#
Unravelling the Potential of Density Functional Theory through Integrated Computational Environments: Recent Applications of the Vienna Ab Initio Simulation Package in the MedeA^{®} Software

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

## 3. Illustrative Cases

#### 3.1. Design of Low-Strain Cathode Materials for Solid-State Li-Ion Batteries

_{4}Ti

_{5}O

_{12}has been shown to operate with nearly zero strain [21], a cathode material with similar properties was sought at the time this study was initiated. Candidate electrode materials as given in Figure 1 show either rather low voltage or large volume change as in the case of the previous benchmark material LiNi

_{0.5}Mn

_{1.5}O

_{4}[22]. In the following, we describe how ab initio calculations based on density functional theory can be utilized to investigate a class of transition-metal oxides based on the spinel structure for their suitability as low-strain cathode materials [23,24]. Indeed, from these calculations, three low-strain materials have been identified within the class of LiMn

_{x}Cr

_{y}Mg

_{z}O

_{4}. The most promising materials have been synthesized and characterized by X-ray diffraction and electrochemical techniques. The results are consistent with the ab initio predictions [23,24].

_{x}M

_{2}O

_{4}, where M = Mg, Al, V, Cr, Mn, Fe, Co, Ni, Cu, on lithiation and delithiation. To this end, three different values for the lithium content were considered, namely, x = 0, 0.5, 1.0. All structures were fully relaxed and different spin-structures were taken into account. In addition, deviations from the cubic symmetry underlying the spinel-structure were allowed for. In a second step, the previous results were used to minimize the volume change of the quasi-ternary compounds LiM

^{1}

_{2-y-z}M

^{2}

_{y}M

^{3}

_{z}O

_{4}in an approach closely following Vegard’s law [25].

_{2}O

_{4}. However, since Mg is electrochemically inactive, Cr was selected as a third component in addition to Mn because of the many oxidation states offered by this element.

_{2-y-z}Cr

_{y}Mg

_{z}O

_{4}in order to minimize the volume change on lithiation and delithiation. To this end, three different approaches were used, namely, (1) full unconstrained optimization to obtain zero-strain behaviour; (2) constrained optimization to find a low-strain material with Mn content 2-y-z larger than 1 in order to ensure structural stability; and (3) constrained optimization to find a low-strain material with Mg content z smaller than 0.2 in order to reduce the amount of highly oxidized Cr necessary to achieve full charge. The results of these three approaches together with the volume change of the benchmark compounds LiNi

_{0.5}Mn

_{1.5}O

_{4}are displayed in Figure 3. As expected, unconstrained optimization indeed leads to zero volume change, however, at a very small Mn and large Cr content. More interesting are the two low-strain configurations LiMn

_{1.1}Cr

_{0.5}Mg

_{0.4}O

_{4}and LiMn

_{0.59}Cr

_{1.21}Mg

_{0.2}O

_{4}with volume changes of 0.6% (3 Å

^{3}) and 1.6% (8 Å

^{3}), respectively, much lower than the value of 6% (30 Å

^{3}) obtained for LiNi

_{0.5}Mn

_{1.5}O

_{4}.

_{0.5}Mn

_{1.5}O

_{4}, LiCu

_{0.25}Ni

_{0.25}Mn

_{1.5}O

_{4}, LiCr

_{0.5}Mn

_{1.5}O

_{4}, LiFe

_{0.5}Mn

_{1.5}O

_{4}, LiCo

_{0.16}Mn

_{1.84}O

_{4}and LiAl

_{0.15}Mn

_{1.85}O

_{4}, which were found in very good agreement with measured values in all cases [23,24].

_{2}O

_{4}has not been observed. In contrast, LiMn

_{1.1}Cr

_{0.5}Mg

_{0.4}O

_{4}could be synthesized in almost pure form and was thus used to build electrochemical cells in order to investigate the volume change upon charging and discharging the cells. As a result, ex situ XRD patterns showed a volume change of 0.8% (4 Å

^{3}) in very good agreement with the calculations. In addition, charge curves taken for this system revealed a performance comparable to that of the benchmark material LiNi

_{0.5}Mn

_{1.5}O

_{4}.

_{x}Mn

_{1.125}Cr

_{0.5}Mg

_{0.375}O

_{4}. With increasing Li concentration, the Mg-O bonds tend to decrease and the Mn-O bonds remain similar, whereas the Cr-O bonds tend to increase. As a result, the overall volume of the crystal structure changes little upon charging and discharging with Li ions. Furthermore, the microscopic analysis reflects the behaviour of the “pure” compounds as displayed in Figure 2. This behaviour is also consistent with observations for the zero-strain mechanism for Li

_{4}Ti

_{5}O

_{12}, where local distortions in the crystal structure similarly allow this material to keep the volume nearly unchanged upon lithium insertion [26].

_{1.1}Cr

_{0.5}Mg

_{0.4}O

_{4}, found a volume change of 0.8% (4 Å

^{3}) in very good agreement with the calculated 0.6% (3 Å

^{3}). Finally, analysis of the DFT results provided an explanation for the observed volume changes based on a microscopic compensation mechanism.

#### 3.2. Embrittlement of Cu Micro-Structures

#### 3.3. Structure and Bonding of Boron Carbide

_{4}C, is one of the hardest materials known, close to diamond and cubic boron nitride. Due to its mechanical properties, B

_{4}C is used in many applications as an abrasive or shielding material. In nuclear reactors boron carbide is used to control the neutron flux due to the high neutron absorption of

^{10}B and the radiation hardness and chemical stability of B

_{4}C. According to the boron-carbon phase diagram displayed in Figure 9, a boron carbide phase exists between approximately 9 at% to 22 at% C with a melting point reaching 2450 °C, which is complemented by carbide phases with excess boron and graphite, respectively, below and above this carbon concentration range [31,32].

_{4}C and boron-rich B

_{13}C

_{2}within the rods from X-ray diffraction data and, hence, boron-pure icosahedra as shown on the left hand side of Figure 10 for the latter compound [33]. Clark and Hoard gave a structure for B

_{4}C with linear C-C-C rods and again icosahedra comprised of only B atoms [34].

_{4}C, Lazzari et al. confirmed the finding of linear C-B-C rods reported by Larson but in addition were able to identify the carbon atom within the B

_{11}C icosahedra at the polar sites, which form the top and bottom triangles [35]. This structure is shown on the right hand side of Figure 10.

^{13}C and

^{11}B NMR chemical shifts [36]. However, these groups considered only small sets of ordered structures.

_{13}C

_{2}(x

_{B}= 0.8667) and B

_{4}C (x

_{B}= 0.8) are identical to those proposed by Larson, Lazzari et al. and Mauri et al. shown in Figure 10, which share the linear C-B-C motifs, whereas additional carbon atoms prefer the polar sites of the icosahedra.

_{9}C and B

_{4}C as expected with the lowering of the enthalpy of formation within this sequence.

_{4}C confirm the thermodynamic stability of this compound and reveal an isolated high-frequency mode, which originates from bond-stretching vibrations of the B atoms in the C-B-C linear rods as illustrated in Figure 14, which is consistent with the analysis of Lazzari et al. [35].

_{B}= 0.8 and x

_{B}= 0.9. For B

_{4}C and B

_{13}C

_{2}the investigation confirmed the C-B-C sequence of the linear rods, whereas additional carbon atoms in B

_{4}C prefer the polar sites of the icosahedra. Computed elastic properties showed a stiffening of boron carbide with increasing carbon concentration, while analysis of vibrational spectra revealed an identifiable high-frequency mode connected to bond-stretching within the linear rods.

#### 3.4. High-Throughput Calculations

#### 3.5. Accurate Band Gaps of Transition-Metal Oxides from Hybrid-Functional Calculations

_{2}at ambient pressure has been the subject of ongoing dispute since its discovery in the 1950s [40,41]. Occurring at the technologically interesting temperature of 340 K this metal-insulator transition is connected with a change in resistivity of several orders of magnitude leading to applications in surface coatings, sensors and imaging systems. Interestingly, this transition temperature is closely related to that of a structural transition from the high-temperature rutile structure to a low-temperature monoclinic M

_{1}structure, which deviates from the former by a pronounced V-V dimerization parallel to the rutile c-axis and a zigzag-like antiferroelectric displacement of the V atoms perpendicular to that axis.

_{6}octahedra and the vertical V-atom chains are displayed on the left hand side. In the monoclinic M

_{1}structure shown on the right hand side of that figure the unit cell is doubled and the vanadium atoms in the chains are dimerized as well as horizontally shifted off the octahedral centres.

_{2}phase with a slightly different pattern of the structural distortions and long-range magnetic ordering [46,47]. Again, for this new phase, density functional theory together with a semi-local approximation failed to fully capture the insulating behaviour [41].

_{2}as well as the antiferromagnetic ordering of its M

_{2}phase and were thus expected to be able to also settle the long-standing dispute on the origin of the metal-insulator transitions of this cornerstone material [48].

_{1}and M

_{2}phases. The partial densities of states (DOS) calculated for the rutile phase using the semi-local GGA and HSE hybrid functional are displayed on the left-hand side of Figure 17.

_{1}phase are displayed on the right-hand side of Figure 17. Again, the O 2p and V 3d states are well separated for both functionals. However, GGA and HSE result in completely different behaviour. While the GGA leads to metallic behaviour as clearly indicated by the presence of electronic states at the Fermi energy, the HSE calculations reproduce the observed optical band gap as indicated by the arrow.

_{2}phase. For this phase, GGA calculations likewise fail to reproduce the insulating behaviour, whereas HSE calculations yield an optical band gap and well localized magnetic moments at the vanadium sites, thus impressively underlining the striking progress offered by hybrid functionals.

_{2}, one of the key representatives of the broad class of electronically correlated materials. Furthermore, the calculations allowed for a deeper understanding of the origin of the metal-insulator transition, which was shown to be intimately connected to the structural changes occurring at the phase transition as suggested by Goodenough and co-workers. Nevertheless, follow-up studies indicated that methods based on density functional theory have difficulties capturing the correct magnetic states of the rutile and M

_{1}phases. A lot of work has been devoted to this issue as well as to investigations regarding the simultaneous occurrence of the electronic and structural transitions leading from the rutile phase to the monoclinic M

_{1}phase [49].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Li chemical potentials and volume changes of candidate cathode materials. The shaded area marks the target materials with a much lower volume change than the benchmark LiNi

_{0.5}Mn

_{1.5}O

_{4}.

**Figure 2.**Computed cell volume as a function of Li concentration in transition-metal oxides with the spinel structure.

**Figure 3.**Computed cell volume as a function of Li concentration in transition-metal oxides with the spinel structure.

**Figure 4.**Computed interatomic distances of the compound Li

_{x}Mn

_{1.125}Cr

_{0.5}Mg

_{0.375}O

_{4}for increasing Li concentration. Li, Mn, Cr, Mg and O atoms are given in dark blue, violet, cyan, light blue and red, respectively.

**Figure 5.**EBSD/SEM images of the surface and cross-section of the tensile-samples. (

**a**–

**c**) Sample type A, (

**d**) sample type A annealed at 1073 K for 5 h, (

**e**–

**g**) sample type B and (

**h**) colour code of the inverse pole figure corresponding to the grain orientations. Reprinted from A. Wimmer, M. Smolka, W. Heinz, T. Detzel, W. Robl, C. Motz, V. Eyert, E. Wimmer, F. Jahnel, R. Treichler and G. Dehm, “Temperature dependent transition of intragranular plastic to intergranular brittle failure in electrodeposited Cu micro-tensile samples,” Mater. Sci. Eng. A 618, 398 (2014), with permission from Elsevier.

**Figure 6.**Stress–strain curves of sample types A and B at 293 K, 473 K and 673 K. Note that at 473 K the yield strength, ultimate tensile strength and elongation to fracture are drastically smaller for the (

**a**) fine-grained sample type A compared to (

**b**) samples of type B with their bamboo-like microstructure. The ultimate tensile strength shows a significant decrease with increasing temperature. Both sample types A and B show a significantly smaller slope during loading at 673 K (apparent Young’s modulus) caused by plastic deformation (settlement) of the sample head. Reprinted from A. Wimmer, M. Smolka, W. Heinz, T. Detzel, W. Robl, C. Motz, V. Eyert, E. Wimmer, F. Jahnel, R. Treichler and G. Dehm, “Temperature dependent transition of intragranular plastic to intergranular brittle failure in electrodeposited Cu micro-tensile samples,” Mater. Sci. Eng. A 618, 398 (2014), with permission from Elsevier.

**Figure 7.**Model of a Σ5(001) twist grain boundary. The adjacent crystal lattices are indicated in green and black. Note that prior to the rotation (twist) the black lattice points have been found above the centres of the squares formed by the green lattice points. For the rotation shown in the figure some of the black lattice points again coincide with the centres of the squares of the green lattice and the square spanned by these points as indicated in purple comprises five points of the green lattice. Within the coincident site lattice theory [30] a maximum number of coincidences of the original and rotated lattice points lets expect a minimal grain-boundary energy.

**Figure 8.**Model of a Cu microstructure with a Cl atom at the centre of the grain (

**left**) and at the grain boundary (

**right**).

**Figure 13.**Enthalpy of formation ($\Delta {\mathrm{H}}_{\mathrm{f}}$) as a function of the boron concentration ${\mathrm{x}}_{\mathrm{B}}$ of a binary cluster expansion of boron carbide. The solid black line indicates the convex hull with the pure phases based on the boron carbide crystal structure fully occupied by carbon and boron atoms, respectively. The solid red line shows the correct ground-state line connecting the graphite and α-boron. The green squares show the DFT enthalpies of formation of the structures in the training-set, green crosses the CE predicted enthalpies of formation of the training set structures and the grey crosses the CE predicted enthalpies of formation of all the other structures considered by the cluster expansion. Note that the true ground state line results throughout from DFT calculations. The green cross found below the ground state line at x

_{B}= 0.8667 corresponds to the green square found slightly above on the ground state line; the difference between both enthalpies may be taken as a measure of the accuracy of the CE predictions.

**Figure 14.**Phonon dispersion of B

_{4}C computed using the direct method (finite displacement method) as implemented in MedeA-Phonon. The high-frequency mode at 48 THz corresponds to the bond-stretching vibrational mode of the B atoms in the C-B-C linear rods as indicated by the arrow shown in the inset.

**Figure 15.**Computed descriptor related to magnesium binding energies (eV) for a range of experimentally known magnesium compounds. This screening survey highlights different classes of materials containing magnesium, thus providing a quantitative comparison of the chemical bonding between host structure and the magnesium within each system.

**Figure 16.**High-temperature rutile structure (

**left**) and low-temperature M

_{1}structure (

**right**) of VO

_{2}. Vanadium and oxygen atoms are given in cyan and red, respectively.

**Figure 17.**Partial DOS of rutile VO

_{2}(

**left**) and M

_{1}VO

_{2}(

**right**) as calculated using the GGA (

**top**) and the HSE (

**bottom**) functional.

ΔE_{seg} [kJ/mol] | ||
---|---|---|

Cl | S | |

GB segregation, Σ5 (001) | −69.9 | −54.3 |

Surface segregation, (001) | −321.5 | −145.3 |

GB segregation, Σ7 (111) | −53.5 | −56.3 |

Surface segregation, (111) | −272.0 | −129.9 |

**Table 2.**Calculated work of separation E

_{sep}of Σ5 and Σ7 grain boundaries of pure Cu and Cu contaminated with Cl and S with planar impurity concentration c

_{imp}in atoms per nm

^{2}.

c_{imp} [1/nm^{2}] | E_{sep} [J/m^{2}] | |||
---|---|---|---|---|

Pure Cu | Cl | S | ||

Σ5 (001) | 0.77 | 1.08 | 0.82 | 1.01 |

Σ7 (111) | 0.62 | 1.13 | 0.89 | 1.04 |

**Table 3.**Computed bulk and Young’s moduli of boron carbides with boron concentrations between x

_{B}= 0.8 and x

_{B}= 0.9.

B_{4}C | B_{5}C | B_{13}C_{2} | B_{9}C | |
---|---|---|---|---|

Bulk modulus [GPa} | 270.16 | 261.43 | 253.31 | 239.35 |

Young’s modulus [GPa] | 487.94 | 436.45 | 405.86 | 383.36 |

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**MDPI and ACS Style**

Eyert, V.; Christensen, M.; Wolf, W.; Reith, D.; Mavromaras, A.; Freeman, C.; Wimmer, E.
Unravelling the Potential of Density Functional Theory through Integrated Computational Environments: Recent Applications of the Vienna Ab Initio Simulation Package in the MedeA^{®} Software. *Computation* **2018**, *6*, 63.
https://doi.org/10.3390/computation6040063

**AMA Style**

Eyert V, Christensen M, Wolf W, Reith D, Mavromaras A, Freeman C, Wimmer E.
Unravelling the Potential of Density Functional Theory through Integrated Computational Environments: Recent Applications of the Vienna Ab Initio Simulation Package in the MedeA^{®} Software. *Computation*. 2018; 6(4):63.
https://doi.org/10.3390/computation6040063

**Chicago/Turabian Style**

Eyert, Volker, Mikael Christensen, Walter Wolf, David Reith, Alexander Mavromaras, Clive Freeman, and Erich Wimmer.
2018. "Unravelling the Potential of Density Functional Theory through Integrated Computational Environments: Recent Applications of the Vienna Ab Initio Simulation Package in the MedeA^{®} Software" *Computation* 6, no. 4: 63.
https://doi.org/10.3390/computation6040063