Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Advanced Laser and Photon Science E
First principles simulations (1) Takeshi Sato http://ishiken.free.fr/english/lecture.html
[email protected]
2018/7/3 No. 1
RESEARCH ARTICLES
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo) bars are reduced at higher intensities, primarily because of the increased spectral shift. Theoretical discussion. First, we show that the measured delay of ~20 as cannot be explained by a delayed onset of streaking, which was the dominant effect in (17). The streaking NIR field may be significantly screened by bound electrons at small distances from the nucleus. After the absorption of an XUV photon, it takes the positive-energy electron a finite time to leave this screened volume, and this time interval may be different for electrons originating from different orbitals. However, for an atom, this difference cannot exceed a few attoseconds. The characteristic scales can be extracted from the classical trajectories shown in Fig. 1B. If we assume that the 2s and 2p electrons are set in motion at the same moment, their classical
High-field & Ultrafast science Ultrafast charge migration
trajectories would acquire a relative delay of 20 as after traveling over 5 Å, whereas significant screening from the streaking field is limited to a distance of less than 1 Å from the nucleus. Furthermore, if screening played a dominant role, the faster 2p electrons would be exposed to the streaking field earlier than the slower 2s ones, whereas measurements and quantum simulations show that the slower electron is emitted first. Now we turn our attention to the quantummechanical description. First of all, we need a definition for the photoemission delay. Consider a photoelectron wave function jyðtÞ〉 created by an XUV pulse centered at t ¼ 0. The motion of the wave packet after photoionization is conveniently described in a basis of continuum states je〉, each of which has a well-defined energy e and describes a wave that propagates in the di-
Attosecond science
Tunneling ionization recombination
Calegari et al, 2014
Schultze et al, 2010
Nonsequential double ionizatoin High-order harmonic generation
rection of th ability ampli 〈ejyðtÞ〉 ¼ c function cðe the wave pac in photoemis tron’s trajecto of cðeÞ. It is group delay et, in accorda d aðeÞ ¼ ℏ de a tions, we av the XUV pu As the fir idate the exp one expects wave packet shift of the st
Fig. 2. Attosecond streaking spectrograms (A and B), evaluated photoelectron wave packets (C), and streaked spectra (D). The spectrograms in (A) are composed of a series of photoelectron energy spectra recorded by releasing 2s and 2p electrons from Ne with an attosecond XUV pulse in the presence of a strong NIR few-cycle laser field, as a function of the delay between the XUV and NIR fields. The spectrogram is processed with a FROG algorithm tailored for streaking measurements (30). (B) shows the spectrogram reconstructed by this algorithm.
1660
The retrieved 2s and 2p spectra, together plotted in (C) (black solid line and red dotte energy spectra are in excellent agreement line). The average difference between the retardation of the 2p emission with resp reconstructed and measured streaked spe largest positive and negative shifts of the
Kaldun et al, 2016, Ossiander et al, 2017 25 JUNE 2010
VOL 328
SCIENCE
www.sciencemag.org
2
Intensity (arb. unit)
10
cutoff
1
10
0
10
Molecular orbital tomography
-1
10
-2
10
-3
10
-4
10
-5
10
plateau
-6
10
-7
10
-8
10
Watson et al, 1997
0
10
20
30
Harmonic order
40
50
Itatani et al, 2004 2018/7/3 No. 2
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Time-dependent Schrödinger Equation (TDSE)
@ ˆ (~r1 , ~r2 , ~r3 , · · ·, ~rN , t) = i H (~r1 , ~r2 , ~r3 , · · ·, ~rN , t) @t ˆ H(t) =
N X
ˆ r i , t) + h(r
i=1
ˆ r , t) = h(r
1 ˆ r h(r , t) = 2
N X X i=1 j>i
1 2 r 2
M X
1 |rr i
rj|
ZA + E (t) · r |rr R A | A=1 M X ZA 2 r A ir + (t) |rr R A |
or
A=1
First principles simulations 2018/7/3 No. 3
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Hydrogen atom i ˙ (~r, t) = φM
・ ・ ・
5 2s,φ2p
continuum
r2 2
1 + zE(t) r
Fixed orbital expansion?
(t) = C1s (t)
iCI (t) =
1s
X
˙Virtual
bound
(~r, t)
J
+ C2s (t)
{✏I
IJ
2s
+ C2p (t)
2p
+···
+ E(t)zIJ } CJ (t)
Occupied Original PDE is transformed into ODE Difficult to include highly-excited/continuum states
φ4 1s φ3 φ2
Directly numerical simulation of TDSE has been proved more advantageous for strong-field phenomena
φ1
TDSE 2018/7/3 No. 4
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Helium atom i ˙ (~r1 , ~r2 , t) =
r21 2
r22 2
2 r1
2 1 + (z1 + z2 )E(t) + r2 r12
(~r1 , ~r2 , t)
1/r12 Z/|r2 |
Z/|r1 |
He atom: already the limit of direct TDSE simulation He
To solve many-electron TDSE within a reasonable approximation 2018/7/3 No. 5
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
First principles simulations ü Time-dependent Z density functional theory (TDDFT) ⇢(rr 1 ) = N drr 2 · · · drr N | (rr 1 , r 2 , · · · r N )|2
Fast. Atoms, molecules, clusters, and solids 計算コストが低い、原子から固体まで Accuracy NOT systematically improvable 計算精度を系統的に向上できない Difficult to treat ionization process イオン化過程の記述が困難 Difficult to extract observables in general 一般の物理量の計算が困難
ü Time-dependent wavefunction theory
(rr 1 , r 2 , · · · r N )
Demanding 計算コストが高い Systematically improvable accuracy 計算精度を系統的に向上できる Can properly describe excitation/ionization process イオン化過程を記述できる Can access to observables 任意の物理量を計算できる
ü Time-dependent reduced density matrix theory ü Time-dependent R-matrix-based approaches
2018/7/3 No. 6
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Time-dependent density functional theory (TDDFT) Electron density as a basic variable, not total wave function 全電子波動関数ではなく電子密度を基本的な変数とする
Exact in principle, approximate in practical 原理的には厳密、実際は近似
N/2
⇢(rr , t) = 2 Electron density
i ˙ i (rr ) =
⇢
X i
| i (rr , t)|2
Kohn-Sham orbital Exchange-correlation potential
ZHartree potential 0 r ⇢(r ) 0 XC ˆ h(rr , t) + drr + V [⇢](rr ) 0 |rr r |
TDDFT equation
r) i (r
2018/7/3 No. 7
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Time-dependent wavefunction theory (rr 1 , r 2 , · · · r N )
2018/7/3 No. 8
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(1) Time-dependent Hartree (TDH) x1 , x 2 , · · · x N ) TDH (x
=
Total wavefunction as a (Hartree) product of orbitals x = {r, σ} r: spatial coordinate σ: spin coordinate
i ˙ i (rr ) =
⇢
x1 ) 2 (x x2 ) · · · 1 (x
x) 2i 1 (x
= x) = 2i (x
ˆ r , t) + h(r
Z
0 r ⇢(r ) 0 drr |rr r 0 |
xN ) N (x
r )↵( i (r
) r) ( ) i (r
r) i (r
TD Hartree equation
Violate Pauli’s antisymmetricity principle
2018/7/3 No. 9
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(2) Time-dependent Hartree-Fock (TDHF) 2
6 6 6 1 x1 , x 2 , · · · x N ) = det 6 TDHF (x 6 N! 6 4
⌘
1
2
···
x1 ) 1 (x
x2 ) 1 (x
x1 ) 2 (x
x2 ) 2 (x
· · · x1 ) N (x
· · · x2 ) N (x
··· ··· · · · ···
xN ) 1 (x xN ) 2 (x · · · xN ) N (x
N
3 7 7 7 7 7 7 5
Single Slater determinant
i ˙ i (rr ) =
⇢
ˆ r , t) + h(r
Z
r0) 0 ⇢(r drr |rr r 0 |
r) i (r
N/2 Z
X j=1
drr 0
⇤ r0 ) i (rr 0 ) j (r |rr r 0 |
r) j (r
Exchange interaction
TD Hartree-Fock equation 2018/7/3 No. 10
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(3) Multiconfiguration TDHF (MCTDHF) Superposition of 多数のスレーター行列式 （多配置） Many Slater determinant の線型結合 (Multi Configurations) X X r 1 , rx22,, ·· ·· ··rxNN)) = (rx Cij···k i j · · · k ⌘ CI = MCTDHF(x i,j,··· ,k TDHF +STDHF +SD +S +SDT +SD
I
I
+SDT
11 10
9 8
7 6
occupied
= occupied C0 +C1= C0 +C2+C1 +C3+C2 +C4+C3 +…+C4
+…
5 4
3
2 1
2018/7/3 No. 11
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
TDHF (and TDDFT) cannot appropriately describe strong ionization process. Why?
2018/7/3 No. 12
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Singlet two electron system (e.g, He)
Spatial part of TDHF wavefunction HF (r1 , r2 )
i ˙ 1 (rr ) =
⇢
=
ˆ r , t) + h(r
Z
1 (r1 ) 1 (r2 ) 0 2 r (r )| 1 drr |rr r 0 | 0|
r) 1 (r
TDHF and TDDFT, with only 1 orbitals for 2 electrons, CANNOT describe bound and ionized electrons 1
2
TDHF 2018/7/3 No. 13
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Singlet two electron system (e.g, He)
At least two spatial orbitals are required 2 GVB (r1 , r2 )
/
1 (r1 ) 2 (r2 )
+
2 (r1 ) 1 (r2 )
Generalized Valence Bond (Chemistry) Extended Hartree-Fock (Physics)
S12 ⌘ h
1| 2i
dS12 /dt 6= 0
= 6 0 2018/7/3 No. 14
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Singlet two electron system (e.g, He)
0.2
Field
40
0.1
20
0.0
0
-0.1
electric field (a.u.)
TDHF
dipole (a.u.)
60
TD-GVB h 2 |z| 2 i TD-GVB h 1 |z| 1 i
-20 -0.2 0
1 2 3 4 5 time (optical cycle)
6
Dipole moment (mean position)
2018/7/3 No. 15
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Singlet two electron system (e.g, He)
TD-GVB GVB (r1 , r2 )
/
1 (r1 ) 2 (r2 )
S12 ⌘ h
1| 2i
+
2 (r1 ) 1 (r2 )
= 6 0
Good for N = 2, but Too complicated for N > 2
2018/7/3 No. 16
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Orbital redundancy Linear transformation of orbitals, that leaves total wavefunction invariant.
1 GVB (r1 , r2 ) = p [ 1 (r1 ) 2 (r2 ) + 2 (r1 ) 1 (r2 )] 2 = A1 1 (r1 ) 1 (r2 ) + A2 2 (r1 ) 2 (r2 )
+ 1
2
+
2
1
= C1
1 1
Equivalent!
+ A2
A1
2 2,
h
1| 2i
=0
+ C3 1{+ |S1 |
2
+
⇤ S12 A1 = p , 2 |S | 2(1 + |S12 | ) 12 12
2 1}
1 |S12 | S12 A2 = p , 2 |S | 2(1 + |S12 | ) 12 1
2
⇢
1 S12 =p 2(1 + |S12 |) |S12 | ⇢ ⇤ 1 S12 =p 2(1 + |S12 |) |S12 |
1
2
+
2
1
2018/7/3 No. 17
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Orbital redundancy All Equivalent
1
2
+
+ B3
B1
+
+
+0 2 1 } 1 C 1+ 2C11 = 1 C 12 12+2 ,C3h {1 | 1 2 i2 = MCTDHF
+ A2
A1
2 2,
h
i= 1 | 2C 1 0 1 1
C1
+ C2
+ C2 2 2,
h
+ C3 1| 2i
+
=0
May be based on the most convenient ansatz2018/7/3 No. 18
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
|h (0)| (t)i|2
1.0
TDSE GVB APSG(2) MCTDHF
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
TDSE GVB APSG(2) MCTDHF
30
dipole moment / au
ground state probability
h (t)|(x1 +x2 )| (t)i
20 10 0 -10 -20
0
(t) =
1
P2
ij
2 3 4 5 6 time / optical cycle
Cij (t) i (1, t)
7
j (2, t)
8
/
0
1
2 3 4 5 6 time / optical cycle
1 (1, t) 2 (2, t)
+
7
8
2 (1, t) 1 (2, t)
MCTDHF with two orbitals and TD-GVB are equivalent, with MCTDHF being much simpler 2018/7/3 No. 19
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
MCTDHF method for two electron systems (t) =
P2
ij
Cij (t) i (1, t)
j (2, t)
/
1 (1, t) 2 (2, t)
+
2 (1, t) 1 (2, t)
ü Propagate both CI coefficients and orbitals ü Can improve accuracy by increasing orbitals n X + C + C + C4 (r1 , r2 , t)(r =1 , r2 )C=ijC(t) 1 i (r1 ,2t) j (r23, t) ij
(r1 , r2 ) = C1
+ C2
TDHF
+ C3
+ C4
+ C5 + C6 TD-GVB
+ C5
+ C6
+ …..
+ ….. 2018/7/3 No. 20
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
PRL, 78, 1884 (1997)
Experiment: “Shoulder” region, or “Knee” region Simulation: with electron-electron correlation neglected
3
2
1
Sequen(al Nonsequen(al
0
-1
-2
2018/7/3 No. 21
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
100
-1
10
10-2
TDSE He+ HF: N = 1 GVB: N = 2 N=4 N=8 N = 16 N = 28
Accuracy can be improved systematically by increasing orbitals
10-3
10-4
-5
10
1.0 × 1015 2 Intensity (W/cm )
(r1 , r2 , t) =
n X
Cij (t) i (r1 , t)
j (r2 , t)
ij
MCTDHF is a powerful tool for multielectron dynamics
2018/7/3 No. 22
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(3) Multiconfiguration TDHF (MCTDHF) Superposition of Many TD Slater determinants r 1 , rx22,, ·· ·· ··rxNN)) = (rx = MCTDHF(x TDHF
X
i,j,··· ,k
Cij···k
i
j
···
+STDHF +SD +S +SDT +SD
k
⌘
X
CI
I
I
+SDT
11 10
9 8
7 6
occupied
= occupied C0 +C1= C0 +C2+C1 +C3+C2 +C4+C3 +…+C4
+…
5 4
3
2 1
2018/7/3 No. 23
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(3) Multiconfiguration TDHF (MCTDHF) Problem:
Factorial cost scaling w.r.t number of electrons 計算コストが電子数の階乗で増加
r 1 , rx22,, ·· ·· ··rxNN)) = (rx = MCTDHF(x TDHF
X
i,j,··· ,k
Cij···k
i
j
···
+STDHF +SD +S +SDT +SD
k
⌘
X
CI
I
I
+SDT
11 10
9 8
7 6
occupied
= occupied C0 +C1= C0 +C2+C1 +C3+C2 +C4+C3 +…+C4
+…
5 4
3
2 1
2018/7/3 No. 24
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(4) More flexible methods TS and K. L. Ishikawa, Phys. Rev. A, 88, 023402 (2013): core & active: TD-CASSCF TS and K. L. Ishikawa, Phys. Rev. A, 91, 023417 (2015): occupation restriction: TD-ORMAS
Flexibly classifying electrons into active , dynamical and frozen core TDHF core,+S +SD +SDT
Active
= C0
+C1
+C2
+C3
+C4
+…
Dynamical core Frozen core
Less demanding (yet accurate), allowing deeper physical insight than fully correlated MCTDHF
2018/7/3 No. 25
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Deriving Equations of motion X X
r1, r 2, · · · r N ) MCTDHF (r
=
Cij···k
i
i,j,··· ,k
j
···
k
⌘
CI
I
I
Time-dependent ✓ variational ◆ principle Z ˆ i@ | i dth | H @t ✓ ◆ ˆ i @ | i + c.c. = 0 | H @t
S[ ] =
S=h
Equations of motion iC˙ I =
X J
h
ˆ
I |H|
2
J iCJ
Field free one-electron terms Electron-laser interaction Z Electron-electron ⇤ interaction 0 0 X
) l (rr ) jl r (r )P j mk (D |rr 3 r 0 | jklm ⇤ r0 0 X r (r ) (r ) l jl j 1 m5 k r r (r )P (D ) + (r )R j j i i mk |rr r 0 | j
ˆ 4{h0 + Vext (t)} i (rr ) + i ˙ i (rr ) = Q
r) ext (t)} i (r
+
XZ
jklm
drr 0
drr 0
r k (r
1 m )i
2018/7/3 No. 26
Z
Advanced Laser and Photon Science grid (Takeshi SATO) for internal use only (Univ. of Tokyo)
X
dxf (x) ⇡ x f (xi ) Real-space implementation i V (x)f (x) ! V (xi )f (xi )
Local Potential
@2 f (xi f (x) ! @x2
Kinetic energy
1)
2f (xi ) + f (xi+1 ) x2 Spherical coordinate for atoms
p (t) !
X
cp,klm (t)
klm
fklm (r) Ylm (✓, ) r
(a)
Multiresolution cartesian grids divide
p (t)
Δxi
divide
6 (b)
6
fi-
y (arb. units)
4
f(xi)
(b) (c)
fi+
f(xi-1)
2
f(xi+1) Δxi
0
y (arb. units)
(a)
6
la
4
lb
-6 -6
-6 -6
4
6
p (t)
cell i
-4
-2 0 2 x (arb. units)
Curvilinear coordinate
fi
0
-4
-4
cell i+1
2
-2
p (xi , yj , zk , t)
fi+1 fi+
-2
!
r -4
-2
0 2 x (arb. units)
4
6
FIG. 1. Illustration of local adaptation around the 12 atoms in the benzene molecule (denoted by the grey circles) using the transformation
! ¯p (⇠i , ⇠j , ⇠k , t),
¯p (⇠⇠ , t) = |J(⇠⇠ )|1/2
r (⇠⇠ , t)) p (r
2018/7/3 No. 27
d ˆ % ψp , ∂ ∂ 1 ∂ ψq %% i −Advanced h % Laser and (4) Photon Science (Takeshi for internal=use only (Univ. (11) of Tokyo) r→ R(r), SATO)→ dt ∂r ∂R(r) q(r) ∂r ' ψr∗ (r⃗2 )ψs (r⃗2 ) r Ws (r⃗1 ) = dr⃗2 (5) |r⃗1 − r⃗2 | Rpq =
Absorbing boundary condition 4.
TD-CASSCF
Mask function, Complex absorbing potential (CAP), Exterior complex scaling (ECS), etc TD-CASSCF 3. Im % &
1 in irECS radial coordinate % &
(3)
scaled region (absorption)
1
!
unscaled region
O
%)
Re % &
i
ordinary radial coordinate
( r r → R(r) = R0 + (r − R0 )eiη
(r < R0 ) (r ≥ R0 )
(6)
0.5 0.5
1#$2 Real part of eigenfunction
! Real part of eigenfunction
Real part of eigenfunction
1.0 1
"#$0 r ≥!外向波 R ! "#$ 外向波
1 0.5
1#$42 ! "#$
⋅!
"#$4 ⋅5!= 57
-0.5 -1
20 0
30 10
40 20
50 30
60 40
70 50
radial coordinate (a.u.) radial coordinate (a.u.)
80 60
70
80
0
-0.5 -1
0
10
"
oqsr
q
qs ˆ ˆ sr U ˆ −1 U ˆ (D−1 )op Por UηR0 W ηR0 ηR0 |ψq ⟩
+
5= + 95:57 + 95:
0.5
-0.5
;
j
q
1 2
(xi
xj ) + d
7.5 fs 0.08
-2
field amplitude
Orbital energy / Hartree
-1
Core
0.04 0.00 -0.04
-3
-4 -15
orbital 1 orbital 2 orbital 3 orbital 4 nuclear -10
-5
0 x / bohr
5
10
-0.08
15
Ground-state
0.4 PW/cm2 750 nm
-0.12 0.0
0.5
Field
1.0 1.5 2.0 time / optical cycle
2.5
3.0
1D “LiH dimer” 4 valence and 4 core electrons 2018/7/3 No. 30
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Applicaions 60
49
dipole moment / au
40
h (t)|x| (t)i
20 0
-40 -60
1
784
-20
-80 0.0
CAS(8e) CAS(4e) CAS(2e) HF 0.5
1.0 1.5 2.0 2.5 time / optical cycle
3.0
44100
TD-CASSCF(4e, 8a) reproduces MCTDHF(8o, 10o) 2018/7/3 No. 31
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Applicaions
dipole acceleration / au
0.8
8e DC+4e FC+4e
0.4
0.0
-0.4
¨ (t)i h (t)|x|
-0.8 0.0
0.5
1.0 1.5 2.0 2.5 time / optical cycle
“Dynamical” or “Frozen” 3.0
2018/7/3 No. 32
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Applicaions
intensity (a.u.)
Cutoff 10-2 3-step model (Koopmans) -4
8e DC+4e FC+4e
10
10-6 10-8 10-10 10-12
FT of h 0
20
“Dynamical” or “Frozen”
¨ (t)i (t)|x|
40 60 harmonic order
80
100
Elucidating roles of valence and core dynamics
2018/7/3 No. 33
