International Journal of Quantum Chemistry

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1 International Journal of Quantum Chemistry First-principles calculation of second-order elastic constants and equations of state for Lithium Azide, LiN, and Lead Azide, Pb(N ) Journal: International Journal of Quantum Chemistry Manuscript ID: QUA R Wiley - Manuscript type: Date Submitted by the Author: Regular Submission - Properties, dynamics and elect structure of condensed systems and clusters 0-May-00 Complete List of Authors: Perger, Warren; Michigan Tech, Physics Keywords: elastic constants, azides, bulk modulus

2 Page of International Journal of Quantum Chemistry First-principles calculation of second-order elastic constants and equations of state for Lithium Azide, LiN, and Lead Azide, P b(n ) Abstract W.F. Perger Physics Department Michigan Tech University First-principles techniques are used to calculate the second-order elastic constants and equations of state for lithium azide, LiN, and lead azide, P b(n ). The bulk modulus is calculated for these systems in two independent ways and results compared. The Hartree-Fock potential and density functional theory are used for the exchange-correlation with different basis sets to examine the effects of each on the elastic constants and bulk modulus. Key words: inorganic azides, elastic constants Introduction Inorganic azides are a valuable class of compounds known to have practical applications in photography and energetic materials [] yet many theoretical problems remain. Younk and Kunz [] presented the band gaps for several azides using experimental values for the lattice constants and also under hydrostatic pressure. More recently Zhu, et al., calculated optical properties for lithium azide using density functional theory []. For lead azide there is little theoretical information on this material, particularly its mechanical properties. This is undoubtedly due, in part, to the computational challenges associated with this orthorhombic system which has electrons in the unit cell. With the advent of enhanced optimization techniques, improvements in the potentials available for the Hamiltonian, and faster computers, it is now possible to use ab initio techniques to calculate the second-order elastic constants address: wfp@mtu.edu (W.F. Perger). URL: (W.F. Perger). Preprint submitted to Elsevier May 00

3 International Journal of Quantum Chemistry Page of (SOECs) for such materials. Elastic constants provide important information on the mechanical properties of materials and on their structural stability [ ]. The work presented here extends prior calculations, using potentials going beyond Hartree-Fock (HF) and more accurate basis sets than were previously practical. Furthermore, optimization has improved to the point where full optimization of more complicated systems is feasible. Therefore, improved estimates of both the atomic positions and lattice parameters can be determined, thereby improving the quality of SOEC calculations. With these capabilities, equation of state (EOS) and SOEC calculations are facilitated and will be reported here for lithium azide and lead azide. For the SOECs, the space group is used to determine which strains are necessary, the strains are applied one at a time, returning to the equilibrium state before each subsequent deformation. The system is re-optimized at each deformation, resulting in a complete set of SOECs. The details of this methodology are described in another work []. The EOS calculations were carried out by selecting a range of volumes around minimum total energy (equilibrium) state, then performing an optimization at each volume, holding the volume constant. This ability to optimize the structure is particularly important for systems such as those studied here, monoclinic (LiN ) and orthorhombic (P b(n ) ), where the symmetry is relatively low. Lithium azide is a monoclinic system C/m and its band structure and electronic properties [] and optical properties [] were previously reported. A figure depicting the crystal structure is given in Fig.. From that figure it is evident why C (requiring a displacement along the z axis) is expected to be much larger than either C or C because a deformation along the z axis would be along the axis containing the nitrogen atoms. Fig.. LiN crystal structure. The nitrogens, in blue, are in the groups of three atoms along the z axis, and the lithium atoms are in the x y plane.

4 Page of International Journal of Quantum Chemistry A lead azide molecule is P b(n ) and in the solid phase there are molecules in the unit cell. The space group is P nma [] (orthorhombic). The crystal structure is depicted in Fig. in a set of projections. From that figure, it can be predicted that C should be much smaller than either C or C as a deformation along the z axis (horizontally in Fig. a) should result in a relatively smaller increase in the total energy due to the relatively greater spacings between atoms. The computational challenges are evident as this is a) b) c) Fig.. P b(n ) crystal structure. Fig. a) is a view in the y z plane, Fig. b) a view in the x z plane, and Fig. c) a view in the x y plane. The nitrogens, in blue, are in the linear groups of three atoms, and the lead atoms are off the planes containing the nitrogens. a system with relatively low symmetry with many electrons per unit cell. In the previous study on bandgaps [], a pseudo-potential was used to replace the P b core, reducing the number of electrons per unit cell to and that is the approach taken in this work. What symmetry exists is exploited to the fullest extent possible, which is especially important because a full optimization is carried out with each deformation. The prior work on lead azide [] did not relax the system at any point, which has been shown to produce large errors in other materials for the pressure-volume curve [], for example. Here, optimization is performed at each deformation of the crystal. Optimization of lattice parameters and atomic positions The first step for calculating either an EOS or the SOECs in a given material is the determination of the equilibrium geometry, in both atomic positions and

5 International Journal of Quantum Chemistry Page of lattice constants, for a given exchange-correlation potential and basis set. For the calculations reported here, the optimizer used was the one implemented in the CRYSTAL0 program []. This is a theoretically important step because it is crucial that any deformation made for the purpose of calculating an EOS or SOEC produce an increase in the total energy. The initial estimates for the atomic positions and lattice constants were taken from experimental values [] because they can often be used to provide reasonable guesses for the optimizer. Table gives the lattice parameters, equilibrium volume, and total energy for LiN using HF and density-functional theory (DFT) exchange-correlation potentials. The effect of optimization can be clearly seen by examining the first and last rows of that table, where the only difference is that the present HF calculation included full optimization. It is observed that the total energy is lowered as a result of optimization, as expected. Furthermore, the DFT- BLYP [] and -PWGGA [] potentials, which include correlation as well as optimization, lower the energy even further. As can be seen in that table, the Hartree-Fock potential tends to overestimate the equilibrium volume, an effect also observed in other systems [,]. In order to establish a connection with prior theoretical work, the bandgap for LiN was determined using Hartree-Fock (HF) and the same basis set of Younk and Kunz [] and agreement is shown in Table to be 0.eV. Also shown in that table, different basis sets and exchange-correlation potentials were used and compared, namely Hartree-Fock (HF), which has the correct exchange but no correlation, and density-functional theory (DFT) choices of BLYP and PWGGA []. These exchange-correlation potentials were chosen because as has been reported in other insulating materials [], HF overestimates the bandgap, PWGGA tends to underestimate it, and BLYP reproduces it more closely to experiment. Note that for consistency with the prior work, the results of Table are before optimization of either lattice or atomic positions. As previously noted, before deformation of the lattice for determination of elastic constants or an equation of state, the system must be optimized, for both lattice parameters and atomic positions. Table shows the lattice constants (a, b, and c) and bond angle (β) for LiN using a variety of both basis sets and exchange-correlation potentials for comparison. As can be seen from that table, the equilibrium volume for the PWGGA calculation is relatively close to the experimental value but that is probably somewhat fortuitous as the lattice constants and bond angle do not show the same relative percent difference from experiment (a, c, and β are larger than experiment, but b is smaller). The optimized lattice parameters for P b(n ) were found and are given in Table with the HF and DFT-PWGGA potentials. The basis set used for all P b(n ) calculations in the present work is that of ref. []. As is evident from that table, the HF potential yields an equilibrium volume greater than when

6 Page of International Journal of Quantum Chemistry using the DFT-PWGGA potential, but both predict a volume less than the experimental value. Equation of state and second-order elastic constant results The pressure-volume relation is obtained by fitting the E(V) curve to an equation of state such as the Murnaghan EOS []: [ E(V ) = B o V o B (B ) ( ) B Vo V + V ] + E B V o B o, () with the fitting parameters V o (volume at minimum energy), B o (zero-pressure bulk modulus), B (pressure derivative of the bulk modulus B at P = 0), and E o (minimum energy). Using CRYSTAL0 [], a program was written which systematically changes the volume around the (optimized) equilibrium state, with a re-optimization at each new volume chosen. The algorithm implemented selects a range of volumes around equilibrium, typically ±%, and a number of volumes, typically, within that range. At each of those volumes, the CRYSTAL0 optimizer was called using the CVOLOPT option, which performs an optimization of the internal co-ordinates and lattice parameters keeping the volume constant (see refs. [,] for a detailed description of the optimization algorithm). Table shows the results of using this program for the calculation of a series of total energies at the chosen volumes and fitted to Eq. () using a Levenburg-Marquardt routine [] as well as to a polynomial of degree. With the equilibrium configuration determined, the second-order elastic constants are then calculated by using a systematic series of deformations (the optimization is performed subject to the crystalline symmetry []). Under a linear elastic deformation, solid bodies are described using Hooke s law that takes the tensorial form σ ij = kl C ijkl ɛ kl () where (i, j, k, l) =,,, σ ij is the stress, ɛ kl is the strain, and C ijkl are the second-order elastic constants (SOECs) []. Evaluation of the elastic constants can be accomplished by using different theoretical approaches that include molecular dynamics simulation through fluctuation formulas (see ref. [0] and references therein) and the use of stress-strain relationships based on total energy calculations (e.g. from ab-initio methods). In the latter approach, SOECs

7 International Journal of Quantum Chemistry Page of are related to the total energy of the crystal through a Taylor expansion in terms of the strain components truncated to the second-order E(V, ɛ) = E(V 0 ) + V α σ α ɛ α + V C αβ ɛ α ɛ β + () αβ where Voigt s notation is used [], α, β =,,..., and V 0 is the equilibrium volume. The strains, ɛ α, are not volume-preserving. The crystalline structure is assumed to be stress-free, so that the second right-hand term in Eq. () is zero. Here, we refer to isoentropic (or adiabatic) elastic constants [], although the differences between adiabatic and isothermal elastic constants are small for temperatures at or below 00K [, p. ]. According to Eq. (), SOECs are related to the strain second derivatives of the total energy by: C αβ = V E. () ɛ α ɛ β 0 The effect of crystalline symmetry is to reduce the number of independent elastic constants. For example, in a cubic crystal, only C, C, and C are required, where C relates the compression stress and strain along the [0] direction, C relates the shear stress and strain in the same direction, and C relates the compression stress in one direction to the strain in another, e.g. the x and y directions (see, for example, ref. [], chap. ). From Eq. (), the calculation of elastic constants for an arbitrary crystal requires the ability to accurately calculate derivatives of the total energy as a function of crystal deformation. For ab-initio methods, this can be done either fully numerically, from total energy curves as a function of the applied strain for different deformations, or from strain first derivatives of the energy [,], or analytically. The bulk modulus is then calculated from the compliance matrix elements []: B = /(S + S + S + (S + S + S )). () Using this procedure and Eqn. (), the SOECs for LiN were obtained and are given in Table for a variety of basis sets and exchange-correlation potentials. As can be seen from that table, there is generally relatively good agreement between values for a given elastic constant using different potentials and basis sets for the elastic constants with larger magnitudes. However, an examination of C, for example, suggests that the elastic constants are not known to better than -GPa. C shows a relatively large spread in values depending on the exchange-correlation potential used. The sensitivity on the choice of potential in this case argues for the development of potentials which better model the intermolecular region.

8 Page of International Journal of Quantum Chemistry The second-order elastic constants for lead azide were likewise determined and are presented in Table. The basis set used was that of ref. [] and the exchange-correlation was again HF, BLYP, and PWGGA. In this case, the HF values are found to be similar to those obtained using the DFT functionals. Comparison of the bulk modulus for LiN using the Murnaghan equation of state Eq. (), Table, and using Eq. (), Table, shows B GPa vs. B GPa. For P b(n ), Table shows B 0 GPa and Table indicates B GPa. The disagreement arises from a variety of sources, both numerical and theoretical, as the two methods are very different in detail. In the EOS approach, a series of volumes are chosen around the equilibrium volume and optimization of the internal co-ordinates is performed for that volume. The energy-volume curve is then fitted to any number of equations of state [] and the bulk modulus extracted from the fit. On the other hand, calculating the bulk modulus from the elastic constants involves a series of displacements along the crystalline axes, with optimization of internal co-ordinates at each displacement, using analytic first-derivatives and numerical second-derivatives of the total energy with respect to displacement taken, resulting in the elastic constants, which are then used to find the compliance matrix elements and the bulk modulus via Eq. (). It is therefore relatively difficult to achieve exact agreement for crystalline systems of this complexity (monoclinic for LiN and orthorhombic for P b(n ) ). A comparison of Tables and shows that the Hartree-Fock elastic constants tend to be a bit smaller than those obtained with DFT. This is consistent with the observation that the lack of correlation in the HF case tends to produce larger lattice constants and larger equilibrium volumes than those found using DFT (see Tables and ). Conclusions The second-order elastic constants and equations of state for lithium azide and lead azide have been presented. Optimization of both lattice parameters and atomic positions was accomplished at each deformation using the CRYS- TAL0 program, with special-purpose extensions written. The bulk modulus was determined in two different ways for each system, and reasonable agreement was observed. Although experimental evidence for these properties of these systems was not available, comparison with prior theory shows a lowering of the total energy for these systems with the use of full optimization, as expected. The systems studied were relatively complicated with lithium azide having a monoclinic structure and lead azide a large number of atoms and electrons per unit cell. Although some consistent trends were observed, such as the Hartree-Fock potential yielding larger optimized volumes and, in general, lower

9 International Journal of Quantum Chemistry Page of elastic constants than observed using DFT, it remains an open question as to which exchange-correlation potential produces the best results for elastic constants. This is due, in part, to the lack of experimental evidence for these azides. For future work on systems with these complexities, it will be important to use different basis sets and exchange-correlation potentials for confidence in the calculated elastic constants. Although elastic constants can likely be determined to within a few GPa for simple systems, for more complicated systems such as those presented here, it is therefore difficult to achieve that same level of precision.

10 Page of International Journal of Quantum Chemistry Acknowledgements The author acknowledges the support of the US Office of Naval Research (ONR) and the MURI grant N The author also gratefully acknowledges the input of Dr. Yogendra Gupta and the suggestions of the referee. References [] H. D. Fair, R. F. Walker, Ed., Physics and Chemistry of Inorganic Azides, in: Energetic Materials, Vol., Plenum Press,. [] E. H. Younk, A. B. Kunz, An ab initio investigation of the electronic structure of lithium azide (LiN ), sodium azide (NaN ) and lead azide (P b(n ) ), Int. J. Quantum Chem. (). [] W. Zhu, J. Xiao, H. Xiao, Density functional theory study of the structural and optical properties of lithium azide, Chem. Phys. Lett. (-) (00). [] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford Univ. Press, Oxford,. [] J. F. Nye, Physical Properties of Crystals, Dover Publications, New York,. [] D. C. Wallace, Thermodynamics of Crystals, Wiley, New York,. [] W. F. Perger, J. Criswell, B. Civalleri, R. Dovesi, Comput. Phys. Commun.(submitted). [] M. Seel, A. B. Kunz, Band structure and electronic properties of lithium azide LiN, Int. J. Quantum Chem. (). [] C. S. Choi, H. P. Boutin, Acta Cryst. B (). [] W. F. Perger, S. Vutukuri, Z. A. Dreger, Y. M. Gupta, K. Flurchick, Firstprinciples vibrational studies of pentaerythritol crystal under hydrostatic pressure, Chem. Phys. Lett. (00) 0. [] R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. J. Bush, P. D Arco, M. Llunell, CRYSTAL00 User s Manual, University of Torino, Torino, Italy, 00. [] A. D. Becke, Density-functional thermochemistry. III. the role of exact exchange, J. Chem. Phys. (). [] J. P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for the exchange-correlation hole of a many-electron system, Phys. Rev. B. ().

11 International Journal of Quantum Chemistry Page of [] W. F. Perger, R. Pandey, M. A. Blanco, J. Zhao, First-principles intermolecular binding energies in organic molecular crystals, Chem. Phys. Lett. /- (00). [] C. S. Choi, Physics and Chemistry of Inorganic Azides, in: H. D. Fair, R. F. Walker (Eds.), Energetic Materials, Vol., Plenum Press, New York,, p.. [] W. F. Perger, Calculation of band gaps in molecular crystals using hybrid functional theory, Chem. Phys. Lett. /- (00). [] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 0 (). [] B. Civalleri, P. D Arco, R. Orlando, V. R. Saunders, R. Dovesi, Hartree-Fock geometry optimisation of periodic systems with the CRYSTAL code, Chem. Phys. Lett. (00). [] D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. (). [0] Z. Zhou, B. Joós, Fluctuation formulas for the elastic constants of an arbitrary system, Phys. Rev. B (00) 0. [] C. Kittel, Introduction to Solid State Physics, th ed., John Wiley & Sons, New York, 00. [] M. A. Omar, Elementary Solid State Physics, Addison-Wesley, Reading, MA,. [] O. H. Nielsen, R. M. Martin, Phys. Rev. Lett. 0 (). [] O. H. Nielsen, R. M. Martin, Phys. Rev. B. (). [] L. Vočadlo, J. P. Poirer, G. D. Price, Grüneisen parameters and isothermal equations of state, American Mineralogist (000) 0.

12 Page of International Journal of Quantum Chemistry Table Lattice parameters (in Å), equilibrium volume (in Å ), and total energy, E (in a.u.), for LiN using Hartree-Fock, DFT-BLYP and DFT-PWGGA potentials. The basis used was the split-valence set (Basis ) of ref. []. For the last row, the geometric values were taken from experiment [] and the total energy HF calculation from ref. []. The numbers in parentheses are the percent differences from the experimental volume. a b c β Vol. E HF o.(.) -. BLYP..0.. o.0(0.) -. PWGGA.... o.(0.) -. refs.[,].... o. -.

13 International Journal of Quantum Chemistry Page of Table Bandgap for LiN using Hartree-Fock, DFT-BLYP, and DFT-PWGGA potentials, and -**, double-zeta plus polarization (DZP), and optimized split-valence (splval) Gaussian set []. All values are in ev. HF HF [] BLYP PWGGA spl-val.... -**... DZP.0..

14 Page of International Journal of Quantum Chemistry Table Lattice parameters (in Å) and volume (in Å ) for LiN using various exchangecorrelation potentials and basis sets. a b c β Volume BLYP-** o. BLYP-DZP.... o. PWGGA-**.0... o. PWGGA-DZP...0. o. Expt [].... o.

15 International Journal of Quantum Chemistry Page of Table Lattice parameters (in Å) and volume (in Å ) for P b(n ) using Hartree-Fock and DFT-PWGGA potentials. a b c Volume HF PWGGA...0. Expt []....

16 Page of International Journal of Quantum Chemistry Table Equation of state data for LiN and P b(n ) using the Murnaghan equation (-M) and fitting to a polynomial of degree (-P). Eleven points in the energy-volume curve were used and the range of volumes used around equilibrium was ±%. B(GPa) V o (Å ) E 0 (a.u.) LiN : BLYP-**-M BLYP-**-P.. -. PWGGA-**-M.. -. PWGGA-**-P.. -. P b(n ) : PWGGA-M PWGGA-P HF-M HF-P

17 International Journal of Quantum Chemistry Page of Table Second-order elastic constants and bulk modulus, B, for LiN using various basis sets and exchange-correlation potentials. HF is Hartree-Fock with the split-valence basis set, B is the BLYP functional, PW is the Perdew-Wang generalized gradient (PWGGA), and Bsv is BLYP with split-valence set of ref. []. The missing entries in the HF column are where convergence could not be obtained. All values are in GPa. HF B -** B DZP Bsv PW -** PW DZP c c c c c c c c c c..... c c c B.....0

18 Page of International Journal of Quantum Chemistry Table Second-order elastic constants and bulk modulus, B, for P b(n ) using Hartree- Fock and DFT BLYP and PWGGA. All values are in GPa. HF DFT-BLYP PWGGA c.0.. c..0.0 c... c..0. c.0..0 c 0... c... c.0.0. c.0.. B 0..0.