EnergeticsEnergetics of of thethe Time-DependentTime-Dependent NaturalNatural OrbitalsOrbitals ofof MoleculesMolecules

1 2 1 3 HirohikoHirohiko Kono Kono11, ,Tsuyoshi Tsuyoshi Kato Kato22 , Takayuki,Takayuki Oyamada Oyamada11 , ,ShiroShiro Koseki Koseki33 1.1. Tohoku Tohoku Univ. Univ. 2. 2. Univ. Univ. of of Tok Tokyoyo 3. 3. Osaka Osaka Prefectural Prefectural Univ. Univ.

Matsushima Bay (from Aoba-yama Campus in Sendai) Matsushima bay Tohoku U. (Sendai)

Tokyo

YITP, Kyoto, Sep. 28 (2011) Interaction of molecules with fs laser pulses (1) Radiative interaction: One-body interaction (fs or as regime) An electron or electrons get energy → Energy sharing by many electrons Development of MCTDHF in grid space T. Kato and H. Kono Characterization of multi-electron dynamics J. Chem. Phys. 128, 184102 (2008); by “time-dependent” chemical potential Chem. Phys. 366, 46 (2009). ・ Classification of adiabatic or nonadiabatic processes ・Quantification of energy exchange between molecular orbitals

(2) Energy transfer to vibrational degrees of freedom (10-100 fs) Coupling of laser-induced ultrafast M. Kanno, H. Kono, Y. Fujimura, -electron rotations with vibrational modes and S. H. Lin in chiral aromatic molecules Phys. Rev. Lett. 104, 108302 (2010). (3)Intramolecular Vibrational Energy Redistribution (IVR) among different vibrational modes (1 ps –) →Rearrangement or reaction processes (ps –ns)

Difficulties in the control of big molecules such as C60; fast IVR e.g., Initial mode selective vibrational excitation HowHow to to overcome: overcome: by intense near-IR pulses > IVR Ultrashort “Intense” Laser Pulses

Pulse length femoto-second (10－15 s) Attosecond (10－18 s)

Light intensity ～1020 W/cm2

Hydrogen atom Electron

Bohr period=24 as us di Atomic Units ra hr Electric field F=5.1x109 V/cm Bo

Light Intensity I=3.5x1016 W/cm2 Proton Tunnel Ionization of H atom in a near-IR field Electronic wave =760 nm Function (1s) 13 2 Ipeak=8.4×10 W/cm (Keldysh parameter =1.2)

Coulomb+ dipole interaction (parallel to the polarization direction)

Ponderomotive radius of a rescattering electron ef  0 2 ~ 100 Bohrs me

 → /2 3D grid points → 64 times Sequential vs. nonsequential double ionization

Tunnel Ionization ti =0.1/ (ejection of one electron)

t =1.4/ Maximum rescattering

energy: 3.17 Up Ponderomotive energy: Rescattering 22 Uftp  ()4 where f () t is the electric field amplitude.

= 800 nm Phys.Rev. Lett. 84, 3546 (2000). Ion Counts Nature 432, 867(2004) HOMO of N2

N2 molecule Angle Experiment Higer-order Harmonic generation Near-IR pulse Theory Harmonic emission intensity Harmonic emission

Harmonic order Electron Dynamics: How to identify multielectron dynamics in ionization by intense near-IR laser pulses Single active electron Ionization due to ionization multielectron dynamics

Experiment*: Continuum Continuum Distinction in C60 ionization by linearly and circularly polarized pulses IonizationIonization by by energyenergy exchange exchange If multielectron dynamics, no difference: Only total pulse energy matters

*I.V. Hertel et al., PRL 102, Linear > Circular Linear = Circular 023003 (2009) . =400 nm  (3.2 eV) electron Single active Multielectron dynamics (1.55 eV) =800 nm  To the doorway state Phys. 50, 219 (2005). . Adv. At. Mol. Opt. et al I. V. Hertel Multielectron of C60 above the doorway state dynamics Multi-configuration time-dependent Hartree-Fock method (MCTDHF) – Beyond the mean field picture – T. Kato and H. Kono, Chem. Phys. Lett. 392, 533 (2004);J. Chem. Phys. 128, 184102 (2008) . MC Time-dependent many-body wave function: a unified way to treat different dynamical processes

 Structure of electronic wave function  t I (t) I Slater determinants with TD MOs ・Ionization dynamics of molecules in intense laser fields Explicit inclusion of continuum states grid point representation for MOs

Dirac-Frenkel variational principle IonizationIonization probabilityprobability(t) (t) Keldysh-Faisal-Reiss theory H2 & N2 Transition amplitude from the initial(strong-field state to approximation) (1) Equations of motion for CI coefficients C (t) a final Volkov state based on the S-matrix formalismI . (2) Equations of motion for MOs (grid representation)

OtherOther MCTD MCTD ・J. Zanghellini, M. Kitzler, T. Brabec, and A. Scrinzi, J. Phys. B 37, 763 (2004). approachesapproaches ・Non-variational approach: T.T. Nguyen-Dang et al., J. Chem. Phys. 127, 174107(2007). iHt  () t

ef  0 Large grid 2 me

Laser Electric Field

Transformation to remove cusps

  Trans

r r (iv) Transformation of the Hamiltonian Electronic Wave function I. Kawata & H. Kono, J. Chem. Phys. 111, 9498 (1999). Application to H2 2.4 fs (R=1.6 a0 , 9 MOs)

13 2 Ipeak=5.8 ×10 W/cm =760 nm

Electron densities of natural orbitals (Log scale)

1g 1u 1g 1u

1g 1g

2g 2g SuppressionSuppression of of ionization ionization of -orbitals due to MainMain ionizing ionizing MOs MOs of -orbitals due to destructivedestructive interference interference inin the the MCTDHF MCTDHF (axis(axis II II polarization) polarization) 1g Ionization from individual natural orbitals N O * Change in the occupation number ()rr,,t  wjj()t ()() r ,t j r ,t , j absorb exchange Δwj (t) = Δwj + Δwj absorbed at the boundaries population exchange Absorbing boundaries

Ionization

MC wave function at time t (a) (a) (a) (a) ① for TDMO after one-step of time propagation  { } (a) wj

Reduction in the MO norm at the (b) (a) (b) (b) ② absorbing boundary  for absorbed { }

Renormalization of MO norms

(c) (c) (c) for ortho- (c) Recalculation of CI coefficients ③  normalized { } (c) absorb (a) (c) wj Δwj = wj － wj 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 )

13 2 = 760 nm, Ipeak = 5.8×10 W/cm Electric field (atomic units) t = 1.5Tc ( = 3.80 fs )

Time (atomic units) Snap shot at t = 1.5 Tc Accumulated reduction through MOs 2 1 g g t = 1.5T 2 c 3g g 2u 1u 1u 1g 3g 1u 2u Ionization 1 1 u g orbitals 1g 1 t t Relative probability absorb absorb wtdtj ()  g>u wtdtj ()  t = 1.5T w (0) 0

Ionization probability Ionization probability (Reduction in population) 0 c j

Time (atomic units) Time (atomic units) 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 Chemical potential j (t) of a two-electron system 2 One-to-one correspondence Occupation number wCj  j for natural orbitals: Chemical potential 4  C1 + C2 + C3 + C4 3

2

1

Chemical potential for natural orbital j 11NO C   hjjjjkjkj   Re k jjj       * 22kj C j One-electron Coulomb Correlation where

N O E()twtt  jj () () j + + MCTDHFChemical 法によるpotentialH change2 と HeH of( RH=1.62 & HeHa0) の軌道ポテンシャルの時間変化の比較in a near-IR field (R=1.6 a0)

+ Potentials of H2 & HeH H2 (R=1.6a0) ：

1 H2： u (excited)

Δ E = 0.4378 (R=1.6a0) Nonadiabatic 1 H2： g (ground) (hartree) potential Chemical =0.06 HeH+： + HeH (R=1.6a0)： 1 (excited)

Δ E = 0.8866 + HeH ： (R=1.6a0) 1 (ground) Degenerate →Adiabatic Chemical potential (hartree) potential Chemical Time (atomic units) Ref. : Light-intensity : I = 1.0×1014 W/cm2 H : W.Kolos and L.Wolniewicz, J. Chem. Phys. 43, 2429 (1965). peak 2 ：λ HeH+ : L.Wolniewicz, J. Chem. Phys. 43, 1087 (1965). Wavelength = 760 nm ( Tc = 2.53 fs ) Energy supply through the interaction with the field

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

ddˆ d d E()ttHtt () ()  ()=()terdj (tt)( () t )  ( t ) dt dtj dt dt ˆˆ where the total electronic Hamiltonian is given by H ()tHe0 rj  () t j

■ Energy gain of the system from the field ttd dtd()  Et() E (0)d ( t )() t dt d()tt () ( t  ) dt 00dtdt  tt  NO  ˆˆdtd()  dwjj() td () t ()tHEt00() () t E (0) (0)d H()tt (0) ()   ( t ) dt  0  ( t) dt ddtt Reference energy 00j1

T. Kato & H. Kono, Chem. Phys. 366, 46 (2009), special issue on “Attosecond chemistry” eds. Andre D. Bandrauk, Jörn Manz, and Mark Vrakking t dtd()  Et() E (0) d()t  (t )   ()tdt   dt Reference energy 0  Introduction of natural orbitals Diagonal representation of the one –electron density: N NO O * ()t ()tN ()rrr,t  wjj()t ()() ,t j ,t , 01wj   wj electron j j where   j () r  are natural orbitals obtained from  () t (unitary transformation of

TD-MOs) and {wj(t)} are occupation numbers. No is the number of spin orbitals. Advantage of natural orbitals: ・uniquely determined for the electron density (i.e., the electronic wave function) ・ the energy of the one-body interaction et  r j   () is given by the sum of the diagonal elements j tt dtd()  NO dw() td () t E()t E (0) d()t (t)  ()tdt jj  () tdt dt dt Reference energy 00j1

Energy injected intoj()t where d j () t are the dipole moments of natural orbitals j()t

d jjj()tet  ()r  () t HowFurther much analysis: energy do the individual MOs gain from the field? HowDefinition much ofenergy the energies is then exchangedof independent among orbitals MOs? t NO dw() td () t Et() E (0)d () t () t  jj  ( t ) dt j1 0 dt t NNO O dtd ( ) wt( ))()ttt (0)d () ()  w (t j  (tdt)  jjjjj   j11j 0 dt  j ()t Sj(t) (1) Injected energy: Energy supply from the field to the natural orbital j through the radiative dipole interaction: Sj(t) (2) Quantification of energy exchange: Total energy decomposition into “chemical

potentials”:  j ()t Natural S (t) orbitals j 1()t 1 through electron-electron Comparison between  ()t interaction 2  ()t S (t)  ()t S (t) 4 j and j 2  ()t S3(t) 3

S4(t) Orbital population

2.4 fs Chemical potential change and Sj(t) Energy supplied from the field 1g 2u Energy supplied from the field 0.02  ()t toto the the natural natural orbital orbital j : : 2.4 fs t dtd () St()j  ( tdt )  0 j  dt 0 1u 1u 0.02  Stj() Spectator orbital 0

 Stj() 2g 1g 0.02 Energy accepting orbital 0 2g is an energy accepting orbital → efficient ionizing MO 0 50 100 150 3g 0.02

QuantificationQuantification ofof energyenergy 0 Net gain: exchange in the sub-fs domain exchange in the sub-fs domain 0 50 100 150 From the field: Stj() Decomposition of the chemical potential

 jj()t j ()tt (0) d j () ()t

 htjj () htjj (=0) d j () t (t ) one-electron 1 + jj   jj  () t jj jj (=0)t 2  two-electron  NO  CCkk +Re kjkj    ( t ) kjkj ( t 0)  CC**  kj jj Correlation Change for1 Correlation Energy

Induced dipole (atomic units) Induced dipole (atomic units) Time (atomic units) Time (atomic Time (atomic units) Time (atomic   = 380nm = 760nm 

g

Electric field field Electric Electric field field Electric (atomic units) (atomic (atomic units) (atomic

Energy (hartree) Energy (hartree) one-body two-body two-body one-body

  Time (atomicunits) Time (atomicunits) () t

11  gg  St () no energyexchange but Nonadiabatic

Change: Correlation energy   () t

1  g : Correlation energy Correlation Energy  ()t : 2g Change: Change for 2g  ()t St() = 760 nm 2g 2 g  two-body (hartree) (hartree)

one-body Energy Energy

Time (atomic units) Time (atomic units) = 380 nm

Energy exchange is dominated by the change intwo-body correlation energy! (hartree)

while the (hartree) Coulomb part of the mean field energy does not change significantly. Energy Energy one-body

Time (atomic units) Time (atomic units) Ionization dynamics in H2 at R =1.6 a0 (MCTDHF by T. Kato) 2 Pz(, z ) d d d (, z z ,  , ,) ReducedReduced density density map map ( (LogLog scale scale)) 12 1 2 12 12 1 2

One electron ionization

NonsequentialNonsequential (Rescattering)(Rescattering)

Sequential two-electron ionization =760 nm I =3.5×1014 W/cm2 Correlation in electron dynamics is well reproducedpeak

cf.cf. Dual Dual Transformation Transformation Method Method K. Harumiya et al. J. Chem. Phys. 113, Electric field Proton (atomic units) 8953(2000) ; K. Harumiya, H. Kono, Y. Fujimura, I. Kawata, A.D. Bandrauk, Phys. Rev. A 66, 043403 (2002). Spatial distribution of 11 Eikj ()()rr ()() rr  the correlation energy: C 2    12rr 12ij , k Correlation energy density ijikijkij 12 11 E ()rrrrrrik ()()j ()() d C 11212,2    ij k 2 ijikijkij rr12

再散乱軌道の特定法 iとkの相関エネルギー

r2

r1 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 potential Ionization of a natural orbital