YITP2011 Hirohiko Kono - Kyoto U
Transcript of YITP2011 Hirohiko Kono - Kyoto U
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