We present how to compute vibrational eigenstates with tree tensor network states (TTNSs), the underlying ansatz behind the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method. The eigenstates are computed with an algorithm that is based on the density matrix renormalization group (DMRG). We apply this to compute the vibrational spectrum of acetonitrile (CH3CN) to high accuracy and compare TTNSs with matrix product states (MPSs), the ansatz behind the DMRG. The presented optimization scheme converges much faster than ML-MCTDH-based optimization. For this particular system, we found no major advantage of the more general TTNS over MPS. We highlight that for both TTNS and MPS, the usage of an adaptive bond dimension significantly reduces the amount of required parameters. We furthermore propose a procedure to find good trees.
REFERENCES
1.
H.-D.
Meyer
, U.
Manthe
, and L.
Cederbaum
, Chem. Phys. Lett.
165
, 73
(1990
).2.
U.
Manthe
, H.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
97
, 3199
(1992
).3.
M. H.
Beck
, A.
Jäckle
, G. A.
Worth
, and H.-D.
Meyer
, Phys. Rep.
324
, 1
(2000
).4.
Multidimensional Quantum Dynamics: MCTDH Theory and Applications
, 1st ed., edited by H.-D.
Meyer
, F.
Gatti
, and G. A.
Worth
(Wiley VCH
, 2009
).5.
T. G.
Kolda
and B. W.
Bader
, SIAM Rev.
51
, 455
(2009
).6.
F.
Huarte-Larrañaga
and U.
Manthe
, J. Phys. Chem. A
105
, 2522
(2001
).7.
G. A.
Worth
, H.-D.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
109
, 3518
(1998
).8.
H.
Wang
, J. Chem. Phys.
113
, 9948
(2000
).9.
M.
Nest
and H.-D.
Meyer
, J. Chem. Phys.
119
, 24
(2003
).10.
H.
Wang
and M.
Thoss
, J. Chem. Phys.
119
, 1289
(2003
).11.
U.
Manthe
, J. Chem. Phys.
128
, 164116
(2008
).12.
O.
Vendrell
and H.-D.
Meyer
, J. Chem. Phys.
134
, 044135
(2011
).13.
H.
Wang
, J. Phys. Chem. A
119
, 7951
(2015
).14.
U.
Manthe
, J. Phys.: Condens. Matter
29
, 253001
(2017
).15.
W.
Hackbusch
and S.
Kühn
, J. Fourier Anal. Appl.
15
, 706
(2009
).16.
L.
Grasedyck
, SIAM J. Matrix Anal. Appl.
31
, 2029
(2010
).17.
W.
Hackbusch
, Tensor Spaces and Numerical Tensor Calculus
(Springer
, 2012
).18.
Y.-Y.
Shi
, L.-M.
Duan
, and G.
Vidal
, Phys. Rev. A
74
, 022320
(2006
).19.
L.
Tagliacozzo
, G.
Evenbly
, and G.
Vidal
, Phys. Rev. B
80
, 235127
(2009
).20.
V.
Murg
, F.
Verstraete
, Ö.
Legeza
, and R. M.
Noack
, Phys. Rev. B
82
, 205105
(2010
).21.
G. K.-L.
Chan
, Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 907
(2012
).22.
R.
Orús
, Ann. Phys.
349
, 117
(2014
).23.
Y.
Xie
, J.
Zheng
, and Z.
Lan
, J. Chem. Phys.
142
, 084706
(2015
).24.
J.
Schulze
, M. F.
Shibl
, M. J.
Al-Marri
, and O.
Kühn
, J. Chem. Phys.
144
, 185101
(2016
).25.
D.
Mendive-Tapia
, E.
Mangaud
, T.
Firmino
, A.
de la Lande
, M.
Desouter-Lecomte
, H.-D.
Meyer
, and F.
Gatti
, J. Phys. Chem. B
122
, 126
(2018
).26.
R.
Orús
, Eur. Phys. J. B
87
, 280
(2014
).27.
U.
Schollwöck
, Ann. Phys.
326
, 96
(2011
).28.
S. R.
White
, Phys. Rev. Lett.
69
, 2863
(1992
).29.
S. R.
White
, Phys. Rev. B
48
, 10345
(1993
).30.
I. V.
Oseledets
, SIAM J. Sci. Comput.
33
, 2295
(2011
).31.
G. K.-L.
Chan
, Phys. Chem. Chem. Phys.
10
, 3454
(2008
).32.
M.
Rakhuba
and I.
Oseledets
, J. Chem. Phys.
145
, 124101
(2016
).33.
S. M.
Greene
and V. S.
Batista
, J. Chem. Theory Comput.
13
, 4034
(2017
).34.
A.
Baiardi
, C. J.
Stein
, V.
Barone
, and M.
Reiher
, J. Chem. Theory Comput.
13
, 3764
(2017
).35.
J.
Ren
, Z.
Shuai
, and G.
Kin-Lic Chan
, J. Chem. Theory Comput.
14
, 5027
(2018
).36.
Y.
Kurashige
, J. Chem. Phys.
149
, 194114
(2018
).37.
A.
Baiardi
, C. J.
Stein
, V.
Barone
, and M.
Reiher
, J. Chem. Phys.
150
, 094113
(2019
).38.
A.
Baiardi
and M.
Reiher
, J. Chem. Theory Comput.
15
, 3481
(2019
).39.
F. A. Y. N.
Schröder
, D. H. P.
Turban
, A. J.
Musser
, N. D. M.
Hine
, and A. W.
Chin
, Nat. Commun.
10
, 1062
(2019
).40.
D.
Iouchtchenko
and P.-N.
Roy
, J. Chem. Phys.
148
, 134115
(2018
).41.
P. S.
Thomas
and T.
Carrington
, J. Phys. Chem. A
119
, 13074
(2015
).42.
T.
Hammer
and U.
Manthe
, J. Chem. Phys.
136
, 054105
(2012
).43.
R.
Wodraszka
and U.
Manthe
, J. Chem. Phys.
136
, 124119
(2012
).44.
H.
Wang
, J. Phys. Chem. A
118
, 9253
(2014
).45.
H. J.
Changlani
, S.
Ghosh
, C. L.
Henley
, and A. M.
Läuchli
, Phys. Rev. B
87
, 085107
(2013
).46.
N.
Nakatani
and G. K.-L.
Chan
, J. Chem. Phys.
138
, 134113
(2013
).47.
M.
Gerster
, P.
Silvi
, M.
Rizzi
, R.
Fazio
, T.
Calarco
, and S.
Montangero
, Phys. Rev. B
90
, 125154
(2014
).48.
K.
Gunst
, F.
Verstraete
, S.
Wouters
, Ö.
Legeza
, and D.
Van Neck
, J. Chem. Theory Comput.
14
, 2026
(2018
).49.
A.
Leclerc
and T.
Carrington
, J. Chem. Phys.
140
, 174111
(2014
).50.
P. S.
Thomas
and T.
Carrington
, J. Chem. Phys.
146
, 204110
(2017
).51.
P. S.
Thomas
, T.
Carrington
, J.
Agarwal
, and H. F.
Schaefer
, J. Chem. Phys.
149
, 064108
(2018
).52.
F.
Otto
, J. Chem. Phys.
140
, 014106
(2014
).53.
O.
Vendrell
, F.
Gatti
, and H.-D.
Meyer
, Angew. Chem., Int. Ed.
46
, 6918
(2007
).55.
V.
Murg
, F.
Verstraete
, R.
Schneider
, P. R.
Nagy
, and Ö.
Legeza
, J. Chem. Theory Comput.
11
, 1027
(2015
).56.
While here we only use real-valued tensors, we hint at complex conjugation in order to be general and to distinguish whether |Ψ⟩ or its dual ⟨Ψ| is represented in a particular diagram.
57.
Another frequently used notation is to use arrows or triangles to specify the matricization.22,109
58.
G. H.
Golub
and C. F. V.
Loan
, Matrix Computations
, 3rd ed. (Johns Hopkins University Press
, 2013
).59.
N. K.
Madsen
, M. B.
Hansen
, A.
Zoccante
, K.
Monrad
, M. B.
Hansen
, and O.
Christiansen
, J. Chem. Phys.
149
, 134110
(2018
).60.
L. J.
Doriol
, F.
Gatti
, C.
Iung
, and H.-D.
Meyer
, J. Chem. Phys.
129
, 224109
(2008
).61.
A.
Jäckle
and H.-D.
Meyer
, J. Chem. Phys.
104
, 7974
(1996
).62.
D.
Peláez
and H.-D.
Meyer
, J. Chem. Phys.
138
, 014108
(2013
).63.
M.
Schröder
and H.-D.
Meyer
, J. Chem. Phys.
147
, 064105
(2017
).64.
S.
Manzhos
and T.
Carrington
, J. Chem. Phys.
125
, 194105
(2006
).65.
W.
Koch
and D. H.
Zhang
, J. Chem. Phys.
141
, 021101
(2014
).66.
The kinetic energy operator very often is already in SOP form.
67.
R.
Wodraszka
and T.
Carrington
, J. Chem. Phys.
148
, 044115
(2018
).68.
T.
Xiang
, Phys. Rev. B
53
, R10445
(1996
).69.
H.-D.
Meyer
, F. L.
Quéré
, C.
Léonard
, and F.
Gatti
, Chem. Phys.
329
, 179
(2006
).70.
F.
Culot
and J.
Liévin
, Theor. Chim. Acta
89
, 227
(1994
).71.
K.
Drukker
and S.
Hammes-Schiffer
, J. Chem. Phys.
107
, 363
(1997
).72.
S.
Heislbetz
and G.
Rauhut
, J. Chem. Phys.
132
, 124102
(2010
).73.
S. R.
White
, Phys. Rev. B
72
, 180403
(2005
).74.
C.
Hubig
, I. P.
McCulloch
, U.
Schollwöck
, and F. A.
Wolf
, Phys. Rev. B
91
, 155115
(2015
).75.
C.
Lubich
, Appl. Math. Res. Express
2015
, 311
.76.
B.
Kloss
, I.
Burghardt
, and C.
Lubich
, J. Chem. Phys.
146
, 174107
(2017
).77.
K.-S.
Lee
and U. R.
Fischer
, Int. J. Mod. Phys. B
28
, 1550021
(2014
).78.
U.
Manthe
, J. Chem. Phys.
142
, 244109
(2015
).79.
H.-D.
Meyer
and H.
Wang
, J. Chem. Phys.
148
, 124105
(2018
).80.
H.
Wang
and H.-D.
Meyer
, J. Chem. Phys.
149
, 044119
(2018
).81.
J. J.
Dorando
, J.
Hachmann
, and G. K.-L.
Chan
, J. Chem. Phys.
127
, 084109
(2007
).82.
W.
Hu
and G. K.-L.
Chan
, J. Chem. Theory Comput.
11
, 3000
(2015
).83.
L.
Zhao
and E.
Neuscamman
, J. Chem. Theory Comput.
12
, 3436
(2016
).84.
L. N.
Tran
, J. A. R.
Shea
, and E.
Neuscamman
, J. Chem. Theory Comput.
15
, 4790
(2019
).85.
R.
Kosloff
and H.
Tal-Ezer
, Chem. Phys. Lett.
127
, 223
(1986
).86.
S.
Wouters
, W.
Poelmans
, P. W.
Ayers
, and D.
Van Neck
, Comput. Phys. Commun.
185
, 1501
(2014
).87.
U.
Manthe
, J. Chem. Phys.
128
, 064108
(2008
).88.
E.
Ronca
, Z.
Li
, C. A.
Jimenez-Hoyos
, and G. K.-L.
Chan
, J. Chem. Theory Comput.
13
, 5560
(2017
).89.
Ö.
Legeza
, J.
Röder
, and B. A.
Hess
, Phys. Rev. B
67
, 125114
(2003
).90.
D.
Mendive-Tapia
, T.
Firmino
, H.-D.
Meyer
, and F.
Gatti
, Chem. Phys.
482
, 113
(2017
).91.
R.
Wodraszka
and T.
Carrington
, J. Chem. Phys.
146
, 194105
(2017
).92.
H. R.
Larsson
and D. J.
Tannor
, J. Chem. Phys.
147
, 044103
(2017
).93.
In practice, the bond dimension is increased by one after every adaption. This avoids oscillations of the bond dimension.
94.
G. K.-L.
Chan
and M.
Head-Gordon
, J. Chem. Phys.
116
, 4462
(2002
).95.
R.
Olivares-Amaya
, W.
Hu
, N.
Nakatani
, S.
Sharma
, J.
Yang
, and G. K.-L.
Chan
, J. Chem. Phys.
142
, 034102
(2015
).96.
97.
D.
Begue
, P.
Carbonniere
, and C.
Pouchan
, J. Phys. Chem. A
109
, 4611
(2005
).98.
G.
Avila
and T.
Carrington
, J. Chem. Phys.
134
, 054126
(2011
).99.
G. A.
Worth
, M. H.
Beck
, A.
Jäckle
, and H.-D.
Meyer
, See http://mctdh.uni-hd.de for the MCTDH Package, Version 8.4.17, 2019
.100.
D. J.
Tannor
, Introduction to Quantum Mechanics: A Time-Dependent Perspective
, 1st ed. (University Science Books
, 2007
).101.
S.
van der Walt
, S. C.
Colbert
, and G.
Varoquaux
, Comput. Sci. Eng.
13
, 22
(2011
).102.
E.
Jones
, T.
Oliphant
, P.
Peterson
et al., SciPy: Open Source Scientific Tools for Python, 2001
, http://www.scipy.org/.103.
H. R.
Larsson
, B.
Hartke
, and D. J.
Tannor
, J. Chem. Phys.
145
, 204108
(2016
).104.
E. R.
Davidson
, J. Comput. Phys.
17
, 87
(1975
).105.
Q.
Sun
, T. C.
Berkelbach
, N. S.
Blunt
, G. H.
Booth
, S.
Guo
, Z.
Li
, J.
Liu
, J. D.
McClain
, E. R.
Sayfutyarova
, S.
Sharma
, S.
Wouters
, and G. K.-L.
Chan
, Wiley Interdiscip. Rev.: Comput. Mol. Sci.
8
, e1340
(2017
).106.
For TTNS, each iteration requires the diagonalization of each tensor and local (from one node to a neighboring node) transformation of the Hamiltonian, whereas for ML-MCTDH with improved relaxation, each iteration requires imaginary time propagation of the single particle functions, followed by a global transformations of the Hamiltonian (creation of the mean-fields), computation of density matrices, and diagonalization of the root node.
107.
Tensor removal and addition is not possible with the current optimization procedure but could be performed manually.
108.
F.
Köhler
, K.
Keiler
, S. I.
Mistakidis
, H.-D.
Meyer
, and P.
Schmelcher
, J. Chem. Phys.
151
, 054108
(2019
).109.
J.
Haegeman
, C.
Lubich
, I.
Oseledets
, B.
Vandereycken
, and F.
Verstraete
, Phys. Rev. B
94
, 165116
(2016
).110.
S.
Paeckel
, T.
Köhler
, A.
Swoboda
, S. R.
Manmana
, U.
Schollwöck
, and C.
Hubig
, Ann. Phys.
411
, 167998
(2019
).111.
D.
Bauernfeind
and M.
Aichhorn
, “Time dependent variational principle for tree tensor networks
,” e-print arXiv:1908.03090 (2019
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
