shell models for alkali metal trimersphysics.usc.edu/shells/talks/ernst_part_ii_shelleffects... ·...

Institute of Experimental Physics

1Wolfgang E. Ernst Shell Models, Erice, July 26-30, 2010

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Shell Models for Alkali Metal Trimers:Electronic Level Structure and Magnetic

Properties from Experimental and Theoretical Investigations

Lecture II■ introduction: helium droplets, doping with foreign

species, spectroscopy■ alkali atoms and molecules on helium droplets■ spectra, absorption and magnetic circular

dichroism■ identification of high spin trimers■ ab initio K3 and Rb3, K2Rb and KRb2 quartet states■ quartet state shell structure, harmonic oscillator

states■ the ultimate resolution: electron spin resonance



EPHelium nanodroplets

cooling clusters evaporative(adiabatic expansion) grow cooling

■ Excellent cryostat, at T~0.4K■ Weak interactions■ Liquid (in fact superfluid)■ Confinement (r = 20-100 Å)■ “Natural selection” of weakly bound species

HeN



EPSuperfluid helium droplets as nanocryostat

Spectroscopic linewidths?

Rotational (microwave) and vibrational (infrared)excitation of a dopantin a superfluid

No friction, only inertia

1 cm-1

300 cm-1

Electronic excitation (visible or UV)of a dopant

Electron repulsionby helium

Broadenedandblue-shifted

K 4s-4p

Details: C. Callegari and W. E. Ernst in: Handbook of High Resolution Spectroscopy, eds. F. Merkt and M. Quack, Wiley 2010

Grebenev, Toennies, Vilesov(Science 1998)

Trot=0.37 K



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KCl:O-2 luminescence spectrum at 4.2K.

From: Freiberg & Rebane in „Zero-Phonon Lines“ (eds. Sild & Haller (Springer))

Comparison of LIF spectra of dopants on/in helium nanodroplets (Na2 singlet vs. glyoxal):

Superfluid helium droplets as nanocryostatSpectroscopic linewidths?



EPExperiment



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Alkali n-mers on HeN: High-spin selectivity

E

+ =

+ =

E



EPHomonuclear & heteronuclear alkali molecules

11500 12000 12500 13000 13500 14000 14500 15000 15500 16000 16500 17000 17500

K2 K2

Rb2

K3 K3

KRb

KRb

KRb

Rb3

Rb3

wavenumber

Rb3

KRb

K3Rb K Rb2

KRb

LIF spectra of K + Rb attached to He nanodroplets

K Rb

1 2 0 0 0 1 3 0 0 0 1 4 0 0 0 1 5 0 0 0 1 6 0 0 0 1 7 0 0 00

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

3 0 0 0

3 5 0 0

w a v e n u m b e r

LIF

sign

al /

arb.

u.

R b 3R b

R b 2 :1 3 g

R b 3

R b 3

R b 2 :2 3 g

Rb only

1 2 0 0 0 1 3 0 0 0 1 4 0 0 0 1 5 0 0 0 1 6 0 0 0 1 7 0 0 0 1 8 0 0 00

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

K 3:

LIF

sign

al

w a v e n u m b e r

K 3:

2 4E ' - 1 4A '2

K

K 2:

1 3 g

K 2 :

2 3 g

K 3 : K only



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12000 13000 14000 15000 16000 170000

1000

2000

3000

4000

5000

6000

7000

8000

wavenumber spektrenKRbcollected

23

33+3

KRb

KRb

K2Rbor

KRb2

K2Rbor

KRb2

Johann Nagl and Carlo Callegari

heteronuclear alkali molecules



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11500 12000 12500 13000 13500 14000 14500 15000 15500 16000 16500 17000 17500

K2 K2

Rb2

K3 K3

KRb

KRb

KRb

Rb3

Rb3

wavenumber

Rb3

KRb

K3Rb K Rb2

KRb




EPAlkali-HeN pseudo diatomic: atomic excitation ns-np

1) Treat He droplet as a giant closed-shell atom

2) Calculate (vdW) molecular potential by integrating vdW pair potentials over He density

3) Approximate spectrum with F-C Factors

23/2

21/2

21/221/2 5s

5p

HeNRb

HeNRb

HeNRb

Rb 5s on HeN surface,DFT calculation (help by F. Toigo) in:Brühl, Trasca, Ernst JCP 115, 10220 (2001)

● ● ● ● ●

●●●●●



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HeN

Key questionsDopants with spin in external B-field

inside

surface

Create spin precession:

Spin relaxation due tohelium environment?

For atoms?

For molecules?

Related topic: helium buffer gas cooling of oriented atoms or molecules

B

LASER

Carlo Callegari



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linear

+

Add magnetic field and exploit dichroism:

Rb 5s-5p

/4

B= 2.9 kG

field free

excitedstate(2P1/2)

with magnetic field

TS TS=0.37K

ground state(2S1/2)

- +linear

0.37 K

T

Spin temperature

Phys. Rev. Lett. 98, 075301-1-4 (2007)



EPHighly excited states and ionization of atomsFor example: two step processes in HeN-Rb

~12817 cm-1

~12579 cm-1

Fixed

~25700 cm-1 52D manifold

52P3/2

52S1/2

52P1/2

2-step excitation

Scanned ~13175cm-1

12601cm-1

12606cm-1

12611cm-1 52S1/2 (2Σ1/2)

52P3/2 (2π3/2, 2Σ1/2)52P1/2 (2π1/2)

12607cm-1

21911cm-1

33691cm-1bare Rb

Moritz Theisen and Florian Lackner



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Florian Lackner

Moritz Theisene-

Rb+

He

Rb+

He

Rb+

Hee-

Goal: High-lying Rydberg state

Rb+

Hee-



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Cold chemistry in confinement

initiate reactions with laserexcited atoms, e.g. Cs 7p + H2

dope dropletwith H2 or HCl

H2

add alkali



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11500 12000 12500 13000 13500 14000 14500 15000 15500 16000 16500 17000 17500

K2 K2

Rb2

K3 K3

KRb

KRb

KRb

Rb3

Rb3

wavenumber

Rb3

KRb

K3Rb K Rb2

KRb




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Rb on He nanodroplets: Rb2 1 3g 1 3u+

13200 13400 13600 138000

100

200

300

400

500

600

700

800

LIF

/ arb

.u.

Wavenumber / cm-1



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Comparison of Frank-Condon factors to LIF spectrum

4 6 8 10 12 14 16

12800

13000

13200

13400

0

500

1000

1500

2000

2500

13200 13300 13400 13500 13600 13700 138000,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

FC

F

Wavenumber / cm-1

LIF

/ ar

b.u.

1g

0g-

0g+

2g

ground state: only v” = 0 populatedexcited state v’=0–9 for each SO substate

+43 cm-1

ab-initio potential energy curves from O. Dulieu (Orsay)



EPSpin-orbit coupling: alkali 3Π on helium

H = Hmol + Hd + HSOBasis: Eigenstates of Hmolcomponents of angular momenta along z: Hd interaction with helium(determined by integrating alkali-He pair potentials(Pascale) weighted by the helium density distribution)

HSO: Well known spin orbit Hamiltonian in basis (approximated R independent)In general: 12x12 matrix

y

z

Gerald Auböck

J. Phys. Chem. A111, 7404 (2007), in memory of Roger Miller

Tendencies:Cs2 at left end,others between 1 and 3 on horizontal scale



EPOur model incl. Zeeman effect

(G. Auböck, J. Nagl, C. Callegari, and W.E. Ernst, J. Phys. Chem. A, 2007)

Rb2: LIF, MCD with Simulation K2: LIF, MCD with Simulation

Zeeman populationscorrespond to 0.37 K

temperature, i.e thesame as inside the

droplet.τ



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Na2 Na3

First observation of trimers on droplets(collaboration with G. Scoles, Princeton)

PRL 74, 3592 (1995)

PRL 77,4532 (1996)



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relaxation

spin flip

h h h

h1850 cmE

trimerzedspinpolari

bind

(~ 700 cm-1three-body int.)

4.4 Å

Photoinduced Spin Dynamics

Science 273, 629 (1996)



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11500 12000 12500 13000 13500 14000 14500 15000 15500 16000 16500 17000 17500

K2 K2

Rb2

K3 K3

KRb

KRb

KRb

Rb3

Rb3

wavenumber

Rb3

KRb

K3Rb K Rb2

KRb




EPDepletion spectra with mass selective detection

(use quadrupole mass spectrometer)



EPAlkali trimer quartet state excitations

(Phys. Rev. Lett. 100, 063001-1-4 (2008))

Calculations:

MOLPRO,Complete Active Space Self Consistent Field(CASSCF) & CASPT2

Andreas W. Hauser



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Quartet trimers: Electronic structure



EPHomonuclear Alkali Trimers K3 and Rb3

3

Geometry optimization at the RHF-CCSD(T) level of theory

strongly boundlow-spin2E´ ground state

K3 Rb3

b = 4.1 A b = 4.4 A= 77°

= 79°

K3 Rb3

b = 5.05 A b = 5.5 A= 60°

= 60°

More weakly bound(van der Waals) high-spin4A2´ ground state



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3

Geometry optimization at the RHF-CCSD(T) level of theory

Introduction of normal coordinates Scan over Qx coordinate



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A borrowed approximation: harmonic oscillator eigenfunctions

yes, but not as good as for doublet

Check CI vectors

MOs are delocalized and show typical s,p,d character

Make density plots of canonical MOs

We observe a close similarity with the energy ordering of a harmonic oscillator

No, it is more complicated

Compare the harmonic osci eigenenergieswith the observed bands of the ab inito calculation

This model offers a qualitative description of state energies

Fit MO energies with 5-parameter formulaincluding anharmonicity and cross terms

Is the state pattern comparable to atomic s,p,d energies?

How localized are the MOs?

Does a single-electron approximation hold?

A shell model for the quartet states



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Time-independend Schrödinger equation in N dimensions:

Apply factorization method:

Choose isotropic harmonic oscillator potential:

Solutions:

dimension !


N=2 (x,y)N=1 (z)



EPA shell model for the quartet states

(x,y) (z)



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(0,0,0)

(0,±2,0)

(1,0,0)

(0,0,1)

(0,±1,1)E=h(2n+l)+hznz+anharm.

(0,±1,0)




EP ×104

(0,±1,0)+

(0,0,1)

(0,±2,0)(1,0,0)

B2

B2

A2 B1

A2

(0,0,0)



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A borrowed approximation: harmonic oscillator eigenfunctions

yes, but not as good as for doublet

Check CI vectors

MOs are delocalized and show typical s,p,d character

Make density plots of canonical MOs

We observe a close similarity with the energy ordering of a harmonic oscillator

No, it is more complicated

Compare the harmonic osci eigenenergieswith the observed bands of the ab inito calculation

This model offers a qualitative description of state energies

Fit MO energies with 5-parameter formulaincluding anharmonicity and cross terms

Is the state pattern comparable to atomic s,p,d energies?

Are the MOs comparable to AOs?

Does a single-electron approximation hold?




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Vibronic spectra



EPFurther refinement: SO-coupling

CASPT2 + ECP-LS



EPJahn-Teller effect theory

Qx Qy

Energy

Two-fold vibrational degeneracy

Two-fold electronic degeneracy



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Adiabatic energy surfaces are obtained by diagonalizationof He +HSO:

Relativistic Jahn-Teller effect theory

Harmonic force term

SO splitting

Linear JT parameter

Quadratic JT parameter

potential surfaces !

= f(Qx,Qy)



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Parameters in dimensionless units:

Jahn-Teller parameters

D



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Vibronic spectra: the 14A2’24E’ transition

V = 0

V = 1V = 2



EPAlkali trimer quartet state excitations

K3 and Rb3(quartet):Hauser, Auböck, Callegari, Ernst,J. Chem.Phys. 132,164310 (2010)

(doublet):Hauser, Callegari,Soldan, Ernst,J. Chem. Phys. 129,044307 (2008)andspectral predictions:Chem.Phys.(in print)

heteronuclear:in preparation



EPLevel Structure and Magnetic Properties

from One-Electron Atoms to Clusterswith Delocalized Electronic Orbitals:

Shell Models for Alkali Trimersby A.W. Hauser, C. Callegari, W.E. Ernst

in: P. Piecuch et al. (eds.), Advances in the Theory of Atomic and Molecular Systems, Progress in Theoretical Chemistry and Physics 20, DOI

10.1007/978-90-481-2985-0 30, Springer Science+Business Media B.V. 2009

Doublet states:Electronic shell model,

See e.g. Cocchini, Upton,Andreoni, J. Chem. Phys. 1989

Quartet states:Our model relating the electronicstructure to the eigenstatesof the harmonic oscillator,cf. single particle states inquantum dots



EPMagnetic Resonance

Superfluid helium nanodroplets

a cold container for single

particles (0.38 K)

study exotic, ‘unstable’ species

(e.g. K3, Rb3)

preparation probingmanipulation

A C

dBdz

dBdz

B

single atoms long spin lifetime create spin polarization

low optical density indirect detection

Rabi: spatial separation



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High resolution mw or rf spectroscopy?

How about polarization methods?

Narrow linewidth on mw and IR transitions

Large linewidth on optical transitions

Molecules in/on helium droplets:

LIF Detection of Microwave Absorption e.g. W. E. Ernst, S. Kindt, and T. Törring, Phys. Rev. Lett. 51, 979(1983)

W. E. Ernst, J. Kändler, C. Noda, J. S. McKillop and R. N. Zare,Hyperfine Structure of BaI, J. Chem. Phys. 85, 3735-3743 (1986).

RTPI Detection of Microwave AbsorptionNa3, W.E. Ernst and O. Golonzka (1999)



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Pumping and probingThe optical 2S1/2 2P1/2 transitions can be used to manipulate and probe spin states

2P1/2

+1/2

-1/2

+3/2

+1/2

-1/2

+1/2-1/2-3/2

+-

mj2P3/2

2S1/2

Zeeman

K can be spin polarized by depletion

Rb can be spin polarized by optical pumping G. Auböck, J. Nagl,

C. Callegari,and W. E. ErnstPRL 98, 075301(2007)PRL 101, 035301(2008)

12600 12800 13000 132000

50

100

150

200

LIF

(103

cou

nts p

er se

cond

)

wavenumber (cm-1)

Rb K

D2

D1

12600 12800 13000 132000

50

100

150

200

LIF

(103

cou

nts p

er se

cond

)

wavenumber (cm-1)

Rb K

D2

D1 D2D1



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A C Bpump mw probe

Markus Koch

Optically Detected ESR

- -



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0 100 200 300 400 500

-8-6-4-202468

Ener

gy /

h (G

Hz)

Magnetic field B0 (mT)

2S1/2F=3

F=2I=5/2

85Rb

linear,|S,I;F,mF ›

linear,|S,I;mS,mI ›

Breit-Rabi formula

ESR on helium droplets

g

aHFS



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0 100 200 300 400 500

-8-6-4-202468

Ener

gy /

h (G

Hz)


2S1/2F=3

F=2I=5/2

85Rb

0 100 200 300 400 50002468

1012141618

Tran

sitio

nene

rgy

/ h (G

Hz)


|∆mS| = 1, ∆mI = 0

trans

ition

ene

rgy

/ h (G

Hz)

9.4

GH

z

free atomdroplet free atom B0droplet

magnetic field shift (µT)

LIF

sign

al (1

03cp

s)

magnetic field (mT)200 300 400




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0 100 200 300 400 500

-8-6-4-202468

Ener

gy /

h (G

Hz)


2S1/2F=3

F=2I=5/2

0 100 200 300 400 50002468

1012141618

Tran

sitio

nene

rgy

/ h (G

Hz)


85Rb

9.4

GH

zfree atom droplet

|∆mS| = 1, ∆mI = 0

trans

ition

ene

rgy

/ h (G

Hz)



LIF

sign

al (1

03cp

s)




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Modeling: Breit-Rabi formula

• droplet: variations δaHFS, δgS free parameters

• 39K: δaHFS/aHFS = 325 ± 40 ppm, δgS/gS = 0 ± 3 ppm

• 85Rb: δaHFS/aHFS = 412 ± 8 ppm, δgS/gS = 0 ± 3 ppm

• δaHFS/aHFS = δ|0|2/|0|2

varies with droplet size!dropletfree atom


LIF

sign

al (1

03cp

s)



LIF

sign

al (1

03cp

s)

200 300 400


LIF

sign

al (1

03cp

s)M. Koch, J. Lanzersdorfer, C. Callegari, J.S. Muenter, and W. E. ErnstJ. Phys. Chem. A 113, 13347-13356(2009)

M. Koch, G. Auböck, C. Callegari and W. E. ErnstPhys. Rev. Lett. 103, 035302-1-4(2009)




EPElectron spin density at alkali nucleus

Following Adrian [J. Chem. Phys. 32 (4), 972–981 (1960)], the relative change of hfs consists of two parts:

Relative change of electron spin density at alkali nucleus in ppm for HeN droplet

see M.Koch, C. Callegari, and W. E. Ernst, Mol. Phys. 108 (7), 1005 (2010), issue in honor of R. N. Zare



EPRabi oscillations

decoherence time T2 > 50 µs

MW magnetic field BMW (µT)

ES

R a

mpl

itude

|c1|2

ES

R a

mpl

itude

cavity position, x=vt

MW magnetic field BMW (µT)

ES

R a

mpl

itude

dephasing time T2 ≥ 50 µs

kxa

x

aAB

sin1

… detuning

… Rabi frequency



EPESR on droplets: conclusions & future• first demonstration of MR (ESR) on doped HeN • hyperfine resolved ESR spectrum of 39K, 85Rb

• shifts (~400 ppm), droplet-size dependent: Fermi contact term

• coherent population transfer: Rabi oscillations

Currently in progress:

Cr:S = 6/2I = 0

vd WaalsXe

129Xe: I = 1/2 131Xe: I = 3/2

rest: I = 0

K, Rb

Poster by Martin Ratschek and Markus Koch (yesterday)



EPThe HeDrop Team

Dr. Andreas W. Hauser

Dr. Carlo Callegarinow Elettra, Trieste Dr. Markus Koch

Johann Naglnow MIBLA Co

Gerald Auböcknow EPFL

Florian Lackner

Moritz Theisen

Martin Ratschek

Research funded by

Fonds zur Förderung der wissenschaftlichen Forschung

& EU Network „Cold Molecules“



EP

WE-Heraeus-Seminar No. 343

Helium Clusters– Finite-Size Superfluid and Nano-Size Cryostat

Physikzentrum Bad Honnef, GermanyMarch 30 to April 1, 2005

Scientific Organizers:Wolfgang E. Ernst and Carlo Callegari

WE-Heraeus-Seminar No. 482

Helium Nanodroplets– Confinement for Cold Molecules and Cold Chemistry

Physikzentrum Bad Honnef, GermanyMay 29 to June 1, 2011

Scientific OrganizersWolfgang E. Ernst and Markus Koch

Institute of Experimental PhysicsGraz University of Technology

Petersgasse 16A-8010 Graz, Austria, Europe

E-mail: [email protected]



EP

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shell models for alkali metal trimersphysics.usc.edu/shells/talks/ernst_part_ii_shelleffects... ·...

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