This article reviews the theory of atomic vibrations in disordered solids, ranging from almost perfect crystal lattices to glassy materials where geometrical regularity is entirely absent. As a preliminary, the basic notations and equations of conventional lattice dynamics are briefly outlined; the essential equivalence of quantal and classical formulations within the harmonic approximation is indicated, as in the considerable simplification which occurs for periodic systems. The classical time-independent Green's function formalism is then introduced and applied to lattices with isolated defects. Extension of the Green's function method to more grossly disordered systems is described, together with other relevant analytical techniques, such as the phase theory and the Anderson approach. Next, the time-independent numerical method, based on the negative eigenvalue theorem, is introduced and applied to two-component mass disordered lattices; here the numerical calculations are related, where possible, to existing analytical results. The most recent applications of the negative eigenvalue approach concern glasses, polymers and orientationally disordered crystals, and the calculated properties of these systems are reviewed in some detail. Finally, attention is drawn to several areas of the vibrational problem which seem ripe for further detailed study.

https://iopscience.iop.org/article/10.1088/0034-4885/35/3/306/pdf

The dynamics of disordered lattices

Koster-Slater Model for Isolated and Paired. Te Isoelectronic Impurities in II-VI Semiconductors

In this thesis, we explore various approaches to modeling electronic properties of large chemical systems, a challenge for theoretical and computational chemistry. We first model experimental systems with conventional electronic structure by excerpting a region of interest: a few amino acids near the reactive site of a protein, an exciplex dimer in a thin-film LED, and a few donor/acceptor molecules at an organic/organic interface. We find that these models work well for explaining experiments provided we can identify a local region that captures the property of interest. Second, we consider a situation in which the property of interest is not local. We investigate the Anderson model of localization in disordered materials, using the tools of random matrix theory. We use free probability to approximately compute the density of states of a Hamiltonian matrix ensemble without exact diagonalization. We develop an error analysis for this method, and show that it is accurate to the eighth moment of the density of states. Finally, we apply the method to various extensions of the Anderson model, to good result. Third, we develop a wavefunction-in-wavefunction embedding theory for strongly correlated systems. We base our work on Density Matrix Embedding Theory (DMET). We extend the original DMET equations to account for correlation in the bath via an antisymmetrized geminal power (AGP) wave function. The resulting formalism has a number of advantages. First, it allows one to properly treat the weak correlation limit of independent pairs, which DMET is unable to do with a mean-field bath. Second, it associates a size extensive correlation energy with a given density matrix (for the models tested), which AGP by itself is incapable of providing. Third, it provides a reasonable description of charge redistribution in strongly correlated but non-periodic systems. We then describe a new electronic embedding method based on DMET, dubbed "Bootstrap Embedding," a self-consistent wavefunction-in-wavefunction embedding theory that uses overlapping fragments to improve the description of fragment edges. We find Bootstrap Embedding converges rapidly with embedded fragment size, overcoming the surface-area-to-volume-ratio error typical of many embedding methods. Fourth, we employ semiempirical neglect of diatomic differential overlap (NDDO) methods as force fields for liquid water. Using force matching, we design a reparameterized NDDO model and find that it qualitatively reproduces the experimental radial distribution function of water, as well as various monomer, dimer, and bulk properties that standard NDDO method do not. This suggests that the apparent limitations of NDDO models are primarily due to poor parameterization and not to the NDDO approximations themselves. We identify the physical parameters that most influence the condensed phase properties. These results help to elucidate the chemistry that a semiempirical molecular orbital picture of water must capture. We conclude that properly parameterized NDDO models could be useful for simulations that require electronically detailed explicit solvent, including the calculation of redox potentials and simulation of charge transfer and photochemistry. Thesis Supervisor: Troy Van Voorhis

Studies of and methods for electronic properties of large chemical systems

We investigate how free probability allows us to approximate the density of states in tight binding models of disordered electronic systems. Extending our previous studies of the Anderson model in one dimension with nearest-neighbor interactions [J. Chen et al., Phys. Rev. Lett. 109, 036403 (2012)], we find that free probability continues to provide accurate approximations for systems with constant interactions on two- and three-dimensional lattices or with next-nearest-neighbor interactions, with the results being visually indistinguishable from the numerically exact solution. For systems with disordered interactions, we observe a small but visible degradation of the approximation. To explain this behavior of the free approximation, we develop and apply an asymptotic error analysis scheme to show that the approximation is accurate to the eighth moment in the density of states for systems with constant interactions, but is only accurate to sixth order for systems with disordered interactions. The error analysis also allows us to calculate asymptotic corrections to the density of states, allowing for systematically improvable approximations as well as insight into the sources of error without requiring a direct comparison to an exact solution.

/pdf/densities-of-states-for-disordered-systems-from-free-4umwlb758c.pdf

Densities of states for disordered systems from free probability

Extension of the single-site coherent potential approximation for random binary alloys to include the effect of off-diagonal randomness and pair scattering. This extension is achieved by analyzing a one-band model of a random binary alloy in terms of a two-sites coherent potential approximation. Numerical results are presented for a number of different alloys. In the overlapping-band case, the presence of off-diagonal randomness is shown to modify the bandwidths to values larger than those obtained from the virtual-crystal approximation. A simple iterative procedure is described for overcoming the convergence difficulties in the split-band case. In this limit, the inclusion of pair scattering and off-diagonal randomness is found to lead to the appearance of structure in the density of states of the minority component band.

Coherent potential theory of a random binary alloy Effects of scattering from two-sites clusters and off-diagonal randomness

The single-site coherent potential approximation (SS-CPA) for a disordered binary alloy is extended in a self-consistent manner to the case of off-diagonal randomness. The density of states for a bcc lattice is calculated in the split band limit for no correlation between diagonal and off-diagonal randomness and compared with the ordered and SS-CPA density of states.

Coherent potential theory of off-diagonal randommess: Binary alloy

A one-band model of a random binary alloy AxB1-x is analysed in terms of a two-sites coherent potential approximation. In the tight-binding hamiltonian, the off-diagonal randomness is introduced via the composition-dependent hopping energies between nearest-neighbour sites. The inclusion of the off-diagonal randomness correlates the scattering from a given site to that from its nearest neighbours. Such a correlation is incorporated in the handling of diagonal randomness (arising from the composition dependence of the atomic potentials) by treating the diagonal randomness in the pair approximation. The theory leads to the wavevector-dependent coherent potentials and previous approximations used in this problem are easily obtained in appropriate limits.

https://iopscience.iop.org/article/10.1088/0022-3719/7/6/010/pdf

Coherent potential theory a random binary alloy: effects of scattering from two-sites clusters and off-diagonal randomness

A two-site coherent potential approximation including both off-diagonal randomness and pair scattering is briefly outlined. Numerical results on disordered binary alloys are obtained and a procedure for resolving convergence difficulties in the split-band case is presented. The single-site coherent potential approximation (SS-CPA) has provided a simple and elegant approach to the study of disordered substitutional alloys [I, 21. A number of extensions of SS-CPA to include the effects of scattering from a pair of atoms have also been discussed [3-91. The need to include off-diagonal randomness within the framework of CPA has been pointed out [lo] and a number of attempts in this direction have recently appeared in the literature [5-7, 11-15]. We have previously formulated a two-site coherent potential approximation which incorporates both off-diagonal randomness (ODR) and pair scattering [5-71. The inclusion of ODR in the formulation makes it suitable for application to the problem of magnetic disorder [16, 171 also, and treating diagonal randomness in the pair approximation is essential whenever (*) Supported in part by Naval Ordnance Systems Command, Contract N00017-72-C-4401 Task A13B. the fluctuations in the single-site energies are large or in the discussion of the high impurity concentration limit when the impurity clustering effects are important. The theory treats z-nearest neighbours of a given site on equal footing, is invariant under the interchange of A and B atoms and in the absence of ODR and pair-scattering reduces to the usual SSCPA. Numerical calculations based on the present method indicated that for the split band limit, convergence difficulties are encountered in the minority band [6]. A thorough description of similar difficulties has been recently given by Nickel and Butler [18] in their study of three-site case (in the absence of ODR) for a onedimensional solid. They showed that the origin of these difficulties lies in the non-analytic behavior of the averaged Green's function and further conjectured that this feature will generally exist in higher order approximations in the coherent potential formulation. In the present paper, a particular example

/pdf/numerical-aspects-of-the-two-sites-coherent-potential-5esifcmepb.pdf

Numerical aspects of the two-sites coherent potential approximation

The pair-theory of the coherent potential approximation in the presence of off-diagonal randomness (CPA-ODR) has been applied to a disordered spin-1/2 Heisenberg ferromag-netic binary alloy. The spin-wave Green’s function has been evaluated in the random phase approximation and the randomly varying exchange interaction has been treated in the CPAODR. The density of spin-wave states (DSWS) has been calculated for a bcc lattice for the entire range of impurity concentrations. It is found that the general effect of disorder is to remove the singularities, modify the band width and introduce an additional structure in the DSWS. The ferromagnetic Curie temperature Tc for all impurity concen-trations has also been investigated. For impurities which are ferromagnetic in nature, the results from the present theory are found to be close to those obtained from the mean field theory.

K. Moorjani

Papers

Coherent potential theory of off-diagonal randommess: Binary alloy

Coherent potential theory of a random binary alloy Effects of scattering from two-sites clusters and off-diagonal randomness

Coherent potential theory a random binary alloy: effects of scattering from two-sites clusters and off-diagonal randomness

Numerical aspects of the two-sites coherent potential approximation

Theory of spin-waves in a disordered heisenberg ferromagnet