Thermodynamics from lattice dynamics with DFT

EPJ Web of Conferences

14

, 02001 (2011)

DOI: 10.1051/epjconf/20111402001

© Owned by the authors, published by EDP Sciences, 2011

“ Fundamentals of Thermodynamic Modelling

of Materials ”

November 15-19, 2010

INSTN – CEA Saclay, France

Organized by

Bo SUNDMAN

[email protected]

Constantin MEIS

[email protected]

PROFESSOR & TOPIC

Stefano BARONI

SISSA, Trieste, Italy

Thermodynamics from

lattice dynamics with DFT

$) *4<0$+%$"&(*0(+%$*%(4

%('0) <(#%$ "0"+%$)

*$%(%$

" "$%

#!

!#

Sunday, November 14, 2010

*'0) <(#%$ &&(%3 #+%$

F

(

_{V, T, λ}

) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

−k

B

T

log

e

−

vλ

(R)

kBT

_dR

−k

B

T

log

n

*'0) <(#%$ &&(%3 #+%$

U

(

_{S, V, λ}

)

=

_F

+

_{T S}

G

(

_{T, p, λ}

)

=

_F

+

_pV

. . .

F

(

_{V, T, λ}

) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

−k

B

T

log

e

−

vλ

(R)

kBT

_dR

−k

B

T

log

n

e

−

EnkBT(V,λ)

Sunday, November 14, 2010

*'0) <(#%$ &&(%3 #+%$

U

(

_{S, V, λ}

)

=

_F

+

_{T S}

G

(

_{T, p, λ}

)

=

_F

+

_pV

. . .

C

V

=

−T

∂

2

F

∂T

2

V

C

p

=

−T

∂

2

G

∂T

2

p

B

T

=

V

∂

2

F

∂V

2

T

B

S

=

V

∂

2

U

∂V

2

S

. . .

F

(

_{V, T, λ}

) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

−k

B

T

log

e

−

vλ

(R)

kBT

_dR

−k

B

T

log

n

*'0) <(#%$ &&(%3 #+%$

U

(

_{S, V, λ}

)

=

_F

+

_{T S}

G

(

_{T, p, λ}

)

=

_F

+

_pV

. . .

C

V

=

−T

∂

2

F

∂T

2

V

C

p

=

−T

∂

2

G

∂T

2

p

B

T

=

V

∂

2

F

∂V

2

T

B

S

=

V

∂

2

U

∂V

2

S

. . .

F

(

_{V, T, λ}

) =

_U

₀

(

_{V, λ}

) +

1

2

q

ν

ω

(

q, ν

|

_{V, λ}

) +

_k

_B

_T

q

ν

log

1

−

e

−

ω(qkBT,ν|V,λ)

F

(

_{V, T, λ}

) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

−k

B

T

log

e

−

vλ

(R)

kBT

_dR

−k

B

T

log

n

e

−

EnkBT(V,λ)

Sunday, November 14, 2010

24 $ +%

24 $ +%

0$)"

((&(%0 "

(("1$*

$0(*

)

) $+ $) *)*#)(%#* " *4*%3*(&%"*%0(0$()*$ $

(%#) #&"#%")*%1(4%#&"3&4) "2%("

$%3*(&%"+%$ )&%)) " %0(!$%2"%*#%") )

Sunday, November 14, 2010

24 $ +%

0$)"

((&(%0 "

(("1$*

$0(*

)

) $+ $) *)*#)(%#* " *4*%3*(&%"*%0(0$()*$ $

(%#) #&"#%")*%1(4%#&"3&4) "2%("

$%3*(&%"+%$ )&%)) " %0(!$%2"%*#%") )

0(

4

)

)# #& ( "

24 $ +%

0$)"

((&(%0 "

(("1$*

$0(*

)

) $+ $) *)*#)(%#* " *4*%3*(&%"*%0(0$()*$ $

(%#) #&"#%")*%1(4%#&"3&4) "2%("

$%3*(&%"+%$ )&%)) " %0(!$%2"%*#%") )

#0#%()%2$*3*(&%"+%$ #&" )

(#+$)%&4) "%$ +%$)

=*#&(*0(8&())0(8*%# %%( $+%$:::>

0(

4

)

)# #& ( "

#%")

Sunday, November 14, 2010

$) *40$+%$"*%(4

E

(

R

) = min

{

ψ

}

$) *40$+%$"*%(4

E

[

{

_ψ

}

_,

R

] =

−

2

2

_m

v

ψ

∗

v

(

r

)

∂

2

_ψ

v

(

r

)

∂

r

2

dr

+

V

(

r

_,

R

)

_ρ

(

r

)

_d

r

+

e

2

2

ρ

(

r

)

_ρ

(

r

)

|

r

−

r

_|

d

r

d

r

+

E

xc

[

ρ

]

E

(

R

) = min

{

ψ

}

E

[

{

_ψ

}

_,

R

]

ρ

(

r

) =

v

|

ψ

v

(

r

)

|

2

ψ

∗

u

(

r

)

ψ

v

(

r

)

d

r

=

δ

uv

Sunday, November 14, 2010

$) *40$+%$"*%(4

E

[

{

_ψ

}

_,

R

] =

−

2

2

_m

v

ψ

∗

_v

(

r

)

∂

2

_ψ

v

(

r

)

∂

r

2

dr

+

V

(

r

_,

R

)

_ρ

(

r

)

_d

r

+

e

2

2

ρ

(

r

)

_ρ

(

r

)

|

r

−

r

_|

d

r

d

r

+

E

xc

[

ρ

]

E

(

R

) = min

{

ψ

}

E

[

{

_ψ

}

_,

R

]

ρ

(

r

) =

v

|

ψ

v

(

r

)

|

2

ψ

∗

_u

(

r

)

_ψ

_v

(

r

)

_d

r

=

_δ

_uv

δE

KS

δψ

∗

v

(

r

)

=

uv

Λ

_vu

_ψ

_u

(

r

)

−

2

2

_m

∇

2

₊

_v

KS

[

ρ

](

r

)

$) *40$+%$"*%(4

E

[

{

_ψ

}

_,

R

] =

−

2

2

_m

v

ψ

∗

v

(

r

)

∂

2

_ψ

v

(

r

)

∂

r

2

dr

+

V

(

r

_,

R

)

_ρ

(

r

)

_d

r

+

e

2

2

ρ

(

r

)

_ρ

(

r

)

|

r

−

r

_|

d

r

d

r

+

E

xc

[

ρ

]

E

(

R

) = min

{

ψ

}

E

[

{

_ψ

}

_,

R

]

ρ

(

r

) =

v

|

ψ

v

(

r

)

|

2

ψ

∗

u

(

r

)

ψ

v

(

r

)

d

r

=

δ

uv

δE

KS

δψ

∗

v

(

r

)

=

uv

Λ

_vu

_ψ

_u

(

r

)

−

2

2

_m

∇

2

₊

_v

KS

[

ρ

](

r

)

ψ

v

(

r

) =

v

ψ

v

(

r

)

v

KS

[

ρ

](

r

) =

V

(

r

,

R

) +

e

2

ρ

(

r

)

|

r

−

r

_|

d

r

+

v

XC

[

ρ

](

r

)

Sunday, November 14, 2010

".4$# )(%#&(*0(+%$*%(4

R’

R

V

(

r

)

=

V

₀

(

r

)

+

R

u

(

R

)

·

∂v

(

r

−

R

)

∂

R

E

=

E

₀

+

1

2

R

,

R

u

(

R

)

·

∂

2

_E

".4$# )(%#&(*0(+%$*%(4

u(R)

u(R’)

R’

R

V

(

r

)

=

V

₀

(

r

)

+

R

u

(

R

)

·

∂v

(

r

−

R

)

∂

R

E

=

E

₀

+

1

2

R

,

R

u

(

R

)

·

∂

2

_E

∂

u

(

R

)

∂

u

(

R

)

·

u

(

R

₎

+

· · ·

Sunday, November 14, 2010

".4$# )(%#&(*0(+%$*%(4

u(R)

u(R’)

R’

R

V

(

r

)

=

V

₀

(

r

)

+

R

u

(

R

)

·

∂v

(

r

−

R

)

∂

R

E

=

E

₀

+

1

2

R

,

R

u

(

R

)

·

∂

2

_E

∂

u

(

R

)

∂

u

(

R

)

·

u

(

R

₎

+

· · ·

det

∂

2

_E

∂u

(

R

)

∂u

(

R

)

−

ω

2

_M

₍

_R

₎

_δR

,

R

V

(

r

) =

_V

₀

(

r

) +

i

u

i

V

i

(

r

)

$) *4<0$+%$"&(*0(+%$*%(4

Sunday, November 14, 2010

V

(

r

) =

_V

₀

(

r

) +

i

u

i

V

_i

(

r

)

$) *4<0$+%$"&(*0(+%$*%(4

E

(u) = min

n

F

[

n

] +

V

u

(r)

n

(r)

V

(

r

) =

V

0

(

r

) +

i

u

i

V

i

(

r

)

$) *4<0$+%$"&(*0(+%$*%(4

∂E

(u)

∂u

i

=

n

u

(r)

V

i

(r)

d

r

E

(u) = min

n

F

[

n

] +

V

u

(r)

n

(r)

n

(

r

)

d

r

=

N

Sunday, November 14, 2010

V

(

r

) =

V

0

(

r

) +

i

u

i

V

_i

(

r

)

$) *4<0$+%$"&(*0(+%$*%(4

∂E

(u)

∂u

i

=

n

u

(r)

V

_i

(r)

d

r

E

(u) = min

n

F

[

n

] +

V

u

(r)

n

(r)

n

(

r

)

d

r

=

N

∂

2

E

(

u

)

∂u

i

∂u

j

=

∂n

u

(

r

)

∂u

j

V

V

(

r

) =

V

0

(

r

) +

i

u

i

V

i

(

r

)

$) *4<0$+%$"&(*0(+%$*%(4

∂E

(u)

∂u

i

=

n

u

(r)

V

i

(r)

d

r

E

(u) = min

n

F

[

n

] +

V

u

(r)

n

(r)

n

(

r

)

d

r

=

N

∂

2

E

(

u

)

∂ui

∂uj

=

∂n

u

(

r

)

∂uj

V

i

(

r

)

d

r

" $(()&%$) )$7

Sunday, November 14, 2010

"0"+$*()&%$)

n

(

r

) =

v

|

φ

v

(

r

)

|

2

n

(

r

) = 2Re

v

φ

_v

=

c

φ

◦

_c

φ

◦

c

|

V

|

φ

◦

v

◦

v

−

◦

c

n

(

r

) = 2Re

v

φ

◦∗

_v

(

r

)

φ

_v

(

r

)

= 2Re

cv

ρ

_vc

φ

◦∗

_v

(

r

)

φ

◦

_c

(

r

)

"0"+$*()&%$)

n

(

r

) =

v

|

φ

v

(

r

)

|

2

n

(

r

) = 2Re

v

φ

◦∗

_v

(

r

)

φ

_v

(

r

)

Sunday, November 14, 2010

φ

_v

=

c

φ

◦

_c

φ

◦

c

|

V

|

φ

◦

v

◦

v

−

◦

c

n

₍

_r

_{) = 2Re}

v

φ

◦∗

v

(

r

)

φ

v

(

r

)

= 2Re

cv

ρ

vc

φ

◦∗

v

(

r

)

φ

◦

c

(

r

)

"0"+$*()&%$)

n

(

r

) =

v

|

φ

v

(

r

)

|

2

n

₍

_r

_{) = 2Re}

v

φ

◦∗

v

(

r

)

φ

v

(

r

)

"0"+$*()&%$)

n

(

r

) = 2Re

v

φ

◦∗

_v

(

r

)

φ

_v

(

r

)

(

H

◦

−

◦

_v

)

φ

_v

=

−

P

c

V

φ

◦

_v

Sunday, November 14, 2010

V

0

(

r

)

n

(

r

)

9*'0+%$)

−

Δ +

V

SCF

(

r

)

φ

v

(

r

) =

v

φ

v

(

r

)

n

(

r

) =

v

<E

F

|

φ

v

(

r

)

|

2

V

SCF

(

r

) =

V

0

(

r

) +

_n

₍

_r

₎

V

0

(

r

)

n

(

r

)

9*'0+%$)

V

(

r

)

n

(

r

)

V

SCF

(

r

) =

V

(

r

) +

_n

₍

_r

₎

|

r

−

r

_|

dr

+

μ

xc

(

r

)

n

₍

_r

_{) = 2 Re}

v

<E

F

φ

∗

v

(

r

)

φ

v

(

r

)

−

Δ +

V

SCF

(

r

)

φ

v

(

r

) =

v

φ

v

(

r

)

n

(

r

) =

v

<E

F

|

φ

v

(

r

)

|

2

V

SCF

(

r

) =

V

0

(

r

) +

_n

₍

_r

₎

|

r

−

r

_|

dr

+

μ

xc

(

r

)

−

Δ +

V

SCF

(

r

)

−

v

φ

v

(

r

) =

P

c

V

SCF

(

r

)

φ

v

(

r

)

7

9

""#337

"

9

'(

7

2'9

/

9

+9

!$

7

DKID=DLKJ>

Sunday, November 14, 2010

monochromatic perturbations

x

2

π

q

V

q

(

x

)

monochromatic perturbations

x

2

π

q

V

q

(

x

)

H

0

−

k

v

φ

_v

k

+

q

(r) =

−

P

c

V

q

φ

k

v

(r)

Sunday, November 14, 2010

monochromatic perturbations

n

q

(

r

) = e

i

q

·

r

v,

k

u

_v

k

∗

(

r

)

u

k

_v

+

q

(

r

)

x

2

π

q

V

q

(

x

)

monochromatic perturbations

n

q

(

r

) = e

i

q

·

r

v,

k

u

_v

k

∗

(

r

)

u

k

_v

+

q

(

r

)

V

q

(

r

) =

V

_ext

q

(

r

) +

e

2

|

r

−

r

_|

+

κ

xc

(

r

,

r

)

n

q

(

r

)

d

r

x

2

π

q

V

q

(

x

)

H

0

−

k

v

φ

_v

k

+

q

(r) =

−

P

c

V

q

φ

k

v

(r)

Sunday, November 14, 2010

&%$%$) $&%"(#*( ")

E

(u

,

E

) =

1

2

Mω

0

2

u

2

−

Ω

&%$%$) $&%"(#*( ")

E

(u

,

E

) =

1

2

Mω

0

2

u

2

−

Ω

8

π

∞

E

2

−

eZ

∗

u

·

E

F

≡ −

∂E

∂

u

=

−

Mω

0

2

u

+

Z

∗

E

D

≡ −

4

π

Ω

∂E

∂

E

=

4

π

Ω

Z

∗

u

+

∞

E

Sunday, November 14, 2010

&%$%$) $&%"(#*( ")

E

(u

,

E

) =

1

2

Mω

0

2

u

2

−

Ω

8

π

∞

E

2

−

eZ

∗

u

·

E

F

≡ −

∂E

∂

u

=

−

Mω

0

2

u

+

Z

∗

E

D

≡ −

4

π

Ω

∂E

∂

E

=

4

π

Ω

Z

∗

u

+

∞

E

&%$%$) $&%"(#*( ")

E

(u

,

E

) =

1

2

Mω

0

2

u

2

−

Ω

8

π

∞

E

2

−

eZ

∗

u

·

E

F

≡ −

∂E

∂

u

=

−

Mω

0

2

u

+

Z

∗

E

D

≡ −

4

π

Ω

∂E

∂

E

=

4

π

Ω

Z

∗

u

+

∞

E

rot

E

∼

i

q

×

E

= 0

u

⊥

q

⇒

E

= 0

=>

Sunday, November 14, 2010

&%$%$) $&%"(#*( ")

E

(u

,

E

) =

1

2

Mω

0

2

u

2

−

Ω

8

π

∞

E

2

−

eZ

∗

u

·

E

F

≡ −

∂E

∂

u

=

−

Mω

0

2

u

+

Z

∗

E

D

≡ −

4

π

Ω

∂E

∂

E

=

4

π

Ω

Z

∗

u

+

∞

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2

−

eZ

∗

u

·

E

F

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u

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2

u

+

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∗

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D

≡ −

4

π

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∂E

∂

E

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Z

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+

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E

rot

E

∼

i

q

×

E

= 0

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D

∼

i

q

·

D

= 0

u

⊥

q

⇒

E

= 0

=>

u

q

⇒

D

= 0

=>

Sunday, November 14, 2010

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(u

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E

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2

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u

2

−

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8

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E

2

−

eZ

∗

u

·

E

F

≡ −

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∂

u

=

−

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0

2

u

+

Z

∗

E

D

≡ −

4

π

Ω

∂E

∂

E

=

4

π

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Z

∗

u

+

∞

E

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E

∼

i

q

×

E

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D

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·

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u

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q

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u

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2

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eZ

∗

u

·

E

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T

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u

F

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4

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∗

M

Ω

∞

u

Sunday, November 14, 2010

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s

Z

t

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α

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R

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R

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D

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d

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q

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s

Z

t

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α

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β

q

2

Sunday, November 14, 2010

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Z

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Sunday, November 14, 2010

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First-Principles Calculation of Vibrational Raman Spectra in Large Systems:

Signature of Small Rings in Crystalline

SiO

₂

Michele Lazzeri and Francesco Mauri

P H Y S I C A L

R E V I E W

L E T T E R S

week ending 24 JANUARY 2003

V

OLUME

90, N

UMBER

3

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Experim. Theory

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200

400

600

800

1000

1200

Raman shift (cm

-1

₎

Coesite

Intensity

200 300 400 500 600 Raman shift (cm-1₎

Vitr.-SiO₂ Coesite

520 cm -1

D₁

490 cm -1

D₂

605 cm-1

Intensity

200 300 400 500 600 Raman shift (cm-1₎

Vitr.-SiO₂ ZSM-18 Zeolite

485 cm -1

D

1

490 cm -1

D₂

605 cm -1 615

cm-1

Sunday, November 14, 2010

)#&"(%($*&&" +%$)

Vibrational Recognition of Adsorption Sites for CO on

Platinum and Platinum

-

Ruthenium Surfaces

Ismaila Dabo,*,†_{Andrzej Wieckowski,}‡_{and Nicola Marzari}†

Published on Web 08/17/2007

11046 J. AM. CHEM. SOC.9VOL. 129, NO. 36, 2007

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Dissociation of MgSiO

₃

in the Cores of

Gas Giants and Terrestrial Exoplanets

Koichiro Umemoto,

1

_{Renata M. Wentzcovitch,}

1

_{* Philip B. Allen}

2

983

www.sciencemag.org SCIENCE VOL 311 17 FEBRUARY 2006

Sunday, November 14, 2010

)#&"(%($*&&" +%$)

Ab Initio

Description of High-Temperature Superconductivity in Dense Molecular Hydrogen

P. Cudazzo,

1

G. Profeta,

1

A. Sanna,

2,3

A. Floris,

3

A. Continenza,

1

S. Massidda,

2

and E. K. U. Gross

3 1_{CNISM - Dipartimento di Fisica, Universita` degli Studi dell’Aquila, Via Vetoio 10, I-67010 Coppito (L’Aquila) Italy}

2_{SLACS-INFM/CNR —Dipartimento di Fisica, Universita` degli Studi di Cagliari, I-09124 Monserrato (CA), Italy} 3_{Institut fu¨r Theoretische Physik, Freie Universita¨t Berlin, Arnimallee 14, D-14195 Berlin, Germany}

(Received 7 December 2007; published 23 June 2008; corrected 27 June 2008)

PRL

100,

257001 (2008)

P H Y S I C A L

R E V I E W

L E T T E R S

27 JUNE 2008week ending

0 1200 2400 3600

ω (

cm

-1)

Z Γ Σ Y Γ S R Z T

0 1000 2000 3000 4000

ω (cm-1

) 0

0.2 0.4 0.6 0.8

α

2F(

ω) 420 440P ( GPa )460 1

1.5 2 2.5

λ

10 15 20

(meV) ₃₀₀

)#&"(%($*&&" +%$)

Ab Initio

Description of High-Temperature Superconductivity in Dense Molecular Hydrogen

P. Cudazzo,

1

G. Profeta,

1

A. Sanna,

2,3

A. Floris,

3

A. Continenza,

1

S. Massidda,

2

and E. K. U. Gross

3

1_{CNISM - Dipartimento di Fisica, Universita` degli Studi dell’Aquila, Via Vetoio 10, I-67010 Coppito (L’Aquila) Italy</sub> 2<sub>SLACS-INFM/CNR —Dipartimento di Fisica, Universita` degli Studi di Cagliari, I-09124 Monserrato (CA), Italy}

3_{Institut fu¨r Theoretische Physik, Freie Universita¨t Berlin, Arnimallee 14, D-14195 Berlin, Germany}

(Received 7 December 2007; published 23 June 2008; corrected 27 June 2008)

PRL

100,

257001 (2008)

P H Y S I C A L

R E V I E W

L E T T E R S

27 JUNE 2008week ending

0 1200 2400 3600 ω ( cm -1)

Z Γ Σ Y Γ S R Z T

0 1000 2000 3000 4000

ω(cm-1)

0 0.2 0.4 0.6 0.8 α 2F ( ω

) 420 440 460 P ( GPa ) 1

1.5 2 2.5 λ

0 20 40 60 80 T(K) 0 5 10 15 20 Δnk (meV)k

425P (GPa)450 75 150 225 300 Tc ( K )

G

D

G

Sunday, November 14, 2010

P

(

V, T

) =

−

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∂V

=

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0

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V

q

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1

2

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1

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Sunday, November 14, 2010

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γ

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q

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1

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e

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V

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q

, ν

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∂ω

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γ

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q

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1

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e

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q, ν

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γ

(

q, ν

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n

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q, ν

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,

Sunday, November 14, 2010

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Sunday, November 14, 2010

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Sunday, November 14, 2010

G

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B

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Sunday, November 14, 2010

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Sunday, November 14, 2010

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2

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kl

Sunday, November 14, 2010

*(#%<")+&(%&(+)

C

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(

p, T

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V

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kl

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T

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T

cV

∂S

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Sunday, November 14, 2010

*(#%<")+&(%&(+)

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(

p, T

|

) =

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V, T

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) +

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F

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ij

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λ

f

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V,

ij

, T

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ij

, λ

)

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p, T

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V

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2

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₌

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T

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c

V

∂S

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λ

f

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V, ij

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)

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V

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2

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∂

ij

∂

kl

C

S

₌

_C

T

₊

T

cV

∂S

∂ij

∂S

∂kl

f

(

V, T

|, λ

) =

U0

(

V

|, λ

) +

1

2

q

ν

ω

(

q

, ν|V, , λ

)

+

k

B

T

q

ν

log

1

−

e

−

ω

(

q

kBT

,ν

|

V,,λ

)

Sunday, November 14, 2010

*(#%")+&(%&(+)

0

50

100

150

P (GPa)

0

200

400

600

800

1000

1200

1400

Elastic Moduli (GPa)

300 K 1000 K 2000 K 3000 K 4000 K

0

50

100

150

P (GPa)

0

50

100

150

P (GPa)

C11

C22 C₃₃

C12

C₁₃ C23

C44 C55 C66

F