# Thermo-Elasticity of Materials from Quasi-Harmonic Calculations

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## Abstract

**:**

## 1. Introduction

_{2}SiO

_{4}: an end-member of the olivine solid solution series that is one of the most abundant silicates in the upper mantle of the Earth. The single-crystal thermo-elasticity of forsterite has been accurately determined experimentally at several temperatures from 300 K to 1700 K so that it represents an ideal system to validate and discuss our methodology [32].

## 2. Theoretical Aspects

#### 2.1. The Athermal Elastic Tensor

#### 2.1.1. Calculation from Analytical Forces

#### 2.1.2. Calculation from the Energy

#### 2.2. The Elastic Tensor at a Finite Temperature

- The determination of the equilibrium structure of the system at temperature T;
- The calculation of the second free-energy derivatives with respect to strain.

#### 2.2.1. The Equilibrium Structure at a Finite Temperature

_{2}O

_{3}[24], forsterite $\alpha $-Mg

_{2}SiO

_{4}[25], calcium stannate CaSnO

_{3}[45], and the molecular crystals of urea, purine and carbamazepine [43,46,47].

#### 2.2.2. Free Energy Derivatives with Respect to Strain

#### 2.2.3. Adiabatic versus Isothermal Elastic Moduli

#### 2.2.4. The Quasi-Static Approximation to Thermo-Elastic Moduli

## 3. The Implemented Algorithm

- A full structural relaxation of the system is performed (both atomic positions and lattice parameters are optimized). The static equilibrium structure, with volume ${V}_{0}$, is obtained.
- A space group symmetry-preserving QHA calculation is performed, which provides the thermal expansion of the system. A fully-automated algorithm is implemented in the Crystal program to perform this task [22,23], where four different volumes are explored (compressed and expanded with respect to ${V}_{0}$). For each volume, a volume-constrained, lattice symmetry-preserving structural relaxation is performed and phonon frequencies computed. By minimizing $F(V;T)$ at several temperatures, the $V\left(T\right)$ relation is determined. As a result, lattice parameters as a function of temperature are obtained.
- A value of temperature T is selected. Starting from the values of the lattice parameters at this temperature obtained at the end of the previous step, a volume-constrained, lattice symmetry-preserving structural relaxation is performed to get the equilibrium structure (also in terms of atomic positions) at the desired temperature.
- A given strain shape $\underline{\eta}$ is chosen, which will provide a linear combination of elastic stiffness constants according to Equation (16).
- The second free energy derivatives with respect to the strain are computed as discussed in Section 2.2.2. A fully-automated algorithm has been implemented in the Crystal program for this task. The starting point is represented by the optimized structure obtained at the end of step 3 above (i.e., the equilibrium structure at temperature T). The structure is deformed, in terms of the strain shape $\underline{\eta}$, into four strained configurations (two with positive and two with negative strain amplitude $\delta $). At each strained configuration, atomic positions are relaxed and phonon frequencies computed. The computed quasi-harmonic free energy as a function of strain amplitude is fitted to a second-order polynomial and the corresponding second-derivative determined.

## 4. Computational Parameters

^{−2}units) of the most diffuse $sp$ shells are 0.32 and 0.13 (Si), 0.59 and 0.25 (O), and 0.68 and 0.22 (Mg); the exponents of the d shells are 0.6 (Si), 0.5 (O), and 0.5 (Mg). The same basis set has already been successfully utilized in a couple of recent studies of vibrational and spectroscopic properties of forsterite [25,54].

**k**-points in the irreducible portion of the Brillouin zone. A pruned grid with 1454 radial and 99 angular points is used to calculate the DFT exchange-correlation contribution through numerical integration of the electron density over the unit cell volume. The self-consistent-field (SCF) convergence on energy was set to a value of 10

^{−10}hartree for all geometry optimizations and phonon frequency calculations. In the calculation of phonon frequencies, the Hessian matrix is computed numerically from finite differences of analytical forces computed at a set of displaced nuclear configurations: a step of 0.003 Å is used.

**k**-points (corresponding to a supercell containing 756 atoms). All quasi-harmonic properties are computed by working in terms of the primitive cell of the system (containing 4 formula units or equivalently 28 atoms), which already ensures convergence.

## 5. Results and Discussion

_{2}SiO

_{4}, that is characterized by nine independent elastic constants: ${C}_{11}$, ${C}_{22}$, ${C}_{33}$, ${C}_{44}$, ${C}_{55}$, ${C}_{66}$, ${C}_{12}$, ${C}_{13}$, ${C}_{23}$. The first step of our algorithm, as sketched in Section 3, consists of a full structural relaxation of the system (i.e., both atomic positions and lattice parameters are optimized by minimizing the static internal energy E of the system). The second step of the algorithm is a lattice symmetry-preserving quasi-harmonic calculation, where harmonic phonon frequencies are evaluated at four volumes (one compressed, one equilibrium and two expanded) according to the fully-automated scheme previously suggested and implemented in the Crystal program by some of the authors [22,23]. From the calculation of phonon harmonic frequencies at different volumes, the Helmholtz free energy of the system as a function of temperature and volume can be evaluated as in Equation (12). By minimizing this function with respect to volume at each temperature, the $V\left(T\right)$ relation is obtained that provides the thermal expansion of the system.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kuehmann, C.J.; Olson, G.B. Computational materials design and engineering. Mater. Sci. Technol.
**2009**, 25, 472–478. [Google Scholar] [CrossRef] - Hafner, J.; Wolverton, C.; Ceder, G. Toward Computational Materials Design: The Impact of Density Functional Theory on Materials Research. MRS Bull.
**2006**, 31, 659–668. [Google Scholar] [CrossRef][Green Version] - Curtarolo, S.; Hart, G.; Nardelli, M.; Mingo, N.; Sanvito, S.; Levy, O. The high-throughput highway to computational materials design. Nat. Mater.
**2013**, 12, 191–201. [Google Scholar] [CrossRef] [PubMed] - Varini, N.; Ceresoli, D.; Martin-Samos, L.; Girotto, I.; Cavazzoni, C. Enhancement of DFT-calculations at petascale: Nuclear Magnetic Resonance, Hybrid Density Functional Theory and Car-Parrinello calculations. Comput. Phys. Commun.
**2013**, 184, 1827–1833. [Google Scholar] [CrossRef] - Corsetti, F. Performance Analysis of Electronic Structure Codes on HPC Systems: A Case Study of SIESTA. PLoS ONE
**2014**, 9, e95390. [Google Scholar] [CrossRef] [PubMed] - Hutter, J.; Iannuzzi, M.; Schiffmann, F.; VandeVondele, J. CP2K: Atomistic simulations of condensed matter systems. WIREs Comput. Mol. Sci.
**2014**, 4, 15–25. [Google Scholar] [CrossRef] - Maniopoulou, A.; Davidson, E.R.; Grau-Crespo, R.; Walsh, A.; Bush, I.J.; Catlow, C.R.A.; Woodley, S.M. Introducing k-point parallelism into VASP. Comput. Phys. Commun.
**2012**, 183, 1696–1701. [Google Scholar] [CrossRef] - Kendall, R.A.; Aprá, E.; Bernholdt, D.E.; Bylaska, E.J.; Dupuis, M.; Fann, G.I.; Harrison, R.J.; Ju, J.; Nichols, J.A.; Nieplocha, J.; et al. High performance computational chemistry: An overview of NWChem a distributed parallel application. Comput. Phys. Commun.
**2000**, 128, 260–283. [Google Scholar] [CrossRef] - Orlando, R.; Delle Piane, M.; Bush, I.J.; Ugliengo, P.; Ferrabone, M.; Dovesi, R. A new massively parallel version of CRYSTAL for large systems on high performance computing architectures. J. Comput. Chem.
**2012**, 33, 2276–2284. [Google Scholar] [CrossRef][Green Version] - Bush, I.J.; Tomic, S.; Searle, B.G.; Mallia, G.; Bailey, C.L.; Montanari, B.; Bernasconi, L.; Carr, J.M.; Harrison, N.M. Parallel implementation of the ab initio CRYSTAL program: Electronic structure calculations for periodic systems. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2011**, 467, 2112. [Google Scholar] [CrossRef] - Wen, S.; Nanda, K.; Huang, Y.; Beran, G.J.O. Practical quantum mechanics-based fragment methods for predicting molecular crystal properties. Phys. Chem. Chem. Phys.
**2012**, 14, 7578–7590. [Google Scholar] [CrossRef] [PubMed] - Moellmann, J.; Grimme, S. DFT-D3 Study of Some Molecular Crystals. J. Phys. Chem. C
**2014**, 118, 7615–7621. [Google Scholar] [CrossRef] - Bucko, T.; Hafner, J.; Lebegue, S.; Angyan, J.G. Improved Description of the Structure of Molecular and Layered Crystals: Ab Initio DFT Calculations with van der Waals Corrections. J. Phys. Chem. A
**2010**, 114, 11814–11824. [Google Scholar] [CrossRef] [PubMed] - Tkatchenko, A.; DiStasio, R.A.; Car, R.; Scheffler, M. Accurate and Efficient Method for Many-Body van der Waals Interactions. Phys. Rev. Lett.
**2012**, 108, 236402. [Google Scholar] [CrossRef] [PubMed] - Civalleri, B.; Zicovich-Wilson, C.; Valenzano, L.; Ugliengo, P. B3LYP augmented with an empirical dispersion term (B3LYP-D*) as applied to molecular crystals. CrystEngComm
**2008**, 10, 405. [Google Scholar] [CrossRef] - Neumann, M.A.; Perrin, M.A. Energy Ranking of Molecular Crystals Using Density Functional Theory Calculations and an Empirical van der Waals Correction. J. Phys. Chem. B
**2005**, 109, 15531–15541. [Google Scholar] [CrossRef] [PubMed] - Dovesi, R.; Orlando, R.; Erba, A.; Zicovich-Wilson, C.M.; Civalleri, B.; Casassa, S.; Maschio, L.; Ferrabone, M.; De La Pierre, M.; D’Arco, P.H.; et al. CRYSTAL14: A Program for the Ab initio Investigation of Crystalline Solids. Int. J. Quantum Chem.
**2014**, 114, 1287–1317. [Google Scholar] [CrossRef] - Dovesi, R.; Erba, A.; Orlando, R.; Zicovich-Wilson, C.M.; Civalleri, B.; Maschio, L.; Rérat, M.; Casassa, S.; Baima, J.; Salustro, S.; et al. Quantum-Mechanical Condensed Matter Simulations with CRYSTAL. WIREs Comput. Mol. Sci.
**2018**, 8, e1360. [Google Scholar] [CrossRef] - Erba, A.; Baima, J.; Bush, I.; Orlando, R.; Dovesi, R. Large Scale Condensed Matter DFT Simulations: Performance and Capabilities of the Crystal Code. J. Chem. Theory Comput.
**2017**, 13, 5019–5027. [Google Scholar] [CrossRef] - Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys.
**2010**, 132, 154104. [Google Scholar] [CrossRef] - Brandenburg, J.; Grimme, S. Dispersion Corrected Hartree-Fock and Density Functional Theory for Organic Crystal Structure Prediction. In Prediction and Calculation of Crystal Structures; Atahan-Evrenk, S., Aspuru-Guzik, A., Eds.; Topics in Current Chemistry; Springer: Berlin/Heidelberg, Germany, 2014; Volume 345, pp. 1–23. [Google Scholar]
- Erba, A. On combining temperature and pressure effects on structural properties of crystals with standard ab initio techniques. J. Chem. Phys.
**2014**, 141, 124115. [Google Scholar] [CrossRef] [PubMed] - Erba, A.; Shahrokhi, M.; Moradian, R.; Dovesi, R. On How Differently the Quasi-harmonic Approximation Works for Two Isostructural Crystals: Thermal Properties of MgO and CaO. J. Chem. Phys.
**2015**, 142, 044114. [Google Scholar] [CrossRef] [PubMed] - Erba, A.; Maul, J.; Demichelis, R.; Dovesi, R. Assessing Thermochemical Properties of Materials through Ab initio Quantum-mechanical Methods: The Case of α-Al
_{2}O_{3}. Phys. Chem. Chem. Phys.**2015**, 17, 11670–11677. [Google Scholar] [CrossRef] [PubMed] - Erba, A.; Maul, J.; De La Pierre, M.; Dovesi, R. Structural and Elastic Anisotropy of Crystals at High Pressure and Temperature from Quantum-mechanical Methods: The Case of Mg
_{2}SiO_{4}Forsterite. J. Chem. Phys.**2015**, 142, 204502. [Google Scholar] [CrossRef] [PubMed] - Allen, R.E.; De Wette, F.W. Calculation of Dynamical Surface Properties of Noble-Gas Crystals. I. The Quasiharmonic Approximation. Phys. Rev.
**1969**, 179, 873–886. [Google Scholar] [CrossRef] - Boyer, L.L. Calculation of Thermal Expansion, Compressiblity, an Melting in Alkali Halides: NaCl and KCl. Phys. Rev. Lett.
**1979**, 42, 584. [Google Scholar] [CrossRef] - Davies, G. Effective elastic moduli under hydrostatic stress. Quasi-harmonic theory. J. Phys. Chem. Solids
**1974**, 35, 1513–1520. [Google Scholar] [CrossRef] - Karki, B.B.; Wentzcovitch, R.M.; de Gironcoli, S.; Baroni, S. High-pressure lattice dynamics and thermoelasticity of MgO. Phys. Rev. B
**2000**, 61, 8793–8800. [Google Scholar] [CrossRef] - Wu, Z.; Wentzcovitch, R.M. Quasiharmonic thermal elasticity of crystals: An analytical approach. Phys. Rev. B
**2011**, 83, 184115. [Google Scholar] [CrossRef] - Karki, B.B.; Wentzcovitch, R.M.; de Gironcoli, S.; Baroni, S. First-Principles Determination of Elastic Anisotropy and Wave Velocities of MgO at Lower Mantle Conditions. Science
**1999**, 286, 1705–1707. [Google Scholar] [CrossRef] - Isaak, D.G.; Anderson, O.L.; Goto, T.; Suzuki, I. Elasticity of single-crystal forsterite measured to 1700 K. J. Geophys. Res. Solid Earth
**1989**, 94, 5895–5906. [Google Scholar] [CrossRef] - Doll, K. Analytical stress tensor and pressure calculations with the CRYSTAL code. Mol. Phys.
**2010**, 108, 223–227. [Google Scholar] [CrossRef][Green Version] - Nye, J.F. Physical Properties of Crystals; Oxford University Press: Oxford, UK, 1957. [Google Scholar]
- Doll, K.; Dovesi, R.; Orlando, R. Analytical Hartree-Fock gradients with respect to the cell parameter for systems periodic in three dimensions. Theor. Chem. Acc.
**2004**, 112, 394–402. [Google Scholar] [CrossRef] - Doll, K.; Dovesi, R.; Orlando, R. Analytical Hartree-Fock gradients with respect to the cell parameter: Systems periodic in one and two dimensions. Theor. Chem. Acc.
**2006**, 115, 354–360. [Google Scholar] [CrossRef] - Perger, W.F.; Criswell, J.; Civalleri, B.; Dovesi, R. Ab-initio calculation of elastic constants of crystalline systems with the CRYSTAL code. Comput. Phys. Commun.
**2009**, 180, 1753–1759. [Google Scholar] [CrossRef] - Erba, A.; Mahmoud, A.; Orlando, R.; Dovesi, R. Elastic Properties of Six Silicate Garnet End-members from Accurate Ab initio Simulations. Phys. Chem. Miner.
**2014**, 41, 151–160. [Google Scholar] [CrossRef] - Erba, A. The Internal-Strain Tensor of Crystals for Nuclear-relaxed Elastic and Piezoelectric Constants: On the Full Exploitation of its Symmetry Features. Phys. Chem. Chem. Phys.
**2016**, 18, 13984–13992. [Google Scholar] [CrossRef] - Erba, A.; Caglioti, D.; Zicovich-Wilson, C.M.; Dovesi, R. Nuclear-relaxed Elastic and Piezoelectric Constants of Materials: Computational Aspects of Two Quantum-mechanical Approaches. J. Comput. Chem.
**2017**, 38, 257–264. [Google Scholar] [CrossRef] - Ashcroft, N.W.; Mermin, N.D. Solid State Physics; Saunders College: Philadelphia, PA, USA, 1976. [Google Scholar]
- Baroni, S.; Giannozzi, P.; Isaev, E. Density-Functional Perturbation Theory for Quasi-Harmonic Calculations. Rev. Miner. Geochem.
**2010**, 71, 39–57. [Google Scholar] [CrossRef] - Erba, A.; Maul, J.; Civalleri, B. Thermal Properties of Molecular Crystals through Dispersion-corrected Quasi-harmonic Ab initio Calculations: The Case of Urea. Chem. Commun.
**2016**, 52, 1820–1823. [Google Scholar] [CrossRef] - Erba, A.; Maul, J.; Itou, M.; Dovesi, R.; Sakurai, Y. Anharmonic Thermal Oscillations of the Electron Momentum Distribution in Lithium Fluoride. Phys. Rev. Lett.
**2015**, 115, 117402. [Google Scholar] [CrossRef] [PubMed] - Maul, J.; Santos, I.M.G.; Sambrano, J.R.; Erba, A. Thermal properties of the orthorhombic CaSnO
_{3}perovskite under pressure from ab initio quasi-harmonic calculations. Theor. Chem. Acc.**2016**, 135, 1–9. [Google Scholar] [CrossRef] - Ruggiero, M.T.; Zeitler, J.; Erba, A. Intermolecular Anharmonicity in Molecular Crystals: Interplay between Experimental Low-Frequency Dynamics and Quantum Quasi-Harmonic Simulations of Solid Purine. Chem. Commun.
**2017**, 53, 3781–3784. [Google Scholar] [CrossRef] [PubMed] - Brandenburg, J.G.; Potticary, J.; Sparkes, H.A.; Price, S.L.; Hall, S.R. Thermal Expansion of Carbamazepine: Systematic Crystallographic Measurements Challenge Quantum Chemical Calculations. J. Phys. Chem. Lett.
**2017**, 8, 4319–4324. [Google Scholar] [CrossRef] [PubMed] - Belousov, R.I.; Filatov, S.K. Algorithm for calculating the thermal expansion tensor and constructing the thermal expansion diagram for crystals. Glass Phys. Chem.
**2007**, 33, 271–275. [Google Scholar] [CrossRef] - Paufler, P.; Weber, T. On the determination of linear expansion coefficients of triclinic crystals using X-ray diffraction. Eur. J. Mineral.
**1999**, 11, 721. [Google Scholar] [CrossRef] - Fortes, A.D.; Wood, I.G.; Knight, K.S. The crystal structure and thermal expansion tensor of MgSO
_{4}–11D2O(meridianiite) determined by neutron powder diffraction. Phys. Chem. Miner.**2008**, 35, 207–221. [Google Scholar] [CrossRef] - Wang, Y.; Wang, J.J.; Zhang, H.; Manga, V.R.; Shang, S.L.; Chen, L.Q.; Liu, Z.K. A first-principles approach to finite temperature elastic constants. J. Phys. Condens. Matter
**2010**, 22, 225404. [Google Scholar] [CrossRef] [PubMed] - Kádas, K.; Vitos, L.; Ahuja, R.; Johansson, B.; Kollár, J. Temperature-dependent elastic properties of α-beryllium from first principles. Phys. Rev. B
**2007**, 76, 235109. [Google Scholar] [CrossRef] - Shang, S.L.; Zhang, H.; Wang, Y.; Liu, Z.K. Temperature-dependent elastic stiffness constants of α- and θ-Al
_{2}O_{3}from first-principles calculations. J. Phys. Condens. Matter**2010**, 22, 375403. [Google Scholar] [CrossRef] - De La Pierre, M.; Orlando, R.; Maschio, L.; Doll, K.; Ugliengo, P.; Dovesi, R. Performance of six functionals (LDA, PBE, PBESOL, B3LYP, PBE0 and WC1LYP) in the simulation of vibrational and dielectric properties of crystalline compounds. The case of forsterite Mg
_{2}SiO_{4}. J. Comp. Chem.**2011**, 32, 1775–1784. [Google Scholar] [CrossRef] [PubMed] - Bouhifd, M.A.; Andrault, D.; Fiquet, G.; Richet, P. Thermal expansion of forsterite up to the melting point. Geophys. Res. Lett.
**1996**, 23, 1143–1146. [Google Scholar] [CrossRef] - Ye, Y.; Schwering, R.A.; Smyth, J.R. Effects of hydration on thermal expansion of forsterite, wadsleyite, and ringwoodite at ambient pressure. Am. Mineral.
**2009**, 94, 899–904. [Google Scholar] [CrossRef] - Hazen, R.M. Effects of temperature and pressure on the crystal structure of forsterite. Am. Mineral.
**1976**, 61, 1280–1293. [Google Scholar] - Anderson, O.L.; Isaak, D.; Oda, H. High-temperature elastic constant data on minerals relevant to geophysics. Rev. Geophys.
**1992**, 30, 57–90. [Google Scholar] [CrossRef] - Howarth, A.; Liu, Y.; Li, P.; Li, Z.; Wang, T.; Hupp, J.; Farha, O. Chemical, thermal and mechanical stabilities of metal-organic frameworks. Nat. Rev. Mater.
**2016**, 1, 15018. [Google Scholar] [CrossRef] - Marmier, A.; Lethbridge, Z.A.; Walton, R.I.; Smith, C.W.; Parker, S.C.; Evans, K.E. ElAM: A computer program for the analysis and representation of anisotropic elastic properties. Comput. Phys. Commun.
**2010**, 181, 2102–2115. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Volumetric thermal expansion of forsterite up to 2000 K. Experimental data are from Bouhifd et al. [55] (filled circles), Ye et al. [56] (empty circles) and Hazen [57] (empty squares). Lines correspond to quasi-harmonic computed values with different functionals of the DFT: LDA (blue), PBE (green), PBEsol (red), and PBE0 (yellow).

**Figure 2.**Thermal dependence of the isothermal ${K}^{\mathrm{T}}$ (left panel) and adiabatic ${K}^{\mathrm{S}}$ (right panel) bulk modulus of forsterite up to 2000 K. Experimental data (filled and empty black circles) are from Anderson et al. [58]. Lines correspond to quasi-harmonic computed values with different functionals of the DFT: LDA (blue), PBE (green), PBEsol (red), and PBE0 (yellow).

**Figure 3.**Adiabatic single-crystal elastic stiffness constants of forsterite as a function of temperature. Circles are experimental data by Isaak et al. [32] while lines correspond to fully quasi-harmonic computed values.

**Figure 4.**Effect of temperature on the Young modulus of forsterite.

**Left**: 3D representation of the spatial dependence of the Young modulus as a function of temperature from experimental elastic constants, quasi-harmonic computed constants, and quasi-static computed constants.

**Right**: 2D projections of the Young modulus in three different planes ($xy$, $xz$ and $yz$) as a function of temperature (blue for 250 K, green for 650 K, yellow for 1050 K and red for 1450 K).

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**MDPI and ACS Style**

Destefanis, M.; Ravoux, C.; Cossard, A.; Erba, A.
Thermo-Elasticity of Materials from Quasi-Harmonic Calculations. *Minerals* **2019**, *9*, 16.
https://doi.org/10.3390/min9010016

**AMA Style**

Destefanis M, Ravoux C, Cossard A, Erba A.
Thermo-Elasticity of Materials from Quasi-Harmonic Calculations. *Minerals*. 2019; 9(1):16.
https://doi.org/10.3390/min9010016

**Chicago/Turabian Style**

Destefanis, Maurizio, Corentin Ravoux, Alessandro Cossard, and Alessandro Erba.
2019. "Thermo-Elasticity of Materials from Quasi-Harmonic Calculations" *Minerals* 9, no. 1: 16.
https://doi.org/10.3390/min9010016