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