1- introduction, overview 2- hamiltonian of a diatomic molecule
DESCRIPTION
1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Hund’s cases; Molecular symmetries 4- Molecular spectroscopy 5- Photoassociation of cold atoms 6- Ultracold (elastic) collisions. Olivier Dulieu Predoc’ school, Les Houches,september 2004. Main steps:. - PowerPoint PPT PresentationTRANSCRIPT
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• 1- Introduction, overview• 2- Hamiltonian of a diatomic
molecule• 3- Hund’s cases; Molecular
symmetries• 4- Molecular spectroscopy• 5- Photoassociation of cold atoms• 6- Ultracold (elastic) collisionsOlivier Dulieu
Predoc’ school, Les Houches,september 2004
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Main steps:
• Definition of the exact Hamiltonian• Definition of a complete set of basis
functions• Matrix representation of finite
dimension+perturbations• Comparison to observations to
determine molecular parameters
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Non-relativistic Hamiltonian for 2 nuclei and n electrons in the lab-fixed frame
Vmmm
H bb
aa
n
ii
2'2
2'2
1
2'2
222
2'
2
2'
2
2'
22'
iiii ZYX
ji ab
ba
ij
n
i ib
bn
i ia
a
r
eZZ
r
e
r
eZ
r
eZV
22
1
2
1
2
04
with
and
electrons nuclei
e-n e-e n-n
Relative distances
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Separation of center-of-mass motion• Origin=midpoint of the axis ≠center of mass• Change of variables
'''
''
1
'''
2
1baii
ba
n
iib
ba
ac
RRRR
RRR
RM
mR
M
mR
M
mR
cii
n
iiRc
bb
n
iiRc
aa
M
m
M
m
M
m
'
1
'
1
'
2
1
2
1
mimmM ba Total mass:
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Second Derivative Operator
n
iiR
n
jiji
n
ii
RRccbb
aa
n
ii
m
Mmmm
11,1
2
2'2'
1
2'
1
4
11
11111
ba
ba
mm
mm
ba
ba
mm
mm
for homonuclear molecules01 reduced mass
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Hamiltonian in new coordinates
22
1
2
1,
2
1
22
22
2
28
22
c
n
iiR
n
jiji
n
iiR
M
Vm
H
Center-of-mass motion
Radial relative motion
Electronic Hamiltonian
Kinetic couplings m/
-Isotopic effect-Origincenter of mass
Study of the internal Hamiltonian…
2
22
2
22
2
222 R
O
RR
RRR
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T in spherical coordinates: rotation of the nuclei
Kinetic momentum of the nuclei
2
22
2
22
2
222 R
O
RR
RRR
X
Y
Z
R Ri e-
22
22
sin
1sin
sin
1
O
RiRO
cos
sinsin
cossin
RZ
RY
RX
R
R
R
iXY
YX
iO
iZX
XZ
iO
iYZ
ZY
iO
RR
RRZ
RR
RRY
RR
RRX
sincotcos
coscotsin
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Rotating or molecular frame
• Specific role of the interatomic axis• Potential energy greatly simplified, independent of the
molecule orientation• Euler transformation with a specific convention: { , /2}
cossin
sinsinsincoscos
cossincoscossin
iii
iiii
iiii
zyZ
zyxY
zyxX
cossinsincossin
sinsincoscoscos
cossin
iiii
iiii
iii
ZYXz
ZYXy
YXx
Molecular lab-fixed
Lab-fixed molecular
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X
Y
Z
R
X ‘’
Y’’
Z
Y
X
Z
Y
X
100
0cossin
0sincos
Z’’=
X’’’
Z’’’
=Y’’’
Z
Y
X
Z
Y
X
cos0sin
010
sin0cos
=0 around Z’’’:x=X’’’,y=Y’’’, z=Z’’’
Oy perp to OZz
=/2 around Z’’’:Ox perp to OZzOR
0
sin
1
z
y
x
O
iO
iO
R 1
R 2
R 3
R 3
R 3 R 2 R 1 O
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X
Y
Z
R
X ‘’
Y’’
Z
Y
X
Z
Y
X
100
0cossin
0sincos
Z’’=
X’’’
Z’’’
=Y’’’
Z
Y
X
Z
Y
X
cos0sin
010
sin0cos
General case: 2/0 and
Z
Y
X
z
y
x
100
0cossin
0sincos
x
y
R 1
R 2
R 3
0
sin
sincos
sin
cossin
z
y
x
O
iO
iO
R 3 R 2 R 1 O
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yz
ZYXzyx
Li
Li
sincos
,,,,
T in the molecular frame (1)
xZYX
n
i ii
ii
ZYX
n
i i
i
i
i
i
i
ZYXzyx
Li
zy
yz
z
z
y
y
x
x
,,
1,,
1,,,,
222
22
2
22
2
sin
1sin
sin
1
222
RRR
RRR
With xi, yi, zi now depending on and .
Total electronic angularmomentum in the molecular frame
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T in the molecular frame (2)
cotsin
122
2
1cot
sin
1cot
2
22
2
2
222
2
2
2
2
2
22
22
2
zyx
yxz
R
Li
Li
Li
R
LLLi
R
RR
RR
vibration
rotation
Electronic spin can be introduced by replacing Lx,y,z with
jx,y,z=Lx,y,z+Sx,y,z
See further on…
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Hamiltonian in the molecular frame
cotsin
122
2
1cot
sin
1cot
2
2
282
2
2
222
2
2
2
2
2
22
2
1
2
1,
2
1
22
zyx
yxz
n
iiR
n
jiji
n
ii
Li
Li
Li
R
LLLi
R
RR
RR
Vm
H
He+H’e
Hv
Hr+H’r
O2 : quite complicated!
Kinetic energy of the nuclei in the molecular frame
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Total angular momentum in the molecular frame
Total angular momentum
Commute with H(no external field)
LOJ
ZYX JJJJ ,,,2
zz
zyyy
xxx
LJ
Li
LOJ
iLOJ
cotsin
1
In the molecular frame0
sin
1
z
y
x
O
iO
iO
xZYX
n
i ii
ii
ZYX
n
i i
i
i
i
i
i
ZYXzyx
Li
zy
yz
z
z
y
y
x
x
,,
1,,
1,,,,
yz
ZYXzyx
Li
Li
sincos
,,,,
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Total angular momentum in the lab frame
In the lab frame
iJ
Li
J
Li
J
Z
zY
zX
sin
sinsincotcos
sin
coscoscotsin
cossin0
sinsinsincoscos
cossincoscossin
molecularlab
222
2
22
2222
sin
1
sin
cot2
sin
1sin
sin
1zz
ZYX
LLi
JJJJ
cot22222 zyx JJJJIn the
molecular frame!!
Depends only on Lz
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Playing further on with angular momenta…
cotsin
122
1cot
sin
1cot
2
222
2
2
222
zyx
yxz
Li
Li
Li
LLLi
O
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Playing further on with angular momenta…
222
2
222
sin
1
sin
cot2
sin
1sin
sin
1zz LLiJ
yzyzxx LLLLiLLii
O
sincossincos2sin
12cot
sin
1sin
sin
1
2
2
2
222
Compare with:
zyx Li
Li
LiLJO
cossin
22222
zyxz Li
Li
LiLJL
cossin
2
zyxyx Li
Li
LiLLOL
cossin
22
Also via a direct calculation:
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Yet another expression for H in the molecular frame….
LJJLLJLJO
LOLJJLLJO
2222
2222 )2(2OLLOalso
:
2
2
2
22
2
2
1
2
1,
2
1
22
2
2
22
282
R
JLL
R
J
RR
RR
Vm
Hn
iiR
n
jiji
n
ii
He H’e
Hv Hr Hc
Coriolis interaction
22
)2(:
R
LOLalso
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What about spin?
Electronic spin Notations:
Nuclear spin
S
I
IJF
LON
SLOJ
If S quantized in the molecular frame (i.e. strong coupling with L), L
should be replaced by j=L+S (with projection ) in all previous equations
But why…?
cossinsincossin
sinsincoscoscos
0cossin
labmolecular
No spatial
representation for S
Rotation matrices:
lablabzz
yyzzmolmol
labzyzmol
labmol
labzyzlabmol
SSLi
SLiSLiS
SiSiSiSS
DHDH
iLiLiLD
)/)(exp(
)/)(exp()2/)(exp(
)/exp()/exp()2/exp(
)/exp()/exp()2/exp(1
1
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Born-Oppenheimer approximation (1)
H=He+H’e+Hv+Hr+Hc.
m/>1800: approximate separation of electron/nuclei motion
BO or adiabatic approximation:factorization
of the total wave function
);()();( RrRURrH iie
);()()(
iBO rRR
Potential curves:R: separated atomsR0: united atom
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Born-Oppenheimer approximation (2)
H=He+H’e+Hv+Hr+Hc.
BO or adiabatic approximation: factorization of the total wave function
);()()()(irRR
BO
)()()(2
1
2)()(2
22
2
2
RERHRUJRR
RRR c
Mean potential
All act on the electronic wave function
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Validity of the BO approximation
RrRCR
rR ii
;
1;
Total wave function with energy EExpressed in the adiabatic basis
022 2
2
2
22
EHH
R
J
R ce
Set of differential coupled equations for C
)(2
)()(22
''
''
2
2'
22
2
2
2
22
RCHRRR
RCEHRUR
J
R
c
c
< | > Integration on
electronic coordinates
J2 diagonalBO approximation
Infinite sum on
non-adiabatic couplings
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Non-adiabatic couplings (1)
• Ex: highly excited potential curves in Na2
)1('
'
R
)()(
0
'
')1('
)1(
RURU
RV
proof
![Page 24: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/24.jpg)
Non-adiabatic couplings (2)
Diagonal elements:
)()()()()()()1(
)2(2
2
RURURURU
RVRV
RURU
RV
R
R
)1('
2
2
2
)()( RURU
RV
proof
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Non-adiabatic couplings (3)
Diagonal elements
zy
yx
z
c
LL
Li
Li
LL
JLLRHR
cot2
sin
22
2
22222
222
zyxz Li
Li
LiLJL
cossin
2
2222
22
1
L
RH c
z
ez
L
HL 0,
0,2 eHL
proof
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« Improved » BO approximation(also « adiabatic » approximation)
Neglect all non-diagonal elements in the adiabatic basis |>
0)(2
)(2
2
2 2
22
2
2222
2
22
RCER
RUR
LJ
R
Unique by definition: Diagonalizes He
![Page 27: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/27.jpg)
Alternative: Diabatic basis
Neglect all (non-diagonal) couplings due to Hc
)(22
)()(2
2
2
''
)1()2(2
2
2222
2
22
RCR
RCERUR
LJ
R
Define a new basis which cancels these
couplings
)()(~
)(
~)(
~
)(~
RMRCRC
RC
RM
)1(: MR
Mif
)(RW
CWCEW
R
~~~~
2 2
22
1~
MWMW
Couplings in the potential matrixproof
![Page 28: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/28.jpg)
Diabatic basis: facts
• Not unique• R-independent
• Definition at R=R0 (ex: R=)
proof
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« Nuclear » wave functions (1)
Adiabatic approximation:
0)(2
)(2
2
22 2
22
2
222
2
2
2
22
RCER
RUR
L
R
J
R
zL
V(R)
CR
1Eigenfunctions of J2, JZ, Lz (ou jz) ( Jz)
0,
0,
0,,2
BOz
cz
Z
HJ
HJ
HJHJ
C.E.C.O
proofWave functions: |JM> ou |JM>
![Page 30: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/30.jpg)
« Nuclear » wave functions (2)
)()(),()(),,( JM
iMJM eRRRC R
0)()()1(22
22
2
2
22
RERVJJRR
)()1()(sin
cos2sin
sin
12
22
JM
JM JJ
MM
),(),(
),(),(
),()1(),( 22
JM
JMz
JM
JMZ
JM
JM
J
MJ
JJJ
RR
RR
RR
![Page 31: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/31.jpg)
Rotational wave functionsPhase convention…
),()1(),();,(),(
),()1()1(),(
)0()0(
)1(2/1
JJMJM
JM
JM
JMYX
YY
MMJJiJJ
RR
RR
(Condon&Shortley 1935, Messiah 1960)
…and normalization convention….!
)(12
4)(
)(),,(
)0,,(4
12),(
),(),(sin2
0 0
JM
JM
iJM
iMJM
JM
JM
MMJJJM
JM
π
Jd
edeD
DJ
dd
R
RR
Up to now: ….
JM
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Vibrational wave functions and energies (1)
0)()(2
)1(
2 2
2
2
22
RERVR
JJ
R
No analytical
solution
2
2
2
)1(
eR
JJ
2)(2
1)( ee RRkRV D
Rigid rotator Harmonic oscillator
)2/1()1( vJJBE eeevJ D
k
RB e
ee ;
2 2
2
eevRRv
v RRHevR e
;)(!2)( 2/)(2/14/12
22
Usefulapproximations
Equilibrium distance
![Page 33: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/33.jpg)
Vibrational wavefunctions and energies (2)
Deviation from the harmonic oscillator approximation: Morse potential
)()(2 2)( ee RRRRe eeDRV
2)21()21()1( vxvJJBDE eeeeevJ
e
ee D
x4
2/12
e
e
D
Deviation from the rigid rotator approximation:
0;)~
(!
1)
~(
2
)1()()(
~2 ~2
2
ee
RR
eff
n
ne
RR
n
effn
eeffeff R
VRR
R
V
nRV
R
JJRVRV
222 )21()21()1()1( vxvJJDJJBE eeeevevJ D
)21( vBB eev 22
34
e
ee
BD
proof
![Page 34: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/34.jpg)
Continuum statesDissociation, fragmentation, collision…
0)(2
:2
2
2
2
RERR
R E
2
2;)sin()(
E
kRkCR EEEEE
Regular solution:
NormalizationInfluence of the potential
)()()(;)sin(2
)(0 EEEEEEE kkdRRRRkR
)()()(;)sin(2
)(02
EEdRRRRkk
R EEEEE
E
In wave numbers
In energyproof
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Matrix elements of the rotational hamiltonian
Easy to evaluate in the BO basis:
But in general, L and S are not good quantum numbers…
…quantum chemistry is needed
cotsin
122
2
1cot
sin
1cot
2
2
2
222
2
2
2
2
2
zyx
yxzrr
Li
Li
Li
R
LLLi
RHH
JLJLJLJL yyxx 22
12
222
1zLL
JMSLJML ou
)1ou (1
Selection rule
![Page 36: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.vdocument.in/reader035/viewer/2022062409/568146b9550346895db3e40f/html5/thumbnails/36.jpg)
Matrix elements of the vibrational hamiltonian
BO basis:
RR
RRH vib
22
2
2
)(1
),( )( RR
Rr v
)()(2
22)()(
2
, 22
vvv
vvvvvvvib RRREH
Quantum chemistry is needed…
Vibrational energy levels Interaction between
vibrational levels