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Supporting information for
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Electron dynamics of solvated Titanium Hydroxide
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Dayton J. Vogel, Dmitri S. Kilin*
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University of South Dakota, Department of Chemistry, Vermillion, SD, 57069
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*Corresponding author email: Dmitri.Kilin@usd.edu
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The electronic nature was calculated using density functional theory (DFT) run
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within the Vienna ab initio simulation package (VASP)[1] using the Perdew-Burke-
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Ernzerhof (PBE)[2] form of the generalized gradient approximation (GGA). The
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electronic nature was implemented with a plane wave basis set and projected
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augmented wave (PAW)[3] potentials. DFT calculations provide data for the
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optimized geometry, ground state electronic structure, absorption spectra, and
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molecular dynamics of the model using a Kohn-Sham (KS) basis set.
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Optical transitions are induced an oscillating electromagnetic field propagating a
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system from a ground state to an excited state. Such transitions are determined by
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matrix elements of the leading term in the electron-photon interaction
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𝑯𝒊𝒏𝒕 =
⃗ )𝟐
(𝒑+𝒆𝑨
𝟐𝒎
𝒑𝟐
= 𝟐𝒎 +
⃗
⃗ 𝒆𝑨
𝒑
𝒎
+
𝐞𝟐 ⃗𝑨𝟐
𝟐𝒎
(S1)
Where 𝑝 is the momentum operator, e is electric charge, 𝐴 is the vector potential.[4]
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S-1
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⃗ 𝑖𝑗 are calculated from Kohn-Sham orbitals (KSO) as
The transition dipole moments 𝐷
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detailed previously.[5] The KSO are eigenfunctions of the KS effective Hamiltonian
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FKS with energies 𝜀𝑗 , given at position r by:
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⟨𝒓|𝒋⟩ = 𝝓𝒋 (𝒓) = ∑𝑮<𝑮𝒄𝒖𝒕 𝑪𝒋,𝑮⃗ 𝒆−𝒊𝑮⃗ 𝒓
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where 𝐶𝑗,𝐺 is a KSO in the momentum representation, G is a grid point in reciprocal
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space and Gcut is the cutoff value for the magnitude of G vectors in a Fourier
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expansion.[6] The transition dipole moment, D jk , for the jk transition is given by:
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̂|𝒌⟩ =
𝑫𝒋𝒌 = ⟨𝒋|𝑫
∑𝑮<𝑮𝒄𝒖𝒕 𝑪∗𝒋,𝑮 𝑫𝑮,𝑮′ 𝑪
𝒌,𝑮′
(S2)
(S3)
Where
′
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𝑫𝑮,𝑮′ = −𝒆 ∫ 𝒅𝟑 𝒓𝒆𝒊𝑮𝒓 𝒓𝒆−𝒊𝑮 𝒓
(S4)
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The dipole matrix elements have been re-expressed in terms of matrix elements of
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the momentum operator:
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⟨𝒋|𝒑̂|𝒌⟩ = 𝒊ℏ ∑𝑮,𝑮′ <𝑮𝒄𝒖𝒕 𝑪∗𝒋,𝑮 𝑮𝑪
𝒌,𝑮′
(S5)
to obtain
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⟨𝒋|𝑫̂|𝒌⟩ = ⟨𝒋|𝒑̂|𝒌⟩𝒊ℏ𝒆/𝒎𝒆 (𝜺𝒌 − 𝜺𝒋 )
(S6)
for calculations.[5]
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The gamma point, Γ, was chosen to represent the k-point sampling based on the
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substantial energy gap of the model. Such an approximation is justified for wide-gap
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insulators and periodic models in vacuum. As the bandgap is larger than 4eV for this
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⃗ ) in TiO2
model it is considered a wide-gap insulator. Dispersion curves of 𝜀𝑖 (𝑘
S-2
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nanowires has been found to be “flat.”[7] Dispersion corrections implemented by
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Grimme are being considered in future calculations.[8]
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Computational details include: the energy cutoff value was chosen to be 300eV, a
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MD trajectory length of 1ps to represent typical nuclear motion, and a time step of
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1fs was chosen for the Ab initio MD. Standard procedure from VASP was used to
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integrate nuclear motion trajectories. Equilibration was reached by modeling
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nuclear interaction with a thermostat at various ambient temperatures according to
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Eq. 6. The temperatures simulated were 100K, 200K, 300K, and 350K.
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Computational details include: the energy cutoff value was chosen to be
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300eV, a MD trajectory length of 1ps to represent typical nuclear motion, and a time
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step of 1fs was chosen for the Ab initio MD. Standard procedure from VASP was
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used to integrate nuclear motion trajectories. Equilibration was reached by
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modeling nuclear interaction with a thermostat at various ambient temperatures
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according to Eq. 6. The temperatures simulated were 100K, 200K, 300K, and 350K.
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Decoherence can be important for very quick initial time, t<10 fs. However,
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after 10fs decoherence is of secondary importance. The equation of motion (EOM)
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for 𝜌ij splits on two independent subsets where populations i=j and 𝑖 ≠ 𝑗.
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Decoherence is important for 𝑖 ≠ 𝑗, but we focus on i=j. The decoherence effects are
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important factors and have been implemented into recent surface hopping
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procedures.[9] Correcting for decoherence is expected to slightly speed up the
S-3
67
relaxation. The transitions with smaller subgap energies will be enhanced to a
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greater measure, thus obeying the energy gap law.
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Excitation States
Oscillator
Excitation
Initial
Final
Strength
Energy
HO-23
LU+2
1.30866221
5.4454
HO-22
LU+2
1.04703001
5.3963
HO-8
LU+2
0.95182345
6.7146
HO-19
LU+2
0.62432678
5.5258
HO-40
134
0.60306177
6.8978
HO-26
LU+2
0.59854254
5.3011
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Table S1. The six highest oscillator strength values are given. Initial and final states for
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each transitions, corresponding to each oscillator strength, are given as well as the energy
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needed to generate the excitation.
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Figure S 1. An insert highlighting the ligand field splitting (close to tetrahedral) found between
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0eV and 2eV of the DOS in Figure 1 C. The non-shaded area under the curve represents the
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electronic states are unoccupied. The vertical lines represent individual state energies. Solid
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lines represent states of Ti 3d character while the dashed line represents a state of oxygen
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character.
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Figure S2. Partial density of states for KSO range of HO to LU+5. Orbital character
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shows Ti 3d nature to LU, LU+1, LU+3, LU+4, LU+5. LU+2 state shows that the main
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orbital character comes from s orbitals. The left panel shows the data points while the
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right panel visualizes the trend between KSOs orbital character.
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The Redfield tensor, Rijlm, expressed in Eq. 12 is comprised of tensor components
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according to
Rijkl = G+ljik + G-ljik - dlj å G+immk - dik å G-lmmj
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m
m
(S7)
where
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G+ijkl = ò dt M ijkl (t ) e-iwklt
(S8)
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G-ijkl = ò dt M ijkl (t ) e
(S9)
S-5
-iwijt
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The Redfield tensor is derived and reported in previous work, according to A.
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Redfield.[10]
A
B
C
D
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Figure S3. The Redfield Tensor is displayed in panels A, B, C, and D for ambient
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temperature simulations at 100K, 200K, 300K, and 350K, respectively. The x and y
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axis are given for orbital indexes j and i, respectively. The z component of the graph
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gives the amplitude of Rijij fs-1.
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Electron Relaxation Time (ps)
Ti(OH)4
2.479
<001> NW
2.812
<001> Surface
0.216
Hole Relaxation Time (ps)
Ti(OH)4
0.237
<001> NW
1.181
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S-6
<001> Surface
0.307
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Table S2. Charge carrier relaxation times for both electron and hole. Relaxation
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times for the solvated Ti(OH)4, <001> nanowire (NW), and <001> anatase surface
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are given in picoseconds.
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Computation of the Ti(OH)4 gas phase molecule were conducted using time
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dependent density functional theory (TDDFT) to use in comparison to the presented
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DFT results.
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Figure S4. Absorption spectra of Ti(OH)4 gas phase molecule computed with
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TDDFT, with B3LYP hybrid functional and LanL2Dz basis.
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Orbitals contributing to
excitation
Initial
Final
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Excitation Energy (eV)
3.319
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3.6866
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3.9426
S-7
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25
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4.0589
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Table S3. Excitation Energies calculated using TDDFT for the Ti(OH)4 gas phase
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molecule.
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The excitation energies calculated using TDDFT are comparable to the band gap
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energies calculated using DFT. The excitation energies calculated using TDDFT are
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expected to be smaller than the band gap energies found using DFT. TDDFT takes
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electron and hole interaction into account when calculating the energies of possible
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excitations.
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References
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