Density Functional Perturbation Theory in FHI-aims · 2018-07-16 · Density Functional Perturbation Theory in FHI-aims Honghui Shang Fritz Haber Institute of the Max Planck Society,

Density Functional Perturbation Theory in FHI-aims

Honghui Shang

Fritz Haber Institute of the Max Planck Society, Berlin

FHI-aims Developers’ and Users’ Meeting, July 10, 2018

H. H. Shang (FHI) DFPT in FHI-aims July 10 1 / 20

Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why perturbations?

neutron

phonon

PRB,43,7231 (1991)

electric field

dielectricconstants

www.insifindia.com

magnetic field

magneticresonance

radiopaedia.org

ice and fire

thermoelectricity

en.wikipedia.org

Perturbations⇓

Physical Properties


Why density-functional perturbations theory?

Physical properties are related to total energy derivatives

atomic displacement⇒ mixed ⇐ electric fielduI εI

1st order∂E

∂uI

∂E

∂εI(force) (dipole moment)

2ed order∂ 2E

∂uI∂uJ

∂ 2E

∂uI∂εJ

∂ 2E

∂εI∂εJ(force constants) (IR intensity) (polarizability)

(Born effective charges) (dielectric constant)

3rd order∂ 3E

∂uI∂uJ∂ ∆V

∂ 3E

∂εI∂εJ∂uK

∂ 3E

∂εI∂εJ∂εK(Gruneisen parameters) (Raman intensity) (hyper-polarizability)



Physical properties are related to total energy derivativesAnalytic evaluation of total energy derivatives

1H. F. Schaefer and Y. Yamaguchi, J. Mol. Struct. THEOCHEM 135, 369 (1986).H. H. Shang (FHI) DFPT in FHI-aims July 10 3 / 20


Physical properties are related to total energy derivatives

Finite Difference1 DFPT/CPSCF234

⊕Easy to implement Hard to implement

Suffer from numerical accuracy ⊕Analytical calculation

Slower ⊕ Faster

1K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).2J. Gerratt and I. M. Mills, J. Chem. Phys. 49, 1719 (1968).3S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515

(2001).4X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997).


DFPT for atomic displacement : concept

Scheme Reciprocal-space12 Real-space

dynamical matrix DIJ(q)

1. dynamical matrix with coarse q grids

DIJ(q) =1√

MIMJ∑m

∑n

d2Etot

dRImdRJnexp(iq · (Rm−Rn)) .

1S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515(2001).

2X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997).H. H. Shang (FHI) DFPT in FHI-aims July 10 4 / 20



dynamical matrix DIJ(q)↓

force constants ΦIm,J(Rm)

2. force constants

ΦIm,J =1

Nq∑q

√MIMJDIJ(q)exp(iq ·Rm) .






force constants ΦIm,J(Rm)↓

dynamical matrix DIJ(q′)

3. dynamical matrix with dense q grids

DIJ(q) =1√

MIMJ∑m

ΦIm,J exp(iq ·Rm) .



DFPT for atomic displacement: concept

equilibrium density

density of distorted system

linear response density

∂n(r)

∂uIis localized

Perturbation inherently localize1F. Giustino, M. L. Cohen, and S. G. Louie . Phys. Rev. B 76, 165108 (2007)



Bloch orbital Wannier orbital1

e-ph gqλ

n′nk = 〈n′k+q|∂λqVscf|nk〉〈χµ0|∂κRpVscf |χνR〉

1F. Giustino, M. L. Cohen, and S. G. Louie . Phys. Rev. B 76, 165108 (2007)H. H. Shang (FHI) DFPT in FHI-aims July 10 5 / 20


Bloch orbital Wannier orbital1

e-ph gqλ


⇓Bloch orbital Atomic orbital

e-ph gqλ


1F. Giustino, M. L. Cohen, and S. G. Louie . Phys. Rev. B 76, 165108 (2007)H. H. Shang (FHI) DFPT in FHI-aims July 10 5 / 20

Real-Space DFPT: atomic displacement

Scheme Reciprocal-space Real-space 1


force constants ΦIm,J(Rm) ΦIm,J(Rm)↓

dynamical matrix DIJ(q′)

2. force constants directly from real-space

ΦIm,J =d2Etot

dRImdRJ=− dFJ

dRIm.

1H. H. Shang, C. Carbogno, P. Rinke and M. Scheffler, Comput. Phys. Commun.215, 26 (2017)



2. force constants directly from real-space

ΦIm,J =d2Etot

dRImdRJ=− dFJ

dRIm.

centers unitcell

auxiliarysupercell

unitcell

Born-von Karman boundary condition




Scheme Reciprocal-space Real-space 1


force constants ΦIm,J(Rm) ΦIm,J(Rm)↓ ↓

dynamical matrix DIJ(q′) DIJ(q′)

3. dynamical matrix with dense q grids

DIJ(q) =1√

MIMJ∑m

ΦIm,J exp(iq ·Rm) .



Real-space DFPT: atomic displacement

1st-order density

1st-order potential

1st-order H matrix

1st-order wave fc

force constants

1 st-order S matrix

0 st-order density

dynimical matrix

DFPT

DFT 1. get DFT density



1st-order density

1st-order potential

1st-order H matrix

1st-order wave fc

force constants

1 st-order S matrix

0 st-order density

dynimical matrix

DFPT


2. get first order overlap

+

+



1st-order density

1st-order potential

1st-order H matrix

1st-order wave fc

force constants

1 st-order S matrix

0 st-order density

dynimical matrix

DFPT



3. begin DFPT cycle

+

+



1st-order density

1st-order potential

1st-order H matrix

1st-order wave fc

force constants

1 st-order S matrix

0 st-order density

dynimical matrix

DFPT



3. begin DFPT cycle



1st-order density

1st-order potential

1st-order H matrix

1st-order wave fc

force constants

1 st-order S matrix

0 st-order density

dynimical matrix

DFPT



3. begin DFPT cycle

4. get force constants

5. get dynamical matrix


Real-space DFPT: results

400 600 800 1000 1200 1400 1600

Inte

nsity

Frequency (cm)-1

finite-differenceDFPT

0 250 500 750 1000 1250 1500

Frequency (cm) 1

0.00

0.02

0.04

0.06

0.08

0.10

Inte

nsi

ty

finite-difference

DFPT

0

500

1000

1500

2000

2500

Γ K M Γ

Freq

uenc

y (c

m-1

)

Graphenefinite difference

DFPT-

0

200

400

600

Γ X W K Γ L

Freq

uenc

y (c

m-1

)Silicon

DFPTfinite-difference


Real-space DFPT: scaling

Parallelization

real-space grid : Ref. [1]

matrix operation : MPI based ScaLapack

1

10

100

1000

10000

100000

8 14 50 98 194 542

Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms

H(C2H4)nH molecules (n=1-90)

n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT)

10

100

1000

6 12 24 48 72

Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms in unit cell

Polyethylene Chain

n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT)

1V. Havu, V. Blum, P. Havu, and M. Scheffler, J. Comput. Phys. 228, 8367 (2009).H. H. Shang (FHI) DFPT in FHI-aims July 10 9 / 20

Real-space DFPT: scaling

1

10

100

1000

10000

100000

8 14 50 98 194 542

Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms


n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT)

10

100

1000

6 12 24 48 72

Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms in unit cell

Polyethylene Chain

n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT)

T ∼ Nα H(C2H4)nH C2H4 chainα(DFT ) = 1.9 N N 6 24 N > 24

n(1) 2.0 1.7 2.0

V(1)es,tot 2.4 1.0 2.8

H(1) 2.0 1.4 2.0

P(1) 3.8 1.2 3.3

Total 2.6 1.3 2.5H. H. Shang (FHI) DFPT in FHI-aims July 10 9 / 20

Real-space DFPT: speedup

Parallel efficiency

efficiency = T (1)/[T (N)/N]

Using 1024 core, the parallel efficiency is 75%

10

100

32 64 128 512 1024Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

for

one

ato

ms

one

coor

d

Number of CPU cores

Extended Si system (1024 atoms in unit cell)

n(1)(r)

Ves,tot(1)(r)

H(1)

P(1)

Total (DFPT)ideal scaling

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

32 64 128 512 1024

Para

llel E

ffic

ienc

y

Number of CPU cores

Total (DFPT)ideal

Open Access


Real-space DFPT: speedup

Parallel efficiency



10

100

32 64 128 512 1024Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

for

one

ato

ms

one

coor

d

Number of CPU cores

Extended Si system (1024 atoms in unit cell)

n(1)(r)

Ves,tot(1)(r)

H(1)

P(1)

Total (DFPT)ideal scaling

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

32 64 128 512 1024

Para

llel E

ffic

ienc

y

Number of CPU cores

Total (DFPT)ideal

Open Access


Real-space DFPT: applications in electronic friction

The nonadiabatic coupling matrix is defined as

Z jkn,n′ =

⟨ψnk

∣∣∣∣ej ∂

∂R

∣∣∣∣ψn′k

⟩(1)

=ejcn′k(H

(1)k − εnkS

(1)Lk − εn′kS

(1)Rk )cnk

εn′k− εnk(2)

Interface with Coolvib by Reinhard J. Maurer(Vibrational cooling of adsorbates on metal surfaces)

1R. J. Maurer, M. Askerka, V. S. Batista, J. C. Tully, Phys. Rev. B. 94, 115432(2016)



Optimized the DFPT code

LDA PBE0

1000

2000

3000

4000

5000

6000

CPU

tim

e (

s) Original

Optimized

Original

Optimized

lapack

21X25X

0

50

100

150

200

250

CPU

tim

e (

s)8X

LDA PBE

OriginalOptimized

Original

Optimized

scalapack



Make it comparable with DFT computation

0.1

1

10

100

1000

14 50 98 194 542 770

CPU

tim

e pe

r ite

ratio

n (s

)

number of atoms

(C2H4)n line (n=8-128) aims.180126.scalacpk.mpi.x at DRACO

DFPTDFT

DFT-force


Real-space DFPT: electron-phonon interaction

finite differences

∆εnk(T ) =1

Nq∑qλ

∂εn

∂nqλ (0)[nqλ (T ) +

1

2] (3)

=1

Nq∑qλ

}2ωqλMqλ

∂ 2εnk

∂z2[nqλ (T ) +

1

2]

(4)

here qλ is phonon index, nk is electric band index, phonon number is

n(qλ ) =1

exp(}ωj(q)/kBT )−1.



analytical method

∆εAHCnk (T ) =

1

Nq∑qλ

∂εn

∂nqλ (0)[nqλ (T ) +

1

2] (5)

=1

Nq∑qλ

}2ωqλMqλ

∂ 2εnk

∂z2[nqλ (T ) +

1

2]

=1

Nq∑qλ ∑n′ [

2|gqλ

n′nk|2

εnk−εn′k+q−i0+ ][nqλ (T ) +1

2]

+1

Nq∑qλ < nk|hks|nk>(2) [nqλ (T ) +

1

2]

1P. B. Allen and V. Heine, J. Phys. C 9, 2305 (1976)2P. B. Allen and M. Cardona, Phys. Rev. B 23, 1495 (1981)



gqλ

n′nk =

⟨ψn′k+q

∣∣∣∣ dvscf

duλq

∣∣∣∣ψnk

⟩= ∑

µ,ν ,κ∑R,Rp

Cn′µ (k+q)Cnν (k)uλκ (q)eiqRpeikR〈χµ0|∂κRpvscf|χνR〉 (6)

Arbitrary q,k points by FT.


DFPT: electric field

1st-order density

1st-order Hamiltonian

1st-order expansion coefficients

electronic density

CPSCF/DFPT

DFT

1st-order density matrix

Polarizability

momentum matrix

1. get DFT density

2. get momentum matrix

3. begin DFPT cycle

4. get Polarizability

αIJ =∫

rI∂n(r)

∂ξJdr (7)

high frequency dielectricconstant:

ε∞IJ = δIJ +

4π

Vuc

∫rI

∂n(r)

∂ξJdr (8)

= δIJ − (− 4π

Vuc

∫rI

∂n(r)

∂ξJdr)



Exp.this work NCPP NCPP PAW PAW

(all 1991 1996 2006 2016electron)

LDA PBE LDA PBE

Si 12.1 13.2 12.9 13.6 - 13.3 13.1AlP 7.5 8.4 8.2 - 8.2 8.3 8.1AlAs 8.2 9.5 9.5 9.2 9.3 - 9.5AlSb 10.24 11.7 11.9 12.2 11.4 - 12.1GaP 9.0 10.6 10.6 - 10.0 - 10.6GaSb 14.44 16.0 15.5 18.1 16.7 - -

Tabelle: Comparison of the high-frequency dielectric constants of varioussemiconductors computed at the LDA/PBE level with literature values:Tight-default settings and basis sets as well as a 16× 16× 16 k-point mesh areused.



0

2

4

6

8

10

12

14

minimal tier 1 tier 2 tier 3 tier 4

|α- α

(tier

4)|

[Boh

r3 ]

αxxαyyαzz

ethylene

0

2

4

6

8

minimal tier 1 tier 2 tier 3

|ε xx

- ε x

x(ti

er 3

)|

Si

Convergence behaviour of the polarizabilities αxx ,αyy ,αzz of ethylene andof the high-frequency dielectric constant ε∞

xx of bulk silicon (16×16×16k-points) with respect to the basis set size



0.01

0.1

1

10

100

50 98 194 542 770Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms


n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT) 0.01

0.1

1

10

100

1000

10000

64 128 256 512 1024Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of atoms

Diamond

n(1)(r)Ves,tot

(1)(r)

H(1)

P(1)

Total (DFPT)

H(C2H4)nH Diamond

n(1) 1.1 1.4

V(1)es,tot 1.6 1.4

H(1) 1.2 1.5

P(1) 2.5 2.6

Total 1.3 1.4



Parallel efficiency



10

100

1000

64 128 512 1024Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of cores

Polyethylene Chain, 768 atoms in unit cell aims.180126.scalapack.mpi.x at DRACO

DFPT

DFTDFT-force



Parallel efficiency



10

100

1000

64 128 512 1024Tim

e pe

r D

FPT

s.c

.f. i

tera

tion

(s)

number of cores

Polyethylene Chain, 768 atoms in unit cell aims.180126.scalapack.mpi.x at DRACO

DFPT

DFTDFT-force


Summary

To summarise, we have derived and implemented DFPT formalism inFHI-aims

1 atomic displacement: vibration, phonon2 electric field: polarizability, dielectric constant

Todo benchmarks:

1 Born effective charge2 electron-phonon coupling


2H. H. Shang, N. Raimbault, P. Rinke, M. Scheffler, M. Rossi and C. Carbogno, NewJ. Phys. (2018) https://doi.org/10.1088/1367-2630/aace6d


Acknowledgement

Many valuable discussions in DFPT@FHI-aims meetingsDr. Nathaniel Raimbault Dr. Raul LaasneDr.Christian Carbogno Dr. Mariana RossiDr. Danilo Brambila Dr. Reinhard J. Maurer

Dr. William Huhn Dr. Heiko AppelProf. Xinguo Ren Prof. Volker Blum

Continued supportsProf. Patrick Rinke Prof. Matthias Scheffler


Summary

Thanks!


Current Status

DFPT electricfield

polar

dielectric

atomicmove

phonon

reduceCPU

reduceme-

mory

vibrationreduceCPU

reduceme-

mory


Current Status

0

100

200

300

minimal tier 1 tier 2 tier 3

|ω- ω

(tie

r 3)|

[cm

-1]

3036 cm-1

2955 cm-1

1421 cm-1

1330 cm-1

785 cm-1

300 cm-1


Documents

Density Functional Perturbation Theory in FHI-aims · 2018-07-16 · Density Functional Perturbation Theory in FHI-aims Honghui Shang Fritz Haber Institute of the Max Planck Society,