Machine learning molecular dynamics for the simulation of infrared spectra

Machine learning has emerged as an invaluable tool in many research areas. In the present work, we harness this power to predict highly accurate molecular infrared spectra with unprecedented computational efficiency. To account for vibrational anharmonic and dynamical effects - typically neglected by conventional quantum chemistry approaches - we base our machine learning strategy on ab initio molecular dynamics simulations. While these simulations are usually extremely time consuming even for small molecules, we overcome these limitations by leveraging the power of a variety of machine learning techniques, not only accelerating simulations by several orders of magnitude, but also greatly extending the size of systems that can be treated. To this end, we develop a molecular dipole moment model based on environment dependent neural network charges and combine it with the neural network potential approach of Behler and Parrinello. Contrary to the prevalent big data philosophy, we are able to obtain very accurate machine learning models for the prediction of infrared spectra based on only a few hundreds of electronic structure reference points. This is made possible through the use of molecular forces during neural network potential training and the introduction of a fully automated sampling scheme. We demonstrate the power of our machine learning approach by applying it to model the infrared spectra of a methanol molecule, n-alkanes containing up to 200 atoms and the protonated alanine tripeptide, which at the same time represents the first application of machine learning techniques to simulate the dynamics of a peptide. In all of these case studies we find an excellent agreement between the infrared spectra predicted via machine learning models and the respective theoretical and experimental spectra.

Static vs dynamic DFT prediction of IR spectra of flexible molecules in the condensed phase: The (ClCF2CF(CF3)OCF2CH3) liquid as a test case

First-principles molecular dynamics (FPMD) simulations in the framework of Density Functional Theory (DFT) are carried out for the prediction of the infrared spectrum of the fluorinated molecule ClCF₂CF(CF₃)OCF₂CH₃ in liquid and gas phase. This molecule is characterized by a flexible structure, allowing the co-existence of several stable conformers, that differ by values of the torsional angles. FPMD computed spectra are compared to the experimental ones, and to Boltzmann weighted IR spectra based on gas phase calculations.

Computing Bulk Phase Raman Optical Activity Spectra from ab initio Molecular Dynamics Simulations

We present our novel methodology for computing Raman optical activity (ROA) spectra of liquid systems from ab initio molecular dynamics (AIMD) simulations. The method is built upon the recent developments to obtain magnetic dipole moments from AIMD and to integrate molecular properties by using radical Voronoi tessellation. These techniques are used to calculate optical activity tensors for large and complex periodic bulk phase systems. Only AIMD simulations are required as input, and no time-consuming perturbation theory is involved. The approach relies only on the total electron density in each time step and can readily be combined with a wide range of electronic structure methods. To the best of our knowledge, these are the first computed ROA spectra for a periodic bulk phase system. As an example, the experimental ROA spectrum of liquid (R)-propylene oxide is reproduced very well.

CO2 Hydration Shell Structure and Transformation

The hydration-shell of CO₂ is characterized using Raman multivariate curve resolution (Raman-MCR) spectroscopy combined with ab initio molecular dynamics (AIMD) vibrational density of states simulations, to validate our assignment of the experimentally observed high-frequency OH band to a weak hydrogen bond between water and CO₂. Our results reveal that while the hydration-shell of CO₂ is highly tetrahedral, it is also occasionally disrupted by the presence of entropically stabilized defects associated with the CO₂-water hydrogen bond. Moreover, we find that the hydration-shell of CO₂ undergoes a temperature-dependent structural transformation to a highly disordered (less tetrahedral) structure, reminiscent of the transformation that takes place at higher temperatures around much larger oily molecules. The biological significance of the CO₂ hydration shell structural transformation is suggested by the fact that it takes place near physiological temperatures.

Local Polar Fluctuations in Lead Halide Perovskite Crystals

Hybrid lead-halide perovskites have emerged as an excellent class of photovoltaic materials. Recent reports suggest that the organic molecular cation is responsible for local polar fluctuations that inhibit carrier recombination. We combine low-frequency Raman scattering with first-principles molecular dynamics (MD) to study the fundamental nature of these local polar fluctuations. Our observations of a strong central peak in the cubic phase of both hybrid (CH_{3}NH_{3}PbBr_{3}) and all-inorganic (CsPbBr_{3}) lead-halide perovskites show that anharmonic, local polar fluctuations are intrinsic to the general lead-halide perovskite structure, and not unique to the dipolar organic cation. MD simulations indicate that head-to-head Cs motion coupled to Br face expansion, occurring on a few hundred femtosecond time scale, drives the local polar fluctuations in CsPbBr_{3}.

Local Mode Analysis: Decoding IR Spectra by Visualizing Molecular Details

Integration of experimental and computational approaches to investigate chemical reactions in proteins has proven to be very successful. Experimentally, time-resolved FTIR difference-spectroscopy monitors chemical reactions at atomic detail. To decode detailed structural information encoded in IR spectra, QM/MM calculations are performed. Here, we present a novel method which we call local mode analysis (LMA) for calculating IR spectra and assigning spectral IR-bands on the basis of movements of nuclei and partial charges from just a single QM/MM trajectory. Through LMA the decoding of IR spectra no longer requires several simulations or optimizations. The novel approach correlates the motions of atoms of a single simulation with the corresponding IR bands and provides direct access to the structural information encoded in IR spectra. Either the contributions of a particular atom or atom group to the complete IR spectrum of the molecule are visualized, or an IR-band is selected to visualize the corresponding structural motions. Thus, LMA decodes the detailed information contained in IR spectra and provides an intuitive approach for structural biologists and biochemists. The unique feature of LMA is the bidirectional analysis connecting structural details to spectral features and vice versa spectral features to molecular motions.

Quantum Chemical Methods for the Prediction of Energetic, Physical, and Spectroscopic Properties of Ionic Liquids

The accurate prediction of physicochemical properties of condensed systems is a longstanding goal of theoretical (quantum) chemistry. Ionic liquids comprising entirely of ions provide a unique challenge in this respect due to the diverse chemical nature of available ions and the complex interplay of intermolecular interactions among them, thus resulting in the wide variability of physicochemical properties, such as thermodynamic, transport, and spectroscopic properties. It is well understood that intermolecular forces are directly linked to physicochemical properties of condensed systems, and therefore, an understanding of this relationship would greatly aid in the design and synthesis of functionalized materials with tailored properties for an application at hand. This review aims to give an overview of how electronic structure properties obtained from quantum chemical methods such as interaction/binding energy and its fundamental components, dipole moment, polarizability, and orbital energies, can help shed light on the energetic, physical, and spectroscopic properties of semi-Coulomb systems such as ionic liquids. Particular emphasis is given to the prediction of their thermodynamic, transport, spectroscopic, and solubilizing properties.

Vibrational Spectroscopy of Ionic Liquids

Vibrational spectroscopy has continued use as a powerful tool to characterize ionic liquids since the literature on room temperature molten salts experienced the rapid increase in number of publications in the 1990's. In the past years, infrared (IR) and Raman spectroscopies have provided insights on ionic interactions and the resulting liquid structure in ionic liquids. A large body of information is now available concerning vibrational spectra of ionic liquids made of many different combinations of anions and cations, but reviews on this literature are scarce. This review is an attempt at filling this gap. Some basic care needed while recording IR or Raman spectra of ionic liquids is explained. We have reviewed the conceptual basis of theoretical frameworks which have been used to interpret vibrational spectra of ionic liquids, helping the reader to distinguish the scope of application of different methods of calculation. Vibrational frequencies observed in IR and Raman spectra of ionic liquids based on different anions and cations are discussed and eventual disagreements between different sources are critically reviewed. The aim is that the reader can use this information while assigning vibrational spectra of an ionic liquid containing another particular combination of anions and cations. Different applications of IR and Raman spectroscopies are given for both pure ionic liquids and solutions. Further issues addressed in this review are the intermolecular vibrations that are more directly probed by the low-frequency range of IR and Raman spectra and the applications of vibrational spectroscopy in studying phase transitions of ionic liquids.

Probing the dynamics of N-methylacetamide in methanol via ab initio molecular dynamics

Two-dimensional infrared (2D IR) spectroscopy of amide 1 vibrational bands provides a valuable probe of proteins as well as molecules such as N-methylacetamide (NMA), which present peptide-like H-bonding possibilities to a solvent.

The significance of the amorphous potential energy landscape for dictating glassy dynamics and driving solid-state crystallisation

We show clear evidence for a theory proposing that the shape and structure of the PES is the fundamental factor underlying the dynamics at temperatures below the glass transition.

Mapping gas phase dipeptide motions in the far-infrared and terahertz domain

Vibrational signatures of Ac-Phe-AA-NH₂ dipeptides are recorded and analysed in the far IR/THz spectral domain (100–800 cm⁻¹, 3–24 THz), with the ‘AA’ amino acid chosen within the series ‘AA’ = Gly, Ala, Pro, Cys, Ser, Val. Phe stands for phenylalanine.

Symmetry and dynamics of FHF− anion in vacuum, in CD2Cl2 and in CCl4. Ab initio MD study of fluctuating solvent–solute hydrogen and halogen bonds

Asymmetric solvation of FHF⁻ by halogen- and hydrogen-bonding solvents breaks the symmetry of the anion.

Sum Frequency Generation of Acetonitrile on a Rutile (110) Surface from Density Functional Theory-Based Molecular Dynamics

We present the calculation of vibrational sum frequency generation for molecules adsorbed on a semiconductor surface via density functional theory-based molecular dynamics. Using an efficient approach relying on the Gaussian and plane waves method and density functional perturbation theory, acetonitrile adsorbed on a rutile (110) surface has been studied, the vapor-solid interface of which has recently been investigated experimentally. Further analysis of the orientation of the acetonitrile molecules directly adsorbed on the rutile (110) surface agrees well with parameters derived from experiment. This opens a promising way for detailed study of semiconductor interfaces, which is of particular importance with respect to numerous applications such as, for instance, in materials design.

Ab Initio Transient Vibrational Spectral Analysis

Pump probe spectroscopy techniques have enabled the direct observation of a variety of transient molecular species in both ground and excited electronic states. Time-resolved vibrational spectroscopy is becoming an indispensable tool for investigating photoinduced nuclear dynamics of chemical systems of all kinds. On the other hand, a complete picture of the chemical dynamics encoded in these spectra cannot be achieved without a full temporal description of the structural relaxation, including the explicit time-dependence of vibrational coordinates that are substantially displaced from equilibrium by electronic excitation. Here we present a transient vibrational analysis protocol combining ab initio direct molecular dynamics and time-integrated normal modes introduced in this work, relying on the recent development of analytic time-dependent density functional theory (TDDFT) second derivatives for excited states. Prototypical molecules will be used as test cases, showing the evolution of the vibrational signatures that follow electronic excitation. This protocol provides a direct route to assigning the vibrations implicated in the (photo)dynamics of several (photoactive) systems.

Resolving the Origins of Crystalline Anharmonicity Using Terahertz Time-Domain Spectroscopy and ab Initio Simulations

Anharmonicity has been shown to be an important piece of the fundamental framework that dictates numerous observable phenomena. In particular, anharmonicity is the driving force of vibrational relaxation processes, mechanisms that are integral to the proper function of numerous chemical processes. However, elucidating its origins has proven difficult due to experimental and theoretical challenges, specifically related to separating the anharmonic contributions from other unrelated effects. While no one technique is particularly suited for providing a complete picture of anharmonicity, by combining multiple complementary methods such a characterization can be made. In this study the role of individual atomic interactions on the anharmonic properties of crystalline purine, the building block of many DNA and RNA nucleobases, is studied by experimental terahertz time-domain spectroscopy and first-principles density functional theory (DFT) and ab initio molecular dynamics simulations (AIMD). In particular, the detailed vibrational information provided by the DFT calculations is used to interpret the atomic origins of anharmonic-related effects as determined by the AIMD calculations, which are in good agreement with the experimental data. The results highlight that anharmonicity is especially pronounced in the intermolecular interactions, particularly along the amine hydrogen bond coordinate, and yields valuable insight into what is similarly observed complex biosystems and crystalline solids.

Colored spectrum characteristics of thermal noise on the molecular scale

Thermal noise is of fundamental importance to many processes. Traditionally, thermal noise has been treated as white noise on the macroscopic scale. Using molecular dynamics simulations and power spectrum analysis, we show that the thermal noise of solute molecules in water is non-white on the molecular scale, which is in contrast to the conventional theory. In the frequency domain from 2 × 10¹¹ Hz to 10¹³ Hz, the power spectrum of thermal noise for polar solute molecules resembles the spectrum of 1/f noise. The power spectrum of thermal noise for non-polar solute molecules deviates only slightly from the spectrum of white noise. The key to this phenomenon is the existence of hydrogen bonds between polar solute molecules and solvent water molecules. Furthermore, for polar solute molecules, the degree of power spectrum deviation from that of white noise is associated with the average lifetime of the hydrogen bonds between the solute and the solvent molecules.

Local electric dipole moments: A generalized approach

We present an approach for calculating local electric dipole moments for fragments of molecular or supramolecular systems. This is important for understanding chemical gating and solvent effects in nanoelectronics, atomic force microscopy, and intensities in infrared spectroscopy. Owing to the nonzero partial charge of most fragments, "naively" defined local dipole moments are origin-dependent. Inspired by previous work based on Bader's atoms-in-molecules (AIM) partitioning, we derive a definition of fragment dipole moments which achieves origin-independence by relying on internal reference points. Instead of bond critical points (BCPs) as in existing approaches, we use as few reference points as possible, which are located between the fragment and the remainder(s) of the system and may be chosen based on chemical intuition. This allows our approach to be used with AIM implementations that circumvent the calculation of critical points for reasons of computational efficiency, for cases where no BCPs are found due to large interfragment distances, and with local partitioning schemes other than AIM which do not provide BCPs. It is applicable to both covalently and noncovalently bound systems. © 2016 Wiley Periodicals, Inc.

Characterizing the local solvation environment of OH(-) in water clusters with AIMD

In this work, we use ab initio molecular dynamics coupled with metadynamics to explore and characterize the glassy potential energy landscape of the OH(-) in a 20 and 48 water cluster. The structural, energetic, and topological properties of OH(-) are characterized for both clusters and the molecular origins of the IR signatures are examined. We find that in both the small and large clusters, the OH(-) can donate or accept a varying number of hydrogen bonds confirming that the amphiphilic character does not depend on cluster size. However, we highlight some important differences found between the energetic and topological properties of both families of clusters which may have implications on understanding the changes in the solvation structure of OH(-) between bulk and interfacial environments. By studying the IR spectra of smaller subsets of molecules within the 20 water molecule cluster, we find that the IR spectrum of the bare OH(-) as well as the water molecule donating a strong hydrogen bond to it exhibits characteristic absorption along the amphiphilic band between 1500 and 3000 cm(-1) at positions very similar to those found for the entire hydroxide cluster. The results presented here will be useful in the calibration and improvement of both ab initio and semi-empirical methods to model this complex anion.

Role of Polar Phonons in the Photo Excited State of Metal Halide Perovskites

The development of high efficiency perovskite solar cells has sparked a multitude of measurements on the optical properties of these materials. For the most studied methylammonium(MA)PbI₃ perovskite, a large range (6–55 meV) of exciton binding energies has been reported by various experiments. The existence of excitons at room temperature is unclear. For the MAPbX₃ perovskites we report on relativistic Bethe-Salpeter Equation calculations (GW-BSE). This method is capable to directly calculate excitonic properties from first-principles. At low temperatures it predicts exciton binding energies in agreement with the reported ‘large’ values. For MAPbI₃, phonon modes present in this frequency range have a negligible contribution to the ionic screening. By calculating the polarization in time from finite temperature molecular dynamics, we show that at room temperature this does not change. We therefore exclude ionic screening as an explanation for the experimentally observed reduction of the exciton binding energy at room temperature and argue in favor of the formation of polarons.

The effect of molecular dynamics sampling on the calculated observable gas-phase structures

In this study, we compare the performance of various ab initio molecular dynamics (MD) sampling methods for the calculation of the observable vibrationally-averaged gas-phase structures of benzene, naphthalene and anthracene molecules. Nose-Hoover (NH), canonical and quantum generalized-Langevin-equation (GLE) thermostats as well as the a posteriori quantum correction to the classical trajectories have been tested and compared to the accurate path-integral molecular dynamics (PIMD), static anharmonic vibrational calculations as well as to the experimental gas electron diffraction data. Classical sampling methods neglecting quantum effects (NH and canonical GLE thermostats) dramatically underestimate vibrational amplitudes for the bonded atom pairs, both C-H and C-C, the resulting radial distribution functions exhibit nonphysically narrow peaks. This deficiency is almost completely removed by taking the quantum effects on the nuclei into account. The quantum GLE thermostat and a posteriori correction to the canonical GLE and NH thermostatted trajectories capture most vibrational quantum effects and closely reproduce computationally expensive PIMD and experimental radial distribution functions. These methods are both computationally feasible and accurate and are therefore recommended for calculations of the observable gas-phase structures. A good performance of the quantum GLE thermostat for the gas-phase calculations is encouraging since its parameters have been originally fitted for the condensed-phase calculations. Very accurate molecular structures can be predicted by combining the equilibrium geometry obtained at a high level of electronic structure theory with vibrational amplitudes and corrections calculated using MD driven by a lower level of electronic structure theory.

The Effects of the Organic-Inorganic Interactions on the Thermal Transport Properties of CH3NH3PbI3

Methylammonium lead iodide perovskite (CH3NH3PbI3), the most investigated hybrid organic-inorganic halide perovskite, is characterized by a quite low thermal conductivity. The rotational motion of methylammonium cations is considered responsible for phonon transport suppression; however, to date, the specific mechanism of the process has not been clarified. In this study, we elucidate the role of rotations in thermal properties based on molecular dynamics simulations. To do it, we developed an empirical potential for CH3NH3PbI3 by fitting to ab initio calculations and evaluated its thermal conductivity by means of nonequilibrium molecular dynamics. Results are compared with model systems that include different embedded cations, and this comparison shows a dominant suppression effect provided by rotational motions. We also checked the temperature dependence of the vibrational density of states and specified the energy range in which anharmonic couplings occur. By means of phonon dispersion analysis, we were able to fully elucidate the suppression mechanism: the rotations are coupled with translational motions of cations, via which inorganic lattice vibrations are coupled and scatter each other.

Water orientation and hydrogen-bond structure at the fluorite/water interface

Water in contact with mineral interfaces is important for a variety of different processes. Here, we present a combined theoretical/experimental study which provides a quantitative, molecular-level understanding of the ubiquitous and important CaF2/water interface. Our results show that, at low pH, the surface is positively charged, causing a substantial degree of water ordering. The surface charge originates primarily from the dissolution of fluoride ions, rather than from adsorption of protons to the surface. At high pH we observe the presence of Ca-OH species pointing into the water. These OH groups interact remarkably weakly with the surrounding water, and are responsible for the "free OH" signature in the VSFG spectrum, which can be explained from local electronic structure effects. The quantification of the surface termination, near-surface ion distribution and water arrangement is enabled by a combination of advanced phase-resolved Vibrational Sum Frequency Generation spectra of CaF2/water interfaces and state-of-the-art ab initio molecular dynamics simulations which include electronic structure effects.

Anharmonic Infrared Spectroscopy through the Fourier Transform of Time Correlation Function Formalism in ONETEP

Density functional theory molecular dynamics (DFT-MD) provides an efficient framework for accurately computing several types of spectra. The major benefit of DFT-MD approaches lies in the ability to naturally take into account the effects of temperature and anharmonicity, without having to introduce any ad hoc or a posteriori corrections. Consequently, computational spectroscopy based on DFT-MD approaches plays a pivotal role in the understanding and assignment of experimental peaks and bands at finite temperature, particularly in the case of floppy molecules. Linear-scaling DFT methods can be used to study large and complex systems, such as peptides, DNA strands, amorphous solids, and molecules in solution. Here, we present the implementation of DFT-MD IR spectroscopy in the ONETEP linear-scaling code. In addition, two methods for partitioning the dipole moment within the ONETEP framework are presented. Dipole moment partitioning allows us to compute spectra of molecules in solution, which fully include the effects of the solvent, while at the same time removing the solvent contribution from the spectra.

Classical Magnetic Dipole Moments for the Simulation of Vibrational Circular Dichroism by ab Initio Molecular Dynamics

We present a new approach for calculating vibrational circular dichroism spectra by ab initio molecular dynamics. In the context of molecular dynamics, these spectra are given by the Fourier transform of the cross-correlation function of magnetic dipole moment and electric dipole moment. We obtain the magnetic dipole moment from the electric current density according to the classical definition. The electric current density is computed by solving a partial differential equation derived from the continuity equation and the condition that eddy currents should be absent. In combination with a radical Voronoi tessellation, this yields an individual magnetic dipole moment for each molecule in a bulk phase simulation. Using the chiral alcohol 2-butanol as an example, we show that experimental spectra are reproduced very well. Our approach requires knowing only the electron density in each simulation step, and it is not restricted to any particular electronic structure method.

Time-Domain Simulations of Transient Species in Experimentally Relevant Environments

Simulating the spectroscopic properties of short-lived thermal and photochemical reaction intermediates and products is a challenging task, as these species often feature atypical molecular and electronic structures. The complex environments in which such species typically reside in practice add further complexity to the problem. Herein, we tackle this problem in silico using ab initio molecular dynamics (AIMD) simulations, employing iso-CHBr3, namely H(Br)C-Br-Br, as a prototypical system. This species was chosen because it features both a nonconventional C-Br-Br bonding pattern, as well as a strong dependence of its spectral features on the local environment in which it resides, as illustrated in recent experimental reports. We simulate the UV-vis and IR spectra of iso-CHBr3 in the gas phase, as well as in a Ne cluster (64 atoms) and in a methylcyclohexane cage (14 solvent molecules) representative of the previously characterized matrix isolated and solvated iso-CHBr3 species. We exclusively perform fully quantum mechanical static and dynamic simulations. By comparing our condensed phase simulations to their experimental analogues, we stress the importance of (i) conformational sampling, even at cryogenic temperatures, and (ii) using a fully quantum mechanical description of both solute and bath to properly account for the experimental observables.

Geometry-dependent distributed polarizability models for the water molecule

Geometry-dependent distributed polarizability models have been constructed by fits to ab initio calculations at the coupled cluster level of theory with up to noniterative triple excitations in an augmented triple-zeta quality basis set for the water molecule in the field of a point charge. The investigated models include (i) charge-flow polarizabilities between chemically bonded atoms, (ii) isotropic or anisotropic dipolar polarizabilities on oxygen atom or on all atoms, and (iii) combinations of models (i) and (ii). For each model, the polarizability parameters have been optimized to reproduce the induction energy of a water molecule polarized by a point charge successively occupying a grid of points surrounding the molecule. The quality of the models is ascertained by examining their ability to reproduce these induction energies as well as the molecular dipolar and quadrupolar polarizabilities. The geometry dependence of the distributed polarizability models has been explored by changing bond lengths and HOH angle to generate 125 molecular structures (reduced to 75 symmetry-unique ones). For each considered model, the distributed polarizability components have been fitted as a function of the geometry by a Taylor expansion in monomer coordinate displacements up to the sum of powers equal to 4.

Role of Presolvation and Anharmonicity in Aqueous Phase Hydrated Proton Solvation and Transport

Results from condensed phase ab initio molecular dynamics (AIMD) simulations suggest a proton transfer reaction is facilitated by "presolvation" in which the hydronium is transiently solvated by four water molecules, similar to the typical solvation structure of water, by accepting a weak hydrogen bond from the fourth water molecule. A new version 3.2 multistate empirical valence bond (MS-EVB 3.2) model for the hydrated excess proton incorporating this presolvation behavior is therefore developed. The classical MS-EVB simulations show similar structural properties as those of the previous model but with significantly improved diffusive behavior. The inclusion of nuclear quantum effects in the MS-EVB also provides an even better description of the proton diffusion rate. To quantify the influence of anharmonicity, a second model (aMS-EVB 3.2) is developed using the anharmonic aSPC/Fw water model, which provides similar structural properties but improved spectroscopic responses at high frequencies.

Ion pairing in ionic liquids

In the present article we briefly review the extensive discussion in literature about the presence or absence of ion pair-like aggregates in ionic liquids. While some experimental studies point towards the presence of neutral subunits in ionic liquids, many other experiments cannot confirm or even contradict their existence. Ion pairs can be detected directly in the gas phase, but no direct method is available to observe such association behavior in the liquid, and the corresponding indirect experimental proofs are based on such assumptions as unity charges at the ions. However, we have shown by calculating ionic liquid clusters of different sizes that assuming unity charges for ILs is erroneous, because a substantial charge transfer is taking place between the ionic liquid ions that reduce their total charge. Considering these effects might establish a bridge between the contradicting experimental results on this matter. Beside these results, according to molecular dynamics simulations the lifetimes of ion-ion contacts and their joint motions are far too short to verify the existence of neutral units in these materials.