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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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