Asymptotic analysis of thin plates under normal load and horizontal edge thrust

We consider the radially symmetric nonlinear von Kármán plate equations for circular or annular plates in the limit of small thickness. The loads on the plate consist of a radially symmetric pressure load and a uniform edge load. The dependence of the steady states on the edge load and thickness is studied using asymptotics as well as numerical calculations. The von Kármán plate equations are a singular perturbation of the Fӧppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions.

We give evidence for the existence of a singular compressive solution for the circular membrane and show by a singular perturbation expansion that the nonsingular compressive solution approach this singular solution as the radial stress at the center of the plate vanishes. In this limit, an infinite number of folds occur with respect to the edge load. Similar behavior is observed for the annular membrane with zero edge load at the inner radius in the limit as the circumferential stress vanishes.

We develop multiscale expansions, which are asymptotic to members of this family for plates with edges that are elastically supported against rotation. At some thicknesses this approximation breaks down and a boundary layer appears at the center of the plate. In the limit of small normal load, the points of breakdown approach the bifurcation points corresponding to buckling of the nondeflected state. A uniform asymptotic expansion for small thickness combining the boundary layer with a multiscale approximation of the outer solution is developed for this case. These approximations complement the well known boundary layer expansions based on tensile membrane solutions in describing the bending and stretching of thin plates. The approximation becomes inconsistent as the clamped state is approached by increasing the resistance against rotation at the edge. We prove that such an expansion for the clamped circular plate cannot exist unless the pressure load is self-equilibrating.

Observational and theoretical advances in cosmological foreground emission

Observational and theoretical work towards the separation of foreground emission from the cosmic microwave background is described. The bulk of this work is in the design, construction, and commissioning of the C-Band All-Sky Survey (C-BASS), an experiment to produce a template of the Milky Way Galaxy's polarized synchrotron emission. Theoretical work is the derivation of an analytical approximation to the emission spectrum of spinning dust grains.

The performance of the C-BASS experiment is demonstrated through a preliminary, deep survey of the North Celestial Pole region. A comparison to multiwavelength data is performed, and the thermal and systematic noise properties of the experiment are explored. The systematic noise has been minimized through careful data processing algorithms, implemented both in the experiment's Field Programmable Gate Array (FPGA) based digital backend and in the data analysis pipeline. Detailed descriptions of these algorithms are presented.

The analytical function of spinning dust emission is derived through the application of careful approximations, with each step tested against numerical calculations. This work is intended for use in the parameterized separation of cosmological foreground components and as a framework for interpreting and comparing the variety of anomalous microwave emission observations.

New optoelectronic devices using GaAs-GaAlAs epitaxy

Three subjects related to epitaxial GaAs-GaAlAs optoelectronic devices are discussed in this thesis. They are:

1. Embedded Epitaxy

This is a technique of selective multilayer growth of GaAs- Ga_1-xAl_xAs single crystal structures through stripe openings in masking layers on GaAs substrates. This technique results in prismatic layers of GaAs and Ga_1-xAl_xAs "embedded" in each other and leads to controllable uniform structures terminated by crystal faces. The dependence of the growth habit on the orientation of the stripe openings has been studied. Room temperature embedded double heterostructure lasers have been fabricated using this technique. Threshold current densities as low as 1.5 KA/cm² have been achieved.

2. Barrier Controlled PNPN Laser Diode

It is found that the I-V characteristics of a PNPN device can be controlled by using potential barriers in the base regions. Based on this principle, GaAs-GaAlAs heterostructure PNPN laser diodes have been fabricated. GaAlAs potential barriers in the bases control not only the electrical but also the optical properties of the device. PNPN lasers with low threshold currents and high breakover voltage have been achieved. Numerical calculations of this barrier controlled structure are presented in the ranges where the total current is below the holding point and near the lasing threshold.

3. Injection Lasers on Semi-Insulating Substrates

GaAs-GaAlAs heterostructure lasers fabricated on semi-insulating substrates have been studied. Two different laser structures achieved are: (1) Crowding effect lasers, (2) Lateral injection lasers. Experimental results and the working principles underlying the operation of these lasers are presented. The gain induced guiding mechanism is used to explain the lasers' far field radiation patterns. It is found that Zn diffusion in Ga_1-xAl_xAs depends on the Al content x, and that GaAs can be used as the diffusion mask for Zn diffusion in Ga_1-xAl_xAs. Lasers having very low threshold currents and operating in a stable single mode have been achieved. Because these lasers are fabricated on semi-insulating substrates, it is possible to integrate them with other electronic devices on the same substrate. An integrated device, which consists of a crowding effect laser and a Gunn oscillator on a common semi-insulating GaAs substrate, has been achieved.

I. Electronic states of heavily-doped molecular crystals--naphthalene. I. Theoretical. II. Experimental. II. Lowest singlet and triplet excited states of pyrazine

PART I

The energy spectrum of heavily-doped molecular crystals was treated in the Green’s function formulation. The mixed crystal Green’s function was obtained by averaging over all possible impurity distributions. The resulting Green’s function, which takes the form of an infinite perturbation expansion, was further approximated by a closed form suitable for numerical calculations. The density-of-states functions and optical spectra for binary mixtures of normal naphthalene and deuterated naphthalene were calculated using the pure crystal density-of-state functions. The results showed that when the trap depth is large, two separate energy bands persist, but when the trap depth is small only a single band exists. Furthermore, in the former case it was found that the intensities of the outer Davydov bands are enhanced whereas the inner bands are weakened. Comparisons with previous theoretical calculations and experimental results are also made.

PART II

The energy states and optical spectra of heavily-doped mixed crystals are investigated. Studies are made for the following binary systems: (1) naphthalene-h₈ and d₈, (2) naphthalene--h₈ and αd₄, and (3) naphthalene--h₈ and βd₁, corresponding to strong, medium and weak perturbations. In addition to ordinary absorption spectra at 4˚K, band-to-band transitions at both 4˚K and 77˚K are also analyzed with emphasis on their relations to cooperative excitation and overall density-of-states functions for mixed crystals. It is found that the theoretical calculations presented in a previous paper agree generally with experiments except for cluster states observed in system (1) at lower guest concentrations. These features are discussed semi-quantitatively. As to the intermolecular interaction parameters, it is found that experimental results compare favorably with calculations based on experimental density-of-states functions but not with those based on octopole interactions or charge-transfer interactions. Previous experimental results of Sheka and the theoretical model of Broude and Rashba are also compared with present investigations.

PART III

The phosphorescence, fluorescence and absorption spectra of pyrazine-h₄ and d₄ have been obtained at 4˚K in a benzene matrix. For comparison, those of the isotopically mixed crystal pyrazine-h₄ in d₄ were also taken. All these spectra show extremely sharp and well-resolved lines and reveal detailed vibronic structure.

The analysis of the weak fluorescence spectrum resolves the long-disputed question of whether one or two transitions are involved in the near-ultraviolet absorption of pyrazine. The “mirror-image relationship” between absorption and emission shows that the lowest singlet state is an allowed transition, properly designated as ¹B_3u ← ¹A_1g. The forbidden component ¹B_2g, predicted by both “exciton” and MO theories to be below the allowed component, must lie higher. Its exact location still remains uncertain.

The phosphorescence spectrum when compared with the excitation phosphorescence spectra, indicates that the lowest triplet state is also symmetry allowed, showing a strong 0-0 band and a “mirror-image relationship” between absorption and emission. In accordance with previous work, the triplet state is designated as ³B_3u.

The vibronic structure of the phosphorescence spectrum is very complicated. Previous work on the analysis of this spectrum all concluded that a long progression of v_6a exists. Under the high resolution attainable in our work, the supposed v_6a progression proves to have a composite triplet structure, starting from the second member of the progression. Not only is the v_9a hydrogen-bending mode present as shown by the appearance of the C-D bending mode in the d₄ spectrum, but a band of 1207 cm^-1 in the pyrazine in benzene system and 1231 cm^-1 in the mixed crystal system is also observed. This band is assigned as 2v_6b and of a_1g symmetry. Its anonymously strong intensity in the phosphorescence spectrum is interpreted as due to the Fermi resonance with the 2v_6a and v_9a band.

To help resolve the present controversy over the crystal phosphorescence spectrum of pyrazine, detailed vibrational analyses of the emission spectra were made. The fluorescence spectrum has essentially the same vibronic structure as the phosphorescence spectrum.

Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media

An exact solution to the monoenergetic Boltzmann equation is obtained for the case of a plane isotropic burst of neutrons introduced at the interface separating two adjacent, dissimilar, semi-infinite media. The method of solution used is to remove the time dependence by a Laplace transformation, solve the transformed equation by the normal mode expansion method, and then invert to recover the time dependence.

The general result is expressed as a sum of definite, multiple integrals, one of which contains the uncollided wave of neutrons originating at the source plane. It is possible to obtain a simplified form for the solution at the interface, and certain numerical calculations are made there.

The interface flux in two adjacent moderators is calculated and plotted as a function of time for several moderator materials. For each case it is found that the flux decay curve has an asymptotic slope given accurately by diffusion theory. Furthermore, the interface current is observed to change directions when the scattering and absorption cross sections of the two moderator materials are related in a certain manner. More specifically, the reflection process in two adjacent moderators appears to depend initially on the scattering properties and for long times on the absorption properties of the media.

This analysis contains both the single infinite and semi-infinite medium problems as special cases. The results in these two special cases provide a check on the accuracy of the general solution since they agree with solutions of these problems obtained by separate analyses.

Dynamic response of hysteretic systems with application to a system containing limited slip

A general class of single degree of freedom systems possessing rate-independent hysteresis is defined. The hysteretic behavior in a system belonging to this class is depicted as a sequence of single-valued functions; at any given time, the current function is determined by some set of mathematical rules concerning the entire previous response of the system. Existence and uniqueness of solutions are established and boundedness of solutions is examined.

An asymptotic solution procedure is used to derive an approximation to the response of viscously damped systems with a small hysteretic nonlinearity and trigonometric excitation. Two properties of the hysteresis loops associated with any given system completely determine this approximation to the response: the area enclosed by each loop, and the average of the ascending and descending branches of each loop.

The approximation, supplemented by numerical calculations, is applied to investigate the steady-state response of a system with limited slip. Such features as disconnected response curves and jumps in response exist for a certain range of system parameters for any finite amount of slip.

To further understand the response of this system, solutions of the initial-value problem are examined. The boundedness of solutions is investigated first. Then the relationship between initial conditions and resulting steady-state solution is examined when multiple steady-state solutions exist. Using the approximate analysis and numerical calculations, it is found that significant regions of initial conditions in the initial condition plane lead to the different asymptotically stable steady-state solutions.

Progress in numerical modeling of non-premixed combustion

Progress is made on the numerical modeling of both laminar and turbulent non-premixed flames. Instead of solving the transport equations for the numerous species involved in the combustion process, the present study proposes reduced-order combustion models based on local flame structures.

For laminar non-premixed flames, curvature and multi-dimensional diffusion effects are found critical for the accurate prediction of sooting tendencies. A new numerical model based on modified flamelet equations is proposed. Sooting tendencies are calculated numerically using the proposed model for a wide range of species. These first numerically-computed sooting tendencies are in good agreement with experimental data. To further quantify curvature and multi-dimensional effects, a general flamelet formulation is derived mathematically. A budget analysis of the general flamelet equations is performed on an axisymmetric laminar diffusion flame. A new chemistry tabulation method based on the general flamelet formulation is proposed. This new tabulation method is applied to the same flame and demonstrates significant improvement compared to previous techniques.

For turbulent non-premixed flames, a new model to account for chemistry-turbulence interactions is proposed. %It is found that these interactions are not important for radicals and small species, but substantial for aromatic species. The validity of various existing flamelet-based chemistry tabulation methods is examined, and a new linear relaxation model is proposed for aromatic species. The proposed relaxation model is validated against full chemistry calculations. To further quantify the importance of aromatic chemistry-turbulence interactions, Large-Eddy Simulations (LES) have been performed on a turbulent sooting jet flame. %The aforementioned relaxation model is used to provide closure for the chemical source terms of transported aromatic species. The effects of turbulent unsteadiness on soot are highlighted by comparing the LES results with a separate LES using fully-tabulated chemistry. It is shown that turbulent unsteady effects are of critical importance for the accurate prediction of not only the inception locations, but also the magnitude and fluctuations of soot.

1. Ultrasonic studies of binary liquid structure in the critical region. Theory and experiment for the 2,6-lutidine/water system. 2. Hartree-Fock calculations of electric polarizabilities of some simple atoms and molecules, and their practicality. 3. Calculation of vibrational transition probabilities in collinear atom-diatom and diatom-diatom collisions with Lennard-Jones interaction

Part 1. Many interesting visual and mechanical phenomena occur in the critical region of fluids, both for the gas-liquid and liquid-liquid transitions. The precise thermodynamic and transport behavior here has some broad consequences for the molecular theory of liquids. Previous studies in this laboratory on a liquid-liquid critical mixture via ultrasonics supported a basically classical analysis of fluid behavior by M. Fixman (e. g., the free energy is assumed analytic in intensive variables in the thermodynamics)--at least when the fluid is not too close to critical. A breakdown in classical concepts is evidenced close to critical, in some well-defined ways. We have studied herein a liquid-liquid critical system of complementary nature (possessing a lower critical mixing or consolute temperature) to all previous mixtures, to look for new qualitative critical behavior. We did not find such new behavior in the ultrasonic absorption ascribable to the critical fluctuations, but we did find extra absorption due to chemical processes (yet these are related to the mixing behavior generating the lower consolute point). We rederived, corrected, and extended Fixman's analysis to interpret our experimental results in these more complex circumstances. The entire account of theory and experiment is prefaced by an extensive introduction recounting the general status of liquid state theory. The introduction provides a context for our present work, and also points out problems deserving attention. Interest in these problems was stimulated by this work but also by work in Part 3.

Part 2. Among variational theories of electronic structure, the Hartree-Fock theory has proved particularly valuable for a practical understanding of such properties as chemical binding, electric multipole moments, and X-ray scattering intensity. It also provides the most tractable method of calculating first-order properties under external or internal one-electron perturbations, either developed explicitly in orders of perturbation theory or in the fully self-consistent method. The accuracy and consistency of first-order properties are poorer than those of zero-order properties, but this is most often due to the use of explicit approximations in solving the perturbed equations, or to inadequacy of the variational basis in size or composition. We have calculated the electric polarizabilities of H₂, He, Li, Be, LiH, and N₂ by Hartree-Fock theory, using exact perturbation theory or the fully self-consistent method, as dictated by convenience. By careful studies on total basis set composition, we obtained good approximations to limiting Hartree-Fock values of polarizabilities with bases of reasonable size. The values for all species, and for each direction in the molecular cases, are within 8% of experiment, or of best theoretical values in the absence of the former. Our results support the use of unadorned Hartree-Pock theory for static polarizabilities needed in interpreting electron-molecule scattering data, collision-induced light scattering experiments, and other phenomena involving experimentally inaccessible polarizabilities.

Part 3. Numerical integration of the close-coupled scattering equations has been carried out to obtain vibrational transition probabilities for some models of the electronically adiabatic H₂-H₂ collision. All the models use a Lennard-Jones interaction potential between nearest atoms in the collision partners. We have analyzed the results for some insight into the vibrational excitation process in its dependence on the energy of collision, the nature of the vibrational binding potential, and other factors. We conclude also that replacement of earlier, simpler models of the interaction potential by the Lennard-Jones form adds very little realism for all the complication it introduces. A brief introduction precedes the presentation of our work and places it in the context of attempts to understand the collisional activation process in chemical reactions as well as some other chemical dynamics.

Numerical methods for ill-posed, linear problems

A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.

In the former method, Tikhonov provides a useful means for incorporating a constraint into numerical algorithms. The analysis suggests that the approach can be generalized to embody constraints other than those employed by Tikhonov. This is effected and the general "T-method" is the result.

A T-method is used on an extended version of the backwards heat equation with spatially variable coefficients. Numerical computations based upon it are performed.

The statistical method developed by Franklin is shown to have an interpretation as a T-method. This interpretation, although somewhat loose, does explain some empirical convergence properties which are difficult to pin down via a purely statistical argument.

Numerical solution of parabolic equations by the box scheme

The box scheme proposed by H. B. Keller is a numerical method for solving parabolic partial differential equations. We give a convergence proof of this scheme for the heat equation, for a linear parabolic system, and for a class of nonlinear parabolic equations. Von Neumann stability is shown to hold for the box scheme combined with the method of fractional steps to solve the two-dimensional heat equation. Computations were performed on Burgers' equation with three different initial conditions, and Richardson extrapolation is shown to be effective.

Nuclear Hartree-Fock calculations on parallel computers

Hartree-Fock (HF) calculations have had remarkable success in describing large nuclei at high spin, temperature and deformation. To allow full range of possible deformations, the Skyrme HF equations can be discretized on a three-dimensional mesh. However, such calculations are currently limited by the computational resources provided by traditional supercomputers. To take advantage of recent developments in massively parallel computing technology, we have implemented the LLNL Skyrme-force static and rotational HF codes on Intel's DELTA and GAMMA systems at Caltech.

We decomposed the HF code by assigning a portion of the mesh to each node, with nearest neighbor meshes assigned to nodes connected by communication· channels. This kind of decomposition is well-suited for the DELTA and the GAMMA architecture because the only non-local operations are wave function orthogonalization and the boundary conditions of the Poisson equation for the Coulomb field.

Our first application of the HF code on parallel computers has been the study of identical superdeformed (SD) rotational bands in the Hg region. In the last ten years, many SD rotational bands have been found experimentally. One very surprising feature found in these SD rotational bands is that many pairs of bands in nuclei that differ by one or two mass units have nearly identical deexcitation gamma-ray energies. Our calculations of the five rotational bands in ^(192)Hg and ^(194)Pb show that the filling of specific orbitals can lead to bands with deexcitation gamma-ray energies differing by at most 2 keV in nuclei differing by two mass units and over a range of angular momenta comparable to that observed experimentally. Our calculations of SD rotational bands in the Dy region also show that twinning can be achieved by filling or emptying some specific orbitals.

The interpretation of future precise experiments on atomic parity nonconservation (PNC) in terms of parameters of the Standard Model could be hampered by uncertainties in the atomic and nuclear structure. As a further application of the massively parallel HF calculations, we calculated the proton and neutron densities of the Cesium isotopes from A = 125 to A = 139. Based on our good agreement with experimental charge radii, binding energies, and ground state spins, we conclude that the uncertainties in the ratios of weak charges are less than 10^(-3), comfortably smaller than the anticipated experimental error.

Three-dimensional models of mantle convection : influence of phase transitions and temperature-dependent viscosity

Two of the most important questions in mantle dynamics are investigated in three separate studies: the influence of phase transitions (studies 1 and 2), and the influence of temperature-dependent viscosity (study 3).

(1) Numerical modeling of mantle convection in a three-dimensional spherical shell incorporating the two major mantle phase transitions reveals an inherently three-dimensional flow pattern characterized by accumulation of cold downwellings above the 670 km discontinuity, and cylindrical 'avalanches' of upper mantle material into the lower mantle. The exothermic phase transition at 400 km depth reduces the degree of layering. A region of strongly-depressed temperature occurs at the base of the mantle. The temperature field is strongly modulated by this partial layering, both locally and in globally-averaged diagnostics. Flow penetration is strongly wavelength-dependent, with easy penetration at long wavelengths but strong inhibition at short wavelengths. The amplitude of the geoid is not significantly affected.

(2) Using a simple criterion for the deflection of an upwelling or downwelling by an endothermic phase transition, the scaling of the critical phase buoyancy parameter with the important lengthscales is obtained. The derived trends match those observed in numerical simulations, i.e., deflection is enhanced by (a) shorter wavelengths, (b) narrower up/downwellings (c) internal heating and (d) narrower phase loops.

(3) A systematic investigation into the effects of temperature-dependent viscosity on mantle convection has been performed in three-dimensional Cartesian geometry, with a factor of 1000-2500 viscosity variation, and Rayleigh numbers of 10^5-10^7. Enormous differences in model behavior are found, depending on the details of rheology, heating mode, compressibility and boundary conditions. Stress-free boundaries, compressibility, and temperature-dependent viscosity all favor long-wavelength flows, even in internally heated cases. However, small cells are obtained with some parameter combinations. Downwelling plumes and upwelling sheets are possible when viscosity is dependent solely on temperature. Viscous dissipation becomes important with temperature-dependent viscosity.

The sensitivity of mantle flow and structure to these various complexities illustrates the importance of performing mantle convection calculations with rheological and thermodynamic properties matching as closely as possible those of the Earth.

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Tools for the study of dynamical spacetimes

This thesis covers a range of topics in numerical and analytical relativity, centered around introducing tools and methodologies for the study of dynamical spacetimes. The scope of the studies is limited to classical (as opposed to quantum) vacuum spacetimes described by Einstein's general theory of relativity. The numerical works presented here are carried out within the Spectral Einstein Code (SpEC) infrastructure, while analytical calculations extensively utilize Wolfram's Mathematica program.

We begin by examining highly dynamical spacetimes such as binary black hole mergers, which can be investigated using numerical simulations. However, there are difficulties in interpreting the output of such simulations. One difficulty stems from the lack of a canonical coordinate system (henceforth referred to as gauge freedom) and tetrad, against which quantities such as Newman-Penrose Psi_4 (usually interpreted as the gravitational wave part of curvature) should be measured. We tackle this problem in Chapter 2 by introducing a set of geometrically motivated coordinates that are independent of the simulation gauge choice, as well as a quasi-Kinnersley tetrad, also invariant under gauge changes in addition to being optimally suited to the task of gravitational wave extraction.

Another difficulty arises from the need to condense the overwhelming amount of data generated by the numerical simulations. In order to extract physical information in a succinct and transparent manner, one may define a version of gravitational field lines and field strength using spatial projections of the Weyl curvature tensor. Introduction, investigation and utilization of these quantities will constitute the main content in Chapters 3 through 6.

For the last two chapters, we turn to the analytical study of a simpler dynamical spacetime, namely a perturbed Kerr black hole. We will introduce in Chapter 7 a new analytical approximation to the quasi-normal mode (QNM) frequencies, and relate various properties of these modes to wave packets traveling on unstable photon orbits around the black hole. In Chapter 8, we study a bifurcation in the QNM spectrum as the spin of the black hole a approaches extremality.

Crystal Coulomb energies: I. Organic donor-acceptor complexes--a review. II. Classical Ewald calculations of the Coulomb binding energy of four donor-acceptor crystals and Wurster's blue perchlorate. III. A rapid-convergence quantum-mechanical formalism for crystal electronic energies

Chapter I

Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε₁ (per DA pair) gained in ionizing a DA lattice exceeds the cost 2ε_o of ionizing each DA pair, ε₁ + ε_o less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D⁺ and A^-). The magnetic properties of the DA crystals are discussed.

Chapter II

A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy E_C due to classical monopole-monopole interactions for crystals of any symmetry. The precision of E_C values obtained is high: the uncertainties, estimated by the effect on E_C of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in E_C due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.

E_C for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. E_C for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.

EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) E_C = -4.0 eV while 2ε_o = 4.6₅ eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) E_C = -4.4 eV, while 2ε_o = 5.0 eV: again EC falls short of 2ε₁. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) E_C = -4.5 eV, 2ε_o = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) E_C = -4.3 eV, 2ε_o = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.

Chapter III

A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]^XT, direct Coulomb [λλ/λ'λ']^XT and exchange Coulomb [λλ'/λ'λ]^XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]^XT, [λλ/λ'λ']^XT, and [λλ/λ'λ]^XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]^XT, [λλ/λ'λ']^XT and [λλ'/λ'λ]^XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]^XT, etc., where some of the portions contain the Gaussian factor.