Ab initio molecular dynamics

Shyue Ping Ong

What is molecular dynamics (MD)?

Moving atomic nuclei by solving Newton’s equation of motion

For N nuclei, we have 3N positions and 3N velocities, i.e., a 6N dimensional phase space.

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M!!RI = −∂E∂RI

= FI[{RJ}]

Where do forces come from?

Lennard-Jones potential •  OK for rare gases •  No chemistry •  Only pair-wise and no directionality

Empirical force-fields •  Fitted parameters based on model function •  Can include many-body terms to incorporate dispersion,

polarization, etc.

Quantum mechanics! •  Expensive…, and surprisingly, not always the most accurate!

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VLJ = 4εσr

!

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$

%&12

−σr

!

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)**

+

,--

2013 Nobel Prize in Chemistry

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Computational Experiment

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Initialize positions and velocities

Compute forces

Determine new positions

Calculate thermodynamic

averages

Repeat until time is reached

Initialization

2nd order PDF -> Initial positions and velocities Positions

•  Reasonable guess based on structure •  No overlap / short atomic distances

Velocities

•  Small initial velocity •  Steadily increase temperature (velocity scaling)

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M!!RI = −∂E∂RI

= FI[{RI}]

Maxwell-Boltzmann distribution

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f (v) = m2πkT!

"#

$

%&3

4πv2e−mv2

2kT

Ensembles

Microcanonical – N, V, E Canonical – N, V, T

•  Temperature control achieved via thermostats •  Examples: velocity rescaling, Andersen thermostat, Nose-Hoover

thermostat, Langevin dynamics, etc.

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Integrating equations of motion – Verlet algorithm

Key properties of Verlet algorithm •  Errors do not accumulate •  Energy is conserved •  Simulations are stable

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M!!RI = FI[{RJ}]

!!RI =1MI

FI[{RJ}]

[RI(t +Δt)−RI(t)]−[RI(t)−RI(t −Δt)](Δt)2

=1MI

FI[{RJ}]

RI(t +Δt) = 2RI(t)+RI(t −Δt)+(Δt)2

MI

FI[{RJ(t)}]

Flavors of quantum MD

Born-Oppenheimer •  Electrons stay in instantaneous ground state as nuclei move •  Perform electronic SCF at each time step •  Compute forces (Hellman-Feynman theorem) •  Move nuclei

Car-Parrinello •  Treat ions and electrons as one unified system •  Accomplished with fictitious kinetic energy for electrons

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Car, R.; Parrinello, M. Unified approach for molecular dynamics and density-functional theory, Phys. Rev. Lett., 1985, 55, 2471–2474, doi:10.1103/PhysRevLett.55.2471.

Coupled equations of motion

The CP Lagrangian

Equations of motion

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L = 12(2µ)

i=1

N

∑ dr !ψi (r)2

∫ + MII=1

NI

∑ !!RI −E[ψi,r]+ Λij drψi*(r)ψ j (r)−δij∫%& '

(ij∑

Imposes orthonormality of electronic states

µ !!ψi (r, t) = −Hψ j (r)+ Λikψk (r, t)k∑

MI!!RI = −

∂E∂RI

Comparison of BO vs CP MD

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CP Avoids need for

costly SCF iteration of electrons at each

step

Time step required is smaller

BO Solves Schrodinger

equation at each step

Time step can be longer

General procedure for MD simulation

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Applications of MD

Thermodynamics from ensemble averages Real-time evolution

•  Reactions •  Interaction of molecules on surfaces •  Trajectories of ions

Ground-state structures

•  Geometry optimization is in fact a form of MD. •  For complex structures (e.g., low symmetry systems, interfaces,

liquids, amorphous solids, etc.), MD can be used to determine low-energy structures

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Nattino, F.; Ueta, H.; Chadwick, H.; Reijzen, M. E. Van; Beck, R. D.; Jackson, B.; Hemert, M. C. Van; Kroes, G. Ab Initio Molecular Dynamics Calculations versus Quantum-State- Resolved Experiments on CHD 3 + Pt(111): New Insights into a Prototypical Gas − Surface Reaction, J. Phys. Chem. Lett., 2014, 5, 1294–1299, doi:10.1021/jz500233n.

Thermodynamic averages

Under ergodic hypothesis

Examples of averages

•  Energy (potential, kinetic, total) •  Temperature •  Pressure •  Mean square displacements (diffusion) •  Radial distribution function

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A =Aie

−βEi (r,p)

i∑

e−βEi (r,p)i∑

= limT→∞

1T

A(t)dt0

T

∫

Correlation Functions from MD

Pair distribution function

Time correlation function

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g(r) = VN 2 δ(r− rij)

j≠i∑

i∑

Einstein relation (valid at large t)

D =13

dt v(t)v(0)0

∞

∫

Example: Diffusion coefficient

D =12dt

(r(t)− r(0))22tγ = (A(t)− A(0))2

γ = dt !A(t) !A(0)0

∞

∫

Structure of Amorphous InP

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Lewis, L.; De Vita, A.; Car, R. Structure and electronic properties of amorphous indium phosphide from first principles, Phys. Rev. B, 1998, 57, 1594–1606, doi:10.1103/PhysRevB.57.1594.

Lithium hydroxide phase transition under high���pressure

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Pagliai, M.; Iannuzzi, M.; Cardini, G.; Parrinello, M.; Schettino, V. Lithium hydroxide phase transition under high pressure: An ab initio molecular dynamics study, ChemPhysChem, 2006, 7, 141–147, doi:10.1002/cphc.200500272.

Dihydrogen oxide

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Izvekov, S.; Voth, G. a. Car-Parrinello molecular dynamics simulation of liquid water: New results, J. Chem. Phys., 2002, 116, 10372–10376, doi:10.1063/1.1473659.

Diffusion in Lithium Superionic Conductors

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ac

ab

1D conductors would be highly sensitive to blocking defects!

X

Trace of Li motion at 900K

Overall Ea (meV)

σ@ 300 K (mS/cm)

First principles1

210 13

Experimental2

240 12

1 S. P. Ong Y. Mo, W. Richards, L. Miara, H. S. Lee, G. Ceder. Energy & Environ. Sci., 2013, 6, 148–156. 2 N. Kamaya et al. Nat. Mater. 2011, 10, 682-686

Cation has a small effect on diffusivity of Li10MP2S12

Isovalent

Aliovalent

Ge Si Sn P Al σ @ 300 K (mS/Cm) 13 23 6 4 33

Ea (meV) 210 200 240 260 180

(Aliovalent substitutions are Li+ compensated)

S. P. Ong, Y. Mo, W. D. Richards, L. Miara, H. S. Lee, G. Ceder, Phase stability, electrochemical stability and ionic conductivity in the Li10±1MP2X12 family of superionic conductors. Energy Environ. Sci., 2012, doi: 10.1039/C2EE23355J

Phonons from MD

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Kim, J.; Yeh, M. L.; Khan, F. S.; Wilkins, J. W. Surface phonons of the Si(111)-7x7 reconstructed surface, Phys. Rev. B, 1995, 52, 14709–14718, doi:10.1103/PhysRevB.52.14709.

Surface reactions with MD

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Arifin, R.; Shibuta, Y.; Shimamura, K.; Shimojo, F.; Yamaguchi, S. Ab Initio Molecular Dynamics Simulation of Ethylene Reaction on Nickel (111) Surface, J. Phys. Chem. C, 2015, 119, 3210–3216, doi:10.1021/jp512148b.