Semiclassical instanton formulation of Marcus-Levich-Jortner theory

Marcus-Levich-Jortner (MLJ) theory is one of the most commonly used methods for including nuclear quantum effects into the calculation of electron-transfer rates and for interpreting experimental data. It divides the molecular problem into a subsystem treated quantum-mechanically by Fermi's golden rule and a solvent bath treated by classical Marcus theory. As an extension of this idea, we here present a "reduced" semiclassical instanton theory, which is a multiscale method for simulating quantum tunnelling of the subsystem in molecular detail in the presence of a harmonic bath. We demonstrate that instanton theory is typically significantly more accurate than the cumulant expansion or the semiclassical Franck-Condon sum, which can give orders-of-magnitude errors and in general do not obey detailed balance. As opposed to MLJ theory, which is based on wavefunctions, instanton theory is based on path integrals and thus does not require solutions of the Schr\"odinger equation, nor even global knowledge of the ground- and excited-state potentials within the subsystem. It can thus be efficiently applied to complex, anharmonic multidimensional subsystems without making further approximations. In addition to predicting accurate rates, instanton theory gives a high level of insight into the reaction mechanism by locating the dominant tunnelling pathway as well as providing information on the reactant and product vibrational states involved in the reaction and the activation energy in the bath similarly to what would be found with MLJ theory.Comment: 21 pages, 4 figure

On-the-fly ab initio semiclassical evaluation of time-resolved electronic spectra

We present a methodology for computing vibrationally and time-resolved pump-probe spectra, which takes into account all vibrational degrees of freedom and is based on the combination of the thawed Gaussian approximation with on-the-fly ab initio evaluation of the electronic structure. The method is applied to the phenyl radical and compared with two more approximate approaches based on the global harmonic approximation - the global harmonic method expands both the ground- and excited-state potential energy surfaces to the second order about the corresponding minima, while the combined global harmonic/on-the-fly method retains the on-the-fly scheme for the excited-state wavepacket propagation. We also compare the spectra by considering their means and widths, and show analytically how these measures are related to the properties of the semiclassical wavepacket. We find that the combined approach is better than the global harmonic one in describing the vibrational structure, while the global harmonic approximation estimates better the overall means and widths of the spectra due to a partial cancellation of errors. Although the full-dimensional on-the-fly ab initio result seems to reflect the dynamics of only one mode, we show, by performing exact quantum calculations, that this simple structure cannot be recovered using a one-dimensional model. Yet, the agreement between the quantum and semiclassical spectra in this simple, but anharmonic model lends additional support for the full-dimensional ab initio thawed Gaussian calculation of the phenyl radical spectra. We conclude that the thawed Gaussian approximation provides a viable alternative to the expensive or unfeasible exact quantum calculations in cases, where low-dimensional models are not sufficiently accurate to represent the full system.Comment: Last 6 pages contain the Supplementary Materia

Unified description of vibronic transitions with coherent states

Vibronic (vibrational-electronic) transition is one of the fundamental processes in molecular physics. Indeed, vibronic transition is essential both in radiative and nonradiative photophysical or photochemical properties of molecules such as absorption, emission, Raman scattering, circular dichroism, electron transfer, internal conversion, etc. A detailed understanding of these transitions in varying systems, especially for (large) biomolecules, is thus of particular interest. Describing vibronic transitions in polyatomic systems with hundreds of atoms is, however, a difficult task due to the large number of coupled degrees of freedom. Even within the relatively crude harmonic approximation, such as for Born-Oppenheimer harmonic potential energy surfaces, the brute-force evaluation of Franck-Condon intensity profiles in a time-independent sum-over-states approach is prohibitive for complex systems owing to the vast number of multi-dimensional Franck-Condon integrals. The main goal of this thesis is to describe a variety of molecular vibronic transitions, with special focus on the development of approaches that are applicable to extended molecular systems. We use various representations of Fermi’s golden rule in frequency, time and phase spaces via coherent states to reduce the computational complexity. Although each representation has benefits and shortcomings in its evaluation, they complement each other. Peak assignment of a spectrum can be made directly after calculation in the frequency domain but this sum-over-states route is usually slow. In contrast, computation is considerably faster in the time domain with Fourier transformation but the peak assignment is not directly available. The representation in phase space does not immediately provide physically-meaningful quantities but it can link frequency and time domains. This has been applied to, herein, for example (non-Condon) absorption spectra of benzene and electron transfer of bacteriochlorophyll in the photosynthetic reaction center at finite temperature. This work is a significant step in the treatment of vibronic structure, allowing for the accurate and efficient treatment of complex systems, and provides a new analysis tool for molecular science.Absorption von Licht und der darauf folgende Elektronentransfer in photosynthetischen Systemen sind entscheidende Prozesse in unserem Alltag. Die Verbesserung von Kontrolle und Effizienz dieser Prozesse ist eine Herausforderung im Hinblick auf die weltweite Nahrungs- und Energieversorgung. Diese Art von Prozessen wird jedoch dadurch kompliziert, dass Absorption, Emission und Lichtstreuung verschiedene strahlungslose molekulare Übergänge wie Ladungswanderung, innere Umwandlung und Interkombinationsübergänge nach sich ziehen können. Ein genaues Verständnis dieser Prozesse auf molekularer Ebene in verschiedenen Systemen ist daher von besonderem Interesse. Molekulare Übergangsprozesse werden durch Wechselwirkungen zwischen Kernen, Elektronen, der Umgebung und äußeren Feldern (z. B. elektromagnetischen) bestimmt. Das Zusammenspiel von vibratorischen und elektronischen (vibronischen) Freiheitsgraden der Moleküle spielt typischerweise eine bedeutende Rolle in molekularen (vibronischen) Übergängen. Ein molekularer vibronischer Übergang wird für gewöhnlich durch Fermis goldene Regel (FGR), die sich aus der zeitabhängigen Störungstheorie ableitet, als eine das absolute Quadrat von Übergangsmomenten enthaltende Übergangsgeschwindigkeitskonstante beschrieben. Laut dem Ausdruck für die Übergangsgeschwindigkeitskonstante in der Basis der Born-Oppenheimer-Wellenfunktionen ist einer der Schlüsselbeiträge zu vibronischen Übergängen der Franck-Condon-Faktor (FCF). Der FCF ist definiert als das Absolutquadrat des Überlappungsintegrals zwischen zu verschiedenen elektronischen Zuständen gehörenden Schwingungswellenfunktionen. Die theoretische Beschreibung vibronischer Übergänge großer polyatomarer Systeme (mehr als 100 Atome) ist jedoch wegen der hohen Dimensionalität eine schwierige Aufgabe. Sogar in einer relativ groben harmonischen Näherung wie den harmonischen Born-Oppenheimerschen Potentialhyperflächen ist die theoretische brute-force-Berechnung der FC-Intensitätsprofile durch eine Summenbildung über die zeitunabhängigen Zustände für komplexe Systeme wegen der gewaltig großen Zahl multi-dimensionaler FC-Integrale ungeeignet. Das Hauptziel dieser Arbeit ist die Beschreibung einer Vielzahl molekularer vibronischer Übergänge, insbesondere der Entwicklung von Herangehensweisen, die auf ausgedehnte molekulare Systeme anwendbar sind. Wir haben verschiedene Darstellungen von FGR in Frequenz-, in Zeit- und, zur Verringerung des Rechenaufwandes über kohärente Zustände, in Phasenräumen verwendet. Jede Darstellung hat Vor- und Nachteile in ihrer Auswertung, aber alle ergänzen einander. Die Signalzuordnung des Spektrums zu verschiedenen Quantenzustandsübergängen kann direkt nach der Berechnung in der Frequenzdomäne vorgenommen werden, doch ist dieser Weg über die Summierung von Zuständen normalerweise zeitintensiv. Im Gegensatz dazu ist die Berechnung über Fouriertransformation in der Zeitdomäne schneller, aber eine Zuordnung der Signale zu verschiedenen Quantenzustandsübergängen ist nicht direkt möglich. Die Darstellung im Phasenraum liefert nicht sofort physikalisch bedeutsamen Größen, kann aber Frequenz- und Zeitdomäne verknüpfen. Folglich können wir die molekularen Übergangsspektren effizient berechnen, einschließlich thermischer und Nicht-Condon-Effekte. Zusätzlich zur Effizienzsteigerung sind wir in der Lage, die einzelnen Dynamiken der Schwingungsfreiheitsgrade während der elektronischen Übergänge für relativ große Systeme zu analysieren. Unsere Methode ist nicht nur auf molekularer Übergänge anwendbar, sondern auf jedes physikalische Problem, das eine Näherung über harmonische Oszillatoren enthält, beispielsweise Nichtgleichgewichtsdynamiken dissipativer Systeme

Dynamics and spectrum of a molecule coupled to a vibrational mode.

El estudio en el campo de la electrodinámica cuántica de cavidad (cavity Quantum Electrodynamics (QED)) durante el último siglo [1] ha permitido la interacción coherente (oscilaciones Rabi) de la luz con la materia, alcanzándose así, el régimen conocido como acoplamiento fuerte o strong coupling.Existen multitud de plataformas experimentales para el estudio de la interacción luz-materia; desde osciladores nanomecánicos (nano-mechanical oscillators) [2] a sistemas de átomos artificiales formados a partir de uniones Josephson. Estos últimos sistemas han permitido la construcción del primer ordenador cuántico por Google en 2019 [3].En este trabajo, nos hemos centrado en un sistema de dos niveles constituido por una molécula orgánica [4]. Un sistema como este, presenta una característica diferenciadora, los modos de vibración. Estos son parametrizados a través del denominado como factor de Huang-Rhys [5] (1950). Así, hemos estudiado una molécula (sistema de dos niveles) con un modo de vibración, embebida en una cavidad electromagnética. En primer lugar, estudiamos el espectro de energías del sistema (capítulo 2) y los efectos ultrastrong en el mismo. Es decir, el efecto en el sistema de los términos que no conservan el número de excitaciones en el modelo de Rabi [6]. Para ello, usamos técnicas tales como la transformación de Polarón y la teoría de perturbaciones; o técnicas numéricas como la diagonalización exacta.Seguidamente, continuamos estudiando la dinámica del sistema (capítulo 3). Así, modelizamos las perdidas energéticas usando el formalismo de la ecuación maestra [7]. Con ello, estudiamos la dependencia de la frecuencia Rabi del sistema y el decaimiento con el acoplo de la molécula con su modo de vibración. Además, analizamos el espectro del Lindbladiano para obtener las transiciones más relevantes en la dinámica.A continuación, calculamos el espectro de ruido (noise spectrum) [8] del sistema (capítulo 4), estudiando de nuevo los efectos ultrastrong en el mismo. Por último, insertamos nuestro sistema en una guía de ondas (capítulo 5) y estudiamos los estados ligados (bound states) [9] que se forman en torno al emisor.[1] Serge Haroche. Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys., 85:1083-1102, Jul 2013.[2] T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab. Preparation and detection of a mechanical resonator near the ground state of motion. Nature, 463(7277):72-75, Jan 2010.[3] Frank Arute, John M. Martinis et al., Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510, Oct 2019.[4] Salvatore Gambino, Marco Mazzeo, Armando Genco, Omar Di Stefano, Salvatore Savasta, Salvatore Patane, Dario Ballarini, Federica Mangione, Giovanni Lerario, Daniele Sanvitto, and Giuseppe Gigli. Exploring light matter interaction phenomena under ultrastrong coupling regime. ACS Photonics, 1(10):1042-1048, 2014.[5] Kun Huang, Avril Rhys, and Nevill Francis Mott. Theory of light absorption and non-radiative transitions in f-centres. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 204(1078):406-423, 1950.[6] I. I. Rabi. On the process of space quantization. Phys. Rev., 49:324-328, Feb 1936.[7] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2003.[8] Jens Jensen and Allan R. Mackintosh. Rare Earth Magnetism - Structures and Excitations. Clarendon Press - Oxford, 2 edition, June 1991.[9] A. Gonzalez-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble. Subwavelength vacuum lattices and atom-atom interactions in two-dimensional photonic crystals. Nature Photonics, 9(5):320-325, May 2015.<br /

Development of low-scaling methods for the description of vibronic transitions and application in gas-phase ion chemistry

Theoretical methods are developed and applied in relation to three gas phase experiments: Neutralization Reionization Mass Spectrometry (NRMS), laser cooling and Cold Target Ion Momentum Spectroscopy (COLTRIMS). All three are important techniques in gas phase ion chemistry research and can be described using similar theoretical models: The molecules under investigation undergo electronic transitions from an initial electronic state to a final electronic state and may at the same time also change their vibrational quantum number. Those vibronic transitions are in the Franck-Condon limit described with Franck-Condon factors (FCFs) which modulate the probability of the different electronic transitions. Whereas in NRMS state transitions are used to generate highly reactive molecular species from stable precursors, laser cooling achieves translational cooling of molecules by repeated electronic transitions and COLTRIMS provides information on electronic and geometrical structure. In this work a time-dependent generating function ansatz for the calculation of FCFs by Huh [PhD thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main, 2010] is extended in order to calculate vibrational population distributions in the final state after one or several subsequent vibronic transitions. The calculated trajectories are discontinuous in some cases, which makes it necessary to implement algorithmic changes to the former developments. The method is applied to the molecule HNSi and combined with reaction rate calculations within the Statistical Adiabatic Channel Model (SACM) to predict signals in the mass spectrum. The results are compared to NRMS experiments. An extended version of the developed method is discussed which includes the description of a laser excitation with specific frequency. It is applied to a system with one vibrational degree of freedom in a laser cooling cycle. In the year 2013 Pitzer et al. [Science 2013, 341 (6150), 1096 1100] have shown that direct assignment of the absolute configuration for the molecule CHBrClF is possible with COLTRIMS. In this dissertation experimental data from synchrotron experiments in the year 2016 by Pitzer et al. [Chem. Phys. Chem. 2016, 17, 24652472] are analysed. The uncertainty of their evaluation due to the presence of different isotopes is discussed. Equilibrium structures and stabilities of multiply charged cations of CHBrClF are determined with quantum chemical methods. Furthermore, the fragmentation pathways of singly charged CDBrFI are investigated and calculated appearance energies are compared to experimental values. This work describes methods for the prediction and analysis of three gas phase experiments that involve vibronic transitions. These theoretical approaches integrate different advantages: The methods developed or applied herein are predominantly of quantum mechanical type, which have inherently high accuracy. The algorithms implemented and the approaches used are computationally efficient as they avoid for instance to explicitly count vibronic transitions or to perform optimizations with overly expensive quantum chemical methods. The methods are versatilely applicable to diverse molecules in different gas phase experiments. Furthermore, this dissertation contributes to the further development of the promising COLTRIMS method for the direct determination of absolute configuration

Development of low-scaling methods for the description of vibronic transitions and application in gas-phase ion chemistry

Theoretical methods are developed and applied in relation to three gas phase experiments: Neutralization Reionization Mass Spectrometry (NRMS), laser cooling and Cold Target Ion Momentum Spectroscopy (COLTRIMS). All three are important techniques in gas phase ion chemistry research and can be described using similar theoretical models: The molecules under investigation undergo electronic transitions from an initial electronic state to a final electronic state and may at the same time also change their vibrational quantum number. Those vibronic transitions are in the Franck-Condon limit described with Franck-Condon factors (FCFs) which modulate the probability of the different electronic transitions. Whereas in NRMS state transitions are used to generate highly reactive molecular species from stable precursors, laser cooling achieves translational cooling of molecules by repeated electronic transitions and COLTRIMS provides information on electronic and geometrical structure. In this work a time-dependent generating function ansatz for the calculation of FCFs by Huh [PhD thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main, 2010] is extended in order to calculate vibrational population distributions in the final state after one or several subsequent vibronic transitions. The calculated trajectories are discontinuous in some cases, which makes it necessary to implement algorithmic changes to the former developments. The method is applied to the molecule HNSi and combined with reaction rate calculations within the Statistical Adiabatic Channel Model (SACM) to predict signals in the mass spectrum. The results are compared to NRMS experiments. An extended version of the developed method is discussed which includes the description of a laser excitation with specific frequency. It is applied to a system with one vibrational degree of freedom in a laser cooling cycle. In the year 2013 Pitzer et al. [Science 2013, 341 (6150), 1096 1100] have shown that direct assignment of the absolute configuration for the molecule CHBrClF is possible with COLTRIMS. In this dissertation experimental data from synchrotron experiments in the year 2016 by Pitzer et al. [Chem. Phys. Chem. 2016, 17, 24652472] are analysed. The uncertainty of their evaluation due to the presence of different isotopes is discussed. Equilibrium structures and stabilities of multiply charged cations of CHBrClF are determined with quantum chemical methods. Furthermore, the fragmentation pathways of singly charged CDBrFI are investigated and calculated appearance energies are compared to experimental values. This work describes methods for the prediction and analysis of three gas phase experiments that involve vibronic transitions. These theoretical approaches integrate different advantages: The methods developed or applied herein are predominantly of quantum mechanical type, which have inherently high accuracy. The algorithms implemented and the approaches used are computationally efficient as they avoid for instance to explicitly count vibronic transitions or to perform optimizations with overly expensive quantum chemical methods. The methods are versatilely applicable to diverse molecules in different gas phase experiments. Furthermore, this dissertation contributes to the further development of the promising COLTRIMS method for the direct determination of absolute configuration

Trajectory-based approaches to vibronic spectroscopy of molecular systems

The goal of this thesis is to bridge the gap between the two standard theoretical approaches to vibronic spectroscopy via trajectory-based methods. As the starting point, a generalized time-correlation function is introduced and the ring-polymer molecular dynamics method is generalized to vibronic transitions, yielding an improvement over known classical approximations. Further, the vibronic spectrum is evaluated via the Matsubara dynamics. Employing an ad-hoc modification, the sign problem of the Matsubara method can be circumvented and the spectra of model systems are adequately simulated

Optical Response and Control of Molecular Systems.

This thesis is comprised of three major parts and is concerned with the theoretical characterization of condensed phase systems within the framework of nonlinear spectroscopy experiments, using both analytical models and numerical approximation schemes. The first part focuses on the chirped-pulse mediated coherent control of electronic population transfer, and investigates the plausibility of control in the presence of pure electronic dephasing. The molecular system is described by a same-frequency shifted harmonic oscillator model, and population transfer was computed using split-operator and direct diagonalization schemes. Dephasing effects were incorporated using a stochastic model that is able to interpolate between the homogeneous and inhomogeneous limits, and results with and without dephasing were compared as functions of the linear chirp parameter and the field intensity. The numerical findings were compared to and found to be consistent with several experimental studies performed on the laser dye LD690 in liquid methanol. The second part is a comparative study of several approximation methods used for computing optical response functions, and is illustrated within the context of two-dimensional electronic spectroscopy. A central theme is the development of a benchmark model that can discriminate between different methods, and consists of a different-frequency shifted harmonic oscillator model. Optical response spectra were computed using four different approximation schemes, which include two distinctly different second-order cumulant approximations, a Linearized Semiclassical method, and a Forward-Backward Semiclassical method. Comparing the spectra as a function of temperature and the oscillator frequency ratio assessed the accuracy and robustness of the methods. The final part concerned a method for computing ab initio optical response tensors in the context of two-dimensional infrared spectroscopy, and was a collaborative effort between the Geva and Kubarych groups. An excitonic Hamiltonian was used to model the photo-active modes of a vibrational system, and a direct diagonalization procedure, which utilized inputs from electronic structure calculations, was used to compute the spectra. Preliminary results for the four-mode system Mn(CO)5 are presented, and the methodology developed here was later continued and extended by other members of the collaboration.Ph.D.PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/78900/1/pmcrobbi_1.pd

Theory of Outer-Sphere Electron-Transfer Reactions

Classical, semiclassical and quantum theories of outer-sphere electron-transfer reactions in polar media are discussed. For each, the Franck-Condon overlap factors for the hexaamminecobalt, hexaaquoiron and hexaammineruthenium self-exchange rates and for the cross-reaction of hexaaquoiron(II) with tris(2,2’-bipyridine)ruthenium(III) are evaluated and compared. The quantum effect on the rates is small in the region of moderate driving force; the "normal" ΔGo region. Direct-sum and saddle-point evaluations of the quantum Franck-Condon factors are made and compared. The saddle-point approximation is shown to be an excellent approximation in the cases considered. Quantum effects in homogeneous outer-sphere electron transfer reactions in the region of large negative ΔGo (the "inverted" region) are considered. The results of quantum, semiclassical and classical calculations on model systems are presented. A sequence of highly exothermic photoinduced reactions of tris(2,2'-bipyridyl) complexes is discussed with regard to the possible importance of quantum effects and of alternate reaction pathways in understanding the failure of the sequence of reactions to exhibit pronounced "inverted" behavior. A mechanism leading to electronically excited products provides a possible explanation for the large discrepancy. The theory of highly exothermic homogeneous outersphere electron-transfer reactions is discussed for transfers occurring over a range of distances. A finite rate of diffusion of reactants and their long-range force are treated by solving the reaction-diffusion equation numerically for the reactant pair distribution function. Steady-state solutions are compared with experimental data. On the basis of short-time solutions it is proposed that experiments which measure electron-transfer rates at short times following the onset of reaction improve the possibility of observing the inverted effect in bimolecular systems. The effect of the reactants' relative orientation on the electron-transfer rate is considered. Reactants are modeled as oblate-spheroidal potential wells of constant, finite depth. Energy levels and wavefunctions are obtained for an electron localized in such a well. The electronic matrix elements that govern electron transfer within a nonadiabatic quantum theory are evaluated. Significant orientational preferences are predicted for electron transfer between nonspherical donor and acceptor sites.</p

Time-Dependent Approaches and Their Utility: Dynamical Formulations of Two-Dimensional Electronic Spectroscopy Signals and Electronic Structure Theory

We present time-dependent reframings of the theory of two-dimensional electronic spectroscopy signals and of electronic structure theory. The dynamical formulation of spectroscopic signals, in particular of two-dimensional wave-packet interferometry (WPI), is used to calculate and interpret signals from a spatially oriented energy transfer dimer. A general study of the detection of electronic energy transfer using WPI is carried out. The signals are interpreted using a semiclassical analysis that considers the paths taken by wave packets through phase space and the conditions required for their phase-space overlap. The dimer is also used to propose a WPI experiment capable of observing electronic intersite and interexciton coherence. Weak-coupling (intersite) and strong-coupling (interexciton) cases are studied, with a variety of systems differing in number of vibrational modes and in excited-state energies of the monomers. The time-dependent framing of electronic structure theory is a spectral filtering technique, where the Fourier transform of the time evolution of an antisymmetrized wave packet to the frequency domain reveals eigenstates and eigenenergies. Direct numerical integration of the time-dependent Schrödinger equation and semiclassical parametrizations are presented and compared as methods of obtaining the time evolution. The method is found to be accurate, and has some benefits; spectral filtering allows for many eigenstates to be obtained at once and includes electron correlation automatically. Future prospects for each of these works are discussed. This dissertation includes previously published and unpublished co-authored material