Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Advanced Laser and Photon Science レーザー・光量子科学特論
First principles simulations 第一原理計算 Takeshi Sato http://ishiken.free.fr/english/lecture.html
[email protected]
7/12 No. 1
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
1. Two electron systems 2. Second quan6za6on 3. Mul6configura6on 6me-dependent Hartree-Fock method
7/12 No. 2
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Time-dependent varia#onal principle S=
Z
t2 t1
ˆ h |(H
S = 0, for (t1 ) = Arbitrary Approximate
Ac#on integral
i@t )| i
!
0
=) =)
TDSE Variational EOMs
=
+
(t2 ) = 0
7/12 No. 3
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Example 1: Time-dependent Configura#on Interac#on Z t2 ˆ i@t )| i Ac#on integral S= h |(H t1
| (t)i = φM
X
Cn (t)|ni,
n
φ5 φ4 φ3 φ2 φ1 7/12 No. 4
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Example 1: Time-dependent Configura#on Interac#on Z t2 ˆ i@t )| i Ac#on integral S= h |(H t1
| (t)i =
X
Cn (t)|ni,
n
Configura#on Interac#on (CI) coefficients: Varia#onal parameters
S=
=
XZ
t2
n,m Zt1 X t2
n,m
t1
⇣
ˆ dt Cn⇤ Cm hn|H|mi
iCn⇤ hn|miC˙ m
ˆ dt Cn⇤ Cm hn|H|mi
iCn⇤ C˙ n
⇣
X S ˆ = hn| H|miC m Cn⇤ (t) m
iC˙ n =
X m
nm
iC˙ n = 0
⌘
⌘
ˆ hn|H|miC m 7/12 No. 5
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Example 2: Time-dependent Hartree-Fock
| (t)i = |11 12 · · · 1N 000 · ··i $ det [
1 2
···
N]
φM
Virtual
φ5 φ4 φ3
Occupied
φ2 φ1
7/12 No. 6
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Example 2: Time-dependent Hartree-Fock
| (t)i = |11 12 · · · 1N 000 · ··i $ det [ S=
Z
t2 t1
2
dt 4
1 2
Orbital func#ons: Varia#onal parameters
N ⇣ X i=1
S ⇤ (t) i
i
⌘
···
N]
3
⌘ From X ⇣ ij 1 hii ih i | ˙ i i + Vij Vjiij 5 Homework 2 ij (2) N ⇣ ⌘ X ˆ i i i+ ˆj i W ˆj j =h W j i N
j=1
i
ˆ =h
i
+
N ⇣ X
ˆj i W j
ˆj j W i
j=1
Wji (r1 ) =
Z
dx2
⇤ i (x2 ) j (x2 )
|r1
⌘
r2 | 7/12 No. 7
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) X | (t)i =
Cn (t)|ni,
n
TD Configura#on Interac#on (CI) with given number of moving orbitals φM
General Complete- orthonormal µ,
⌫,
, ··
φ5
Virtual
a,
b
Occupied
i,
j,
k,
l
φ4 φ3 φ2 φ1 7/12 No. 8
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) X | (t)i =
Cn (t)|ni,
n
Both CI coefficients & orbital func#ons: Varia#onal parameters
Working with Slater determinant is, in general, extremely tedious: Need techniques of second quan#za#on (1) Matrix (operator) exponen#al 1 X An exp(A) ⌘ n! n=0 exp(A)† = exp(A† ), B
1
exp(A)B = exp(B
1
AB)
exp(A + B) = exp(A) exp(B) ( [A, B] = 0
1 1 exp(A)B exp( A) = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + · · · 2! 3! Baker-Campbell-Hausdorff (BCH) expansion
7/12 No. 9
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) (2) Exponen#al parameteriza#on of unitary matrix (operator)
U: X:
U = exp(X) unitary U †U = U U † = 1 anti-Hermitian X† = X
An#-Hermi#an matrix can be parameterized more easily than unitary one
(3) Unitary transforma#on of orbitals X X µ (t) = ⌫ (0)U⌫µ = ⌫ (0) exp(X)⌫µ , ⌫ ⌫ X X () a†µ (t) = a†⌫ (0)U⌫µ = a†⌫ (0) exp(X)⌫µ ⌫ ⌫ X X ⇤ aµ (t) = a⌫ (0)U⌫µ = a†⌫ (0) exp(X)⇤⌫µ ⌫
⌫
7/12 No. 10
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) (3) Unitary transforma#on of orbitals X X µ (t) = ⌫ (0)U⌫µ = ⌫ (0) exp(X)⌫µ , ⌫ ⌫ X X () a†µ (t) = a†⌫ (0)U⌫µ = a†⌫ (0) exp(X)⌫µ ⌫ ⌫ X X ⇤ aµ (t) = a⌫ (0)U⌫µ = a†⌫ (0) exp(X)⇤⌫µ ⌫
⌫
ˆ †µ (0) exp( X) ˆ () a†µ (t) = exp(X)a ˆ µ (0) exp( X) ˆ aµ (t) = exp(X)a ˆ= X
X µ⌫
X⌫µ a†µ (0)a⌫ (0)
⌘
X
ˆµ⌫ (0) E
µ⌫
Proof: From BCH expansion 7/12 No. 11
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) (4) Unitary transforma#on of Slater determinants X X µ (t) = ⌫ (0)U⌫µ = ⌫ (0) exp(X)⌫µ , ⌫
⌫
ˆ †µ (0) exp( X) ˆ () a†µ (t) = exp(X)a ˆ µ (0) exp( X) ˆ aµ (t) = exp(X)a ˆ= X
X
X⌫µ a†µ (0)a⌫ (0)
µ⌫
⌘
X
ˆµ⌫ (0) E
µ⌫
†n2 †n3 1 |n(0)i = a†n (0)a (0)a 1 2 3 (0) · · · |i
†n2 †n3 1 |n(t)i = a†n (t)a (t)a 1 2 3 (t) · · · |i
ˆ = exp(X)|n(0)i
7/12 No. 12
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) (5) Unitary transforma#on of total wave func#on
| (t)i =
X
Cn (t)|n(t)i n X ˆ = exp(X) Cn (t)|n(0)i n
(6) Varia#on and Time deriva#ve of total wave func#on
|
X
X
ˆ⌫µ | (t)i Cn (t)|n(t)i + X⌫µ E n µ⌫ X X ˆ⌫µ | (t)i | ˙ (t)i = C˙ n (t)|n(t)i + X˙ ⌫µ E (t)i =
n
X˙ ⌫µ = h
˙
µ (t)| ⌫ (t)i
µ⌫
† ˆµ ⌘ a ˆ⌫ ) ˆ (E µa ⌫
7/12 No. 13
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) X | (t)i =
Cn (t)|ni,
n
Both CI coefficients & orbital func#ons: Varia#onal parameters Insert previous results into TDVP and require
S/ Cn⇤ (t) = S/ Xµ⌫ (t) = 0
iC˙ n = i
X
"
h
ˆ⌫µ |E
1
X
X
ˆ i hn| H m ! µ⌫ X ˆ | i h |nihn| E
ˆ⌫µ = h |E
n
1
X n
!
ˆ⌫µ X˙ ⌫µ E ˆ |E
ˆ i |nihn| H|
!
1
|miCm X n
ˆ h |H
1
General equa#ons of mo#on
!
#
ˆ⌫µ | i X˙ |nihn| E X n
!
ˆ⌫µ | i |nihn| E
7/12 No. 14
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Mul#configura#on TD Hartree-Fock (MCTDHF) X | (t)i =
Cn (t)|ni,
n
In case of complete CI expansion within the given orbitals
iC˙ n =
0
X m
0
ˆ hn| @H
ˆ ii + ˆ @h| i| ˙ i i = Q
occ X
˙
j | ii
ij
(D
jklm
ˆ=1 Q
Rij ⌘ ih
X
1
Eji Rji A |miCm
1
1 i ˆ lk | j iA )m Pjlmk W
Pocc j
|
j ih j |
+
occ X j
j iR j i
|
: Arbitrary Hermitian matrix
† † ik ˆ ik = a Dji = h |Eji | i, Pjlik = h |Ejl | i (E ˆ ˆk a ˆl a ˆj ) ia jl
One (D) and two (P) par#cle reduced density matrices
7/12 No. 15
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Importance of non-complete CI expansions φM
φM
Virtual
Virtual
Occupied φ5
Ac/ve
φ4
φ5 φ4
Dynamical-core
φ3 φ2
φ3
Occupied
φ2
Frozen-core
φ1
MCTDHF NDet = 784
φ1
TD-CASSCF (complete-ac#ve-space selfconsistent-field): core and ac#ve subspaces NDet = 36 7/12 No. 16
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
8 N < X 1 @2 H= : 2 @x2i i=1 0
Valence
Applicaions
M X
q a=1 (xi
Za
xi E(t)
2
Xa ) + c 0.12
9 = ;
+
N X i>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
-0.12 0.0
0.4 PW/cm2 750 nm 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 7/12 No. 17
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 -20 -40 -60
1
784
-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) 7/12 No. 18
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
Core is important for higher-order response 7/12 No. 19
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 10
8e DC+4e FC+4e
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
Core is important for higher-order response 7/12 No. 20
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Applicaions
Core Valence DC+4e: Net
intensity (a.u.)
10-2 10-4 10-6 10-8 10-10 10-12
FT of h 0
20
¨ (t)i (t)|x|
40 60 harmonic order
80
100
Core and Valence contribu#ons: Deeper physical understanding 7/12 No. 21
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
Report Submission due: July 31. Place to submit: the office of the Nuclear Engineering & Management, 2nd floor of the Bldg. 3. Language: English or Japanese.
(1) MCTDHF includes single determinant TDHF as a special case. Derive the TDHF equations of motion (given in p. 7) starting from the MCTDHF equations (p. 15) by ignoring CI equations and inserting HF wave function,
| i = |11 12 13 · · · 1N 0000i
in the definition of one and two particle reduced density matrices. Here N is the number of electrons. The resultant equations will still look different from those in p. 7. Choose the appropriate Hermitian matrix R in order to obtain exactly the same equations as those in p. 7.
7/12 No. 22
Advanced Laser and Photon Science (Takeshi SATO) for internal use only (Univ. of Tokyo)
(2) Derive the transformation (Expressions for A1 , A2 , 1 , 2 below) from GVB wave function to the MCTDHF wave function for the two-electron singlet system, and explicitly show that MCTDHF orbitals are orthonormal. Assume that GVB orbitals are normalized. See J. Phys. B: At. Mol. Opt. Phys. 47 , 204031 (2014). 1 GVB (r1 , r2 )
+ 1
A1
2
+
2
1
=
+ A2
i= 2A21, 1h 1 |+2 A 2 02 2
= p [ 1 (r1 ) 2 (r2 ) + 2 (r1 ) 1 (r2 )] 2 = A1 1 (r1 ) 1 (r2 ) + A2 2 (r1 ) 2 (r2 )
⇤ S12 A1 = p , 2 |S | C1 1 1 +2(1C+3|S{12 | 1) 212 + 1 |S12 | S12 A2 = p , 2 |S | 2(1 + |S12 | ) 12
1 + |S12 |
1
2
⇢
1 S12 =p 2(1 + |S12 |) |S12 | ⇢ ⇤ 1 S12 p = 2(1 + |S12 |) |S12 |
S12 = h
1| 2i
2 1}
1
2
+
2
1
7/12 No. 23
