1.
Ab initio molecular
dynamics
Shyue Ping Ong

2.
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}]

3.
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 = 4ε
σ
r
!
"
#
$
%
&
12
−
σ
r
!
"
#
$
%
&
6(
)
*
*
+
,
-
-

4.
2013 Nobel Prize in Chemistry
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5.
Computational Experiment
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Initialize positions
and velocities
Compute forces
Determine new
positions
Calculate
thermodynamic
averages
Repeat until time is reached

6.
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}]

7.
Maxwell-Boltzmann distribution
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f (v) =
m
2πkT
!
"
#
$
%
&
3
4πv2
e
−
mv2
2kT

8.
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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9.
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 =
1
MI
FI [{RJ}]
[RI (t + Δt)− RI (t)]−[RI (t)− RI (t − Δt)]
(Δt)2
=
1
MI
FI [{RJ}]
RI (t + Δt) = 2RI (t)+ RI (t − Δt)+
(Δt)2
MI
FI [{RJ (t)}]

10.
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.

11.
Coupled equations of motion
The CP Lagrangian
Equations of motion
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L =
1
2
(2µ)
i=1
N
∑ dr !ψi (r)
2
∫ + MI
I=1
NI
∑ !!RI − E[ψi,r]+ Λij drψi
*
(r)ψj (r)−δij∫%
&
'
(
ij
∑
Imposes orthonormality of
electronic states
µ !!ψi (r,t) = −Hψj (r)+ Λikψk (r,t)
k
∑
MI
!!RI = −
∂E
∂RI

12.
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

13.
General procedure for MD simulation
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14.
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.

15.
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
∑
= lim
T→∞
1
T
A(t)dt
0
T
∫

16.
Correlation Functions from MD
Pair distribution function
Time correlation function
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g(r) =
V
N2
δ(r − rij)
j≠i
∑
i
∑
Einstein relation
(valid at large t)
D =
1
3
dt v(t)v(0)
0
∞
∫
Example: Diffusion coefficient
D =
1
2dt
(r(t)− r(0))22tγ = (A(t)− A(0))2
γ = dt !A(t) !A(0)
0
∞
∫

17.
Structure ofAmorphous 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.

18.
Lithium hydroxide phase transition under high