Chimica Fisica dei Materiali e laboratorio - unito.it 3 B. Civalleri Chimica Fisica del Materiali...

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Chimica Fisica del Materiali – a.a. 2013/2014 B. Civalleri

A.A. 2013-14

Chimica Fisica dei Materiali e laboratorio

Bartolomeo Civalleri Dip. Chimica IFM – Via P. Giuria 5 – 10125 Torino

[email protected]

Vibrazioni nei solidi

z

x

y

Libration (B1) Rz: 61 cm-1

2


Atomic motion

• The crystal lattice is never rigid.

• Atoms actually move around their equilibrium positions inside the crystalline structure.

• Motions of atoms in solids provide the key to understand many physical phenomena mainly related to thermal effects, phase transitions, transport properties, and so forth.

• Theoretical calculation of atom vibrations then gives access to a number of properties (see next slides)

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3


Molecular dynamics Fourier transformation of the atomic

velocity autocorrelation function

Atomic trajectories

Disordered systems, high atomic

mobility

Better at high temperature

Include anharmonic effects

Accuracy depends on simulation

time (supercells)

Lattice dynamics Taylor expansion of the potential

energy surface harmonic approx.

Dynamical matrix

Crystalline systems

Better at low temperature

Anharmonic corrections (quasi-

harmonic approximation)

Thermodynamics through statistical

mechanics (supercells)

The two approaches provide complementary information

Vibrations in solids: computational tools

4


LD and Potential Energy Surface

Taylor expansion of the potential energy around the equilibrium configuration:

0

1 1 1...

2 3! 4!ij i j ijk i j k ijkl i j k l

ij ijk ijkl

E E H u u H u u u H u u u u

For an equilibrium structure first-derivatives are zero (stationary point)

Index i labels the triplet (G,t,a) with G as a translation vector of the primitive

lattice, t as an atom within the primitive unit cell and a as the Cartesian

coordinate of the atomic displacement u.

Usually, truncated at the second order terms harmonic approximation

Hij, Hijk and Hijkl are derivatives of the energy with respect to atomic

displacements. They are the harmonic, cubic and quartic force constants,

respectively

3

5


Dynamical Matrix

In lattice dynamics the central role is played by the Dynamical Matrix

Where:

Mi is the mass of the atom associated to the i-th coordinate;

ui is the cartesian atomic displacements of the i-th coordinate;

R(G) = xi(0) - xj(G)

21

( ) exp ( )ij

i ji j

ED i

M M

0 GG

0

k k R Gu u

As for electronic energy levels, translation symmetry leads to a band structure

for vibrational energy levels (phonons). E.g. Silicon band structure and vDOSs

6


Phonons: matter-radiation interaction

• Vibrational modes can be considered as being particle-like

(phonons)

• Phonons can interact with radiation and matter

• Phonon - Photon interaction

• Optic modes at k0

• Absorption: Infrared

• Scattering: Raman

• Acoustic modes at k0

• Scattering: Brillouin (elastic constants)

• Phonon-Neutron interaction

• Inelastic Neutron Scattering (INS)

Optic branch

Acoustic branch

4

7


Dynamical Matrix and Phonon Dispersion

21

( ) exp ( )ij

SC i ji j

ED i

M M

0 G

G0

k k R Gu u

The dynamical matrix can be computed by using:

Linear response methods

Density Functional Perturbation Theory (S. Baroni, et al. Rev. Mod. Phys. (2001))

Finite displacements

Numerical derivatives (supercells must be used) (CRYSTAL)

A supercell calculation at G permits to map some k-points in the reciprocal space.

Number and kind of k-points depends on shape and size of the supercell

Covalent solids: reasonable approximation, fast decay of the 2nd derivatives, interpolation

schemes

Ionic and semi-ionic (polar) solids: slow decay, long-range contribution important, approximate

electrostatic models

Results can be compared with Inelastic Neutron Scattering (INS) experiments

8


Frequencies Calculation in CRYSTAL - I

21( 0)ij

i ji j

ED

u uM M

0 0

0

k

CRYSTAL computes vibrational frequencies at G point (k=0)

The second-derivatives matrix is computed by numerical differentiation

of the analitical first-derivatives (gradients)

Special properties of the G point (k=0):

• D(0) is simple to calculate

• Three modes have zero frequency (acoustic branch – “translations”)

• D(0) possesses the point symmetry of the crystal (factorization)

• G point modes give rise to infrared and Raman spectra

LO/TO splitting, relevant to polar crystals, can be also computed by

using e and Z*.

2, ,,

0,..., ,...,0 0,..., ,...,0

2

i j i ji

i j j j

g u g ugE

u u u u

0 0

0 00

0 0 0 0

0 0

5

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Frequencies Calculation in CRYSTAL - II

LO-TO splitting is computed by including a non-analytical term which

depends on the electronic dielectric tensor e and on the Born effective

charge tensor associated to each atom.

4( 0)

i jna

ijDV

k Z k Zk

k ε kWhere:

V is the volume of the unit cell;

Z* is the Born effective-charge tensor (analogous to the molecular GAPT

charges);

e is the electronic dielectric-constant tensor (CPHF/KS, see Bernasconi’s lecture)

All those quantities can be computed by CRYSTAL

In polar crystals, long range Coulomb effects give rise to macroscopic electric

fields for longitudinal optic modes (LO) at k0 (LO-TO splitting):

( 0) ( 0) ( 0)an na

ij ij ijD D Dk k k

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The atomic Born tensors are key quantities for :

calculation of the IR intensities

calculation of the static (low-frequency) dielectric tensor, e0

calculation of the Longitudinal Optical (LO) modes

They are defined, in the cartesian basis, as (for atom a):

a

a a

*

ij i

j i j

EZ

u u

*i=component of an applied external field

**μ=cell dipole moment (polarization per unit cell)

The Born tensor

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The IR intensity of the p-th mode:

2

p p

p

A dQ

a

a a

*

ij i

j i j

EZ

u uThe Born charge tensor:

,

2

p jp pA d Z

*dp=degeneracy of the p-th mode

The IR intensity - II

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ε0 → static (low-frequency) dielectric constant

ε → electronic (high-frequency) dielectric constant

ωp → p-th frequency eigenvalue

e e

, ,0 4 p i p j

ij ij

p p

Z Z

Ω → unit cell volume

Ionic contribution

Only one component for each Zp is

non null

e e

2

,0 4 p i

ii ii

p p

Z

The static dielectric constant

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, ,

2

4 p i p j

ij ij

p p p

Z Z

i

e e

2

1

1R

e

e

Spessartine …… Rcalc

___ Rexp

Reflectance Spectrum

Spessartine (garnet)

Mn3Al2 (SiO4)3

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Applications of Lattice Dynamics - I

Interpretation of vibrational spectra analysis of the normal modes assignment visualization/animations

symmetry analysis IR/Raman active/inactive

isotopic substitution

Thermodynamics calculation of thermodynamic functions phonon density of state

pressure- and temperature-dependent properties free energy

harmonic approximation, quasi-harmonic approximation simple models

Equations of state (p-V-T) phase diagrams phase stability

phase transitions pt

solid-state reactions

kinetics of transformation simple models

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Applications of Lattice Dynamics - II

Calculation of the Atomic Displacement Parameters (ADPs) computed from the eigeinvectors of the dynamical matrix

anisotropic thermal ellipsoids diffraction data

thermal motion corrections e.g. bond distances

Debye-Waller thermal factors dynamic X-ray structure factors

related to the intensity of Inelastic Neutron Scattering measurements

Characterization of the PES structure stability no imaginary frequencies

characterization of minima, transition states, higher-order saddle points

Isotopic equilibria isotope enrichment in minerals

Ab-initio derived semiempirical interatomic potentials basic information: E, X, g, i, Cij, B, ... construction (benchmark) of new (existing) interatomic potentials

transfer from the electronic to the atomic scale better transferability (?)

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Thermodynamics

, lnstG T E pV RT Z exp( 2 )

exp( ) 1i B

i

i i i B

h k TZ Z

h k T

, ln lnV

S T R Z RT Z T

Vibrational partition function

Phonon density of state

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22

ln,

1V

V

R ZC T

T T

i i pi ipBZ

g d d k k k k 1g d

max

0

,V VC T C T g d

max

0

,f T f T g d

e.g.

;

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17


Thermodynamic functions: Pyrope

Specific Heat CV Entropy

M. Catti, F. Pascale and R. Dovesi, unpublished

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From lattice dynamics to MSDs and ADPs

• e(q|j,k) corresponds to the atomic displacement (eigenvector component) of the atom q in the mode j along the wavevector k.

2

1( ) | |

Tj

atom

jq j

EB q q j q j

M

k

ke k e k

k

B

1 1

2 exp /k 1j j

j

ET

k kk

Atomic Anisotropic Displacement Parameters (ADPs, U(q)) can be

readily obtained from Batom(q) tensors

They can be compared with ADPs from X-ray or neutron diffraction

Visualized in terms of thermal ellipsoids

• Ej(k) is the energy of the vibrational mode

The atomic Mean Square Displacement (MSD) tensors (symmetric 3x3

tensor) can be computed as

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From internal to external modes: supercell approach

B3LYP/6-31G(d,p)

ADPs strongly depend on the supercell size

A supercell of 2x2x2 size gives a reasonable agreement with experiment

ADPs of N atom show a great variability, with a large contribution from low frequency modes

123 K 0.13

0.39 0.98 3.36 0.96

Equal-probability ellipsoids (50%) Wavenumbers (cm-1)

Isotr

opic

MSD

(Å

2)

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0.83

0.43

0.56

0.53

0.53 0.68

ADPs of Benzene, Urotropine and L-Alanine

B3LYP/6-31G(d,p) [2x2x2]

Equal-probability ellipsoids (50%) with similarity index

0.38

1.38

15 K 15 K

Equal-probability ellipsoids (75%)

23 K

0.24

0.06

0.17

0.46 0.37

0.31 0.22

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0

2

4

6

8

10

12

14

16

18

MA

D (

cm-1

)

Accuracy: DFT methods vs experiment

Less than 20 cm-1 • Four different

systems: pyrope,

forsterite, quartz and

alumina

• Dataset of 134

vibrational frequencies

(IR and Raman data)

• 11 DFT methods:

LDA, GGA (standard

and for solids), hybrids

• Hybrid methods, in

particular, B3LYP and

WC1LYP give the

lowest MAD

Demichelis, Civalleri, Ferrabone, Dovesi, IJQC (2010)

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Interpretation of vibrational spectra

How to do that?

Scaling factors: Comparison between computed and experimental frequencies

Symmetry analysis IR/Raman active/inactive

Direction of transition moment vectors (TMV) (IR active modes)

Analysis of the normal modes assignment visualization/animations

Isotopic substitution

Known problems:

Anharmonicity (in particular: H-X vibrations and low-frequency modes)

Combination of modes: overtones and Fermi mixing

Approximations in the structural model

Deficiencies of the adopted level of theory

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F. J. Torres, B. Civalleri, C. Pisani, P. Musto, A. R. Albunia, G. Guerra, J. Phys. Chem. B 111 (2007) 6327

Polystyrene: trans-Planar and s(2/1)2 Helix

F. J. Torres, B. Civalleri, A. Meyer, P. Musto, A. R. Albunia, P. Rizzo, G. Guerra, J. Phys. Chem. B 113 (2007) 5059

A. R. Albunia, P. Rizzo, G. Guerra, J. Torres, B. Civalleri, C. M. Zicovich-Wilson, Macromolecules 40 (2007) 3895

Trans-planar s(2/1)2 Helix

Pm2a (C2v)

P2122 (D2h)

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3000 1500 1000 500

am

a

a

Ab

so

rba

nce

Wavenumber (cm-1)

am

calc

12201260130013401380

840880920

14521379

906

840

1440

IR spectrum of trans-planar sPS: spectrum

B3LYP/6-31G(d,p) scaled frequencies (scale factor: 0.9614)

Calc.

Exp.

IR intensities as % fraction of the max. computed intensity of 89 km/mol ( = 681 cm−1).

Lorentzian profile was used with a FWMH of 10 cm-1

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Trans-planar polystyrene: normal modes animation

B3LYP/6-31G(d,p) scaled frequencies (scale factor: 0.9614)

Animations of the normal modes:

http://www.crystal.unito.it/vibs/alpha-ps/

Main spectral regions:

3200 - 2800 cm-1

n(C-H) aromatic and alkyl

groups

1600 - 1350 cm-1

phenyl and alkyl groups:

n(CC) (1575 cm-1) and (CH)

1350 - 1000 cm-1

phenyl (CH) and alkyl

(CH) (1329 cm-1)

1000 - 500 cm-1

deformation aromatic rings

and CH groups (976 cm-1)

below 500 cm-1

collective vibrations

(torsions) (70 cm-1)

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Trans-planar sPS: exp. vs calc.

Full assignment of 50 IR and Raman frequencies

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sPS s(2/1)2 helical chain: Transition moment vectors

In CRYSTAL polymers are

oriented along the x-axis.

Therefore:

x calc. = z exp.

y calc. = x exp.

z calc. = y exp.

BORN TENSOR COMPONENTS IN THE NORMAL MODE BASIS

MODE X Y Z

1 -0.00501 0.00000 0.00000

2 0.00000 0.00000 -0.00339

3 0.00000 -0.00212 0.00000

4 0.00087 0.00000 0.00000

5 0.00000 -0.00049 0.00000

6 0.00000 0.00000 0.00000

7 -0.00950 0.00000 0.00000

8 0.00000 0.00000 0.00523

9 0.01023 0.00000 0.00000

10 0.00000 0.00000 -0.00408

11 0.00000 0.00000 0.00000

12 0.00000 0.00166 0.00000

13 0.00000 0.00000 -0.00651

14 0.00000 -0.00726 0.00000

15 0.00000 0.00000 0.00000

16 -0.00235 0.00000 0.00000

17 -0.01972 0.00000 0.00000

18 0.00000 0.00000 0.00000

19 0.00000 0.00000 0.00000

20 0.00000 0.00000 0.00019

...

187 0.00000 -0.15817 0.00000

188 0.00000 0.00000 0.00000

189 0.00000 0.00000 0.00000

190 -0.13278 0.00000 0.00000

191 0.00000 0.00000 -0.21353

192 0.00000 -0.19065 0.00000 x y

z

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sPS s(2/1)2 helical chain: Transition moment vectors

nexp TMVexp calc TMVcalc Symmetry

1378 Y 1383 y B2u

1364 Z 1366 z B1u

1354 Z 1358 z B1u

1329 X 1329 x B3u

1320 X 1320 x B3u

1232 Z 1231 z B1u

1169 Y 1167 y B2u

1117 Z 1098 z B1u

1078 X 1078 x B3u

977 X 969 x B3u

944 Z 934 z B1u

934 X 921 x B3u

858 Z 853 z B1u

780 X 769 x B3u

766 Z 765 z B1u

750 X 743 x B3u

601 X 594 x B3u

581 Y 577 y B2u

572 Z 566 z B1u

548 X 540 x B3u

534 Z 530 z B1u

503 X 498 x B3u

1400 1200 1000 800 600 400

z

z

x

xx

x

x

x

z

z

z

z

z

z

z y

y

x

y

y

y

y

xx

x

x

x

xx

xxx

C

calc

B

A+am

Ab

sorb

an

ce

Wavenumber (cm-1)

z

x

FTIR spectra: (A) of an unoriented form film

of s-PS; (B) spectrum A, after subtraction of the

spectrum of the amorphous phase; (C) ab-initio

simulated spectrum of a s(2/1)2 helix of s-PS

Frequencies scaled by 0.972. Only the most relevant spectral region (1400 - 500 cm-1) is shown

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• As a tool for the assignment of the modes and for the

interpretation of the spectrum

• One atom at a time (e.g. 29Al for 27Al)

(experimental data available for comparison)

• In some cases also infinite mass:

Advantages with respect to subunits investigated with clusters

a) the atoms move in the field created by the infinite system.

b) and in the presence of the other atoms

c) and the hessian matrix is the correct one

Isotopic substitution and isotopic shift

21( 0)ij

i ji j

ED

u uM M

0 GG

0

k

30


B3LYP

Isotopic

substitution 62

Dn(exp.t)

13C 18O

12÷8 28÷27

9

15 38

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37

Exp. Data: P. Gillet, et al. Geochim. Cosmochim. Acta 60 (1996) 3471; M.E. Böttcher, et al. Solid State Ion. 101-103 (1997) 1379

Vibrational frequencies of Calcite (CaCO3)

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E2

E1

E0

02

01

exe=(2 01- 02) / 2

X-H stretching fully decoupled

from any other normal modes

A wide range (0.5 Å) of X-H

distances must be explored to

properly evaluate E1 and E2

Direct comparison with

experiment for fundamental

frequency, first overtone and

anharmonicity constant

(ANHARM)

X-H stretching modes are highly anharmonic. How to deal with that?

Anharmonicity: the problem of X-H modes

32


Isolated OH groups in crystals: model structures/1

M O

H

M=Mg Brucite M=Ca Portlandite

Edingtonite surface

Chabazite

All calculations with 6-31G(d,p) basis set

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B3LYP vs experimental OH frequencies

System 01 Raman 01 IR

Brucite Calc 3663 3694

Exp 3654 3698

Portlandite Calc 3637 3650

Exp 3620 3645

Edingtonite Calc -- 3742

Exp -- 3747

Chabazite Calc -- 3648

Exp 3603

Hydroxylated amorphous silica surfaces

MCM-41 mesoporous material model

3000330036003900

A300/423 K

B3LYP

30003200340036003800

MTS/423 K

B3LYP

B3LYP, P1, 200 atoms, 3000 AO

B3LYP, P1, 580 atoms, 7800 AO

unit cell

13 Å

13 Å

41 Å

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