eigenstate-analysis using sakurai-sugiura method with o(n)-dft … · 2018-08-22 ·...
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Eigenstate-analysis
using Sakurai-Sugiura method
with O(N)-DFT code CONQUEST
2018/8/17 @ ELSI workshop
Ayako Nakata1,2, Yasunori Futamura2, Tetsuya Sakurai2,
David R Bowler3, Tsuyoshi Miyazaki1
1 National Institute for Materials Science (NIMS)2 University of Tsukuba3 University College London
Today’s topics
2
Multi-site method + SS
⚫ Electronic structure analysis of
large materials using
Sakurai-Sugiura (SS) method and CONQUETS
Introduction of CONQUEST
O(N)-DFT + SS
O(N) DFT
Multi-site method
Large-scale DFT calculations
3
Large-scale model is needed
to represent specific
atomic and electronic structures
About 8,000 atoms
Scales cubically to
system size (# of atoms N) [O(N3)]
Usually ≤ 1,000 atoms
(N)
O(N3)
dopantstep
defect
Large-scale DFT with CONQUEST
4
Concurrent O(N) QUantum Electronic Structure Technique [1]
DFT code for large-scale systems
[1] http://www.order-n.org
Parallel efficiency in K-computer
Bulk Si,
constant # of atoms / node
Million atom systems: 5 min./MD step
[88,128 CPU = 705,024core]
Large-scale DFT with CONQUEST
5
Concurrent O(N) QUantum Electronic Structure Technique [1]
DFT code for large-scale systems
[1] http://www.order-n.org
⚫ Local orbital functions
⚫ Order-N method (Density matrix minimization)
⚫ High parallel efficiency
Local orbital functions
6
( ) ( )r c r
=Local orbitals
(Support function=SF)
Real-space basis functions
z … Radial functions
l … Angular momentum number
m … Magnetic quantum number
Radial functions Spherical harmonic functions
⚫ Finite-element (B-spline) basis (akin to plane-wave functions)
⚫ Pseudo atomic orbital (PAO) basis
( ) ( ) ( ) ( )lm l lm
i i ir r R r Yz z
= = r
Construct from basis functions
A. S. Torralba et al., J. Phys. Cond. Matt. 8, 20, 294206 (2008)
Oxygen PAOs
s-orbitals in TZDP
How to optimize electronic density
7
⚫ Density matrix minimization
⚫ Diagonalization
=HC εSC
O(N3)
H
S
C
e
: Electronic Hamiltonian
: Overlap matrix
: Eigenvectors (Bands/MOs)
: Eigenvalues (Band/MO energies)
→ O(N)
by using Scalapack
O(N) in CONQUEST
8
( ) ( )tot
6 4E
SLH HLS SLSLH SLHLS HLSLSL
= + − + +
tot T ps H xc ME E E E E E= + + + +
( ) ( )H
1' ' / '
2E d d n n= − r r r r r r
( ) ( )xc xcE d n e n= r r r
( )2
2
T'
, 'E dm
=
= − r rr r r
( ) ( )ps 2 ' ', , 'psE d d V = r r r r r r
( ) ( )2 ,n =r r r
( ) ( ) ( )
( ) ( )
*
*
, ' '
'
i i i
i
f
K
=
=
r r r r
r r
⚫ Density matrix minimization (DMM)
[3] E. Hernandez and M. J. Gillan, PRB_51_10157 (1995) [4] E. Hernandez, M. J. Gillan and C. M. Goringe, PRB_53_7147 (1996)
L = Auxiliary density matrix
O(N) in CONQUEST
9
⚫ Density matrix minimization (DMM)
3 2K LSL LSLSL= −
Spacial cutoff for L⇒ O(N)
E
L
K = Density matrix
L = Auxiliary density matrix
S = Overlap matrix
Energy minimization
keeping idempotency condition (LNV)
2 3( ) 3 2f x x x= −
Capability of CONQUEST
10
STM of Ge hut cluster on Si surface[Y.-W. Mo et al. PRL 65, 1020 (1990).]
Optimized geometry by CONQUEST[Miyazaki et al., JPSJ, 77, 123706 (2008).]
About 200,000 atoms Hydrated DNA (3,439 atoms) PBE/SZP[T. Otsuka et al, in preparation.]
Ab initio MDGeometry optimization
Si (32,768 atoms) LDA/DZP
[M. Arita et al, JCTC, 10, 5419 (2014).]
STM, STSSTM picture of Si (100) surface (-1.5eV)
Exptl CONQUEST
4.6 nm
TDDFT,
Constraint DFT, EXX,
Blue-moon ensemble
simulation,
…
Combination with extended Lagrangian method
Today’s topics
11
Multi-site method + SS
⚫ Electronic structure analysis of
large materials using
Sakurai-Sugiura (SS) method and CONQUETS
Introduction of CONQUEST
O(N)-DFT + SS
O(N) DFT
Multi-site method
Eigenstates with O(N) in CONQUEST
12
No 1-e- wave functions (Bands, MOs)
O(N) method☺ Low cost: O(N)
☺ Energies and geometries
Electronic states:Orbital pictures
Density of states
STM images …
(IPs, EAs, excitation energies)
Sakurai-Sugiura method: Projection method to obtain
eigenstates in a finite range
Metals??
Sakurai-Sugiura method
13
(1) N simultaneous linear equations(number of integration points on the curve)
Im
Re××× ××××× ××
( )j j − =B A y v
Projection method to obtain the eigenstates in a finite range
0 1 1
1 2
1 2 2
:
m
m
m
m m m
−
− −
=
H
1 2
2 3 1
1 2 1
:
m
m
m
m m m
+
+ −
=
H
m m −H H
( )
( )1
1
2k k
f zdz
i z
+
=−
( ) ( ) ( )1
1
0
1ˆ : , 0,1,
Nk
k k j j
j
f kN
−
+
=
= − =
( )2
exp , 0,1, , 1k D
iR k k N
N
= + = −
( )H , 0, , 1j jf y j n = −u
(2) m×m diagonalization(number of states in the region)
T. Sakurai, H. Sugiura, J. Comput. Appl. Math. 159, 119 (2003).
Sakurai-Sugiura method
14
Re××× ××××× ××× × × ××
Energy region of interest
Estimate the number of eigenstates in the region stochastically
by using Sylvester’ s law of inertia
NSS = number of quadrature points
MSS = number of complex moments
LSS = number of columns of the source matrix V (MSSLSS > m)
( )j j − =B A Y BV
T. Sakurai et al., Journal of Algorithms & Computational Technology, 7, 249 (2013).
Procedure
15
Construct Hamiltonian and overlap matrices with CONQUEST
Eigenstate calculation with Sakurai-Sugiura calculations
with zPares
(sum over k-points)
(save only lower-triangle elements, larger than threshold)
Read eigenvectors and eigenvalues with CONQUEST
MO / Band information Electronic properties
STM pictures
A. Nakata, Y. Futamura, T. Sakurai, D. R. Bowler, T. Miyazaki, J. Chem. Theory. Comput., 13, 4146 (2017)
[CQ] http://www.order-n.org [Z-pares] http://zpares.cs.tsukuba.ac.jp/
STM images
16
P2 on Si(100)
-2 eV +2 eV
6.1
nm
4 node
5,064 s
924 s
64 node
629 s
85 sSakurai-Sugiura
Scalapack (PDSYGVX)
(GGA/TDZP)
by PACS-IX (Intel Xeon E5-2670v2) in Univ of Tsukuba
(NSS=16, MSS=8, LSS=100)
Large systems
17
Ge hut cluster on Si(001)
⚫ Charge distributions
±0.01eV around Fermi level
43 states
213,633 states
20.6 nm
15.2
nm
2.1
nm
(GGA/SZP)
23,737 64 146
194,573 6,400 2,399
atoms node time
with K computer
STM of Ge hut cluster on Si surface[Y.-W. Mo et al. PRL 65, 1020 (1990).]
S. Arapan, D. R. Bowler, T. Miyazaki, arXiv:1510.00526.
(NSS=16, MSS=8, LSS=64)
Today’s topics
20
Multi-site method + SS
⚫ Electronic structure analysis of
large materials using
Sakurai-Sugiura (SS) method and CONQUETS
Introduction of CONQUEST
O(N)-DFT + SS
O(N) DFT
Multi-site method
Large-scale DFT with CONQUEST
22
Concurrent O(N) QUantum Electronic Structure Technique [1]
DFT program for large scale systems
⚫ High parallel efficiency
⚫ Order-N method (Density matrix minimization)
⚫ Local orbital function (support function)
Localized in finite region around each atom → sparse
Cost scales cubically to the number of functions
3aN
bN
Diagonalization
Order-N
N = number of atoms
( ) ( )lm lm
i i i
i
r c r
=
Multi-site functions
24
Single-site contraction
Linear combination of AOs on each atom
No contraction Primitive functions
Multi-site contraction ( ) ( )neighbors
k lm lm
i i k
k lm k
r C r
=
Contract to be minimal size
keeping primitive AOs’ accuracy
rMS
Linear combination of AOs in cutoff region
A. Nakata, D. R. Bowler, T. Miyazaki, Phys. Chem. Chem. Phys., 17, 31427 (2015)
Gradient wrt coefficients
25
( )( )ˆ4 j
i
EK H G
= +
rr
, ,
ik
i k i i k i
E E E
C C
= =
*
n n n n
n
G f u u e=Diag.
*
n n n
n
K f u u =
( ) ( )3 2G LHL LSLHL LHLSL = − +O(N)
( ) ( )3 2K LSL LSLSL = −
Advantages of multi-site method
26
⚫ Applicable to metallic systems
Cutoff is introduced for support functions
but not for wave functions
⚫ Improve accuracy as much as we want
Primitive Multi-site support functions
DZP
(ex) Si atom
TZDP
13
22
4
4
Always to be minimal size
Computational time
27
DNA in water
27883 4744
Multi-site SFs
(8 bohr) (16 bohr)
4744
Primitive AO
(DZP)
PBE / DZP
HA8000t (Kyushu Univ)
96 cores parallel
# of functions
28.0 34.5 484.8Time [sec.]
Matrix construction
12317.4 627.7 566.3Diagonalziation
Large reduction! (about 1/12)
3439 atoms
Gradient of coefficients 4.3 25.7ー
DOS of DNA in water (8 bohr)
28
Cutoff rMS = 8.0 bohr
[hartree]
[/h
art
ree]
☺ Occupied space
Unoccupied space
Black
Red
: Primitive AO (DZP)
: Multi-site SFs
DOS of DNA in water (8 bohr)
29
Black
Red
: Primitive AO (DZP)
: Multi-site SFs
[hartree]
[/h
art
ree] One-shot recalculation
using multi-site charge
with primitive AOs(by Sakurai-Sugiura method)
Cutoff rMS = 8.0 bohrBlue : Primitive AO (DZP) with
charge of multi-site SFs
MOs of DNA in water
30
HOMO-1 HOMO
SS
Exact
Diag.
LUMO LUMO+1
-1.4458947736
-1.4458947736 -1.2563299879 0.9071560148 0.9438977188
-1.2563299881 0.9071560149 0.9438977187
[eV]
Summary
31
Investigation of eigenstates (1e- wave functions)
becomes available even with O(N) method
by using Sakurai-Sugiura method
MO pictures of DNA
LUMOHOMO
Multi-site method makes available
accurate calculation with low computational cost
simulation of large metallic systems
Metallic nanoparticles