An introduction to nonadiabatic molecular dynamics

Hands on workshop on DFT, Beijing

Aug. 3, 2018

Sheng Meng (孟胜) Institute of Physics,

Chinese Academy of Sciences 2018.8.3

An introduction to nonadiabatic molecular dynamics

I. Motivation

II. Theory

III. Implementation

IV. Applications

- NV center dynamics

OUTLINE

I. Background: What is nonadiabatic dynamics?

Adiabatic process Thermodynamics: A process occuring without transfer of heat or matter

between a system and its surroundings.

Quantum mechanics: In the quasi-static change of a parameter, the system stays in the same (eigen)state (no quantum transitions).

“A physical system remains in its instantaneous eigenstate if a

given perturbation is acting on it slowly enough and if there is a gap

between the eigenvalue and the rest of the Hamiltonian's spectrum”

-Born and Fock, 1928

• parameter = ? hν; R

• how slow? (relative to gap)

• the same state ? • approximation !

nonadiabatic

adiabatic

Adiabatic process

Nonadiabatic process

• Excited states

• Metallic systems

• Transport

• Electron-phonon coupling

• Superconductivity

• Chemical reactions

• Conical intersection

• …

Nonadiabatic effects widely exist

Courtesy: A. Rubio

George Wald, Nobel Prize 1967

Arieh Warshel, Nobel Prize 2013

Understanding vision

A light-driven worm-like nanocar

Sasaki & Tour, Org. Lett (2008).

Pisana et al., Nat. Mater. 6, 198 (2007).

II. Theory: where the story really starts …

• Two-component quantum system: electrons + nuclei / ions

Ψ 𝑟, 𝑅, 𝑡 ≡ Φ𝑅 𝑟, 𝑡 𝜒 𝑅, 𝑡 ≅ Φ𝑅0(𝑡) 𝑟 𝛿 𝑅(𝑡) − 𝑅0(𝑡)

• Born-Oppenheimer (BO) approximation (classical & adiabatic):

Classical ion trajectory;

Coupled electron-ion dynamics is neglected.

• Consequence of BO approximation:

Non-adiabatic effects!

Ψ 𝑟, 𝑅, 𝑡 ≡ Φ𝑅 𝑟, 𝑡 𝜒 𝑅, 𝑡 ≅ Φ𝑅0(𝑡) 𝑟 , 𝑡 𝛿 𝑅(𝑡) − 𝑅0(𝑡)

Ψ 𝑟, 𝑅, 𝑡

Full quantum dynamics

Potential energy surface (PES)

(Born-Oppenheimer (BO) approximation) （Assuming ）

Born-Oppenheimer (BO) dynamics

(nonadiabatic couplings)

(BOMD/CPMD; AIMD)

(Born-Huang expansion)

Nonadibatic dynamics:

Full quantum treatment

Nuclear-electronic orbitals (Quantum nuclei)

Hammes-Schiffer et al., JPCA 110, 9983 (2006); JPCL 9, 1765 (2018).

Wavefunction:

Hartree-Fock eq.:

• e-N correlation?

• many e, many N?

Time dependent (TD)

Time-dependence: Exact factorization

Time independent /BO

EKU Gross et al. PRL (2010); PRL (2015).

Nonadiabatic dynamics

(BO)PES

Nuclear wavefunction

EKU Gross et al. PRL (2010); PRL (2015).

TD Potential Energy Surface (PES) of H2+

TDPES

(BO)PES

Wavefunction

Next goals: TD + DFT ?

EKU Gross et al. PRL (2010); PRL (2015).

Dashed: 1014 W/cm2

Solid: 2.5× 1014 W/cm2

Semiclassical methods

1. Wavepacket propagation

Multi-configuration time-dependent Hartree (MCTDH) method

Advantages: Combines the efficiency of a mean-field

method with the accuracy of a numerically exact solution

Challenges: Global potential energy surfaces are required

HD Meyer et al., CPL 165, 73 (1990); H Wang, M Thoss, JCP119, 1289 (2003); GA Worth, I Burghardt, CPL 368, 502 (2003).

Quantum Dynamics of Multi-component Systems →

Semiclassical methods

2. Time evolution of density matrix

Mixed quantum-classical Liouville approaches

Advantages: Describes well the dynamics of nonlinear quantum systems for

quite long time

Challenges: fails to describe quantum dynamics if a part of the Hamiltonian

does not preserve irreducible subspaces of the symmetry group

W. H. Miller, J. Chem. Phys. 53, 3578 (1970); P. Huo and D. F. Coker, J. Chem. Phys. 137, 22A535 (2012).

Semiclassical methods

3. Trajectory-based approaches

• Mean-field Ehrenfest dynamics

• Trajectory surface hopping (TSH)

• Bohmian dynamics

• Wentzel–Kramers–Brillouin (WKB) approximation

• Dephasing representation (DR) framework

• Pechukas’ path integrals method

• …

Ehrenfest Theorem ( ↔ Schrӧdinger equation )

Ehrenfest Dynamics

Is this newton's second law?

No. Since 𝐹 ≠ 𝐹 𝑥 .

Transition probability

a is the off-diagonal element

C Wittig, JPCB109, 8428 (2005); M Desouter-Lecomte, JC Lorquet, JCP 71,4391 (1979).

In diabats:

In adiabats:

H12

XS Li, JCP123, 084106(2005).

Trajectory Surface Hopping

Tully, JCP 93, 1061 (1990).

, a jump occurs: k → j

Fewest Switch Surface Hopping

Detailed balance

Tully et al., JCTC 2, 229 (2006).

SH

Ehrenfest

Comparison

Ehrenfest dynamics

Fewest Switch Surface Hopping

Tully, JCP (1991), JPCC(2009); Prezhdo et al. PRL (2005); ...

• Coherence

• Detailed balance?

• Final state?

)()( tVtF

• Detailed balance

• Decoherence?

• Frustrated hop?

O3 dissociation

A special case: conical intersections (CI)

Baloitcha et al., JCP123, 014106 (2005).

BG Levine, TJ Martinez, Annu. Rev. Phys. Chem. 58, 613 (2007).

Ryabinkin et al. JCP 140, 214116 (2014).

Effect of Berry Geometry Phases (GP)

po

pu

latio

n

Ehrenfest dynamics: From DFT to TDDFT

)...,,,( 21 Nrrr

N

j

jN rdrrrr2

2

2 ),...,,()(

)()(]['

)'(')(

2

22

rErVrr

rdrrV

miiixcexternal

Theorem II.

Theorem I. )...,,,( 21 Nrrr

)(r

Time-dependent density functional theory (TDDFT) (Runge-Gross, 1984)

Density functional theory (DFT) and single-particle approximation (Kohn-Sham, 1965)

E Rouge & EKU Gross, PRL 52, 997 (1984).

III. Implementation

“Electron-nuclear” density functional theory

j

tot j j

( , , t), , t ( , , t)

J

J J

r Ri H r R r R

t

2 2 22 2

,

1 1

2 2 2 2

, ,

JI

tot j J

j J i j I J I Ji j

Jext j J

j J j J

Z ZeH

m m R Rr r

eZU r R t

r R

ext xc

, r ,, , ,I I

s

I

Z R t tv r t v r t dR dr v r t

r R r r

ext xc

, ,, , ,J J JI I

s J J J

I

r t Z R tV R t V R t Z dr Z dR V R t

R r R R

Ψ 𝑟, 𝑅, 𝑡 ≡ Φ𝑅 𝑟, 𝑡 𝜒 𝑅, 𝑡 ≅ Φ𝑅0(𝑡) 𝑟 , 𝑡 𝛿 𝑅(𝑡) − 𝑅0(𝑡)

Meng & Kaxiras, JCP (2008).

Coupled electron-ion dynamics Beyond Born-Oppenheimer

Gross 1984’

A new implementation:

• Real time (nonlinear, dynamics)

• Local bases: numeric atomic orbitals

• Paralleling over Kohn-Sham orbitals

• External field, spin excitation, large scale,…

Ehrenfest dynamics combined with (TD)DFT

Time-dependent density functional theory (TDDFT)

Time-Dependent

Ab-initio Package

Crank-Nicholson propagator

Propagating wavefunctions

Taylor polynomial

Splitting techniques

Other propagators

Castro et al. JCP 121, 3425(2004).

●

●

●

●

rc

r0

pseudopotential + numerical atomic orbitals

0 20 40 6010

1

102

103

104

Our method

Real space grid

Com

pute

r T

ime (

s)

Number of Valence Electrons

101

102

MQPyrazineO3

COH2

Mem

ory

(M

B)

C. Lian, M.X. Guan, S.Q. Hu, J. Zhang, S. Meng, Adv. Theo. Simul. (2018).

W. Ma, J. Zhang,…, S. Meng, Comp. Mater. Sci. 112, 478 (2016).

Computational efficiency

timestep 1 as → 24 as ~50 as

Quantum dynamics with real time TDDFT

with rt-TDDFT

• fixed ions : pure electron dynamics (one-component)

photoabsorption; nonlinear optics; transport…

• coupled e-ion dynamics:

e-phonon coupling; strong laser field; photo reactions…

• DFT-MD → Ab initio MD

ion dynamics driven by DFT forces (PES)

Current challenges

blocking wide use of TDDFT • Lack of non-adiabatic fXC(ω)

• Charge transfer excitation: nonlocal exchange

• Double excitation; Rydberg states

• Tiny timestep ~ 1 as;

• Inefficient propagation: stability; convergence

• Heavy computation: 103× heavier than AIMD; 106× than static DFT

• How to prepare physically-sound initial states?

• Calculation of time-dependent properties ?

• Beyond Ehrenfest dynamics ?

• Open systems ? …

Implementing Trajectory Surface Hopping

NA AD

Prezhdo et al.

H2

IV. Application: some examples

O3

Photodynamics in a molecule

Clouds = e density in excited state

Meng & Kaxiras, Biophys. J. 95,4396 (2008). Meng & Kaxiras, Biophys. J. 94, 2095 (2008). Kaxiras, Tsolakidis, Zonios, Meng, Phys. Rev. Lett. (2006).

e-proton concerted dynamics

Femtosecond dynamics of ion-molecule collision

Burnus et al. PRA 71, R10501 (2005).

-2

0

2

1

2

3

0 10 20 30 40

1

2

3

ћeV

E (

V/Å

)

O15

H29

O49

H98

dO

H (

Å)

dO

H (

Å)

Without Au20

t (fs)

E

Water Photosplitting Dynamics

Yan et al., ACS Nano 10, 5452(2016)；J. Phys. Chem. Lett. 9, 63 (2018).

Ultrafast evolution of water orbitals

Occupation changes of KS orbitals

Red : Increase

Blue: Decrease

Generation of H2 “bubles”

“Chain reaction” mechanism

Yan et al., ACS Nano 10, 5452(2016)；J. Phys. Chem. Lett. 9, 63 (2018).

An introduction to nonadiabatic molecular dynamics

I. Motivation

II. Theory

III. Implementation

IV. Applications

- NV center dynamics

OUTLINE

Prof. E.G. Wang (PKU/CAS)

Prof. Efthimios Kaxiras (Harvard)

Prof. Z.Y. Zhang (USTC)

Prof. S.W. Gao (CSRC)

Prof. S.B. Zhang (RPI)

Prof. X.C. Zeng (UNL)

Prof. G. Lu (CSUN)

Prof. X.F. Guo (PKU)

Prof. F. W. Wang (IOP-CAS)

Prof. X.H Lu (IOP-CAS)

Prof. K.H. Wu (IOP-CAS)

Prof. X.Z. Li (PKU)

Dr. Junyeok Bang (RPI)

Prof. Maria Fyta (Stuttgart)

Prof. Tomas Frauenheim (Bremen)

…

Funding:

Collaborators:

http://everest.iphy.ac.cn smeng@iphy.ac.cn

Chao Lian

Lei Yan

Jin Zhang

Hang Liu

Jiyu Xu

Mengxue Guan

Shiqi Hu

Peiwei You

...

Dr. Jiatao Sun

Team members:

THANK YOU