamiltonian:

perturbations:perturbations: HarmonicHarmonic ffff

anharmonicanharmonic correctionscorrections

CoriolisCoriolis couplingcoupling BeyondBeyond thethe RigidRigid--RotatorRotator ApproximationApproximation vibrationalvibrational correctionscorrections toto rotationalrotational constants:constants:

vibrationvibration--rotationrotation interactioninteraction constants:constants: ComputationComputation ofof CubicCubic andand QuarticQuartic ForceForce FieldsFields

•• cubiccubic forceforce fields:fields:

single numerical differentiation along qr

•• quarticquartic forceforce fields:fields:

double numerical differentiation along qr

Schneider & Thiel, Chem.Chem. Phys.Phys. LettLett. 157, 367 (1989) Stanton et al., J.J. Chem.Chem. PhysPhys. 108, 7190 (1998) AccurateAccurate forceforce fieldfield

>>>>>>>> MainMain requirementsrequirements:: -- ““correlatedcorrelated”” methodmethod -- cccc basisbasis setset

-- harmonicharmonic ffff:: analyticanalytic 2nd2nd derivderiv.. ofof EE -- anharmonicanharmonic partpart:: numericalnumerical differdiffer..

Schneider & Thiel, Chem.Chem. Phys.Phys. LettLett. 157, 367 (1989) Stanton et al., J.J. Chem.Chem. PhysPhys. 108, 7190 (1998) COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule  EquilibriumEquilibrium RotationalRotational ConstantsConstants r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  VibrationalVibrational CorrectionsCorrections toto RotationalRotational ConstantsConstants

B3LYP/N07D MP2/cc-pVTZ 1.9811(29) 1.9811(29) 1.9255(24) 1.5273(32) 0.1055(23) 1.9255(24) 1.5273(32) 0.1055(23) 0.4530(32) 0.02623(18) 0.00680(13) 0.4530(32) 0.02623(18) 0.00680(13) 1.7600 (25) Experiment Experiment 1.7600 (25) 0.06336(44) 0.06336(44) - - - - 1330.928108(33) 1330.928108(33) 3883.873021(60) 2023.732581(45) 3883.873021(60) 2023.732581(45) % % , 7158 (2011) <0.2 <0.2 0.026 0.006 0.026 0.006 1.871 1.491 1.871 1.491 1.739 1.952 0.061 0.107 1.739 1.952 0.061 0.107 0.447 0.447 - - - - 1332.761 1332.761 3885.475 2027.763 Calculated Calculated 3885.475 2027.763 Barone, PCCP 13 & Puzzarini kHz kHz kHz kHz kHz kHz kHz kHz kHz kHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz URACIL URACIL

0 0 J J JK JK K K 0 0 0 0 1 1 2 2 aa aa bb bb aa aa bb bb        A A B B C C D D D D D D d d d d  AccuracyAccuracy ofof TheoreticalTheoretical RotationalRotational ConstantsConstants

STATISTICALSTATISTICAL ANALYSISANALYSIS forfor •• 1616 moleculesmolecules ((9797 isotopologues)isotopologues) •• 180180 rotationalrotational constantsconstants

ReferenceReference values:values: BBee ,, BB00 fromfrom experimentexperiment

HF, N2, CO, F2, HCN, HNC, O=C=O, H2O, NH3,

CH4, HCCH, HOF, HNO, NH=NH, CH2=CH2, H2C=O

C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VTZCCSD(T)/VTZ

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/VTZ

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V5Z

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5Z

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5Z ++ CVCV

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5Z ++ CVCV + +++ CV CVfTfT

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5Z ++ CVCV + +++ CV CVfTfT ++ fQfQ

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VCCSD(T)/VCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5ZZZ ++++ CVCVCVCV + +++++ CV CVfTfTfTfT ++++ fQfQfQfQ

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VCCSD(T)/VCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5ZZZ ++++ CVCVCVCV + +++++ CV CVfTfTfTfT ++ fQfQ++ ++++ fQfQ fQfQvibvib

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008) calccalc expexp BB vsvs BB00 CCSD(T)/VCCSD(T)/VCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/VQZCCSD(T)/VQZCCSD(T)/VTZCCSD(T)/V6ZCCSD(T)/VTZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V5ZZZ ++++ CVCVCVCV + +++++ CV CVfTfTfTfT ++ fQfQ++ ++ fQfQ vibvib++ ++++ + fQ+fQ fQfQvibvib eleele

-4-3-2-101234 normal distributions of relative errors C.C. PuzzariniPuzzarini,, M.M. HeckertHeckert,, J.J. GaussGauss JCPJCP 128128,, 194108194108 (2008)(2008)   ElectronicElectronic contributioncontribution toto BB  B0  Be  Bvib  Bel 

 =x,y,z me  princ. inertia system  B el  g Be mp

g = rotational g tensor

me = mass of the electron mp = mass of the proton CCSDCCSD((TT)) calc:calc: GGauss,auss, Ruud,Ruud, KallaKallayy,, JCPJCP 127,127, 074101074101 ((20072007)) 15 NN2--URACIL:URACIL: rotationalrotational spectrumspectrum inin thethe 55--1212 GHzGHz rangerange

Prediction using B e

Experiment

51 50 5 200 77 50 780 0 7 850 11200 11250 11300 11350 11400 11450

FREQUENCY (MHz)

Experimental data from: V. Vaquero, M. E. Sanz, J. C. Lopeź́ and J. L. Alonso, JPCA 111, 3443 (2007). Simulation from: Puzzarini, PCCP 15, 6595 (2013) 15 NN2--URACIL:URACIL: rotationalrotational spectrumspectrum inin thethe 55--1212 GHzGHz rangerange

Prediction using B e Prediction using B 0 Experiment

51 50 520 0 775 0 7 80 0 78 50 1120 0 1125 0 113 00 11 350 1 1400 11450 FREQUENCY (MHz)

Experimental data from: V. Vaquero, M. E. Sanz, J. C. Lopeź́ and J. L. Alonso, JPCA 111, 3443 (2007). Simulation from: Puzzarini, PCCP 15, 6595 (2013) CENTRIFUGALCENTRIFUGAL--DISTORTIONDISTORTION CONSTANTSCONSTANTS CentrifugalCentrifugal--distortiondistortion constantsconstants requiresrequires forceforce fieldfield calculationscalculations

HarmonicHarmonic forceforce fieldfield:: quarticquartic centrifugalcentrifugal--distortiondistortion constantsconstants

CubicCubic forceforce fieldfield:: sexticsextic centrifugalcentrifugal--distortiondistortion constantsconstants

…… …… …… QuarticQuartic centrifugalcentrifugal--distortiondistortion constants:constants:

combinationscombinations ofof  ’’ss

 1 r  1 r    r  2 r

LinearLinear MoleculesMolecules  D   xxxx J 4 QuarticQuartic centrifugalcentrifugal--distortiondistortion constants:constants: effecteffect onon rotationalrotational spectrumspectrum SexticSextic centrifugalcentrifugal--distortiondistortion constants:constants: combinationscombinations ofof ’’ss

AlievAliev && Watson,Watson, J.J. Mol.Mol. Spectrosc.Spectrosc. 61,61, 2929 (1976)(1976) QuarticQuartic && sexticsextic centrifugalcentrifugal--distortiondistortion constantsconstants

HF-SCF/ CCSD/ CCSD(T)/ CCSD(T)/ CCSD(T)/ Experiment 17 HF-SCF/ CCSD/ CCSD(T)/ CCSD(T)/ CCSD(T)/ Experiment DD2 OO augCVTZ augCVTZ augCVTZ augCVQZ augCV5Z

J / MHz 8.127 8.730 8.818 8.826 8.818 9.2889(14)

JK / MHz -40.525 -40.374 -40.303 -40.872 -40.959 -45.2241(51)

K / MHz 255.112 227.951 224.317 229.674 231.368 271.0554(57)

J / MHz 3.155 3.431 3.470 3.472 3.467 3.69282(21)

K / MHz 5.486 4.820 4.722 4.781 4.807 9.9004(51)

J / kHz 1.451 1.708 1.747 1.731 1.728 2.445(53)

JK / kHz -9.069 -10.633 -10.880 -10.740 -10.734 -10.89(30)

KJ / kHz -47.771 -40.950 -39.905 -42.413 -42.871 -71.02(38)

K / kHz 414.678 347.717 338.834 354.604 359.120 535.07(44)

J / kHz 0.715 0.845 0.864 0.856 0.854 0.9938(40)

JK / kHz -1.233 -1.631 -1.699 -1.681 -1.673 -4.05(13)

K / kHz 60.687 55.733 55.113 56.253 56.737 95.31(73)

Puzzarini,Puzzarini, Cazzoli,Cazzoli, Gauss,Gauss, J.J. Chem.Chem. Phys.Phys. 137137,, 154311154311 (2012)(2012) COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule  EquilibriumEquilibrium RotationalRotational ConstantsConstants r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  VibrationalVibrational CorrectionsCorrections toto RotationalRotational ConstantsConstants

B3LYP/N07D MP2/cc-pVTZ

 CentrifugalCentrifugal--DistortionDistortion ConstantsConstants D(best)  D(CCSD(T)/TZ) D(MP2/CVTZ,all) D(MP2/CVTZ,fc) D(MP2/aVTZ) D(MP2/VTZ)

CV diffuse

Puzzarini & Barone, PCCP 13, 7158 (2011) 1.9811(29) 1.9811(29) 1.9255(24) 1.5273(32) 0.1055(23) 1.9255(24) 1.5273(32) 0.1055(23) 0.4530(32) 0.02623(18) 0.00680(13) 0.4530(32) 0.02623(18) 0.00680(13) 1.7600 (25) Experiment Experiment 1.7600 (25) 0.06336(44) 0.06336(44) - - - - 1330.928108(33) 1330.928108(33) 3883.873021(60) 2023.732581(45) 3883.873021(60) 2023.732581(45) % % , 7158 (2011) ~1 ~1 0.026 0.006 0.026 0.006 1.871 1.491 1.871 1.491 1.739 1.952 0.061 0.107 1.739 1.952 0.061 0.107 0.447 0.447 - - - - 1332.761 1332.761 3885.475 2027.763 Calculated Calculated 3885.475 2027.763 Barone, PCCP 13 & Puzzarini kHz kHz kHz kHz kHz kHz kHz kHz kHz kHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz URACIL URACIL

0 0 J J JK JK K K 0 0 0 0 1 1 2 2 aa aa bb bb aa aa bb bb        A A B B C C D D D D D D d d d d  HYPERFINEHYPERFINE STRUCTURESTRUCTURE AccurateAccurate hyperfinehyperfine parametersparameters

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

-- accurateaccurate methodmethod [CCSD(T)][CCSD(T)] -- cccc basisbasis setset [n[nQ]Q] -- CVCV correctioncorrection [[additivityadditivity]] -- vibrationalvibrational correctioncorrection [[ffff:: correlcorrel methmeth..]] QuantumQuantum--ChemicalChemical CalculationCalculation ofof HyperfineHyperfine ParametersParameters

• NuclearNuclear quadrupolequadrupole couplingcoupling ELECTRIC FIELD GRADIENT

qK

first-order property

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

-- firstfirst derivativederivative ofof EE wrtwrt QQKK computedcomputed atat QQ=0=0 -- expectationexpectation valuevalue ofof thethe correspondingcorresponding operatoroperator NuclearNuclear quadrupolequadrupole--couplingcoupling constants:constants: fromfrom electricelectric fieldfield gradientsgradients ij-th element of the nuclear quadrupole-coupling tensor of the K-th nucleus: K ij  eQK qij

-eQK = quadrupole moment (known!!) qij = ij-th element of the electric field-gradient tensor

IK  1 NuclearNuclear quadrupolequadrupole--couplingcoupling constants:constants: effecteffect onon rotationalrotational spectrumspectrum

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

EXP.

CALC.

(~2 GHz freq. Shift!) CALC. (without Cl quadrupole coupling)

462980 462990 463000 463010 FREQUENCY (MHz) COMPOSITECOMPOSITE APPROACHAPPROACH extendedextended toto largelarge moleculemolecule  EquilibriumEquilibrium RotationalRotational ConstantsConstants r(CBS CV  diff  T)  r(CBS)  r(CV)  r(diff)  r(T)

MP2/cc-pV(T,Q)Z MP2/aug-cc-pVTZ

MP2/cc-pCVTZ CCSD(T)/cc-pVTZ  VibrationalVibrational CorrectionsCorrections toto RotationalRotational ConstantsConstants

B3LYP/N07D MP2/cc-pVTZ

 NitrogenNitrogen qudrupolequdrupole--couplingcoupling ConstantsConstants (best)  (CCSD(T)/CVTZ)  (TZ  QZ)  (diff)

MP2/cc-pCV(T,Q)Z

MP2/aug-cc-pVTZ

Puzzarini & Barone, PCCP 13, 7158 (2011) 1.9811(29) 1.9811(29) 1.9255(24) 1.5273(32) 0.1055(23) 1.9255(24) 1.5273(32) 0.1055(23) 0.4530(32) 0.02623(18) 0.00680(13) 0.4530(32) 0.02623(18) 0.00680(13) 1.7600 (25) Experiment Experiment 1.7600 (25) 0.06336(44) 0.06336(44) - - - - 1330.928108(33) 1330.928108(33) 3883.873021(60) 2023.732581(45) 3883.873021(60) 2023.732581(45) % % 2 2 - - , 7158 (2011) 1 1 0.026 0.006 0.026 0.006 1.871 1.491 1.871 1.491 1.739 1.952 0.061 0.107 1.739 1.952 0.061 0.107 0.447 0.447 - - - - 1332.761 1332.761 3885.475 2027.763 Calculated Calculated 3885.475 2027.763 Barone, PCCP 13 & Puzzarini kHz kHz kHz kHz kHz kHz kHz kHz kHz kHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz URACIL URACIL

0 0 J J JK JK K K 0 0 0 0 1 1 2 2 aa aa bb bb aa aa bb bb        A A B B C C D D D D D D d d d d  QuantumQuantum--ChemicalChemical CalculationCalculation ofof HyperfineHyperfine ParametersParameters

• SpinSpin--rotationrotation interactioninteraction

second-order property NuclearNuclear spinspin--rotationrotation tensortensor

ElectronicElectronic contributioncontribution NuclearNuclear contributioncontribution ++

ˆl = electronic angular momentum J = rotational angular momentum I = nuclear spin angular momentum

K = gyromagnetic ratio of the K-th nucleus ˆ lK = electronic angular momentum defined wrt RK QuantumQuantum--ChemicalChemical CalculationCalculation ofof SpectroscopicSpectroscopic ParametersParameters

• SpinSpin--spinspin couplingcoupling

DIPOLAR SPIN-SPIN COUPLING TENSOR

requires equilibrium geometry: no „electronic property“

addditional contribution due to:  vibrational corrections (anharmonic force field) VIBRATIONALVIBRATIONAL CORRECTIONCORRECTION DifferenceDifference betweenbetween vibrationallyvibrationally averagedaveraged valuevalue andand equilibriumequilibrium valuesvalues (same(same level:level: i.e.,i.e., samesame methodmethod andand samesame basisbasis setset)) P  P P vib ave eq VIBRATIONALVIBRATIONAL AVERAGINGAVERAGING ExpansionExpansion ofof thethe expectationexpectation valuevalue overover thethe vibvib wfwf aroundaround thethe equilequil wrtwrt normalnormal--coordinatecoordinate displacementsdisplacements

 P  1  2P  P P    Q    QQ ... eq Q  r 2 QQ  r s rs r eq r,  r s eq

wherewhere r s  rs Q  rss Q Q   r  2  4r s s  2 r

A.A. Auer, J. Gauss & J.F. Stanton, JCP 118, 10407 (2003)  transtrans--HCOOD:HCOOD: hyperfinehyperfine structurestructure duedue toto DD

188308.35 188308.40 188308.45 188308.50

F = 27 1 F = 28,26 1

J = 27 - 27 F = 0 3,24 3,25 1

Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011) transtrans--HCOOD:HCOOD: hyperfinehyperfine structurestructure duedue toto DD andand HH

188308.35 188308.40 188308.45 188308.50

F ,F = 27,53/2 F ,F = 27,55/2 1 1 F ,F = 28,57/2 1 F ,F = 28,55/2 1 26,53/2 26,51/2

J = 27 - 27 F , F = 0 3,24 3,25 1

Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011) transtrans--HCOOD:HCOOD: hyperfinehyperfine structurestructure duedue toto DD andand HH

188308.35 188308.40 188308.45 188308.50

F ,F = 27,53/2 F ,F = 27,55/2 1 1 F ,F = 28,57/2 1 F ,F = 28,55/2 1 26,53/2 26,51/2

J = 27 - 27 F , F = 0 3,24 3,25 1

Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011) ROTATIONALROTATIONAL SPECTROSCOPY:SPECTROSCOPY: InterplayInterplay ofof ExperimentExperiment andand TheoryTheory

CristinaCristina PuzzariniPuzzarini Dip.Dip. ChimicaChimica ““GiacomoGiacomo CiamicianCiamician”” UniversitUniversitàà didi BolognaBologna

CODECSCODECS SummerSummer SchoolSchool 20132013 THEORETICALTHEORETICAL SPECTROSCOPYSPECTROSCOPY PREDICTINGPREDICTING ROTATIONALROTATIONAL SPECTRASPECTRA PuzzariniPuzzarini ,, Biczysko,Biczysko, Barone,Barone, Pena,Pena, Cabezas,Cabezas,Alonso,Alonso, PCCPPCCP acceptedaccepted ObservationObservation ofof thethe rotationalrotational spectrumspectrum ofof thiouracilthiouracil:: CanCan compositecomposite schemesschemes provideprovide thethe spectroscopicspectroscopic parametersparameters withwith thethe properproper accuracy?accuracy? COMPOSITECOMPOSITE APPROACHAPPROACH

1)1) rre (B(Be),), D’s,D’s, q’sq’s:: “cheap”“cheap” geomgeom schemescheme BestBest == CBS(MP2/TZCBS(MP2/TZ--QZ)QZ) ++ CV(MP2/CVTZ)CV(MP2/CVTZ) ++ diff(MP2/AVTZ)diff(MP2/AVTZ) ++ pertT(CCSD(T)/VTZ)pertT(CCSD(T)/VTZ)

2)2) alphasalphas:: DFTDFT

1 B B0  Be   r DFTDFT == B3LYP/SNSDB3LYP/SNSD 2 r Parameter Main 34S Exp Theo Exp Theo

A0 [MHz] 3555.18805(64) 3555.458 3545.6594(11) 3545.945

B0 [MHz] 1314.86002(27) 1315.287 1276.1741(51) 1276.569

C0 [MHz] 960.03086(16) 960.200 938.57117(54) 938.732

14N(1) χ [MHz] 1.634(10)   1.616(13) aa 1.6090.1%0.1% 1.614

χbb [MHz] 1.777(12) 1.813 1.755(17) 1.807

χcc [MHz] -3.411(12) -3.422 -3.371(17) -3.422

χab [MHz] - 0.314 - 0.316

14N(3)

χaa [MHz] 1.726(10) 1.739 1.732(13) 1.733

χbb [MHz] 1.399(13) 1.384 1.429(19) 1.390

χcc [MHz] -3.125(13) -3.123 -3.161(19) -3.123

χab [MHz] - -0.336 - -0.339 EXPERIMENT THEORY

32S

6000 7000 8000 9000 10000 11000 12000 FREQUENCY (MHz) EXPERIMENT THEORY

5 - 4 0,5 1,4 4 - 3 1,4 0,3 5 - 5 2,4 1,5

9500 9600 9700 9800 9900 10000 10100 FREQUENCY (MHz) EXPERIMENT THEORY (only B's) THEORY (B's + D's)

5 - 4 0,5 1,4 4 - 3 1,4 0,3 5 - 5 2,4 1,5

9500 9600 9700 9800 9900 10000 10100 FREQUENCY (MHz) EXPERIMENT THEORY (only B's) 5 - 4 THEORY (B's + D's) 0,5 1,4

~3~3 MHzMHz

9601.0 9601.5 9602.0 9602.5 9603.0 9603.5 9604.0 9604.5 FREQUENCY (MHz) Parameter Main 34S Exp Theo Exp Theo

A0 [MHz] 3555.18805(64) 3555.458 3545.6594(11) 3545.945

B0 [MHz] 1314.86002(27) 1315.287 1276.1741(51) 1276.569 C [MHz] 0 960.03086(16) 960.200~~1%1%938.57117(54) 938.732

14N(1)

χaa [MHz] 1.634(10) 1.609 1.616(13) 1.614

χbb [MHz] 1.777(12) 1.813 1.755(17) 1.807

χcc [MHz] -3.411(12) -3.422 -3.371(17) -3.421

χab [MHz] - 0.314 - 0.316

14N(3)

χaa [MHz] 1.726(10) 1.739 1.732(13) 1.733

χbb [MHz] 1.399(13) 1.384 1.429(19) 1.390

χcc [MHz] -3.125(13) -3.123 -3.161(19) -3.123

χab [MHz] - -0.336 - -0.339 2,3  2,2 theory experiment

1,2  1,1 2,2  1,1 1,2  2,2 2,2  2,2

1,1  0,0 1,1  1,1 1,1  2,2 0,1  0,0 0,1  1,1 ~0.4 MHz 0,1  2,2

2,1  0,0 2,1  1,1 1,0  1,1 2,1  2,2

4514.5 4515.0 4515.5 4516.0 FREQUENCY (MHz) Cazzoli,Cazzoli, Puzzarini,Puzzarini, Stopkowicz,Stopkowicz, Gauss,Gauss, AA &AA 520520,, A64A64 (2010)(2010) HCOOH: J = 18 - 18 2,16 2,17 LaboratoryLaboratory ofof MillimetreMillimetreExperiment:--wavewave Lam b-dip

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

RF data: only SR

107638.20SpectroscopySpectroscopy 107638.25ofof 107638.30 BolognaBologna 107638.35 FREQUENCY (MHz) HCOOH: J = 18 - 18 2,16 2,17 LaboratoryLaboratory ofof MillimetreMillimetreExperiment:--wavewave Lam b-dip

RF data: only SR

107638.20SpectroscopySpectroscopy 107638.25ofof 107638.30 BolognaBologna 107638.35 FREQUENCY (MHz) J.-C. Chardon, C. Genty, D. Guichon, & J.-G. Theobald, J. Chem. Phys. 64, 1434 (1976) HCOOH: J = 18 - 18 LaboratoryLaboratory ofof MillimetreMillimetre2,16 2,17 --wavewave Experiment

Theory: SR and SS

Theory:

only SR

RF data: only SR

107638.20SpectroscopySpectroscopy 107638.25ofof 107638.30 BolognaBologna 107638.35 FREQUENCY (MHz) HCOOH: J = 18 - 18 LaboratoryLaboratory ofof MillimetreMillimetre2,16 2,17 --wavewave Experiment

Theory: SR and SS

Theory:

only SR

RF data: only SR

107638.20SpectroscopySpectroscopy 107638.25ofof 107638.30 BolognaBologna 107638.35 FREQUENCY (MHz) HyperfineHyperfine parametersparameters ofof transtrans--HCOOHHCOOH ExperimentExperiment TheoryTheory RFRF resultsresults

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

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

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

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

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

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

1.51.5DDaa 4.49(12)4.49(12) 4.624.62 ----

((DDbb –– DDcc)/4)/4 --3.53(35)3.53(35) --3.473.47 ---- Equil: CCSD(T)/CV5Z + Vib. Corr: CCSD(T)/CVTZ HyperfineHyperfine parametersparameters ofof transtrans--HCOOHHCOOH ExperimentExperiment TheoryTheory RFRF resultsresults

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

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

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

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

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

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

1.51.5DDaa 4.49(12)4.49(12) 4.624.62 ----

((DDbb –– DDcc)/4)/4 --3.53(35)3.53(35) --3.473.47 ---- Cazzoli, Puzzarini, Stopkowicz, Gauss, A &A 520, A64 (2010) MOLECULARMOLECULAR PROPERTIESPROPERTIES ElectricElectric andand magneticmagnetic propertiesproperties fromfrom RotationalRotational SpectroscopySpectroscopy

-- ELECTRIC:ELECTRIC: • Dipole moment • Nuclear quadrupole coupling -- MAGNETIC:MAGNETIC: • Spin-rotation interaction • Spin-spin interaction ELECTRICELECTRIC PROPERTIESPROPERTIES ElectricElectric dipoledipole momentmoment  AnalysisAnalysis ofof thethe spectraspectra completed:completed: 1.1. TransitionsTransitions assignedassigned (transition(transition frequenciesfrequencies retrieved)retrieved) 2.2. FrequeciesFrequecies fittedfitted (with(with thethe properproper Hamiltonian)Hamiltonian)  3.3. SpectroscopicSpectroscopic parameters:parameters: -- rotationalrotational constantsconstants BB -- centrifugalcentrifugal--distortiondistortion constantsconstants D,D, H,H, …… -- hyperfinehyperfine parametersparameters (if(if thethe case)case) -- dipoledipole momentmoment (if(if StarkStark spectroscopy)spectroscopy) UnknownUnknown molecularmolecular dipoledipole momentmoment …… ExperimentExperiment:: StarkStark spectroscopyspectroscopy ……

J = 5 - 5 64.8 V 2,3 1,4 74.0 V CHCH2FBrFBr 82.2 V 91.1 V F = 11 - 11 106.6 V

F = 9 - 9

106780 106790 106800 106810 Frequency (MHz) UnknownUnknown molecularmolecular dipoledipole momentmoment …… ExperimentExperiment:: StarkStark spectroscopyspectroscopy ……

-- POSITIVEPOSITIVE JPEAKS:PEAKS: = 5 - unperturbedunperturbed5 transitionstransitions 64.8 V 2,3 1,4 74.0 V CHCH2FBrFBr 82.2 V -- NEGATIVENEGATIVE PEAKS:PEAKS: StarkStark componentscomponents 91.1 V F = 11 - 11 106.6 V

F = 9 - 9

106780 106790 106800 106810 Frequency (MHz) J = 5 - 5 64.8 V CH2FBr 2,3 1,4 74.0 V 82.2 V 91.1 V F = 11 - 11 106.6 V ValuesValues inin debyedebye F = 9 - 9

a b aVQZaVQZ -0.341 -1.696 106780 106790 106800 106810 Frequency (MHz) 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 115115 (2007)(2007) DIPOLEDIPOLE MOMENTMOMENT ofof CHCH2FIFI Values in debye

-0.022

2nd-order Direct Perturbation Theory spin-free Dirac Coulomb approach

importanceimportance ofof relativisticrelativistic effectseffects forfor heavyheavy elementselements  AnalysisAnalysis ofof thethe spectraspectra completed:completed: 1.1. TransitionsTransitions assignedassigned (transition(transition frequenciesfrequencies retrieved)retrieved) 2.2. FrequeciesFrequecies fittedfitted (with(with thethe properproper Hamiltonian)Hamiltonian)  3.3. SpectroscopicSpectroscopic parameters:parameters: -- rotationalrotational constantsconstants BB -- centrifugalcentrifugal--distortiondistortion constantsconstants D,D, H,H, …… -- hyperfinehyperfine parametersparameters (if(if thethe case)case) -- dipoledipole momentmoment (if(if StarkStark spectroscopy)spectroscopy) NuclearNuclear QuadrupoleQuadrupole CouplingCoupling

DETERMINATIONDETERMINATION ofof thethe NUCLEARNUCLEAR QUADRUPOLEQUADRUPOLE MOMENTMOMENT BromineBromine NuclearNuclear QuadrupoleQuadrupole MomentMoment

yearyear eQeQ Lederer,Lederer, ShirleyShirley 1978 293 TaqquTaqqu 1978 331(4) Kellö,Kellö, SadlejSadlej 1990 304.5 Kellö,Kellö, SadlejSadlej 1996 298.9 Hass,Hass, PetrilliPetrilli 2000 305(5); 308.7 VanVan Lenthe,Lenthe, BaerendsBaerends 2000 300(10) BieronBieron etet al.al. 2001 313(3) valuesvalues inin mbarnmbarn forfor 79BrBr RevisionRevision ofof thethe 79BrBr QuadrupoleQuadrupole MomentMoment experimental quadrupole coupling

nuclear computed quadrupole moment electric field gradient HBrHBr

StoStoppkowicz,kowicz, ChenChengg,, HardinHardingg,, Puzzarini,Puzzarini, GGauss,auss, Mol.Mol. PhPhyys.s. 111111,, 13821382 ((20132013)) Bromine Quadrupole Coupling in CH2FBr

Theory:

includingincluding relativisticrelativistic effectseffects & usingusing newnew Q:Q: exp rel+vib χij χij 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 goodgood agreementagreement betweenbetween theorytheory andand experimentexperiment Stopkowicz,Stopkowicz, Cheng,Cheng, Harding,Harding, Puzzarini,Puzzarini, Gauss,Gauss, Mol.Mol. Phys.Phys. 111111,, 13821382 (2013)(2013) MAGNETICMAGNETIC PROPERTIESPROPERTIES NMRNMR MWMW connectionconnection nuclear quadrupole nuclear quadrupole

coupling CQ coupling 

nuclear magnetic shielding  chemical nuclear shift absolute shielding spin-rotation  scales C Ramsey-Flygare equations direct dipolar tensor spin-spin coupling coupling (rank 2)

D form of Hamiltonians: C3 coupling mechanism indirect spin-spin vs scalar spin-spin coupling tensor rank coupling (rank 0)

J C4

BryceBryce && WasylishenWasylishen,, AccAcc 3636,, 327327 (2003)(2003) .. ChemChem .. Res.Res. DIATOMICDIATOMIC oror LINEARLINEAR MOLECULESMOLECULES

nuclearnuclear magneticmagnetic shieldingshielding 

 == dd ++ pp  DIAMAGNETICDIAMAGNETIC PARTPART PARAMAGNETICPARAMAGNETIC PARTPART

 2 3  mp c  3  e  Z p p   I 0             2 2megN  B 24 3me  r  DIATOMICDIATOMIC oror LINEARLINEAR MOLECULESMOLECULES

nuclearnuclear magneticmagnetic shieldingshielding 

 == dd ++ pp  DIAMAGNETICDIAMAGNETIC PARTPART PARAMAGNETICPARAMAGNETIC PARTPART

 2 3  mp c  3  e  Z p p   I 0             2 2megN  B 24 3me  r  ASYMMETRICASYMMETRIC--TOPTOP MOLECULESMOLECULES

nuclearnuclear magneticmagnetic shieldingshielding 

 == dd ++ pp

DIAMAGNETICDIAMAGNETIC PARTPART PARAMAGNETICPARAMAGNETIC PARTPART AbsoluteAbsolute NMRNMR shieldingshielding scalescale

1717 1717 && 22

Puzzarini,Puzzarini, Cazzoli,Cazzoli, Harding,Harding, Vázquez,Vázquez, Gauss,Gauss, workwork inin progressprogress ………… LaboratoryLaboratory ofof MillimetreMillimetre--wavewave TheThe beginningbeginning ofof thethe storystory ….….

SpectroscopySpectroscopy ofof BolognaBologna 17 J = 4 - 3 HH2 OO:: 1,4 2,1

Experiment

Real+Ghost

Real

Ghost

385784 385786 385788 385790 FREQUENCY (MHz) PuzzariniPuzzarini,, Cazzoli,Cazzoli, HardingHarding ,, VázquezVázquez & GausGauss,s, JCPJCP 131131,, 234304234304 (2009)(2009) ResultsResults …….……. SRSR ofof 1717OO

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 resultsresults inin kHzkHz Method:Method: Equil.Equil. Vib.Vib. Vib.Vib. TotalTotal

CCSD(T)CCSD(T) (exp(exp rre)) Corr.Corr. Corr.Corr. (Eq+Vib)(Eq+Vib) (VPT2)(VPT2) (DVR)(DVR) 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 AbsoluteAbsolute 1717OO NMRNMR scalescale [ppm] isotropicisotropic

 (dia)(dia) 416.4416.4 calculatedcalculated  (para)(para) --78.578.5 fromfrom expexp  (equil)(equil) 338.1(3)338.1(3)  (vib)(vib) --11.711.7  (T)(T) --0.40.4  (300K)(300K) 326.2(3)326.2(3) BestBest theoreticaltheoretical estimateestimate 325.6325.6 ppmppm InIn searchsearch ofof confirmationconfirmation ….….

DeterminationDetermination ofof thethe 1717OO spinspin--rotationrotation constantsconstants 1717 1717 forfor DD22 OO andand HDHD OO 1717 22 EXPERIMENTEXPERIMENT THEORYTHEORY 17 eQqeQqaa (( O)O) // MHzMHz --8.8717(28)8.8717(28) --8.88.811 Equilibrium: CCSD(T)/augCV6Z eQqeQq ((17O)O) // MHzMHz --1.2716(68)1.2716(68) --1.21.233 bb Vibrat. Corr.: CCSD(T)/augCV5Z 17 eQqeQqcc (( O)O) // MHzMHz 10.1433(68)10.1433(68) 10.010.0VPT244 DVR 17 CCaa (( O)O) // kHzkHz --114.574.57(2(211)) --14.6714.67 --14.8014.80 17 CCbb (( O)O) // kHzkHz --113.343.34(2(255)) --13.6113.61 --13.13.6060 17 CCcc (( O)O) // kHzkHz --9.669.66((2828)) --9.49.411 --9.419.41 eQqeQqaa (D)(D) // MHzMHz 0.1479(26)0.1479(26) 0.150.15 eQqeQqbb (D)(D) // MHzMHz 0.041(11)0.041(11) 0.020.02 eQqeQqcc (D)(D) // MHzMHz --0.189(11)0.189(11) --0.10.188

CCaa (D)(D) // kHzkHz ------2.92.944

CCbb (D)(D) // kHzkHz ------2.42.411

CCcc (D)(D) // kHzkHz ------2.612.61 17 SSaa (D(D-- O)O) // kHzkHz 2.11(65)2.11(65) 2.42.444 SS (D(D ------1.611.61 aa --D)D) // kHzkHz AbsoluteAbsolute 1717OO NMRNMR scalescale [ppm] 17 17 [ppm] HH2 OO DD2 OO  (dia)(dia) 416.4416.4 416.4416.4 calculatedcalculated  (para)(para) --79.0(3)79.0(3) --78.6(9)78.6(9) fromfrom expexp  (equil)(equil) 337.4(3)337.4(3) 337.8(9)337.8(9)  (vib)(vib) --11.711.7 --8.48.4  (T)(T) --0.40.4 --0.40.4  (300K)(300K) 325.3(3)325.3(3) 329.0(9)329.0(9) MOLECULARMOLECULAR STRUCTURESTRUCTURE DETERMINATIONDETERMINATION  RotationalRotational constantconstant

 InertiaInertia tensortensor

 IsotopicIsotopic substitutionsubstitution TYPESTYPES ofof MOLECULARMOLECULAR STRUCTURESTRUCTURE

 EFFECTIVEEFFECTIVE STRUCTURE:STRUCTURE: rr00

 SUBSTITUTIONSUBSTITUTION STRUCTURE:STRUCTURE: rrss

 MASSMASS--DEPENDENCEDEPENDENCE STRUCTURE:STRUCTURE: rrmm

 EQUILIBRIUMEQUILIBRIUM STRUCTURE:STRUCTURE: rree EFFECTIVEEFFECTIVE STRUCTURESTRUCTURE rr00

StructureStructure calculatedcalculated directlydirectly fromfrom BB0:: leastleast--squaressquares fitfit ofof thethe molecularmolecular structuralstructural parametersparameters toto thethe momentsmoments ofof inertiainertia II0

 I calc  I exp  I calc   i p i i   j j  p j  i runs over inertial moments (isotopic substitution) j runs over structural parameters Accuracy:Accuracy: limitedlimited Approximation = zero-point vibrational effects are the same for different isotopic species

rr00 >> rree FCP SO2: r(S–O) r(F–C) r(C–P)

rre = 1.4308 Å rre (Å) 1.27547 1.54476 rr0 = 1.4336 Å rr0 (Å) 1.28456 1.54097 Morino et al. J. Mol. Spectrosc. 13, 95 (1964) 0 Bizzocchi, Degli Esposti, Puzzarini Mol. Phys. 104, 2627 (2006) SUBSTITUTIONSUBSTITUTION STRUCTURESTRUCTURE rrss MakeMake useuse ofof isotopicisotopic substitutionsubstitution forfor derivingderiving thethe positionposition (coordinates)(coordinates) ofof thethe substituedsubstitued atom:atom: I '  I  ( y 2  z 2 ) xx  x I '  I  ( x 2  z 2 ) yy  y ' 2 2 I zz  I z  ( x  y ) '  Kraitchman’sKraitchman’s equationsequations I xy   xy  [C.C.[C.C. Costain,Costain, J.J. Chem.Chem. Phys.Phys. 2929,, 864864 (1958)](1958)] I '   xz xz  ' I yz   yz  M  m   M   m 1)1) Accuracy:Accuracy: rree  rrss  rr00 Approximation = zero-point vibrational effects tend to cancel using Kraitchman’s equation 2)2) EachEach nonnon--equivalentequivalent atomsatoms bebe substitutedsubstituted  mi zsi When not feasible: z   m firstfirst--momentmoment equationsequations ClCl BB SS

(Å) Cl–B B=S

rr0 1.6819(22) 1.6063(22)

rrs 1.6815(10) 1.6040(10)

rre 1.680567(89) 1.604923(90)

Bizzocchi, Degli Esposti, Puzzarini J. Mol. Spectrosc. 216, 177 (2002) leastleast--squaressquares treatmenttreatment toto obtainobtain rrs structures:structures: PlanarPlanar momentmoment ofof inertiainertia 2 2 2 Px   mi xi Py   mi yi Pz   mi zi i i i

Pxy   mi xi yi Pxz   mi xi zi Pyz   mi yi zi i i i

exp calc  Px   Px   Px  Px  Px    x    y    z  x 0  y 0  z 0

- similar equations for Py and Pz -(x0 ,y0 ,z0) coordinate of the atom in the parent molecule [Mostly[Mostly usedused forfor asymmetricasymmetric--toptop molecules]molecules] MASSMASS--DEPENDENCEDEPENDENCE STRUCTURESTRUCTURE rrmm ExtensionExtension ofof thethe substitutionsubstitution method:method: toto firstfirst--order,order, thethe massmass dependencedependence ofof thethe vibrationalvibrational contributionscontributions areare determineddetermined  first-order approx 1   2 M  I m  I e   m  b b   2  i  M i  mi  I m mass-dependence moment of inertia b  e I b b    s d s Linear molecule case 2Be s AccuracyAccuracy Validity of the first-order approximation I m  I e MajorMajor problems:problems: - light atoms (as H) - missing isotopic substitution (as F)

ImprovementsImprovements (1) (2) (1L) (2L) rm rm rm rm r (1) model  rm model  1/ 2 0 m  m I  I  c I    a,b,c

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

(2)  rrm modelmodel    0 m m 1/ 2 1/ (2N 2) I  I  c I  d m1 mN / M    a,b,c

Suitable correction function based on appropriate reduced masses MolecularMolecular structurestructure ofof OCSOCS

OCS r(C–O) r(C–S)

r0 1.15638(113) 1.56488(92) rs 1.15842(76) 1.56150(93)

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

(2) rm 1.15619(12) 1.56120(5) re 1.155386(21) 1.562021(17) Watson et al. J. Mol. Spectrosc. 196, 102 (1999) Foord et al. Mol. Phys. 29, 1685 (1975) To solve anomalies due to light atoms … (1L) (2L) rrm andand rrm modelsmodels 1/ 2 eff  M  rm (XH)  rm (XH)  H   mH M  mH  Laurie-type correction: introduced by using an effective bond length

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

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

(2) rm 1.06163(24) 1.15404(15) 1.06531(92) 1.15310(24) re 1.06501(8) 1.15324(2) Comparison & Accuracy EXAMPLESEXAMPLES TheThe failurefailure ofof thethe rrs structurestructure

H CC CC F r 1.2221 1.2854 s 1.0573 1.2079 1.3525 r 1.0558 1.2078 1.3713 1.2031 1.2729 0 r 1.0614 1.2080 1.3731 1.2013 1.2735 e

H CC

r 1.0457 1.2193 r0 1.0465 1.2165 s r 1.0651 1.2075 e 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). TheThe failurefailure ofof thethe rrm structuresstructures Br re in black (1) rm in red 1.92854(12) (1L) 1.9274(10) rm in blue 1.9286(8) 107.233(8) 107.36(5) 107.19(5) 110.151(32) 110.36(16) 110.24(20)

1.35757(13) 1.3641(19) CH 1.3674(15) 1.08302(8) 1.0854(4) F 109.552(10) H 1.0699(37) 109.13(7) 109.28(7) C. Puzzarini, G. Cazzoli, A. Baldacci, A. Baldan, C. Michauk, and J. Gauss, J. Chem. Phys. 127, 164302 (2007) 0 0 r r , (2006) 054307 125 JCP s s r r (2) (2) m m r r G. Gauss, Gambi, J. Cazzoli, A. (emp) e e r r 123.07(1) 123.1(2)120.74(9) 122.9(6) 121.9(8)122.61(6) 123.2(2) 126.4(6) 122.8(4)123.50(2) 126.7(3) 122.8(5) 123.8(2) 122.1(2) 124.0(6) 124.6(3) 1.7128(6) 1.715(4)1.0776(4) 1.721(5) 1.077(6) 1.729(2) 1.108(5)1.3317(3) 1.110(2) 1.327(8)1.0802(6) 1.330(5) 1.081(5) 1.345(3) 1.088(6) 1.083(2) 1.3240(14) 1.330(7) 1.323(4) 1.314(2) 2 2 1 1 C , 4189 (2001) // C. Puzzarini, C C 1 C 3 2 1 2 2 –Cl –H –C –F –H HC HC FC ClC

1 1 1 2 2

PCCP    C C C C  C

e e

n

1.0787 n e

e 1.3317(3) 1.3310 l

F l

y

1.0802(6) y

H h h

t

t

e

2 e o

o

r

C r

o

122.53 122.61(6) o

lu lu

f

123.50(2) 123.43 f

-

-

2 2

-

-

o

cis o

1.3249 r

r

o

1.3240(14) o l

1 l

h

h

c C c

123.10 - -

1

1

-

123.07(1) - s s 120.74(9) 120.43 i

i c H c Cl ) 6 C. Puzzarini, G. Cazzoli, L. Dore, A. Gambi ( 1.0764 1.0776(4) 107 128 7 7 EQUILIBRIUMEQUILIBRIUM STRUCTURESTRUCTURE rree

-- StructureStructure calculatedcalculated fromfrom BBe:: leastleast--squaressquares fitfit ofof thethe molecularmolecular structuralstructural parametersparameters toto thethe momentsmoments ofof inertiainertia IIe -- ClearClear physicalphysical meaning:meaning: minimumminimum ofof thethe BornBorn--OppenheimerOppenheimer PES,PES, trulytruly isotopicisotopic independentindependent   

  dr  Bv  Be   r  vr     a,b,c r  2  r runs over vibrational normal modes MainMain limitation:limitation:

AAvv,, BBvv,, CCvv for each vibrational state v

InvestigationInvestigation ofof eithereither purepure--rotationalrotational oror vibrovibro-- rotationalrotational spectraspectra ofof eacheach fundamentalfundamental modemode

ApproachApproach limitedlimited toto smallsmall (2(2--44 atoms)atoms) moleculesmolecules IMPOSSIBILITYIMPOSSIBILITY OFOF GETTINGGETTING ALLALL VIBRATIONVIBRATION--ROTATIONROTATION INTERACTIONINTERACTION CONSTANTSCONSTANTS NEEDED:NEEDED: HOWHOW TOTO SOLVESOLVE THETHE PROBLEM?PROBLEM?

THETHE SEMISEMI--EXPERIMENTALEXPERIMENTAL APPROACHAPPROACH P.P. Pulay,Pulay, W.W. Meyer,Meyer, J.E.J.E. Boggs,Boggs, J.J. Chem.Chem. Phys.Phys. 68,68, 50775077 (1978)(1978) EquilibriumEquilibrium structurestructure::

needneed ofof BBee forfor variousvarious isotopicisotopic speciesspecies

1 B Be  B0   r 2 r

RotationalRotational constantconstant ofof vibrationalvibrational groundground statestate VibrationalVibrational correctioncorrection EXPERIMENTEXPERIMENT THEORYTHEORY P. Pulay, W. Meyer, J.E. Boggs, J. Chem. Phys. 68, 5077 (1978). BB00 fromfrom EXPERIMENTEXPERIMENT ((variousvarious isotopicisotopic speciesspecies))

ActualActual FIT:FIT: momentsmoments ofof inertiainertia

VibrationalVibrational CorrectionsCorrections ffromrom THEORYTHEORY ((cubiccubic forceforce fieldfield)) TypicalTypical accuracyaccuracy:: betterbetter thanthan 0.0010.001 ÅÅ RequirementsRequirements forfor accurateaccurate structure:structure:  computedcomputed fromfrom forceforce fieldfield obtainedobtained withwith correlatedcorrelated methodmethod and,and, atat least,least, tripletriple--zetazeta basisbasis setset 125.83 125.768(48) 8 I 1 I .3 118.8(7) 97 11..3 4 397 1. 793 11 4 38 3((44 (2 0) )) ) 9 10 5 5 ( 5 6 )) 3 33 9 4 33 9 (4( 4 44 1 9 91 ) 4 ..9 0) 33 1 0 .. 7 21 (1 3 12 4( . 1 24 11 3 92 . .9 1 21 1 112 ) .33((6 6 11222 12 2 1 85 1.3.37 3)) 75(5 7 ..3381 CC HH 1 )) 86(5 1.38 9 FF HH

SS

CC HH1 HH1 NN NN

HH2 HH2 EquilibriumEquilibrium structurestructure determinationdetermination:: reviewreview

1)1) ExperimentallyExperimentally:: rr00,, rrss,, rrmm,, …… rree(?)(?)

2)2) ComputationallyComputationally:: rree

3)3) MixedMixed expexp--calccalc:: rree ((empiricalempirical)) a)a) ExpExp data:data: rotationalrotational constantsconstants b)b) ExpExp data:data: spinspin--spinspin constantsconstants SPINSPIN--SPINSPIN INTERACTIONINTERACTION

 DIRECTDIRECTKL spinspin--spinspin interactioninteraction constantconstant:: ij L  D K HH SS == ++ IILL DD IIKK g g N 0 KL 2 i 4   3(R ) (RKL )  R ε j 2 D    ij c 5 KL   R LK  INDIRECTINDIRECT spinspin--spinspin interactioninteraction constantconstant:: J HHSS == ++ IILL JJ IIKK PROCEDUREPROCEDURE

1)1) SubtractionSubtraction ofof thethe computedcomputed vibrationalvibrational correctioncorrection inin orderorder toto getget equilibriumequilibriumK DDKL::

KL L KL Deq  Dexp   Dvib KL ij L  2)2) DeterminationDeterminationK ofof thethe molecularmolecular structurestructure byby invertinginverting g g N 0 KL 2 i 4   3(R ) (RKL )  R ε j 2 D    ij c 5 KL   R LK

PuzzariniPuzzarini,, MetzrothMetzroth & GaussGauss unpublishedunpublished EquilEquil.. structurestructure fromfrom onlyonly 11 isotopologueisotopologue

rre [DC][DC] rre [semi[semi--expexp BB]] 14 NHNH3 r(N-H) HNH r(N-H) HNH

Dzz (N-H)

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

Dzz (H-H)

semi-exp B: Pawlowski et al. JCP 116, 6482 (2002) PuzzariniPuzzarini,, MetzrothMetzroth & GaussGauss unpublishedunpublished PartialPartial equilibriumequilibrium structurestructure

rre[DC][DC] rre[exp][exp] rre[abi][abi] HH13CNCN HH--CC 1.064(52)1.064(52) 1.06501(8)1.06501(8) 1.06551.0655

XBOXBO (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 abiabi=(=(allall)CCSD(T)/)CCSD(T)/cccc--pwCVQZpwCVQZ PuzzariniPuzzarini,, MetzrothMetzroth & GaussGauss unpublishedunpublished