CLUSHIQ2022
November 2, 2022

Real-Time Dynamics of Hydrogen and Its Isotopic
Molecules Revealed by the Non-Empirical Quantum
Molecular Dynamics Method

Kim Hyeon-Deuk (Kyoto University)

Table of Contents

1. Nuclear Quantumness Plays a Role.
2. The Nuclear and Electron Wave Packet Molecular Dynamics (NEWPMD) method-Nuclear quantumness (Zero Point Energy and Nuclear Delocalization)-No empirical parameter-No external model-Real-time dynamics- Many-body system
3. Examples-Liquid-Solid-Supercooled Liquid-Supercritical Fluid-Gas-Non-Equilibrium Heat Flux-Non-Equilibrium Flow
4. The Nuclear and Electron Wave Packet Molecular Dynamics method with the Quantum
RotationElectron Distribution in H2

Hydrogen nuclei are treated as classical mass points

Silvera, Rev. Mod. Phys. 52 (1980) 393

Nuclear Quantumness ①

Zero Point Energy
Nuclear Delocalization

O-H・・・O

proton displacement

Tuckerman et al., Nature 417(2002) 925

Nuclear Quantumness ②

RPMD=Ring Polymer Molecular Dynamics Methodwith the Nuclear Quantumness

Diffusion
RPMD

Radial Distribution Function
Classical MD

Classical MD
RPMDMiller and Manolopoulos, J. Chem. Phys. 122 (2005) 184503

Yonetani and Kinugawa, J. Chem. Phys. 120 (2004) 10624

Table of Contents

1. Nuclear Quantumness Plays a Role.
2. The Nuclear and Electron Wave Packet Molecular Dynamics (NEWPMD) method-Nuclear quantumness (Zero Point Energy and Nuclear Delocalization)-No empirical parameter-No external model-Real-time dynamics- Many-body system
3. Examples-Liquid-Solid-Supercooled Liquid-Supercritical Fluid-Gas-Non-Equilibrium Heat Flux
-Non-Equilibrium Flow
4. The Nuclear and Electron Wave Packet Molecular Dynamics method with the QuantumRotation

Nuclear and Electron Wave Packet Molecular Dynamics (NEWPMD)
Electron Wave Packet(EWP)
HH

Nuclear Wave Packet (NWP)
Radial Distribution Function
Diffusion Coefficient

Freezing Temperature

Viscosity

Vibrational Displacement

Vibrational Frequency Shift

Solid Phonon Mode~40cm-1

Isotope Effects (D2 and T2)

Discrete H-H frequency~ a few cm-1 in 4000cm-1

J. Chem. Phys. (Commun.) 140 (2014) 171101
Phys. Rev. B 90 (2014) 165132
J. Phys. Chem. B, 122 (2018) 8233
Phys. Chem. Chem. Phys. 23 (2021) 22110

25 K

LiquidSolid

2.5 K

J. Chem. Phys. (Commun.) 143 (2015) 171102

Supercooled Nonequilibrium

Boson Peak ~30~40cm-1

COLDHOT

LiquidNano Flow inCarbon Nano Tube> >Vibrational Frequency
Solid

H2 jetOptimal Mixing Rate for CoolingSupercooling Exp.

Phys. Chem. Chem. Phys. (Commun.) 18 (2016) 2314
J. Phys. Chem. Lett. 11 (2020) 4186

Heat Flux
Thermal Conductivity

J. Phys. Chem. Lett. 8 (2017) 3595
J. Phys. Chem. Lett. 13 (2022) 3579

CPU time: 6 minutes for 1 ps dynamics of 1200 molecules with 16 cores

Time-Dependent Nuclear and Electron Wave Function
a,bJ. Chem. Phys. (Communication) 140 (2014) 171101
Chem. Phys. Lett. 532 (2012) 124

c,d

HC－HD

Time-Dependent Nuclear and Electron Wave Function

(a,b)

HA－HB

(c,d)

HC－HD

αβ: Antisymmetrizer
Perfect-Pairing Valence Bond Theory : (a,b) (c,d) in Singlet

Gaussian Nuclear Wave Packet (NWP)

ħ=1electron charge =1
electron-mass scaled

4 log
ln 2 K
( RK , x , RK , y , RK , z )Nuclear Quantumness
Zero Point Energy
Nuclear Delocalization

Thawed EWP
Time-Dependent Wave Function

(a,b)

HA－HB

(c,d)

HC－HD

Gaussian Electron Wave Packet (EWP)
Isolated HA－HB➢ Thawed EWP

RHH(t)

and➢ Born-Oppenheimer Approximation
anda,bHA

HB

RHH(t)

Time-Dependent Variational Principle

Action Integral

Kinetic energy of elec.
Coulomb energies
electron-electron
nucleus-nucleus
nucleus-electron

Hamiltonian

NWP

4 log
ln 2 K

Equations of Motion( RK , x , RK , y , RK , z )Kinetic Energy of Electrons Electrostatic Energies
electron-electron
nucleus-nucleus
nucleus-electron

Many-Body H2 Hamiltonian

Nmol-Body Hamiltonian

Subtracting Multiple Counting: Number of Nuclei: Number of Molecules: Single Hydrogen Molecule Energy

HC－HD

Centroid Molecular Dynamics
Ring Polymer Molecular Dynamics

Semiquantum Path Integral MethodrQuantization
H2H2
H2
Silvera–Goldman Model

H2
Ring Polymer

Silvera-Goldman Model

Path Integral

8 Empirical Parameters
No Intramolecular Degree of Freedom
Limited Thermodynamic States

Equilibrium Thermal StateNo Real-Time Trajectory
Unstable for Long TimeHigh Computational Cost
Not Applicable for T→0+Silvera, Rev. Mod. Phys. 52 (1980) 393
Miller and Manolopoulos, J. Chem. Phys. 122 (2005) 184503

Table of Contents

1. Nuclear Quantumness Plays a Role.
2. The Nuclear and Electron Wave Packet Molecular Dynamics (NEWPMD) method-Nuclear quantumness (Zero Point Energy and Nuclear Delocalization)-No empirical parameter-No external model-Real-time dynamics- Many-body system
3. Examples-Liquid-Solid-Supercooled Liquid-Supercritical Fluid-Gas-Non-Equilibrium Heat Flux
-Non-Equilibrium Flow
4. The Nuclear and Electron Wave Packet Molecular Dynamics method with the QuantumRotation

T-Shape

H－H

2D Radial Distribution Function

number of NWPs within a small region ofradii r~r +dr and angle θ~ θ+d θθ・・・ H－H
total number density

𝑹 𝑡 − 𝑹(0) 2 𝑁mol
𝐷(𝑁mol) = lim𝑡→∞6𝑡Transport CoefficientR(t) : Position Vector of H2 Center

1.65

25 K

(exp. ~1.6)

14 K

0.345
(exp. ~0.35)

System Size Scaling : D(Nmol)
Molar Length :Viscosity :

3.55

3.76

2.3X10-5

1.10X10-5(exp. ~2.5X10-5)(exp. ~0.94X10-5)

H－H

H－H

H－H

H-H Bond in H2 LiquidElongated Bond0.7659Å

0.7655ÅIsolated～0.7643Å
Exp. ～0.755Å

Two Structures-14611.9 cm

Motional Narrowing
Red-Shift-14604.7 cm

Isolated～4629.2 cm -1-1

exp. ～4161.13 cm

Mean Square Displacement in Supercooled Liquid

Plateau & Relaxation

Mean Square Displacement

Partially Broken
Phonons

Cluster
Deformation

exp. E2g mode~36 cm-1

Boson Peak

StablePhonons
RK(t) : Position Vector of K-th H2 Center

Partially Broken Phonons
in Mesoscale Clusters

Solid

Frequency

Translational

Supercooled
Boson Peak

LiquidRadial Distribution Function (RDF)@ D2 Vapor-Pressure Density

H2:D2 Mixtures
3:1

1:0

1:1

0:1

1:3

number of NWPs withinradius r~r +drRDF

total number density30 K

25 Kr (Å)

18 K

Isotope Effects –Bond Length-

25 K @ D2 Vapor-Pressure DensityDiffusion of Isotope Mixtures@ D2 Vapor-Pressure Density

H2:D2 Mixtures
0:1

1:0>

>

3:1

1:1

1:3>

>

𝑹 𝑡 − 𝑹(0) 2 𝑁mol
𝐷(𝑁mol) = lim𝑡→∞
6𝑡
Diffusion Coefficient (Å2/ps)

R(t) : Position Vector of H2 Center

Mass

Supercooling v.s. Mass

D2 Ratio(%)

D2 Ratio(%)

Non-Equilibrium Heat Conduction

Heat Flux

𝐽 = −𝜆𝛻T
Current Result~0.16 Wm-1K-1

Acknowledgments
PRESTO, Japan Science and Technology Agency
Grant No.22713147
Grant-in-Aid for Scientific Research on Innovative Areas
No.18H05407
Grants-in-Aid for Scientific Research
No.20K05419
KUSUNOKI・125
Interview & Article :https://research.kyoto-u.ac.jp/kusunoki125/

Mr. Shutaro Yamaoka

Dr. I-Ya Chang