Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
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)
inte
nsity
(a.u
.)
Coordinate Coordinate SSyystemstemss
Molecule-fixed coordinate system Lab-fixed coordinate system
ROTATING RIGID BODYROTATING RIGID BODY
IωωT
21
T
= = angolar velocityangolar velocityI = inertia tensorI = inertia tensor
(CLASSIC VIEW)(CLASSIC VIEW)
Angolar Angolar VelocitVelocityy
ii rωv RIGID BODY:RIGID BODY:
ri
vi
z
y
x
ω
zzzyzx
yzyyyx
xzxyxx
III
III
III
I
n
iiiixy
n
iiii
n
iiiixx
yxmI
zymxrmI
1
1
22
1
22
Inertia TensorInertia Tensor
zzzyzx
yzyyyx
xzxyxx
III
III
III
I
z
y
x
I
II
00
0000
I
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
2T
21
21
T
= angular velocy= angular velocyI = inertia tensorI = inertia tensor
(CLASSIC VIEW)(CLASSIC VIEW)
IJIωω
2T
21
21
T
•• Potential energy?Potential energy?•• From classic mechanics to quantum mechanicsFrom classic mechanics to quantum mechanics
z
z
y
y
x
xrotrot I
JIJ
IJTE
222
21
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
222
222
22 1)(
KMKJJMKJ
MMKJJMKJ
JJMKJJMKJ
z
Z
,,ˆ,,
,,ˆ,,
,,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
z
z
y
y
x
xrot I
JIJ
IJH
222
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
m1 m2
R
DIATOMIC/LINEAR MOLECULE:DIATOMIC/LINEAR MOLECULE:RIGID ROTOR (approx)RIGID ROTOR (approx)
m1 m2
RCM
r1 r2
i
iirmI 2
2RI 21
21
mmmm
where
reduced mass
zIx = Iy = IIz = 0
2231231
22332
21221
1 rrmmrmmrmmM
I
1 32rr1212 rr2323
Ix = Iy = IIz = 0
222
21
21
21 J
IJ
IJ
IH y
yx
xrot
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
2
JBJJJI
Erot
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
J=0
J=1
J=2
J=3
Erot=0
Erot=2B
Erot=6B
Erot=12B
ROTATIONAL ENERGY LEVELS:ROTATIONAL ENERGY LEVELS:Diatomic and Linear moleculesDiatomic and Linear molecules
J=0
J=1
J=2
J=3
Erot=0
Erot=2B
Erot=6B
Erot=12B
ROTATIONAL ENERGY LEVELS:ROTATIONAL ENERGY LEVELS:Diatomic and Linear moleculesDiatomic and Linear molecules
)1(2)1( JBJJE
ROTATIONAL ENERGY LEVELSROTATIONAL ENERGY LEVELS
2222222ZYXzyx JJJJJJJ ˆˆˆˆˆˆˆ
x,y,z molecule-fixed coordinate systemX,Y,Z space-fixed coordinate system
022 zJJ ˆ,ˆ 022 ZJJ ˆ,ˆ
222
22 1)(
MMKJJMKJ
JJMKJJMKJ
Z
,,ˆ,,
,,ˆ,,
M=J,J-1 … -J
Rotational energy levels: Rotational energy levels: (2(2JJ+1) fold degenerate in +1) fold degenerate in MM
“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)
inte
nsity
(a.u
.)
SELECTION RULESSELECTION RULES
Transition moment: Transition moment: 00
Approx BO: tot=rotvibele
elevibrotele
ivib
irot
irotf
vibf
elef ddd
dipole moment in the space-fixed coordinate system
FF
FFF
cos where=x,y,z (molecule-fixed)F=X,Y,Z (space-fixed)
Xy
Xz
Xx
X
Z=direction cosines
F
elevibelei
vibi
vibf
elefrot
rotiF
rotf ddd
molecular dipole moment components
F
F
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
SELECTION RULESSELECTION RULES
ifR FF
ij
(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
“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)
inte
nsity
(a.u
.)
Rotational energy levelsRotational energy levels++
Selection rulesSelection rules
Rotational transition frequenciesRotational transition frequencies(rotational spectrum: (rotational spectrum: xx axis) axis)
)1( JBJErot
1J
J=0
J=1
J=2
J=3
Erot=0
Erot=2B
Erot=6B
Erot=12B
++
)1(2)1( JBJJE
)1(2 JBh rot
frequency
Inte
nsity
???
Int e
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
kTE
JJrot
egg
NN
00
2J+1
kTE
Jrot
eJNN
)12(
0
Jmax
Nf /
N0
J
gf/g0=2J+1 exp(-Erot/kT) Nf/N0=(2J+1)exp(-Erot/kT)
kTE
Jrot
eJNN
)12(
0
221 mnNT
I mni μ
Intensity of Rotational TransitionsIntensity of Rotational Transitions
0 1 2 3 4 5 6 7 8 9 10
NJ/N
0
I abs
J
Intensity Population
“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)
inte
nsity
(a.u
.)
0 20 40 60 80 100
Rotational spectrum of CO
wavenumbers (cm-1)
Inte
nsity
( u. a
.)
iMMJ
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
J
M
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'
2
34
eJ
BD > 0 !!> 0 !!
J=0
J=1
J=2
J=3
Erot=0
Erot=2hB
Erot=6hB
Erot/h=12hB
J=0
J=1
J=2
J=3
centrifugal distortioncentrifugal distortion
22 )1()1( JJDJBJhE Jrot /
frequency
Inte
nsity
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
22 11
21
zrot
z
yx
JIII
JH
II
III
ˆˆˆ//
// (z = symmetry axis)
SYMMETRIC TOPSSYMMETRIC TOPS
22 11)1(
2K
IIIJJErot
//
K=J,J-1 … -J
SYMMETRIC TOPSSYMMETRIC TOPS
22 11)1(
2K
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
<0
SYMMETRIC TOP: rotational energy levelsSYMMETRIC TOP: rotational energy levels
PROLATE PROLATE OBLATEOBLATEAA > > BB = = CC AA < < BB = = CC
SYMMETRIC TOP: rotational energy levelsSYMMETRIC TOP: rotational energy levels
PROLATE PROLATE OBLATEOBLATEAA > > BB = = CC AA < < BB = = CC
J J = 6, = 6, K K = = 44
SELECTION RULESSELECTION RULES
In addition to In addition to JJ = = 11::
KK = 0= 0
SYMMETRIC TOP: rotational energy levelsSYMMETRIC TOP: rotational energy levels
PROLATE PROLATE OBLATEOBLATEAA > > BB = = CC AA < < BB = = CC
SELECTION RULESSELECTION RULES
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))
SELECTION RULESSELECTION RULES
In addition to In addition to JJ = = 11::
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 = 9
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
K = 6
F r e q u e n c y ( M H z )
K = 3
1 4 N F 3 : J = 2 6 - 2 5
Rotational spectrum of NF3: a small portion
JKM,,
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
z
z
y
y
x
xrot I
JIJ
IJH
222
21 ˆˆˆˆ
No longer possible to rearrange the Hamiltonian sothat it is comprised soley of and one componentof
2JJ
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.
ASYMMETRIC ROTORASYMMETRIC ROTOR
z
z
y
y
x
xrot I
JIJ
IJH
222
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
ASYMMETRIC ROTORASYMMETRIC ROTOR
ASYMMETRIC ROTORASYMMETRIC ROTOR
+1
-101 1
1
1
00
PROLATEPROLATE OBLATEOBLATE
J Ka JKc(-J +J)
near oblatenear oblatenear prolatenear prolate
2B A C
A C
Asymmetric parameter Asymmetric parameter
= -1 = +1
= 0
2 notation scheme: JKa,Kc or J
ca KK
SELECTION RULESSELECTION RULES
In addition to In addition to JJ = 0,= 0, 11::
KKa a ,, KKcc = 0, = 0, 11
Ka Kc
Symmetric Rotor
Asymmetric Rotor
ASYMMETRIC ROTORASYMMETRIC 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
Rotational constantsRotational constants
Nuclear quadrupole Nuclear quadrupole couplingcoupling
K KK
KJK
JJIIqeQ 222
233
)12()12(221 JIJIJI
222CBA CBA JJJ
frequency
J=1-0
F=-1F=+1
unperturbed
F=0
nuclear quadrupole coupling
unperturbed
F = 1/2
F = 5/2
F = 3/2
F = 3/2
J = 1
J = 0
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
1;0 F
frequency
J=1-0
F=-1F=+1
unperturbed
F=0
nuclear quadrupole coupling
unperturbed
F = 1/2
F = 5/2
F = 3/2
F = 3/2
J = 1
J = 0
frequency
J=1-0
F=-1F=+1
unperturbed
F=0
[[IIKK 1]1] IIKK=3/2; =3/2; eQq eQq 00
LINEAR MOLECULELINEAR MOLECULEF = J+I, J+I-1, …, |J-I |
hyperfine structurehyperfine structure
Hyperfine structureHyperfine structureRotational HamiltonianRotational Hamiltonian
Rotational constantsRotational constants
Nuclear quadrupole Nuclear quadrupole couplingcoupling
K KK
KJK
JJIIqeQ 222
233
)12()12(221 JIJIJI
SpinSpin--rotation interactionsrotation interactions
K
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
J = 2
J = 1
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
Hyperfine structureHyperfine structureRotational HamiltonianRotational Hamiltonian
Rotational constantsRotational constants
SpinSpin--spin (direct)spin (direct)interactionsinteractions
LK
LKLK IDI
222CBA CBA JJJ
Nuclear quadrupole Nuclear quadrupole couplingcoupling
K KK
KJK
JJIIqeQ 222
233
)12()12(221 JIJIJI
SpinSpin--rotation interactionsrotation interactions
K
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
1/2
direct spin-spin interaction
unperturbed
F =F'+I2
0211
10
F' =J+I1
3/2
1/2
J
1
0
frequency
J=1-0
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
ROTATIONAL ENERGY LEVELSROTATIONAL ENERGY LEVELS
2222222ZYXzyx JJJJJJJ ˆˆˆˆˆˆˆ
x,y,z molecule-fixed coordinate systemX,Y,Z space-fixed coordinate system 022 zJJ ˆ,ˆ 022 ZJJ ˆ,ˆ
222
22 1)(
MMKJJMKJ
JJMKJJMKJ
Z
,,ˆ,,
,,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
1J
1JM
0JM
0J
J = 1
J = 0
MJ = 0
MJ = ±1
MJ = 0
0E0
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())((
2
)1(
3
2222
3
222222(2)
(1)
JJJMJKJ
JJJMJKJ
hBE
JJKME
Stark
Stark
STARK EFFECT: the SYMMETRIC TOP caseSTARK EFFECT: the SYMMETRIC TOP case
NO FIELDNO FIELD
|000JKM
|100|10-1
|101
|110|11-1
|111|1-10|1-1-1
|1-11
1st ORDER1st ORDER
|000
|101 |100|10-1
|111 |1-1-1|110 |1-10|11-1 |1-11
2B2B
AA--BB
2nd ORDER2nd ORDER
|000
|111 |1-1-1|110 |1-10|11-1 |1-11
|101|100
|10-1
STARK EFFECT: the SYMMETRIC TOP caseSTARK EFFECT: the SYMMETRIC TOP case
NO FIELDNO FIELD
|100|10-1
|101
|110|11-1
|111|1-10|1-1-1
|1-11
1st ORDER1st ORDER
|000
|101 |100|10-1
|111 |1-1-1|110 |1-10|11-1 |1-11
2B2B
AA--BB
2nd ORDER2nd ORDER
|000
|111 |1-1-1|110 |1-10|11-1 |1-11
|101|100
|10-1
shift Stark:shift Stark: ==’’--
((’’ > > ))
’’
|000JKM
CODECS Summer School 2013 CODECS Summer School 2013 THEORETICAL SPECTROSCOPY THEORETICAL SPECTROSCOPY
ROTATIONAL SPECTROSCOPY:ROTATIONAL SPECTROSCOPY:Computational RequirementsComputational Requirements
&&
AccuracyAccuracy
Cristina PuzzariniCristina PuzzariniDip. Chimica Dip. Chimica ““Giacomo CiamicianGiacomo Ciamician””
UniversitUniversitàà di Bolognadi Bologna
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
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
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
H2
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, … …
1) 1) Principal error sourcesPrincipal error sources in in ab initio calculationsab initio calculations::
-- 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
mT
kji cba
abcijkm
2)(1
...!!
32
31
211)exp( TTTT
HFCC T )exp(
.... 321 TTTT
energyenergy
amplitudesamplitudes
coupledcoupled--cluster equationscluster equations
CoupledCoupled--Cluster TheoryCluster Theory
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
CoupledCoupled--Cluster TheoryCluster Theory
CoupledCoupled--Cluster TheoryCluster Theory
CCSD(T) T=T1 + T2 + (T) N6 + N7 (no iter)
....)(
dx
Eddx
Eddx
dEdx
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
1) 1) Principal error sourcesPrincipal error sources in in ab initio calculationsab initio calculations::
-- 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)
>> ………>> ………
22
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 ))(()(
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))
COMPOSITE APPROACHCOMPOSITE APPROACH
2) 2) “Minor” “Minor” error sourceserror sources in in ab initio calculationsab initio calculations::
-- corecore--valencevalence (CV) (CV) correlationcorrelation
-- …………-- scalar scalar relativityrelativity (SR)(SR)
… … 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
n
ii BB
n
1
1
Standard deviation:
n
iistd n 1
2
11
2
21exp)(
stdcN
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZ
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZ
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5Z
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6Z
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6ZCCSD(T)/6Z+ core
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6ZCCSD(T)/6Z+ core
CCSD(T)/6Z+ core
+T
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6ZCCSD(T)/6Z+ core
CCSD(T)/6Z+ core
+T
CCSD(T)/6Z+ core
+T+Q
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
Accuracy of Theoretical Rotational Constants:Accuracy of Theoretical Rotational Constants:StatisticsStatistics
-3 -2 -1 0 1 2 3
[MHz]
CCSD(T)/TZCCSD(T)/QZCCSD(T)/5ZCCSD(T)/6ZCCSD(T)/6Z+ core
CCSD(T)/6Z+ core
+T
CCSD(T)/6Z+ core
+T+Q
CCSD(T)/Z+ core
+T+Q
normal distributions of relative errorsnormal distributions of relative errors
C. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
BBeecalccalc vs Bvs Bee
expexp
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))
COMPOSITE APPROACH: COMPOSITE APPROACH: the “geometry scheme”the “geometry scheme”
2) 2) “Minor” “Minor” error sourceserror sources in in ab initio calculationsab initio calculations::
-- corecore--valencevalence (CV) (CV) correlationcorrelation
-- …………-- scalar scalar relativityrelativity (SR)(SR)
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)
>> ………>> ………
22
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)
2) at “PARAMETERS” level:2) at “PARAMETERS” level:
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
2) at “PARAMETERS” level:2) at “PARAMETERS” level:
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)
SiHSiH33FF SiSi--F / F / ÅÅ SiSi--H / H / ÅÅ FSiH / deg.FSiH / deg.
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)
SiHSiH33FF SiSi--F / F / ÅÅ SiSi--H / H / ÅÅ FSiH / deg.FSiH / deg.
gradient schemegradient scheme +0.0004+0.0004 +0.0001+0.0001 +0.00+0.00
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.
gradient schemegradient scheme +0.0014+0.0014 +0.0013+0.0013 +0.03+0.03
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)
(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
COMPOSITE APPROACHCOMPOSITE APPROACH
1) 1) “cheap” geom scheme“cheap” geom scheme
2) 2) “accurate” grad scheme“accurate” grad scheme
dxCEd
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 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
(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?
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
r
Bre BB
21
0
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
r
Bre BB
21
0
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
C2C4
C5C6
N1
N3O7O8
H11H12
H9
bb
aa
H10
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).
(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
COMPOSITE APPROACHCOMPOSITE APPROACH
1) 1) “cheap” geom scheme“cheap” geom scheme
2) 2) “accurate” grad scheme“accurate” grad scheme
dxCEd
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
(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
Vibrational Corrections to Rotational ConstantsVibrational Corrections to Rotational Constants
Vibrational corrections to rotational constants:Vibrational corrections to rotational constants:
r
BreBB
21
0
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..
COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule Equilibrium Rotational ConstantsEquilibrium Rotational Constants
(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
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)
Puzzarini & Barone, PCCP 13, 7158 (2011)
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
C. 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)/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
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)/VQZCCSD(T)/V5ZCCSD(T)/V5ZBBcalccalc 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)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZBBcalccalc 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)/VQZCCSD(T)/V5ZCCSD(T)/V5ZCCSD(T)/V6ZCCSD(T)/V6ZCCSD(T)/V6Z + CVCCSD(T)/V6Z + CVBBcalccalc 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)/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
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)/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
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)/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
CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQfQ
BBcalccalc 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)/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
CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQfQ
CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQ fQ + + vibvib
BBcalccalc 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)/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
CCSD(T)/VCCSD(T)/VZ + CV + fTZ + CV + fT+ + fQfQ
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
BBcalccalc vs vs BB00expexp
normal distributions of relative errorsC. C. PuzzariniPuzzarini, M. , M. HeckertHeckert, J. Gauss , J. Gauss JCP JCP 128128, 194108, 194108 (2008)(2008)
Electronic contribution to Electronic contribution to BBelvibe BBBB 0
e
p
eel Bg
mmB
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
r
rr
r 1
21
4xxxx
JD
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)
COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule Equilibrium Rotational ConstantsEquilibrium Rotational Constants
(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
Vibrational Corrections to Rotational ConstantsVibrational Corrections to Rotational ConstantsB3LYP/N07D MP2/cc-pVTZ
CentrifugalCentrifugal--Distortion ConstantsDistortion Constants(MP2/VTZ)(MP2/aVTZ)fc)(MP2/CVTZ,all)(MP2/CVTZ,Z)(CCSD(T)/T(best) DDDDDD
CV diffuse
Puzzarini & Barone, PCCP 13, 7158 (2011)
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)
~1~1%%
URACILURACIL
Puzzarini & Barone, PCCP 13, 7158 (2011)
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
qK
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)
CALC.
EXP.
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)
CALC.
EXP.
FREQUENCY (MHz)
(~2 GHz freq. Shift!)(~2 GHz freq. Shift!)
Nuclear quadrupoleNuclear quadrupole--coupling constants:coupling constants:effect on rotational spectrumeffect on rotational spectrum
COMPOSITE APPROACH extended to large moleculeCOMPOSITE APPROACH extended to large molecule Equilibrium Rotational ConstantsEquilibrium Rotational Constants
(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
Vibrational Corrections to Rotational ConstantsVibrational Corrections to Rotational ConstantsB3LYP/N07D MP2/cc-pVTZ
Nitrogen qudrupoleNitrogen qudrupole--coupling Constantscoupling Constants
Puzzarini & Barone, PCCP 13, 7158 (2011)
(diff)QZ)(TZVTZ)(CCSD(T)/Cest)( b
MP2/cc-pCV(T,Q)Z
MP2/aug-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)
11--22%%
URACILURACIL
Puzzarini & Barone, PCCP 13, 7158 (2011)
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
r sr
sreqsr
reqr
eq QQQQPQ
QPPP
,...
2
21
s s
rss
rrQ
24
wherewhere
rrssrQQ
2
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
Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011)
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
Cazzoli, Puzzarini, Stopkowicz, Gauss, Astrophys. J. Suppl. 196, 10 (2011)
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
CODECS Summer School 2013 CODECS Summer School 2013 THEORETICAL SPECTROSCOPY THEORETICAL SPECTROSCOPY
ROTATIONAL SPECTROSCOPY:ROTATIONAL SPECTROSCOPY:Interplay Interplay ofof
Experiment Experiment andand TheoryTheory
Cristina PuzzariniCristina PuzzariniDip. Chimica Dip. Chimica ““Giacomo CiamicianGiacomo Ciamician””
UniversitUniversitàà di Bolognadi Bologna
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?
COMPOSITE APPROACHCOMPOSITE APPROACH
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)
r
BreBB
21
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
32S
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)
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
F R E Q U E N C Y (M H z)
J.-C. Chardon, C. Genty, D. Guichon, & J.-G. Theobald, J. Chem. Phys. 64, 1434 (1976) “rf spectrum and hyperfine structure of formic acid”
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
F R E Q U E N C Y (M H z)
J.-C. Chardon, C. Genty, D. Guichon, & J.-G. Theobald, J. Chem. Phys. 64, 1434 (1976)
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
T heo ry:on ly S R
T heo ry: S R and S S
E xpe rim en t
F R E Q U E N C Y (M H z)
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
T heo ry:on ly S R
T heo ry: S R and S S
E xpe rim en t
F R E Q U E N C Y (M H z)
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
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 ……
-- 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
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
aa bb
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
CC
chemical chemical shiftshift
tensor spintensor spin--spinspincouplingcoupling ((rank rank 2)2)
CC33
scalar scalar spinspin--spinspincouplingcoupling ((rank rank 0)0)
CC44
direct direct dipolardipolarcouplingcoupling
DD
indirect spinindirect spin--spinspincouplingcoupling
JJ
nuclear magnetic shielding nuclear magnetic shielding
DIATOMIC or LINEAR MOLECULESDIATOMIC or LINEAR MOLECULES
r
Zme
Bc
gmm
e
I
Ne
ppp
3423
223 2
0
= = dd ++ pp
DIAMAGNETIC PARTDIAMAGNETIC PART PARAMAGNETIC PARTPARAMAGNETIC PART
nuclear magnetic shielding nuclear magnetic shielding
DIATOMIC or LINEAR MOLECULESDIATOMIC or LINEAR MOLECULES
r
Zme
Bc
gmm
e
I
Ne
ppp
3423
223 2
0
= = dd ++ pp
DIAMAGNETIC PARTDIAMAGNETIC PART PARAMAGNETIC PARTPARAMAGNETIC PART
nuclear magnetic shielding nuclear magnetic shielding
ASYMMETRICASYMMETRIC--TOPTOP MOLECULESMOLECULES
= = dd ++ pp
DIAMAGNETIC PARTDIAMAGNETIC PART PARAMAGNETIC PARTPARAMAGNETIC PART
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
LaboratoryLaboratory of of MillimetreMillimetre--wavewave
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
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
j
jj
calcicalc
ii pp
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
mMmM
yzI
xzI
xyI
yxII
zxII
zyII
yz
xz
xy
zzz
yyy
xxx
'
'
'
'
'
'
)(
)(
)(
22
22
22
[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)
zzPy
yPx
xPPP xxxcalc
xx
000
exp
- 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
iiiz
iiiy
iiix
zymPzxmPyxmP
zmPymPxmP
222
[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
ss
bs
e
eb
ii
i
eb
mb
dBI
mmM
MII
2
12
2
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)
em II
ImprovementsImprovementsL)2(L)1()2()1(
mmmm rrrr
cbaIcII mm ,,/
210
rrmm(1)(1) modelmodel
It can be used for molecules that contains atoms such as F
cbaMmmdIcII NN
mm ,,/ //
)22(11
210
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)
21
(XH)(XH)/
HH
Hmeff
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
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
rs
r0
re
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).
C H
HF
Br
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
och
loro--2
2--flu
oroe
thyle
ne
fluor
oeth
ylene
H H
ClF
C C
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
cis
21
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 ,,
2vrv
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
r
Bre BB
21
0
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
1122
33
4455
66
77
88
99
10101111
1212
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)
1.34
331.
3433
1.34
496(
59)
1.34
496(
59)
1.37
9(4)
1.37
9(4)
125.83125.83125.768(48)125.768(48)118.8(7)118.8(7)
121.91121.91
121.924(10)
121.924(10)
122.3(6)122.3(6)
SS
CCNN NN
HH11 HH11
HH22 HH22
II
FFCC
HHHH
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
5LK
2KLijjKLiKLN
0
KLKLij
3cε4
ggR
RRRD
)()(2
1)1) SubtractionSubtraction of the of the computedcomputed vibrationalvibrational correctioncorrection in in orderorder to getto get equilibriumequilibrium DDKLKL::
PROCEDUREPROCEDURE
KLvib
KLKLeq DDD exp
2) 2) DeterminationDetermination of the of the molecularmolecular structurestructure byby invertinginverting
5LK
2KLijjKLiKLN
0
KLKLij
3cε4
ggR
RRRD
)()(2
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