1
Chimica Fisica dei Materiali
e laboratorio
Vibrazioni nei solidi
z
y
x
Libration (B1) Rz: 61 cm-1
A.A. 2013-14
Bartolomeo Civalleri
Dip. Chimica IFM – Via P. Giuria 5 – 10125 Torino
bartolomeo.civalleri@unito.it
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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)
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
1
3
Vibrations in solids: computational tools
Molecular dynamics
Lattice dynamics
 Fourier transformation of the atomic
velocity autocorrelation function
 Taylor expansion of the potential
energy surface  harmonic approx.
 Atomic trajectories
 Dynamical matrix
 Disordered systems, high atomic
mobility
 Crystalline systems
 Better at high temperature
 Better at low temperature
 Include anharmonic effects
 Anharmonic corrections (quasiharmonic approximation)
 Accuracy depends on simulation
time (supercells)
 Thermodynamics through statistical
mechanics (supercells)
The two approaches provide complementary information
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
4
LD and Potential Energy Surface
Taylor expansion of the potential energy around the equilibrium configuration:
E  E0 
1
1
1
Hij ui u j   Hijk ui u j uk   Hijkl ui u j uk ul  ...

2 ij
3! ijk
4! ijkl
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.
For an equilibrium structure first-derivatives are zero (stationary point)
Hij, Hijk and Hijkl are derivatives of the energy with respect to atomic
displacements. They are the harmonic, cubic and quartic force constants,
respectively
Usually, truncated at the second order terms  harmonic approximation
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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5
Dynamical Matrix
In lattice dynamics the central role is played by the Dynamical Matrix
Dij (k ) 
  2E 
  0 G  exp  ik  R(G)
Mi M j G  ui u j  0
1

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)
As for electronic energy levels, translation symmetry leads to a band structure
for vibrational energy levels (phonons). E.g. Silicon band structure and vDOSs
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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 k0
• Absorption: Infrared
• Scattering: Raman
• Acoustic modes at k0
• Scattering: Brillouin (elastic constants)
• Phonon-Neutron interaction
• Inelastic Neutron Scattering (INS)
B. Civalleri
Optic branch
Acoustic branch
Chimica Fisica del Materiali – a.a. 2013/2014
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7
Dynamical Matrix and Phonon Dispersion
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
Dij (k ) 
  2E 
  0 G  exp  ik  R(G)
Mi M j GSC  ui u j  0
1
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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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Frequencies Calculation in CRYSTAL - I
CRYSTAL computes vibrational frequencies at G point (k=0)
  2E 
Dij (k  0) 
 0 0
Mi M j  ui u j  0
1
The second-derivatives matrix is computed by numerical differentiation
of the analitical first-derivatives (gradients)



g i ,0 0,..., u 0j ,...,0  gi ,0 0,...,  u 0j ,...,0
  2E 
 gi ,0 
 0 0  0  
2 u 0j
 ui u j  0  u j  0

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*.
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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Frequencies Calculation in CRYSTAL - II
In polar crystals, long range Coulomb effects give rise to macroscopic electric
fields for longitudinal optic modes (LO) at k0 (LO-TO splitting):
Dij (k  0)  Dijan (k  0)  Dijna (k  0)
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 k  Z i k  Z
D (k  0) 
V
k  ε  k
na
ij

j
Where:
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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
10
The Born tensor
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):
Za* ij 

ua j
 E

  i


i

 ua j
*i=component of an applied external field
**μ=cell dipole moment (polarization per unit cell)
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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The IR intensity - II

Ap  d p
Qp
The IR intensity of the p-th mode:
Za* ij 
The Born charge tensor:

ua j
 E

  i
2


i


u

aj
2
Ap  d p Z p, j
*dp=degeneracy of the p-th mode
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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The static dielectric constant
ε0 → static (low-frequency) dielectric constant
ε → electronic (high-frequency) dielectric constant
ωp → p-th frequency eigenvalue
Ω → unit cell volume
4
e e 

0
ij

ij
4
e ii0  e ii 

B. Civalleri

Z p,i Z p, j
Ionic contribution
p
p
2
Z p,i

p

p
Only one component for each Zp is
non null
Chimica Fisica del Materiali – a.a. 2013/2014
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Reflectance Spectrum
R   
e    1
2
e    1
4
e ij    e 


ij
Z p,i Z p, j
p     2  i
p
p
Spessartine
(garnet)
Mn3Al2 (SiO4)3
Spessartine
…… Rcalc
___ Rexp
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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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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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 (?)
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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Thermodynamics
Vibrational partition function
Z   Zi  
i
i
CV ,T  
exp(hi 2kBT )
exp(hi kBT )  1
R   2 ln Z 


T 2   1 T 2 

V
G ,T   Est  pV  RT ln Z
S ,T   R ln Z  RT   ln Z T V
Phonon density of state
g   
     k  dk      k  dk
i
BZ
f T  
max

i
i
f ,T  g   d
e.g.
CV T  
0
B. Civalleri
p
ip
max

;
 g    d  1
CV ,T  g   d
0
Chimica Fisica del Materiali – a.a. 2013/2014
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Thermodynamic functions: Pyrope
Specific Heat CV
Entropy
M. Catti, F. Pascale and R. Dovesi, unpublished
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
From lattice dynamics to MSDs and ADPs
18
The atomic Mean Square Displacement (MSD) tensors (symmetric 3x3
tensor) can be computed as
Batom (q ) 
1
Mq
E j k 
  k e q | jk  e  q | jk  
jk

T
2
j
• e(q|j,k) corresponds to the atomic displacement (eigenvector component)
of the atom q in the mode j along the wavevector k.
• Ej(k) is the energy
of the vibrational
mode
1

1

E j k    j k   
 2 exp  j k  / kBT  1 




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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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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
0.39
3.36
0.98
0.13
Isotropic MSD (Å2)
ADPs of N atom show a great
variability, with a large contribution
from low frequency modes
0.96
123 K
Equal-probability ellipsoids (50%)
B. Civalleri
Wavenumbers (cm-1)
Chimica Fisica del Materiali – a.a. 2013/2014
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ADPs of Benzene, Urotropine and L-Alanine
B3LYP/6-31G(d,p) [2x2x2]
15 K
0.43
0.83
0.53
1.38
23 K
15 K
0.38
0.46
0.37
0.06
0.56
0.53
0.24
0.68
0.17
Equal-probability ellipsoids (50%) with similarity index
B. Civalleri
0.31
0.22
Equal-probability ellipsoids (75%)
Chimica Fisica del Materiali – a.a. 2013/2014
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Accuracy: DFT methods vs experiment
Less than 20 cm-1
• 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
MAD (cm -1 )
• Four different
systems: pyrope,
forsterite, quartz and
alumina
18
16
14
12
10
8
6
4
2
0
Demichelis, Civalleri, Ferrabone, Dovesi, IJQC (2010)
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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Polystyrene: trans-Planar and s(2/1)2 Helix
Trans-planar
s(2/1)2 Helix
Pm2a
(C2v)
P2122
(D2h)
F. J. Torres, B. Civalleri, C. Pisani, P. Musto, A. R. Albunia, G. Guerra, J. Phys. Chem. B 111 (2007) 6327
A. R. Albunia, P. Rizzo, G. Guerra, J. Torres, B. Civalleri, C. M. Zicovich-Wilson, Macromolecules 40 (2007) 3895
F. J. Torres, B. Civalleri, A. Meyer, P. Musto, A. R. Albunia, P. Rizzo, G. Guerra, J. Phys. Chem. B 113 (2007) 5059
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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IR spectrum of trans-planar sPS: spectrum
B3LYP/6-31G(d,p) scaled frequencies (scale factor: 0.9614)
1452
906
1379
840

a
am
Absorbance
1440
1380
1340
1300
1260
1220
920
880
840
Calc.
calc

a
Exp.
am
3000
1500
1000
-1
500
Wavenumber (cm )
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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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25
Trans-planar polystyrene: normal modes animation
Animations of the normal modes:
http://www.crystal.unito.it/vibs/alpha-ps/
B3LYP/6-31G(d,p) scaled frequencies (scale factor: 0.9614)
B. Civalleri
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)
Chimica Fisica del Materiali – a.a. 2013/2014
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Trans-planar sPS: exp. vs calc.
Full assignment of 50 IR and Raman frequencies
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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27
sPS s(2/1)2 helical chain: Transition moment vectors
BORN TENSOR COMPONENTS IN THE NORMAL MODE BASIS
MODE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
...
187
188
189
190
191
192
X
-0.00501
0.00000
0.00000
0.00087
0.00000
0.00000
-0.00950
0.00000
0.01023
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
-0.00235
-0.01972
0.00000
0.00000
0.00000
Y
0.00000
0.00000
-0.00212
0.00000
-0.00049
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00166
0.00000
-0.00726
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
Z
0.00000
-0.00339
0.00000
0.00000
0.00000
0.00000
0.00000
0.00523
0.00000
-0.00408
0.00000
0.00000
-0.00651
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00019
0.00000
0.00000
0.00000
-0.13278 z
0.00000
0.00000
-0.15817
0.00000
0.00000
0.00000
0.00000
-0.19065
0.00000
0.00000
0.00000
0.00000
-0.21353
0.00000
B. Civalleri
x
In CRYSTAL polymers are
oriented along the x-axis.
Therefore:
x calc. = z exp.
y calc. = x exp.
z calc. = y exp.
y
Chimica Fisica del Materiali – a.a. 2013/2014
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sPS s(2/1)2 helical chain: Transition moment vectors
calc TMVcalc Symmetry
1383
y
B2u
1366
z
B1u
1358
z
B1u
1329
x
B3u
1320
x
B3u
1231
z
B1u
1167
y
B2u
1098
z
B1u
1078
x
B3u
969
x
B3u
934
z
B1u
921
x
B3u
853
z
B1u
769
x
B3u
765
z
B1u
743
x
B3u
594
x
B3u
577
y
B2u
566
z
B1u
540
x
B3u
530
z
B1u
498
x
B3u
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
x
z
Absorbance
nexp TMVexp
1378
Y
1364
Z
1354
Z
1329
X
1320
X
1232
Z
1169
Y
1117
Z
1078
X
977
X
944
Z
934
X
858
Z
780
X
766
Z
750
X
601
X
581
Y
572
Z
548
X
534
Z
503
X
z
A
+am
x
z
y xx
x
y
x
x
z
z
y
1400
xx
x
z x
x
x
y
1000
800
-1
z x
B

z
y
x
zx
1200
y
600
xz
x
C
calc
400
Wavenumber (cm )
Frequencies scaled by 0.972. Only the most relevant spectral region (1400 - 500 cm-1) is shown
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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Isotopic substitution and isotopic shift
Dij (k  0) 
  2E 
 0 G 
Mi M j G  ui u j  0
1
•
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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
30
Vibrational frequencies of Calcite (CaCO3)
Dn(exp.t)
13C
Isotopic
substitution
18O
62
B3LYP
28÷27
12÷8
9
15
38
16
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
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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31
Anharmonicity: the problem of X-H modes
X-H stretching modes are highly anharmonic. How to deal with that?
X-H stretching fully decoupled
from any other normal modes
exe=(2 01- 02) / 2
A wide range (0.5 Å) of X-H
distances must be explored to
properly evaluate E1 and E2
E2
E1
E0
02
01
Direct comparison with
experiment for fundamental
frequency, first overtone and
anharmonicity constant
(ANHARM)
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
32
Isolated OH groups in crystals: model structures/1
H
O
M
Edingtonite surface
Chabazite
M=Mg Brucite
M=Ca Portlandite
All calculations with
6-31G(d,p) basis set
B. Civalleri
Chimica Fisica del Materiali – a.a. 2013/2014
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33
B3LYP vs experimental OH frequencies
01 Raman
01 IR
Calc
3663
3694
Exp
3654
3698
Calc
3637
3650
Exp
3620
3645
Calc
--
3742
Exp
--
3747
Calc
--
3648
System
Brucite
Portlandite
Edingtonite
Chabazite
Exp
B. Civalleri
3603
Chimica Fisica del Materiali – a.a. 2013/2014
Hydroxylated amorphous silica surfaces
B3LYP, P1, 200 atoms, 3000 AO
13 Å
A300/423 K
B3LYP
13 Å
3900
MCM-41 mesoporous material model
3600
3300
3000
B3LYP, P1, 580 atoms, 7800 AO
MTS/423 K
41 Å
B3LYP
unit cell
3800
3600
3400
3200
3000
17