Supporting Information for:
Matrix product states with large sites
Henrik R. Larsson,
1,
a)
Huanchen Zhai,
1
Klaas Gunst,
1, 2, 3,
b)
and Garnet Kin-Lic Chan
1,
c)
1)
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125,
USA
2)
Center for Molecular Modeling, Ghent University, Technologiepark 46, B-9052 Zwijnaarde,
Belgium
3)
Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, B-9000 Ghent,
Belgium
(Dated: 7 December 2021)
I. ADDITIONAL DATA FOR THE CHROMIUM DIMER
See Table I for MPS-MRCISD+
Q
P
energetics of the chromium dimer for various bond distances and basis sets.
Table II shows various MPS-MRCISD+
Q
and MPS-MRCISDT energies at different bond distances for the cc-pVDZ-
DK basis.
TABLE I. Energies in
E
H
for the chromium dimer at the MPS-MRCISD+Q
P
level of theory for various cc-p
n
Z-DK bases.
“5Z(extrap)” denotes the energy for the cc-pV5Z-DK basis after extrapolating the bond dimension to
∞
.
R/
Å
DZ
TZ
QZ
5Z
5Z(extrap) error 5Z extrap
1.5 -2099.8945 -2100.1811 -2100.3137 -2100.3861 -2100.3932
0.0016
1.6 -2099.9174 -2100.2023 -2100.3328 -2100.4045 -2100.4123
0.0018
1.7 -2099.9215 -2100.2051 -2100.3334 -2100.4046 -2100.4119
0.0017
1.8 -2099.9186 -2100.2013 -2100.3277 -2100.3981 -2100.4055
0.0017
2.0 -2099.9105 -2100.1929 -2100.3165 -2100.3856 -2100.3919
0.0015
2.1 -2099.9075 -2100.1906 -2100.3133 -2100.3822 -2100.3873
0.0012
2.2 -2099.9054 -2100.1897 -2100.3116 -2100.3794 -2100.3839
0.0011
2.4 -2099.9024 -2100.1895 -2100.3105 -2100.3786 -2100.3818
0.0009
2.5 -2099.9010 -2100.1894 -2100.3102 -2100.3787 -2100.3819
0.0008
2.7 -2099.8980 -2100.1882 -2100.3085 -2100.3769 -2100.3809
0.0010
2.9 -2099.8936 -2100.1854 -2100.3058 -2100.3738 -2100.3747
0.0004
3.1 -2099.8886 -2100.1818 -2100.3022 -2100.3695 -2100.3706
0.0004
3.5 -2099.8793 -2100.1745 -2100.2949 -2100.3629 -2100.3629
0.0001
∞
-2099.8648 -2100.1626 -2100.2800 -2100.3503
atom -1049.9316 -1050.0835 -1050.1442 -1050.1785
II. CLUSTER MPS FOR A TWO-DIMENSIONAL HYDROGEN LATTICE
In Fig. A we compare the performance of a cluster MPS with that of an ordinary MPS for a 4x4 hydrogen lattice with
lattice spacing of
1
.
1
Å and the minimal STO-3G basis set. The molecular basis is based on Löwdin-orthogonalized
atomic orbitals. The orbitals are ordered by their lattice position. The 2D lattice is mapped column by column onto
the one-dimensional MPS lattice. We identify each of the four columns as one cluster. To simulate various levels of
approximation, we optimize the cluster MPS for different combinations of inter- and intracluster bond dimensions.
We estimate the relative computational cost by
∑
i
D
3
i
, where
D
i
is the bond dimension for site
i
(such a cost estimate
becomes increasingly accurate in the asymptotically large limit). Fig. A shows that the cluster MPS never achieves a
smaller error, for a given computational cost.
a)
Electronic mail: larsson [a t] caltech . e
δ
u
b)
Present address: Quantum Simulation Technologies, Inc., Cambridge, MA 02139; Present address: Quantum Simulation Technologies,
Inc., Cambridge, MA 02139
c)
Electronic mail: garnetc [a t] caltech . e
δ
u
2
TABLE II. Energies in
E
H
for the chromium dimer using the cc-pVDZ-DK basis.
R/
Å MPS-MRCISD +Q
D
+Q
RD
+Q
M
+Q
P
MPS-MRCISDT
1.5 -2099.8150 -2099.8824 -2099.8911 -2099.8805 -2099.8945 -2099.9311
1.6 -2099.8366 -2099.9048 -2099.9139 -2099.9030 -2099.9174 -2099.9536
1.7 -2099.8400 -2099.9086 -2099.9178 -2099.9069 -2099.9215 -2099.9568
1.8 -2099.8372 -2099.9056 -2099.9148 -2099.9040 -2099.9186 -2099.9529
2.0 -2099.8317 -2099.8979 -2099.9068 -2099.8963 -2099.9105 -2099.9426
2.1 -2099.8310 -2099.8955 -2099.9041 -2099.8939 -2099.9075 -2099.9385
2.2 -2099.8312 -2099.8940 -2099.9022 -2099.8923 -2099.9054 -2099.9352
2.4 -2099.8326 -2099.8922 -2099.8997 -2099.8903 -2099.9024 -2099.9294
2.5 -2099.8328 -2099.8912 -2099.8985 -2099.8892 -2099.9010 -2099.9272
2.7 -2099.8319 -2099.8886 -2099.8956 -2099.8867 -2099.8980 -2099.9217
2.9 -2099.8295 -2099.8848 -2099.8914 -2099.8828 -2099.8936 -2099.9157
3.1 -2099.8263 -2099.8802 -2099.8866 -2099.8781 -2099.8886 -2099.9094
3.5 -2099.8199 -2099.8717 -2099.8776 -2099.8695 -2099.8793 -2099.8985
∞
-2099.8093 -2099.8582 -2099.8636 -2099.8559 -2099.8648 -2099.8899
atom -1049.9167 -1049.9323 -1049.9333 -1049.9287 -1049.9316 -1049.9416
10
−
5
10
−
4
10
−
3
10
−
2
10
−
1
10
5
10
6
10
7
10
8
10
9
10
10
Absolute error per H atom/
E
H
computational cost
ordinary MPS
cluster MPS D=2000
cluster MPS D=1000
cluster MPS D= 800
cluster MPS D= 600
cluster MPS D= 200
FIG. A. Comparison of accuracy vs. (relative) computational cost of a cluster MPS with an ordinary MPS for a 4x4 hydrogen
lattice.
D
in the legend refers to the intracluster
D
, while for a given symbol-line, the intercluster
D
is varied (i.e. each data
point along a given line is for a different intercluster bond dimension). See text for details.