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

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