Tensor Network States: Algorithms and Applications
Dec. 5, 2014 @ Beijing
Tensor Network in Chemistry:
Recent DMRG/TTNS Studies and Perspectives for
Catalysis Research
Naoki Nakatani
Catalysis Research Center, Hokkaido University, Japan
Catalysts
Without Catalyst
Ea
E'a
Catalyst promotes chemical reaction by lowering the energy barrier
However, reaction mechanism becomes much complicated…
With Catalyst
What’s
happened?
Reaction Coordinate
Reaction rate is given by
k e

Ea
RT
Theoretical understanding plays an important role for designing new catalysts
What Catalyst is desired?
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
Se
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Fr
Ra
Ac
Difficult to use but very cheap
Widely used but very expensive
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Th
Pa
U
Np
Pu
Am Cm
Bk
Cf
Es
Fm
Md
No
Lr
Designing a new iron catalyst: You can get much money!!
DMRG-related Activity in Chemistry
• Why we need DMRG?
• Providing a convenient tool to doing DMRG for Chemistry
• Get starting practical applications
Single-reference Ansatz (post-HF approach)
virtual
Common molecules, Mean-field approximation (i.e. Hartree-Fock theory) works very well
1 electron configuration covers more than 99% of electronic energy for the
ground state
The rest can be treated by a small perturbation
occupied
Many-body wavefunction can be spanned by taking particle excitations
from a reference configuration
Ψ  Ψ H F  c i E i Ψ H F  c ij E i E j Ψ H F 
a
Ψ HF
dominant
E i Ψ HF
a
ab
a
b
a
dominant
small perturbation
Multi-reference Ansatz (CASSCF approach)
Secondary
Transition metal complex, Mean-field approximation doesn’t work anymore
Active Space
Defining “Active Space” (AS), orbital subspace
which can describe important contributions
Inactive
Computational cost: O(N!) for AS
AS is limited up to 14~16 orbs.
Ψ CA SSCF 

n1
Inside AS:
dominant
Ψ CA SSCF
Outside AS:
small perturbation
C
n1
nk
nk
1,  , k   A S
n1
nk
Practical DMRG Calculations in Quantum Chemistry
DMRG can be applied to solve CASSCF wavefunction
C
12 34
β
α
β
α
 T r A1 A 2 A 3 A 4
σ
where A i : M  M
Computational cost: O(M3k3+M2k4)/sweep
α
Memory cost: O(M2k2)
β
1234
Disk storage: O(M2k3)
QC-DMRG code can approach ca 100 sites (orbitals) calculation
AS limitation is considerably eased
DMRG-CASSCF Implementations
Several implementations of DMRG-CASSCF are available
Used by only
few groups
Self-involved packages:
• ORZ package: DMRG-CASSCF/CASPT2 in T. Yanai group (obtain upon request)
• ORCA package: DMRG-CASSCF/NEVPT2 in F. Neese group (free for academic use)
• Most users (including me) don’t like to use a new code because reading the manual is tough!!
• It’s so hard for small groups to implement everything from scratch
(e.g. integrals, SCF, geometry opt., relativistic effect, etc…)
Integration to the conventional (i.e. well-established) package is important
DMRG on Molcas package
It’s not free software, but they already have a lot of users
&rasscf
symmetry= 1
spin= 1
nactel= 16 0 0
inactive= 49 0
ras2= 0 16
lumorb
ciroot= 1 1 1
dmrg= 100
Normal CASSCF input in Molcas
Only 1 line is required to carry out DMRG-CASSCF calc.
All other settings are automatically determined for novice users
Hopefully, we can get many DMRG/TNS people in near future!!
Benchmark: DMRG-CASSCF/cu4-CASPT2
NC dependency of polyene ground state (AS : full π-valence)
1.E+05
1.00
only CASSCF
0.80
0.60
EDMRG − ECAS / mEh
Wall time / sec.
1.E+04
CASSCF/CASPT2
1.E+03
DMRG-CASSCF/cu4-CASPT2
(M = 100)
1.E+02
0.40
DMRG-CASSCF
(M = 100)
0.20
0.00
-0.20
-0.40
cu4-CASPT2
-0.60
-0.80
1.E+01
-1.00
6
8
10
12
14
16
18
NC
20
6
8
10
12
14
16
18
NC
DMRG-CASSCF/CASPT2 shows polynomial scaling
Energy agrees very well with CAS
DMRG-CASSCF scales better for (14e, 14o) and larger
Negative error in CASSCF might be a numerical error
20
Benchmark: DMRG-CASSCF Geometry Optimization
[Fe2S2(CH3)4]2−
Density Functional Theory (BP86)
S
Fe
Fe
S
HS
Fe-Fe = 3.003 Å / S-S = 3.514 Å
LS
Fe-Fe = 2.716 Å / S-S = 3.526 Å
HS-LS Gap = 0.86 eV
CASSCF(10 elec., 10 orbs.)
HS
Fe-Fe = 3.076 Å / S-S = 3.567 Å
LS
Fe-Fe = 3.045 Å / S-S = 3.575 Å
HS-LS Gap = 0.06 eV
CASSCF(22 elec., 16 orbs.) – performed by DMRG-CASSCF
HS
Fe-Fe = 3.065 Å / S-S = 3.551 Å
LS
Fe-Fe = 3.011 Å / S-S = 3.557 Å
HS-LS Gap = 0.12 eV
Tree Tensor Network States for QC
• What’s the entanglement structure of molecule?
• Efficient QC-DMRG algorithm on Tree graph lattice
• Illustrative calculations
TTNS: Note for Structure
Two different types of TTNS were proposed: Here, focused on TPS type TTNS
“Layered” TTNS
“TPS-type” TTNS
TTNS where each site has
physical index
Ψ 

n1
nk
tr A
n1
A
ni
A
nk
n1
nk
Entanglement Structure of Molecule
H2O / cc-pVDZ
24
1
23
2
Plotted exchange interaction b/w two orbitals
3
22
(main source of entanglement)
4
21
5
20
6
7
19
Hamiltonian is highly non-local
Hˆ 

ij σ
18
†
h ij aˆ i σ aˆ j σ 
1
2

†
†
ij lk aˆ i σ aˆ j σ aˆ k σ aˆ l σ
ijkl σ σ 
8
9
17
10
16
15
14
13
12
11
Hard to map entanglement onto 1D-lattice!
Somehow to map onto 1D-lattice for DMRG
Heuristic algorithm to minimize a cost function
F 
K
2
ij
R ij
ij
7
17
6
13 12
8
24
9
18 22
3
19 10
4
2
23 16 21 14
5
15 20 11
Exchange interactions to be closer as possible
Even though, there still be a lot of non-local interaction!!
Can any Tensor Network approach give the better computational scaling?
1
TTNS: Renormalization along with Tree Graph
Biparticity of tree graph (i.e. no cycles) is useful to keep “Canonical Form” of wavefunction
MPS
O(M3k3+M2k4) per sweep
Having the same super-block structure
TTNS
O(M4k3+M2k5) per sweep
LEFT
RIGHT
TTNS is one of natural ND-generalizations of MPS
• Non-local interaction can be considered without much
increase of computational scaling
• Many algorithms for MPS can be reused for TTNS
DMRG Algorithm on TTNS
DMRG on MPS
aˆ i
aˆ j
DMRG on TTNS
aˆ i aˆ j
aˆ j
aˆ i
O(M3k2) per site
O(M4k2) per site
aˆ i aˆ j
2-Site Algorithm
Half-Renormalization (HR): To reduce the variational space to be searched for each step
M
4
4
n1
n2
M
M
M
M
M2 → 4M
M2 → 4M
4
4
n1
n2
4M
4M
M
Reduced computational cost without loss of accuracy!
What “shape of tree” should be used?
Minimum Spanning Tree (MST)
Minimum Entangled(?) Tree (MET)
Mmax = 410
21
Mmax = 48
18
14
12
19
15
24
1
7
23
3
24
11
20
1
12
7
6
23
2
5
9
22
4
11
5
2
16
4
3
10
8
13
9
19
17
20
18
22
E = −76.243652 (M = 200)
CPU: 1210 sec.
16
8
21
15
6
10
14
13
17
E = −76.243491 (M = 200)
CPU: 827 sec.
N2 molecule: Case where TTNS works better
Large entanglement due to triple bond breaking
R = 1.1208 Å
R = 1.4288 Å
R = 1.9050 Å
The same accuracy can be obtained from
TTNS with less than half # states in MPS
Computational scaling is also better in TTNS
Cr2 molecule: Case where MPS works better
Strong correlation from sextuple bond in Cr2
Number of states can be reduced in TTNS
but the total performance scales better in MPS
Densritic molecule: Case where TTNS works extremely fine
Stilbenoid dendrimer: Model for Light-Harvesting system in Photosystem
M
100
MPS
ΔE / Eh
CPU / swp. sec.
-1.9437
43146
M
50
TTNS
ΔE / Eh
CPU / swp. sec.
-2.0214
19162
2-4 times faster in TTNS
g = 2 (110e, 110o)
Summary
DMRG on Molcas package
•
•
•
•
DMRG code (G. Chan) has integrated into conventional QC program so that novice users can carry out
DMRG calculation much easier
Large active space (~ 100 orbs.) can be taken into account
Geometry optimization, perturbation treatment, etc. can be used without further coding
Hoping we can get a lot of DMRG users in Chemistry
•
For me, now I’ve got the useful tool to do catalysis research
Tree Tensor Network States (TTNS)
•
•
•
Introduced and developed efficient DMRG algorithm on TTNS for Quantum Chemistry
Performance is highly depending on system
So far, it can introduce another freedom to chose a lattice where entanglement structure of molecule
is to be mapped