Vibration
Papers
 [2024  npj Comput Mater  Automated allfunctionals infrared and Raman spectra]: finitedisplacement and finitefield approach for computing harmonic IR and Raman spectra of solids
 [2023  JPCL  Proton transfer in water complexs revealed by IRMPD and AIMD]
 Pure AIMD simulations in B3LYP and CP2K
 [2023  JCTC  A Periodic VSCF study of OH stretching in ice XI]: CRYSTAL23 code
 [2023  Nat comm.  Twodimensional infraredRaman spectroscopy as a probe of water’s tetrahedrality]
 Peerreview report, scripts and raw data are included
 MD vs TRPMD, nuclear quantum effects and temperature dependence, using qTIP4P/F
 The librations (400 ~ 1000 cm $^{1}$) of water molecules, carries rich information about liquid water
 [2022  JPCC  Ceotto Group  IR of water on TiO2]: there are some discussions on zeropoint energy and one must provide kinetic energy higher than zeropoint energy in a classical MD simulation.
 [2022  J. Mol. Spectrosc.  Spectroscopy with Free Electron Lasers and Synchrotron Radiation]
 [2021  JCC  Non longrange corrected density functionals incorrectly describe the intensity of the CH stretching band in polycyclic aromatic hydrocarbons]
 [2021  J. Phys. D: Appl. Phys.  Raman spectra of hydrocarbons under extreme conditions of pressure and temperature: a firstprinciples study]
 autocorrelation of CH bonds and HCH angles
 [2021  J. Chem. Inf. Model.  Predicting Infrared Spectra with Message Passing Neural Networks]
 [2021  JCTC  Transfer Learning to CCSD(T): Accurate Anharmonic Frequencies from Machine Learning Models]
 [2021  My Trajectory in Molecular Reaction Dynamics and Spectroscopy]
 [2021  JCTC  Christian Ochsenfeld et.al.]: Quantitative Comparison of Experimental and Computed IRSpectra Extracted from Ab Initio Molecular Dynamics
 [2020  Faraday Discuss.  temperature dependence of the vibrational spectrum of porphycene]: MD vs TRPMD for NH streching mode
 Neural network PES and KRR dipole moment surface
 1D doublewell to explain the underlying physics
 [2020  Faraday Discuss.  Which quantum statistics–classical dynamics method is best for water?]
 MD, TRPMD, CMD, QCMD, LSCIVR
 Gas, liquid and ice, using the qTIP4P/F model potential
 150K and 300K
 [2020  PCCP  Raman spectrum and polarizability of liquid water from deep neural networks]
 Deepkit and DPgen
 Reducing the statistical error in a spectrum calculation by MLMD
 Temperature dependence of the OH/OD stretch
 [2020  JCP  Onthefly ab initio semiclassical evaluation of vibronic spectra at finite temperature]
 [2020  JPCA  Snakes on the Rungs of Jacob’s Ladder: Anomalous Vibrational Spectra from DoubleHybrid DFT Methods]
 The double hybrid functional predicts very unphysical frequencies for some CHNO species
 [2020  JPCA  Comprehensive Benchmark Results for the Accuracy of Basis Sets for Anharmonic Molecular Vibrations]
 The accuracy and convergence of Gaussin's splitvalence basis sets (631G, 6311+G(d, p), ...) are systemically investigated. def2TZVPP is a large basis set and considered as one reference basis set.
 Benchmark on def2 basis sets and Jensen's basis sets are coming soon
 [2020 JCTC  Dual Basis Approach for Ab Initio Anharmonic Calculations of Vibrational Spectroscopy: Application to Microsolvated Biomolecules]
 [2020  JPCL  Glycolic Acid as a Vibrational Anharmonicity Benchmark]
 Harmonic and VPT2 studies of Glycolic Acid (C2H4O3) using B3LYPD3, B2PLYPD3, MP2, and CCSD(T)
 B3LYPD3 has a good error cancellation and thus shows good harmonic frequencies.
 With the VPT2 approach, CCSD(T) achieves the lowest error compared with the experiment.
 [2020  PCCP  On the separability of largeamplitude motions in anharmonic frequency calculations]
 [2019  PNAS  Nanoscale infrared imaging analysis of carbonaceous chondrites to understand organicmineral interactions during aqueous alteration]
 AFMbased IR measurements of minerals with a high spatial resolution. The cantilever of AFM has a characteristic thermal expansion when the sample absorbs light. The oscillation amplitude of the cantilever is directly proportional to the amount of light absorbed.
 Slices (70100nm) Meteorites
 Why different substrates for different meteorites
 [2019  Faraday Discussions]:Conformational assignment of gas phase peptides and their Hbonded complexes using farIR/THz: IRUV ion dip experiment, DFTMD spectroscopy, and graph theory for mode assignment
 [2018  ACS Omega  André F. RodriguesOliveira et.al.]: Evaluation of Common Theoretical Methods for Predicting Infrared Multiphotonic Dissociation Vibrational Spectra of Intramolecular HydrogenBonded Ions [IRMPDvibxc]
 [2018  PCCP  Spinstate dependence of the structural and vibrational properties of solvated iron(ii) polypyridyl complexes from AIMD simulations: aqueous [Fe(bpy)3]Cl2, a case study]
 [2017  JCTC  Simulation of Vibronic Spectra of Flexible Systems: Hybrid DVRHarmonic Approaches]
 Treating largeamplitude motion (LAM) within the static anharmonic approach
 [2017  JCTC  Christian Ochsenfeld et.al. Efficient and Accurate Born–Oppenheimer Molecular Dynamics for Large Molecular Systems]
 [2017  JCTC  Daria R. Galimberti et.al.]: Combining Static and Dynamical Approaches for Infrared Spectra Calculations of Gas Phase Molecules and Clusters
 [2017  PhD thesis  Martin Thomas]: Theoretical Modeling of Vibrational Spectra in the Liquid Phase
 As the intensity ratio of the two bands clearly depends on the energy, classical dynamics provides again not a quantitative estimation of the quantum spectrum, but a qualitative insight even into special effects like Fermi resonance is possible.
 In general, MD simulations with classical particles contain a description of anharmonicity effects in qualitative agreement with the corresponding quantum particles. This allows to observe overtones and combination bands in the power spectra.
 [2016  JPCA  Huan Wang et.al. Complete Assignment of the Infrared Spectrum of the GasPhase Protonated Ammonia Dimer]
 AIMD vs VPT2
 Combination bands, overtone, resonances from AIMD
 [2015  Chemistry  Interplay of Exciton Coupling and LargeAmplitude Motions in the Vibrational Circular Dichroism Spectrum of Dehydroquinidine]
 Boltzman weighting of different conformers
 Largeamplitude motions in dihedral angles
 [2014  JPCA  Approximate FirstPrinciples Anharmonic Calculations of Polyatomic Spectra Using MP2 and B3LYP Potentials: Comparisons with Experiment]
 [2014  PCCP  Dispersion corrected DFT approaches for anharmonic vibrational frequency calculations: nucleobases and their dimers]
 The most promising method to get anharmonic IR of large systems is VPT2 with the hybrid functional (B3LYP) and medium size (SNSD) basis sets.
 Dispersion interactions should be properly treated in order to compute IR of weakly bound molecular complexes.
 Empirical scaling factors
 simple/universal scaling factors for all fundamental frequencies
 sophisticated scaling factors: different scaling factors are applied for different modes without the guarantee of transferability.
 Zeropoint energies need different scaling factors than vibrational frequencies.
 vibrational frequencies can be scaled but not intensities.
 In order to assign vibrational modes, one has to visualize the atomic displacements along normal modes and compare with experiments.
 CC/DFT hybrid approaches for calculating anharmonic IR for mediumsized molecules: harmonic at couple cluster level and anharmonic corrections at B3LYP level
 CCSD(T)/CBS is limited to 1015 atoms for harmonic calculations while B2PLYP can be used for larger systems.
 Uracil test case
 a good test system which does not have the problem in outofplane NH2 vibration, has highly accurate theoretical results (CCSD) available.
 VPT2 with B3LYP provides accurate frequencies because of the good quality of both harmonic and anharmonic contributions, not a lucky error cancellations.
(more proofs?)
 B3LYPD3 provides almost identical results as B3LYP for both harmonic and anharmonic calculations.
 B3LYPDCP, M062X and wB97XD yield less accurate harmonic frequencies.
 M062X and wB97XD are worsened for anharmonic corrections.
 B3LYP and B3LYPD3 provide nearly equivalent harmonic and anharmonic frequencies for 6 different nucleobases.
 [2013  JCTC  Anharmonic Vibrational Frequency Calculations Are Not Worthwhile for Small Basis Sets]
 [2013  PCCP  Sergei D. Ivanov, Dominik Marx el.al.]:Theoretical spectroscopy using molecular dynamics: theory and application to CH5+ and its isotopologues
 Theory behind getting IR from MD
 CH$_5^+$ is chosen as it is a typical floppy system with an unusually flat PES and undergoes intricate largeamplitude motion "hydrogen scrambling"
 Mode assignment from AIMD trajectories (difficult for floppy molecules)
 Conformerweighted AIMD spectra
 [2012  JCTC  Gerald Mathias et.al. Infrared Spectroscopy of Fluxional Molecules from (ab Initio) Molecular Dynamics: Resolving LargeAmplitude Motion, Multiple Conformations, and Permutational Symmetries]
 Band assignments (largeamplitude motion) from MD trajectories, taking CH$_5^+$ as an example
 [2011  JCTC  Gerald Mathias et.al. Generalized Normal Coordinates for the Vibrational Analysis of Molecular Dynamics Simulations]
 [2011  JCTC  Direct Calculations of Mid and NearIR Absorption and Circular Dichroism Spectra of Chiral Molecules Using QM/MM Molecular Dynamics Simulation Method]
 first overtone and combination bands
 [2010  JCTC  Harmonic and Anharmonic Vibrational Frequency Calculations with the DoubleHybrid B2PLYP Method: Analytic Second Derivatives and Benchmark Studies]
 [2009  JCP  Dominik Marx et.al. On the applicability of centroid and ring polymer path integral molecular dynamics for vibrational spectroscopy]
 [2006  JACS  Popular Theoretical Methods Predict Benzene and Arenes To Be Nonplanar]
 MP2 produces imaginary frequencies for planar benzene.
 MP2, MP3, and CISD levels favor nonplanar benzene, while RHF, B3LYP, and BLYP methods do not have this problem.
 Atomic natural orbital (ANO) basis sets can minimize basis set superposition errors
 [2006  PCCP  Dominik Marx et.al. Understanding hydrogen scrambling and infrared spectrum of bare CH5+ based on ab initio simulations]
 Massive NoseHoover chain thermostat guarantees that all modes are properly thermalized at the target temperature
 [2004  JCP  Dominik Marx et.al. Quantum corrections to classical timecorrelation functions: Hydrogen bonding and anharmonic floppy modes]
 [2004  JCPA  Accurate Vibrational Spectra of Large Molecules by Density Functional Computations beyond the Harmonic Approximation: The Case of Azabenzenes]
 The inclusion of HartreeFock exchanges in exchange correlation in necessary for vibrational properties of organic molecules.
 For azabenzenes, anharmonic corrections do not have huge impacts on zeropoint energies, but cannot be ignored in quantitative studies.
 [2003  JPCB  Ab Initio Molecular Dynamics Computation of the Infrared Spectrum of Aqueous Uracil]
 Fermi resonances are not well described in our classical approach
 IR in solution, contribution from solvent and solute
Water
 [2024  Faraday Discuss.  Firstprinciples spectroscopy of aqueous interfaces using machinelearned electronic and quantum nuclear effects]
 MD, CMD, TRPMD, Te PIGS
 IR, isotropic Raman, anisotropic Raman
 liquid water, polycrystalline ice at 150 K, water–air interface at 300 K, confined water
 [2020  Faraday Discuss. Which quantum statistics–classical dynamics method is best for water?]
 classical, TRPMD, CMD, QCMD, LSCIVR
 gas phase water, liquid water at 300 K, ice Ih at 150 K
 [2019  JCP  Pathintegral dynamics of water using curvilinear centroids]
 CMD, TRPMD, QCMD
 qTIP4P/f
CO2
SiO2
Researcher
Basic:
fundamental band
: the vibration from the vibrational ground state to the vibrational first excited state
overtone band
: the vibration from the vibrational ground state to the highorder (> 1) excited vibrational states
combination band
: the superposition of two fundamental bands. The combination band is forbidden by harmonic oscillator selection rules.
hot band
: the transition between two excited vibrational states
resonance
: a resonance occurs when the energy of the normal modes or the sum of modes are close to one another in energy.
 Harmonic approximation will overestimate the energies of the fundamentals due to the neglect of anharmonicity.
floppy molecule
: the structure is not at all rigid, but rather constantly changing. It is also called fluctuating
and fuxional
. The bond connectivity has not been modified during the dynamical changes.
 CH$_5^+$: 2005ScienceDominik Marx
 Various cis/trans conformations by thermally driven isomerization transitions
 Methyl groups can easily undergo internal rotations
 VPT2 needs
6N11
Hessian calculations for a nonlinear system where 2*(3N6)+1 = 6N12+1 = 6N11
. The 6N11
comes about due to the use of a twosided finite difference approach to find the derivatives of the force constants with respect to displacements along each of the normal modes, plus one calculation for the unperturbed geometry.
 IR of molecular systems with increasing size and in different environments.
VPT2
Packages
MD
Phonons
Phonons are collective atomic motions in solids (either crystaline or amorphous).
They are normally described in reciprocal space using wave vectors.
Due to the periodic boundary conditions, the Hessian is in principle a matrix of infinite size.
The finitedisplacement supercell approach is a common method to calculate phonon related properties.
If longrange vibrations are important, a large supercell is needed.
 Quantum espresso's phonon module
ph.x
has not yet supported hybrid functional for DFPT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Error in routine phq_readin (1):
The phonon code with hybrid functionals is not yet available
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 DFTD3 Hessian available in
ph.x
when version >= 7.2
 In a planewave DFT code, HellmannFeynman force already contains contributions from neighboring images (cells): $F_I(l, u) = \sum_{l^\prime, J, v}\Phi_{I,J}(l, u, l^\prime, v)U_j(l^\prime, v)$. The sum is taken over all supercell images, all atoms and displacement directions. (Check P. Blaha's Wien2K tutorial on phonons)
 It is possible to compute anharmonic phonon through higher order energy derevatites as described in K. Parlinski, Phys. Rev. B 98(5), 054305 (2018)
 The wavevector $\mathbf{q}$ is commensurate with the super cell if $e^{i\mathbf{q}\mathbf{R}} = 1$
 Gamma point is commensurate with any choice of supercell
 Check more discussions from Krzysztof Parlinski's papers
 At gamma point, dynamic matrix calculation is exactly the same as obtaining Hessian matrix of a molecule without PBC.
 Unstable crystalline structures show soft modes with imaginary frequencies.
 For a $d$dimensional crystal with $n$ atoms in the unit cell, there are ($nd$) phonons in total
 $d$ acoustic phonons
 $(n1)d$ optical phonons which normally have larger energies than acoustic phonons
 Phonon unit conversion and PHONONPY unit
 PHONON Software by Krzysztof Parlinski
 PHON by Dario Alfè and PHON tutorial
 CASTEP Phonon tutorial 2021
 FHIAIMs Phonon by Christian Carbogno 2021
Supercell
Theory of direct method to calculate phonons from HellmannFeynman forces is documented in details in Phys. Rev. Lett. 78(21), 4063–4066 (1997) with cubic ZrO2 as an example
IR and Raman spectroscopy
Energies for differnt objects are
 Phonon: $E_\text{phonon} = \hbar \omega(\mathbf{q}, j)$. $\mathbf{q}$ is the wave vector and $j$ th branch of the phonon
 Phontons: $E_\text{photon} = \hbar c \mathbf{k}$. $c$ is the speed of light and $\mathbf{k}$ is the wave vector
 $E_i = h \nu_i = \dfrac{h\omega_i}{2\pi} = \hbar \omega _ i$ ($\nu _ i$ is the frequency and $\omega _ i$ is the angular frequency)
Some important gradients for computing IR spectra of solids are

Born effective charge on atoms $I$ is $Z _ {I,\alpha\beta}^{\star} = \dfrac{\Omega}{e} \dfrac{\partial P _ \alpha}{\partial R _ {I,\beta}}  _ {\Epsilon=0} = \dfrac{\partial F _ {I, \beta}}{\partial \Epsilon_\alpha}$

IR cross section of the vibrational mode $\nu$ is computed as
$$
I_{IR}(\nu) = \sum_{\alpha=1}^3 \sum_{I=1}^{N_{atoms}} \sum _ {\beta=1}^3 Z _ {I,\alpha\beta}^{\star} u_{\nu, I, \beta}^2
$$
 $u _ {\nu, I, \beta}$ is massweighted eigenvector of the vibrational mode $\nu$ at atom $I$ along the direction $\beta$.
d
 Github  IR intensity unit
 Github  IR VASP
 Github  PhonopySpectroscopy and Phys. Chem. Chem. Phys., 2017,19, 1245212465
 Raman and IR spectroscopy in materials science. Raman and IR spectroscopy in materials science. Symmetry analysis of normal phonon modes 2009
 Infrared Spectroscopy from Phonons
 2017  Quantum Espresso workshop  Ab Initio Simulation of Infrared and Raman Spectroscopy
Phonopy
A workflow to do phonon calculations at q=0 is (do everything in the same directory)
1. Optimize structure with very low fmax and convert it to POSCAR
2. Generate displacements via phonopy d dim="1 1 1"
(add v
if you want more structure/symmetry information printed) and the outputs are POSCAR001 POSCAR002 ...
and phonopy_disp.yaml
3. Compute atomic force for each POSCARXXX
with any DFT code (e.g. GPAW)
4. Prepare the force constant file FORCE_SETS
 Construct by hand (use eV/Å as the force unit)
 phonopy f OUT1 OUT2 ... OUT3
depending on the DFT calculator
 Get dynamical matrix
 Command line tool:
phonopy qpoints="0 0 0" writedm c phonopy_disp.yaml
or rename POSCAR
to soemthing else and run phonopy qpoints="0 0 0" writedm
 Python API
import phonopy
# force constants will be loaded from FORCE_SETs by default
phonon = phonopy.load('phonopy_disp.yaml')
q = [0, 0, 0]
# already massweighted, with a unit of eV/Å^2/amu
dynamical_matrix = phonon.get_dynamical_matrix_at_q(q)
# harmonic frequencies in THz
freqs = phonon.get_frequencies(q)
Practical accuracy
 Single point calculation
 Grid setting for calculating exchangecorrelation term
 SCF energy convergence criteria (Accurate force needs wellconverged SCF)
 Number of kpoints
 Equilibrium structure optimization
 Max atomic force
 Cell volume (Investigate effect of pressure on the vibrational spectra of solids)
 Numerical calculation of 2nd derivative based on finite displacements
 Displacement step (0.001 Å is normally okay)
Dielectric property
The static dielectric tensor $\epsilon_{ij}$ (a 3x3 matrix) is computed as a sum over the ionic $\epsilon^0_{ij}$ and electronic high frequency $\epsilon^\infty_{ij}$ contributions
$$
\epsilon_{ij} = \epsilon^0_{ij} + \epsilon^\infty_{ij}
$$
MISC
 For medium size PAHs, the addition of
D3
has minor effects on band positions and intensities. There might be divergences for some vibrational peaks.
SPECTRO program
not available online: Gaw, J. F.; Willets, A.; Green, W. H.; Handy, N. C. in Advances in Molecular Vibrations and Collision Dynamics; Bowman, J. M.; Ratner, M. A., Eds.; JAI Press, Inc.: Greenwich, CT, 1991; pp 170– 185.
 IR from AIMD intrinsically contains some anharmonic effects. However, other anharmonic effects like mode coupling leading to combination bands and Fermi resonances are still not included because the nuclear motion is treated classically.
 J. Horníček, P. Kaprálová, and P. Bouř, J. Chem. Phys. 127, 084502 (2007).
 J. Hudecová, K.H. Hopmann, and P. Bouř, J. Phys. Chem. B 116, 336 (2012).
 M. Thomas, M. Brehm, R. Fligg, P. Vöhringer, and B. Kirchner, Phys. Chem. Chem. Phys. 15, 6608 (2013).
 S.A. Fischer, T.W. Ueltschi, P.Z. ElKhoury, A.L. Mifflin, W.P. Hess, H.F. Wang, C.J. Cramer, and N. Govind, J. Phys. Chem. B 120, 1429 (2016).