Rigorous numerical approaches in electronic structure theory

Electronic structure theory concerns the description of molecular properties according to the postulates of quantum mechanics. For practical purposes, this is realized entirely through numerical computation, the scope of which is constrained by computational costs that increases rapidly with the size of the system. The significant progress made in this field over the past decades have been facilitated in part by the willingness of chemists to forego some mathematical rigour in exchange for greater efficiency. While such compromises allow large systems to be computed feasibly, there are lingering concerns over the impact that these compromises have on the quality of the results that are produced. This research is motivated by two key issues that contribute to this loss of quality, namely i) the numerical errors accumulated due to the use of finite precision arithmetic and the application of numerical approximations, and ii) the reliance on iterative methods that are not guaranteed to converge to the correct solution. Taking the above issues in consideration, the aim of this thesis is to explore ways to perform electronic structure calculations with greater mathematical rigour, through the application of rigorous numerical methods. Of which, we focus in particular on methods based on interval analysis and deterministic global optimization. The Hartree-Fock electronic structure method will be used as the subject of this study due to its ubiquity within this domain. We outline an approach for placing rigorous bounds on numerical error in Hartree-Fock computations. This is achieved through the application of interval analysis techniques, which are able to rigorously bound and propagate quantities affected by numerical errors. Using this approach, we implement a program called Interval Hartree-Fock. Given a closed-shell system and the current electronic state, this program is able to compute rigorous error bounds on quantities including i) the total energy, ii) molecular orbital energies, iii) molecular orbital coefficients, and iv) derived electronic properties. Interval Hartree-Fock is adapted as an error analysis tool for studying the impact of numerical error in Hartree-Fock computations. It is used to investigate the effect of input related factors such as system size and basis set types on the numerical accuracy of the Hartree-Fock total energy. Consideration is also given to the impact of various algorithm design decisions. Examples include the application of different integral screening thresholds, the variation between single and double precision arithmetic in two-electron integral evaluation, and the adjustment of interpolation table granularity. These factors are relevant to both the usage of conventional Hartree-Fock code, and the development of Hartree-Fock code optimized for novel computing devices such as graphics processing units. We then present an approach for solving the Hartree-Fock equations to within a guaranteed margin of error. This is achieved by treating the Hartree-Fock equations as a non-convex global optimization problem, which is then solved using deterministic global optimization. The main contribution of this work is the development of algorithms for handling quantum chemistry specific expressions such as the one and two-electron integrals within the deterministic global optimization framework. This approach was implemented as an extension to an existing open source solver. Proof of concept calculations are performed for a variety of problems within Hartree-Fock theory, including those in i) point energy calculation, ii) geometry optimization, iii) basis set optimization, and iv) excited state calculation. Performance analyses of these calculations are also presented and discussed

Density functional theory for large molecular systems

Nøyaktige simuleringer av kjemiske og biologiske prosesser på molekylært nivå har lenge vært uoppnålig for en rekke molekylære systemer, og har nå blitt mulig for mange av disse systemene gjennom ny metodeutvikling av Simen Reine, Trygve Helgaker og medarbeidere ved Universitetet i Oslo. Datasimuleringer er utbredt innen kjemi og relaterte felt som biologi, farmasi og medisin. Kvantekjemiske metoder er fundamentale for de mest nøyaktig simuleringsteknikkene, og er til stor hjelp ved bestemmelse og prediksjon av molekylære egenskaper, som for eksempel molekylers struktur, og gir i tillegg viktig og detaljert innsikt i kjemiske reaksjoner - både kvalitativt og kvantitativt. Anvendelsesområdet er nært knyttet til metodenes nøyaktighet, effektivitet og brukervennelighet. Utviklingen av nye og forbedrede metoder gjør oss i stand til å studere molekylære systemer som foreløpig har vært utenfor rekkevidde, og gir oss mer nøyaktig beskrivelse av de systemene vi allerede behandler idag. Som en konsekvens vil man kunne redusere bruken av kostbare og tidkrevende eksperimenter og samtidig hjelpe forskere verden over til bedre å forstå kjemiske mekanismer. De fleste kvantekjemiske beregninger som utføres idag benytter tetthetsfunksjonalteori (DFT), da denne metoden utgjør et godt kompromiss mellom nøyaktighet og beregningstid. Selv om DFT er meget nyttig, er dagens metoder begrenset til systemer bestående av noen få hundre atomer, og utelukker derfor en rekke systemer, for eksempel proteiner. I doktorgraden "Tetthetsfunksjonalteori for store molekylære systemer" har nye metoder innen DFT blitt utviklet med tanke på rutinemessige beregninger for store systemer. Beregninger for systemer med 1400 atomer er rapportert og metodene er i etterkant blitt benyttet for systemer med opp til 4000 atomer

A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry

We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.Comment: 64 pages, 3 figures, tMPH2e.cls style file, doublesp, mathbbol and subeqn package

Distributed Memory, GPU Accelerated Fock Construction for Hybrid, Gaussian Basis Density Functional Theory

With the growing reliance of modern supercomputers on accelerator-based architectures such a GPUs, the development and optimization of electronic structure methods to exploit these massively parallel resources has become a recent priority. While significant strides have been made in the development of GPU accelerated, distributed memory algorithms for many-body (e.g. coupled-cluster) and spectral single-body (e.g. planewave, real-space and finite-element density functional theory [DFT]), the vast majority of GPU-accelerated Gaussian atomic orbital methods have focused on shared memory systems with only a handful of examples pursuing massive parallelism on distributed memory GPU architectures. In the present work, we present a set of distributed memory algorithms for the evaluation of the Coulomb and exact-exchange matrices for hybrid Kohn-Sham DFT with Gaussian basis sets via direct density-fitted (DF-J-Engine) and seminumerical (sn-K) methods, respectively. The absolute performance and strong scalability of the developed methods are demonstrated on systems ranging from a few hundred to over one thousand atoms using up to 128 NVIDIA A100 GPUs on the Perlmutter supercomputer.Comment: 45 pages, 9 figure

Performance Models for Electronic Structure Methods on Modern Computer Architectures

Electronic structure codes are computationally intensive scientic applications used to probe and elucidate chemical processes at an atomic level. Maximizing the performance of these applications on any given hardware platform is vital in order to facilitate larger and more accurate computations. An important part of this endeavor is the development of protocols for measuring performance, and models to describe that performance as a function of system architecture. This thesis makes contributions in both areas, with a focus on shared memory parallel computer architectures and the Gaussian electronic structure code. Shared memory parallel computer systems are increasingly important as hardware man- ufacturers are unable to extract performance improvements by increasing clock frequencies. Instead the emphasis is on using multi-core processors to provide higher performance. These processor chips generally have complex cache hierarchies, and may be coupled together in multi-socket systems which exhibit highly non-uniform memory access (NUMA) characteristics. This work seeks to understand how cache characteristics and memory/thread placement affects the performance of electronic structure codes, and to develop performance models that can be used to describe and predict code performance by accounting for these effects. A protocol for performing memory and thread placement experiments on NUMA systems is presented and its implementation under both the Solaris and Linux operating systems is discussed. A placement distribution model is proposed and subsequently used to guide both memory/thread placement experiments and as an aid in the analysis of results obtained from experiments. In order to describe single threaded performance as a function of cache blocking a simple linear performance model is investigated for use when computing the electron repulsion integrals that lie at the heart of virtually all electronic structure methods. A parametric cache variation study is performed. This is achieved by combining parameters obtained for the linear performance model on existing hardware, with instruction and cache miss counts obtained by simulation, and predictions are made of performance as a function of cache architecture. Extension of the linear performance model to describe multi-threaded performance on complex NUMA architectures is discussed and investigated experimentally. Use of dynamic page migration to improve locality is also considered. Finally the use of large scale electronic structure calculations is demonstrated in a series of calculations aiming to study the charge distribution for a single positive ion solvated within a shell of water molecules of increasing size

Calculation of Intermolecular Interactions via Diffusion Monte Carlo

Quantum chemistry is a useful tool that provides insight into the properties and behavior of chemical systems. Modern software packages have made quantum chemistry methods more easily accessible, and the continued increase in available computational resources has allowed them to be applied to larger systems at higher levels of theory. Two significant problems that the field faces are the high computational complexity of high-level methods and the shift toward parallelism in high performance computing architectures. This work examines the treatment of weakly interacting molecular systems with the fixed-node diffusion Monte Carlo (DMC) method. DMC and other quantum Monte Carlo (QMC) methods offer a possible solution to both of the aforementioned problems: they can produce near-exact results with a lower scaling (with respect to problem size) than other similarly-accurate methods, and they are inherently parallel, so there is little additional cost associated with distributing the work of a single QMC calculation across a large number of processing units. The only error in DMC that is not systematically improvable is the constraint of a fixed nodal surface of the many-particle wave function of the system being studied. There are many cases in which a single Slater determinant trial function is sufficient to obtain accurate results, but there are others in which more sophisticated multi-determinant trial functions are necessary. Furthermore, it is non-trivial to generate nodal surfaces of similar quality for isolated and interacting molecules, so cancellation of errors is not guaranteed. We examine the use of different single- and multi-determinant trial functions in DMC calculations on small chemical systems with the goal of further understanding how to construct appropriate trial functions for molecules and clusters

The IPS fidelity scale as a guideline to implement Supported Employment

info:eu-repo/semantics/publishe

Single-Determinant Theory of Electronic Excited States and Many-Electron Integrals for Explicitly Correlated

The aim of this thesis is twofold. Its first part, Part A, is concerned with the development and assessment of a single-determinant theory for electronic excited states. The theory is based on two simple algorithms for finding excited-state solutions to self-consistent field (SCF) equations, the Maximum Overlap Method (MOM) and the Initial Maximum Overlap Method (IMOM). The extent to which these higher SCF solutions are useful approximations to excited states is examined in diverse case studies, including challenging instances such as double excitations, conical intersections and charge-transfer states. Results indicate that single-determinant models yield, in most cases, accurate approximations to electronic excited states, even for di cult excitations where other low-cost excited-state methods either perform poorly or fail completely. In Part B, we present efficient methods for the accurate evaluation of many-electron integrals arising in the explicitly correlated electronic structure theory. In our computational schemes e cient screening techniques, which adopt newly developed upper bounds, are used to sift out the tiny fraction of integrals which are significant. Then, non-negligible integrals are evaluated via recurrence relations that represent the generalization to three and four-electron integrals of two-electron integrals contraction-e cient schemes such as the Head-Gordon-Pople and PRISM algorithms. In this way, we developed general computational schemes for integrals arising from the use of a wide class of multiplicative correlation factors of the form f12 = f(|r1 r2|) and more specific methods for many electron integrals involving Gaussian Geminals. Our results support the evidence that our Gaussian-Geminal-based schemes yield a dramatic reduction of the computational complexity of these integral