Quantum Molecular Dynamics Basics

Transcription

1 Quantum Molecular Dynamics Basics Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological Sciences University of Southern California

2 Objective Derive quantum molecular dynamics (QMD) equations, which follow classical-mechanical trajectories of atoms, while computing interatomic interactions quantum mechanically: 1. Ehrenfest molecular dynamics (EMD) Solves the time-dependent Schrödinger equation for electrons Attosecond (10-18 s) to femtosecond (10-15 s) electron dynamics P. Ehrenfest, Zeit. Phys. 45, 455 ( 27) 2. Born-Oppenheimer molecular dynamics (BOMD) Obtain the electronic ground state at every time instance Electron & nucleus dynamics above femtosecond (10-15 s) M. Born & R. Oppenheimer, Annal. Phys. 84, 457 ( 27) Paul Ehrenfest ( ) Max Born ( ) Robert Oppenheimer ( )

3 Electron-Nucleus Dynamics Consider a system of N electrons & N atom nuclei, with their position operators, r " i = 1,, N & R + I = 1,, N 7 -./0 89:; H = 2 P h r 2M ", R :; = 2 P + 4 +<= +<= 2M + + V AB. R e4 r " r I "KI electron-electron interaction ħ4 2m 4 r + v AB. r " " "<= 2 Z Me 4 r " R M ",M electron-nuclei interaction Z +Z M e 4 R + R M +KM nuclei-nuclei interaction Here, ħ is the Planck constant, P I, M I & Z I are the momentum, mass & charge of the I-th nucleus, and m & e are the electron mass & charge; V ext & v ext are external potentials (like external electric field) acting on nuclei & electrons, respectively We focus on the system dynamics described by the time-dependent Schrödinger equation in non-relativistic quantum mechanics, where Ψ AOP is the electron-nucleus wave function & t is the time iħ t Ψ AOP r ", R +, t = HΨ AOP r ", R +, t

4 Separation of Length Scales Due to the much larger nuclei masses (M I ) compared to the electron mass (m), the quantum-mechanical nature of nuclei is negligible except in extreme cases like nuclear fusion More specifically, the length scale below which a particle s quantummechanical nature becomes appreciable at a given temperature T (i.e., thermal de Broglie wave length) is much smaller for nuclei than for electron Λ + = ħ 2πM + k U T λ ~ Λ AZA_.b/P = ZAP[.\ ^_-ZA /` ap.aba^. ap cde ħ 2πmk U T Element electron proton C O Zn 1 a.u. (bohr) = Å Λ (a.u.) at 300 K Motivate classical & quantum-mechanical descriptions of nuclei & electrons, respectively

5 Ehrenfest Molecular Dynamics (EMD) Small ħ expansion, applied to the nucleus degrees-of-freedom, leads to mixed quantum (for electrons) & classical (for nuclei) dynamical equations Classical Newton s equation of motion for nucleus positions d 4 M + dt 4 R + = Ψ(t) h r ", R + R + Ψ(t) iħ t Ψ(t) = h r ", R + Ψ(t) Time-dependent Schrödinger equation for the electronic wave function Ψ See notes on: (1) QMD summary & (2) QMD equation

6 Derivation of EMD Equations (1) Dynamics of the electron-nucleus system is encoded in the scattering matrix (or S matrix) in the closed-time path integral form Unitary time propagator remote future S = 1 ρ s R s 2 ρ s R kr U u t v, t w U O t w, t v kr s Initial probability of (k, R) state at time t 0 k-th electronic state with nucleus positions R U ± t, t = T ± exp i ħ } dτh ±(τ) Time-ordering operator remote past See notes on: (1) unitary time propagation; (2) closed-time path integral; (3) QMD equation K.-c. Chou et al., Equilibrium & nonequilibrium formalisms made unified, Phys. Rep. 118, 1 ( 85)

7 Derivation of EMD Equations (2) Path-integral w.r.t. nucleus trajectories: H(t) = P 4 + h r, R, t T R(t) = R <R S = } D R(t) exp i ħ S v R(t) T R(t) 1 ρ s R s -.\, S v R(t) = } dt M 2 dr dt 2 ρ s R kr T exp i ħ dth r, R t, t s 4 kr See note on: QMD equation

8 Derivation of EMD Equations (3) Keep the leading term of the ħ expansion (i.e. saddle-point approximation) of the path integral δ S v R(t) + ħ i lnt R(t) = 0 which amounts to M d4 dt 4 R 1 + = s ρ s R 2 ρ s R k(t)r s h r, R t, t R t iħ k t, R = h r, R t, t k t, R t k(t)r See notes on (1) QMD equation & (2) functional derivative

9 EMD Application: Electron Mobility Electron transport in condensed matter under electric field E iħ t ψ r, t = h r, R +(t), t h r, R + (t), t = 1 2m ħ i 4 r eet + v r, R + (t) For the computation of electronic conductivity & associated gauge transformation, see the note on quantum dynamical computation of electronic conductivity A. Nakano, P. Vashishta & R. K. Kalia, Electron transport in disordered systems: a nonequilibrium quantum molecular dynamics approach, Phys. Rev. B 43, ( 91) A. Nakano, P. Vashishta & R. K. Kalia, Probing localization & mobility of an excess electron in a-si by quantum molecular dynamics, Phys. Rev. B 45, 8363 ( 92)

10 EMD Application: Attosecond Dynamics Electric field-induced polarization in silica Positive (red) & negative (blue) change in charge density A. Sommer, K. Yabana et al., Attosecond nonlinear polarization & light-matter energy transfer in solids, Nature 534, 86 ( 16) Ehrenfest dynamics codes by Prof. Kazuhiro Yabana s group ARTED: SALMON:

11 Born-Oppenheimer Molecular Dynamics Due to the much larger nuclei masses (M I ) compared to the electron mass (m), the quantum-mechanical wave function of the system is separable to those of the electrons & nuclei At ambient conditions, the electronic wave function remains in its ground state ( Ψ v ) with the energy eigenvalue ε v, corresponding to the instantaneous nuclei positions ({R I }), with the latter following classical mechanics M + d 4 dt 4 R + = R + Ψ v h r ", R + Ψ v h r ", R + Ψ v = ε v Ψ v See notes on: (1) QMD summary & (2) adiabatic approximation

12 Born-Oppenheimer (BO) MD Derivation (1) Expand the wave function in terms of the complete set of eigenstates, {ψ k (r, R)}, with fixed nuclei position R (i.e., adiabatic basis) ψ r, R, t = 2 χ s (R, t)ψ s (r, R) s h r, R ψ s (r, R) = E s (R)ψ s (r, R) Resulting time-dependent Schrödinger equation iħ t + 2 ħ4 2M + 4 R T ss R = k ħ " E s R T ss R + k ħ R " R k ħ See notes on: (1) QMD summary & (2) adiabatic approximation χ s (R, t) = 2 T ss R χ s (R, t) 4 R Nonadiabatic coupling due to nuclei motion + k s Ks

13 Born-Oppenheimer (BO) MD Derivation (2) Born-Oppenheimer approximation neglects all T kk terms; when in the electronic ground state (k = 0), off-diagonal transition is negligible if T sv R E s R E v R diagonal term T kk was shown to be O(m/M I ) Classical limit of the resulting equation for nuclei can be derived using the same ħ expansion as in the derivation of Ehrenfest MD iħ t + 2 ħ4 2M + 4 R E s R χ s (R, t) = 0 M + d 4 dt 4 R + = R + E s R = k h r, R, t R Hellmann-Feynman theorem See notes on: (1) QMD summary & (2) adiabatic approximation k

14 Hellmann-Feynman Theorem Consider a Hamiltonian that include a parameter λ (in our case, nuclei positions R) ψ ψ = 1

15 BOMD Application: H 2 Production from Water 16,661-atom BOMD simulation of Li 441 Al 441 in water on 786,432 IBM Blue Gene/Q cores K. Shimamura et al., Nano Lett. 14, 4090 ( 14) 21,140 time steps (129,208 self-consistent-field iterations); unit time-step = fs

16 Berry Phase The adiabatic basis (electronic eigenstates with fixed nucleus positions R(t) at each instance of time t) with energy E k (R), used here, plays a role in the discussion of Berry (or geometric) phase of electronic wave function during adiabatic turningon/off of external potential k(t) = exp iγ s (t) exp " ħ dt E s R(t ) v γ s = } dr i k(r) R k(r) -.\ k(t = 0) Berry phase Integration of the Berry phase along a closed path can be nonzero, which is observable (e.g. Aharonov-Bohm effect) QXMD uses it to compute electronic polarizability D. Xiao et al., Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 ( 10)

17 Quantum-Mechanical Calculation of Polarization While polarization P = drr ψ(r) 4 is ill-defined under periodic boundary condition, its change P = dt j is well-defined, with a proper gauge to v compute current j (note on quantum dynamical computation of electronic conductivity) Change of polarization upon adiabatic switching of finite electric field E with periodic boundary condition R. Resta, Phys. Rev. Lett. 80, 1800 ( 98); P. Umari & A. Pasquarello, ibid. 89, ( 02) P AZ, = L π Im ln det ψ exp i2πx/l ψ m, n {occupied} ψ = argmin E ³/\Pu \-0 ψ EΔP AZ, ψ The above formula is equivalent to a sum of valence-band Berry phases R. D. King-Smith & D. Vanderbilt, Phys. Rev. B 47, 1651( 93); I. Souza, J. Iniguez & D. Vanderbilt, Phys. Rev. Lett. 89, ( 02) E P el = } dλ P AZ λ = ie v 2π 2 } dk ψ k k ψ k / º aa¼ UbaZZ/ºaP»/PA Above a critical field E _ ~ (energy gap)/(simulation cell size), the energy functional has no minimum, indicating Zener breakdown (i.e. tunneling from valence to conduction bands)