Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why the isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to nonspherical particles. We develop first- and second-order conditions for jamming, and demonstrate that packings of nonspherical particles can be jammed even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.Comment: 33 pages, third revisio

Scalable computational chemistry: new developments and applications

The computational part of the thesis is the investigation of titanium chloride (II) as a potential catalyst for the bis-silylation reaction of ethylene with hexaclorodisilane at different levels of theory. Bis-silylation is an important reaction for producing bis(silyl) compounds and new C-Si bonds, which can serve as monomers for silicon containing polymers and silicon carbides. Ab initio calculations on the steps involved in a proposed mechanism are presented. This choice of reactants allows us to study this reaction at reliable levels of theory without compromising accuracy. Our calculations indicate that this is a highly exothermic barrierless reaction. The TiCl 2 catalyst removes a 50 kcal/mol activation energy barrier required for the reaction without the catalyst. The first step is interaction of TiCl 2 with ethylene to form an intermediate that is 60 kcal/mol below the energy of the reactants. This is the driving force for the entire reaction. Dynamic correlation plays a significant role because RHF calculations indicate that the net barrier for the catalyzed reaction is 50 kcal/mol. We conclude that divalent Ti has the potential to become an important industrial catalyst for silylation reactions.;In the programming part of the thesis, parallelization of different quantum chemistry methods is presented. The parallelization of code is becoming important aspect of quantum chemistry code development. Two trends contribute to it: the overall desire to study large chemical systems and the desire to employ highly correlated methods which are usually computationally and memory expensive. In the presented distributed data algorithms computation is parallelized and the largest arrays are evenly distributed among CPUs. First, the parallelization of the Hartree-Fock self-consistent field (SCF) method is considered. SCF method is the most common starting point for more accurate calculations. The Fock build (sub step of SCF) from AO integrals is also often used to avoid MO integral computation. The presented distributed data SCF increases the size of chemical systems that can be calculated by using RHF and DFT. The important ab initio method to study bond formation and breaking as well as excited molecules is CASSCF. The presented distributed data CASSCF algorithm can significantly decrease computational time and memory requirements per node. Therefore, large CASSCF computations can be performed. The most time consuming operation to study potential energy surfaces of reactions and chemical systems is Hessian calculations. The distributed data parallelization of CPHF will allow scientists carry out large analytic Hessian calculations

Variational determination of the two-particle density matrix as a quantum many-body technique

In this thesis the variational optimisation of the density matrix is discussed as a method in many-body quantum mechanics. This is a relatively unknown technique in which one tries to obtain the two-particle reduced density matrix directly in a variational approach. In the first Chapter we introduce the subject in the broader context of many-body quantum mechanics, and briefly sketch the history of the field. The second Chapter tries to summarise what is known about N-representability of reduced density matrices, and derives these results in a unified framework. The optimisation problem can be formulated as a semidefinite program. Three different algorithms are discussed in Chapter three, and their performance is compared. In Chapter four we show how symmetry can be exploited to reduce the computational cost considerably. Several applications of the method are discussed in Chapter five. We show that the standard two-index constraints fail in the strong-correlation limit of the one-dimensional Hubbard model. In Chapter six we identify the spin-adapted lifting conditions as the most compact constraints that can correctly describe this limit.Comment: PhD thesis; 196 page

In- and out-of-equilibrium {\em ab initio} theory of electrons and phonons

We lay down the {\em ab initio} many-body quantum theory of electrons and phonons in- and out-of-equilibrium at any temperature. We begin by addressing a fundamental issue concerning the {\em ab initio} Hamiltonian in the harmonic approximation, which we show must be determined {\em self-consistently} to avoid inconsistencies. After identifying the most suitable partitioning into a ``noninteracting'' and an ``interacting'' part we embark on the Green's function diagrammatic analysis. We single out key diagrammatic structures to carry on the expansion in terms of dressed propagators and screened interaction. The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, featuring the appearance of the time-local Ehrenfest diagram in the electronic self-energy. The Hedin equations have limited practical utility for real-time simulations of driven systems. We leverage the versatility of diagrammatic expansion to generate a closed system of integro-differential equations for the Green's functions and nuclear displacements. These are the Kadanoff-Baym equations for electrons and phonons. Another advantage of the diagrammatic derivation is the ability to use conserving approximations, which ensure the satisfaction of all fundamental conservation laws during the time evolution. As an example we show that the adiabatic Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electron-phonon coupling. We also derive the formal solution of the Kadanoff-Baym equations in the long time limit and at the steady state. The expansion of the phononic Green's function around the quasi-phonon energies points to a possible correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.Comment: 33 pages, 28 figures (8 of which with caption

A numerical approach for calculating exact non-adiabatic terms in quantum dynamics

Understanding how non-adiabatic terms affect quantum dynamics is fundamental to improving various protocols for quantum technologies. We present a novel approach to computing the Adiabatic Gauge Potential (AGP), which gives information on the non-adiabatic terms that arise from time dependence in the Hamiltonian. Our approach uses commutators of the Hamiltonian to build up an appropriate basis of the AGP, which can be easily truncated to give an approximate form when the exact result is intractable. We use this approach to study the AGP obtained for the transverse field Ising model on a variety of graphs, showing how the different underlying graph structures can give rise to very different scaling for the number of terms required in the AGP.Comment: 28 pages, 6 figures, comments welcom