1- introduction, overview 2- hamiltonian of a diatomic molecule
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1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of cold atoms 6- Ultracold (elastic) collisions. Olivier Dulieu Predoc’ school, Les Houches,september 2004. - PowerPoint PPT PresentationTRANSCRIPT
• 1- Introduction, overview• 2- Hamiltonian of a diatomic
molecule• 3- Molecular symmetries; Hund’s
cases• 4- Molecular spectroscopy• 5- Photoassociation of cold atoms• 6- Ultracold (elastic) collisionsOlivier Dulieu
Predoc’ school, Les Houches,september 2004
Generalities on molecular symmetries
• Determine the spectroscopy of the molecule• Guide the elaboration of dynamical models• Allow a complete classification of molecular
states by:– Solving the Schrödinger equation– Looking at the separated atom limit (R)– Looking at the united atom limit (R0)– Adding electron one by one to build electronic
configurations
Symmetry properties of electronic functions (1)
Axial symmetry: 2 rotation
p
S 12Planar symmetry
Central symmetry
1
~
111
,,2
1
~;~),,(
1
iiiii
ii
iiii
re
rr
...
...43210
zz LL
i;~
1
),,(),,( iiiiiixzv rr
0,0, xzv
,2,,,
,,
xzv
xzv
xzv
),,(),,(ˆ iiiiii rri
uu
gg
i
i
ˆ
ˆ gerade
ungerade
spin
is not a good quantum number (precession around the axis)
is a good quantum number if electrostatic interaction is dominant
Symmetry properties of electronic functions (2)
2L
2S
,,,,,:12 SpS
,...,,,,,,, 53311312 guguguggEx:
2S+1: multiplicity
states: spin fixed in space, 2S+1 degenerate components states: precession around the axis, multiplet structure, almost equidistant in energy
Symmetry properties of electronic functions (3)
Otherwise: ,: p
,...1,1,0,0,0,0 uguugg
Hund’s cases for a diatomic molecule (1)
Rules for angular momenta couplingsDetermine the appropriate choice of basis functionsThis choice depends on the internuclear distance
(recoupling)
F. Hund, Z. Phys. 36, 657 (1926); 40, 742 (1927); 42, 93 (1927)
Hund’s cases (2): vector precession model
Hund’s case aHerzberg 1950
L S
JN
Hund’s cases (2): vector precession model
Hund’s case bHerzberg 1950
L
S
J N
K
not defined: state-Spin weakly coupled
Hund’s cases (2): vector precession model
Hund’s case cHerzberg 1950
L
S
JN
j
Hund’s cases (2): vector precession model
Hund’s case dHerzberg 1950
L
JN
K
S
Hund’s cases (2): vector precession model
Hund’s case eHerzberg 1950
L
S
JN
j
Hund’s case (3): interaction ordering
(adapted from Lefebvre-Brion&Field)
E.E. Nikitin & R.N. Zare, Mol. Phys. 82, 85 (1994)
SOre HHHH
He HSO Hr
(a) strongintermediat
eweak
(b) strong weakintermediat
e
(c) intermediate
strong weak
(d) intermediate
weak strong
(e) weakintermediat
estrong
Rotational energy for (a)-(e) cases
(d), (e) cases: useful for Rydberg electrons (see Lefebvre-Brion&Field)
2
2
2 R
OH r
)( SNSLOJ
22)( )1()1( SSJJBE va
rot
SJM
2)( )1( NNBE vb
rot
2)( 2)1( JJBE vc
rot
JMCase (c) Case (b)
Case (a)
Parity(ies) and phase convention(s) (1)On electron coordinates
in the molecular frame:
),,(),,( iiiiiixzv rr ,, xz
v
Convention of ab-initio calculations
Convention of molecular spectroscopy
,1, xzv« Condon&Shortley »
)(1 A
AA
xzv AMAM
labmol
)1()1()1(
)1()1()1(
AAAMiAAAM
MMAAAMiAAMA
AyxA
AAAYXA
One-electron orbital
Many-electron wave function
,21,1,21, 21 ssxzv
SJMSJM
SS
sSJxzv
sSxzv
1
1
With s=1 for - states, s=0 otherwise
Parity(ies) and phase convention(s) (2)
Parity of the total wavefunction: +/-
states :)1(or )1(
states :)1(or )1(211
21
f
eJJ
JJ
0,0
0,,0,0,10,,0,0, SJSJ ssSJsxzv
,...,,: even
,...,,: odd:,
,...,,: odd
,...,,: even :,
05
03
01
05
03
01
012
05
03
01
05
03
01
012
J
J
J
J
sS
sS
,...,,:,,
,...,,:,,
05
03
01
012
05
03
01
012
fJ
eJ
sS
sS
Total parity:
Parity(ies) and phase convention(s) (3)
Parity of the total wavefunction: +/-
states :)1(or )1(
states :)1(or )1(211
21
f
eJJ
JJ
0,0
SJSJSJSJ sssSJssxzv 1
21,,12
SJSJJ ssSJssS
Total parity:
All states except
21,,12
SJSJf
eJ ssSssS
Or –S+s+1/2
Radiative transitions (1)
Absorption cross section:2
. if
n
iire
1
In the mol frame
ZZYYXX
.In the lab frame
2)(; 10 YXZ i 2)(; 10 yxz i
1,0,
*1 )0,,()1(.qp
pqqpp D
RrDJ
R
Ri
JM
vMJv ;)0,,(
4
12)(
BO approximation
Radiative transitions (2)
Dipole transition moment
Absorption cross section:
2
1,0,
11)12)(12()1(
qp if
if
if
iffivqvp
p
q
JJ
MpM
JJJJ
i
if
f
1
1
12
0)0,,()0,,()0,,()(cos
4
)12)(12(
i
ii
f
ff
JMpq
JM
fiDDDdd
JJ
ifif
if
qqkkkq RrRrrdR ;;)( 3
Hönl-London factor
Selection rules for radiative transitions (1)
Parallel transition
f=i
ifif ifzL 00 0,
0cos;,11
0
n
kkk
n
k kz re
iL
if if 0
111
1 sin2
;,
n
k
ikk
n
k kz
kere
iL
ififif ifzL 111,
1 if 0 if Perpendiculartransition
f=i±11,0
Selection rules for radiative transitions (2)
= 0 otherwise
0 if 22
0 fi
iMipiM MpMeeed if
1,0 M
011
if
if
if
if
q
JJ
MpM
JJ1,0 J
00 fi JJ X
line 1
line 0
line 1
R
Q
P
JJJ if
0000
1
if JJ
If Jf+Ji+1 odd
No Q line for transition
Selection rules for radiative transitions (3)
Allowed
Forbidden
if
if
i
if
f vveqvqv R )( Franck-Condon factor
00 xzv
X
Allowed
Forbidden
gguu
gui qq ,ˆ
X X