1- introduction, overview 2- hamiltonian of a diatomic molecule

• 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