© nec corporation 2004 1 社会人再教育プログラム（コース１ -...

© NEC Corporation 2004 1

社会人再教育プログラム（コース１ - ナノマテリアル・ナノデバイスデザイン学）

励起状態 MD シミュレーション１励起状態のシミュレーション手法

宮本良之NEC 基礎･環境研究所


　　　　　　　　　　　　材料設計の方針1. ある特定元素を特定位置に配置→安定性、電子構

造2. 反応経路の特定→活性化障壁の最小パスの探査3. 効率のよい反応探査→光励起反応


普段、理論屋がなにげなくおこなっていること。。。

波動関数 |ψ> が求められている場合。

電流密度 J= 　 <ψ| |ψ>iħ ∂∂r

1m

何故だったか？　期待値の時間依存方程式

J= 　 <ψ| r |ψ>= <ψ| [H, r]|ψ>dd t ħ

i

何故だったか？　時間依存 Schrödinger 方程式

iħ |ψ>=H| ψ >ddt

しかし、通常は永年方程式 H|ψ>=ε|ψ> を解いている。密度汎関数理論で時間依存計算をして良いのか？

+ <ψ|[Vnl, r]|ψ>iħ


　　　　　　　　　　　　　　　　　　 Motivation (Nobody is doing!)Motivation (Nobody is doing!) Ab-initio approach on non-equilibrium electron dynamics

hν

e-

e-

e-

excited states

ground states

transport

1. Optical spectroscopy2. Electron-phonon coupling (adiabatic/nonadiabatic)3. Photo-induced dynamics (Ultrafast dynamics)


Ultrafast dynamics in femtosecond time constant.

Why fast? Absence/lowering of the activation barrierDiffusion on |g> takes ps, while on |e> takes fs.

F. Gai et al. Science 279, 1886 (1998)

|g>

|e>

An example: photo-isomerization of retinal


Can we do CP-like MD on |e> with the same manner as MD on |g>? Sometimes, yes.

Why yes?HOMO-LUMO long living excitation and no level alternation among differently occupied states throughout the simulation.

Mauri,Car [PRL 74, 3166 (1995)]: Diamond graphitization by optical excitation.

But these are NOT always the case.But these are NOT always the case.But these are NOT always the case.But these are NOT always the case.


密度汎関数理論→全エネルギー最小の原理

Etot=∑ifi<ψi|-Δ|ψi>+1/2∫∫ drdr’+∫Exc{ρ}ρ(r)dr

　　　 -∫ dr +∑ifi<ψi|Vnl|ψi>+Eion

| r-r’|ρ(r)ρ(r’)

|r-R|

ρ(r)Z(R)

=0 → HKSψi=εiψi 永年方程式δψi

δEtot

これを時間発展方程式に拡張するには量子力学での作用最小原理の考え方に戻る。　 L=<ψ|iħ ψ> 　 - 　 <ψ|H|ψ>d

dt

ただし DFT では <ψ|H|ψ> の換わりに Etot が来る。

LDFT ＝ =∑ifi <ψ|iħ ψ> - Etot

=0 　→　 iħ ψi=HKSψi

ddt

ddt

δLDFT

δψi


Hohenberg-Kohn :Physical Review 136, B864 (1964) 基底状態において電子状態と電荷密度は 1 対 1 対応

Runge-Gross, Physical Review Letters 52, 997 (1984)時間発展する電子状態は時間発展する電荷密度と 1 対 1 対応

正しい交換相関作用を示す汎関数さえ見つかれば時間発展問題も精度良く計算できる。


|e>

|g>

Reaction coordinate

Pot

entia

l How to do this? (Pseudopotentials, Plane-waves, LDA)

Time-dependent DFT for dynamics undeTime-dependent DFT for dynamics under electronic excitation.r electronic excitation.

More details:More details: Sugino & Miyamoto PRB Sugino & Miyamoto PRB 5959, 2579 (1999) , 2579 (1999) ; ;

ibid, Bibid, B6666, 89901(E) (2002)., 89901(E) (2002).

First-Principles Simulation tool for Electron-Ion DynamicsFirst-Principles Simulation tool for Electron-Ion Dynamics

First principles molecular dynamics under electronic excitation.

TM


|g>

MD under electronic excitation [Sugino, Miyamoto,PRB 59, 2579, (1999)]

t = 0: Promote the electronic occupations to mimic the excited states. Then perform the static SCF calculation.

No Observation of the nonradiative decay! lifetime,

decay path

Yes Do MD.

Hellmann-Feynman theorem works

|e>

t > 0: Solve ψn(t+Δt)=exp{- i ΔtH(t)} ψn(t).

Is Matrix of HKohn-Sham diagonal?

1. No need of level assignment for a hole and an excited electron except at the beginning.

2. Automatic monitoring of the nonradiative decay (lifetime, decay path) without experiences.


時間依存 Schrödinger 方程式

i |ψ>=H|ψ> 　をどう解く？ddt

両辺を時間積分！　 ψ (t0) がわかっている場合

ψ(t0+dt)=ψ(t0)+∫(-idt)H(t1)ψ(t1)dt1t0

t+dt

= ψ(t0)+∫(-idt) H(t1){ψ(t0)+∫(-idt) H(t2)ψ(t2)dt2} dt1t0

t+dt

t0

t1

= …….


Integral of the time-dependent Schrödinger equation, time evolution of wave functions is obtained as follows.

So must be smaller than typical period of plasma oscillation.

Potential V(t) depends on the charge density.

t

Here, ignorance of T means ignoring the following commutation relation.


How to treat exp(-idtH)ψ?

Taylor expansion: 1 -idtH + (-idtH)2 + (-idtH)3 + (-idtH)4 + ….2!1

3!1

4!1

Use of eigen vectors and eigen values:

exp(-idtε1)exp(-idtε2)

exp(-idtεn)

...0

0

UU†

Numerical instability beyond 100 fs.

Diagonalization at every timestep is needed


Split operator method (Trotter formula)

When H = Δ + Vexp (-idtH)≈exp( Δ)exp(idtV)exp( Δ)

2-idt

Accurate in the second order.

1. Needed more higher order of accuracy.2. Pseudopotentials: H = Δ + ∑τVnl(τ) + V

2-idt


The fourth order decomposition is

Here, and

.

.

Suzuki-Trotter formula (split operator for e-iΔtH)M. Suzuki, and T. Yamauchi, J. Math. Phys. 34, 4892 (1993).

The second order decomposition of exponential of H is

.

Suppose H consists of non-commutable operators as

.


Adapted Hamiltonian(Kohn-Sham - Pseudopotentials)

Wavefunction expression is

or

.

Exponential of each operator of H acts as

????????

< Application for the plane-wave method >

FFT

.

,

,


Here the nonlocal pseudopotentials are adopted as

(separable K-B type) .

Then we know

and .

So the exponential is

which can directly be operated to .

,


Gcut

Gcut

Gcut: for charge

Gcut: for Wfs

∑Gρ(G)exp(iGr)=∑G,G’ψ*(G)ψ （ G‘)exp{-i(G-G’)r}: convolution

Full grid is necessary for time-evoluving WFs to avoid numerical noise.

or FFT

exp(-iΔ/2dt)exp(-iV/2dt).....ψ(r)


For charge density and local potential

(G) → (G)f(G,Gcut) and V

H xc(G) → V H xc(G) f(G,Gcut).

G2-Gcut2

f(G,Gcut)= 1/[1 + exp( )].Gcut

2/10

Car-Parrinello-like smoothing technique in G-space

For kinetic energy operator

- /2 = max { G2/2, Gcut2/2 }.

Ekin

GGcut

G2

f

Gcut2


Railway curve interpolation scheme for SCF predictor-corrector routine.

t+Δt

t-Δt

st

Repeat this part until theself-consistency betweenn,k and VHxc

is achieved.

VH xc(s)=( )2 [3VHxc(t)+ΔtVH xc(t)+ ( 2VHxc (t)+ΔtVHxc (t) )]Δt Δt

s -t-Δts -t-Δt ..

. .s - tΔt Δt

s - t

Iri and Fujimoto, Common sense for numerical calculations(Kyoritsu, Tokyo, ‘85), p116 (Japanese)H. Akima, J. Assoc. Comput. Machinery, Vol. 17, 589 (1970).

Eq. 14 in Sugino Miyamoto (PRB) was mistyped!

+( )2 [ 3VHxc (t+Δt)-ΔtVHxc (t+Δt) - ( 2VHxc (t+Δt) - ΔtVHxc (t+Δt) )]


CPU 0:

ρ(t+Δt)=Σ|Ψi(t+Δt)|2

HKS (t+Δt)=HKS{ρ(t+Δt)}

CPU 1: Ψ1(t+Δt)=exp{-i/ħΔtHKS(t)} Ψ1(t)

CPU 2: Ψ2(t+Δt)=exp{-i/ħΔtHKS(t)} Ψ2(t)

CPU n: Ψn(t+Δt)=exp{-i/ħΔtHKS(t)} Ψn(t)

|Ψ1(t+Δt)|2

|Ψ2(t+Δt)|2

|Ψn(t+Δt)|2

HKS (t+Δt)

HKS (t+Δt)

HKS (t+Δt)

Simple communication!Simple communication!

Parallel computing


Advantage of the split-operator methodAdvantage of the split-operator method

1. Ψn(t+Δt)=exp{-i/ħHKS(t)Δt}Ψn(t) can be parallelized. (No need of orthonormalization.)

2. No need to identify occupied and unoccupied bands throughout the simulation, except the beginning.

Disadvantage of the split-operator methodDisadvantage of the split-operator method

1. Full grids of FFT should be used for extending plane waves to express Ψn(t). (Large core memory is needed.)

2. Many FFT operations are necessary.

Suitable computer system:Suitable computer system:Many processors, each of which has large memory size and high FFT performance. (Vector-parallel computer system.)


Applications of the TDDFT-MD schemeApplications of the TDDFT-MD scheme1. Re-activation of Si donor in GaAs by electronic excitation – Mi

yamoto, Sugino, and Mochizuki, Appl. Phys. Lett. 75, 2915 (1999).

2. H-desorption from Si(111) by electronic excitation - Miyamoto, and Sugino, Phys. Rev. B 62, 2039 (2000).

3. Si-H bond weakening around defects in SiO2 by electronic excitation – Yokozawa and Miyamoto, J. Appl. Phys. 88, 4542 (2000).

4. Br-desorption from Si(111) by photoemission – Miyamoto, Solid State Comm. 117, 727 (2001). (TDLDA and TDGGA, Cutoff 8 Ry, t=0.5 a.u.=1.21x10-2 fs)

5. Self-healing of vacancies in nanotubes, Miyamoto, Berber, Yoon, Rubio, and Tománek, Chem. Phys. Lett. 392, 209 (2004).

6. Induction of localized vibration on SW defects in nanotubes, Miyamoto, Rubio, Berber, Yoon, and Tománek, Phys. Rev. B69, 121413(R ) (2004).


御清聴ありがとうございました

次回 12 月 1 日（水）：励起状態 MD シミュレーション２「ナノスケール物質の高速現象（事例）」

© nec corporation 2004 1 社会人再教育プログラム（コース１ -...

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