Ab Initio and Molecular Dynamics-Based Pair ... - ACS Publications

Article pubs.acs.org/JPCC

Ab Initio and Molecular Dynamics-Based Pair Potentials for Lanthanum Hexaboride Kevin M. Schmidt,† Olivia A. Graeve,‡ and Victor R. Vasquez*,† †

Chemical and Materials Engineering Department, University of Nevada−Reno, Reno, Nevada 89557, United States Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California 92093, United States

‡

ABSTRACT: Lanthanum hexaboride (LaB6) is a well-known refractory ceramic with unique mechanical and electrochemical behavior, leading to a diverse array of possible attractive applications. In this work, we present and discuss the development of interatomic potentials for LaB6 using a combination of density functional theory with molecular dynamics simulations. Density functional theory is employed to acquire energetic and dynamic data of the atoms in various configurations and environments. Lattice inversion techniques are then combined with other optimization procedures to yield potentials which can accurately capture the lattice dynamics−mean-square displacements and equilibrium energetics of the system as a function of temperature.

■

INTRODUCTION Metal hexaborides (MB6) are a class of boron-rich solids characterized by high values of hardness and melting points, low density and thermal expansion coefficients, low work functions, chemical stability, and unique electrochemical behavior.1−10 Current usage of these materials includes areas such as field-electron emitters, electrical coatings for resistors, transition metal catalysts, high-energy optical systems, and sensors for high-resolution detectors.11−13 Utilizing their unique properties may open the door to exciting new applications, including gas storage and separation technologies, near-infrared sensors,14 solar energy harvesting,15 and hightemperature sensors. However, an understanding of the largescale effects associated with the materials is required before these can be realized. From a fundamental modeling perspective, most efforts have been focused on predicting equilibrium structure and calculation of electronic properties mainly using density functional theory (DFT) methods.4,8,16−18 If one wants to understand the behavior of this type of material at larger time and length scales using molecular dynamics (MD), for example, it is necessary to have interatomic potentials coupled with an appropriate modeling framework to gain insight into aspects related to mass transport, vacancy effects, and dopants, among others. The potential new applications aforementioned require this type of analysis in combination with experiments. Although there is much interest in studying these materials, there are currently no known interaction potentials or MD simulations in published literature. In this work, we present and discuss a modeling framework to develop interatomic potentials using a combination of DFT and MD methods. The proposed model uses relative fixed units for boron atoms in the octahedral units in LaB6, and the inter boron−boron, lanthanum−boron, and © 2015 American Chemical Society

lanthanum−lanthanum interactions are described with the potentials developed in this work. The results show very good predictions of the cohesive lattice energies as well as a very good description of the mean-square displacements (MSD) in the system. The methodology can be easily extended to any other MB6 material, in general. Hexaborides have a cubic crystalline structure with octahedral space group Pm3m symmetry. 3 The crystal structures are fully characterized by the lattice constant (a) and structural parameter (ω) (see Figure 1), with the structural parameter defined by ω=

B − B{interOh} 2·a

(1)

A metal ion occupies the 1a (0, 0, 0) Wyckoff site of the unit cell while the six boron atoms form an octahedral unit located at the 6f (0.5, 0.5, ω) Wyckoff sites4 with the octahedra bonded together at their apexes.6 The cesium chloride-like structure is relatively rigid and constant with respect to metal substitution, owing to the stability of the covalently linked boride octahedra.7,19 Electron donation from the metal stabilizes the electron-deficient boron framework,20 ideally requiring two electrons to fill the valence orbitals of the octahedra.21 Barium, calcium, strontium, and most lanthanoids have been found to form hexaboride structures, producing systems which exhibit various magnetic and super-, semi-, and electrically conductive properties dependent on the cation types.11,22 Divalent metal cations typically form semiconductors, and electrical conductors result Received: February 27, 2015 Revised: May 22, 2015 Published: May 27, 2015 14288

DOI: 10.1021/acs.jpcc.5b01962 J. Phys. Chem. C 2015, 119, 14288−14296

Article

The Journal of Physical Chemistry C

8 × 8 × 8 Monkhorst−Pack grid for k-points with energy and electron density cutoffs of 80 and 960 Ry, respectively. Thermodynamic properties derived from the QHA utilize an 8 × 8 × 8 q-point mesh. All parameters are checked for convergence before full-scale calculations are carried out. Equilibrium lattice constant values for bulk lanthanum and LaB 6 crystals have been calculated with the ultrasoft pseudopotentials and tested against published experimental data3,34 to validate their usage (shown in Table 1). The LaB6 Table 1. Experimental and Calculated Lattice Constants for Bulk Lanthanum and LaB6

Figure 1. General structure of metal hexaborides with central cation (yellow) and boron atoms (purple). ① = B−BinterOh bond, ② = B− BintraOh bond, ③ = lattice constant (a).

crystal

LPa

calc (Å)

exptl (Å)

error

α-La α-La β-La γ-La LaB6

a c a a a

3.805 12.271 5.302 4.212 4.154

3.774b 12.171b 5.303b 4.26b 4.1563c

0.82% 0.82% 0.02% 1.13% 0.06%

LP = lattice parameter; error = |1 − calc/exp|. bFrom ref 34. cFrom ref 3.

a

when the metal has additional electrons upon saturation of the boron framework.23−25 For rare-earth metals, small deviations to the lattice constant occur with changes in the 4f electron number of the metal (known as lanthanide contraction), generally found within the interoctahedral bonds connecting the clusters and not within the octahedra themselves.5 A sample of 12 different hexaborides showed that experimental intraoctahedral and interoctahedral bonds have standard deviations of 0.0159 and 0.0331 Å, respectively.3,7,24,26−29 LaB6 is a well-known refractory compound characterized by high electrical conductivity,30 superconductivity (Tc = 0.45 K),4 and a low work function31,32 with widespread usage as a highbrightness emitter in scanning and transmission electron microscopes. This allows a larger data set of experimental results to be incorporated for validation. Further work will apply this method to develop potentials describing other hexaborides systems to understand their behavior for possible future applications.

lattice constant and structural parameter have been optimized to aeq ± 0.005 Å and ωeq ± 0.0001, and the bulk modulus obtained from a Birch−Murnaghan35 fit (Bo = 171 GPa) shows good agreement with published data.16,36 In addition, MSD’s from the QHA are congruent with values obtained by Booth et al.37 Lattice Inversion. The total energy for a molecular or atomic system can be approximated by a sum of quantities representing zeroth and higher-order interactions based upon particle positions qi ,tot =

∑ ϕ1(q i) + ∑ ∑ ϕ2(q i, q j) + ··· i

i

(2)

j>i

Assuming that pairwise interactions are sufficient to capture the energetics, the complexity of the problem is significantly reduced, as all terms involving three or more particles are neglected. This allows one to write the lattice energy for a system interacting through pair potentials as 1 ,tot(a) = ∑ ϕk ° + ∑ ∑ ϕk (di(k)(a)) 2 k i (3) k

■

COMPUTATIONAL DETAILS OVERVIEW The interaction potentials are optimized through a series of stages utilizing various computational techniques. The system and its sublattices are first modeled with DFT to determine the equilibrium structural parameters and energetic data. A lattice inversion technique is then applied to extract initial pairwise potentials which are further modified to reflect the correct static equilibrium data. To capture the correct motion of the atoms, MD simulations are utilized to produce a system of interactions which can correctly reproduce the lattice dynamics at varying temperatures for equilibrium structures. The latter are also obtained from DFT using the quasi-harmonic approximation (QHA) and provide the reference data for optimization. Density Functional Theory. Electronic structure calculations are performed with the Quantum ESPRESSO integrated suite of open-source computer codes,33 using plane-wave basis sets and pseudopotentials to self-consistently solve the Kohn− Sham equations. Ultrasoft pseudopotentials are chosen for all atoms in this work. The generalized-gradient approximation with Perdew−Burke−Ernzerhof exchange-correlation functionals is used to handle many-body interactions, and a width of 0.02 Ry is given for Marzari−Vanderbilt smearing functions. Self-consistent field calculations over the Brillouin zone use an

in which the isolated atomic energies for each sublattice “k” are represented by ϕk° and the summations for interactions run over all observed atom−atom distances d(k) i (a) for a specific sublattice and lattice constant a. This relationship can also be cast in terms of the degeneracy in particle displacements, similar in nature to a discrete radial distribution. Thus, one can produce an infinite series for each sublattice dependent only on the lattice constant when the lattice geometry and interaction potential are fixed. ,k(a) = ϕk ° +

1 2

∞

∑ rk(n)·ϕ(bk(n)a) n=1

(4)

The two terms rk(n) and bk(n) are solely defined by the crystal structure and refer to the degeneracy and unit displacements from a reference particle for each type of interaction, respectively. Utilizing a technique proposed by Carlsson, Gelatt, and Ehrenreich,38 it is possible to invert a system’s cohesive energy to produce interaction potentials, incorporating knowledge about the lattice geometry to perform this 14289


Article

The Journal of Physical Chemistry C deconvolution. Further work by Chen39 has shown that the successive Gaussian elimination from a multiplicative semigroup in the Chen−Mö bius inversion method has the advantage of fast convergence along with simplicity.40 Optimization Algorithms. The goal of this work is to produce pairwise potentials for use in large-scale molecular dynamics calculations; therefore, the potentials obtained should be simple enough for fast computation yet still yield the correct properties for the system in varying environments. Although there is a multitude of software packages available with this capability, they are generally limited to a small subset of potential forms. Typically, pairwise interactions are described by Lennard-Jones, Morse, Buckingham, or Born−Huggins− Mayer potentials within solids. As potentials generated from lattice inversion are not guaranteed to accurately fit these specific curves, a function which represents the inversion well will not necessary reproduce the cohesive energy. Even small deviations from the inverted potential give rise to large errors in lattice characteristics. Two methods are proposed below to address this discrepancy.

lattice energy. The 15 sets giving the lowest error are then allowed to randomly change by a maximum of ±50% for hundreds of iterations, keeping the modified parameter only if the error in lattice energies has been reducedsimilar to the Metropolis Monte Carlo algorithm. Once a pseudostationary state is found, each parameter is then altered by ±0.05%, producing 6 new potentials to test for each of the 15 sets. Of the six potentials, the change that produces the lowest in error is then chosen, and this “annealing” process is continued until a local optimum is found. Typically, two-thirds of the 15 trial functions will relax to similar stationary states, and that which gives the best fit is chosen. Molecular Dynamics Relaxation. The last step of the optimization process consists of further refinement of potential parameters obtained using DFT data by performing MD simulations to reproduce the dynamics of boron and lanthanum atoms given by the QHA results. MD runs using DL_POLY41 are performed on an 8 × 8 × 8 lattice of unit cells (a = 4.154 Å) for a total of 9 ps after an initial 1 ps for equilibration during the fitting process. The microcanonical ensemble is integrated over 1 fs intervals with a Velocity−Verlet algorithm, having temperature controlled by a Nosé−Hoover thermostat (τT = 0.5). Radial cutoffs of 12 Å are set in accordance with the development of pair potentials, and temperature is variable. Electrostatics are calculated using the smooth-particle mesh Ewald (SPME) method with a precision of 10−8. DL_POLY allows one to perform this calculation on a periodic atomic configuration, and it is the standard Ewald method used for electrostatic calculations within this software. For the molecular system being described, which contains boron atoms connected intraoctahedrally through the use of rigid bonds, the full electrostatic calculation is amended to exclude these mutual Coulombic interactions specifically, and they form the group of excluded atoms. Total electrostatics are calculated via the formula41

■

OPTIMIZATION APPROACH Method I. We use Morse potentials to represent the interactions within this system, allowing the particles the freedom to dissociate while still producing a realistic set of vibrational modes. This function has the form ϕM(rij) = E0(e−2k 0(rij − req) − 2e−k 0(rij − req))

(5)

where E0 represents the dissociation energy, k0 is related to the stiffness of the potential, and req is the interatomic separation at which the force acting between the particles reaches zero. As a first approximation, the equilibrium distance and dissociation energy are constrained to the values given by the Möbius inversion. The value of k0 is then fit to the nearest thousandth through minimization of an objective function defined by N

Eobj =

∑ i=0

⎛ ϕ (ri) ⎞ exp⎜ − I ⎟(ϕI(ri) − ϕM(ri, E0 , k 0))2 ⎝ E* ⎠

,elec =

(6)

where k0 is the parameter being fit, E* dampens the values within the exponential weighting function, ϕI(·) is the inverted potential, and ϕM(·) refers to the trial Morse potential. The form of the weighting function allows a greater relevance to be given to regions near the equilibrium distance and extending outward toward infinite separation. Regions of high repulsion have weights that quickly taper off to zero, as these are described by energies much larger than a typical simulation would encounter. Method II. It is found that potentials obtained using Method I are very sensitive to the stability and equilibrium interatomic separation within lattices. This is due to the inability of general pairwise potential forms to properly describe the lattice inversion. To handle this effect, an additional optimization scheme is introduced to produce a potential which would relax to the correct cohesive energies and equilibrium configuration of the crystal. We use Method I to obtain initial values for the potentials. Lattice energies are recalculated for the entire system with a radial cutoff value of 12 Å for potentials. A group of 1200 parameter sets are generated by randomly changing any or all of the three parameters by up to ±120%. An objective function similar to that given for Method I is incorporated, which allows for a weighted optimization of

1 2Voε0ε +

∞

∑

k2

k≠0

1 4πε0ε

N*

∑ n La−La > La−B. In addition, the optimization process achieved larger dissocation energies for boron than could be extracted from bulk boron DFT calculations. Baseline potentials were necessary to parametrize the boron homatomic and heteroatomic interactions, as no inversion process has yet been determined. However, the lanthanum interaction is easily inverted and can be parametrized using a more suitable function. The high repulsion and “stiffness” of the recalculated LaB6 lattice energy is likely due to the form of the interaction potentials. A more appropriate functional form may alleviate this issue and produce higher quality predictions to energetics and dynamics.

Figure 8. Lanthanum MSD values generated through DFT and MD compared with experimental results from ref 37.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: +775 784 6060. Fax: +775 784 4764. E-mail: victor. [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We acknowledge funding support from NSF Award No. 1246792 (“SNM: Scalable Manufacturing of Unique Hexaboride Nanomaterials for Advanced Energy Generation and Gas Storage Applications”).

Figure 9. Cohesive energies for LaB6 from DFT and lattice summations with pair potentials from Table 2 and the SPME method.

■

Å. Calculated cohesive energies from DFT and the lattice summation resulted in −20.130 eV and −20.177 eV, respectively, a remarkably close value considering how many nonequilibrium structures and dynamic optimization steps were incorporated. The repulsive part of the potential matches the

REFERENCES

(1) Lemis-Petropoulos, P.; Kapaklis, V.; Peikrishvili, A.; Politis, C. Characterization of B4C and LaB6 by Ultrasonics and X-rays Diffraction. Int. J. Mod. Phys. B 2003, 17, 2781−2788. (2) Mukherji, U. Engineering Physics; Alpha Science Int. Ltd.: Oxford, 2006. 14294


Article

The Journal of Physical Chemistry C (3) Loboda, P.; Kysla, H.; Dub, S.; Karasevs’ka, O. Mechanical Properties of the Monocrystals of Lanthanum Hexaboride. Mater. Sci. 2009, 45, 108−113. (4) Bai, L.; Ma, N.; Liu, F. Structure and Chemical Bond Characteristics of LaB6. Phys. B (Amsterdam, Neth.) 1009, 404, 4086−4089. (5) Kimura, S.; Nanba, T.; Tomikawa, M.; Kunii, S.; Kasuya, T. Electronic Structure of Rare-Earth Hexaborides. Phys. Rev. B.: Condens. Matter Mater. Phys. 1992, 46, 12196−12204. (6) Lafferty, J. Boride Cathodes. J. Appl. Phys. 1951, 22, 299−309. (7) Chen, C.-H.; Aizawa, T.; Iyi, N.; Sato, A.; Otani, S. Structural Refinement and Thermal Expansion of Hexaborides. J. Alloys Compd. 2004, 366, L6−L8. (8) Guo-Liang, X.; Jing-Dong, C.; Yao-Zheng, X.; Xue-Feng, L.; YuFang, L.; Xian-Zhou, Z. First-Principles Calculations of Elastic and Thermal Properties of Lanthanum Hexaboride. Chin. Phys. Lett. 2009, 26, 156201. (9) Craciun, V.; Craciun, D. Pulsed Laser Deposition of Crystalline LaB6 Thin Films. Appl. Surf. Sci. 2005, 247, 384−389. (10) Ordan’yan, S.; Paderno, Y.; Nikolaeva, E.; Khoroshilova, I. Interaction in the LaB6-CrB2 System. Powder Metall. Met. Ceram. 1984, 23, 387−389. (11) Ji, X.; Zhang, Q.; Xu, J.; Zhao, Y. Rare-Earth Hexaborides Nanostructures: Recent Advances in Materials, Characterization, and Investigations of Physical Properties. Prog. Solid State Chem. 2011, 39, 51−69. (12) Uribe, F.; Garzon, F.; Brosha, E.; Johnston, C.; Conradson, S.; Wilson, M. Spontaneous Deposition of Noble Metal Films onto Hexaboride Surfaces. J. Electrochem. Soc. 2007, 154, D623−D628. (13) Kher, S.; Spencer, J. Chemical Vapor Deposition of Metal Borides. VII. The Relatively Low Temperature Formation of Crystalline Lanthanum Hexaboride Thin Films from Boron Hydride Cluster Compounds by Chemical Vapor Deposition. J. Phys. Chem. Solids 1998, 59, 1343−1351. (14) Jiang, F.; Leong, Y.-K.; Saunders, M.; Martyniuk, M.; Faraone, L.; Keating, A.; Dell, J. M. Uniform Dispersion of Lanthanum Hexaboride Nanoparticles in a Silica Thin Film: Synthesis and Optical Properties. ACS Appl. Mater. Interfaces 2012, 4, 5833−5838. (15) Hasan, M. M.; Sugo, H.; Kisi, E. H. Low Temperature Synthesis of Rare-Earth Hexaborides for Solar Energy Conversion. MATEC Web Conf. 2014, 13, 06005. (16) Baranovskiy, A.; Grechnev, G.; Fil, V.; Ignatova, T.; Logosha, A.; Panfilov, A.; Svechkarev, I.; Shitsevalova, N.; Filippov, V.; Eriksson, O. Electronic Structure, Bulk and Magnetic Properties of MB6 and MB12 Borides. J. Alloys Compd. 2007, 442, 228−230. (17) Monnier, R.; Delley, B. Properties of LaB6 Elucidated by Density Functional Theory. Phys. Rev. B.: Condens. Matter Mater. Phys. 2004, 70, 193403. (18) Ammar, A.; Ménétrier, M.; Villesuzanne, A.; Matar, S.; Chevalier, B.; Etourneau, J. Investigation of the Electronic and Structural Properties of Potassium Hexaboride, KB6, by Transport, Magnetic Susceptibility, EPR, and NMR Measurements, TemperatureDependent Crystal Structure Determination, and Electronic Band Structure Calculations. Inorg. Chem. 2004, 43, 4974−4987. (19) Oshima, C.; Umeuchi, M.; Nagao, T. Surface Atomic Structure and Surface Force Constants: Cu(100)2√2 × √2-R45°-O and LaB6(100). Surf. Sci. 1993, 298, 450−455. (20) Trenary, M. Surface Science Studies of Metal Hexaborides. Sci. Technol. Adv. Mater. 2012, 13, 023002. (21) Börrnert, C.; Grin, Y.; Wagner, F. Position-Space Bonding Indicators for Hexaborides of Alkali, Alkaline-Earth, and Rare-Earth Metals in Comparison to Molecular Crystal K2[B6H6]. Z. Anorg. Allg. Chem. 2013, 639, 2013−2024. (22) Lundström, T. Structure, Defects and Properties of Some Refractory Borides. Pure Appl. Chem. 1985, 57, 1383−1390. (23) Getty, S. Electron Transport Studies of the Ferromagnetic Semiconductor Calcium Hexaboride. Ph.D. Thesis, University of Florida, 2001.

(24) Bat’ko, I.; Bat’ková, M.; Flachbart, K.; Filippov, V.; Paderno, Y.; Shicevalova, N.; Wagner, T. Electrical Resistivity and Superconductivity of LaB6 and LuB12. J. Alloys Compd. 1995, 217, L1−L3. (25) Mandrus, D.; Sales, B.; Jin, R. Localized Vibrational Mode Analysis of the Resistivity and Specific Heat of LaB6. Phys. Rev. B.: Condens. Matter Mater. Phys. 2001, 64, 012302. (26) Tanaka, K.; O̅ nuki, Y. Observation of 4f Electron Transfer from Ce to B6 in the Kondo Crystal CeB6 and Its Mechanism by MultiTemperature X-ray Diffraction. Acta Crystallogr., Sect. B: Struct. Sci. 2002, 58, 423−436. (27) Blomberg, M.; Merisalo, M.; Korsukova, M.; Gurin, V. SingleCrystal X-ray Diffraction Study of NdB6, EuB6, and YbB6. J. Alloys Compd. 1995, 217, 123−127. (28) Funahashi, S.; Tanaka, K.; Iga, F. X-ray Atomic Orbital Analysis of 4f and 5d Electron Configuration of SmB6 at 100, 165, 230, and 298 K. Acta Crystallogr., Sect. B: Struct. Sci. 2010, 66, 292−306. (29) Konrad, T.; Jeitschko, W.; Danebrok, M. E.; Evers, C. Preparation, Properties, and Crystal Structures of the Thorium Chromium Borides ThCrB4 and ThCr2B6; Structure Refinements of CeCr6B6, ThB4, and ThB6. J. Alloys Compd. 1996, 234, 56−61. (30) Otani, S.; Ishizawa, Y. Single Crystals of Carbides and Borides as Electron Emitters. Prog. Cryst. Growth Charact. 1991, 23, 153−177. (31) Aono, M.; Oshima, C.; Tanaka, T.; Bannai, E.; Kawai, S. Structure of the LaB6(001) Surface Studied by Angle-Resolved XPS and LEED. J. Appl. Phys. 1978, 49, 2761−2764. (32) Nishitani, R.; Aono, M.; Tanaka, T.; Oshima, C.; Kawai, S.; Iwasaki, H.; Nakamura, S. Surface Structures and Work Functions of the LaB6 (100), (110), and (111) Clean Surfaces. Surf. Sci. 1980, 93, 535−549. (33) Giannozzini, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.; Cococcioni, M.; Dabo, I.; et al. Quantum ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials. J. Phys.: Condens. Matter 2009, 395502, 21. (34) Gschneider, K. In CRC Handbook of Chemistry and Physics, 87th ed.; Lide, D., Ed.; CRC Press: Boca Raton, FL, 2005; Chapter: Physical Properties of the Rare Earth Metals. (35) Birch, F. Finite Elastic Strain of Cubic Crystals. Phys. Rev. 1947, 71, 809−824. (36) Tanaka, T.; Yoshimoto, J.; Ishii, M.; Bannai, E.; Kawai, S. Elastic Constants of LaB6 at Room Temperature. Solid State Commun. 1977, 22, 203−205. (37) Booth, C.; Sarrao, J.; Hundley, M.; Cornelius, A.; Kwei, G.; Bianchi, A.; Fisk, Z.; Lawrence, J. Local and Average Crystal Structure and Displacements of La11B6 and EuB6 as a Function of Temperature. Phys. Rev. B.: Condens. Matter Mater. Phys. 2001, 63, 224302. (38) Carlsson, A.; Gelatt, C.; Ehrenreich, H. An ab Initio Pair Potential Applied to Metals. Philos. Mag. A 1980, 41, 241−250. (39) Chen, N.-X.; bao Ren, G. Carlsson−Gelatt−Ehrenreich Technique and the Möbius Inversion Theorem. Phys. Rev. B.: Condens. Matter Mater. Phys. 1992, 45, 8177−8180. (40) Chen, N.-X. Möbius Inversion in Physics; World Scientific: Hackensack, NJ, 2010. (41) Todorov, I., Smith, W. The DL_POLY User Manual; STFC Daresbury Laboratory: Cheshire, England, 2014. (42) Ying, L.; Chen, N.-X.; Yan-Mei, K. Virtual Lattice Technique and the Interatomic Potentials of Zinc-Blend-type Binary Compounds. Mod. Phys. Lett. B 2002, 16, 187−194. (43) Zhang, S.; Chen, N.-X. Ab Initio Interionic Potentials for NaCl by Multiple Lattice Inversion. Phys. Rev. B.: Condens. Matter Mater. Phys. 2002, 66, 064106. (44) Brown, D. The Chemical Bond in Inorganic Chemistry: The Bond Valence Model; Oxford University Press: Oxford, 2002. (45) Greenwood, N.; Earshaw, A. Chemistry of the Elements; Pergamon Press: Oxford, 1985. (46) Gaydon, A. Dissociation Energies and Spectra of Diatomic Molecules; Chapman & Hall: London, 1953. 14295


Article

The Journal of Physical Chemistry C (47) Herzberg, G. Molecular Spectra and Molecular Structure. Vol 1: Spectra of Diatomic Molecules, 2nd ed.; D. Van Nostrand: New York, 1950. (48) Kittel, C. Introduction to Solid State Physics; John Wiley & Sons: New York, 2005. (49) Jensen, F. Introduction to Computational Chemistry; John Wiley & Sons: New York, 1999. (50) Nemkevich, A.; Burgi, H.; Spackman, M.; Corry, B. Molecular Dynamics Simulations of Structure and Dynamics of Organic Molecular Crystals. Phys. Chem. Chem. Phys. 2010, 12, 14916−14929. (51) Belonoshko, A.; Ramzan, M.; Mao, H.; Ahuja, R. Atomic Diffusion in Solid Molecular Hydrogen. Sci. Rep. 2013, 3, 2340. (52) Teredesai, P.; Muthu, D.; Chandrabhas, N.; Meenakshi, S.; Vijayakumar, V.; Modak, P.; Rao, R.; Godwal, B.; Sikka, S.; Sood, A. High Pressure Phase Transition in Metallic LaB6: Raman and X-ray Diffraction Studies. Solid State Commun. 2004, 129, 791−796. (53) Giustino, F. Materials Modelling Using Density Functional Theory: Properties & Predictions; Oxford University Press: Oxford, 2014. (54) Zherlitsyn, S.; Wolf, B.; Lüthi, B.; Lang, M.; Hinze, P.; Uhrig, E.; Assmus, W.; Ott, H.; Young, D.; Fisk, Z. Elastic Properties of Ferromagnetic EuB6. Eur. Phys. J. B 2001, 22, 327−333.

14296


Ab Initio and Molecular Dynamics-Based Pair ... - ACS Publications

Recommend Documents