Lattice dynamics molecular solids

N.L. Allan, G.D. Barrera, J.A. Purton, C.E. Sims M.B. Taylor (2000). Phys. Chem. Chem. Phys., 2, 1099-1111. Ionic solids at elevated temperatures and/or high pressures lattice dynamics, molecular dynamics, Monte Carlo and ab initio studies. 

Murthy CS, Singer K, Klein ML et al (1983) Electrostatic interactions in molecular crystals. Lattice dynamics of solid nitrogcm and carbon dioxide. Mol Phys 50 531-541 Hirschfelder JO, Curtiss CF, Bird RB (1954) Molecular theory of gases and liquids. Wiley, New York... 

The lattice dynamics of solids consisting of atoms or ions is a well developed field in physics. The work of Born and Huang (1954) is the classic which deals with the theory of small vibrations as applied to periodic solids. Another standard reference on the subject is by Ziman (1960). However, a new element is introduced when molecular solids are treated. Librational motions occur only in these solids and in some ionic solids which contain polyatomic ions. 

Walmsley and Pople (1964) discussed the lattice dynamics of solid COj. In their description of the librations, they chose as displacement coordinates the changes in the direction cosines of the molecular axis in the crystal (space-fixed) coordinate system. If A,(ft ) is the direction cosine for the symmetry axis of a molecule at site k in unit cell / with respect to crystal coordinate axis j x, y, or z), then we call A,(ft) the change of the direction cosine such that... 

While in Chaps.2 and 3 a straightforward formalism is developed, the philosophy in Chap.4 is different in this chapter we are concerned with an interpretation of measured phonon dispersion curves and the information they provide for the interatomic forces. It is an important chapter and certainly not an easy one the difficulties are intrinsic and arise from the complicated nature of the different types of chemical bonds. The chapter contains the study of the lattice dynamics of solid inert gases, ionic crystals, covalent solids, molecular crystals and a qualitative discussion of the lattice dynamics of metals. 

Molecular Motions and Dynamic Structures. Molecular motions are of quite general occurrence in the solid state for molecules of high symmetry (22,23). If the motion does not introduce disorder into the crystal lattice (as, for example, the in-plane reorientation of benzene which occurs by 60° jumps between equivalent sites) it is not detected by diffraction measurements which will find a seemingly static lattice. Such molecular motions may be detected by wide-line proton NMR spectroscopy and quantified by relaxation-time measurements which yield activation barriers for the reorientation process. In addition, in some cases, the molecular reorientation may be coupled with a chemical exchange process as, for example, in the case of many fluxional organometallic molecules. ... 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.These early applications have been well reviewed elsewhere.In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion, lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[l] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. 

Conventional infrared spectra of powdery materials are very often used for studying solid hydrates in terms of sample characterization (fingerprints), phase transitions, and both structural and bonding features. For the latter objects mostly deuteration experiments are included. However, it must be born in mind that the band frequencies observed (except those of isotopically dilute samples (see Sect. 2.6)) are those of surface modes rather than due to bulk vibrations, i.e., the transverse optical phonon modes, and, hence, not favorably appropriate for molecular and lattice dynamic calculations. 

Pawley GS (1972) Analytic formulation of molecular lattice dynamics based on pair potentials. Physica Statns Solid 49b 475-488... 

In most cases, the crystal potential is not known a priori. The usual procedure is to introduce some model potential containing several parameters, which are subsequently found by fitting the calculated crystal properties to the observed data available. This procedure has the drawback that the empirical potential thus obtained includes the effects of the approximations made in the lattice dynamics model, which is mostly the harmonic model. It is very useful to have independent and detailed information about the potential from quantum-chemical ab initio calculations. Such information is available for nitrogen (Berns and van der Avoird, 1980) and oxygen (Wormer and van der Avoird, 1984), and we have chosen the results calculated for solid nitrogen and solid oxygen to illustrate in Sections V and VI, respectively, the lattice dynamics methods described in Sections III and IV. Nitrogen is the simplest typical molecular crystal as such it has received much attention from theorists and... 

The molecular rearrangements encountered in a solid state reaction are governed not only by the static properties (topochemical) of the lattice but also by the dynamical features of the lattice. The molecular motions, also called lattice phonon motions, describe the dynamical response of the lattice and can be expected to play an important role in determining the reaction course. 

Green s function approach. A simple model of the low-temperature phases of diatomic molecular solids has been examined. The calculations show that ortho-H2 and N2 have the optimum quadrupole structure, Pa3. The self-consistent phonon approximation of anharmonic lattice dynamics has been applied to solid 3-N2. The phonon spectrum and thermal expansion as a function of temperature at zero pressure were calculated and the a-/3 f.c.c.-h.c.p. transition temperature has been estimated."... 

The temperatureHlepeodence data give effective lattice temperatures of = 166 K and = 85 K, showing how markedly molecular solids differ from simple lattice theories. The four vibrational frequencies of Snl4 arc known from I.R./Raman data to be 47, 63,149, and 216 cm, the tin participating only in the 216 cm mode. Thus in the temperature range used for the measurements ( 80-200 K) the iodine 47 cm (68 K) vibration is almost fully excited whereas the tin 216 cm" (311 K) mode is not. This accounts for the lower lattice temperature of iodine. More detailed molecular-dynamical calculations for SnU have introduced the intermolecular translational and rotational vibrations [100]. 

Hendrickson and co-workers have continued to probe the dynamics of electron transfer in molecular systems in the solid state. Mossbauer and specific-heat data on biferrocenium [(C5H5)Fe(C5H4 C5H4)Fe(C5H5)] salts indicate that intramolecular electron transfer is controlled by lattice dynamics. The tri-iodide salts show valence localization up to 350 K by Mbssbauer data. The room-temperature crystal structure is centrosymmetric and evidently disordered. 

The basic tools for the modeling of the solid and liquid states belong to three main categories. We can mention first the Molecular Mechanics which rely on site-site or covalent potentials and which are used to study in particular defect formations, to calculate accoustic and optical phonon modes by lattice dynamics and to estimate mechanical and thermodynamical properties. The easy implementation of the Molecular Mechanics scheme supports its intensive use in the past and its success in commercial softwares. 

Energy minimization produces a structure that is thermodynamically stable at 0 K (not necessarily the lowest energy one, however). In order to produce results that are meaningful at higher temperatures, one must consider the ways in which the system will vary as the temperature is increased. The first approximation utilized in this context (from the solid-state perspective) is lattice dynamics [6]. Equivalently, for molecular approaches, one considers the entropie and enthalpic contributions arising from the rotational, translational, and vibrational degrees of freedom [5]. In solid-state systems, the effect of temperature... 

Unlike the lattice dynamics technique, it is often more difficult to obtain reliable thermodynamic data from molecular dynamics. This is partly due to the large number of configurations that need to be sampled. However, such calculations have been undertaken widely within the biochemistry community (see review by Kollman 1993 Osguthorpe and Dauber-Osguthorpe 1992). Except for a few notable exceptions (Harding 1989 Matsui 1989) there are surprisingly few applications to solids. 

Organic Molecular Solids m. Quantum Lattice Dynamics... 

More recently, interest in the lattice vibrations of molecular solids has centered around the elucidation of intermolecular potential functions. If a pair potential is assumed, it can be tested by calculating the observables by application of the appropriate lattice dynamics. Dows (1962) was the first to attempt a calculation of lattice vibrational frequencies from an assumed potential. He treated solid ethylene and used a model which represented the pair interaction by repulsions between hydrogen atoms on neighboring molecules. 

The lattice vibrations of molecular solids have received brief consideration in reviews dealing with the infrared spectra of these solids (Dows, 1963,1965, 1966). ThereviewbySchnepp (1969) provides a good summary of the field. A recent review by Venkataraman and Sahni (1970) of the lattice dynamics of complex crystals contains much subject matter related to the present review. A number of good reviews are available on the lattice motions of ionic, covalent, and metallic solids (Mitra and Gielisse, 1964 Martin, 1965 Cochran and Cowley, 1969). 

In Section II of this review we discuss the different forms of classical lattice dynamical treatments which have been applied to molecular solids. The applications to specific systems and comparison of results with experiment will then be taken up. In Section III we give a short treatment of quantum lattice dynamics, which has been developed to deal with quantum solids as helium and hydrogen. Classical approaches in the harmonic approximation fail for these systems. In Section IV, intensities of infrared and Raman spectra in the lattice vibration region are discussed. A group theoretical appendix has been added for the reader who is not familiar with this aspect. 

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