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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
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
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
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:
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
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
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
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
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
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
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
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…
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
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
,,,,
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
Playing further on with angular momenta…
cotsin
122
1cot
sin
1cot
2
222
2
2
222
zyx
yxz
Li
Li
Li
LLLi
O
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:
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
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
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
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
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
Non-adiabatic couplings (1)
• Ex: highly excited potential curves in Na2
)1('
'
R
)()(
0
'
')1('
)1(
RURU
RV
proof
Non-adiabatic couplings (2)
Diagonal elements:
)()()()()()()1(
)2(2
2
RURURURU
RVRV
RURU
RV
R
R
)1('
2
2
2
)()( RURU
RV
proof
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
« 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
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
Diabatic basis: facts
• Not unique• R-independent
• Definition at R=R0 (ex: R=)
proof
« 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>
« 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
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
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
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
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
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
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
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