Finite Size Scaling and Stability of

Atomic and Molecular Systems

in Superintense laser fields

Sabre Kais

Department of Chemistry

Birck Nanotechnology Center

Purdue University

West Lafayette, IN 47907

Finite Size Scaling and Quantum Criticality

(1) Introduction: Criticality and Finite Size Scaling

(2) Finite Size Scaling in Quantum Mechanics:

How it works

(3) Stability of Matter:

(a) Atoms

(b) Molecules

(c) Quantum Dots

(4) Stabilization in Superintense Laser Fields

(5) Future Work: Open Problems

Phase Transitions

Classical: Classical phase transitions are driven by thermal energy fluctuations

Like the melting of an ice cube:

Solid GasLiquid

P

T

SolidLiquid

Gas

CP

Quantum: Quantum phase transitions, at T=0, are driven by the Heisenberg uncertainty principle

Like the melting of a Wigner crystal: Two dimensional electron layer trapped in a quantum well

crystalWigner liquid Fermi

Quantum Phase Transitions

Transitions that take place at the absolute zero of temperature, T=0, where crossing the phase boundary means that the quantum ground state energy , of the system changes in some fundamental way.

This is accomplished by changing some parameter in the Hamiltonian of the system .

We shall identify any point of non-analyticity in the ground state energy , as a quantum phase transition.

)(0 E

)(0 H

C

Phase Transition

Free energy: f[K]=-kBT log[Z]

Coupling Constants: {K1, K2; KD}

As a function of [K], f[K] is analytic almost everywhere

Possible non-analyticities of f[K] are points (DS=0), lines (DS=1), planes (DS=2), …

Regions of analyticity of f[K] are called phases

Partition Function

Phase Transitions

First-order:

is discontinuous across a phase boundaryiKf

Continuous Phase Transition:

All are continuous across the phase boundary. But, second derivatives or higher derivatives are discontinuous or divergent

iKf

G

T

T

GS

T

CTT

2

2

T

GTCV

T

T

T

First-order transition Continuous transition

Phase Transitions

P

TCT

CP

SolidLiquid

Gas

CP

T

CT

Liquid

Phase One

Gas

Phases Two

327.0

CTTCritical

ExponentOrder

Parameter

Critical Exponents

The critical exponents describe the nature of the singularities in various measurable

quantities at the critical point ],,,,,[

In the limit CTT

Heat Capacity:

Order Parameter:

Susceptibility:

Equation of State:

Correlation Length:

Scaling Laws:

Fisher:

Rushbrooke:

Widom:

Josephson:

)2(

22

)1(

2d

CTTC ~

CTTM ~

CTT~

1~ HM

CTT~

Universality Classes

Near a second-order phase transitionmacroscopic quantities show a universal

scaling behavior that is characterized by critical exponents that depend only on general properties of the

system, such as

its dimensionality, symmetry of the order parameter, or

range of interaction.

Accordingly, phase transitions are classified in terms of universality classes.

Critical Exponents],,,,,[

Exponent TH EXPT MFT ISING2 ISING3 HEIS3

0-0.14 0 0 0.12 -0.14

0.32-0.39 ½ 1/8 0.31 0.3

1.3-1.4 1 7/4 1.25 1.4

4-5 3 15 5

0.6-0.7 ½ 1 0.64 0.7

0.05 0 ¼ 0.05 0.04

2 2.00 ±0.01 2 2 2 2

1 0.93 ±0.08 1 1 1

1 1.02 ±0.05 1 1 1 1

1 4/d 1 1 1

2

)(

)2(

d )2(

TH. Theoretical values (from scaling laws); EXPT. Experimental values (from a variety of systems); MFT. Mean field theory; ISINGd. Ising model in d dimension; HEIS3. classical Heisenberg model. D=3

Kenneth G. Wilson (1982)

Renormalization Group

The Nobel Prize in Physics 1982

"for his theory for critical phenomena in connection with phase transitions"

Kenneth G. Wilson

Cornell University

Finite Size Scaling

In statistical mechanics, the finite size scaling method provides a

systematic way to extrapolate information obtained from a finite

system to the thermodynamic limit

Importance

The existence of phase transitions is associated with singularities

of the free energy. These singularities occur only in the

thermodynamic limit.

Yang and Lee Phys. Rev. 87, 404 (1952)

The Nobel Prize in

Physics 1957

Finite-size effects in Statistical Mechanics

Cv

TC T

N'''

N

N'

N''

manifested are effects size-Finite1 y

behavior like-Bulk1 y

occur toexpected are effects Critical1~y

system. infinite theoflength n correlatio

theis where,)(N y variablescaled FSS

form in the offunction a asy singularit

a developsK quantity amical thermodyna If

cN

NKK ~)(lim

lengthn correlatio for the particularin and

cN

N~)(lim)(

thatassumes ansatz FSS the

)()(~)( yfKK KN

bemust )( of

behavior theso ,analytical also is K N, finite

aFor function. analyticalan is )( where

K

N

K

yf

yf

yyfy

K0

~)(

cat that followsIt

thereforeand

NNK cN ~)(

c

(q)(q)

at singular also is

)(K ),K( of derivativeqth theis )(K If

q

cq

q

q

KKd

d

~)()( )(

N

NK qc

qN

;

~)( )()(

.length n correlatio the toFSSapply can We

. around

regular ishich function w scaling a is where K

c

as expressed becan

)(K and aroundexpansion Taylor a has

it ,in function analyticalan is )(K Since

Nc

N

NNNK cKN ),(~)( 1

bygiven is N and N sizes of

systems finitefor equation (PR)ation renormaliz

ogicalphenomenol thedeveloped eNightingal

NNN cN ),(~)( 1

NN

NN

)()(

as written becan point fixed at thethen

length,n correlatio theof definition theUsing

. true the toconverge tosizes infinite oflimit

in the }{ points of succession that the

expected isIt ).(at point fixed a has and

c

)(

NN,

NN,

N

NNN

NNNN

NNN

NNN

E

E

E

E

)(

)(

)(

)(),()(

0

),()(1

),()(0

),()(1

Statistical Mechanics

Classical Quantum

Phase Transitions

Free Energy F(Ki)=-KBT log(Z)

Phase Transitions

T 0Ground State ……: )(0 iE

Critical Phenomena

Correlation Length )(~ CTT

Critical Phenomena

Mass Gap of H

)(~1

CE

Finite Size ScalingThermodynamic Limit

N

Finite Size ScalingNumber of Basis Functions

M

Applications Applications

In the present approach, the finite size corresponds not to the

spatial dimension, as in statistics, but to the number of elements

in a complete basis set used to expand the exact eigenfunction of a

given Hamiltonian.

Quantum Mechanics

M

n

nn

n

nn aa00

(Variational Calculations)

)(

Classical

N )(

Quantum

M

Phys. Rev. Letters 79, 3142 (1997)

Finite Size Scaling: Quantum Mechanics

In order to apply FSS to quantum mechanics problems, let us consider the following Hamiltonian of the form

VHH 0

termdependent - is V andt independen- is where 0 H

For a given complete orthonormal λ-independent basis set , the ground state eigenfunction has the following expansion

}{ n

n

nna )(

The Nth-order approximation for the energies are given by the eigenvalues of the matrix H(N),}{

)(Ni

(N)

i{i}

(N)

λ ΛE min

The corresponding eigenfunction are given by

)(

)()()(

NM

n

nN

nN

a

The expectation value of any operator O

N

mn

mnN

mN

nN

OaaO,

,)()()(

)( )(

The FSS ansatz

CO

NNFOO ~

)(

)ln(

ln),;(

)()(

NN

OONN

NN

O

The curves intersect at the critical point

),;(),;( NNNN COCO

In order to obtain the critical exponent for the energy

),;( NNCH

Hellmann-Feynman Theorem

VHE

)1(for equation an gives VO

Finite Size Scaling: Quantum Mechanics

VHH 0

)(

)()()(

NM

n

nN

nN

a

The FSS ansatz

CO

NNFOO ~

)(

)ln(

ln),;(

)()(

NN

OONN

NN

O

The curves intersect at the critical point

),;(),;( NNNN COCO

Short Range Potentials Yukawa Potential

r

eH

r

2

2

1)(

Basis Set

)()()2)(1(

1),( ,

)2(2/

mllnr

n YrLelnln

r

Where is the Laguerre polynomial of degree n and order 2 and are the spherical harmonic functions of the solid angle

)(, mlY

)()2( rL ln

Hamiltonian

839903.0C

Yukawa

VH

H 839908.0exact

C

Phys. Rev. A 57, R1481 (1998)

Finite Size Scaling with Gaussian Basis Sets

The main idea is to use Gaussian basis sets to do FSS calculations

for large atomic and molecular systems.

The basis-set is an over-complete set of Gaussian functions:

kji

ijkkji zyxrCx )exp();( 2

,,

Where Cijk are the normalization constants and βis a free

parameter.

GaussiansSlater

Finite Size Scaling Data Collapse

)(~

CC

O

CON

NFOO

~

/

0~)(

xxF

x

/~

NO

NN

)(~ /1/CN

NGNO

)(~ /1/0 CNGNE

Data Collapse

1 ,2

/1NC

Chem. Phys. Letters 319, 273 (2000)

~/

0 GNE /1NC

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.82 0.86 0.90 0.94 0.98

840.0c

)Re(

)Im(

56

4N

11

Yukawa 4N

Chem. Phys. Letters 423, 45 (2006)

Quantum Mechanics

Complex Angular

Momentum

Complex Time

Complex Energy

Complex Charge

ReggeTheory

InstantonMethod

Rotation Complex

Coordinates

Variation Method

Scattering Amplitude

Tunneling ResonancesStability

of Ions

Two Electrons Atoms

1221

22

21

11

2

1

2

1)(

rrrH

2112 rrr r12

r2

r1

e-

e-

Z

ZC=0.911

ET=-0.5

He like system

)(gsE

H like +e-

ZC=0.911

H- He Z

Z

1

Finite Size Scaling procedure

Hamiltonian:

Basis Set:

Hamiltonian Matrix:

Renormalization Equation:

ZrrrH

1 ;

11

2

1

2

1

1221

2

2

2

1

kji

krrijrrjikji rerrerrC

,,

12

21

21,, 2

12121

Nkji

10 E ,E s,Eigenvalue Leading

1

0

1

0

1

)1(

)1(

)(

)(

NN

NE

NE

NE

NE

097.1C

Phys. Rev. Letters 80, 5293 (1998)

-0.10

-0.06

-0.02

0.02

0.06

0.10

1.08 1.10 1.12 1.14 1.16 1.18 1.20

096.1c

)Re(

)Im(

5

67

4N

11

Helium

4N

C

C

C

Conditional Probability

Critical Charges and Stable Atoms and Ions

N=2

0.91 Z

N=3

2.08 Z

N=4

2.85 Z

H- He

H-- He- Li

He-- Li- Be

Int. J. Quantum Chem. 75, 533 (1999)

Surcharge Se=N-ZC

Do doubly charged negative atomic ions exist

in the gas phase?

NO

What is the smallest object that can bind two

extra electrons?

This is a challenge for

experiment and theory!

The two electrons must be

separated by at least 5.6 Å

Model Potential for Spherical Molecular Dianions

Stability of Spherical Carbon Cluster Dianions

C60

The interaction potential of an electron at a

distance from a singly charged negative sphere of radius R and dielectric

constant ε

2

60C

1

5.3

4.4

l

AR

NC=79

RC~5.5

smsC 1.0~1)(2

60

Molecular Physics 100, 475 (2002)

Stability of Dianions of Fullerenes

C 60

C84

C 79

EA1 = + 2.65 eV

EA2 = - 0.3 eV

EA1 = + 3.14 eV

EA2 = + 0.44 eV

R = 5.6 Ac

_

_ _

_ _

Advances in Chemical Physics, Volume 125, 1 (2003).

Stability of Linear Dianions

RC=5.6 Å

-- CC-Be-CC

e- e-

g

1

BeC4--

BeC4-- BeC2

- + C2-

e- e-

Phase Transitions and Stability of Three Body Coulomb Systems

(M,Q)

(m,q) (m,q)(m,q)

(M,Q) (M,Q)

12

2

21

2122

21

1

2

1

2

1

r

Q

r

qQ

r

qQ

mH

12

21

21

22

21 111

22 rrrH

p H H

eHe He

2

-

qqQQ

Mm

mMu

Qquf

fuHH

frr

2

110

0

1

Mm

qQ

or?

Molecule Atom

e- e-

e+

R

r1r2

e- e-

e+

r1r2

r12

D

Three Body Coulomb Systems (ABA)

Particles Mass

e: electron 1

p: proton 1836.15

u: muon 206.76

d: deuteron 3670.5

t: tritium 5476.92

Stable )p( Stable )e(e ---

S. Kais, Phys. Rev. A 62, 06050 (2000)

Phase Transitions and Stability of Three Body Coulomb Systems

H2+

Charge density probability

2402.124.1 20 NC

States Bound

2402.1241.1 20 NC

Explosion Coulomb

FSS for Critical Conditions for Stable Dipole Bound

AnionsThe Hamiltonian

Slater Basis Set:

Chem. Phys. Lett. 372,205 (2003)

-Z +ZR

e-

z

H3C-CN

H2CCC

C3H4O3

C3H2O3

µ= 4.3 D

µ= 4.34 D

µ= 5.5 D

µ = 4.5 D

Ea=108 cm-

Ea=173 cm-

Ea=40 meV

Ea=20 meV

µc=0.655 a.u =1.625D without B.O =2.5D

Electron will be trapped with µc>2.5D

µc=0.655

a.u

FSS for Critical Conditions for Stable Quadrupole Bound Anions

Hamiltonian: consists of a charge q at the origin and two charges –q/2 at

z = +1 & -1

Where β is the variational parameter used to optimize the numerical results

and is the Legendre polynomial of order l.

Slater Basis Set:

)(2 lP

q q=QR

Journal of Chemical Physics, 120, 8412 (2004)

Journal of Chemical Physics, 120, 8412 (2004)

qc=1.46 a.u

qc=3.9 a.u

2,,

2

qq

q

2,,

2

qq

q

KCl2- Qzz=10 a.u

K2Cl- Qyy=27 a.u

(BeO)2- (13,13,0.5)

CS2- Qzz=4 a.u

FSS for Critical Conditions for Stable Dipole-Bond

DianionsThe Hamiltonian

Basis Set:

J. Chem. Phys. (in press, 2007)

-Q +Q

R

e- e-

Prolate spheroidal coordinates

ar br

QR1

R2

Stability diagram for two electron dipole

-Q +Q

R

e- e-

Atomic & Molecular Stabilization by

Superintense Laser Fieldswith

Prof. Dudley HerschbachHarvard University

Multiply charged negative

atomic ions

in superintense laser fields

The peak-power of pulsed lasers has increased by 12 ordersof magnitude during the past 4 decades.

QUESTION: What is the highest-level intensity presently possible?

ANSWER: The highest possible focused laser intensity =1019 W/cm2

This level of intensity can be achieved with femtosecond laser based on Chirped Pulse Amplification (CPA).

Superintense Laser Fields (I > a.u.)

The electric field of a monochromatic plane wave can be written

)sinˆ tancosˆ()( 0 teteEtE yx

ti

rrtrp

N

i

i

j jii

i

1

1

1

2 1

)(

1

2

1

With ex and ey orthogonal to each other and to the propagation direction

)()()( 00 tEEtwhere

α0=E0/ω2, where E0 and ω are the amplitude and frequency of the

laser field.

Laser Atom Interaction

Moving frame of reference which follows the quiver motion of the classical electron

on.polarizaticircular toscorrespond π/4δ

onpolarizatilinear toscorrespond 0δ

High Frequency Floquet Theory

Vo is the “dressed” Coulomb potential, is the time average of the shifted

Coulomb over one period of the laser

2

0 0

00)sinˆ tancosˆ(2

1),(

yx eear

draV

Hamiltonian of atomic system in super-intense laser fields

N

i

i

j ji

ioirr

arVpH1

1

1

02 1

,2

1

e-e termm laser tererm coulomb trm kinetic teH

frequency. theis andintensity beam averaged- time theis I where,2 Io

Ψ ) α ( E Ψ H 0Structure Equation:

Hydrogen atom in super-intense linear laser fields

J. H. Eberly, et al Science 262, 1229 (1993)M. Pont, et al. Phys. Rev. Lett. 61, 939 (1988)

For linear polarization, the

“dressed” potential is the same

as that generated by a “linear

charge”: the trajectory α(t)

For circular polarization, the

“dressed” potential is the same

as that generated by a “circular

charge”.

Circular charge

0a

Charge density

0a

Linear charge

Linear Charge and Circular Charge

Prolate Spheroidal Coordinates

for Linear Polarization

z

YX ,

Basis Set (Linear Polarization)

eeimmqpmqp

222,, )1)(1()1(),,(

energy theoptimize toparameter a is

integers. negative-non are and ,

mqp

One-electron basis functions in elliptical

coordinates

:becomes basis theso 0,m state, groundFor

)1(

,, )1(),,( eqp

mqp

We used about 100 basis functions

The electronic orbitals for the ground states. The presentation is given in a plane

passing through the axis of the field taken as polar axis z

Negative of the detachment energy of the ground state of He-, He2−, Li-,and

Li2− in a linearly polarized high-frequency laser field as a function of

α0=E0/ω2, where E0 and ω are the amplitude and frequency of the laser field.

electrons N theof onedetach toneededenergy The

01 NN- EE energyDetachment

J. Chem. Phys. 124, 201108 (2006).

For example, when ultra-high-power KrF laser (5 eV photons) are used, the peak

intensity in the experiments should be I ≈ 1016W/cm2.

Oblate Spheroidal Coordinates

for Circular Polarization

z

YX ,

Basis Set (Circular Polarization)

0m

0m

0m

)sin(

)cos(

)1(),,(

)()1(),,(

12

1

1

,,

,,

m

m

e

Se

qpmqp

mqp

mqp

energy theoptimize toparameter a is

1;;1

,...l-l,-l mml qlnp

We used over 200 basis functions

Large-D stability in Linearly polarized

superintense laser fields

N

i

N

ij jiji

N

i

ii

N

i i zzzV

1 1222

1

0

12

)(

1),(

1

2

1

2

0 20

20

)(2),(

Sinz

dZzV

ii

ii

0 2z

2

1z

1

2e

1e

Dudley Herschbach, Harvard

Large D stability in super-intense circular

polarized laser fields

N

i

N

ij jiji

N

i

ii

N

i i rrrV

1 1222

1

0

12

)ˆˆ(

1)ˆ,(

1

2

1

2

0 20

20

20

)ˆˆ()ˆˆ(2)ˆ,(

SinerCoser

dZrV

yixii

ii

02̂r

2

1̂r

1

2e

1e

Negative of the detachment energy of the ground state of He-,H2− and He2− in a circularly

polarized high-frequency laser field as a function of α0=E0/ω2, where E0 and ω are the

amplitude and frequency of the laser field.

Linear

Circular

Do atoms in superintense laser fields behavelike diatomic molecules ?

Large-D stability of molecules in linearly polarized superintense laser field

RzVzV

),(),(

1

2

12,01,02

2

0 202

2

2

21,0

)cos()sin(2

1),(

Sinzx

dzV

RR

2

0 202

2

2

22,0

)cos()sin(2

1),(

Sinzx

dzV

RR

0

1e

z

x

z

0e

R

0E

00

8.00

0.20

The laser polarization

along the molecular

axis (θ=0)

2H

00

8.00

0.20

The laser polarization

is vertical to the

molecular axis (θ=90)

2H

At the D→∞ limit

N

i

N

ijjijiji

N

i

iii

N

i ii zzyyxxzyxV

yxH

1222

1

0

122

1,,

1

2

1

2

02

222

0

)2sin()cos(2

),,(

dxydz

dqzyxV

As per D-Scaling method:

This is the problem to be minimized

2

d

Our Potential

• Potential generated by application of

Relativistic Trajectory in HFFT

• ν = 400

Alternate View

Trajectory

•The new relativistic trajectory used:

jifinejirel xxwheretxtxt ,)2sin()()cos( 00

•This trajectory was used within theHFFT potential, and the potentialtraces the equivalent parametric equation:

2/

)2,0(,)cos(,0),2sin(

0

2

d

ttdtd fine

•The curve reaches it’s maximum at t=π/4; the potential and electron density reach their extremes as well

Morphology

Summary

Symmetry breaking of electronic structure

configurations resemble classical phase transitions

Quantum phase transitions can be used to explain and

predict the stability of atoms, molecules and quantum

dots.

Atomic dianions are unstable in the gas phase

Multiply charged negative ions are stable in

superintense laser fields

(Intensity > 1 a.u. ~ 1016 W/cm2)

Future WorkCombining FSS with Ab Initio and DFT

Combining FSS with Finite Element Methods

New Classification of Chemical Reactions

FSS and Efimov Systems

Stability of Matter in Superintese Laser Fields

For more details See the review article:

• Sabre Kais and Pablo Serra, "Quantum Critical Phenomena and Stability of

Atomic and Molecular Ions", Int. Rev. Phys. Chem. Vol 19, 97-121 (2000).

• S. Kais and P. Serra, "Finite Size Scaling for Atomic and Molecular Systems",

Advances in Chemical Physics, Volume 125, 1-100 (2003).

Further Directions

Combinations of linearly polarized lasers can deliver more exotic potentials with the chance of binding even more electrons.

Two lasers fired in theY direction, linearly polarizedin the X and Z directions

Two lasers fired, in the X and Y directions, both polarized in the Z

Efimov States

Efimov State is a quantum mechanical stable bound state of three

particles, with any two particle subsystem is unstable. It was

proposed by Vitaly Efimov in 1970 theoretically

and was observed experimentally in 2006

for ultracold gas of caesium atoms.

Remove any one ring and the other two will fall apart.

Letter to: Nature 440, 315-318 (16 March 2006)

Evidence for Efimov quantum states in an ultracold gas of caesium atoms

T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange,

K. Pilch, A. Jaakkola, H.-C. Nägerl and R. Grimm

Acknowledgments

Coworkers:

Future Projects:

Funding:

Pablo Serra, Juan Pablo Neirotti, Jiaxiang Wang,

Ricardo Sauerwien, Alexei Sergeev , Qicun Shi and

Dudley Herschbach

Ross Hoehn, Jiaxiang Wang

(1) Office of Naval Research (ONR)

(2) American Chemical Society (ACS)

(3) National Science Foundation (NSF)

(4) Army Research Office (ARO)

Thank You!