Multireference approaches for excited states of molecules

Understanding the properties of electronically excited states is a challenging task that becomes increasingly important for numerous applications in chemistry, molecular physics, molecular biology, and materials science. A substantial impact is exerted by the fascinating progress in time-resolved spectroscopy, which leads to a strongly growing demand for theoretical methods to describe the characteristic features of excited states accurately. Whereas for electronic ground state problems of stable molecules the quantum chemical methodology is now so well developed that informed nonexperts can use it efficiently, the situation is entirely different concerning the investigation of excited states. This review is devoted to a specific class of approaches, usually denoted as multireference (MR) methods, the generality of which is needed for solving many spectroscopic or photodynamical problems. However, the understanding and proper application of these MR methods is often found to be difficult due to their complexity and their computational cost. The purpose of this review is to provide an overview of the most important facts about the different theoretical approaches available and to present by means of a collection of characteristic examples useful information, which can guide the reader in performing their own applications

The density matrix renormalization group for ab initio quantum chemistry

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. The MPS ansatz naturally captures exponentially decaying correlation functions. Therefore DMRG works extremely well for noncritical one-dimensional systems. The active orbital spaces in quantum chemistry are however often far from one-dimensional, and relatively large virtual dimensions are required to use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its computational cost, and its properties are discussed. Two important aspects to reduce the computational cost are given special attention: the orbital choice and ordering, and the exploitation of the symmetry group of the Hamiltonian. With these considerations, the QC-DMRG algorithm allows to find numerically exact solutions in active spaces of up to 40 electrons in 40 orbitals.Comment: 24 pages; 10 figures; based on arXiv:1405.1225; invited review for European Physical Journal

Development and Validation of the REMP and OO-REMP Hybrid Perturbation Theories

In dieser Arbeit werden die hybriden Störungstheorien REMP und OO-REMP zur Be- rechnung der elektronischen Korrelationsenergie von Atomen und Molekülen eingeführt und validiert. Es handelt sich dabei um quantenchemische Methoden im Formalismus der Rayleigh-Schrödinger-Störungstheorie, für die hier die Energie 2. Ordnung untersucht wird. Basierend auf den Partitionierungen der Møller-Plesset- (MP) und der Anregungsgrader- haltenden Störungstheorie (Retaining the excitation Degree=RE) wird ein ungestörter Hamiltonoperator mit zugehörigem Störoperator definiert, der sich aus einer gewichteten Summe der vorgenannten Methoden zusammensetzt, wodurch die REMP-Methode defi- niert ist. Die neuartige Partitionierung nutzt komplementäre Fehler der zugrunde liegenden Methoden zur internen Fehlerkompensation. In dieser Arbeit werden Energien bis zur 2. Ordnung der Störungstheorie untersucht. Es wird gezeigt, dass die REMP-Partitionierung des elektronischen Hamiltonoperators zu systematisch besseren Ergebnissen führt als jede der Einzelmethoden allein, wobei die Parametrisierung der Mischung universell und praktisch systemunabhängig ist. Dies wird am Beispiel unterschiedlicher Typen von Reaktionsenergien und Gleichgewichtsstrukturen, Schwingungswellenzahlen und elektri- schen Dipolmomenten kleiner Moleküle demonstriert. Es wird außerdem ein variationelles Energiefunktional definiert, das auf der Hybridpartitionierung basiert. Dabei wird die Form der besetzen Molekülorbitale variiert und so optimiert, dass die Gesamtenergie minimal wird. Die Minimierung dieses Funktionals bezüglich aller variationellen Para- meter liefert Ergebnisse, die die der kanonischen Methode systematisch übertreffen. Die vollständig variationelle Methode zeichnet sich zudem durch hervorragende rechnerische Effizienz bei der Vorhersage molekularer Eigeschaften aus. Es wird gezeigt, dass insbeson- dere die vollständig variationelle, orbitaloptimierte Variante (OO-REMP) den Kriterien allgemein anwendbarer Quantenchemiemethoden genügt und hochgenaue Ergebnisse produziert. Die Validierungen legen nahe, dass OO-REMP für single-reference-Systeme für die meisten Thermochmie-Testsätze chemische Genauigkeit erreicht (Root mean square-Fehler ⩽1 kcal mol−1 ). Die neu entwickelten Methoden wurden in ein quelloffenes Quantenchemieprogramm implementiert und stehen nun jedermann zur Verfügung.In this work, the hybrid perturbation theories REMP and OO-REMP for the calculation of electronic correlation energies of atoms and molecules are introduced and validated. These are quantum chemical methods in the framework of Rayleigh-Schrödinger perturbation theory, whose second order energy is investigated here. Based on the partitionings of the Møller-Plesset (MP) and the Retaining the Excitation Degree (RE) perturbation theory, an unperturbed Hamiltonian with a corresponding perturbation operator is defined, which is a weighted sum of the previous methods, thereby defining the REMP method. The novel partitioning has the property to exploit complementary errors of the parent methods for internal error compensation. In this work, energies up to 2nd order in perturbation theory are investigated. It is shown that the REMP partitioning of the electronic Hamiltonian leads to systematically better results than each of the original methods, with the important aspect that the parameterization of the mixture is universal and practically independent of the system considered. This is demonstrated with the example various types of reaction energies and equilibrium structures, vibrational wavenumbers, and electric dipole moments of small molecules. Furthermore, a variational energy functional based on the hybrid partitioning is defined. Here, the shape of the occupied molecular orbitals is varied and optimized such, that the total energy becomes minimal. The minimization of this functional with respect to all variational parameters provides results which systematically surpass those of the canonical method. The fully variational method is furthermore characterized by outstanding computational efficiency regarding the prediction of molecular properties. It is shown that especially the fully variational, orbital-optimized variant suffices the criteria of a generally applicable quantum chemical method and does produce highly accurate results. The validations imply that for single-reference systems OO-REMP reaches chemical accuracy (root mean square error ⩽1 kcal mol−1) for most of the thermodynamic test sets. The newly developed methods were implemented in an open-source quantum chemistry program package and are now available to everyone

Optische Anregungen in Biologischen Systemen: Multiskalen-Simulations-Strategien und Anwendungen an Rhodopsinen

In this work, novel computational approaches are developed for the quantitative calculation of optical properties of chromophores in biologic systems. To investigate the influence of interactions between a chromophore and its molecular environment (protein, membrane, solvent), different theoretical approaches, which describe the matter at different length- and timescales, in a "multi-scale approach", which is tested on proteins of the rhodopsin family. In these proteins, acting as visual pigments or light-driven proton pumps in halobacteria, the optical absorbtion maximum of retinal is shifted over a wide spectral range. For the quantum mechanical (QM) description of the chromophore, appropriate approximations are found, which correctly describe its response (geometry, optical spectrum) to steric and electrostatic interactions. Here, shortcomings of time-dependent density functional theory and the role of static and dynamic correlation are analyzed. The conventional QM/MM scheme, which combines the QM description of the chromophore with a molecular-mechanical (MM) description of the environment, is extended to achieve a self-consistent mutual polarization of QM and MM fragments. For this purpose, an interacting atomic induced dipole model is implemented and assessed for application in polypeptides. This polarization model is integrated in a multi-state QM/MM scheme for the calculation of excitation energies, which incorporates the instantaneous polarization response of the protein/solvent environment to the excitation-induced charge redistribution on the chromophore. Moreover, the effects of charge transfer and dispersive interactions on the optical spectrum is investigated, and simple tests are proposed to predict the relevance of these effects and make an optimal choice for the QM region in a particular system.In dieser Arbeit werden neue computergestützte Ansätze für die quantitative Berechnung der optischen Eigenschaften von Chromophoren (CHR) in biologischen Systemen entwickelt. Um den Einfluss der Wechselwirkungen (WW) zwischen CHR und dessen molekularer Umgebung (Protein, Membran, Solvent) zu untersuchen, werden verschiedene theoretische Ansätze, welche die Materie auf unterschiedlicher Längen- und Zeitskala beschreiben, in einem "Multiskalen-Ansatz" kombiniert, der an Proteinen der Rhodopsin Familie getestet wird. In diesen Proteinen, die als Sehpigmente oder Licht-getriebene Ionenpumpen in Halobakterien dienen, wird das Absorptionsmaximum des Retinals im optischen Spektrum über einen weiten Wellenlängenbereich verschoben. Für die quantenmechanische (QM) Beschreibung des CHR werden Näherungen gesucht, welche die Reaktion des CHR (Geometrie, Absorptionsspektrum) auf sterische und elektrostatische WW korrekt beschreiben. Dabei werden u.a. Defizite der zeitabhängigen Dichtefunktionaltheorie und die Rolle statischer und dynamischer Elektronenkorrelation analysiert. Das übliche QM/MM-Schema, welches die QM Beschreibung des CHR mit einer molekular-mechanischen (MM) Beschreibung der Umgebung verbindet, wird erweitert, um eine selbst-konsistente gegenseitige Polarisation von QM und MM Fragment zu erreichen. Zu diesem Zweck wird ein Modell wechselwirkender atomarer induzierter Dipole implementiert und für die Anwendung an Polypeptiden getestet. Dieses Polarisationsmodell wird in ein QM/MM-Schema zur Berechnung vertikaler Anregungsenergien integriert, welches die instantane Polarisations-Antwort des Protein-Solvent-Systems auf die anregungsinduzierte Ladungsumverteilung auf dem CHR einbezieht. Ferner wird der Einfluss von Ladungstransfer und dispersiven WW auf das optische Spektrum untersucht und einfache Tests vorgeschlagen, um die Relevanz dieser Effekte vorherzusagen und die optimale Wahl der QM-Zone für das zu untersuchende System zu finden

Quantum Mechanical Simulations of Defect Dynamics in DNA and Model Systems

In dieser Arbeit wurde die Dynamik von elektronischen Defekten untersucht, die in DNA durch die Einwirkung von UV Strahlung oder oxidativem Stress entstehen können. Ultraschnelle dynamische Prozesse bestimmen, ob diese Defekte zu Schäden in der DNA führen oder ohne negative Effekte deaktiviert werden können. Die Simulation dieser Dynamik stellt wegen der ausgedehnten Struktur von DNA, wegen des Einflusses von nicht-adiabatischen Effekten, und wegen komplexer offenschaliger Wellenfunktionen der Elektronen einige Herausforderungen dar. Um all diese Phänomene zuverlässig beschreiben zu können, wurde ein systematischer Zugang gewählt, der von kleineren Modellsystemen zu realistischen DNA Fragmenten führte. Durch Simulationen von Ladungstransfer Dynamik an kleineren Modellsystemen konnte der Einfluss von nicht-adiabatischen Effekten und Polarisation der Umgebung untersucht werden. In dieser Phase konnte schon viel interessante Einsicht in die zugrunde liegenden physikalischen Phänomene gewonnen werden. Darüberhinaus konnte die Anwendbarkeit von verschiedenen methodischen Strategien untersucht werden, wobei einige kritische Punkte identifiziert wurden. Ausgiebige Methodenentwicklung in Bezug auf SA-MCSCF Gradienten, lokale Diabatisierung für Surface Hopping Dynamik und QM/MM Dynamik wurde durchgeführt um diese Probleme zu behandeln und verlässliche Simulationen an größeren Systemen zu ermöglichen. Zusätzlich wurde eine Analyse von angeregten Zuständen in Systemen mit mehreren gekoppelten Chromophoren ausgearbeitet und implementiert. Durch Rechnungen an größeren Systemen konnte Einsicht in verschiedene komplexe Phänomene gewonnen werden. Zuerst wurden die Prozesse, die in pi-stacks zur Bildung eines Excimers führen untersucht. Dabei konnte vor allem gezeigt werden, dass ein stark stabilisierter kohärenter Zustand durch Wechselwirkungen von Ladungstransfer- und excitonischen Konfigurationen entsteht. Weiters wurden in einem Modellsystem für Wasserstoff gebundene DNA Basenpaare Energietransfer und Protonen gekoppelte Elektronentransfer Prozesse untersucht, wobei vor allem der Einfluss von nicht-adiabatischen Effekten aufgezeigt werden konnte. Schließlich wurden realistische Simulationen der absorbierenden Zustände in DNA durchgeführt. Diese eröffnen neue Perspektiven zu verschiedenen Fragen bezüglich deren excitonischem und Ladungstransfer Charakter, die bisher in der Literatur kontrovers diskutiert wurden.The purpose of this study was the examination of the dynamics of electronic defects introduced into DNA by UV irradiation or oxidative stress. Such defects may either lead to damage in the DNA structure or to deactivation without adverse effects, as determined by ultrafast processes. The simulation of such dynamics presents several challenges, related to the extended structure of DNA, the presence of non-adiabatic interactions between electronic and nuclear degrees of freedom, and complex open-shell electronic wavefunctions requiring accurate computation and a meaningful analysis. To allow for a reliable description of all these phenomena a systematic approach going from smaller model systems to realistic DNA fragments was chosen. Simulations of charge transfer dynamics on smaller systems were performed to give a detailed examination of the influence of non-adiabatic effects and of environmental polarization. Already at this stage very interesting insight into the underlying physics of such processes could be obtained. Moreover, the applicability of the available methodologies could be assessed and several critical points were identified. Significant method development, concerned with state-averaged MCSCF gradients, with a local diabatization method for surface hopping dynamics, and with QM/MM simulations, was performed to address these points leading the way to more extended simulations. In addition an analysis procedure for excited states in systems with several coupled chromophores was devised and implemented. By considering larger systems new insight into several complex phenomena could be obtained. Firstly, the processes leading to excimer formation in stacked pi-systems were examined in the naphthalene dimer. It was found that a strongly stabilized coherent excited state was formed through interactions between excitonic and CT configurations. Secondly, excitation energy transfer as well as proton coupled electron transfer in a hydrogen bonded base-pair analog were analyzed highlighting in particular non-adiabatic effects, which are involved in these processes. Finally, a realistic simulation of the absorbing states in DNA was performed, shedding new light onto several questions regarding excitonic and charge transfer character of these states that had been discussed controversially in literature

Digging Deeper into the Methods of Computational Chemistry

This dissertation applies a skeptical but hopeful analytical paradigm and the tools of linear algebra, numerical methods, and machine learning to a diversity of problems in computational chemistry. When the foundation underlying a project is undermined, the primary purpose of the project becomes digging into the nature and structure of the problem. A common theme emerges in which assumptions in an area are challenged and a deeper understanding of the problem structure leads to new insights. In chapter 2, this approach is exploited to approximate derivative coupling vectors, which together with the difference gradient span the branching planes of conical intersections between electronic states. While gradients are commonly available in many electronic structure methods, the derivative coupling vectors are not always implemented and ready for use in characterizing conical intersections. An approach is introduced which computes the derivative coupling vector with high accuracy (direction and magnitude) using energy and gradient information. The new method is based on the combination of a linear-coupling two-state Hamiltonian and a finite-difference Davidson approach for computing the branching plane. Benchmark cases are provided showing these vectors can be efficiently computed near conical intersections. In chapter 3, this approach yields a countercultural explanation for what machine learning algorithms have learned in modeling a chemical reactivity dataset. Data-driven models of chemical reactions, a departure from conventional chemical approaches, have recently been shown to be statistically successful using machine learning. These models, however, are largely black box in character and have not provided the kind of chemical insights that historically advanced the field of chemistry. The chapter examines the knowledgebase of machine learning models—what does the machine learn?—by deconstructing black box machine learning models of a diverse chemical reaction dataset. Through experimentation with chemical representations and modeling techniques, the analysis provides insights into the nature of how statistical accuracy can arise, even when the model lacks informative physical principles. By peeling back the layers of these complicated models we arrive at a minimal, chemically intuitive model (and no machine learning involved). This model is based on systematic reaction type classification and Evans-Polanyi relationships within reaction types which are easily visualized and interpreted. Through exploring this simple model, we gain deeper understanding of the dataset and uncover a means for expert interactions to improve the model’s reliability. In chapter 4, human - algorithm interaction is explored as a paradigm for generating representative ensembles of conformers for organic compounds, a challenging problem in computational chemistry with implications on the ability to understand and predict reactivity. The approach utilizes the molecular editor IQmol as an interface between chemists and reinforcement learning algorithms with the cheminformatics package RDKit as a backbone. Conformer ensembles are evaluated by uniqueness and the approximation they yield of the partition function. Prototype results are presented for a standard reinforcement learning algorithm tested on linear alkanes and chemist manipulation of a fragment of the biomolecule lignin. Future aims and directions for this young project are discussed. The concluding chapter reflects on the broader lessons learned from conducting the dissertation. It discusses open questions and potential paradigms for pursuing them.PHDChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155137/1/joshkamm_1.pd

Beschreibung angeregter Molekülzustände in komplex strukturierter Umgebung durch einen effizienten, individuell selektierenden MRCI-Algorithmus gekoppelt an ein molekularmechanisches Kraftfeld

Atomistic theoretical descriptions of thermal chemical recations in complex environments are well achieved by combined quantum and molecular mechanical (QM/MM) methods. The goal of this work is to extend such techniques to enable the description of photochemical reactions and to carry out case studies subsequently. In order to tackle the enormous computational demand due to the involvement of excited electronic states, we (i) largely speed up the individually selecting multi-refernce configuration interaction (IS/MRCI) scheme by Tavan and Schulten (1980) by a new grahical algorithm and (ii) take use of recently developed semimempirical valence shell models (Thiel 1997) which are well suited for excited electronic states. The efficiency and accuracy of the resulting IS/MRCI-algorithm is demonstrated by its application to the first electronic excited states of butadiene. The new QM/MM method is used to calculate absorption energies along a molecular dynamics trajectory of a small Schiff base in isotonic solution.Für die theoretische Beschreibung thermochemischer Reaktionen in komplexer Umgebung auf atomarem Niveau ist die Kombination aus quanten- und molekülmechanischen Verfahren (QM/MM) bereits etabliert. Ziel dieser Arbeit ist, für die Beschreibung photochemischer Prozesse ebenso ein QM/MM-Verfahren zu entwickeln und es exemplarisch zu validieren. Dem extremem Rechenaufwand bei der QM-Beschreibung der nun auftretenden elektronisch angeregten Molekülzustände wird hier begegnet durch (i) Beschleunigung der individuell selektierenden Multireferenz-Konfigurationswechselwirkungs (IS/MRCI)- Methode von Tavan und Schulten (1980) mittles eines neuen Graphenalgorithmus und (ii) den Einsatz von neuen semiempirischen Valenzschalenmodellen (Thiel 1997) mit besonderer Eignung für angeregte elektronische Zustände. Die Effizienz und Genauigkeit des neuen IS/MRCI-Algorithmus wird anhand der Beschreibung der ersten angeregten Zustände von Butadien demonstriert. Das neue QM/MM-Verfahren findet Anwendung bei der Berechnung der Anregungsenergien entlang einer molekulardynamischen Trajektorie einer kleinen Schiff-Base in isotonischer Lösung

Accurate variational electronic structure calculations with the density matrix renormalization group

During the past fifteen years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low­-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many­-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. Whereas the MPS ansatz can only capture exponentially decaying correlation functions in the thermodynamic limit, and will therefore only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for finite­-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. For hydrogen chains, accurate longitudinal static hyperpolarizabilities were obtained in the thermodynamic limit. In addition, the low-lying states of the carbon dimer were accurately resolved. DMRG and Hartree-­Fock theory have an analogous structure. The former can be interpreted as a self­-consistent mean­-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-­DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single­-particle excitations for a Slater determinant leads to the Thouless theorem for Hartree-­Fock theory, an explicit nonredundant parameterization of the entire manifold of Slater determinants. For an MPS with open boundary conditions, exponentiation of the single-site excitations leads to the Thouless theorem for DMRG, an explicit nonredundant parameterization of the entire manifold of MPS wavefunctions. This gives rise to the configuration interaction expansion for DMRG. The Hubbard-­Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-­lattice Hamiltonians allows to formulate a promising variant for matrix product states