Tokyo

Tohoku U.(Sendai)

Matsushima Bay (from Aoba-yama Campus in Sendai)

YITP, Kyoto, Sep. 28 (2011)

Matsushima bay

Energetics of the Time-Dependent Natural Orbitalsof Molecules

Hirohiko Kono1, Tsuyoshi Kato2 ,Takayuki Oyamada1 ,Shiro Koseki3

1. Tohoku Univ. 2. Univ. of Tokyo 3. Osaka Prefectural Univ.

Energetics of the Time-Dependent Natural Orbitalsof Molecules

Hirohiko KonoHirohiko Kono11, Tsuyoshi Kato, Tsuyoshi Kato2 2 ,,Takayuki OyamadaTakayuki Oyamada1 1 ,,Shiro KosekiShiro Koseki3 3

1. Tohoku Univ. 2. Univ. of Tokyo 3. Osaka Prefectural Univ.

Interaction of molecules with fs laser pulses

(1) Radiative interaction: One-body interaction (fs or as regime)

(2) Energy transfer to vibrational degrees of freedom (10-100 fs)

→Rearrangement or reaction processes (ps –ns)

Development of MCTDHF in grid space→ Energy sharing by many electrons

(3) Intramolecular Vibrational Energy Redistribution (IVR)among different vibrational modes (1 ps –)

How to overcome:How to overcome:

An electron or electrons get energy

e.g., Initial mode selective vibrational excitation by intense near-IR pulses > IVR

Difficulties in the control of big molecules such as C60; fast IVR

Phys. Rev. Lett. 104, 108302 (2010).

M. Kanno, H. Kono, Y. Fujimura, and S. H. Lin

Coupling of laser-induced ultrafast -electron rotations with vibrational modes in chiral aromatic molecules

J. Chem. Phys. 128, 184102 (2008);Chem. Phys. 366, 46 (2009).

T. Kato and H. KonoCharacterization of multi-electron dynamicsby “time-dependent” chemical potential

・ Classification of adiabatic or nonadiabatic processes・Quantification of energy exchange between molecular orbitals

～1020 W/cm2

Light Intensity I=3.5x1016 W/cm2

Light intensity

Atomic UnitsElectric field F=5.1x109 V/cm

Hydrogen atom

Proton

Ultrashort “Intense” Laser Pulses

Attosecond (10－18 s)

Pulse length

Electron

femoto-second(10－15 s)

Bohr period=24 as

Bohr radius

Tunnel Ionization of H atom in a near-IR field

=760 nmIpeak=8.4×1013W/cm2

(parallel to the polarization direction)

Electronic waveFunction (1s)

(Keldysh parameter=1.2)

02

e

efm

Ponderomotive radius of a rescattering electron

~ 100 Bohrs

→ /23D grid points → 64 times

Coulomb+ dipole interaction

t =1.4/

Rescattering

ti =0.1/

Sequential vs. nonsequentialdouble ionization

Maximum rescatteringenergy: 3.17 Up

Tunnel Ionization(ejection of one electron)

Ion

Cou

nts

= 800 nm

2 2( ) 4pU f t Ponderomotive energy:

( )f twhere is the electric field amplitude.

Phys.Rev. Lett. 84, 3546 (2000).

Nature 432, 867(2004)

Experiment

Theory

Harmonic order

N2 molecule

Near-IR pulse

Angle

Har

mon

ic e

mis

sion

inte

nsity

HOMO of N2

Higer-orderHarmonic generation

Electron Dynamics: How to identify multielectron dynamics in ionization by intense near-IR laser pulses

Single active electron ionization

Ionization due to multielectron dynamics

Continuum Continuum

*I.V. Hertel et al., PRL 102,023003 (2009) .

Distinction in C60 ionization by linearly and circularly polarized pulses

Experiment*:

If multielectron dynamics,no difference: Only total pulse energy matters

Linear > Circular Linear = Circular

Ionization by energy exchange

Ionization by energy exchange

=800 nm(1.55 eV)

=400 nm(3.2 eV)

I. V. Hertel et al. Adv. At. Mol. Opt. Phys. 50, 219 (2005).

Multielectron dynamics of C60 above the doorway state

To the doorway state

Single activeelectron

Multielectron dynamics

Slater determinants with TD MOs

Multi-configuration time-dependent Hartree-Fock method(MCTDHF) – Beyond the mean field picture –

Ionization probability Ionization probability Keldysh-Faisal-Reiss theory

Transition amplitude from the initial state to a final Volkov state based on the S-matrix formalism.

(strong-field approximation)

Dirac-Frenkel variational principle

(1) Equations of motion for CI coefficients CI(t) (2) Equations of motion for MOs (grid representation)

T. Kato and H. Kono, Chem. Phys. Lett. 392, 533 (2004);J. Chem. Phys. 128, 184102 (2008) .

Structure of electronic wave function t

MC Time-dependent many-body wave function: a unified way to treat different dynamical processes

・J. Zanghellini, M. Kitzler, T. Brabec, and A. Scrinzi, J. Phys. B 37, 763 (2004).・Non-variational approach: T.T. Nguyen-Dang et al., J. Chem. Phys. 127, 174107(2007).

Other MCTDapproaches

Other MCTDapproaches

I(t)I

(t)(t) H2 & N2

・Ionization dynamics of molecules in intense laser fieldsExplicit inclusion of continuum states grid point representation for MOs

Laser Electric Field

( )i H tt

r rElectronic Wave function

Transformation to remove cusps

Trans

02

e

efm

Large grid

I. Kawata & H. Kono, J. Chem. Phys. 111, 9498 (1999).

(iv) Transformation of the Hamiltonian

Application to H2(R=1.6 a0 , 9 MOs)

Electron densities of natural orbitals (Log scale)

Ipeak=5.8 ×1013 W/cm2

=760 nm

2.4 fs

1u

1g

1g

2g

1u

1g

1g

2g

Main ionizing MOsin the MCTDHF

Main ionizing MOsin the MCTDHF

Suppression of ionization of -orbitals due to destructive interference(axis II polarization)

Suppression of ionization of -orbitals due to destructive interference(axis II polarization)

1g

Reduction in the MO norm at the absorbing boundary

Renormalization of MO norms

Recalculation of CI coefficients

{ }②(b)(a)(b) (b)

for absorbed

{ }③(c)(c)(c) (c)

normalizedfor ortho-

Δwj (t) = Δwjabsorb + Δwj

exchange

{ }①(a)(a)(a) (a)

for TDMO

Δwjabsorb = wj

(a) － wj(c)

MC wave function at time t after one-step of time propagation

wj(a)

wj(c)

Ionization from individual natural orbitals

Absorbing boundaries

Ionization

population exchange

Change in the occupation number

absorbed at the boundaries

O( ) ,*( ) ( ) ( )

j

N

j j jt, ,t w ,t ,t r r r r

Accumulated reduction through MOs

H2分子 ( R = 1.6a0 ) の自然軌道からのイオン化率およびイオン化の様子 (時刻 t = 1.5Tc )

Snap shot at t = 1.5 Tc

1g2g

3g

1u

2u1u1g

Time (atomic units)

Ioni

zatio

n pr

obab

ility

(R

educ

tion

in p

opul

atio

n)

t = 1.5Tc

1g

2g3g

1u2u

1u

1g

t = 1.5Tc

Rel

ativ

e p

roba

bilit

y

Time (atomic units)

Time (atomic units)

Ele

ctri

c fie

ld

(ato

mic

uni

ts)

= 760 nm, Ipeak = 5.8×1013 W/cm2

t = 1.5Tc ( = 3.80 fs )

Ionizationorbitals

g>u0

( )t absorb

jw t dt 0

1 ( )(0)

t absorbj

j

w t dtw

Road to “time-dependent” chemical potential

T. Kato & H. Kono, Chem. Phys. 366, 46 (2009), special issue on “Attosecond chemistry”eds. Andre D. Bandrauk, Jörn Manz, and Mark Vrakking

CoulombOne-electron

C1 + C2 + C3 + C4

Chemical potential j (t) of a two-electron system

Chem

ical potential1

3

2

4

O

*

1 1 Re2 2

Nk

j jjjk j

Ch j j j j k j k jC

O

( ) ( ) ( )N

j jj

E t w t t

One-to-one correspondence for natural orbitals:

2j jw COccupation number

Correlation

where

Chemical potential for natural orbital j

H2：1u (excited)

H2：1g (ground)

HeH+：1 (excited)

HeH+：1 (ground)

Δ E = 0.4378(R=1.6a0)

Δ E = 0.8866(R=1.6a0)

MCTDHF法による H2 と HeH+ (R=1.6a0) の軌道ポテンシャルの時間変化の比較

Potentials of H2 & HeH+H2 (R=1.6a0) ：

HeH+ (R=1.6a0)：

H2 : W.Kolos and L.Wolniewicz, J. Chem. Phys. 43, 2429 (1965).HeH+ : L.Wolniewicz, J. Chem. Phys. 43, 1087 (1965).

Ref. : Light-intensity : Ipeak = 1.0×1014 W/cm2

Wavelength ： λ = 760 nm ( Tc = 2.53 fs )

Chemical potential change of H2 & HeH+ in a near-IR field (R=1.6 a0)

Che

mic

al p

oten

tial (

hartr

ee)

Che

mic

al p

oten

tial (

hartr

ee)

Time (atomic units)

Degenerate→Adiabatic

Nonadiabatic

=0.06

■ The total energy supplied to the system from the field per unit time within the dipole approximation in the length gauge:

( )t

■ Energy gain of the system from the field

O

10 0

( ) ( )( )( ) (0) ( ) (( ) ( ) )

t tNj j

j

d w t td tE t E t dt t dt

d dtt t

t

d

dd

Reference energy

where the total electronic Hamiltonian is given by 0ˆ ˆ( ) ( )j

j

H t H e t r

0 0ˆ ˆ( ) ( ) (0) (0)t H t H

Energy supply through the interaction with the field

0

T. Kato & H. Kono, Chem. Phys. 366, 46 (2009), special issue on “Attosecond chemistry”eds. Andre D. Bandrauk, Jörn Manz, and Mark Vrakking

( ) (ˆ( ) ( ) ( ) ( ) = ( ) )) () (jj

t ed d d dE t t H t t t tdt dt dt

t tdt

r d

0 0

( )( ) (0) ( ) ( ) (( ) )) (

t t d tdE t E t t dt t dtd

t tt dt

d

dd

O( ) ,*( ) ( ) ( )

j

N

j j jt,t w ,t ,t r r r

Diagonal representation of the one –electron density:

where are natural orbitals obtained from (unitary transformation of TD-MOs) and {wj(t)} are occupation numbers.

( )j r

( )0 1j tw O

electron( )j

N

j t Nw

( )j td ( )j t

0

( ) (0( )

() ( )( ) )t d t

t dtdt

tE t E t

d

d

where are the dipole moments of natural orbitals( ) ( ) ( )j j jt e t t d r

( )t

Reference energy

O

10 0

( ) ( )( ) ( ( ) (

( )( ) ( ))0)

t tNj j

j

Ed w t td t

t dt t dtd

t tdt

tt

E

dd

d

Reference energy

( )jj

e t r

Introduction of natural orbitals

Advantage of natural orbitals: ・uniquely determined for the electron density (i.e., the electronic wave function) ・ the energy of the one-body interaction is given

by the sum of the diagonal elements

No is the number of spin orbitals.

Energy injected into ( )j t

Further analysis: Definition of the energies of independent orbitals

O

1 0

( ) ( )( ) (0) ( ) ( ) ( )

tNj j

j

d w t tE t E t t t dt

dt

dd

How much energy do the individual MOs gain from the field?How much energy is then exchanged among MOs?

O O

1 1 0

( ) (0) ( ) ((

))

( )) ) ((t

j j

N N

j jj

j

j

t t td t

t dtw tt

wd

t

jd

d

2 ( )t

3( )t4 ( )t

1( )t

through electron-electron interaction

S2(t) S3(t)

S1(t)

S4(t)

Naturalorbitals j

(1) Injected energy: Energy supply from the field to the natural orbital j through the radiative dipole interaction:

(2) Quantification of energy exchange: Total energy decomposition into “chemical potentials”: ( )j t

Sj(t)

Comparison betweenand Sj(t) ( )j t

( )j t Sj(t)

2.4 fs

Orbital population

1g

1u

2u

1u

2g

3g

1g

50 100 1500

0

0.02

0

0.02

0

0.02

0

0.02

50 100 15002g is an energy accepting orbital

→ efficient ionizing MO

Quantification of energy exchange in the sub-fs domainQuantification of energy exchange in the sub-fs domain

( )jS t

Energy accepting orbital

( )jS tFrom the field:

Net gain:

2.4 fs

Spectator orbital

( )jS t

Chemical potential change and Sj(t)

Energy supplied from the fieldto the natural orbital :

Energy supplied from the fieldto the natural orbital :

0

( )( ) ( )jj

t d tdt

S t t dt

d

( )j t

( ) ( ) (0) ( ) ( )

+

( ) ( =0)

1 ( ) ( =0)2

( ( ) )jj

j

j

j j

j

j j j j t j j j

h t h t t

t t

t

t

t

t

j

j

jd

d

Decomposition of the chemical potential

O

* *+ Re ( ) ( 0)N

k k

k j j j

C Ck j k j t k j k j tC C

one-electron

two-electron

Correlation

Indu

ced

dipo

le (a

tom

ic u

nits

)

Time (atomic units)

Electric field (atom

ic units)

Indu

ced

dipo

le (a

tom

ic u

nits

) Electric field (atom

ic units)

Time (atomic units)

1 1( ) ( )g g

t S t

= 760 nm

= 380 nm

Correlation Energy Change for 1 g

two-body

one-body

two-body

one-body

Time (atomic units)

Time (atomic units)

1( ) :g

t

Ene

rgy

(har

tree)

Ene

rgy

(har

tree)

Correlation energy Change:

Nonadiabatic but no energy exchange

= 760 nm

= 380 nm

2 2( ) ( )g g

t S t

Correlation Energy Change for 2g

two-body

one-body

two-body

one-body

Time (atomic units)

Time (atomic units) Time (atomic units)

Time (atomic units)

Ene

rgy

(har

tree)

2( ) :g

t Correlation energy Change:

Ene

rgy

(har

tree)

Ene

rgy

(har

tree)

Ene

rgy

(har

tree)Energy exchange is dominated by the change in correlation energy!

while the Coulomb part of the mean field energy does not change significantly.

Ionization dynamics in H2 at R =1.6 a0 (MCTDHF by T. Kato)

One electron ionization

Sequential two-electron ionization

=760 nmI peak=3.5×1014 W/cm2

Ele

ctric

fiel

d(a

tom

ic u

nits

)

Proton

Nonsequential(Rescattering)

Nonsequential(Rescattering)

Correlation in electron dynamics is well reproduced

Reduced density map (Log scale)Reduced density map (Log scale)2

1 2 1 2 1 2 1 2 1 2( , ) ( , , , , )P z z d d d z z

K. Harumiya et al. J. Chem. Phys. 113, 8953(2000) ; K. Harumiya, H. Kono, Y. Fujimura, I. Kawata, A.D. Bandrauk, Phys. Rev. A 66, 043403 (2002).

cf. Dual Transformation Methodcf. Dual Transformation Method

Spatial distribution of the correlation energy:Correlation energy density

1 1 2 1 2 , 21 2

1 1( ) ( ) ( ) ( ) ( )2C ij k

i j i k i j k i j

E i k j d

r r r r r r

r r

1 2 1 2 ,1 2

1 1( ) ( ) ( ) ( )2C ij k

i j i k i j k i j

E i k j

r r r r

r r

1r

2r

iとkの相関エネルギー

再散乱軌道の特定法

Concluding Remarks

Analysis of the electronic dynamics by using the “time-dependent” chemical potential

◆For stationary states or adiabatic processes：The chemical potentials of natural orbitals are all degenerate.

◆Chemical potential vs. Energy supplied from the field

Small correlation energy Spectator orbital (both are equal)

Energy exchange between natural orbitals through Correlation Energy donating, accepting orbitals

Large increase in the chemical potentialof a natural orbital

Ionization