Author Archives: Jeremy Hu

The effects of DFT+U on the DFT density of states of anatase TiO2 (001)

Abstract

Density functional theory (DFT) calculations of TiO₂ anatase, a commonly used catalyst support, typically requires a DFT+U correction term to account for the electron self-interaction error in Ti. DFT as implemented in the Vienna Ab Initio Simulation Package (VASP) was used to determine the minimum value of the Hubbard’s U parameter to accurately represent the electronic density of states of TiO₂ and partially-reduced TiO₂. This study determined that a minimum Hubbard’s parameter of U = 3 eV is sufficient to accurately determine the density of states of TiO₂ and partially-reduced TiO₂.

Introduction

Titanium dioxide (TiO₂) is a common metal oxide for depositing catalytically-active metals onto, such as those used in biochemical production [1]. The metal oxide support provides structure for the active metal sites to be deposited onto. TiO₂ supported-catalysts for biochemical syntheses generally undergo reduction conditions, with the formation of oxygen vacancy sites in the presence of H₂ [2]. The anatase polymorph of TiO₂ most readily forms oxygen vacancies, with the (001) facet widely considered the most catalytically active [1]. Understanding the chemistry of TiO₂using DFT typically requires a DFT+U correction term to account for the self-interaction error between electrons. This correction term is typically necessary for Ti due to its numerous d-orbital valence electrons. Thus, the DFT+U correction penalizes the delocalization of d-orbital electrons in Ti [3]. Previous literature suggests that the U parameter can range anywhere from 2-10 eV, with the U parameter “sufficient” when the density of states of TiO₂ behaves as an insulator with KS orbitals appearing in the band gap in the presence of oxygen vacancies [3]. Thus, understanding the effects of the U parameter on the density of states of TiO₂ could bring insight into the minimum value of U necessary to confirm the localization of electrons in bare TiO₂ and TiO₂ under oxidation conditions.

Methods

Electronic Methods

Density functional theory (DFT) analysis of TiO₂anatase (001) was calculated using the plane-wave basis set in the Vienna Ab Initio Simulation Package (VASP) [4]. The Perdew–Burke-Ernzerhof (PBE) exchange correlation functional was used [5]. PBE+D3 was used for dispersion corrections and the projector augmented-wave method (PAW) corrected for core-valence interactions [6] [7]. The Hubbard’s parameter (U) for the DFT+U correction was iterated for Ti from U = 0-3 eV [8]. Each structure was reoptimized for each value of U before calculating the density of states. The forces on the atoms from the geometric optimizations used a convergence criteria of < 0.05 eV/Å. The self-consistent field tolerance for all calculations was 10^-5eV.

Similar work on TiO₂suggest that a Monkhorst-Pack k-point mesh of 3 x 3 x 1 and a cutoff energy of 450 eV are above the minimum for convergence [9] [10]. The valence electrons considered for each atom type were O (2s² 2p⁴) and Ti (3s² 3p⁶ 4s² 3d²).

A vacuum space of 15 Å between slabs was used to minimize dipole interactions in the z-direction (i.e., normal to the surface). A 2 x 2 supercell of anatase TiO₂(001) was used for the DFT calculations, with the bottom three atomic layers fixed while the rest of the atoms were allowed to freely relax (Fig. 1). The slab’s thickness was two layers thick, with a single layer defined as the minimum thickness of atoms to have a stoichiometric ratio of Ti_xO_2x. Additionally, the termination of the surface was chosen to be the same as the termination on the bottom of the slab (i.e., oxygen atoms on the top and bottom) to minimize any large dipole moments that would otherwise occur through asymmetry.Figure 1. A 2 x 2 supercell of TiO₂anatase (001) used for the density of states calculation.

The second-coordinated surface oxygen atom on the surface, which required the lowest amount of energy to remove, was removed for the density of states calculations of anatase TiO₂with an oxygen vacancy (Fig. 2)

Figure 2. A 2 x 2 supercell of TiO₂anatase (001) with a second-coordinated bridging oxygen (O_2C) removed used for the density of states calculation. The symmetrically-equivalent O_2C behind the removed atom was removed in this figure for clarity.

Results and Discussion

The density of states for the bare TiO₂anatase (001) was plotted for values of U from 0 to 3 (Fig. 3a-d).

Figure 3a. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=0. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3b. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=1. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3c. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=2. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3d. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=3. The blue line represents the spin up states and the black line denotes the spin down states.

There was slight noise and variation in the density of states when U was iterated from 0 to 3. However, the band gap of approximately ~2 eV stayed relatively constant in all cases, which relatively agrees with the experimentally determined band gap of ~3 eV in TiO₂[11]. The nuances in the density of states at U=3 (i.e., the clearer differentiation of peaks) suggest that a higher U value may be more appropriate for the calculation of TiO₂density of states.

The density of states for the partially-reduced TiO₂(i.e., TiO₂with a surface oxygen vacancy) was plotted as well from U = 0 to 3 (Fig. 4a-d).

Figure 4a. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=0. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4b. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=1. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4c. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=2. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4d. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=3. The blue line represents the spin up states and the black line denotes the spin down states.

As expected, there was noise and variation in the density of states ranging from U = 0 to 3. The peak in the band gap only clearly appeared in the cases where U=0 and U=3. As mentioned previously, the density appearing in the band gap for TiO₂with an oxygen vacancy likely represents occupied KS orbitals following reduction. Recent experimental and DFT studies of TiO₂with an oxygen vacancy suggest a band gap of approximately 3.0 eV with a gap state appearing around 0.7 eV below the conduction band [12]. In the case where U was set to 0, the gap state was ~0.50 eV below the conduction band, whereas when U=3 the gap state was ~0.70 eV below the conduction band. Since the peak in the band gap more closely matches experimental results in the case where U=3, it is likely that a U parameter of 3 and above is necessary to correctly determine the density of states of TiO₂with an oxygen vacancy.

Conclusion

The density of states was calculated for anatase TiO₂(001) and anatase TiO₂(001) with a surface oxygen vacancy at varying U correction values from 0 to 3. The data suggest that for bare TiO₂, a Hubbard’s U correction of 3 eV and above may prove appropriate for clearer peak density. However, the band gap of around 2 eV stayed relatively constant in all four cases. In the case of TiO₂(001) with an oxygen vacancy, the appearance of a gap state near the Fermi level correlates to occupied KS orbitals following the reduction of TiO₂. Although gap states appeared in both cases where U=0 and U=3, the gap state peak which occurred when U=3 more closely agreed with experimental data. Thus, the results suggest that a Hubbard’s U correction of 3.0 eV and above is appropriate for calculating the density of states of TiO₂(001) and TiO₂(001) with an oxygen vacancy.

Citations

[1] Liu et al., Journal of Catalysis 2019, 369, 396-404

[2] Wan et al., J. Phys. Chem. C 2018, 122, 17895-17916

[3] Z. Hu, H. Metiu, J. Phys. Chem. C 2011, 115, 13, 5841-5845

[4] Kresse, G.; Furthmueller, J., Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186

[5] J. P. Perdew, K. Burke, M. Enzerhof, Phys, Rev. Lett., 1996, 77, 3865.

[6] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys., 132 (2010), Article 154104

[7] P.E. Blöchl, Phys. Rev. B, 50 (1994), pp. 17953-17979.

[8] B. Himmetoglu et al., International Journal of Quantum Chemistry (2014), 114, 14–49

[9] H. J. Monkhorst, J.D. Pack, Physical Review B, 1976, 13, 5188-5192.

[10] H.V. Thang et al., Journal of Catalysis, 367 (2018), 104-114

[11] G. Rocker, J.A. Schaefer, W. Göpel, Phys. Rev. B, 30 (1984), p. 3704

[12] B.J. Morgan, G.W. Watson, Surface Science, 601 (2007) 5034–5041

Surface Energy Calculations and Binding Sites of Ag on TiO2 anatase (001)

Introduction

Titanium dioxide (TiO2) is a frequently-used metal oxide for depositing metal catalysts onto, such as those for hydrodeoxygenation (HDO) and hydrogenation reactions [1]. Since catalytically-active metals are generally expensive, depositing them onto a metal oxide support such as TiO2 helps to stabilize and maximize the usage of the active metal sites. Although several polymorphs and surfaces of TiO2 exist, the anatase form has been shown to be favorable under reduction conditions, while the (001) surface has demonstrated increased catalytic activity [1] [2]. Additionally, silver (Ag) single atom catalysts (SACs) on TiO2 anatase (001), which have highly dispersed Ag in the single-atom limit on the surface of TiO2, have been of interest due to their increased selectivity and activity in HDO reactions [1]. In this study, the surface energy of TiO2 anatase (001) with 1, 2, and 3 layered slabs was calculated. Additionally, the preferred binding sites of Ag single atoms on TiO2 anatase (001) were identified.

Methods

Electronic Methods

Density functional theory (DFT) analysis of TiO2 anatase (001) and Ag on TiO2 anatase (001) was calculated using the plane-wave basis set in the Vienna Ab Initio Simulation Package (VASP) [3]. The Perdew–Burke-Ernzerhof (PBE) exchange correlation functional was used as well as the Generalized Gradient Approximation (GGA) for the exchange correlation functional type [4]. PBE+D3 was used for dispersion corrections, along with the projector augmented-wave method (PAW) to correct for core-valence interactions [5] [6]. The Ti d-electrons were accounted for using the DFT+U approach, which corrects for the self-interaction error and underestimation of the band gap [5]. The Hubbard’s parameter (U) was set for Ti to be U = 4 eV. The self-consistent field tolerance of the calculations used a convergence criteria of < 0.05 eV/Å.

The Monkhorst-Pack k-point mesh for the reciprocal space was 3 x 3 x 1 and the cutoff energy was 450 eV, both of which are above the minimum for convergence of similar TiO2 DFT calculations in the literature [7] [8]. The valence electrons considered for each atom type were O (2s² 2p⁴), Ti (3s² 3p⁶ 4s² 3d²), and Ag (5s¹ 4d¹⁰). To minimize dipole interactions between periodic slabs in the z-direction (i.e., normal to the surface), a vacuum space of 10 Å was used between slabs. For the surface energy calculations, a 1 x 1 slab was used. One “layer” was considered to be the smallest thickness of TiO2 with the stoichiometric number of atoms (i.e., Ti_xO_2x). The adsorption energy of Ag was calculated using a 2 x 2 supercell of TiO2 anatase (001).

Calculation Methods

The surface energy for the different number of layers was calculated as follows, where A = the surface area of the slab (i.e., a² where a is the lattice parameter) and n is the number of TiO2 units in the slab [9]:

$$E_{surf} = \frac{1}{2A} (E_{slab} – n E_{bulk})$$ (Eqn. 1)

The 2A in the denominator accounts for the fact that there are two surface areas of interest from the slab (i.e., the top and bottom of the cell). All the atoms in the slab were allowed to fully relax during the geometric optimization. The bulk energy of TiO2 anatase was taken as -25.539 eV/unit of TiO2 [11]. The surface area of each slab (A) was 14.26 Å².

For the calculation of adsorption energy of Ag on various sites on TiO2 anatase (001), the adsorption energy was calculated as follows, with the isolated energy of the Ag atom taken under vacuum:

$$E_{ads} = E_{Ag,TiO2} – E_{TiO2} – E_{Ag}$$ (Eqn. 2)

Results and Discussion

1. Slab models and surface energy calculations

The geometry optimization of a 1 x 1 single-layered slab of TiO2 anatase (001) was performed, with one layer defined as the smallest thickness for stoichiometric TiO2 (Fig. 1).

Figure 1. Single-layered slab of TiO2 anatase (001).

Next, a geometry optimization (i.e., energy minimization) of a 1 x 1 two-layered cell of TiO2 anatase (001) was performed (Fig. 2).

Figure 2. Two-layered slab of TiO2 anatase (001).

Finally, the surface energy of a 1 x 1 three-layered slab was calculated (Fig. 3).

Figure 3. Three-layered slab of TiO2 anatase (001).

The surface energy for various layered slabs is displayed in Table 1.

Table 1. Surface energy (eV/Å²) with respect to number of layers in a 1 x 1 slab of TiO2 anatase (001)

Number of layers	# of TiO2 sub-units	Surface Energy (eV/Å^2)
1	4	0.08309
2	8	0.08199
3	12	0.08315

It would appear that the single, two, and three-layered slab models of TiO2 have reasonably consistent surface energies. Additionally, the three calculations relatively approximate the literature values of the surface energy of TiO2 anatase (001) of 0.0562 eV/Å²[10]. The similarity of surface energy values suggest that slab models beyond the single-layered slab are sufficient for the calculation of the surface energy of TiO2 anatase (001).

2. Ag adsorption sites

The following sites of Ag single atoms on TiO2 anatase (001) were identified as possible energy minima (Fig. 4).

Figure 4. Four possible adsorption sites of Ag single atoms on TiO2 anatase (001) (top view).

The O2C and O3C refer to the coordination number of the oxygen atom of interest. The following adsorption energies of Ag were calculated using Eqn. 2 (Table 2).

Table 2. Adsorption energy (eV) of Ag single atoms on various sites on the TiO2 anatase (001) surface

Position	Adsorption energy (eV)
1	-1.474
2	-0.680
3	-0.783
4	-1.003

Regardless of the binding site, the binding of Ag single atoms on the surface of TiO2 anatase (001) is energetically favorable, since all the adsorption energies are negative. Furthermore, the DFT calculations suggest that position 1 is the most energetically favorable position for Ag single atoms to bind, since its adsorption energy is the lowest. Thus, we propose that deposition of Ag on TiO2 anatase (001) in the single-atom limit likely results in a majority of positions where the Ag atom is bridged between the two second-coordinated oxygen atoms on the surface (Fig 5).

Figure 5. Most energetically favorable position of Ag SAC on TiO2 anatase (001) (side view).

Conclusion

The surface energy calculations of single, two, and three-layered slabs of TiO2 anatase (001) suggest that models beyond a single-layered slab are sufficient to approximate the surface energy. The adsorption of Ag single atoms on a 2 x 2 supercell of TiO2 anatase (001) demonstrated possible energy minima of adsorption sites. The adsorption study suggests that the deposition of Ag in the single-atom limit results in a state where Ag is bridged between two O2C oxygen atoms. These results are promising for understanding catalyst behavior and structure of Ag single atoms on TiO2 anatase (001) supports.

Citations

[1] Liu et al., Journal of Catalysis 2019, 369, 396-404

[2] Wan et al., J. Phys. Chem. C 2018, 122, 17895-17916

[3] Kresse, G.; Furthmueller, J., Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186

[4] J. P. Perdew, K. Burke, M. Enzerhof, Phys, Rev. Lett., 1996, 77, 3865.

[5] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys., 132 (2010), Article 154104

[6] P.E. Blöchl, Phys. Rev. B, 50 (1994), pp. 17953-17979.

[7] H. J. Monkhorst, J.D. Pack, Physical Review B, 1976, 13, 5188-5192.

[8] H.V. Thang et al., Journal of Catalysis, 367 (2018), 104-114

[9] Sholl, D.S. Steckel, J.A. (2009) “Density Functional Theory: A Practical Introduction.” Wiley, 96-98.

[10] Lazzeri, M.; Vittadini, A.; Selloni, Phys. Rev. B 2001, 63, 155409

[11] A. Kiejna, T. Pabisiak, S.W. Gao, J. Phys.: Condens. Matter 18 (2006) 4207-4217

DFT Optimization of Ag in sc, fcc, and hcp Geometries

Introduction

Silver (Ag) has been used for multiple chemical engineering applications, most notably in the field of catalysis. In fact, Ag catalysts are used in a wide field of applications including electrocatalysis and oxidation/reduction reactions due to their high activity and stability [1]. Understanding the crystal structure of Ag may aid in explaining the structure and behavior of Ag in various applications. Experimental studies on the crystal structure of Ag suggest that Ag prefers the fcc crystal structure with a lattice constant of a = 4.079 Å [2]. To offer a comparison to the experimental data on Ag, plane-wave density functional theory (DFT) was used to characterize and optimize three crystal structures of Ag using CASTEP in Materials Studio [3]. The structure and energy of Ag in simple cubic (sc), face-centered cubic (fcc), and hexagonal close-packed (hcp) crystal structures were analyzed [4].

Methods

Electronic Methods

DFT analysis of Ag was run using the plane-wave basis set in CASTEP in Materials Studio using the following parameters:

Table 1. Parameters for DFT Calculations in CASTEP

Exchange correlation functional type	Generalized Gradient Approximation (GGA) [5]
Exchange correlation Functional	Perdew–Burke-Ernzerhof (PBE) [5]
Pseudopotentials	On-the-fly generated (OTFG) ultrasoft
Relativistic treatment	Koelling-Harmon
Core radius	1.6 a.u.
Valence electron configuration	4s2 4p6 4d10 5s1 (19 valence electrons)
SCF Tolerance	2E-06 eV/atom

First, the energy of the sc, fcc, and hcp crystal structure of Ag was iterated with respect to the energy cutoff (ENCUT) and the irreducible number of k-points. The energy cutoffs and irreducible number of k-points were considered converged when the energy did not vary significantly from the energy at the infinite energy cutoff or irreducible k-point limit (i.e., the energy at the highest ENCUT or number of k-points). For each convergence check, the relative energy was reported with respect to the energy value at the highest number of irreducible k-points or ENCUT. For the initial convergence check for the minimum number of irreducible k-points, the default ENCUT in CASTEP of 489.8 eV was used. The assumption that the default ENCUT was high enough to ensure convergence was confirmed during the ENCUT convergence check.

After a minimum ENCUT and irreducible number of k-points were identified (i.e., the values at which the energy was considered converged), lattice parameters were varied for three different crystal structures of Ag (e.g., sc, fcc, and hcp) and the relative energy was plotted as a function of the lattice parameters. The energy minimum of each plot corresponds to the preferred (i.e., energetically favorable) lattice parameter.

Model

Crystal structures of Ag were tested with the following parameters:

Table 2. Iteration ranges of parameters for three possible crystal structures of Ag

Lattice type	Lattice parameter (a) range	Irreducible k-points (Monkhorst [6])	Energy cutoff (eV)
Simple cubic (sc)	a: 2.0 - 3.2 Å	1 - 120	200 - 600
Face-centered cubic (fcc)	a: 3.5 - 4.5 Å	1 - 182	200 - 600
Hexagonal close-packed (hcp)	c/a: 1.5 - 1.7	15 - 343	200 - 600

Results and Discussion

1. K-point convergence

The energy of Ag in the sc structure was converged after 20 irreducible k-points, with the relative energy reaching within 0.01 eV/atom (Fig. 1).

Figure 1. Relative energy (eV/atom) as a function of irreducible k-points for the simple cubic (sc) structure of Ag at a = 2.7 Å and ENCUT = 489.8 eV.

Next, the energy of the fcc structure of Ag was converged after 28 irreducible k-points, where the relative energy was less than 0.01 eV/atom than the energy at the infinite k-point limit (Fig. 2).

Figure 2. Relative energy (eV/atom) as a function of irreducible k-points for the face-centered cubic (fcc) structure of Ag at a = 4.1 Å and ENCUT = 489.8 eV.

Finally, the energy of the hcp structure was converged after 216 irreducible k-points, with a relative energy of 0.007 eV/atom (Fig. 3).

Figure 3. Relative energy (eV/atom) as a function of irreducible k-points for the hexagonal close-packed (hcp) structure of Ag at a = 2.8 Å, c/a = 1.7, and ENCUT = 489.8 eV.

For the purposes of brevity, it is assumed that the irreducible k-points necessary for convergence at the chosen arbitrary lattice constants is valid for lattice constants close to the chosen values.

2. Cutoff energy convergence

The energy of the fcc structure was determined at various cutoff energies from 200 eV to 600 eV. The minimum cutoff energy was 450 eV, where the corresponding energy had less than a 0.1% difference than the energy at the highest cutoff energy (Fig. 4). Since the cutoff energy should be kept constant for all the crystal structures for optimization, it was assumed that the same minimum cutoff energy would be appropriate for the other crystal structures.

Additionally, Fig. 4 confirms the assumption that the default ENCUT value of 489.8 eV used for the previous k-points convergence was sufficient to ensure convergence with respect to the ENCUT value.

Figure 4. Relative energy (eV/atom) as a function of cutoff energy for the face-centered cubic (fcc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid (i.e., 110 irreducible k-points) and a = 4.1 Å.

3. Lattice parameter optimization

In the following lattice parameter optimization calculations, the lattice parameter (a) was varied for each structure. Each energy value was taken relative to the lowest calculated energy in the most stable structure.

For the sc structure of Ag, the energy minimum was observed at around a = 2.7 Å (Fig. 5), which corresponds to a minimum relative energy of 0.325 eV/atom.

Figure 5. Relative energy (eV/atom) as a function of lattice parameter (Å) for the simple cubic (sc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid and ENCUT = 489.8 eV.

Since the hexagonal close-packed structure of Ag has two lattice parameters (i.e., a and c), the ratio of c/a was kept fixed from c/a = 1.5 – 1.7 for each iteration of a. The relative energy minimum was found to be 0.237 eV/atom at a = 2.8 Å and c/a = 1.7 (Fig. 6).

Figure 6. Energy (eV/atom) as a function of lattice parameter (Å) for the hexagonal close-packed (hcp) structure of Ag at a constant 12 x 12 x 6 Monkhorst k-point grid and ENCUT = 489.8 eV.

Finally, iterations were performed for the fcc structure of Ag, where the lowest energy out of all systems (i.e., the state where the relative energy was set to zero) was observed at a = 4.1 Å (Fig. 7).

Figure 7. Energy (eV/atom) as a function of lattice parameter (Å) for the face-centered cubic (fcc) structure of Ag at a constant 10 x 10 x 10 Monkhorst k-point grid and ENCUT = 489.8 eV.

Conclusion

The convergence checks for each crystal structure with respect to k-points and cutoff energy confirm that the values of irreducible k-points and ENCUT used in the crystal structure calculations were converged with respect to energy. The final parameter values for each crystal system is shown below in Table 3.

Table 3. Final lattice parameters and parameter values for Ag in the sc, fcc, and hcp crystal structures.

Lattice structure	Cutoff energy	Monkhorst k-point grid	Irreducible k-points	Optimized lattice parameter	Minimum relative energy
sc	489.8 eV	10 x 10 x 10	35	a = 2.7 Å	0.325 eV
fcc	489.8 eV	10 x 10 x 10	110	a = 4.1 Å	0 eV
hcp	489.8 eV	12 x 12 x 6	216	a = b = 2.8 Å c/a = 1.7	0.237 eV

Thus, our DFT calculations suggest that Ag prefers the fcc crystal structure with a lattice parameter of a = 4.1 Å since it has the minimum energy (i.e., the most thermodynamically favorable crystal structure). These results are consistent with the experimentally determined results, which show that Ag exists as the fcc crystal structure with a lattice constant of a = 4.079 Å [2]. These results on the crystal structure of Ag are promising for understanding Ag-based catalysts, such as catalysts with Ag nanoparticles.

Citations

[1] G. Liao, J. Fang, Q. Li, S. Li, Z. Xu, B. Fang, “Ag-Based nanocomposites: synthesis and applications in catalysis,” Nanoscale, 2019, 11, 7062-7096
[2] W. Davey, “Precision Measurements of the Lattice Constants of Twelve Common Metals,” Physical Review, 1925, 25 (6), 753-761.
[3] Clark Stewart J et al., “First principles methods using CASTEP ,” Zeitschrift für Kristallographie – Crystalline Materials, 2005, 220, 567
[4] D. Sholl, J. Steckel, “Density Functional Theory,” Somerset: Wiley, 2008
[5] J. P. Perdew, K. Burke, M. Enzerhof, “Generalized Gradient Approximation Made Simple,” Phys, Rev. Lett., 1996, 77, 3865.
[6] H. J. Monkhorst, J.D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B, 1976, 13, 5188-5192.

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Author Archives: Jeremy Hu

The effects of DFT+U on the DFT density of states of anatase TiO2 (001)

Abstract

Introduction

Methods

Electronic Methods

Results and Discussion

Conclusion

Citations

Surface Energy Calculations and Binding Sites of Ag on TiO2 anatase (001)

Introduction

Methods

Electronic Methods

Calculation Methods

Results and Discussion

1. Slab models and surface energy calculations

2. Ag adsorption sites

Conclusion

Citations

DFT Optimization of Ag in sc, fcc, and hcp Geometries

Introduction

Methods

Electronic Methods

Model

Results and Discussion

1. K-point convergence

2. Cutoff energy convergence

3. Lattice parameter optimization

Conclusion

Citations