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When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk UNIVERSITY OF SOUTHAMPTON SCHOOL OF CHEMISTRY Computational methods for density functional theory calculations on insulators and metals based on localised orbitals by Álvaro Ruiz-Serrano Thesis for the degree of Doctor of Philosophy August 2013 Sueña el rico en su riqueza, que más cuidados le ofrece; sueña el pobre que padece su miseria y su pobreza; sueña el que a medrar empieza, sueña el que afana y pretende, sueña el que agravia y ofende, y en el mundo, en conclusión, todos sueñan lo que son, aunque ninguno lo entiende. Yo sueño que estoy aquí de estas prisiones cargado, y soñé que en otro estado más lisonjero me vi. ¿Qué es la vida? Un frenesí. ¿Qué es la vida? Una ilusión, una sombra, una ficción, y el mayor bien es pequeño; que toda la vida es sueño, y los sueños, sueños son. La vida es sueño, acto segundo. Pedro Calderón de la Barca, 1635. 3 4 Declaration of authorship I, Álvaro Ruiz-Serrano, declare that the thesis entitled “Computational methods for density functional theory calculations on insulators and metals based on localised orbitals” and the work presented in the thesis are both my own, and have been generated by me as the result of my own original research. I confirm that: • this work was done wholly or mainly while in candidature for a research degree at this University; • where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated; • where I have consulted the published work of others, this is always clearly attributed; • where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; • I have acknowledged all main sources of help; • where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself; • parts of this work have been published as: – Pulay forces from localized orbitals optimized in situ using a psinc basis set. Á. Ruiz-Serrano, N. D. M. Hine, and C.-K. Skylaris. J. Chem. Phys. 136(23), 234101 (2012). – A variational method for density functional theory calculations on meta- llic systems with thousands of atoms. Á. Ruiz-Serrano and C.-K. Sky- laris. J. Chem. Phys. 139(5), 054107 (2013). – Variationally-localized search direction method for constrained optimi- zation of non-orthogonal, localized orbitals in electronic structure cal- culations. Á. Ruiz-Serrano and C.-K. Skylaris. J. Chem. Phys. 139(16), 164110 (2013). – DIIS and Hamiltonian diagonalisation for total-energy minimisation in the ONETEP program using the ScaLAPACK eigensolver. Á. Ruiz- Serrano. M. Sc. in High Performance Computing. University of Edin- burgh (2010). Signed: . Date:................................................. 5 6 UNIVERSITY OF SOUTHAMPTON SCHOOL OF CHEMISTRY Doctor of Philosophy Computational methods for density functional theory calculations on insulators and metals based on localised orbitals by Álvaro Ruiz-Serrano Abstract Kohn-Sham density functional theory (DFT) calculations yield reliable accu- racy in a wide variety of molecules and materials. The advent of linear-scaling DFT methods, based on locality of the electronic matter, has enabled calculations on systems with tens of thousands of atoms. Localisation constraints are imposed by expanding the Kohn-Sham states in terms of a set of atom-centred, spherically- localised functions. Chemical accuracy is then achieved via a self-consistent opti- misation using a high-resolution basis set. This formalism reduces the size of, and brings predictable sparsity patterns to, the matrices expressed in this representa- tion, such as the Hamiltonian matrix. In this work, we used the ONETEP program for DFT calculations, which is based on the abovementioned principles. The vi- sion behind our research is to advance the method by developing new and robust algorithms to enable novel applications based on localised orbitals. We investigated the consequences of strict spatial localisation constraints on the energy minimisation problem. This part of our research led to new theoretical realisations and approaches to calculate the total energy and atomic forces. We show that the self-consistent energy minimisation process is in fact a constrained optimisation problem where the localisation constraints must be taken into account for consistency. With this important point in mind, we were able to show that the atomic forces must be corrected with extra terms that account for the localisa- tion constraints. We developed the code to determine the correction terms, known as Pulay forces, which provide consistency between the total energy and atomic forces. Weakly-bound systems, such as protein-ligand complexes, are now accu- rately described after the total forces were corrected. We took a step forward in the formulation of the optimisation problem with lo- calisation constraints. Using an analytic manipulation of the equations that govern the energy minimisation procedure, we were able to design a new algorithm for constrained optimisation of non-orthogonal, localised functions. This new method is exact and accounts for the localisation constraints in a fully variational man- ner. We named this new algorithm the variationally-localised search direction (VLSD) method. We implemented a prototypical version of the VLSD method 7 in the ONETEP program. We were able to compare the convergence speed to- wards the solution between the VLSD and the standard method, based on an un- constrained optimisation process that does not account for localisation constraints in a variational fashion. Calculations with the VLSD method typically show faster convergence, especially in systems with a certain degree of expected natural delo- calisation. We not only studied the effects of localisation, we also used the localised- functions formalism to develop two new types of algorithms for calculations on insulators (various density mixing methods) and metals (a direct minimisation method for finite-temperature DFT). These methods use the SCALAPACK dense algebra package to distribute memory and computational requirements over a large number of processors. In these approaches, the Hamiltonian matrix must be diag- onalised for consistency. The formalism based on localised functions makes this step efficient, despite the cubic-scaling cost, and allowed us to perform calculations on systems with thousands of atoms. Density mixing methods are well-established techniques for total energy min- imisation in self-consistent field approaches. Their implementation is very useful for profiling and benchmarking other approaches that are fundamental for achiev- ing linear-scaling cost. Validation calculations on a set of insulators and semi- conductors show that accuracy in the description of the electronic structure is con- sistently achieved with the density mixing schemes. The new algorithm that we have implemented in the ONETEP program for cal- culations on metallic systems is an important new functionality for the code. Typ- ically, standard DFT methods for metals, based on plane-waves or other forms of delocalised orbitals, are limited to a few tens of atoms. Calculations on thousands of atoms are possible at the expense of a high computational cost, requiring tens of thousands of processors. With our new approach, calculations on large metallic systems with thousands of atoms can be executed employing for the task only a few hundreds of processors. This method can be used in studies of metallic complexes that are not viable with traditional approaches, such as the kind of nanoparticles used in innovative catalytic processes. 8 Acknowledgements The last few years of research activity have been very intense and productive, both professionally and personally. It is a pleasure for me to finally be able to see all our scientific contributions put together into a single document, to form this Ph. D. dissertation. I am grateful to all the people that contributed to make it a reality: • To my thesis supervisor, Chris-Kriton Skylaris, for his direct participation in all our projects, for his much-appreciated scientific advise, and for his support. • To the Engineering and Physical Sciences Research Council (EPSRC) for the financial support via a High End Computing Studentship through the UKCP consortium (Grant No EP/F038038/1). • To the members of the Edinburgh Parallel Computing Centre (EPCC) for sharing their expertise with me during my studies at the University of Edin- burgh. In particular, I would like to thank my M. Sc. dissertation supervisor, Bartosz Dobrzelecki, for his advice. • To Nicholas Hine for his active involvement in the work regarding the cal- culation of the Pulay forces, as well as for being a reliable source of advise regarding high-standards computing programming. • To the members of the Skylaris group, past and present, for creating a friendly environment to work and coexist. Ben Lowe, Chris Cave-Ayland, Chris Pit- tock, Chris Sampson, Jacek Dziedzic, Karl Wilkinson, Max Phipps, Nawzat Saadi, Nicholas Zonias, Quintin Hill, Peter Cherry, Stephen Fox and Valerio Vitale. It was great sharing the office with you. 9 • To those that I can proudly call my friends in Southampton, and whose com- pany always made feel at home.