Tosio Kato (1917–1999)

Tosio Kato was born August 25, 1917, in Kanuma City, Tochigi-ken, Japan. His early training was in physics. He obtained a B.S. in 1941 and the degree of Doctor of Science in 1951, both at the University of Tokyo. Between these events he published papers on a variety of subjects, including pair creation by gamma rays, motion of an object in a fluid, and results on spectral theory of operators arising in quantum mechanics. His dissertation was entitled “On the convergence of the perturbation method”. Kato was appointed assistant professor of physics at the University of Tokyo in 1951 and was promoted to professor of physics in 1958. During this time he visited the University of California at Berkeley in 1954–55, New York University in 1955, the National Bureau of Standards in 1955–56, and Berkeley and the California Institute of Technology in 1957–58. He was appointed professor of mathematics at Berkeley in 1962 and taught there until his retirement in 1988. He supervised twenty-one Ph.D. students at Berkeley and three at the University of Tokyo. Kato published over 160 papers and 6 monographs, including his famous book Perturbation Theory for Linear Operators [K66b]. Recognition for his important work included the Norbert Wiener Prize in Applied Mathematics, awarded in 1980 by the AMS and the Society for Industrial and Applied Mathematics. He was particularly well known for his work on Schrödinger equations of nonrelativistic quantum mechanics and his work on the Navier-Stokes and Euler equations of classical fluid mechanics. His activity in the latter area remained at a high level well past retirement and continued until his death on October 2, 1999

On dynamical low-rank integrators for matrix differential equations

This thesis is concerned with dynamical low-rank integrators for matrix differential equations, typically stemming from space discretizations of partial differential equations. We first construct and analyze a dynamical low-rank integrator for second-order matrix differential equations, which is based on a Strang splitting and the projector-splitting integrator, a dynamical low-rank integrator for first-order matrix differential equations proposed by Lubich and Osedelets in 2014. For the analysis, we derive coupled recursive inequalities, where we express the global error of the scheme in terms of a time-discretization error and a low-rank error contribution. The first can be treated with Taylor series expansion of the exact solution. For the latter, we make use of an induction argument and the convergence result derived by Kieri, Lubich, and Walach in 2016 for the projector-splitting integrator. From the original method, several variants are derived which are tailored to, e.g., stiff or highly oscillatory second-order problems. After discussing details on the implementation of dynamical low-rank schemes, we turn towards rank-adaptivity. For the projector-splitting integrator we derive both a technique to realize changes in the approximation ranks efficiently and a heuristic to choose the rank appropriately over time. The core idea is to determine the rank such that the error of the low-rank approximation does not spoil the time-discretization error. Based on the rank-adaptive pendant of the projector-splitting integrator, rank-adaptive dynamical low-rank integrators for (stiff and non-stiff) first-order and second-order matrix differential equations are derived. The thesis is concluded with numerical experiments to confirm our theoretical findings

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange–correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear–electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an “open teamware” model and an increasingly modular design

Dielectric and Spin Susceptibilities using Density Functional Theory

The response of a system to some external perturbation is almost ubiquitous in Physics. The application of perturbation theory through an electronic structure method such as Density Functional Theory has had significant contributions over the last few decades. Its implementation, aptly named Density Functional Perturbation Theory has seen use in a number of ab initio calculations on a variety of physical properties of materials which depend on their lattice-dynamical behaviour. Specific heats, thermal expansion, infrared, Raman and optical spectra are to name just a few. Understanding the complex phenomena has significantly corroborated the current understanding of the quantum picture of solids. The Sternheimer scheme falls under the umbrella of methods to compute response functions in Time-Dependent Density Functional Theory. Initially developed to study the electronic polarisability, it is now commonly utilised in the field of lattice dynamics to study phonons and related crystal properties. The Sternheimer equation has also been used to model spin wave excitations by computation of the magnetic susceptibility. The poles of the susceptibility are known to correspond to magnon excitations and these computations have been corroborated by experimental inelastic neutron scattering data. These excitations are of a transverse nature, in that they involve fluctuations of the magnetisation perpendicular to a chosen z axis. The lesser-known longitudinal excitations involve fluctuations of the magnetisation along z, an investigation of collective modes present in transition metals may be carried out from self-consistent computations of the Sternheimer equation. The dielectric response is an important linear response function in solid-state physics. Its computation from first principles provides an invaluable tool in the characterisation of optical properties and can be compared to the experimental method of spectroscopic ellipsometry.The work in this thesis concerns the implementation of the Sternheimer method in computing the dynamical response from either an external plane wave or spin-polarised perturbation. These response functions are the dielectric and spin (magnetisation) susceptibilities respectively.The scheme to compute the frequency-dependent dielectric response is implemented in a plane-wave pseudopotential DFT package. Calculations are performed on the semiconducting systems of Silicon, Gallium Arsenide, Zinc Oxide and perovskite Methylammonium Lead Triiodide. The overall shape of the dielectric spectra is in good agreement with spectroscopic ellipsometry data, however, there is a shift which is attributed to the limitations of DFT.The scheme developed to compute longitudinal spin dynamics is applied to the transitionmetal systems of body-centred cubic Iron and face-centred cubic Nickel. In a similar manner to another first principles approach, a single dominant peak is shown to be present in the magnetisation channel with the charge dynamics being effectively null in comparison. However, the exact position of these peaks is not in agreement with the other approach, a discussion is made regarding difficulties pertaining to self-consistent optimisation

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Software for the frontiers of quantum chemistry : An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange–correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear–electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an “open teamware” model and an increasingly modular design.This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.Peer reviewe

Software for the frontiers of quantum chemistry:An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange–correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear–electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an “open teamware” model and an increasingly modular design