Anharmonic Vibrational States of Solids from DFT Calculations. Part I: Description of the Potential Energy Surface

Unbiased Comparison between Theoretical and Experimental Molecular Structures and Properties: Toward an Accurate Reduced-Cost Evaluation of Vibrational Contributions

The tremendous development of hardware and software is constantly increasing the role of quantum chemical (QC) computations in the assignment and interpretation of experimental results. However, an unbiased comparison between theory and experiment requires the proper account of vibrational averaging effects. In particular, high-resolution spectra in the gas phase are now available for molecules containing up to about 50 atoms, which are too large for a brute-force approach with the available QC methods of sufficient accuracy. In the present paper, we introduce hybrid approaches, which allow the accurate evaluation of vibrational averaging effects for molecules of this size beyond the harmonic approximation, with special attention being devoted to rotational constants. After the validation of new tools for relatively small molecules, the β-estradiol hormone and a prototypical molecular motor have been considered to witness the feasibility of accurate computations for large molecules.

First-Principles Anharmonic Infrared and Raman Vibrational Spectra of Materials: Fermi Resonance in Dry Ice

We introduce a computational tool for the quantum-mechanical simulation of anharmonic infrared and Raman vibrational spectra of materials. The approach, implemented in the CRYSTAL software, stems from Taylor's expansion of the potential energy surface (PES) on the basis of normal modes up to cubic and quartic terms. The PES can be sampled with four different numerical schemes at the level of density functional theory (DFT), with local, generalized-gradient, and hybrid density functional approximations. Anharmonic states are obtained by solving Shrödinger's nuclear equation with either the vibrational self-consistent field (VSCF) or vibrational configuration interaction (VCI) methods. Nuclear quantum effects (NQEs) are thus fully accounted for. Infrared intensities are computed numerically through a Berry phase approach or analytically through a coupled-perturbed (CP) approach. Raman intensities are computed analytically via the CP approach. A variety of anharmonic features of vibrational spectra of materials can be simulated, including band shifts, combination bands, overtones, resonances (first-order Fermi, second-order Darling-Dennison), and hot bands. We showcase the effectiveness of the approach on the description of a first-order Fermi resonance (FR) in CO2 dry ice: a challenging test-case given that the FR occurs in the Raman spectrum, requires NQEs, and involves two- and three-mode couplings. Fundamental mechanistic differences with respect to the well-known FR in molecular CO2 are addressed. This application represents the first quantum-mechanical, periodic description of FR in dry ice.

A Critical Assessment on Calculating Vibrational Spectra in Nanostructured Materials

Vibrational spectroscopy is an omnipresent spectroscopic technique to characterize functional nanostructured materials such as zeolites, metal-organic frameworks (MOFs), and metal-halide perovskites (MHPs). The resulting experimental spectra are usually complex, with both low-frequency framework modes and high-frequency functional group vibrations. Therefore, theoretically calculated spectra are often an essential element to elucidate the vibrational fingerprint. In principle, there are two possible approaches to calculate vibrational spectra: (i) a static approach that approximates the potential energy surface (PES) as a set of independent harmonic oscillators and (ii) a dynamic approach that explicitly samples the PES around equilibrium by integrating Newton's equations of motions. The dynamic approach considers anharmonic and temperature effects and provides a more genuine representation of materials at true operating conditions; however, such simulations come at a substantially increased computational cost. This is certainly true when forces and energy evaluations are performed at the quantum mechanical level. Molecular dynamics (MD) techniques have become more established within the field of computational chemistry. Yet, for the prediction of infrared (IR) and Raman spectra of nanostructured materials, their usage has been less explored and remain restricted to some isolated successes. Therefore, it is currently not a priori clear which methodology should be used to accurately predict vibrational spectra for a given system. A comprehensive comparative study between various theoretical methods and experimental spectra for a broad set of nanostructured materials is so far lacking. To fill this gap, we herein present a concise overview on which methodology is suited to accurately predict vibrational spectra for a broad range of nanostructured materials and formulate a series of theoretical guidelines to this purpose. To this end, four different case studies are considered, each treating a particular material aspect, namely breathing in flexible MOFs, characterization of defects in the rigid MOF UiO-66, anharmonic vibrations in the metal-halide perovskite CsPbBr3, and guest adsorption on the pores of the zeolite H-SSZ-13. For all four materials, in their guest- and defect-free state and at sufficiently low temperatures, both the static and dynamic approach yield qualitatively similar spectra in agreement with experimental results. When the temperature is increased, the harmonic approximation starts to fail for CsPbBr3 due to the presence of anharmonic phonon modes. Also, the spectroscopic fingerprints of defects and guest species are insufficiently well predicted by a simple harmonic model. Both phenomena flatten the potential energy surface (PES), which facilitates the transitions between metastable states, necessitating dynamic sampling. On the basis of the four case studies treated in this Review, we can propose the following theoretical guidelines to simulate accurate vibrational spectra of functional solid-state materials: (i) For nanostructured crystalline framework materials at low temperature, insights into the lattice dynamics can be obtained using a static approach relying on a few points on the PES and an independent set of harmonic oscillators. (ii) When the material is evaluated at higher temperatures or when additional complexity enters the system, e.g., strong anharmonicity, defects, or guest species, the harmonic regime breaks down and dynamic sampling is required for a correct prediction of the phonon spectrum. These guidelines and their illustrations for prototype material classes can help experimental and theoretical researchers to enhance the knowledge obtained from a lattice dynamics study.

CRYSTAL23: A Program for Computational Solid State Physics and Chemistry

The Crystal program for quantum-mechanical simulations of materials has been bridging the realm of molecular quantum chemistry to the realm of solid state physics for many years, since its first public version released back in 1988. This peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals (LCAO) approach and from the corresponding efficiency in the evaluation of the exact Fock exchange series. In particular, this has led to the implementation of a rich variety of hybrid density functional approximations since 1998. Nowadays, it is acknowledged by a broad community of solid state chemists and physicists that the inclusion of a fraction of Fock exchange in the exchange-correlation potential of the density functional theory is key to a better description of many properties of materials (electronic, magnetic, mechanical, spintronic, lattice-dynamical, etc.). Here, the main developments made to the program in the last five years (i.e., since the previous release, Crystal17) are presented and some of their most noteworthy applications reviewed.

Efficient calculation of derivatives of integrals in a basis of non-separable Gaussians

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched.

Machine learning in computational chemistry: interplay between (non)linearity, basis sets, and dimensionality

Machine learning (ML) based methods and tools have now firmly established themselves in physical chemistry and in particular in theoretical and computational chemistry and in materials chemistry. The generality of popular ML techniques such as neural networks or kernel methods (Gaussian process and kernel ridge regression and their flavors) permitted their application to diverse problems from prediction of properties of functional materials (catalysts, solid state ionic conductors, etc.) from descriptors to the building of interatomic potentials (where ML is currently routinely used in applications) and electron density functionals. These ML techniques are assumed to have superior expressive power of nonlinear methods, and are often used "as is", with concepts such as "non-parametric" or "deep learning" used without a clear justification for their need or advantage over simpler and more robust alternatives. In this Perspective, we highlight some interrelations between popular ML techniques and traditional linear regressions and basis expansions and demonstrate that in certain regimes (such as a very high dimensionality) these approximations might collapse. We also discuss ways to recover the expressive power of a nonlinear approach and to help select hyperparameters with the help of high-dimensional model representation and to obtain elements of insight while preserving the generality of the method.

Anharmonic Coupling of Stretching Vibrations in Ice: A Periodic VSCF and VCI Description

The anharmonicity of O-H stretching vibrations of water ice is characterized by use of a periodic implementation of the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, which take phonon-phonon couplings explicitly into account through numerical evaluation of high-order terms of the nuclear potential. The low-temperature, proton-ordered phase of water ice (namely, ice XI) is investigated. The net effect of a coupled anharmonic treatment of stretching modes is not just a rigid blue-shift of the respective harmonic spectral frequencies but rather a complex change of their relative spectral positions, which cannot be captured by simple scaling strategies based on harmonic calculations. The adopted techniques allow for a hierarchical treatment of anharmonic terms of the nuclear potential, which is key to an effective identification of leading factors. We show that the anharmonic independent-mode approximation─only describing the "intrinsic anharmonicity" of the O-H stretches─is unable to capture the correct physics, and that couplings among O-H stretches must be described. Inspection of harmonic normal coordinates allows identification of specific features of the O-H stretching motions which most likely enable strong mode-mode couplings. Finally, by coupling O-H stretches to all other possible modes of ice XI (THz collective vibrations, molecular librations, bendings), we identify specific types of motion which significantly affect O-H stretching states: in particular, molecular librations are found to affect the stretching states more than molecular bendings.

Phonon anharmonicity in binary chalcogenides for efficient energy harvesting

Thermoelectric (TE) materials have received much attention due to their ability to harvest waste heat energy. TE materials must exhibit a low thermal conductivity (κ) and a high power factor (PF) for efficient conversion. Both factors define the figure of merit (ZT) of the TE material, which can be increased by suppressing κ without degrading the PF. Recently, binary chalcogenides such as SnSe, GeTe, and PbTe have emerged as attractive candidates for thermoelectric energy generation at moderately high temperatures. These materials possess simple crystal structures with low κ in their pristine forms, which can be further lowered through doping and other approaches. Here, we review the recent advances in the temperature-dependent behavior of phonons and their influence on the thermal transport properties of chalcogenide-based TE materials. Because phonon anharmonicity is one of the fundamental contributing factors for low thermal conductivity in SnSe, Sb-doped GeTe, and related chalcogenides, we discuss complementary experimental approaches such as temperature-dependent Raman spectroscopy, inelastic neutron scattering, and calorimetry to measure anharmonicity. We further show how data gathered using multiple techniques helps us understand and engineer better TE materials. Finally, we discuss the rise of machine learning-aided efforts to discover, design, and synthesize TE materials of the future.

On the use of a volume constraint to account for thermal expansion effects on the low-frequency vibrations of molecular crystals

A volume-constraint method is presented as a means to capture the influence of thermal expansion on the low-frequency vibrations in molecular crystals. In particular, the room-temperature terahertz absorption spectra of L-tartaric acid, α-lactose monohydrate, and α-para-aminobenzoic acid (PABA) have been simulated using dispersion-corrected, solid-state density functional theory (DFT-D). By comparing the normal modes obtained with a unit cell optimised without constraints to those obtained with a unit cell optimised while constrained to keep its experimental volume, wholesale improvements to the resultant spectrum is achieved when using the constrained geometry by inhibiting cell contraction. These improvements are demonstrated over a range of popular density functionals and basis sets up to triple-zeta complexity. A correlation method is then presented as a means to quantitatively compare the vibrational pattern of normal modes obtained from both unit cells. This analysis reveals that thermal expansion can effect the character and relative frequency of normal modes, with the choice of geometry ultimately affecting the assignment of the experimental absorptions. The sensibility of using the experimental volume as an approximation is then discussed, where it is speculated that large basis sets or hybrid functionals are necessary to ensure that the thermal expansion effect is not overestimated. The low-frequency absorption spectrum of PABA is then fully characterised using the PBE-D3BJ/6-311G(2d,2p) method.

Computational vibrational spectroscopy of molecule-surface interactions: what is still difficult and what can be done about it

Interactions of molecules with solid surfaces are responsible for key functionalities for a range of currently actively pursued technologies, including heterogeneous catalysis for synthesis or decomposition of molecules, sensitization, surface...

An ab initio anharmonic approach to IR, Raman and SFG spectra of the solvated methylammonium ion

The methylammonium ion (CH₃NH₃⁺, or noted as MA-H⁺) is one of the smallest organic ammonium ions that play important roles in organic-inorganic halide perovskites. Despite the simple structure, the vibrational spectra of MA-H⁺ exhibit complicated features in the 3 μm region which are sensitive to the solvation environment. In the present work, we have applied the ab initio anharmonic algorithm at the CCSD/aug-cc-pVDZ level to simulate the IR and Raman spectra of the solvated methylammonium ion, MA-H⁺⋯X₃, where X denotes the solvent molecules, to understand the Fermi resonance mechanism in which the overtones of NH bending modes are coupled with the fundamentals of NH stretching modes. The spectral features of the solvated clusters with proper solvent species resemble those observed in the perovskite crystal, indicating that they have similar solvation environments and hydrogen bond interactions. Therefore, a linkage between the gas-phase cluster models and the condensed-phase materials can be established, and our simulations and anharmonic analyses help in interpreting the spectral assignments of the observed IR and Raman spectra of perovskites reliably. Furthermore, we have extended this approach to the SFG spectra to demonstrate the selective appearance of bands depending on both the beam polarization configurations and the symmetry of vibrational modes.

Advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase

The purpose of this review is to demonstrate advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase. Molecular spectroscopy, particularly vibrational spectroscopy and electronic spectroscopy, has been used extensively for a wide range of areas of chemical sciences and materials science as well as nano- and biosciences because it provides valuable information about structure, functions, and reactions of molecules. In the meantime, quantum chemical approaches play crucial roles in the spectral analysis. They also yield important knowledge about molecular and electronic structures as well as electronic transitions. The combination of spectroscopic approaches and quantum chemical calculations is a powerful tool for science, in general. Thus, our article, which treats various spectroscopy and quantum chemical approaches, should have strong implications in the wider scientific community. This review covers a wide area of molecular spectroscopy from far-ultraviolet (FUV, 120-200 nm) to far-infrared (FIR, 400-10 cm-1)/terahertz and Raman spectroscopy. As quantum chemical approaches, we introduce several anharmonic approaches such as vibrational self-consistent field (VSCF) and the combination of periodic harmonic calculations with anharmonic corrections based on finite models, grid-based techniques like the Numerov approach, the Cartesian coordinate tensor transfer (CCT) method, Symmetry-Adapted Cluster Configuration-Interaction (SAC-CI), and the ZINDO (Semi-empirical calculations at Zerner's Intermediate Neglect of Differential Overlap). One can use anharmonic approaches and grid-based approaches for both infrared (IR) and near-infrared (NIR) spectroscopy, while CCT methods are employed for Raman, Raman optical activity (ROA), FIR/terahertz and low-frequency Raman spectroscopy. Therefore, this review overviews cross relations between molecular spectroscopy and quantum chemical approaches, and provides various kinds of close-reality advanced spectral simulation for condensed phases.

Thermoelasticity of Flexible Organic Crystals from Quasi-harmonic Lattice Dynamics: The Case of Copper(II) Acetylacetonate

A computationally affordable approach, based on quasi-harmonic lattice dynamics, is presented for the quantum-mechanical calculation of thermoelastic moduli of flexible, stimuli-responsive, organic crystals. The methodology relies on the simultaneous description of structural changes induced by thermal expansion and strain. The complete thermoelastic response of the mechanically flexible metal-organic copper(II) acetylacetonate crystal is determined and discussed in the temperature range 0-300 K. The elastic moduli do not just shrink with temperature but they do so anisotropically. The present results clearly indicate the need for an explicit account of thermal effects in the simulation of mechanical properties of elastically flexible organic materials. Indeed, predictions from standard static calculations on this flexible metal-organic crystal are off by up to 100%.

Understanding the solubilization of Ca acetylide with a new computational model for ionic pairs

The unique reactivity of the acetylenic unit in DMSO gives rise to ubiquitous synthetic methods. We theoretically consider CaC2 solubility and protolysis in DMSO and formulate a strategy for CaC2 activation in solution-phase chemical transformations. For this, we use a new strategy for the modeling of ionic compounds in strongly coordinating solvents combining Born-Oppenheimer molecular dynamics with the DFTB3-D3(BJ) Hamiltonian and static DFT computations at the PBE0-D3(BJ)/pob-TZVP-gCP level. We modeled the thermodynamics of CaC2 protolysis under ambient conditions, taking into account its known heterogeneity and considering three polymorphs of CaC2. We give a theoretical basis for the existence of the elusive intermediate HC[triple bond, length as m-dash]C-Ca-OH and show that CaC2 insolubility in DMSO is of thermodynamic nature. We confirm the unique role of water and specific properties of DMSO in CaC2 activation and explain how the activation is realized. The proposed strategy for the utilization of CaC2 in sustainable organic synthesis is outlined.

Resolving Anharmonic Lattice Dynamics in Molecular Crystals with X-Ray Diffraction and Terahertz Spectroscopy

Molecular crystals are increasingly being used for advanced applications, ranging from pharmaceutics to organic electronics, with their utility dictated by a combination of their three-dimensional structures and molecular dynamics-with anharmonicity in the low-frequency vibrations crucial to numerous bulk phenomena. Through the use of temperature-dependent x-ray diffraction and terahertz time-domain spectroscopy, the structures and dynamics of a pair of isomeric molecular crystals exhibiting nearly free rotation of a CF_{3} functional group at ambient conditions are fully characterized. Using a recently developed solid-state anharmonic vibrational correction, and applying it to a molecular crystal for the first time, the temperature-dependent spatial displacements of atoms along particular terahertz modes are obtained, and are found to be in excellent agreement with the experimental observations, including the assignment of a previously unexplained absorption feature in the low-frequency spectrum of one of the solids.

Calculation of Anharmonic IR and Raman Intensities for Periodic Systems from DFT Calculations: Implementation and Validation

An extension of the CRYSTAL program is presented allowing for calculations of anharmonic infrared (IR) intensities and Raman activities for periodic systems. This work is a follow-up of two papers devoted to the computation of anharmonic vibrational states of solids from DFT (density functional theory) calculations (Erba et al. J. Chem. Theory Comput. 2019, 15, 3755-3765 and Erba et al. J. Chem. Theory Comput. 2019, 15, 3766-3777). The approach presented here relies on the evaluation of integrals of the dipole moment and polarizability operators over anharmonic wave functions obtained from either VSCF (vibrational self-consistent field) or VCI (vibrational configuration interaction) calculations. With this extension, the program now allows for a more complete characterization of the vibrational spectroscopic features of solids within the density functional theory. In particular, it is able (i) to provide reliable positions and intensities for the most intense spectral features and (ii) to check whether a first overtone or a combination band has a nonvanishing IR intensity or Raman activity. Therefore, it becomes possible to assign the transition(s) corresponding to satellite peak(s) around a fundamental transition or the overtones or combination bands that may be as intense as their corresponding fundamental transitions through the strongest mode-mode couplings, as in so-called Fermi resonances. The present method is assessed on two molecular systems, H₂O and H₂CO, as well as on two solid state cases, boron hydrides BH₄ and their deuterated species BD₄ in a crystalline environment of alkali metals (M = Na, K). The solid state cases are particularly insightful as, in the B-H (or B-D) stretching region here considered, they exhibit many spectral features entirely due to anharmonic effects: two out of three in the IR spectrum and four out of six in the Raman spectrum. All IR and Raman active overtones and combination bands experimentally observed are correctly predicted with our approach. The effect of the adopted quantum-chemical model (DFT exchange-correlation functional/basis set) for the electronic structure calculations on the computed spectra is discussed and found to be significant, which suggests some special care is needed for the analysis of subtle spectral features.

Elucidating the structure and dynamics of CO ad-layers on MgO surfaces

The combination of quantum-mechanical simulations and infrared absorption spectroscopy measurements provides a clear picture for a long standing puzzle in surface science: the actual structure and vibrational dynamics of the low-temperature ordered CO monolayer adsorbed on (001) MgO surfaces. The equilibrium structure of the commensurate (4 × 2) adsorbed phase consists of three CO molecules per primitive cell (surface coverage of 75%) located at two inequivalent sites: one molecule seats upright on top of a Mg site while two molecules, tilted off the normal to the surface, are symmetrically positioned relative to the upright one with anti-parallel projections on the surface. This configuration, long believed to be incompatible with measured polarization infrared spectra, is shown to reproduce all observed spectral features, including a new, unexpected one: the vanishing anharmonicity of CO stretching modes in the monolayer.

Negative thermal expansion of Cu2O studied by quasi-harmonic approximation and cubic force-constant method

Cubic cuprous oxide, Cu₂O, is characterized by a peculiar structural response to temperature: it shows a relatively large negative thermal expansion below 250 K, then followed by a positive thermal expansion at higher temperatures. The two branches of its thermal expansion (negative and positive) are almost perfectly symmetric at low temperatures, with the minimum of its lattice parameter at about 250 K and with the lattice parameter at 500 K almost coinciding with that at 0 K. We perform lattice-dynamical quantum-mechanical calculations to investigate the thermal expansion of Cu₂O. Phonon mode-specific Grüneisen parameters are computed, which allows us to identify different spectral regions of atomic vibrations responsible for the two distinct regimes of thermal expansion. Two different computational approaches are explored, their results compared, and their numerical aspects critically assessed: a well-established method based on the quasiharmonic approximation, where harmonic frequencies are computed at different lattice volumes, and an alternative approach, where quadratic and cubic interatomic force-constants are computed at a single volume. The latter scheme has only recently become computationally feasible in the context of lattice thermal conductivity simulations. When proper numerical parameters are used (phonon sampling, tolerances, etc.), the two approaches are here shown to provide a very consistent description, yet at a rather different computational cost. All of the experimentally observed features of the complex thermal expansion of Cu₂O are correctly reproduced up to 500 K, with a slight overall underestimation of the volume contraction.