A new approximation for post-Hartree–Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to

$\mathcal O({d \cdot R \cdot n})$
O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to
$\mathcal O({R^{2} \cdot n})$
O(R2·n)
. We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO–MO transformation, the Møller–Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be
$\mathcal O({N^{1.8}})$
O(N1.8)
for the AO integrals,
$\mathcal O({N^{1.4}})$
O(N1.4)
for the MO integrals, and
$\mathcal O({N^{1.2}})$
O(N1.2)
for the MP2 t2-amplitudes (N denotes a measure of system size) if the upper bound of the error in the ℓ2-norm is chosen as ε = 10−2. This leads to an error in the MP2 energy in the order of mHartree.

1.
F.
Hitchcock
,
J. Math. Phys.
6
,
164
(
1927
).
2.
L. R.
Tucker
,
Psychometrika
31
,
279
(
1966
).
3.
J. D.
Carroll
and
J. J.
Chang
,
Psychometrika
35
,
283
(
1970
).
4.
R. A.
Harshman
and
M. E.
Lundy
,
Comput. Stat. Data Anal.
18
,
39
(
1994
).
5.
C. J.
Appellof
and
E. R.
Davidson
,
Anal. Chem.
53
,
2053
(
1981
).
6.
R.
Henrion
,
J. Chemom.
7
,
477
(
1993
).
7.
A. K.
Smilde
,
Y. D.
Wang
, and
B. R.
Kowalski
,
J. Chemom.
8
,
21
(
1994
).
8.
R.
Bro
,
Crit. Rev. Anal. Chem.
36
,
279
(
2006
).
9.
L.
De Lathauwer
,
B.
De Moor
, and
J.
Vandewalle
,
SIAM J. Matrix Anal. Appl.
21
,
1253
(
2000
).
10.
A. N.
Langville
and
W. J.
Stewart
,
Numer. Linear Algebra Appl.
11
,
723
(
2004
).
11.
L.
De Lathauwer
and
J.
Vandewalle
,
Linear Algebra Appl.
391
,
31
(
2004
).
12.
L.
De Lathauwer
and
J.
Castaing
,
Signal Process.
87
,
322
(
2007
).
13.
M. A. O.
Vasilescu
,
Human motion signatures: analysis, synthesis, recognition, Proceedings
.
16th International Conference on Pattern Recognition
2002, 2002,
3
,
456
460
, Quebec City, Canada,
11-15 August 2002
, 1051-4651, http://doi.ieeecomputersociety.org/10.1109/ICPR.2002.1047975.
14.
M. A. O.
Vasilescu
and
D.
Terzopoulos
,
Multilinear image analysis for facial recognition, Proceedings
.
16th International Conference on Pattern Recognition, 2002
. 2002,
2
,
511
514
, Quebec City, Canada,
11-15 August 2002
, 1051-4651, http://doi.ieeecomputersociety.org/10.1109/ICPR.2002.1048350.
15.
D.
Vlasic
,
M.
Brand
,
H.
Pfister
, and
J.
Popovic
,
ACM Transactions On Graphics
24
,
426
(
2005
).
16.
E.
Acar
,
S. A.
Camtepe
,
M. S.
Krishnamoorthy
, and
B.
Yener
,
Proceedings of the Intelligence and Security Informatics
3495
,
256
(
2005
).
17.
J. M.
Sun
,
S.
Papadimitriou
, and
P. S.
Yu
, Window-based Tensor Analysis on High-dimensional and Multi-aspect Streams, Data Mining, IEEE International Conference on, 2006,
0
,
1076
1080
,
18-22 December 2006
, Hong Kong, China, 1550-4786, http://doi.ieeecomputersociety.org/10.1109/ICDM.2006.169.
18.
L.
Omberg
,
G. H.
Golub
, and
O.
Alter
,
P. Natl. Acad. Sci. U.S.A.
104
,
18371
(
2007
).
19.
E.
Martínez-Montes
,
P. A.
Valdés-Sosa
,
F.
Miwakeichi
,
R. I.
Goldman
, and
M. S.
Cohen
,
Neuroimage
22
,
1023
(
2004
).
20.
F.
Miwakeichi
,
E.
Martínez-Montes
,
P. A.
Valdés-Sosa
,
N.
Nishiyama
,
H.
Mizuhara
, and
Y.
Yamaguchia
,
Neuroimage
22
,
1035
(
2004
).
21.
R.
Bellman
,
Adaptive Control Processes: A Guided Tour
(
Princeton University
,
Princeton, NJ
,
1961
), p.
255
.
22.
S.
Wilson
,
Comput. Phys. Commun.
58
,
71
(
1990
).
23.
I.
Røeggen
and
E.
Wisløff-Nilssen
,
Chem. Phys. Lett.
132
,
154
(
1986
).
24.
R.
Lindh
,
U.
Ryu
, and
B.
Liu
,
J. Chem. Phys.
95
,
5889
(
1991
).
25.
H.
Koch
,
A. S.
de Merás
, and
T. B.
Pedersen
,
J. Chem. Phys.
118
,
9481
(
2003
).
26.
F.
Aquilante
,
T. B.
Pedersen
, and
R.
Lindh
,
J. Chem. Phys.
126
,
194106
(
2007
).
27.
F.
Aquilante
and
T. B.
Pedersen
,
Chem. Phys. Lett.
449
,
354
(
2007
).
28.
L.
Boman
,
H.
Koch
, and
A. S.
de Merás
,
J. Chem. Phys.
129
,
134107
(
2008
).
29.
I.
Røeggen
and
T.
Johansen
,
J. Chem. Phys.
128
,
194107
(
2008
).
30.
S.
Schweizer
,
J.
Kussmann
,
B.
Doser
, and
C.
Ochsenfeld
,
J. Comput. Chem.
29
,
1004
(
2008
).
31.
F.
Aquilante
,
L.
Gagliardi
,
T. B.
Pedersen
, and
R.
Lindh
,
J. Chem. Phys.
130
,
154107
(
2009
).
32.
F.
Weigend
,
M.
Kattannek
, and
R.
Ahlrichs
,
J. Chem. Phys.
130
,
164106
(
2009
).
33.
J.
Zienau
,
L.
Clin
,
B.
Doser
, and
C.
Ochsenfeld
,
J. Chem. Phys.
130
,
204112
(
2009
).
34.
J. L.
Whitten
,
J. Chem. Phys.
58
,
4496
(
1973
).
35.
F. R.
Manby
,
J. Chem. Phys.
119
,
4607
(
2003
).
36.
H. J.
Werner
,
F. R.
Manby
, and
P. J.
Knowles
,
J. Chem. Phys.
118
,
8149
(
2003
).
37.
M.
Schütz
and
F. R.
Manby
,
Phys. Chem. Chem. Phys.
5
,
3349
(
2003
).
38.
R.
Kendall
and
H.
Früchtl
,
Theor. Chem. Acc.
97
,
158
(
1997
).
39.
O.
Vahtras
,
J.
Almlöf
, and
M.
Feyereisen
,
Chem. Phys. Lett.
213
,
514
(
1993
).
40.
M.
Feyereisen
,
G.
Fitzgerald
, and
A.
Komornicki
,
Chem. Phys. Lett.
208
,
359
(
1993
).
41.
R.
Ahlrichs
,
Phys. Chem. Chem. Phys.
6
,
5119
(
2004
).
42.
W.
Kutzelnigg
,
Theor. Chim. Acta
68
,
445
(
1985
).
43.
E. F.
Valeev
and
H. F.
Schaefer
,
J. Chem. Phys.
113
,
3990
(
2000
).
44.
S.
Ten-No
,
Chem. Phys. Lett.
398
,
56
(
2004
).
45.
W.
Klopper
,
F. R.
Manby
,
S.
Ten-No
, and
E. F.
Valeev
,
Int. Rev. Phys. Chem.
25
,
427
(
2006
).
46.
T. B.
Adler
,
H. J.
Werner
, and
F. R.
Manby
,
J. Chem. Phys.
130
,
054106
(
2009
).
47.
T. B.
Adler
and
H. J.
Werner
,
J. Chem. Phys.
130
,
241101
(
2009
).
48.
M.
Häser
and
J.
Almlöf
,
J. Chem. Phys.
96
,
489
(
1992
).
49.
M.
Häser
,
Theor. Chim. Acta.
87
,
147
(
1993
).
50.
A. K.
Wilson
and
J.
Almlöf
,
Theor. Chim. Acta
95
,
49
(
1997
).
51.
P. Y.
Ayala
and
G. E.
Scuseria
,
J. Chem. Phys.
110
,
3660
(
1999
).
52.
D.
Braess
and
W.
Hackbusch
,
IMA J. Numer. Anal.
25
,
685
(
2005
).
53.
D. S.
Lambrecht
,
B.
Doser
, and
C.
Ochsenfeld
,
J. Chem. Phys.
123
,
184102
(
2005
).
54.
A.
Takatsuka
,
S.
Ten-No
, and
W.
Hackbusch
,
J. Chem. Phys.
129
,
044112
(
2008
).
55.
G. E.
Scuseria
and
P. Y.
Ayala
,
J. Chem. Phys.
111
,
8330
(
1999
).
56.
P. Y.
Ayala
,
K. N.
Kudin
, and
G. E.
Scuseria
,
J. Chem. Phys.
115
,
9698
(
2001
).
57.
B.
Doser
,
D. S.
Lambrecht
,
J.
Kussmann
, and
C.
Ochsenfeld
,
J. Chem. Phys.
130
,
064107
(
2009
).
58.
W.
Meyer
,
Int. J. Quantum Chem.
5
,
341
(
1971
).
59.
W.
Meyer
,
J. Chem. Phys.
58
,
1017
(
1973
).
60.
R.
Ahlrichs
and
F.
Driessler
,
Theor. Chim. Acta
36
,
275
(
1975
).
61.
R.
Ahlrichs
,
F.
Driessler
,
H.
Lischka
,
V.
Staemmler
, and
W.
Kutzelnigg
,
J. Chem. Phys.
62
,
1235
(
1975
).
62.
L.
Adamowicz
and
R. J.
Bartlett
,
J. Chem. Phys.
86
,
6314
(
1987
).
63.
L.
Adamowicz
,
R. J.
Bartlett
, and
A. J.
Sadlej
,
J. Chem. Phys.
88
,
5749
(
1988
).
64.
P.
Neogrády
,
M.
Pitoňák
, and
M.
Urban
,
Mol. Phys.
103
,
2141
(
2005
).
65.
M.
Pitoňák
,
P.
Neogrády
,
V.
Kellö
, and
M.
Urban
,
Mol. Phys.
104
,
2277
(
2006
).
66.
M.
Pitoňák
,
F.
Holka
,
P.
Neogrády
, and
M.
Urban
,
J. Mol. Struct. THEOCHEM
768
,
79
(
2006
).
67.
F.
Neese
,
F.
Wennmohs
, and
A.
Hansen
,
J. Chem. Phys.
130
,
114108
(
2009
).
68.
F.
Neese
,
A.
Hansen
, and
D. G.
Liakos
,
J. Chem. Phys.
131
,
064103
(
2009
).
69.
T.
Kinoshita
,
O.
Hino
, and
R. J.
Bartlett
,
J. Chem. Phys.
119
,
7756
(
2003
).
70.
O.
Hino
,
T.
Kinoshita
, and
R. J.
Bartlett
,
J. Chem. Phys.
121
,
1206
(
2004
).
71.
T. G.
Kolda
,
SIAM J. Matrix Anal. Appl.
23
,
243
(
2001
).
72.
T.
Zhang
and
G. H.
Golub
,
SIAM J. Matrix Anal. Appl.
23
,
534
(
2001
).
73.
M.
Ishteva
,
L.
De Lathauwer
,
P. A.
Absil
, and
S.
Van Huffel
,
Numer. Analy. Appl. Math.
1048
,
274
(
2008
).
74.
M.
Ishteva
,
L.
De Lathauwer
,
P. A.
Absil
, and
S.
Van Huffel
,
Numer. Algor.
51
,
179
(
2009
).
75.
T. G.
Kolda
and
B. W.
Bader
,
SIAM Rev.
51
,
455
(
2009
).
76.
L.
De Lathauwer
,
B.
De Moor
, and
J.
Vandewalle
,
SIAM J. Matrix Anal. Appl.
21
,
1324
(
2000
).
77.
G.
Beylkin
and
M. J.
Mohlenkamp
,
P. Natl. Acad. Sci. U.S.A.
99
,
10246
(
2002
).
78.
G.
Beylkin
and
M. J.
Mohlenkamp
,
SIAM J. Sci. Comput.
26
,
2133
(
2005
).
79.
S. R.
Chinnamsetty
,
M.
Espig
,
B. N.
Khoromskij
,
W.
Hackbusch
, and
H. J.
Flad
,
J. Chem. Phys.
127
,
084110
(
2007
).
80.
M.
Espig
, ‘Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen,” Ph.D. thesis (Universität Leipzig,
2008
).
81.
E.
Acar
and
B.
Yener
,
IEEE Trans. Knowl. Data Eng.
21
,
6
(
2009
).
82.
R.
Bro
,
Chemom. Intell. Lab. Syst.
38
,
149
(
1997
).
83.
R.
Bro
and
H. A. L.
Kiers
,
J. Chemom.
17
,
274
(
2003
).
84.
M.
Espig
and
W.
Hackbusch
, “A Regularized Newton method for the Efficient Approximation of Tensors Represented in the Canonical Tensor Format,” submitted to: Numerische Mathematik.
85.
This expansion is exact as long as the rank R is large enough. As has been shown, any tensor in d dimensions can be represented exactly with a rank of R = nd − 1, where n denotes the number of entries in each dimension.
86.
J.
Gauss
,
J. F.
Stanton
, and
R. J.
Bartlett
,
J. Chem. Phys.
95
,
2623
(
1991
).
87.
J. F.
Stanton
and
J.
Gauss
,
Int. Rev. Phys. Chem.
19
,
61
(
2000
).
88.
W.
Hackbusch
and
B.
Khoromskij
,
J. Complex.
23
,
697
(
2007
).
89.
W.
Hackbusch
and
B. N.
Khoromskij
,
SIAM J. Matrix Anal. Appl.
30
,
1233
(
2008
).
90.
M.
Espig
,
L.
Grasedyck
, and
W.
Hackbusch
,
Constr. Approx.
30
,
557
(
2009
).
91.
M.
Espig
,
W.
Hackbusch
,
T.
Rohwedder
, and
R.
Schneider
, “
Variational Calculus with Sums of Elementary Tensors of Fixed Rank
,”
Numer. Math.
, (to be published).
92.
L.
Armijo
,
Pac. J. Math.
16
,
1
(
1966
).
93.
P.
Kosmol
,
Methoden zur numerischen Behandlung nichtlinearer Gleichungen und Optimierungsaufgaben
(
Teubner Verlag; Auflage: 2
,
1993
).
94.
R. J.
Bartlett
,
Ann. Rev. Phys. Chem.
32
,
359
(
1981
).
95.
CFOUR, coupled cluster techniques for Computational Chemistry, a quantum-chemical program package by
J. F.
Stanton
,
J.
Gauss
,
M. E.
Harding
,
P. G.
Szalay
with contributions from
A. A.
Auer
,
R. J.
Bartlett
,
U.
Benedikt
,
C.
Berger
,
D. E.
Bernholdt
,
O.
Christiansen
,
M.
Heckert
,
O.
Heun
,
C.
Huber
,
D.
Jonsson
,
J.
Jusélius
,
K.
Klein
,
W. J.
Lauderdale
,
D.
Matthews
,
T.
Metzroth
,
D. P.
O’Neill
,
D. R.
Price
,
E.
Prochnow
,
K.
Ruud
,
F.
Schiffmann
,
S.
Stopkowicz
,
A.
Tajti
,
M. E.
Varner
,
J.
Vázquez
,
F.
Wang
,
J. D.
Watts
and the integral packages MOLECULE (J. Almlöf and
P. R.
Taylor
), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by
A. V.
Mitin
and
C.
van Wüllen
. For the current version, see http://www.cfour.de.
96.
The antisymmetrized two electron integrals were used as described in J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, J. Chem. Phys. 94, 4334 (1991). All integrals smaller than a threshold of 10−14 were neglected.
97.
All calculations were carried out on the “Chemnitz High performance Linux Cluster” (CHiC); F. Mietke, T. Mehlan, T. Hoefler, and W. Rehm, “Design and Evaluation of a 2048 Core Cluster System,” Proceedings of the 3rd Workshop KiCC (Kommunikation in Clustern und Clusterverbundsystemen), 2007, Aachen, RWTH Aachen University, Germany, 12 December, 2007,
40
50
.
98.
It should be noted that this is the current performance using a pilot implementation—an optimized scheme for two-electron integrals and wavefunction parameters is the subject of current work and will be presented in forthcoming publications.
99.
The compression has been calculated as: compression =
$100 \cdot \left(1-\left(\protect \frac{\mbox{reduced rank}}{\mbox{initial rank}}\right)\right)$
100·1reducedrankinitialrank
.
100.
With the current pilot implementation that is a general-purpose, BLAS-level 2 based C++ development platform, only initial ranks up to 10 000 are computationally feasible. For an example of this size, the rank reduction algorithm takes in the order of a few weeks using a single core on a modern workstation computer (AMD Opteron 2218 Stepping 2, 2.6 GHz, 4 GB Memory).
101.
The geometry for LiH monomer was taken from K. P. Huber and G. H. Herzberg, Molecular Spectra and Moleculare Structure IV: Constants of Diatomic Molecules (Van Nostrand-Reinhold, New York, 1979): RLiH = 159.5 pm. The LiH chain was build up as a linear chain LiH–LiH–
$\protect \ldots$
...
using a distance of 300 pm between the H-atom of one molecule and the Li-atom of the next.
102.
It should be noted that the initial scaling with system size is actually not
$\protect \mathcal O({n^{3}})$
O(n3)
due to the fact that integrals smaller than 10−14 are neglected.
106.
A. R.
Hoy
,
I. M.
Mills
, and
G.
Strey
,
Mol. Phys.
24
,
1265
(
1972
).
107.
J. L.
Duncan
and
I. M.
Mills
,
Spectrochim. Acta
20
,
523
(
1964
).
108.
D. L.
Gray
and
A. G.
Robiette
,
Mol. Phys.
37
,
1901
(
1979
).
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