Vibrational frequencies in solids: tools and tricks

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1 Vibrational frequencies in solids: tools and tricks Roberto Dovesi Gruppo di Chimica Teorica Università di Torino Torino, 4-9 September 2016

2 This morning 3 lectures: R. Dovesi Generalities on vibrations Frequency calculations in molecules and solids Isotopic substitution Something about anharmonicity L. Maschio IR and Raman spectra A. Erba Phonon dispersion and Thermodynamics

3 The harmonic approximation (1) A molecule with N atoms has 3N degrees of freedom (coordinates that must be defined at each time t (classical mechanics) if we want to follow the dynamic of the nuclei). The electronic motion has been separated thanks to the Born-Oppenheimer approximation. A system of 3N equations (to be solved simultaneously: the movement of the atoms is correlated). The separation of the variables (if possible!) permits to transform this messy problem in a much simpler problem: 3N independent problems, each one involving a single variable. The condition: truncate the Taylor expansion of the energy of the system with respect to the equilibrium geometry to second order: Harmonic Approximation.

4 The harmonic approximation (2) Within the Harmonic Approximation the strongly correlated motion of the N nuclei is described as a set of N - Independent harmonic oscillators (this means: when one oscillator is excited, the others do not care) - Whose wavefunction is the product of the wavefunctions of the harmonic oscillators ( the only difference among them being the frequency parameter resulting from the diagonalization of the Dynamical Matrix to give new coordinates: the Normal Modes) - Whose energy is the SUM of the energies of the independent harmonic oscillators

5 Number of vibrational modes for molecules A molecule with N atoms has 3N-5 vibrational modes if linear, 3N-6 in the other cases -5 : 3 translations, 2 rotations -6 : 3 translations, 3 rotations e.g. H 2 O; we expect 3*3-6=3 modes:

6 Are all modes visible in the spectrum? Experimental or simulated? Selection rules (Fermi Golden Rule) The crucial role of Symmetry (active or silent mode?) Selection rules depend on the way the experiment is performed: IR and Raman spectra. :

7 The case of methane: 3x5-6=9 modes (peaks) expected Actually there are only 4 peaks of degeneracy 3, 2, 1, 3 (the total is 9, as expected), of symmetry F 2, E, A, F 2 cm -1 SYM IR RAMAN 1693 F E A F Two active in IR, four in Raman IR intensities: Km/mole Raman intensities: arbitrary units Data refer to a B3LYP calculation with minimal basis set

8 But you should be aware that: a) A non forbidden mode can have very low intensity b) Peaks can be close to each other and merge in experiment in a single larger band c) Vibrations ARE anharmonic, then forbidden peaks will appear, in general with low intensity d) And then. Impurities, background, temperature effects, isotopic mixing. :

9 How to know how many modes are active in IR and RAMAN? The analysis is performed automatically by CRYSTAL. You can do it by yourself by using the Character Table of the point group of your molecule. :

10 What about solids? Apparently, a large molecule with an infinite number of atoms Infinite number of modes But the translational symmetry permits to organize the infinite number of modes in a finite set (3N, where N of the number atoms in the unit cell) of continuous bands (each point of the band corresponds to a k point of the Brillouin zone, or to an Irreducible Representation of the infinite Translation Group) The good point is that eigenvalues and eigenvectors of the dynamical matrix are continuous in reciprocal space, so that the calculation can be limited to a few k points followed by interpolation. :

11 Two main cases: A) For IR and RAMAN spectra, as a consequence of symmetry and momentum conservation, only the frequencies at the Γ point can be measured (and then need to be computed). So, at the end, the analysis is extremely similar to the one performed for the molecular case, if N is meant to be the number of atoms in the unit cell instead of the number of atoms in the molecule. B) For the thermodynamic analysis (and the neutron scattering spectra), ALL the infinite set of frequencies should be computed. As anticipated above, and in a very parallel way to what happens for the electronic states, the Dynamical Matrix is solved only in a subset of points of the BZ, and the integral is substituted by a finite sum.

12 Phonons in solids in short

13 Harmonic frequency in solids with CRYSTAL Page 13 João Pessoa, 2014

14 The crucial step of the analytical gradient in a gaussian basis set K. Doll, V. R. Saunders, N. M. Harrison, Int. J. Quant. Chem. 82, 1 (2001) K. Doll, Comp. Phys. Comm. 137, 74 (2001) The cost of the gradient with respect to all the 3N coordinates (and the 9 cell coordinates) is a small fraction of the full SCF!! Frequency option: F. Pascale, C.M. Zicovich-Wilson, F. López Gejo, B. Civalleri, R. Orlando and R. Dovesi, J. Comput. Chem. 25, 888 (2004)

15 Modes can be Some more complication. longitudinal (LO): amplitude is parallel to wave propagation Transverse (TO, transverse Optical): amplitude is perpendicular to wave propagation TO modes obtained diagonalizing the Dynamical Matrix LO modes are present only when the polarization is changing during the vibration (IR modes), and require an additional term

16 The dynamical matrix The behavior of the phonons of a wave vector k close to the Γ point can be described as follows: Center-zone phonons: ANALYTICAL α, β: atoms in the primitive cell i,j: cartesian coordinates Dependence on the direction of k: limiting cases k 0 NON ANALYTICAL

17 The non-analytical contribution and the LO modes Analytical contribution: Non-analytical contribution:

18 The Born charges Page 18 The atomic Born tensors are the key quantities for calculating: the Longitudinal Optical (LO) modes (see previous slide) the IR intensities the static dielectric tensor They are defined, in the cartesian basis, as: *E i =component of an applied external field **µ=cell dipole moment João Pessoa, 2014

19 µ depends on the choice of the cell Page 19 BUT the dipole moment difference between two geometries of the same periodic system (polarization per unit cell) is a properly defined observable. The partial second derivatives appearing in the Born tensors are estimated numerically from the polarizations generated by small atomic displacements (the same as for the second energy derivative) João Pessoa, 2014

20 Page 20 The IR intensity of the p-th mode: The Born charge tensor: *d p =degeneracy of the p-th mode João Pessoa, 2014

21 The oscillator strengths Page 21 Oscillator strength of the p-th mode: In relationship with the IR intensity A p : Can be adopted in a classic damped harmonic oscillator model for the dielectric function (see later) João Pessoa, 2014

22 The static dielectric constant Page 22 ε 0 static dielectric constant ε electronic dielectric constant ω p p-th frequency eigenvalue Ω unit cell volume Ionic contribution Only one component for each Z p is non null João Pessoa, 2014

23 IR reflectance spectrum Page 23 Reflectivity is calculated from dielectric constant by means of: (θ is the beam incident angle) The dielectric function is obtained with the classical dispersion relation (damped harmonic oscillator): João Pessoa, 2014

24 IR reflectance spectrum Page 24 Reflectivity is calculated from dielectric constant by means of: (θ is the beam incident angle) The dielectric function is obtained with the classical dispersion relation: Comparison of computed and experimental IR reflectance spectra for garnets: a) pyrope b) grossular c) almandine. London, 2013

25 The problem of the comparison between experiment and simulation: model model model model model

26 In the experiment a best fit is performed to generate from the Reflectance spectrum: frequencies and intensities and FWHM of the peak In garnets: 3 parameters times 17 peaks = best fit with 51 parameters!!!!!! Maybe strongly correlated?!?!

27 Raman Intensities: see next talk

28 Example: garnets: X 3 Y 2 (SiO 4 ) 3 Space Group: Ia-3d 80 atoms in the primitive cell (240 modes) Γ rid = 3A 1g + 5A 2g + 8E g + 14 F 1g + 14 F 2g + 5A 1u + 5 A 2u + 10E u + 18F 1u + 16F 2u 17 IR (F 1u ) and 25 RAMAN (A 1g, E g, F 2g ) active modes

29 Structure of pyrope: Mg 3 Al 2 (SiO 4 ) 3 Si O Mg tetrahedra O Cubic Ia-3d 160 atoms in the crystallographic unit cell (80 in the primitive) Al octahedra Animations of the vibrational modes at

30 Page 30 Università di Torino Torino Italy, 4th Agoust 2006

31 Page 31 DIFFICULT? A single keyword: FREQCALC Università di Torino Torino Italy, 4th Agoust 2006

32 Page 32 Aim of what follows Document the numerical stability of the computational process. Document the accuracy (with respect to experiment, when experiment is accurate). Interpret the spectrum and attribute the modes. Università di Torino Torino Italy, 4th Agoust 2006

33 Page 33 Dependence on the parameters Many parameters Let s check the most important a) The step for the numerical derivative b) The number of points along a coordinate c) The SCF conbergence d) The grid for the Numerical Integration of the XC term e) The basis set Università di Torino Torino Italy, 4th Agoust 2006

The step for the hessian: u, the number of displacements for each cartesian coordinate: N the SCF convergence: t E Dependence of B3LYP vibrational frequencies of pyrope at Γ point on t E (SCF

34 The step for the hessian: u, the number of displacements for each cartesian coordinate: N the SCF convergence: t E Dependence of B3LYP vibrational frequencies of pyrope at Γ point on t E (SCF convergence tolerance), u (atomic displacement for the numerical evaluation of the Hessian from the analytical energy gradients) and N (number of total energy calculations along each nuclear cartesian coordinates for the evaluation of the Hessian matrix). Statistical values calculated with respect to the case where t E = 11, u = Å and N = 2. Page 34 Università di Torino Torino Italy, 4th Agoust 2006

35 Page 35 Effect of the DFT integration grid size on vibrational frequencies. Statistical indices refer to the (99,1454)p grid. Università di Torino Torino Italy, 4th Agoust 2006

An example: the effect of the basis set Page 36 Description of the three basis sets adopted for the calculation of the vibrational frequencies of pyrope.

36 An example: the effect of the basis set Page 36 Description of the three basis sets adopted for the calculation of the vibrational frequencies of pyrope G(d) means that a 8G contraction is used for the 1s shell; a 5G contraction for the 2sp, and a single G for the 3sp and 4sp shells, plus a single G d shell ( =18 AOs per Mg or Al atom). +sp and +d means that a diffuse sp or d shell has been added to basis set A. Università di Torino Torino Italy, 4th Agoust 2006

υ and Δυ in cm -1. Animation of the Red modes will be shown in the following.

37 IR-TO modes (F 1u ) of pyrope as a function of the basis set size. Frequency differences (Δυ) are evaluated with respect to experimental data. υ and Δυ in cm -1. Animation of the Red modes will be shown in the following. Page 37 a) Hofmeister et. al. Am. Mineral , 418 Università di Torino Torino Italy, 4th Agoust 2006

38 Statistical analysis (17 data used) of calculated Infrared modes of pyrope with respect to experimental data for different Basis Set sizes. Number in parentheses obtained by excluding the frequency indicated as uncertain by Hofmeister et al. Numbers in cm -1 Page 38 Università di Torino Torino Italy, 4th Agoust 2006

39 Page 39 COST and Symmetry 1. Each SCF+Gradient calculation provides one line of H ik atoms = SCF+G calculations with low (null) symmetry 3. Point symmetry is used to generate lines of atoms symmetry related 4. Other symmetries (among x, y, z lines; translational invariance) further reduce the required lines 5. At the end only 9 (nine) SCF+G calculations are required, instead of 241!! Università di Torino Torino Italy, 4th Agoust 2006

Spessartine: IR-TO frequencies Page 40 IR-TO modes (F 1u ) of spessartine. Frequency differences (Δν) are evaluated with respect to experimental data.

40 Spessartine: IR-TO frequencies Page 40 IR-TO modes (F 1u ) of spessartine. Frequency differences (Δν) are evaluated with respect to experimental data. (ν and Δν in cm -1 ) EXP Hofmeister and Chopelas, Vibrational spectoscopy of end-member silicate garnets, Phys. Chem. Min., 17, (1991). Δν Nancy, March 2011

Spessartine: IR-LO frequencies Page 41 IR-LO modes (F 1u ) of spessartine. Frequency differences (Δν) are evaluated with respect to experimental data.

41 Spessartine: IR-LO frequencies Page 41 IR-LO modes (F 1u ) of spessartine. Frequency differences (Δν) are evaluated with respect to experimental data. (ν and Δν in cm -1 ) EXP Hofmeister and Chopelas, Vibrational spectoscopy of end-member silicate garnets, Phys. Chem. Min., 17, (1991). Δν Nancy, March 2011

42 Spessartine: oscillator strengths Page 42 Oscillator strengths corresponding to the IR-TO modes (F 1u ) of spessartine. Differences (Δf) are evaluated with respect to experimental data. f and Δf dimensionless. EXP Hofmeister and Chopelas, Vibrational spectoscopy of end-member silicate garnets, Phys. Chem. Min., 17, (1991). Δf Nancy, March 2011

43 Spessartine: RAMAN modes Page 43 Frequency differences (Δν) are evaluated with respect to experimental data. (ν and Δν in cm -1 ) a) Hofmeister &Chopelas, Phys. Chem Min b) Kolesov &Geiger, Phys. Chem. Min Nancy, March 2011

44 Spessartine: RAMAN modes - 2 Page 44 Frequency differences (Δν) are evaluated with respect to experimental data. (ν and Δν in cm -1 ) a) Hofmeister &Chopelas, Phys. Chem Min b) Kolesov &Geiger, Phys. Chem. Min Nancy, March 2011

45 Garnets: statistics Page 45 IR frequencies Raman frequencies (e) a) Hofmeister et al., Phys. Chem. Min , 418 b) McAloon et. al., Phys. Chem. Min , 1145 c) Hofmeister et. al., Phys. Chem. Min , 503 d) Hofmeister, private comm. e) Kolesov et. al., Phys. Chem. Min , 142 Statistical analysis of calculated IR and Raman modes of garnets compared to experimental data. Nancy, March 2011

46 The isotopic shift As a tool for the assignement of the modes and for the interpretation of the spectrum. Natural isotopes, any mass, also infinite (to study subunits) Cost is negligible (no recalculation of the hessian matrix) Synergic with animations!!

Page 47 Δν (cm -1 ) 100 350 ν (cm -1 ) Isotopic shift on the

Note the gap at about 700 cm -1 separating the stretching modes from

47 Page 47 Δν (cm -1 ) ν (cm -1 ) Isotopic shift on the vibrational frequencies of pyrope when 26 Mg is substituted for 24 Mg. Note the gap at about 700 cm -1 separating the stretching modes from the rest of the spectrum. Università di Torino Torino Italy, 4th Agoust 2006

48 Page 48 Δν (cm -1 ) ν (cm -1 ) Isotopic shift on the vibrational frequencies of pyrope when 29 Al is substituted for 27 Al. Università di Torino Torino Italy, 4th Agoust 2006

49 Pyrope : 28 Si 30 Si Δν (cm -1 ) Low ν : rotations and bending of tetrahedra and octahedra (involving by connectivity also Si) High ν: stretching of tetrahedra ν (cm -1 ) Isotopic shift on the vibrational frequencies of pyrope when 30 Si is substituted for 28 Si.

50 Page 50 Isotopic shift on the vibrational frequencies of pyrope when 18 O is substituted for 16 O. Δν (cm -1 ) ν (cm -1 ) Università di Torino Torino Italy, 4th Agoust 2006

51 How are moving the atoms? In the litterature very often a classification id used that applies only to small molecules: stretching bending translation of a group rotation of a group If the cell does contain more that, say, 4 atoms this classification is partial, misleading, often completely wrong. Can we we the atoms when moving?

52 Page 52 υ cm -1 υ 4,T(Al) 674 cm -1 Università di Torino Torino Italy, 4th Agoust 2006

53 Page 53 T(Al) cm -1 R(SiO 4 ) cm -1 The complete set of animations is available in the CRYSTAL s web site: Università di Torino Torino Italy, 4th Agoust 2006

54 Page 54 T(SiO 4 ),T(Mg) 215 cm -1 T(Mg) cm -1 Università di Torino Torino Italy, 4th Agoust 2006

55 PASI June 2007, Zacatecas, Mexico Page 55

56 The animation of the full set of modes, that can be «tailored» selecting various options (stick and balls, size of the set of atoms, rotating the frame.) Easily generated from the output of the FREQCALC option.

57 Anharmonicity effects

58 The problem of H Stretching modes involving hydrogen atoms are strongly anharmonic: the O-H stretching anharmonicity can be as large as 180 cm -1. This difficulty is compensated by the full separability of this mode.

59 Anharmonic correction for hydroxyls ω e x e =(2 ω 01 - ω 02 ) / 2 E 2 E 1 ω 02 E 0 ω 01

60 Isolated OH groups in crystals: model structures/1

61 Hydrogen bonded OH groups: model structures/2 b O2 O1 H2 O2 Be H1 O1 a

62 Is the choice of the Hamiltonian critical? Brucite: fundamental OH stretching frequencies. All data in cm -1 Be(OH) 2 : Fundamental OD stretching frequencies. All data in cm -1 P. Ugliengo, F. Pascale, M. Mérawa, Pierre Labéguerie, S. Tosoni, and R. Dovesi, J. Phys. Chem. B 108, (2004)

63 B3LYP frequencies for brucite: a test case F. Pascale, S. Tosoni, C. Zicovich-Wilson, P. Ugliengo, R. Orlando, R. Dovesi, Chem. Phys. Lett. 396, 308 (2004)

64 Symmetric OH bending modes in brucite Antisymmetric Bending Mode Bending Mode (803 cm -1 ) (459 cm -1 )

65 Symmetric Stretching Mode (3847 cm -1 ) OH stretching modes in brucite Anti-symmetric Stretching Mode (3873 cm -1 )

66 OH stretching in 50% deuterated-brucite Deuterium Hydrogen Stretching Stretching (2817 cm -1 ) (3860 cm -1 )

Electronic Supplementary Information

Electronic Supplementary Material (ESI) for CrystEngComm. This journal is The Royal Society of Chemistry 2014 Electronic Supplementary Information Configurational and energetical study of the (100) and