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1 University of Torino
Calculation of Equations of State and Elastic Constants
W. F. Perger wfp@mtu.edu
Michigan Tech University Houghton, Michigan, USA
B. Civalleri and R. Dovesi University of Torino
MSSC2009 9 September, 2009
2 University of Torino
Outline Application of pressure (hydrostatic) Basic concepts of mechanical deformation
i Stress/strain i Role of total energy in calculations
Equations of state i Practical use of CRYSTAL for creating Energy vs. Volume
curve i Use of equation of state to determine pressure-volume curve
Calculation of elastic constants i Crystalline structure and choice of deformations i Use of symmetry i Use of analytic first derivative of total energy; numerical
second derivative Tests to determine CRYSTAL09 default parameters Application to organic molecular crystals
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Applying pressure to a crystal
To theoretically apply pressure, obviously must change cell parameters
Cell parameter changes can, in principle, be both positive and negative
With each change in the cell parameter(s), total energy is recalculated
This is relatively simple for cubic systems Increasingly difficult for systems of lower symmetry Hydrostatic pressure is the application of uniform stress, not
strain Therefore, in general, a scheme must be used which, for a given
volumetric change, gives the cell parameters minimizing the total energy of the crystal
An “equation of state” is used to fit Energy vs. Volume (often Murnaghan EOS)
Pressure vs. volume is then given by:
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Application of hydrostatic pressure
Choose volumes around equilibrium Use CRYSTAL09 program with full optimization (both atomic
positions and lattice constants) at that volume Recalculate total energy at that volume Fit to Murnaghan EOS:
F. D. Murnaghan, Proc. Natl. Acad. Sci USA 30 (1944) 244
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Calculation of Elastic Constants Credit: Y. Noël, R. Dovesi, M. Catti… Computer Physics Comm., vol 180, p. 1753-1759, 2009 A solid body deforms when subjected to stress Below the elastic limit, the strain is recoverable (body returns to
original state with stress removed) Hooke’s Law: For sufficiently small stresses, the amount of strain is
proportional to the magnitude of applied stress:
Alternatively:
σ ij = cijklεkl
kl
1,3
∑
Where ε is the strain, c is the elastic constant (matrix) and σ is the stress. Expressed in component form, it becomes:
6 University of Torino
Elastic constants…
Because there only 6 independent components, a more compact, one-index (Voigt) notation is used:
Where the cijkl are the 81 elastic constants of the crystal The strain tensor can be decomposed into two parts: an
antisymmetric part corresponding to pure rotations and a symmetric part corresponding to pure strain (of interest here):
ε1 ≡ ε11 ε4 ≡ ε23 + ε32
ε2 ≡ ε22 ε5 ≡ ε13 + ε31
ε3 ≡ ε33 ε6 ≡ ε12 + ε21
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Elastic constants…
A Taylor expansion of the total energy of the unit cell as a function of strain gives:
The strain tensor is then:
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Elastic constants…
Where V is the volume of the cell.
If E(0) is an equilibrium, i.e. previously-optimized, state, then the first derivatives are zero. The 21 possible elastic constants are then:
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Elastic constants…
When only the total energy is available: 1. Optimize structure 2. Select elastic constant of interest; choose appropriate strain 3. Deform the crystal 4. Re-optimize structure 5. Repeat step 3. for a set of similar deformations with different amplitudes,
creating a Total Energy vs. δ curve (a parabola) 6. Fit the curve to extract the second derivative of the total energy and
calculate elastic constant Scripts by Yves Noel, et al, multdef and elastic to simplify process (see
previous tutorial) Procedure for finding all elastic constants begins with identifying the
structure and determining the complete set. Cubic: c11, c12, c44 Hexagonal: c11, c12, c13, c44 Etc….
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Elastic constant matrix element for several crystal classes
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Elastic constants…
Example: ZnO, hexagonal
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Elastic constants…
Example: ZnO, hexagonal
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Elastic constants…
Example: ZnO, hexagonal
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Elastic constants…
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Elastic constants…
If analytic first derivatives of the total energy are available, as in CRYSTAL09, then fewer displacements are required because we need only fit to a linear function
“ELASTCON” option ELASTCON option uses NUMDERIV points, default=3: (-δ,0,δ) An optimized input is assumed, so the central-point calculation is simply to
calculate the cell gradients at the equilibrium state
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Using ELASTCON option in CRYSTAL09
Minimum of two keywords: i ELASTCON i ENDELAST (or just END)
Possible options: i TOLDEG (or GTOL), RMS gradient tolerance; default = 6x10-5 i TOLDEX (or XTOL), RMS displacement; default = 0.00012 i TOLDEE (or ETOL), Total energy convergence; default = 8 i STEPSIZE; default = 0.005 i NUMDERIV; default = 3 i PRINT (gives additional printing) i NOPREOPT i RESTART
Internal diagnostics: i Total energy is monitored at each displacement; warning printed
if ever lower than central-point value i If total energy change at any displacement is less than
100*TOLDEE, warning is printed
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Elastic constants for ZnO
Present Prior Crystal Expt
c11 246.5 246.2 209.0
c12 127.7 127.7 120.0
c13 104.6 106.2 104.4
c33 242.0 242.5 216.0
c44 56.3 56.0 44.2
c66 59.4 59.3 44.5
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Example: LiH Input file: …. ELASTCON NUMDERIV 3 STEPSIZE 0.005 PRINT ENDELAST ….
Banner: ELASTCON OPTION, W.F. Perger, B. Civalleri, R. Dovesi, July 2007 ***IT IS ASSUMED THAT THE INPUT IS FROM AN OPTIMIZED RUN*** Convergence criterion for energy, TOLDEE = 1.0000E-08 Convergence criterion for RMS gradient, TOLDEG = 6.0000E-05 Convergence criterion for RMS displacement, TOLDEX = 1.2000E-04 Magnitude of stepsize for adimensional strains = 5.0000E-03 Number of points (total) in numerical 2nd derivative= 3
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Output for LiH
Energies = DISPL Energy Delta E -0.0050 -8.0617947883E+00 1.1632111830E-05 0.0000 -8.0618064204E+00 0.0000000000E+00 0.0050 -8.0617947883E+00 1.1632111786E-05
slopes, order = 2 | -1.51546E-12 -6.93889E-18 -2.08167E-17 | | -5.55311E-15 -1.92348E-12 1.19667E-01 | | -1.11042E-14 1.19667E-01 -1.91237E-12 |
c44 = 58.252671585 GPa
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Output for LiH
Elastic constants for CUBIC case, in GPa
| 69.22 17.08 17.08 . . . | | 69.22 17.08 . . . | | 69.22 . . . | | 58.25 . . | | 58.25 . | | 58.25 |
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Elastic moduli (compliance tensor), in pPa-1
| 1601.2 -317.0 -317.0 0.0 0.0 0.0 | | 1601.2 -317.0 0.0 0.0 0.0 | | 1601.2 0.0 0.0 0.0 | | 1716.7 0.0 0.0 | | 1716.7 0.0 | | 1716.7 |
Eigenvalues = 1 2 3 4 5 6 52.1344 52.1344 58.2527 58.2527 58.2527 103.3871
Eigenvectors = 1 2 3 4 5 6 -0.7071 -0.4082 0.0000 0.0000 0.0000 0.5774 0.7071 -0.4082 0.0000 0.0000 0.0000 0.5774 0.0000 0.8165 0.0000 0.0000 0.0000 0.5774 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 Bulk modulus = 34.46235GPa
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Elastic constants (in GPa) for LiH and ZnO, as function of displacement (HF). Mean and standard deviation include 0.0025 to 0.03. Range of stability given by italics.
LiH ZnO δ C11 C12 C44 C11 C12 C13 C33 C44 C66
0.0005 69.281 17.137 58.247 294.41 129.47 98.75 240.00 56.30 59.97
0.001 69.268 17.143 58.246 247.87 129.74 100.12 240.60 56.57 59.07
0.0025 69.143 17.023 58.246 246.62 128.30 102.98 241.34 56.42 59.16
0.005 69.219 17.084 58.253 246.41 127.20 104.40 241.32 56.40 59.6
0.01 69.309 17.115 58.269 246.73 126.66 104.60 241.26 56.48 60.04
0.02 69.566 17.132 58.339 246.09 126.92 104.74 240.70 56.53 59.59
0.03 69.974 17.139 58.457 245.75 126.88 104.77 239.80 56.62 59.44
µ 69.442 17.099 58.313 246.32 127.19 104.30 240.88 56.49 59.57
σ 0.3373 0.0473 0.0887 0.4012 0.6485 0.7511 0.6611 0.0889 0.3193
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Elastic constants (in GPa) of LiH and ZnO (using HF) as a function of NUMDERIV (Np). Suggested default is given in italics.
LiH ZnO Np C11 C12 C44 C11 C12 C13 C33 C44 C66 2 67.637 17.121 58.253 244.99 126.33 103.03 240.91 56.40 59.33
3 69.219 17.084 58.253 246.41 127.20 104.40 241.32 56.40 59.61
5 69.188 17.074 58.247 246.31 127.38 104.34 241.34 56.38 59.46
7 69.184 17.070 58.247 246.24 127.48 104.31 241.33 56.37 59.38
9 69.182 17.068 58.248 246.20 127.54 104.28 241.33 56.36 59.33
11 69.181 17.066 58.248 246.20 126.84 104.69 241.33 56.36 59.68
µ 69.191 17.072 58.269 246.30 127.15 104.48 241.25 56.40 59.57
σ 0.016 0.0071 0.0448 0.089 0.282 0.166 0.007 0.017 0.137
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Elastic constants (in GPa) and the order of the fit (HF)
fit order NUMDERIV
3 5 7 9
LiH 2 3 4 5 6 7 8
69.22 69.29 69.29 69.19
69.37 69.37 69.21 69.21 69.18
69.48 69.48 69.22 69.22 69.20 69.20 69.18
ZnO 2 3 4 5 6 7 8
246.41 246.67 246.67 246.31
246.59 246.59 246.68 246.68 246.24
246.32 246.32 246.90 246.90 246.55 246.55 246.20
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Effect of DFT integration grid on elastic constants (in GPa) for α-quartz at B3LYP level. M.D. is the mean, maximum (Max) and minimum (Min) deviations referred to largest adopted grid.
Grid (55,434)p (75,434)p (75,974)p (75,1454)p (99,1454)p (99,2030)p (125,2702)p
NP 9773 21728 44827 68058 88546 123963 213069
a 4.9699 4.9700 4.9715 4.9716 4.9715 4.9716 4.9716
c 5.4774 5.4774 5.4755 5.4756 5.4754 5.4755 5.4755
C11 93.64 92.97 93.88 9510 94.85 94.73 94.75
C12 10.99 10.88 10.17 10.50 10.44 10.58 10.58
C13 19.71 18.71 18.29 19.67 19.55 19.48 19.52
C14 9.40 9.53 9.59 8.97 8.99 9.03 9.01
C33 116.35 116.17 115.35 116.13 115.97 115.70 115.86
C44 61.95 61.34 61.23 61.45 61.36 61.34 61.33
C66 41.32 41.05 41.85 42.30 42.21 42.08 42.09
M.D. 0.57 0.68 0.56 0.17 0.08 0.04
Max 0.62 0.52 0.58 0.35 0.12 0.02
Min -1.11 -1.78 -1.23 -0.08 -0.14 -0.16
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Second-order elastic constants (in GPa) for α-SiO2, using the HF potential, with different tolerances: Tn(2n), where n=6…9. NUMDERIV =3, δ =0.01 and the order of the fit was NUMDERIV -1.
Tol T6(12) T7(14) T8(16) T9(18) a 4.9491 4.9562 4.9553 4.9551 c 5.4415 5.4479 5.4469 5.4476
C11 109.907 115.347 114.475 114.354 C12 5.540 7.175 7.057 7.376 C13 17.273 21.410 20.990 20.905 C14 11.945 11.214 11.276 11.381 C33 127.194 131.956 131.233 131.147 C44 69.627 70.873 70.645 70.145 C66 52.183 54.086 53.709 53.489
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Elastic constants (in GPa) for different displacements and number of points in the fit (HF)
δ NUMDERIV 3
0.005 5 7 3
0.01 5 7
MgO C11 C12 C44
336.87 105.64 186.69
336.69 105.64 186.69
336.68 105.63 186.70
337.40 105.63 186.69
336.72 105.66 186.62
336.73 105.67 186.62
LiF C11 C12 C44
127.52 50.25 76.14
127.45 50.25 76.14
127.45 50.25 76.14
127.73 50.25 76.16
127.44 50.25 76.11
127.44 50.25 76.11
NaCl C11 C12 C44
45.50 11.19 15.60
45.47 11.19 15.60
45.47 11.19 15.60
45.58 11.19 15.59
45.47 11.19 15.58
45.47 11.19 15.58
KBr C11 C12 C44
35.41 4.688 6.688
35.38 4.688 6.694
35.38 4.687 6.697
35.49 4.688 6.671
35.37 4.690 6.667
35.37 4.690 6.669
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Elastic constants (in GPa) for ZnO and α-SiO2 using the HF Hamiltonian and sets of NUMDERIV and δ.
δ 0.005 0.01
NUMDERIV 3 5 7 3 5 7
ZnO C11 246.41 246.31 246.24 246.73 246.22 245.92
C12 127.20 127.38 127.48 126.66 126.87 126.88
C13 104.40 104.34 104.31 104.60 104.71 104.75
C33 241.32 241.34 241.33 241.26 241.45 241.45
C44 56.40 56.38 56.47 56.48 56.47 56.47
C66 59.61 59.46 59.38 60.04 59.68 59.52
α-SiO2 C11 109.55 109.80 110.08 110.22 110.81 111.75
C12 5.51 5.46 5.50 5.96 5.92 5.86
C13 17.10 17.21 17.33 17.40 17.61 17.93
C14 11.99 11.98 11.98 11.93 11.96 11.96
C33 127.17 127.09 127.07 126.97 126.84 126.81
C44 69.88 69.95 69.96 69.67 69.83 69.82
C66 52.02 52.17 52.29 52.13 52.45 52.95
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Elastic constants (in GPa) for α-SiO2 and TiO2 (rutile) using B3LYP potential for sets of NUMDERIV and δ.
δ 0.005 0.01
NUMDERIV 3 5 7 3 5 7
α-SiO2 C11 94.85 95.40 95.59 95.54 95.89 96.50
C12 10.44 11.10 11.25 11.27 11.32 11.32
C13 19.55 19.95 20.05 20.05 20.18 20.44
C14 9.00 8.98 8.97 9.05 9.06 9.05
C33 115.97 115.98 116.02 115.94 115.78 115.77
C44 61.36 61.40 61.41 61.24 61.35 61.36
C66 42.21 42.15 42.17 42.13 42.29 42.59
TiO2 C11 279.48 279.46 279.40 279.52 280.01 280.05
C12 175.99 176.05 176.08 175.82 175.81 175.77
C13 161.19 161.23 161.22 161.06 161.30 161.29
C33 507.04 506.95 506.91 507.31 507.17 507.16
C44 125.09 125.11 125.12 125.03 125.08 125.08
C66 229.91 229.88 229.88 230.01 229.88 229.88
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Elastic constants (in GPa) as a function of displacement and number of points for more complicated (orthorhombic) systems
δ 0.005 0.01
NUMDERIV 3 5 7 3 5 7
NaNO2
C11 C22 C33 C12 C13 C23 C44 C55 C66
46.32 76.11 99.89 18.76 31.60 24.27 17.92 10.35 9.31
46.37 76.07 99.76 18.59 31.90 24.74 17.88 10.43 9.40
46.37 76.07 99.76 18.59 31.90 24.74 17.88 10.43 9.40
46.17 76.23
100.27 19.29 30.70 22.87 18.06 10.12 9.04
46.12 76.07
100.14 19.25 30.56 22.79 18.10 10.05 9.06
46.12 76.09
100.16 19.23 30.51 22.76 18.11 10.08 9.09
MgSiO3
C11 C22 C33 C12 C13 C23 C44 C55 C66
437.09 491.36 415.27 115.95 122.19 132.13 180.70 160.41 133.84
437.12 491.40 415.17 115.96 122.16 132.10 180.68 160.42 133.76
437.12 491.40 415.17 115.96 122.16 132.10 180.68 160.42 133.76
437.01 491.22 415.55 115.92 122.28 132.22 180.77 160.35 134.09
437.16 491.28 415.13 115.91 122.23 132.17 180.63 160.34 133.84
437.17 491.26 415.13 115.92 122.23 132.20 180.62 160.31 133.83
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Test of ZnO with symmetry removed. Elastic constants in GPa.
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Modeling the deformation of organic molecular crystals Effects of compression, via hydrostatic pressure and uniaxial strain:
i on mechano-chemistry? i on shock initiation-to-detonation transition?
OMC’s often anisotropic, e.g. orthorhombic RDX Computational difficulties with molecular crystals; often many atoms
per unit cell and low symmetry Accurate calculation of organic molecular crystal (OMC) properties
requires consideration of 3 structural levels i Electronic: responsible for optical properties i Intra-molecular: relatively strong binding i Inter-molecular: relatively weak, van der Waals interaction.
Changes in vibrational frequencies as a function of deformation carry information on inter-molecular potential
Studies on PE predict pressures which are too high if a rigid-molecule approximation is taken: “relaxation” of atomic positions is required
BOTH intramolecular and intermolecular potentials must be accurately accounted for
This requires an n-parameter optimization procedure (n=168 for RDX)
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Inter-molecular interaction responsible for mechanical and elastic properties; characterization of the inter-molecular potential is valuable for understanding the shock initiation-to-detonation transition.
Accurate description of inter-molecular interaction necessary for ambient volume/density
Understanding the intra-molecular potential is required for crystal deformations which invalidate the rigid-molecule approximation.
Atomistic calculations; require quantum mechanics (non-relativistic) and complexity of exchange-correlation theory
The intermolecular potential, often van der Waals, yields ~ 1/r6 and commonly-used exchange-correlation density-functional theory technique is not well-suited
OMC’s can have many electrons in the unit cell: e.g. RDX has 912.
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First molecular crystal example: PE
Van der Waal’s bonded between layers Hydrogen bonded within a layer
ab plane ac plane
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Energy-volume curve for PE, HF/6-31G
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Variation of internal geometry with pressure in PE, to 4GPa
Hydrogen Oxygen Carbon
Note the relatively small changes in bond lengths and angles!
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HF 631G
HF 631G**
B3LYP 631G**
Expt*
C11 56.4 36.8 50.1 40.5(1)
C12 39.3 27.8 37.0 26.6(8) C13 3.73 3.43 4.49 10.5(2.5) C33 6.68 3.10 12.7 13.9(1) C44 3.52 3.10 4.16 2.74(1) C16 0.13 1.40 2.92 0.0(0.5) C66 15.9 10.3 15.8 2.5(1)
Elastic constants (in GPa) for Penta-erythritol (PE). Difference from Expt* in parentheses.
*Nomura, et al, Jpn. J. Appl. Phys. 11, 304, 1972
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Elastic constants for Lithium Azide, LiN3
Monoclinic, space group is C2/m Expect c33 to be larger than c11 or c22 (nitrogens in blue)
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B3LYP DZP
PWGGA DZP
Expt
a 5.711 5.719 5.627
b 3.275 3.261 3.319 c 5.229 5.220 4.979 β 113.9 113.5 107.4
Vol 89.41 88.47 88.73
Lattice constants and equilibrium volume for Lithium Azide.
W. F. Perger, Int. J. Quant. Chem, in press Expt: Choi, in Fair and Walker, “Energetic Materials”, vol 1, New York, 1977.
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B3LYP 6-311**
B3LYP DZP
PWGGA 6-311**
PWGGA DZP
C11 29.8 29.1 30.2 28.5
C22 29.8 30.4 31.0 30.5 C12 12.6 12.9 11.5 12.3 C13 20.4 23.1 19.9 23.2 C33 180.8 181.1 183.2 180.0 C44 4.10 6.51 3.83 4.92 C55 4.82 4.60 4.75 4.87 C66 2.00 0.28 11.35 10.04 B 21.6 22.7 21.1 24.2
Elastic constants and bulk modulus (in GPa) for Lithium Azide.
W. F. Perger, Int. J. Quant. Chem, in press, B~34GPa using Murnaghan equation
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Elastic constants for Lead Azide, Pb(N3)2
Orthorhombic, space group is Pnma 12 molecules in unit cell Pseudo-potential was used for Pb, reducing number of
electrons in the unit cell to 552
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Lead Azide, Pb(N3)2, views in y-z, x-z, and x-y planes. Nitrogen in blue, Pb in grey.
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HF PWGGA Expt
a 6.517 6.432 6.63
b 16.24 15.97 16.25 c 11.00 11.02 11.31
Vol 1164.1 1131.5 1218.5
Lattice constants and equilibrium volume for Lead Azide.
W. F. Perger, Int. J. Quant. Chem, in press Expt: Choi and Boutin, Acta Cryst. B25, 982 (1969)
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HF B3LYP PWGGA
C11 74.9 104.9 103.5
C22 98.2 113.0 126.0 C12 31.6 47.8 46.19 C13 26.1 38.25 37.3 C33 49.3 48.4 48.0 C44 15.5 14.2 17.5 C55 22.0 26.1 28.6 C66 4.60 23.4 25.1 B 40.27 46.06 45.81
Elastic constants and bulk modulus (in GPa) for Lead Azide.
W. F. Perger, Int. J. Quant. Chem, in press, B~50GPa Using Murnaghan equation
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Top view along [001] Side view along [100]
Tetragonal lattice, space group , 2 molecules per unit cell Intermolecular: weak Van der Waals interaction
Intramolecular: strong covalent bonding
Next example: PETN
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Ambient pressure lattice constants and volume for PETN
Exper. Theory
Cady & Larson
Gan, et al
Brand CRYSTAL03 CASTEP
HF-STO B3LYP PW91 X3LYP HF LDA PW91
a(Å) 9.383 9.425 9.2546 9.439 9.431 9.568 9.677 8.887 9.868
c(Å) 6.711 6.758 6.6636 6.762 6.746 6.881 7.048 6.392 6.925
V(Å3) 590.8 600.3 570.72 602.4 599.0 630.6 660.0 504.9 674.3
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PETN Elastic Constants (in GPa) Theory and Experiment
HF 631G**
B3LYP 631G**
HF STO*
Expt
C11 11.23 18.95 C12 4.53 7.71 C13 4.09 9.01 7.06 7.99 C33 8.35 12.8 13.12 12.2 C44 2.64 4.98 C66 3.92 4.51
*Brand, Chem. Phys. Lett., 418, 428, 2005
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Final example: RDX (work done with WSU)
RDX (C3H6N6O6), organic molecular crystal
Orthorhombic, space group Pbca, 8 molecules per unit cell (168 atoms)
Molecular Cs point group, AAE configuration (A: axial, E: equatorial)
Intramolecular: covalent bonds; Intermolecular: van der Waals bonding
axial
axial
equatorial�
equatorial�
Objective: examine compression-induced changes to lattice and molecular geometry; role of non-hydrostaticity
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RDX crystal at ambient conditions
a (Å) b (Å) c (Å) V (Å3) Experiment (Choi, 1972) 13.182 11.574 10.709 1633.86
Experiment (Olinger, 1978) 13.200 11.600 10.720 1641.45
B3LYP (Crystal06) 13.53 11.78 11.09 1767.39
ReaxFF Strachan, et al, Phys. Rev. Lett., vol 91, 09301-1 (2003)
13.78 12.03 10.96 1816.8
GGA (CASTEP) 13.490 11.643 11.105 1744.20
CASTEP GGA calculations overestimate the cell volume by ~6.3%
CRYSTAL06 B3LYP overestimates the cell volume by ~7.7%
Experimental intramolecular geometry reasonably reproduced, e.g., average
deviation of bond lengths: 0.015 Å ( ~1.2%)
Computed lattice energy is 0.5 eV per molecule (experiment: ~1.35 eV)
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Changes in intramolecular geometry
N-N bond is most compressible Small deformation of C3N3 six-member ring
(change of bond length <0.01 Å, bond angle <1°) Angle of NO2 group with respect to C2N plane
changes slightly (~ 1° to 3°)
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P-V relation under hydrostatic loading
Discrepancy between theory and experiment for unit cell volume decreases from ~6.3% at 0 GPa to ~2.7% at 3.65 GPa
CASTEP GGA calculation underestimated the overall stiffness of RDX crystal
B0 (GPa) B0’
Experiment 12.61 6.95
GGA 8.04 7.97
Fit P-V response using Murnaghan equation
Experiment from Olinger et al. (1978) �
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Our GGA calculations describe the b
axis reasonably well, results for
other two axes not as good.
c axis is most compressible,
consistent with experiment
Experimental data from Olinger et al. (1978) �
Change of lattice constants under hydrostatic loading
a (%)
b (%)
c (%)
Experiment 4.02 5.78 6.44
GGA 5.67 5.55 7.38
Reduction of lattice constants of
RDX crystal up to 3.65 GPa
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Concluding Remarks and Future Work Defaults must always be watched, but calculation of
elastic constants for various systems appears reliable and robust
Explore different DFT functionals and dispersion correction (Civalleri, et al, CrystEngComm, 2008)
Use better basis sets Third-order elastic constants?
Recommended