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Supporting Information for
Classical and Quantum Modeling of Li and Na
Diffusion in FePO4
Mudit Dixit,1 Hamutal Engel, ‡1 Reuven Eitan, ‡1 Doron Aurbach,1 Michael D. Levi,1 Monica
Kosa,1 and Dan Thomas Major*1
1 Department of Chemistry, Institute of Nanotechnology and the Lise Meitner-Minerva Center of
Computational Quantum Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
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Experimental
For the preparation of thin-coated composite electrodes, we used commercially available
carbon-coated LiFePO4 powder (SudChemie). As follows from scanning electron microscopy
(SEM) images, the pristine powder consists of several hundred nanometer size ellipsoid-shape
particles, and some larger agglomerates thereof. The BET specific surface area of 19 m2/g
indicate that the particle size << 1 µm. The content of carbon was determined by elemental
analysis to be 1.9 wt.%. XRD patterns of pristine LixFePO4 powder clearly show an
orthorhombic phospho-olivine type structure (space group No. 62 [Pmna], PDF file #01-081-
1173). PVDF binder (10 wt.%) in N-methyl pyrrolidone was added to the slurry.
The diluted composite slurry was spray-coated onto the heated surface of 1-inch 5 MHz gold-
coated Maxtek crystals (electrochemical surface area 1.27 cm2). The low active mass density (ca.
50 µg cm-2) allowed the system to reach “quasi-equilibrium conditions” on charging by cyclic
voltammetry (CV) in the range of a few mV s-1. Based on quasi-equilibrium conditions we infer
practical independence of the intercalation/deintercalation charge on the scan rate.
The CV measurements were carried out using Schlumberger 1287 electrochemical interface
driven by Corrware software (Scribner). Electrochemical potentials were measured and reported
versus a Ag/AgCl/KCl(sat.) reference electrode.
Li2SO4 (Sigma-Aldrich purity ≥99.99% on trace metals basis) and Na2SO4 (high purity quality
from Fluka) were used to prepare solutions of different concentrations in double-distilled water.
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Figure S1. Spin polarized diffusion profile of Li in Li0.93FePO4 using a 1×2×2 supercell. Blue
dots represent NEB images. Violet spheres represent Li atoms, brown spheres represent Fe
atoms, yellow sphere represent P atoms and red spheres represent O atoms. Blue dots represent
NEB images.
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Figure S2(a). Potential energy grids (meV) of the initial states and the transition states in
Li0.25FePO4 and Na0.25FePO4.
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Figure S2(b). Potential energy slices of Li0.25FePO4 (TS) with grid length of 1.4 Å and grid
spacing of 0.05 Å with maximum potential energy isovalue of 20 kcal/mol.
.
Figure S2(c). Slices of the potential energy grid of Li0.25FePO4 (IS) along (a) YX (b) YZ (c) ZX
planes, and slices of the potential energy grid of Li0.25FePO4 (TS) along (d) YX (e) YZ (f) ZX
planes.
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Figure S2(d). Slices of the potential energy grid of Na0.25FePO4 (IS) along (a) YX (b) YZ (c) ZX
planes, and slices of the potential energy grid of Na0.25FePO4 (TS) along (d) YX (e) YZ (f) ZX
planes.
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Methods.
Confinement of the wavefunction at the transition state.
An interesting question relating to the construction of the grids used in the wavefunction
calculations is the nature of the potential of the Li and Na ions in the vicinity of the TS. For the
wavefunction to be localized at the TS, the potential should be bound. In the current calculations,
the Li and Na ions are weakly bound at the TS due to the way in which the grid is constructed.
This is so, as we move only the Li or Na ion during the grid construction (i.e. we do not move
the set of collective coordinates composing the unstable normal mode at the TS), the potential is
weakly bound. Inspection of the figures below (Fig. S3-5), underscores this point. In these
figures the Li or Na ion were displace along the NEB path, while keeping the olivine framework
fixed at the TS configuration. This grid construction approach is reasonable as the motion of the
Li or Na ions is expected to be considerably faster than the olivine framework reorganization.
We do note that if we do try move the ion (within the fixed olivine TS-framework) down to the
initial state or final state, the ion will be unbound (with respect to the TS) and this is also clear
from the figures (S3-5). However, the bound region of the TS is sufficiently wide and the De
Broglie wavelength of the ions sufficiently small to keep the ion there without the wavefunction
escaping (this cannot be checked in 3D due to the need for a very large grid). This view is
consistent with a physical picture in which the ions reach the TS region via largely classical
thermal hopping. Furthermore, tunneling is unlikely due to the same reasons as above: the width
of the barrier and the small De Broglie wavelength. To further inspect this assumption, we
performed centroid Monte Carlo Path-Integral calculations with a Li-ion located at the TS of the
1D potential using methods described in Ref. 1 and 2 (below). In Fig. S6 we compare the
position distribution of the Feynman paths at 300K around the TS for 4 cases (using 32 beads):
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(1) A free Li particle (2,3) Li in Eckart potentials (4) Li spin polarized diffusion potential in a
fixed olivine TS-framework. This shows that at this temperature, the Li-ion in the diffusion
potential is localized at the TS and does not show increased (or decreased) spread in the diffusion
direction compared to the free particle. On the other hand in an Eckart potential the distribution
is smeared out. This simulation also shows that the grid we employed was sufficient as the
distribution decays relatively fast around the barrier.
In reality, the potential experience by the Li or Na ion in our PI-EV calculations is 3D and the
ion will have significant zero-point energy due to confinement in the directions orthogonal to the
diffusion direction (as shown in the manuscript). This zero-point energy will presumably not
invalidate our assumption of centroid localization at the TS, because the vibration energy is not
in the diffusion direction.
Figure S3. Change in potential energy (Non Spin Polarized) of Li0.25FePO4 along the classical
diffusion path with fixed FePO4 framework atoms.
Li0.25FePO4 (Non Spin Polarized)
TS
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Figure S4. Change in potential energy (Spin Polarized) of Li0.25FePO4 along the classical
diffusion path with fixed FePO4 framework atoms.
Figure S5. Change in potential energy (Spin Polarized) of Na0.25FePO4 along the classical
diffusion path with fixed FePO4 framework atoms.
Na0.25FePO4 (Spin Polarized)
Li0.25FePO4 (Spin Polarized)
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Figure S6. Position distribution of the Feynman paths at T=300K around the TS (1) A free Li
particle (2,3) Li in Eckart potentials (4) Li spin polarized diffusion potential in a fixed olivine
TS-framework. The calculations employed Monte Carlo Path-Integral simulations with 32 beads.
Further details may be found in Ref. 1 and 2.
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Table S1. The first three nuclear energy levels (meV) for the initial and transition states (Spin-
polarized DFT) in Li0.25FePO4 and Na0.25FePO4.
Nuclear
Energy levels
(kcal/mol)
Grid specification LiFePO4 (IS) LiFePO4 (TS) NaFePO4 (IS) NaFePO4 (TS)
E0
l=0.8, d=0.2 44.6 37.3 15.7 14.3
l=1.0, d=0.1 53.7 65.0 40.5 33.9
l=1.0, d=0.05 54.2 68.0 (42.4) [36.1]
E1
l=0.8, d=0.2 81.0 44.6 84.5 23.7
l=1.0, d=0.1 81.5 68.0 58.9 41.1
l=1.0, d=0.05 81.9 72.4 (60.2)* [43.1]
E2 l=0.8, d=0.2 88.0 55.9 84.9 27.1
l=1.0, d=0.1 87.1 77.6 71.5 46.8
l=1.0, d=0.05 88.0 83.2 (72.4) [48.5]**
* (…) corresponds to the grid l=0.98 d=0.06.** […] corresponds to the grid l=0.96 d=0.07
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Table S2. The first three nuclear energy levels (meV) for the initial and transition states (Non-
spin polarized DFT) in Li0.25FePO4 and Na0.25FePO4.
System Grid Nuclear Energy Levels
E0 E1 E2
LiFePO4 (IS)
l=0.8, d=0.2 42.9 73.2 79.3
l=0.8, d=0.05 65.0 75.4 78.0
l=1.0, d=0.05 49.8 75.0 77.6
l=1.04, d=0.04 49.8 75.0 78.0
l=1.2, d=0.05 49.8 75.0 77.6
LiFePO4 (TS)
l=0.8, d=0.02 38.5 50.7 64.6
l=0.8, d=0.05 54.2 66.7 84.5
l=1.0, d=0.05 53.3 63.7 76.3
l=1.04, d=0.04 53.3 63.7 75.4
l=1.2, d=0.05 53.3 62.4 71.9
NaFePO4 (IS) l=0.8, d=0.02 5.9 7.5 7.5
l=1.0, d=0.1 37.7 55.5 64.1
NaFePO4 (TS) l=0.8, d=0.02 15.3 35.0 37.9
l=1.0, d=0.1 34.7 47.2 58.5
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Table S3. Calculated free energy and partition function ratios using normal mode analysis.
System Na0.25FePO4 Li0.25FePO4 Expression
(QTS/QIS)Q1 0.302 0.130
3 2
((1 )
i
i
N X
q
i
eQ
e
ε β
ε β
−
−
−=
−∏
(QTS/QIS)Cl 0.411 0.198 3 6
3 7( ) *
'
N
i
TS i
N
ISi
i
vQ h
clQ KT
v
−
−=
∏
∏
QG∆2 0.030 eV 0.052 eV
( , ) ln(1 )2
ii
i i
hG q k T e
ε ββ
νν −= + −∑ ∑
ClG∆2 0.022 eV 0.041 eV
ln( )iClassical
i
hG K T
K Tβ
β
ν=∑
Pre-factor3 2.57*1012/Sec 1.24x1012/Sec 3 6
3 7*
'
N
i
i
N
i
i
v
v
v
−
−=∏
∏
(QTS/QIS)NQE 1.66 0.84 Our wave function method, i
i
Q eε β=∑
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Figure S7. First excited state nuclear wavefunctions for initial and transition states in
Li0.25FePO4 and Na0.25FePO4.
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Figure S8. Second excited state nuclear wavefunctions for initial and transition states in
Li0.25FePO4 and Na0.25FePO4.
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Figure S9. Spin polarized and non-spin polarized DFT diffusion barriers for Li0.25FePO4 and
Na0.25FePO4.
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Table S4: The distance between the Metal (Li/Na) and oxygen atoms in the corresponding coordination spheres.
References:
1. Major, D.T.; Gao, J. J. Chem. Theory Comput., 3, 949 (2007).
2. Major, D.T.; Gao, J. J. Mol. Graph. Mod., 24, 121 (2005).
System M-O distance M-O distance M-O distance M-O distance M-O distance M-O distance
Li0.25FeP4 (IS) 2.15 2.15 2.16 2.16 2.19 2.19
Li0.25FeP4 (TS) 1.92 1.92 2.25 2.25 - -
Na0.25FeP4 (IS) 2.27 2.27 2.30 2.30 2.34 2.34
Na0.25FeP4 (TS) 2.19 2.14 2.36 2.36 - -
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