For Peer Review

First-principles calculation of second-order elastic constants

and

equations of state for Lithium Azide, LiN3, and Lead Azide,

Pb(N3)2

Journal: International Journal of Quantum Chemistry

Manuscript ID: QUA-2009-0080.R2

Wiley - Manuscript type: Regular Submission - Properties, dynamics and elect structure of condensed systems and clusters

Date Submitted by the Author:

01-May-2009

Complete List of Authors: Perger, Warren; Michigan Tech, Physics

Keywords: elastic constants, azides, bulk modulus

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

For Peer Review

First-principles calculation of second-order

elastic constants and equations of state for

Lithium Azide, LiN3, and Lead Azide, Pb(N3)2

W.F. Perger

Physics DepartmentMichigan Tech University

Abstract

First-principles techniques are used to calculate the second-order elastic constantsand equations of state for lithium azide, LiN3, and lead azide, Pb(N3)2. The bulkmodulus is calculated for these systems in two independent ways and results com-pared. The Hartree-Fock potential and density functional theory are used for theexchange-correlation with different basis sets to examine the effects of each on theelastic constants and bulk modulus.

Key words: inorganic azides, elastic constants

1 Introduction

Inorganic azides are a valuable class of compounds known to have practicalapplications in photography and energetic materials [1] yet many theoreticalproblems remain. Younk and Kunz [2] presented the band gaps for severalazides using experimental values for the lattice constants and also under hy-drostatic pressure. More recently Zhu, et al., calculated optical properties forlithium azide using density functional theory [3]. For lead azide there is littletheoretical information on this material, particularly its mechanical proper-ties. This is undoubtedly due, in part, to the computational challenges associ-ated with this orthorhombic system which has 1488 electrons in the unit cell.With the advent of enhanced optimization techniques, improvements in thepotentials available for the Hamiltonian, and faster computers, it is now pos-sible to use ab initio techniques to calculate the second-order elastic constants

Email address: wfp@mtu.edu (W.F. Perger).URL: http://www.ee.mtu.edu/faculty/wfp.html (W.F. Perger).

Preprint submitted to Elsevier 1 May 2009

Page 2 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

(SOECs) for such materials. Elastic constants provide important informationon the mechanical properties of materials and on their structural stability[4–6]. The work presented here extends prior calculations, using potentialsgoing beyond Hartree-Fock (HF) and more accurate basis sets than were pre-viously practical. Furthermore, optimization has improved to the point wherefull optimization of more complicated systems is feasible. Therefore, improvedestimates of both the atomic positions and lattice parameters can be deter-mined, thereby improving the quality of SOEC calculations. With these ca-pabilities, equation of state (EOS) and SOEC calculations are facilitated andwill be reported here for lithium azide and lead azide. For the SOECs, thespace group is used to determine which strains are necessary, the strains areapplied one at a time, returning to the equilibrium state before each subse-quent deformation. The system is re-optimized at each deformation, resultingin a complete set of SOECs. The details of this methodology are describedin another work [7]. The EOS calculations were carried out by selecting arange of volumes around minimum total energy (equilibrium) state, then per-forming an optimization at each volume, holding the volume constant. Thisability to optimize the structure is particularly important for systems suchas those studied here, monoclinic (LiN3) and orthorhombic (Pb(N3)2), wherethe symmetry is relatively low.

Lithium azide is a monoclinic system C2/m and its band structure and elec-tronic properties [8] and optical properties [3] were previously reported. Afigure depicting the crystal structure is given in Fig. 1. From that figure it isevident why C33 (requiring a displacement along the z−axis) is expected tobe much larger than either C11 or C22 because a deformation along the z−axiswould be along the axis containing the 3 nitrogen atoms.

Fig. 1. LiN3 crystal structure. The nitrogens, in blue, are in the groups of threeatoms along the z−axis, and the lithium atoms are in the x− y plane.

2

Page 3 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

A lead azide molecule is Pb(N3)2 and in the solid phase there are 12 moleculesin the unit cell. The space group is Pnma [9] (orthorhombic). The crystalstructure is depicted in Fig. 2 in a set of 3 projections. From that figure, itcan be predicted that C33 should be much smaller than either C11 or C22 asa deformation along the z−axis (horizontally in Fig. 2a) should result in arelatively smaller increase in the total energy due to the relatively greaterspacings between atoms. The computational challenges are evident as this is

a) b) c)

Fig. 2. Pb(N3)2 crystal structure. Fig. a) is a view in the y− z plane, Fig. b) a viewin the x− z plane, and Fig. c) a view in the x− y plane. The nitrogens, in blue, arein the linear groups of three atoms, and the lead atoms are off the planes containingthe nitrogens.

a system with relatively low symmetry with many electrons per unit cell. Inthe previous study on bandgaps [2], a pseudo-potential was used to replacethe Pb core, reducing the number of electrons per unit cell to 552 and thatis the approach taken in this work. What symmetry exists is exploited to thefullest extent possible, which is especially important because a full optimiza-tion is carried out with each deformation. The prior work on lead azide [2]did not relax the system at any point, which has been shown to produce largeerrors in other materials for the pressure-volume curve [10], for example. Here,optimization is performed at each deformation of the crystal.

2 Optimization of lattice parameters and atomic positions

The first step for calculating either an EOS or the SOECs in a given materialis the determination of the equilibrium geometry, in both atomic positions and

3

Page 4 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

lattice constants, for a given exchange-correlation potential and basis set. Forthe calculations reported here, the optimizer used was the one implemented inthe CRYSTAL06 program [11]. This is a theoretically important step becauseit is crucial that any deformation made for the purpose of calculating an EOSor SOEC produce an increase in the total energy. The initial estimates for theatomic positions and lattice constants were taken from experimental values [1]because they can often be used to provide reasonable guesses for the optimizer.Table 1 gives the lattice parameters, equilibrium volume, and total energyfor LiN3 using HF and density-functional theory (DFT) exchange-correlationpotentials. The effect of optimization can be clearly seen by examining thefirst and last rows of that table, where the only difference is that the presentHF calculation included full optimization. It is observed that the total energyis lowered as a result of optimization, as expected. Furthermore, the DFT-B3LYP [12] and -PWGGA [13] potentials, which include correlation as wellas optimization, lower the energy even further. As can be seen in that table,the Hartree-Fock potential tends to overestimate the equilibrium volume, aneffect also observed in other systems [14,10].

In order to establish a connection with prior theoretical work, the bandgap forLiN3 was determined using Hartree-Fock (HF) and the same basis set of Younkand Kunz [2] and agreement is shown in Table 2 to be 0.1eV. Also shown inthat table, different basis sets and exchange-correlation potentials were usedand compared, namely Hartree-Fock (HF), which has the correct exchangebut no correlation, and density-functional theory (DFT) choices of B3LYPand PWGGA [11]. These exchange-correlation potentials were chosen becauseas has been reported in other insulating materials [16], HF overestimates thebandgap, PWGGA tends to underestimate it, and B3LYP reproduces it moreclosely to experiment. Note that for consistency with the prior work, the resultsof Table 2 are before optimization of either lattice or atomic positions.

As previously noted, before deformation of the lattice for determination ofelastic constants or an equation of state, the system must be optimized, forboth lattice parameters and atomic positions. Table 3 shows the lattice con-stants (a, b, and c) and bond angle (β) for LiN3 using a variety of both basissets and exchange-correlation potentials for comparison. As can be seen fromthat table, the equilibrium volume for the PWGGA calculation is relativelyclose to the experimental value but that is probably somewhat fortuitous asthe lattice constants and bond angle do not show the same relative percentdifference from experiment (a, c, and β are larger than experiment, but b issmaller).

The optimized lattice parameters for Pb(N3)2 were found and are given inTable 4 with the HF and DFT-PWGGA potentials. The basis set used for allPb(N3)2 calculations in the present work is that of ref. [2]. As is evident fromthat table, the HF potential yields an equilibrium volume greater than when

4

Page 5 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

using the DFT-PWGGA potential, but both predict a volume less than theexperimental value.

3 Equation of state and second-order elastic constant results

The pressure-volume relation is obtained by fitting the E(V) curve to an equa-tion of state such as the Murnaghan EOS [17]:

E(V ) = BoVo

[1

B′(B′ − 1)

(Vo

V

)B′−1

+V

B′Vo

− 1

B′ − 1

]+ Eo, (1)

with the fitting parameters Vo (volume at minimum energy), Bo (zero-pressurebulk modulus), B′ (pressure derivative of the bulk modulus B at P = 0), andEo (minimum energy). Using CRYSTAL06 [11], a program was written whichsystematically changes the volume around the (optimized) equilibrium state,with a re-optimization at each new volume chosen. The algorithm implementedselects a range of volumes around equilibrium, typically ±12%, and a num-ber of volumes, typically 10, within that range. At each of those volumes,the CRYSTAL06 optimizer was called using the CVOLOPT option, whichperforms an optimization of the internal co-ordinates and lattice parameterskeeping the volume constant (see refs. [11,18] for a detailed description of theoptimization algorithm). Table 5 shows the results of using this program forthe calculation of a series of total energies at the chosen volumes and fitted toEq. (1) using a Levenburg-Marquardt routine [19] as well as to a polynomialof degree 3.

With the equilibrium configuration determined, the second-order elastic con-stants are then calculated by using a systematic series of deformations (theoptimization is performed subject to the crystalline symmetry [18]). Under alinear elastic deformation, solid bodies are described using Hooke’s law thattakes the tensorial form

σij =∑kl

Cijklεkl (2)

where (i, j, k, l) = 1, 2, 3, σij is the stress, εkl is the strain, and Cijkl are thesecond-order elastic constants (SOECs) [5]. Evaluation of the elastic constantscan be accomplished by using different theoretical approaches that includemolecular dynamics simulation through fluctuation formulas (see ref. [20] andreferences therein) and the use of stress-strain relationships based on total en-ergy calculations (e.g. from ab-initio methods). In the latter approach, SOECs

5

Page 6 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

are related to the total energy of the crystal through a Taylor expansion interms of the strain components truncated to the second-order

E(V, ε) = E(V0) + V∑α

σαεα +V

2

∑αβ

Cαβεαεβ + · · · (3)

where Voigt’s notation is used [5], α, β = 1, 2, . . . 6, and V0 is the equilibriumvolume. The strains, εα, are not volume-preserving. The crystalline structureis assumed to be stress-free, so that the second right-hand term in Eq. (3) iszero. Here, we refer to isoentropic (or adiabatic) elastic constants [6], althoughthe differences between adiabatic and isothermal elastic constants are smallfor temperatures at or below 300K [21, p. 73]. According to Eq. (3), SOECsare related to the strain second derivatives of the total energy by:

Cαβ =1

V

∂2E

∂εα∂εβ 0

. (4)

The effect of crystalline symmetry is to reduce the number of independentelastic constants. For example, in a cubic crystal, only C11, C44, and C12 arerequired, where C11 relates the compression stress and strain along the [100]direction, C44 relates the shear stress and strain in the same direction, andC12 relates the compression stress in one direction to the strain in another,e.g. the x− and y−directions (see, for example, ref. [22], chap. 3). From Eq.(4), the calculation of elastic constants for an arbitrary crystal requires theability to accurately calculate derivatives of the total energy as a functionof crystal deformation. For ab-initio methods, this can be done either fullynumerically, from total energy curves as a function of the applied strain fordifferent deformations, or from strain first derivatives of the energy [23,24], oranalytically. The bulk modulus is then calculated from the compliance matrixelements [5]:

B = 1/(S11 + S22 + S33 + 2(S12 + S13 + S23)). (5)

Using this procedure and Eqn. (4), the SOECs for LiN3 were obtained and aregiven in Table 6 for a variety of basis sets and exchange-correlation potentials.As can be seen from that table, there is generally relatively good agreementbetween values for a given elastic constant using different potentials and basissets for the elastic constants with larger magnitudes. However, an examinationof C25, for example, suggests that the elastic constants are not known to betterthan 2-3GPa. C66 shows a relatively large spread in values depending on theexchange-correlation potential used. The sensitivity on the choice of potentialin this case argues for the development of potentials which better model theintermolecular region.

6

Page 7 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

The second-order elastic constants for lead azide were likewise determinedand are presented in Table 7. The basis set used was that of ref. [2] and theexchange-correlation was again HF, B3LYP, and PWGGA. In this case, theHF values are found to be similar to those obtained using the DFT functionals.

Comparison of the bulk modulus for LiN3 using the Murnaghan equation ofstate Eq. (1), Table 5, and using Eq. (5), Table 6, shows B ∼ 18 GPa vs. B ∼21 GPa. For Pb(N3)2, Table 5 shows B ∼ 50 GPa and Table 7 indicates B ∼46 GPa. The disagreement arises from a variety of sources, both numericaland theoretical, as the two methods are very different in detail. In the EOSapproach, a series of volumes are chosen around the equilibrium volume andoptimization of the internal co-ordinates is performed for that volume. Theenergy-volume curve is then fitted to any number of equations of state [25]and the bulk modulus extracted from the fit. On the other hand, calculatingthe bulk modulus from the elastic constants involves a series of displacementsalong the crystalline axes, with optimization of internal co-ordinates at eachdisplacement, using analytic first-derivatives and numerical second-derivativesof the total energy with respect to displacement taken, resulting in the elasticconstants, which are then used to find the compliance matrix elements and thebulk modulus via Eq. (5). It is therefore relatively difficult to achieve exactagreement for crystalline systems of this complexity (monoclinic for LiN3 andorthorhombic for Pb(N3)2). A comparison of Tables 6 and 7 shows that theHartree-Fock elastic constants tend to be a bit smaller than those obtainedwith DFT. This is consistent with the observation that the lack of correlationin the HF case tends to produce larger lattice constants and larger equilibriumvolumes than those found using DFT (see Tables 1 and 4).

4 Conclusions

The second-order elastic constants and equations of state for lithium azideand lead azide have been presented. Optimization of both lattice parametersand atomic positions was accomplished at each deformation using the CRYS-TAL06 program, with special-purpose extensions written. The bulk moduluswas determined in two different ways for each system, and reasonable agree-ment was observed. Although experimental evidence for these properties ofthese systems was not available, comparison with prior theory shows a lower-ing of the total energy for these systems with the use of full optimization, asexpected.

The systems studied were relatively complicated with lithium azide having amonoclinic structure and lead azide a large number of atoms and electronsper unit cell. Although some consistent trends were observed, such as theHartree-Fock potential yielding larger optimized volumes and, in general, lower

7

Page 8 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

elastic constants than observed using DFT, it remains an open question asto which exchange-correlation potential produces the best results for elasticconstants. This is due, in part, to the lack of experimental evidence for theseazides. For future work on systems with these complexities, it will be importantto use different basis sets and exchange-correlation potentials for confidencein the calculated elastic constants. Although elastic constants can likely bedetermined to within a few GPa for simple systems, for more complicatedsystems such as those presented here, it is therefore difficult to achieve thatsame level of precision.

8

Page 9 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Acknowledgements

The author acknowledges the support of the US Office of Naval Research(ONR) and the MURI grant N00014-06-1-0459. The author also gratefullyacknowledges the input of Dr. Yogendra Gupta and the suggestions of thereferee.

References

[1] H. D. Fair, R. F. Walker, Ed., Physics and Chemistry of Inorganic Azides, in:Energetic Materials, Vol. 1, Plenum Press, 1977.

[2] E. H. Younk, A. B. Kunz, An ab initio investigation of the electronic structureof lithium azide (LiN3), sodium azide (NaN3) and lead azide (Pb(N3)2), Int.J. Quantum Chem. 63 (1997) 615–621.

[3] W. Zhu, J. Xiao, H. Xiao, Density functional theory study of the structural andoptical properties of lithium azide, Chem. Phys. Lett. 422 (1-3) (2006) 117 –121.

[4] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford Univ. Press,Oxford, 1954.

[5] J. F. Nye, Physical Properties of Crystals, Dover Publications, New York, 1957.

[6] D. C. Wallace, Thermodynamics of Crystals, Wiley, New York, 1972.

[7] W. F. Perger, J. Criswell, B. Civalleri, R. Dovesi, Comput. Phys.Commun.(submitted).

[8] M. Seel, A. B. Kunz, Band structure and electronic properties of lithium azideLiN3, Int. J. Quantum Chem. 39 (1991) 149–157.

[9] C. S. Choi, H. P. Boutin, Acta Cryst. B25 (1969) 982.

[10] W. F. Perger, S. Vutukuri, Z. A. Dreger, Y. M. Gupta, K. Flurchick, First-principles vibrational studies of pentaerythritol crystal under hydrostaticpressure, Chem. Phys. Lett. 422 (2006) 397–401.

[11] 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, CRYSTAL2006 User’s Manual, University of Torino, Torino, Italy,2006.

[12] A. D. Becke, Density-functional thermochemistry. III. the role of exactexchange, J. Chem. Phys. 98 (1993) 5648.

[13] J. P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for theexchange-correlation hole of a many-electron system, Phys. Rev. B. 45 (1992)13244.

9

Page 10 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

[14] W. F. Perger, R. Pandey, M. A. Blanco, J. Zhao, First-principles intermolecularbinding energies in organic molecular crystals, Chem. Phys. Lett. 388/1-3 (2004)175–180.

[15] C. S. Choi, Physics and Chemistry of Inorganic Azides, in: H. D. Fair, R. F.Walker (Eds.), Energetic Materials, Vol. 1, Plenum Press, New York, 1977, p.102.

[16] W. F. Perger, Calculation of band gaps in molecular crystals using hybridfunctional theory, Chem. Phys. Lett. 368/3-4 (2003) 319–323.

[17] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244.

[18] B. Civalleri, P. D’Arco, R. Orlando, V. R. Saunders, R. Dovesi, Hartree-Fockgeometry optimisation of periodic systems with the CRYSTAL code, Chem.Phys. Lett. 348 (2001) 131–138.

[19] D. Marquardt, An algorithm for least-squares estimation of nonlinearparameters, SIAM J. Appl. Math. 11 (1963) 431–441.

[20] Z. Zhou, B. Joos, Fluctuation formulas for the elastic constants of an arbitrarysystem, Phys. Rev. B 66 (2002) 054101.

[21] C. Kittel, Introduction to Solid State Physics, 8th ed., John Wiley & Sons, NewYork, 2005.

[22] M. A. Omar, Elementary Solid State Physics, Addison-Wesley, Reading, MA,1975.

[23] O. H. Nielsen, R. M. Martin, Phys. Rev. Lett. 50 (1983) 697.

[24] O. H. Nielsen, R. M. Martin, Phys. Rev. B. 32 (1985) 3792.

[25] L. Vocadlo, J. P. Poirer, G. D. Price, Gruneisen parameters and isothermalequations of state, American Mineralogist 85 (2000) 390–395.

10

Page 11 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 1Lattice parameters (in A), equilibrium volume (in A3), and total energy, E (in a.u.),for LiN3 using Hartree-Fock, DFT-B3LYP and DFT-PWGGA potentials. The basisused was the split-valence set (Basis 4) of ref. [8]. For the last row, the geometricvalues were taken from experiment [15] and the total energy HF calculation from ref.[8]. The numbers in parentheses are the percent differences from the experimentalvolume.

a b c β Vol. E

HF 5.613 3.403 5.039 107.9o 91.59(3.2) -170.69237

B3LYP 5.692 3.250 5.265 113.7o 89.20(0.53) -171.62114

PWGGA 5.696 3.235 5.263 113.5o 88.95(0.25) -171.66769

refs.[15,8] 5.627 3.319 4.979 107.4o 88.73 -170.69182

11

Page 12 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 2Bandgap for LiN3 using Hartree-Fock, DFT-B3LYP, and DFT-PWGGA potentials,and 6-311**, double-zeta plus polarization (DZP), and optimized split-valence (spl-val) Gaussian set [8]. All values are in eV.

HF HF [2] B3LYP PWGGA

spl-val 11.5 11.4 4.77 3.16

6-311** 10.9 5.31 3.63

DZP 12.0 5.14 3.46

12

Page 13 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 3Lattice parameters (in A) and volume (in A3) for LiN3 using various exchange-correlation potentials and basis sets.

a b c β Volume

B3LYP-6311** 5.720 3.271 5.236 114.0o 89.46

B3LYP-DZP 5.711 3.275 5.229 113.9o 89.41

PWGGA-6311** 5.706 3.241 5.232 113.9o 88.47

PWGGA-DZP 5.719 3.261 5.220 113.5o 88.47

Expt [15] 5.627 3.319 4.979 107.4o 88.73

13

Page 14 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 4Lattice parameters (in A) and volume (in A3) for Pb(N3)2 using Hartree-Fock andDFT-PWGGA potentials.

a b c Volume

HF 6.517 16.24 11.00 1164.3

PWGGA 6.432 15.97 11.02 1131.5

Expt [9] 6.63 16.25 11.31 1218.5

14

Page 15 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 5Equation of state data for LiN3 and Pb(N3)2 using the Murnaghan equation (-M)and fitting to a polynomial of degree 3 (-P). Eleven points in the energy-volumecurve were used and the range of volumes used around equilibrium was ±12%.

B(GPa) Vo(A3) E0(a.u.)

LiN3: B3LYP-6311**-M 17.63 89.70 -171.79

B3LYP-6311**-P 17.99 89.66 -171.79

PWGGA-6311**-M 17.33 88.75 -171.84

PWGGA-6311**-P 17.62 88.72 -171.84

Pb(N3)2: PWGGA-M 50.13 1130.3 -3961.13

PWGGA-P 50.36 1130.9 -3961.13

HF-M 50.76 1190.4 -3937.03

HF-P 50.48 1189.8 -3937.03

15

Page 16 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 6Second-order elastic constants and bulk modulus, B, for LiN3 using various basissets and exchange-correlation potentials. HF is Hartree-Fock with the split-valencebasis set, B3 is the B3LYP functional, PW is the Perdew-Wang generalized gradient(PWGGA), and B3sv is B3LYP with split-valence set of ref. [8]. The missing entriesin the HF column are where convergence could not be obtained. All values are inGPa.

HF B3 6-311** B3 DZP B3sv PW 6-311** PW DZP

c11 28.22 29.75 29.05 29.52 30.23 28.53

c22 27.65 29.78 30.42 30.88 30.97 30.48

c33 180.8 181.1 193.5 183.2 180.0

c12 4.73 12.61 12.93 13.35 11.53 12.27

c13 10.80 20.38 23.13 19.97 19.94 23.23

c23 21.24 8.813 10.44 4.192 10.43 13.02

c44 5.25 4.087 6.507 2.864 3.829 4.924

c15 0.689 0.340 0.806 1.417 0.673 1.709

c25 6.925 -0.965 0.258 -2.988 -1.602 0.809

c35 32.34 31.62 33.97 32.71 31.98

c55 17.16 4.827 4.597 4.313 4.747 4.868

c46 0.188 8.394 10.82 7.967 4.760 5.603

c66 8.665 2.001 0.283 1.422 11.35 10.04

B 21.58 22.68 21.74 21.14 24.20

16

Page 17 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review

Table 7Second-order elastic constants and bulk modulus, B, for Pb(N3)2 using Hartree-Fock and DFT B3LYP and PWGGA. All values are in GPa.

HF DFT-B3LYP PWGGA

c11 74.90 104.9 103.5

c22 98.23 113.0 126.0

c33 49.32 48.38 47.98

c12 31.65 47.80 46.19

c13 26.07 38.25 37.30

c23 30.39 36.59 37.16

c44 15.46 14.23 17.53

c55 22.01 26.08 28.59

c66 4.601 23.37 25.14

B 40.27 46.06 45.81

17

Page 18 of 18

John Wiley & Sons, Inc.

International Journal of Quantum Chemistry

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960