Free energy barriers
for local H diffusion in
sodium alanates
S. Bonella
Universita’ di Roma “La Sapienza”
Sodium alanates are prototypical materials for solid state
hydrogen storage.
3NaAlH4 ⇔ Na3 AlH6 + 2Al + 3H2
T = 350Kand 3.7wt%
2Na3 AlH6 ⇔ 6NaH + 2Al + 3H2
T = 423Kand 1.9wt%
2NaH ⇔ 2Na + H2
T = 698Kand 1.9wt%
3NaAlH4 ⇔ Na3 AlH6 + 2Al + 3H2
Experiments cannot identify activated mobility mechanism, barrier
of 0.126 eV (~4KBT)
Hypothesis:
Local H diffusion
Short range H diffusion
Na vacancy diffusion
1 Voss, J., Shi, Q., Jacobsen, H.S., Zamponi, M., Lefmann, K.,Vegge, T., J. Phys. Chem. B 111 (2007), 3886!
2 Palumbo, O., Paoloni, A., Cantelli R., Jensen A. J. Phys. Chem. B 110 (2008), 9105!
Preliminary ab initio MD run shows that the local diffusion process
is not activated
Short range diffusion
Collective variable: Al-H coordination number
θα =
NH
!
i=1
1
1 + ek(riα −r0 )
riα = |ri − rα |
α = 1, ..., 8
F (z) = −β −1 lnZ −1
!
dx e−βV (x)
m
"
δ(θj (x) − zj )
j=1
Run a long MD trajectory to explore the configuration
space of the system, histogram of the values of the
CV to reconstruct F(z), and compute the free energy
barriers
(a) Metastabilities
(b) Exponential scaling of the number of grid point with
dimensions
Single sweep method
L. Maragliano, E.Vanden Eijnden, J. Chem. Phys., 128 (2008) 184110!
(a) Use Temperature Accelerated MD to generate a trajectory in the
space of the collective variables that efficiently explores relevant
regions of the free energy even in the presence of significant barriers
(b) Points along the trajectory are chosen as centers for an
interpolation grid and the free energy is represented as an optimized
linear combination of radial basis functions centered on the grid
Temperature Accelerated MD (TAMD)
L. Maragliano L., E. Vanden-Eijnden Chem. Phys. Letters 426 168-175
U. Roethlisberger et al. (Canonical Adiabatic Free Energy Sampling)
J.Abrams, M. Tuckerman (Adiabatic Free Energy Dynamics)
Consider the extended system
x = (x1 , ..., xN )
z = (z1 , ..., zl )
l
!
k
Uk (x, z) = V (x) +
(θj (x) − zj )
2 j=1
new potential
coupled evolution equations:
l
!
∂V (x)
∂θj (x)
mẍi = −
−k
(θj (x) − zj )
+ thermostat@T
∂xi
∂x
i
j=1
M z̈j = k(θj (x) − zj ) + thermostat@T̄
(any dynamics that generates the canonical
distribution will do)
B
theTAMD
physical and new variables. Adiabatic separation of the motion of the physica
titious variables can be induced by increasing the value of m̄ and γ̄ (with the cond
at m̄ = O(γ̄ 2 )). In these conditions, the z variables evolve, on the slower time scale of
Adiabatic
separation
of
the
motions
can
be
induced
via
the
tion, according to an effective force which is obtained by averaging the right hand
parameters
associated
to the evolution
the z-variables.
E.g. varia
eq. (2.4)
with respect
to the distribution
to which of
thermalize
the, fast, physical
γ̄
>>
γ
is distribution
the conditional
function
for eq. (2.3) at z(t) = z fi
Mprobability
>> m density
Langevinisdynamics
$
en by ρ(x|z) = Zκ−1 (z)e−βUκ (x,z) with Zκ (z) = dxe−βUκ (x,z) . The only term affecte
e average
in the evolution
the z variables
the one proportional
The z-variables
thenequation
evolvefor
according
to theis effective
force to κ an
ective evolution equation obtained after performing this average can be written as
!
e−βUk (x,z)#
M z̈j = k dx(θ
− zzFj κ)(z) − γ̄ ż + +2β̄thermostat@
T̄
−1 γ̄η z
m̄z̈j (x)
= −∇
Z (z)
k
∂Fk (z)
%
=−
+ thermostat@
T̄
−1
−βUκ (x,z)
∂z
−∇
F
(z)
=
Z
(z)
dxκ(z
−
θ(x))e
j
z κ
κ
ere
where
d we
have implicitly defined
th Zκ =
Fκ (z) = −β −1 lnZκ−1
= −β
$
−1
lnZκ−1
%
%
dxe−βUκ (x,z)
dxe
−βV (x)
− βκ
2
e
Pν
α (x)−z α )2
(θ
α=1
dxdze−βUκ (x,z) . In the limit βκ → ∞, the Gaussian function in the qua
∂Fk (z)
M z̈j = −
+ thermostat@T̄
∂zj
Holds for all values of fictitious temperature
If T̄ >> T the trajectory of the auxiliary variables can overcome
barriers and quickly sweep the relevant regions of the original free
energy landscape
Explore the free energy (find the metastable states), many
collective variables
Points along the trajectory can be used as centers for an
interpolation grid
θα =
NH
!
i=1
1
1 + ek(riα −r0 )
Radial basis reconstruction
F̃ (z) =
J
!
aj φσ (|z − zj |)
j=1
u2
− 2
2σ
e.g. φσ (u) = e
Coefficients and variance can be determined by minimizing the
object function
Input:
J
J
!
!
E(a, σ) =
|
aj ! ∇z φσ (|zj − zj ! |) + fj |2
j=1 j ! =1
(a) location of the centers, distance criterion along the TAMD
trajectory
(b) fj = −∇zj F (z)
average
mean force at the centers, local conditional
k
fj =
T
!
0
T
(zj − θj (x(t)))dt
Minimization of the object function
(a) For fixed value of the variance, solve
J
!
Bj,j ! (σ)aj ! = cj (σ)
j ! =1
where
Bj,j ! (σ) =
cj (σ) =
J
!
j !! =1
J
!
∇z φσ (|zj − zj !! |)∇z φσ (|zj !! − zj ! |)
∇z φσ (|zj − zj ! |)fj !
j ! =1
(b) The optimal value of the variance, satisfies
and is found by computing the residual for a (1d) grid of values
of the variance
Variations on single sweep
Modify definition of the basis elements
− 21 (z−zk )T Σ−1
k (z−zk )
ΦΣk (|z − zk |) = e
(Σk )α,β = E[(z α − z̄kα )(z β − z̄kβ )] = σk,α σk,β (δα,β + ρk,α,β )
Modify definition of the object function
Er (a, Σ) =
ν
K !
!
k=1 α=1
!αk [fkα + ∇α F̃ (zk )]2
!αk = 1/fkα
2
6
5.8
5.6
5.4
C2
5.2
5
4.8
4.6
4.6
4.8
5
5.4
5.2
5.6
5.8
6
C1
(a) Original method
5.6
5.6
5.4
5.4
C2
5.8
C2
5.8
5.2
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5
5
4.8
4.8
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5
5.2
C
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1
(b) 1xCV, standard objective function
6
4.6
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4.8
5
5.2
C
5.4
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5.8
1
(c) 1xCV, relative objective function
6
Conclusions and future work:
- Single Sweep is a promising method to reconstruct free
energy landscapes of rather complex systems
- Results point to the “Non-local” hydrogen vacancy diffusion
as the mobile species that appears during the first dissociation
reaction of sodium alanates
Many thanks to:
Michele Monteferrante
Simone Meloni
Giovanni Ciccotti
Eric Vanden-Eijnden
Funding:
Ministero dell’Ambiente e della Tutela del Territorio e del Mare