vlab development team university of minnesota indiana university florida state louisiana state...
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VLab development team
UNIVERSITY OF MINNESOTAIndiana University
Florida StateLouisiana State University
Thermoelastic Properties within VLabThermoelastic Properties within VLab
Method--- Density Functional Theory
(Hohenberg and Kohn, 1964)
--- LDA and GGA(Ceperley and Alder, 1985)
(Perdue at al.,1996)
--- Plane wave basis – pseudopotential (Troullier and Martins, 1991
von Bar and Car)
--- Variable Cell Shape Molecular Dynamics (Wentzcovitch, 1991)
--- Density Functional Perturbation Theory for phonons +QHA (Baroni et al., 1987)
--- Quantum ESPRESSO package (DEMOCRITOS)
Thermodynamic Method
∑∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡−−++=
qj B
qjB
qj
qj
Tk
VTk
VVUTVF
)(exp1ln
2
)()(),(
ωω hh
• VDoS and F(T,V) within the QHA
PVTSFG +−=TV
FP ⎥⎦
⎤⎢⎣⎡∂∂
−=VT
FS ⎥⎦
⎤⎢⎣⎡∂∂
−=
N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
IMPORTANT: crystal structure and phonon frequencies depend on volume alone!!….
Typical Computational Experiment
Damped dynamics (Wentzcovitch, 1991)
)(~ PI−Πε&&),(~ int ε&&&& rffr +
P = 150 GPa
(Wentzcovitch, Martins, and Price, PRL 1993)
Summation (integration) over the Brillouin Zone
In general:
1) Compute and diagonalize the dynamical matrix at few ’s (CPU intensive procedure)
2) Extract “force constants”
3) Recompute dynamical matrices at several points using those force constants
4) Summation over tetrahedral volume elements is very accurate for DoSs
qr
1ln 1
2
i
Bk Ti B
i i
F U k T eω
ω−⎛ ⎞
= + + −⎜ ⎟⎜ ⎟⎝ ⎠
∑ ∑h
h
3,
3
( )
( )q n
ni
i
d q f
fd q
ωω
⎛ ⎞⎜ ⎟⎝ ⎠→∑∫
∑∫
r XΓ
MEx: square BZ
,i q n→r
( ) ( )i j ji j i
f w fω ω<<
→∑ ∑
iw is the “multiplicity” of a pointdetermined by symmetry
qr
Phonon dispersions in MgO
Exp: Sangster et al. 1970
(Karki, Wentzcovitch, de Gironcoli and Baroni, PRB 61, 8793, 2000)
-
cubic2 atoms/cell
Zero Point Motion Effect
Volume (Å3)
F (Ry)
MgO
Static 300K Exp (Fei 1999)V (Å3) 18.5 18.8 18.7K (GPa) 169 159 160K´ 4.18 4.30 4.15K´´(GPa-1) -0.025 -0.030
1ln 1
2
i
Bk Ti B
i i
F U k T eω
ω−⎛ ⎞
= + + −⎜ ⎟⎜ ⎟⎝ ⎠
∑ ∑h
h
ZP
Karki, Wentzcovitch,de Gironcoli, Baroni, Science 1999
equilibrium structure
εkl
re-optimize
Adiabatic thermoelastic constant tensor CijS(T,P)
Pji
Tij
GPTc
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂∂
=εε
2
),(
V
jiTij
Sij C
VTPTcPTc
λλ+= ),(),(
Tii
S
ελ
∂∂
=
11x8x6=528 runs for MgO11x6x16=1056 runs for MgSiO3-pv
cij
(Wentzcovitch, Karki, Cococciono, de Gironcoli, Phys. Rev. Lett. 2004)
300 K1000K2000K3000 K4000 K
(Oganov et al,2001)
Cij(P,T)
MgSiO3-pv
Today
• Demo (real run) to fit the allocated time
• Regatta, Altix, and Macs at MSI
• Thermodynamic properties of MgO
• Parameter sampling 11 pressures, 4x4x4 q-grid (8 q in the IBZ)
• Plots of thermodynamic properties
MgSiO3-perovskite and MgO
ρ (gr/cm -3)
V (A 3)
K T (GPa)
d K T /dP
d K T
2/dP 2 (GPa -1)
d K T/dT (Gpa K -1)
α 10 -5 K-1
3.580
18.80
159
4.30
-0.030
-0.014
3.12
Calc.
pc
3.601
18.69
160
4.15
~
-0.0145
3.13
Exp.
pc
4.210
164.1
247
4.0
-0.016
-0.031
2.1
Calc.
Pv
4.247
162.3
246 | 266
3.7 | 4.0
~
-0.02 | -0.07
1.7 | 2.2
Exp.
Pv
Exp.: [Ross & Hazen, 1989; Mao et al., 1991; Wang et al., 1994; Funamori et al., 1996; Chopelas, 1996; Gillet et al., 2000; Fiquet et al., 2000]
4.8
(256)