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
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 ),...,,()(
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rErVrr
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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
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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 [email protected]
Chao Lian
Lei Yan
Jin Zhang
Hang Liu
Jiyu Xu
Mengxue Guan
Shiqi Hu
Peiwei You
...
Dr. Jiatao Sun
Team members:
THANK YOU