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

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

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

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)

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)

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

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”)

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

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

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

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

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

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.

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`

}

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`

}

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`

}

Standard Method and MCTDH trees

q1 q2 q3 q4 q5 q6

StandardMethod

q1 q2 q3 q4 q5 q6

MCTDH

MCTDH and ML-MCTDH trees

q1 q2 q3 q4 q5 q6

MCTDHcombined

q1 q2 q3 q4 q5 q6

ML-MCTDH

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

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

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

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.

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

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!

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!

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!

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!

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

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).

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).

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

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〉

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〉

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!

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!

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!

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).

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).

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).

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).

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.

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.

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)

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)

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

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

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

Tunneling splitting in malonaldehyde

9 Atoms, 21 degrees of freedom

J. Chem. Phys. 134 (2011), 234307

J. Chem. Phys. 141 (2014), 034116

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

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

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.

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.

The End

Thank you!http://mctdh.uni-hd.de/