UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.

1. ModelingTransition
States
Shyue Ping Ong

2. What is a transition state?
Particular configuration along a reaction
coordinate corresponding to the highest potential
energy along the reaction coordinate.
NANO266
2

3. Applications
Reaction rates
• Activated states
• Barriers to reactions
• Lowest energy pathway from reactant to products
Diffusion
• Activation energies for migration
• E.g., migration of defects (radiation damage), Li migration for
batteries
NANO266
3

4. The potential energy surface
NANO266
4

5. The rate equation
Atomic vibrations are on the order of 0.1-1ps,
which implies ν ~ 1012-1013. This is a frequently
used approximation to avoid having to undertake
a potentially complex calculation of the frequency
directly.
~60 meV change in ΔEa leads to an order of
magnitude change in the rate.
NANO266
5
r =νe
−
ΔEa
kbT

6. The Nudged Elastic Band Method
Method for finding the minimum
energy path (MEP) when initial and
final endpoints are known.
Initial guesses for MEP typically
given by linear interpolations
between the start and end points.
The interpolated points are known as
images.
NANO266
6
Source: http://theory.cm.utexas.edu/henkelman/research/saddle/
{rx} = x{ri}+(1− x){rf }

7. Forces in NEB
Conceptually, this is like adding
springs between images.
Images are then moved in
accordance to the force using an
optimization algorithm.
NANO266
7
Fi = Fi
s
||
− ∇E(Ri ) ⊥
∇E(Ri ) ⊥
= ∇E(Ri )− ∇E(Ri )⋅ ˆti
Fi
s
||
= k Ri+1 − Ri − Ri − Ri-1( )⋅ ˆti
Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy
paths, J. Chem. Phys., 2000, 113, 9901, doi:10.1063/1.1329672.

8. Climbing Image NEB
Slight modification of NEB
Effect:
• Moves the potential energy surface along the elastic band and
down the potential surface perpendicular to the band.
• Other images in the band define the one degree of freedom for
which a maximization of the energy is carried out.
• Good approximation of saddle point.
• No additional cost.
NANO266
8
Fimax
= −∇E(Rimax
)+ 2∇E(Rimax
)
||
= −∇E(Rimax
)+ 2∇E(Rimax
).ˆtimax
ˆtimax
Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy
paths, J. Chem. Phys., 2000, 113, 9901, doi:10.1063/1.1329672.

9. CI vs Regular NEB
NANO266
9
Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy
paths, J. Chem. Phys., 2000, 113, 9901, doi:10.1063/1.1329672.

10. Variable springs
Use stronger springs near saddle point
NANO266
10
ki
! =
kmax − Δk
Emax − Ei
Emax − Eref
$
%
&&
'
(
)) if Ei > Eref
kmax − Δk if Ei < Eref
*
+
,,
-
,
,
Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged
elastic band method for finding saddle points and minimum energy
paths, J. Chem. Phys., 2000, 113, 9901, doi:10.1063/1.1329672.

11. General NEB calculation procedure
NANO266
11
Full relaxation of start and
end points (good force
convergence is critical!)
Choose # of images, and
generate guesses
Perform NEB calculation

12. Challenges in NEB calculations
Stability of calculation depends on number of images. Too
few images -> lack of resolution. Too many -> unstable
convergence.
Linear interpolation between start and end points may
lead to very bad guesses for MEP
Convergence towards MEP may be extremely slow.
Force convergence criteria typically much more stringent.
Guess for the MEP may bias the final solution.
NANO266
12

13. Oxygen vacancy diffusion inYSZ
NANO266
13
Kushima, A.; Yildiz, B. Oxygen ion diffusivity in strained yttria stabilized zirconia: where is the fastest strain?, J. Mater. Chem., 2010, 20, 4809, doi:
10.1039/c000259c.

14. Hydrogen diffusion onTM surfaces
NANO266
14
Kristinsdóttir, L.; Skúlason, E. A systematic DFT study of hydrogen diffusion on transition metal surfaces, Surf. Sci., 2012, 606, 1400–
1404, doi:10.1016/j.susc.2012.04.028.

15. Methanol oxidation on Cu surfaces
NANO266
15
Sakong, S.; Groß, A. Density functional theory study of the partial
oxidation of methanol on copper surfaces, J. Catal., 2005, 231,
420–429, doi:10.1016/j.jcat.2005.02.009.

16. Spinel compounds for multivalent electrodes
NANO266
16
Liu, M.; Rong, Z.; Malik, R.; Canepa, P.; Jain, A.; Ceder, G.; Persson, K. a. Spinel compounds as multivalent battery cathodes: a systematic
evaluation based on ab initio calculations, Energy Environ. Sci., 2014, 00, 1–11, doi:10.1039/C4EE03389B.

17. NEB calculations -> KMC
Kinetic Monte Carlo – MC simulation method for the time
evolution of natural processes.
• For studies of diffusivity or reaction rates, requires as input environment-
dependent activation barriers(usually from NEB)
Basic algorithm:
• Start with an equilibrium atomic arrangement
• Determine migration probabilities Γ based on the paths available. Γ is calculated
based on Arrhenius form.
• Sample a random number in (0,1], and the migration event k is chosen such that:
• Time is updated by Δt given by:
NANO266
17

18. Concentration Dependent Diffusion in LiCoO2
NANO266
18
Van Der Ven, A.; Ceder, G. Lithium Diffusion in Layered Li[sub
x]CoO[sub 2], Electrochem. Solid-State Lett., 2000, 3, 301–304.

19. NEB + PercolationTheory
NANO266
19
Lee, J.; Urban, A.; Li, X.; Su, D.; Hautier, G.; Ceder, G. Unlocking the
potential of cation-disordered oxides for rechargeable lithium batteries.,
Science, 2014, 343, 519–22, doi:10.1126/science.1246432.

20. Growing string method
NANO266
20
Peters, B.; Heyden, A.; Bell, A. T.; Chakraborty, A. A growing string method for determining
transition states: comparison to the nudged elastic band and string methods., J. Chem. Phys.,
2004, 120, 7877–86, doi:10.1063/1.1691018.
Begins as two string
fragments, one associated
with the reactants and
the other with the products.
Each string fragment is
grown separately until the
fragments converge. Once
the two fragments join, the
full string moves toward the
MEP.
Typically finds saddle
points much more quickly
when linearly interpolated
guess is very far from MEP.

21. Dimer Method
Use to finding saddle
points when final state
is not known.
Involves two replicas of
the system, a ‘dimer’,
which is used to
transform the force in
such a way that
optimization leads to
convergence to a
saddle point rather
than a minimum.
NANO266
21