We present an efficient implementation of dynamically pruned quantum dynamics, both in coordinate space and in phase space. We combine the ideas behind the biorthogonal von Neumann basis (PvB) with the orthogonalized momentum-symmetrized Gaussians (Weylets) to create a new basis, projected Weylets, that takes the best from both methods. We benchmark pruned time-dependent dynamics using phase-space-localized PvB, projected Weylets, and coordinate-space-localized DVR bases, with real-world examples in up to six dimensions. For the examples studied, coordinate-space localization is the most important factor for efficient pruning and the pruned dynamics is much faster than the unpruned, exact dynamics. Phase-space localization is useful for more demanding dynamics where many basis functions are required. There, projected Weylets offer a more compact representation than pruned DVR bases.
REFERENCES
1.
D. J.
Tannor
, Introduction to Quantum Mechanics: A Time-Dependent Perspective
, 1st ed. (University Science Books
, 2007
).2.
M.
Tizniti
, S. D.
Le Picard
, F.
Lique
, C.
Berteloite
, A.
Canosa
, M. H.
Alexander
, and I. R.
Sims
, Nat. Chem.
6
, 141
–145
(2014
).3.
T.
Wu
, H.-J.
Werner
, and U.
Manthe
, Science
306
, 2227
–2229
(2004
).4.
R.
Otto
, J.
Ma
, A. W.
Ray
, J. S.
Daluz
, J.
Li
, H.
Guo
, and R. E.
Continetti
, Science
343
, 396
–399
(2014
).5.
T.
Westermann
, J. B.
Kim
, M. L.
Weichman
, C.
Hock
, T. I.
Yacovitch
, J.
Palma
, D. M.
Neumark
, and U.
Manthe
, Angew. Chem., Int. Ed.
53
, 1122
–1126
(2014
).6.
L.
Chen
, K.
Shao
, J.
Chen
, M.
Yang
, and D. H.
Zhang
, J. Chem. Phys.
144
, 194309
(2016
).7.
H.
Song
, J.
Li
, M.
Yang
, Y.
Lu
, and H.
Guo
, Phys. Chem. Chem. Phys.
16
, 17770
(2014
).8.
J.
Palma
and D. C.
Clary
, J. Chem. Phys.
112
, 1859
(2000
).9.
W.
Zhang
, Y.
Zhou
, G.
Wu
, Y.
Lu
, H.
Pan
, B.
Fu
, Q.
Shuai
, L.
Liu
, S.
Liu
, L.
Zhang
, B.
Jiang
, D.
Dai
, S.-Y.
Lee
, Z.
Xie
, B. J.
Braams
, J. M.
Bowman
, M. A.
Collins
, D. H.
Zhang
, and X.
Yang
, Proc. Natl. Acad. Sci. U. S. A.
107
, 12782
(2010
).10.
E.
Baloïtcha
, B.
Lasorne
, D.
Lauvergnat
, G.
Dive
, Y.
Justum
, and M.
Desouter-Lecomte
, J. Chem. Phys.
117
, 727
(2002
).11.
F.
Huarte-Larrañaga
and U.
Manthe
, J. Phys. Chem. A
105
, 2522
(2001
).12.
A.
Vikár
, T.
Nagy
, and G.
Lendvay
, J. Phys. Chem. A
120
, 5083
(2016
).13.
I.
Burghardt
, H.-D.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
111
, 2927
(1999
).14.
S.
Römer
, M.
Ruckenbauer
, and I.
Burghardt
, J. Chem. Phys.
138
, 064106
(2013
).15.
G. A.
Worth
, M. A.
Robb
, and I.
Burghardt
, Faraday Discuss.
127
, 307
(2004
).16.
G.
Richings
, I.
Polyak
, K.
Spinlove
, G.
Worth
, I.
Burghardt
, and B.
Lasorne
, Int. Rev. Phys. Chem.
34
, 269
–308
(2015
).17.
Y.
Wu
and V. S.
Batista
, J. Chem. Phys.
118
, 6720
(2003
).18.
T. J.
Martínez
, M.
Ben-Nun
, and R. D.
Levine
, J. Phys. Chem.
100
, 7884
(1996
).19.
M.
Ben-Nun
and T. J.
Martínez
, J. Chem. Phys.
108
, 7244
(1998
).20.
D. V.
Shalashilin
and M. S.
Child
, Chem. Phys.
304
, 103
(2004
).21.
H.-D.
Meyer
, U.
Manthe
, and L.
Cederbaum
, Chem. Phys. Lett.
165
, 73
(1990
).22.
M. H.
Beck
, A.
Jäckle
, G. A.
Worth
, and H.-D.
Meyer
, Phys. R.ep
324
, 1
(2000
).23.
H.
Wang
and M.
Thoss
, J. Chem. Phys.
119
, 1289
(2003
).24.
U.
Manthe
, J. Chem. Phys.
128
, 164116
(2008
).25.
O.
Vendrell
and H.-D.
Meyer
, J. Chem. Phys.
134
, 044135
(2011
).26.
R.
Welsch
and U.
Manthe
, J. Chem. Phys.
137
, 244106
(2012
).27.
Q.
Meng
and H.-D.
Meyer
, J. Chem. Phys.
141
, 124309
(2014
).28.
Z.
Bačić
and J. C.
Light
, J. Chem. Phys.
85
, 4594
(1986
).29.
Z.
Bačić
and J. C.
Light
, J. Chem. Phys.
86
, 3065
(1987
).30.
J. M.
Bowman
and B.
Gazdy
, J. Chem. Phys.
94
, 454
(1991
).31.
M. J.
Bramley
and N. C.
Handy
, J. Chem. Phys.
98
, 1378
(1993
).32.
A.
Viel
and C.
Leforestier
, J. Chem. Phys.
112
, 1212
(2000
).33.
34.
X.-G.
Wang
and T.
Carrington
, J. Chem. Phys.
117
, 6923
(2002
).35.
A.
Shimshovitz
, ‘‘Phase space approach to solving the Schrödinger equation
,’’ Ph.D. thesis, Weizmann Institute of Science
, 2015
.36.
X.-G.
Wang
and T.
Carrington
, J. Chem. Phys.
129
, 234102
(2008
).37.
R.
Dawes
and T.
Carrington
, J. Chem. Phys.
122
, 134101
(2005
).38.
A.
Maynard
, R. E.
Wyatt
, and C.
Iung
, J. Chem. Phys.
106
, 9483
(1997
).39.
D.
Bégué
, N.
Gohaud
, C.
Pouchan
, P.
Cassam-Chenaï
, and J.
Liévin
, J. Chem. Phys.
127
, 164115
(2007
).40.
T.
Halverson
and B.
Poirier
, J. Phys. Chem. A
119
, 12417
–12433
(2015
).41.
A.
Shimshovitz
, Z.
Bačić
, and D. J.
Tannor
, J. Chem. Phys.
141
, 234106
(2014
).42.
R.
Dawes
and T.
Carrington
, J. Chem. Phys.
121
, 726
(2004
).43.
I.
Degani
, J.
Schiff
, and D. J.
Tannor
, Numer. Math.
101
, 479
(2005
).44.
I.
Degani
and D. J.
Tannor
, J. Phys. Chem. A
110
, 5395
(2006
).45.
G.
Avila
and T.
Carrington
, J. Chem. Phys.
131
, 174103
(2009
).46.
D.
Lauvergnat
and A.
Nauts
, Spectrochim. Acta, Part A
119
, 18
(2014
).47.
D. T.
Colbert
and W. H.
Miller
, J. Chem. Phys.
96
, 1982
(1992
).48.
M.
McLeod
and T.
Carrington
, Chem. Phys. Lett.
501
, 130
–133
(2010
).49.
50.
D. H.
Zhang
and J. Z. H.
Zhang
, J. Chem. Phys.
101
, 1146
(1994
).51.
B.
Hartke
, Phys. Chem. Chem. Phys.
8
, 3627
(2006
).52.
J.
Sielk
, H. F.
von Horsten
, F.
Krüger
, R.
Schneider
, and B.
Hartke
, Phys. Chem. Chem. Phys.
11
, 463
–475
(2009
).53.
D. A.
McCormack
, J. Chem. Phys.
124
, 204101
(2006
).54.
L. R.
Pettey
and R. E.
Wyatt
, Chem. Phys. Lett.
424
, 443
(2006
).55.
L. R.
Pettey
and R. E.
Wyatt
, Int. J. Quantum Chem.
107
, 1566
(2007
).56.
M. J.
Davis
and E. J.
Heller
, J. Chem. Phys.
71
, 3383
(1979
).57.
B.
Poirier
, J. Theor. Comput. Chem.
02
, 65
(2003
).58.
B.
Poirier
and A.
Salam
, J. Chem. Phys.
121
, 1690
(2004
).59.
B.
Poirier
and A.
Salam
, J. Chem. Phys.
121
, 1704
(2004
).60.
A.
Shimshovitz
and D. J.
Tannor
, Phys. Rev. Lett.
109
(2012
).61.
R.
Lombardini
and B.
Poirier
, J. Chem. Phys.
124
, 144107
(2006
).62.
T.
Halverson
and B.
Poirier
, J. Chem. Phys.
137
, 224101
(2012
).63.
T.
Halverson
and B.
Poirier
, J. Chem. Phys.
140
, 204112
(2014
).64.
T.
Halverson
and B.
Poirier
, Chem. Phys. Lett.
624
, 37
(2015
).65.
N.
Takemoto
, A.
Shimshovitz
, and D. J.
Tannor
, J. Chem. Phys.
137
, 011102
(2012
).66.
E.
Assémat
, S.
Machnes
, and D.
Tannor
, “Double ionization of Helium from a phase space perspective
,” e-print arXiv:1502.05165 (2015
).67.
J.
Brown
and T.
Carrington
, Jr., “Assessing the utility of phase-space-localized basis functions: Exploiting direct product structure and a new basis function selection procedure
,” J. Chem. Phys.
144
, 244115
(2016
).68.
O.
Christiansen
, Phys. Chem. Chem. Phys.
14
, 6672
(2012
).69.
A.
Leclerc
and T.
Carrington
, J. Chem. Phys.
140
, 174111
(2014
).70.
A.
Jäckle
and H.
Meyer
, J. Chem. Phys.
104
, 7974
(1996
).71.
D.
Peláez
and H.-D.
Meyer
, J. Chem. Phys.
138
, 014108
(2013
).72.
F.
Otto
, J. Chem. Phys.
140
, 014106
(2014
).73.
S.
Manzhos
and T.
Carrington
, J. Chem. Phys.
125
, 194105
(2006
).74.
W.
Koch
and D. H.
Zhang
, J. Chem. Phys.
141
, 021101
(2014
).75.
G.
Avila
and T.
Carrington
, J. Chem. Phys.
143
, 044106
(2015
).76.
B.
Ziegler
and G.
Rauhut
, J. Chem. Phys.
144
, 114114
(2016
).77.
78.
J.
von Neumann
, Mathematische Grundlagen der Quantenmechanik
, 2nd ed. (Springer
, 1996
).79.
R.
Balian
, C. R. Acad. Sci. Paris
292
, 1357
(1981
).80.
F.
Low
, “Complete sets of wave packets
,” in A Passion for Physics: Essays in Honor of Geoffrey Chew
(World Scientific
, 1985
), pp. 17
–22
.81.
G.
Battle
, Lett. Math. Phys.
15
, 175
–177
(1988
).82.
C. C.
Marston
and G. G.
Balint-Kurti
, J. Chem. Phys.
91
, 3571
(1989
).83.
I.
Daubechies
and A. J. E. M.
Janssen
, IEEE Trans. Inf. Theory
39
, 3
(1993
).84.
S.
Machnes
, E.
Assémat
, H. R.
Larsson
, and D. J.
Tannor
, J. Phys. Chem. A
120
, 3296
(2016
).85.
U.
Manthe
and H.
Köppel
, J. Chem. Phys.
93
, 345
(1990
).86.
M. J.
Bramley
and T.
Carrington
, J. Chem. Phys.
99
, 8519
(1993
).87.
X.-G.
Wang
and T.
Carrington
, Jr., J. Phys. Chem. A
105
, 2575
(2001
).88.
J.
Brown
and T.
Carrington
, J. Chem. Phys.
143
, 044104
(2015
).89.
90.
I.
Daubechies
, S.
Jaffard
, and J.-L.
Journé
, SIAM J. Math. Anal.
22
, 554
(1991
).91.
92.
S.
Machnes
, E.
Assémat
, and D.
Tannor
, “Quantum dynamics in phase space using the biorthogonal von Neumann bases: Algorithmic considerations
,” e-print arXiv:1603.03963 (2016
).93.
We use std::unordered_set of the C++ programming language standard library as implemented in the GNU compiler collection.112
94.
95.
T. J.
Park
and J. C.
Light
, J. Chem. Phys.
85
, 5870
(1986
).96.
G. A.
Worth
, M. H.
Beck
, A.
Jäckle
, and H.-D.
Meyer
, The MCTDH Package, Version 8.4.10, 2014
, See http://mctdh.uni-hd.de.97.
V.
Engel
, Chem. Phys. Lett.
189
, 76
(1992
).98.
U.
Manthe
, H.-D.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
97
, 9062
(1992
).99.
G. A.
Worth
, H.-D.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
109
, 3518
(1998
).100.
A.
Shimshovitz
and D. J.
Tannor
, J. Chem. Phys.
137
, 101103
(2012
).101.
T.
Westermann
and U.
Manthe
, J. Chem. Phys.
136
, 204116
(2012
).102.
G.
Hirsch
, R. J.
Buenker
, and C.
Petrongolo
, Mol. Phys.
73
, 1085
–1099
(1991
).103.
E.
Leonardi
, C.
Petrongolo
, V.
Keshari
, G.
Hirsch
, and R. J.
Buenker
, Mol. Phys.
82
, 553
–565
(1994
).104.
Brown and Carrington noted problems with the condition number of G, namely numbers after an adaption of the width of each von Neumann function for a grid of size 48.88 We do not experience such problems.
105.
A.
Raab
, G. A.
Worth
, H.-D.
Meyer
, and L. S.
Cederbaum
, J. Chem. Phys.
110
, 936
(1999
).106.
A.
Zanchet
, O.
Roncero
, and N.
Bulut
, Phys. Chem. Chem. Phys.
18
, 11391
(2016
).107.
J.
Ma
, J.
Li
, and H.
Guo
, J. Phys. Chem. Lett.
3
, 2482
(2012
).108.
109.
E. D.
Napoli
, D.
Fabregat-Traver
, G.
Quintana-Ortí
, and P.
Bientinesi
, Appl. Math. Comput.
235
, 454
(2014
).110.
K.
Goto
and R. A. v. d.
Geijn
, ACM Trans. Math. Software
34
(3
), 1
(2008
).111.
J.
Cooper
and T.
Carrington
, J. Chem. Phys.
130
, 214110
(2009
).112.
R. M.
Stallman
et al, Using The Gnu Compiler Collection: A Gnu Manual For GCC Version 5.3.0, 2015
.© 2016 Author(s).
2016
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