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The Multiconfiguration Time-Dependent Hartree(MCTDH) Method and its Multi-Layer
(ML-MCTDH) Extension
Hans-Dieter Meyer
Theoretische ChemieUniversitat Heidelberg
Quantum Days, Bilbao, July 13/14, 2015
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Outline
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Content
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Multiconfiguration time-dependent Hartree, MCTDH
The ansatz for the MCTDH wavefunction reads
Ψ(q1, · · · , qf , t) =
n1∑j1=1
· · ·nf∑
jf =1
Aj1,··· ,jf (t)f∏
κ=1
ϕ(κ)jκ
(qκ, t)
=∑J
AJ ΦJ
Single-particle functions:
ϕ(κ)jκ
(qκ, t) =Nκ∑l=1
c(κ)jκl
(t) χ(κ)l (qκ)
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Multiconfiguration time-dependent Hartree, MCTDH
The ansatz for the MCTDH wavefunction reads
Ψ(q1, · · · , qf , t) =
n1∑j1=1
· · ·nf∑
jf =1
Aj1,··· ,jf (t)f∏
κ=1
ϕ(κ)jκ
(qκ, t)
=∑J
AJ ΦJ
Single-particle functions:
ϕ(κ)jκ
(qκ, t) =Nκ∑l=1
c(κ)jκl
(t) χ(κ)l (qκ)
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MCTDH equations of motion
MCTDH equations of motion:
i AJ =∑L
〈ΦJ |H|ΦL〉AL
iϕ(κ)j =
(1− P(κ)
)∑k,l
ρ(κ)−1
j ,k 〈H〉(κ)k,l ϕ(κ)l
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Improved Relaxation
Time-independent Schrodinger equation
Applying a variational principle leads to an eigenvalue problem forthe coefficients ∑
L
〈ΦJ |H|ΦL〉AL = E AJ
and a relaxation procedure for the single-particle functions
∂
∂τϕ(κ)j := −
(1− P(κ)
) nκ∑k,l=1
(ρ(κ)
)−1
jk
⟨H⟩(κ)klϕ(κ)l → 0.
Equations must be fulfilled simultaneously
Start with a guess wavefunction
Solve iteratively until self-consistency (”Improved relaxation”)
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MCTDH with Mode Combination
(q1, q2︸ ︷︷ ︸, q3, q4, q5︸ ︷︷ ︸, q6︸︷︷︸, · · · , qf−1, qf︸ ︷︷ ︸)Q1 Q2 Q3 · · · Qp
MCTDH wavefunction
Ψ(q1, · · · , qf , t) ≡ Ψ(Q1, · · · ,Qp, t)
=
n1∑j1
· · ·np∑jp
Aj1,··· ,jp(t)
p∏κ=1
ϕ(κ)jκ
(Qκ, t)
Single-particle functions:
ϕ(κ)jκ
(Qκ, t) =
N1,κ∑l1=1
· · ·Nd,κ∑ld=1
c(κ)jκl1···ld (t) χ
(κ)l1
(q1,κ) · · ·χ(κ)ld
(qd ,κ)
Exponential Scaling:
Standard : N f , MCTDH : nf , combined :(n1/d
)f
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MCTDH with Mode Combination
(q1, q2︸ ︷︷ ︸, q3, q4, q5︸ ︷︷ ︸, q6︸︷︷︸, · · · , qf−1, qf︸ ︷︷ ︸)Q1 Q2 Q3 · · · Qp
MCTDH wavefunction
Ψ(q1, · · · , qf , t) ≡ Ψ(Q1, · · · ,Qp, t)
=
n1∑j1
· · ·np∑jp
Aj1,··· ,jp(t)
p∏κ=1
ϕ(κ)jκ
(Qκ, t)
Single-particle functions:
ϕ(κ)jκ
(Qκ, t) =
N1,κ∑l1=1
· · ·Nd,κ∑ld=1
c(κ)jκl1···ld (t) χ
(κ)l1
(q1,κ) · · ·χ(κ)ld
(qd ,κ)
Exponential Scaling:
Standard : N f , MCTDH : nf , combined :(n1/d
)f
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MCTDH with Mode Combination
(q1, q2︸ ︷︷ ︸, q3, q4, q5︸ ︷︷ ︸, q6︸︷︷︸, · · · , qf−1, qf︸ ︷︷ ︸)Q1 Q2 Q3 · · · Qp
MCTDH wavefunction
Ψ(q1, · · · , qf , t) ≡ Ψ(Q1, · · · ,Qp, t)
=
n1∑j1
· · ·np∑jp
Aj1,··· ,jp(t)
p∏κ=1
ϕ(κ)jκ
(Qκ, t)
Single-particle functions:
ϕ(κ)jκ
(Qκ, t) =
N1,κ∑l1=1
· · ·Nd,κ∑ld=1
c(κ)jκl1···ld (t) χ
(κ)l1
(q1,κ) · · ·χ(κ)ld
(qd ,κ)
Exponential Scaling:
Standard : N f , MCTDH : nf , combined :(n1/d
)f
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MCTDH with Mode Combination
(q1, q2︸ ︷︷ ︸, q3, q4, q5︸ ︷︷ ︸, q6︸︷︷︸, · · · , qf−1, qf︸ ︷︷ ︸)Q1 Q2 Q3 · · · Qp
MCTDH wavefunction
Ψ(q1, · · · , qf , t) ≡ Ψ(Q1, · · · ,Qp, t)
=
n1∑j1
· · ·np∑jp
Aj1,··· ,jp(t)
p∏κ=1
ϕ(κ)jκ
(Qκ, t)
Single-particle functions:
ϕ(κ)jκ
(Qκ, t) =
N1,κ∑l1=1
· · ·Nd,κ∑ld=1
c(κ)jκl1···ld (t) χ
(κ)l1
(q1,κ) · · ·χ(κ)ld
(qd ,κ)
Exponential Scaling:
Standard : N f , MCTDH : nf , combined :(n1/d
)f
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Content
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Multi-Layer MCTDH
Mode-combination has proved to be very helpful
But mode-combination orders larger than 3 or 4 makethe propagation of the SPFs infeasible
Use MCTDH to propagate the SPFs of an underlyingMCTDH calculation
H. Wang and M. Thoss J.Chem.Phys. 119 (2003), 1289.
U. Manthe J.Chem.Phys. 128 (2008), 164116.
O. Vendrell and H.-D. Meyer J.Chem.Phys. 134 (2011), 044135.
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ML-MCTDH expansion of wavefunction
Ψ(Q11 , . . . ,Q
1p) =
n11∑j1=1
· · ·n1p∑
jp=1
A11; j1,...,jp
p∏κ1=1
ϕ(1;κ1)jκ1
(Q1κ1)
ϕ(1;κ1)m (Q1
κ1) =
n21∑j1=1
· · ·n2pκ1∑jpκ1
A2;κ1m; j1,...,jpκ1
pκ1∏κ2=1
ϕ(2;κ1,κ2)jκ2
(Q2;κ1κ2 )
ϕ(2;κ1,κ2)m (Q2;κ1
κ2︸ ︷︷ ︸) =Nα∑j=1
A3;κ1,κ2m;j χ
(α)j (qα)
qα
Q`;κ1,··· ,κ`−1κ` = {Q`+1;κ1,··· ,κ`
1 , . . . ,Q`+1;κ1,··· ,κ`pκ`
}
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ML-MCTDH expansion of wavefunction
Ψ(Q11 , . . . ,Q
1p) =
n11∑j1=1
· · ·n1p∑
jp=1
A11; j1,...,jp
p∏κ1=1
ϕ(1;κ1)jκ1
(Q1κ1)
ϕ(1;κ1)m (Q1
κ1) =
n21∑j1=1
· · ·n2pκ1∑jpκ1
A2;κ1m; j1,...,jpκ1
pκ1∏κ2=1
ϕ(2;κ1,κ2)jκ2
(Q2;κ1κ2 )
ϕ(2;κ1,κ2)m (Q2;κ1
κ2︸ ︷︷ ︸) =Nα∑j=1
A3;κ1,κ2m;j χ
(α)j (qα)
qα
Q`;κ1,··· ,κ`−1κ` = {Q`+1;κ1,··· ,κ`
1 , . . . ,Q`+1;κ1,··· ,κ`pκ`
}
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ML-MCTDH expansion of wavefunction
Ψ(Q11 , . . . ,Q
1p) =
n11∑j1=1
· · ·n1p∑
jp=1
A11; j1,...,jp
p∏κ1=1
ϕ(1;κ1)jκ1
(Q1κ1)
ϕ(1;κ1)m (Q1
κ1) =
n21∑j1=1
· · ·n2pκ1∑jpκ1
A2;κ1m; j1,...,jpκ1
pκ1∏κ2=1
ϕ(2;κ1,κ2)jκ2
(Q2;κ1κ2 )
ϕ(2;κ1,κ2)m (Q2;κ1
κ2︸ ︷︷ ︸) =Nα∑j=1
A3;κ1,κ2m;j χ
(α)j (qα)
qα
Q`;κ1,··· ,κ`−1κ` = {Q`+1;κ1,··· ,κ`
1 , . . . ,Q`+1;κ1,··· ,κ`pκ`
}
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Standard Method and MCTDH trees
q1 q2 q3 q4 q5 q6
StandardMethod
q1 q2 q3 q4 q5 q6
MCTDH
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MCTDH and ML-MCTDH trees
q1 q2 q3 q4 q5 q6
MCTDHcombined
q1 q2 q3 q4 q5 q6
ML-MCTDH
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ML-MCTDH tree for naphthalene (48D)
ν∗1
10
ν∗29
10
9
ν∗2
8
ν∗18
8
9
ν∗6
17
ν40
8
9
8
ν∗4
6
ν∗39
9
12
ν∗5
7
ν∗7
12
12
ν∗41
12
ν∗42
18
12
8
ν∗3
7
ν38
8
9
ν∗10
8
ν∗25
9
9
ν∗17
8
ν∗37
9
9
8
8
ν∗11
11
ν∗14
6
4
ν21
8
ν∗22
9
4
ν∗45
9
ν∗46
8
4
4
ν∗12
7
ν∗26
8
4
ν13
8
ν28
12
4
ν15
8
ν∗16
7
ν∗27
9
4
ν∗19
11
ν20
7
4
4
ν30
9
ν31
10
ν∗32
9
4
ν33
10
ν∗34
9
4
4
4
ν8
9
ν43
9
3
ν9
8
ν44
8
3
ν47
9
ν48
8
3
3
ν23
8
ν24
7
3
ν35
11
ν36
8
3
3
4
8
6 6electronicvibration
system
bath
elq
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PE-spectrum of naphthalene (48D) Q. Meng
8.0 9.0 10.0 11.0 12.0
8.0 9.0 10.0 11.0
Photon Energy, eV
Inte
nsi
ty, A
rbitr
ary
un
i tsIn
ten
sity
, Arb
itra
ry u
ni ts
0
200
X 2Au
A 2B3u
B 2B2g
C 2B1g
~ D 2Ag ~ E 2B
3g
Gas-phase photoelectron spectrum
X 2Au
A 2B3u
B 2B2g
C 2B1g
~ D 2Ag
Theoretical spectrum
48D ML-MCTDH
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Problems studied with the Heidelberg ML-MCTDHpackage
Henon-Heiles: 6D, 18D, 1458D
Pyrazine: 24D, 2E
Difluorobenzene cation: 30D, 5E
Naphtalene cation: 48D, 6E
Antracene cations: 66D, 6E
Formaldehyde Oxide: 9D, 5E
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ML-Conclusions
ConclusionsML-MCTDH
ML-MCTDH is capable to treat very large systems withhundreds of degrees of freedom.
ML-MCTDH is very suitable for studying system/bathproblems.
ML-MCTDH is most useful when using model Hamiltonians.However, model Hamiltonians like the VC-Hamiltonian can bevery helpful to investigate real chemical systems.
ML-MCTDH is very fast in a low accuracy mode butmay become costly if a high accuracy is asked for.
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Content
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Expansion of coefficients
Standard Method
Ψ(q1, · · · , qf ) =
N1∑i1
· · ·Nf∑if
Ψi1,··· ,if χ(1)i1
(q1) · · ·χ(f )if
(qf )
MCTDHΨi1,··· ,if =
∑j1,··· ,jf
Aj1,··· ,jf c(1)j1,i1· · · c(f )jf ,if
MCTDH combined
Ψi1,··· ,if =∑
j1,··· ,jp
Aj1,··· ,jp c(1)j1,i1···id · · · c
(p)jp ,i..···if
MCTDH is a decomposition of the wave-function tensor into a(time-dependent) Tucker form!
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Expansion of coefficients
Standard Method
Ψ(q1, · · · , qf ) =
N1∑i1
· · ·Nf∑if
Ψi1,··· ,if χ(1)i1
(q1) · · ·χ(f )if
(qf )
MCTDHΨi1,··· ,if =
∑j1,··· ,jf
Aj1,··· ,jf c(1)j1,i1· · · c(f )jf ,if
MCTDH combined
Ψi1,··· ,if =∑
j1,··· ,jp
Aj1,··· ,jp c(1)j1,i1···id · · · c
(p)jp ,i..···if
MCTDH is a decomposition of the wave-function tensor into a(time-dependent) Tucker form!
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Expansion of coefficients
Standard Method
Ψ(q1, · · · , qf ) =
N1∑i1
· · ·Nf∑if
Ψi1,··· ,if χ(1)i1
(q1) · · ·χ(f )if
(qf )
MCTDHΨi1,··· ,if =
∑j1,··· ,jf
Aj1,··· ,jf c(1)j1,i1· · · c(f )jf ,if
MCTDH combined
Ψi1,··· ,if =∑
j1,··· ,jp
Aj1,··· ,jp c(1)j1,i1···id · · · c
(p)jp ,i..···if
MCTDH is a decomposition of the wave-function tensor into a(time-dependent) Tucker form!
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Expansion of coefficients
Standard Method
Ψ(q1, · · · , qf ) =
N1∑i1
· · ·Nf∑if
Ψi1,··· ,if χ(1)i1
(q1) · · ·χ(f )if
(qf )
MCTDHΨi1,··· ,if =
∑j1,··· ,jf
Aj1,··· ,jf c(1)j1,i1· · · c(f )jf ,if
MCTDH combined
Ψi1,··· ,if =∑
j1,··· ,jp
Aj1,··· ,jp c(1)j1,i1···id · · · c
(p)jp ,i..···if
MCTDH is a decomposition of the wave-function tensor into a(time-dependent) Tucker form!
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Expansion of coefficients in ML-MCTDH form
ML-MCTDH (one extra layer)
Ψi1,··· ,if =∑
j1,··· ,jpA(1)j1,··· ,jp
( ∑k1,··· ,kp1
A(2;1)j1,k1,··· ,kp1
A(3;1,1)k1,i1
· · ·A(3;1,p1)kp1 ,ip1
)× · · ·
· · · ×( ∑
k1,··· ,kpκ1
A(2;κ1)jκ1,k1,··· ,kpκ1
A(3;κ1,1)k1,iα
· · ·A(3;κ1,pκ1)
kpp ,if
)× · · ·
· · · ×( ∑
k1,··· ,kpp
A(2;p)jp,k1,··· ,kpp
A(3;p,1)k1,iα
· · ·A(3;p,pp)kpp ,if
)
Hierachical Tucker format
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Other decomposition methods
CANDECOMP, CP
Ψ(q1, · · · , qf ) =∑r
ar ϕ(1)r (q1) · · ·ϕ(f )
r (qf )
Ψi1,··· ,if =∑r
ar c(1)r ,i1· · · c(f )r ,if
Tensor Train (TT) format. Similar to matrix product states.TT can be viewed as a simplified, restricted form of the HierachicalTucker format (i.e. ML-MCTDH).
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Other decomposition methods
CANDECOMP, CP
Ψ(q1, · · · , qf ) =∑r
ar ϕ(1)r (q1) · · ·ϕ(f )
r (qf )
Ψi1,··· ,if =∑r
ar c(1)r ,i1· · · c(f )r ,if
Tensor Train (TT) format. Similar to matrix product states.TT can be viewed as a simplified, restricted form of the HierachicalTucker format (i.e. ML-MCTDH).
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Content
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Product representation of the Hamiltonian
The computation of the Hamiltonian matrix 〈ΦJ | H | ΦL〉 and the
mean-fields 〈H〉(κ)k,l requires the evaluation of multi-dimensionalintegrals. It is essential that these integrals are done fast.To this end we require the Hamiltonian to be in product form
H =s∑
r=1
cr
p∏κ=1
h(κ)r
where h(κ)r operates on the κ-th particle only.
The multi-dimensional integrals can then be written as a sum ofproducts of one- or low-dimensional integrals
〈ΦJ | H | ΦL〉 =s∑
r=1
cr 〈ϕ(1)j1| h(1)r | ϕ(1)
l1〉 . . . 〈ϕ(p)
jp| h(p)r | ϕ(p)
lp〉
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Product representation of the Hamiltonian
The computation of the Hamiltonian matrix 〈ΦJ | H | ΦL〉 and the
mean-fields 〈H〉(κ)k,l requires the evaluation of multi-dimensionalintegrals. It is essential that these integrals are done fast.To this end we require the Hamiltonian to be in product form
H =s∑
r=1
cr
p∏κ=1
h(κ)r
where h(κ)r operates on the κ-th particle only.
The multi-dimensional integrals can then be written as a sum ofproducts of one- or low-dimensional integrals
〈ΦJ | H | ΦL〉 =s∑
r=1
cr 〈ϕ(1)j1| h(1)r | ϕ(1)
l1〉 . . . 〈ϕ(p)
jp| h(p)r | ϕ(p)
lp〉
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Potfit
The most direct way to the product form is an expansion in aproduct basis. Hence we approximate some given potential V by
V app (Q1, . . . ,Qp) =
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)j1
(Q1) . . . v(p)jp
(Qp)
working with grids:
V appi1...ip
=
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)i1j1
. . . v(p)ip jp
Tucker format!
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Potfit
The most direct way to the product form is an expansion in aproduct basis. Hence we approximate some given potential V by
V app (Q1, . . . ,Qp) =
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)j1
(Q1) . . . v(p)jp
(Qp)
working with grids:
V appi1...ip
=
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)i1j1
. . . v(p)ip jp
Tucker format!
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Potfit
The most direct way to the product form is an expansion in aproduct basis. Hence we approximate some given potential V by
V app (Q1, . . . ,Qp) =
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)j1
(Q1) . . . v(p)jp
(Qp)
working with grids:
V appi1...ip
=
m1∑j1=1
. . .
mp∑jp=1
Cj1...jp v(1)i1j1
. . . v(p)ip jp
Tucker format!
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Potfit
The coefficients are given by overlap
Cj1...jp =
N1∑i1=1
. . .
Np∑ip=1
v(1)i1j1· · · v (p)ip jp
Vi1...ip
More difficult is to find optimal single-particle potentials (SPPs).We define the SPPs as eigenvectors of the potential densitymatrices
%(κ)kk ′ =
∑I
κVi1...iκ−1kiκ+1...ip Vi1...iκ−1k ′iκ+1...ip
POTFIT is feasible for at most 109 grid points (7 DOF, say).
POTFIT (1996), HOSVD (2000).
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Potfit
The coefficients are given by overlap
Cj1...jp =
N1∑i1=1
. . .
Np∑ip=1
v(1)i1j1· · · v (p)ip jp
Vi1...ip
More difficult is to find optimal single-particle potentials (SPPs).We define the SPPs as eigenvectors of the potential densitymatrices
%(κ)kk ′ =
∑I
κVi1...iκ−1kiκ+1...ip Vi1...iκ−1k ′iκ+1...ip
POTFIT is feasible for at most 109 grid points (7 DOF, say).
POTFIT (1996), HOSVD (2000).
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Potfit
The coefficients are given by overlap
Cj1...jp =
N1∑i1=1
. . .
Np∑ip=1
v(1)i1j1· · · v (p)ip jp
Vi1...ip
More difficult is to find optimal single-particle potentials (SPPs).We define the SPPs as eigenvectors of the potential densitymatrices
%(κ)kk ′ =
∑I
κVi1...iκ−1kiκ+1...ip Vi1...iκ−1k ′iκ+1...ip
POTFIT is feasible for at most 109 grid points (7 DOF, say).
POTFIT (1996), HOSVD (2000).
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Potfit
The coefficients are given by overlap
Cj1...jp =
N1∑i1=1
. . .
Np∑ip=1
v(1)i1j1· · · v (p)ip jp
Vi1...ip
More difficult is to find optimal single-particle potentials (SPPs).We define the SPPs as eigenvectors of the potential densitymatrices
%(κ)kk ′ =
∑I
κVi1...iκ−1kiκ+1...ip Vi1...iκ−1k ′iκ+1...ip
POTFIT is feasible for at most 109 grid points (7 DOF, say).
POTFIT (1996), HOSVD (2000).
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Multi-grid Potfit (MGPF) and Monte Carlo Potfit (MCPF)
MGPF
Chose a fine (Nκ) and a coarse (nκ) product grid. The coarsegrid should be part of the fine grid.
Perform a full (i.e. exact) POTFIT on the coarse grid.
Interpolate the SPPs to the fine grid (˜= fine-grid):
v(κ) = ρ(κ)ρ(κ)−1v(κ)
MCPF
Perform all ”integrations” over the grid by Monte Carlo.
To be accurate, the determination of the coefficients requiresnow the inversion of a huge matrix.
A Boltzmann weighting is easy to include.
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Multi-grid Potfit (MGPF) and Monte Carlo Potfit (MCPF)
MGPF
Chose a fine (Nκ) and a coarse (nκ) product grid. The coarsegrid should be part of the fine grid.
Perform a full (i.e. exact) POTFIT on the coarse grid.
Interpolate the SPPs to the fine grid (˜= fine-grid):
v(κ) = ρ(κ)ρ(κ)−1v(κ)
MCPF
Perform all ”integrations” over the grid by Monte Carlo.
To be accurate, the determination of the coefficients requiresnow the inversion of a huge matrix.
A Boltzmann weighting is easy to include.
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Further size reduction, CANDECOMP and ML-Potfit
Potfit and its MG and MC variants express the potential tensor ina Tucker format. But MCTDH does not require this structure, aCANDECOMP is sufficient. As the latter can be more compact,we want to further decrease the size of the potential representationby reducing the Tucker format generated by MG- or MC-Potfit to aCANDECOMP. But how to do that?
As there is ML-MCTDH, one may think of ML-POTFIT.This will lead to a more compact representation, but not to afaster evaluation, because MCTDH cannot make use of thehirarchical Tucker format strucure.
However, ML-MCTDH can do!
See: F. Otto, J.Chem.Phys. 140, 014106 (2014)
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Further size reduction, CANDECOMP and ML-Potfit
Potfit and its MG and MC variants express the potential tensor ina Tucker format. But MCTDH does not require this structure, aCANDECOMP is sufficient. As the latter can be more compact,we want to further decrease the size of the potential representationby reducing the Tucker format generated by MG- or MC-Potfit to aCANDECOMP. But how to do that?
As there is ML-MCTDH, one may think of ML-POTFIT.This will lead to a more compact representation, but not to afaster evaluation, because MCTDH cannot make use of thehirarchical Tucker format strucure.
However, ML-MCTDH can do!
See: F. Otto, J.Chem.Phys. 140, 014106 (2014)
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High dimensional model representation, HDMR
Hierarchical representation of a multidimensional function
V (q) = V (0)+f∑
α=1
V (1)α (qα)+
f∑α<β
V(2)αβ (qα, qβ)+
f∑α<β<γ
V(3)αβγ(qα, qβ, qγ) · · ·
The component functions (clusters) are determined as:
V (0) = V (a)
V (1)α (qα) = V (qα; aα)− V (0)
V(2)αβ (qα, qβ) = V (qα, qβ; aαβ)− V (1)
α (qα)− V(1)β (qβ)− V (0)
Unfortunately, the number of clusters increases strongly with order.
Possible improvements:
Perform the cluster expansion in combined modes
One may use more than one reference point
One may use a reference path rather than a reference point
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High dimensional model representation, HDMR
Hierarchical representation of a multidimensional function
V (q) = V (0)+f∑
α=1
V (1)α (qα)+
f∑α<β
V(2)αβ (qα, qβ)+
f∑α<β<γ
V(3)αβγ(qα, qβ, qγ) · · ·
The component functions (clusters) are determined as:
V (0) = V (a)
V (1)α (qα) = V (qα; aα)− V (0)
V(2)αβ (qα, qβ) = V (qα, qβ; aαβ)− V (1)
α (qα)− V(1)β (qβ)− V (0)
Unfortunately, the number of clusters increases strongly with order.
Possible improvements:
Perform the cluster expansion in combined modes
One may use more than one reference point
One may use a reference path rather than a reference point
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High dimensional model representation, HDMR
Hierarchical representation of a multidimensional function
V (q) = V (0)+f∑
α=1
V (1)α (qα)+
f∑α<β
V(2)αβ (qα, qβ)+
f∑α<β<γ
V(3)αβγ(qα, qβ, qγ) · · ·
The component functions (clusters) are determined as:
V (0) = V (a)
V (1)α (qα) = V (qα; aα)− V (0)
V(2)αβ (qα, qβ) = V (qα, qβ; aαβ)− V (1)
α (qα)− V(1)β (qβ)− V (0)
Unfortunately, the number of clusters increases strongly with order.
Possible improvements:
Perform the cluster expansion in combined modes
One may use more than one reference point
One may use a reference path rather than a reference point
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Tunneling splitting in malonaldehyde
9 Atoms, 21 degrees of freedom
J. Chem. Phys. 134 (2011), 234307
J. Chem. Phys. 141 (2014), 034116
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Content
1 Multiconfiguration time-dependent Hartree, MCTDH
2 Multi-Layer MCTDH
3 MCTDH and ML-MCTDH viewed as tensor decompositionmethods
4 Compact representations of the PES
5 Highlights and Conclusions
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Highlights
Highlights and Breakthroughs
1990, Very first MCTDH publication, Meyer, Manthe, Cederbaum
1999, pyrazine, 24D, 2E, Raab, Worth, Meyer, Cederbaum
2003, Dissipative quantum dynamics, 61D, Nest, Meyer
2005, Vibronic spectrum of C5H+4 , 21D 5E, Markmann et al
2007, IR spectrum of H5O+2 , (15D) Vendrell et al
2008, Tunneling dynamics of bosons, Zollner et al
2009, Isotopologues of H5O+2 , (15D) Vendrell et al
2011, 2014, Tunnelling splittings in malonaldehyde, 21D,Schroder, Meyer
2013, Vibronic dynamics of naphthalene (48D,6E) andanthracene (66D,6E) cations, Meng, Meyer
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Conclusions
ConclusionsMCTDH, realistic problems with 5 to 9 atoms
Search for good coordinates.
Deriving the KEO can be cumbersome, but it is a solvedproblem.
Finding a compact representation for the PES is a majorproblem for molecules with 5 or more atoms.
The PES representation is often the source of largest errors.
Work on improving PES-representations is in progress.
Finally, the MCTDH calculation as such may take aconsiderable amount of CPU-time, but MCTDH is stable andwe can check its accuracy.
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People, who made the Heidelberg MCTDH package
Graham Worth, Birmingham (MCTDH, pyrazine)
Fabien Gatti, Montpellier (Kinetic energy operators)
Oriol Vendrell, Hamburg (ML-MCTDH, Zundel-cation)
Michael Brill (Parallelization of MCTDH)
Andreas Raab (Density operator propagation)
Markus Schroder, Heidelberg (Malonaldehyde, MC-Potfit)
Frank Otto, Hong Kong (ML-MCTDH, ML-Potfit)
Daniel Pelaez-Ruiz, Lille (MG-Potfit, H3O−2 )
Qingyong Meng, Dalian (ML calculations with VCH)
M. Beck, A. Jackle, M.-C. Heitz, S. Wefing, S. Sukiasyan,Ch. Cattarius, P. S. Thomas, K. Sadri and others.
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The End
Thank you!http://mctdh.uni-hd.de/