Advances in quantum chemistry

EDITORIAL BOARD Guillermina Estiu´ (University Park, PA, USA) Frank Jensen (Aarhus, Denmark) Mel Levy (Greensboro, NC, USA) Jan Linderberg (Aarhus, Denmark) William H Miller (Berkeley, CA, USA) John W Mintmire (Stillwater, OK, USA) Manoj Mishra (Mumbai, India) Jens Oddershede (Odense, Denmark) Josef Paldus (Waterloo, Canada) Pekka Pyykko (Helsinki, Finland) Mark Ratner (Evanston, IL, USA) Dennis R Salahub (Calgary, Canada) Henry F Schaefer III (Athens, GA, USA) John Stanton (Austin, TX, USA) Harel Weinstein (New York, NY, USA) Advances in QUANTUM CHEMISTRY UNSTABLE STATES IN THE CONTINUOUS SPECTRA, PART II: INTERPRETATION, THEORY AND APPLICATIONS VOLUME 63 Special Editors CLEANTHES A NICOLAIDES Theoretical and Physical Chemistry Institute National Hellenic Research Foundation Athens, Greece ă ERKKI BRANDAS Department of Quantum Chemistry Uppsala University Uppsala, Sweden Editors JOHN R SABIN Quantum Theory Project University of Florida Gainesville, Florida ă ERKKI BRANDAS Department of Quantum Chemistry Uppsala University Uppsala, Sweden Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA 32 Jamestown Road, London NW1 7BY, UK Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2012 Copyright c 2012 Elsevier Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting: Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein ISBN: 978-0-12-397009-1 ISSN: 0065-3276 For information on all Academic Press publications visit our web site at www.elsevierdirect.com Printed and bounded in USA 12 13 14 15 10 PREFACE Since the late 1920s, most of the many thousands of publications contributing to quantum chemistry have dealt with issues and problems that essentially concern, or are applicable to, the ground or the low-lying discrete states of atoms and molecules and of electronic matter in general In this context, samples of topics that have been examined are many-faceted formalisms, analysis and computation of various features of the many-electron problem, computational methodologies and techniques, results of computation of properties and of low-energy chemical reactions, computation of spectroscopic data involving mainly discrete states, etc On the other hand, significant advances have also been made in the broader domain of quantum chemistry, a prime example being areas of research that involve the continuous spectrum and, as such, are more complex, conceptually, formally, and computationally When the continuous spectrum of a quantum system acquires physical significance, a plethora of special and challenging physical and mathematical features and questions emerge that are absent in problems involving just the discrete spectrum In the variety of excitation or de-excitation processes that allow the preparation and/or observation of the system via the participation of the continuous spectrum, the dominant and most interesting characteristics are generated by the transient formation of nonstationary or unstable states For example, the excitation may be caused by the absorption of one or of many photons during the interaction of an initial atomic or molecular state with pulses of long or of short duration Or, the transient formation and influence on the observable quantity may occur during the course of electron–atom scattering or of chemical reactions In principle, the physics involving unstable states ought to engage descriptions that are time dependent Yet, in the formulation and practical solution of related problems, both time-dependent and time-independent treatments are pertinent and necessary Furthermore, in certain theoretical approaches, the phenomenologies as well as the computational methodology are based on constructions that are non-Hermitian We add that the Hamiltonians may vii viii Preface or may not include the coupling of atomic or molecular states to external electromagnetic fields The two volumes of Unstable States in the Continuous Spectra, which we have edited (Part I is AQC volume 60 and Part II is the present volume, 63), contain a total of 15 review articles on topics covered by the general theme The invitation of the contributing experts had as one of its purposes to create a book on the above theme where the spectrum of the information contained in it is wide, authoritative, and relevant to quantum chemistry The invited authors were free to choose their topic(s) and style of presentation Before final acceptance, their manuscripts were subjected to “friendly yet critical” review by referees suggested by the authors, aiming at improving the contents as much as possible The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced manyelectron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing The present volume of the Advances in Quantum Chemistry is the sequel of the first volume, mentioned above, i.e., Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications It contains six chapters with contents varying from a pedagogical introduction to the notion of unstable states to the presence and role of resonances in chemical reactions, from discussions on the foundations of the theory to its relevance and precise limitations in various fields, from electronic and positronic quasi-bound states and their role in certain types of reactions to applications in the field of electronic decay in multiply charged molecules and clusters, as well Given the plurality of the aforementioned discussions in both volumes, we hope that both senior and young quantum chemists and physicists with an interest in the specific theme of “unstable states in the continuous spectra” and in quantum theory, in general, will find the present set of two volumes resourceful, innovative, and helpful Cleanthes A Nicolaides Athens, Greece Erkki J Brăandas Uppsala, Sweden CONTRIBUTORS Erkki J Brăandas, Quantum Chemistry, Department of Physical and Analytical Chemistry, Uppsala University, Uppsala, Sweden Eva Lindroth, Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden Ido Gilary, Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Isao Shimamura, Atomic Physics Laboratory, RIKEN, Wako, Saitama 3510198, Japan ´ ´ Luca Argenti, Dept Qu´ımica, Modulo 13, Universidad Autonoma de Madrid, 28049 Madrid, Spain Pˇremysl Kolorenˇc, Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holeˇsoviˇck´ach 2, 180 00 Prague, Czech Republic Rex T Skodje, Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309, USA Shachar Klaiman, Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Vitali Averbukh, Department of Physics, Imperial College London, Prince Consort Road, SW7 2AZ London, UK ix CHAPTER On Resonance: A First Glance into the Behavior of Unstable States Shachar Klaimana and Ido Gilarya Contents Abstract a Introduction A Quantum Mechanical Resonance State from a Time-Dependent Perspective 2.1 From bound state to metastable state 2.2 Evolution of the resonance wavefunction 2.3 Dynamics inside the interaction region 2.4 Dynamics outside the interaction region A Stationary Analysis of Resonance States 3.1 Expansion of localized functions in terms of scattering states 3.2 Stationary solutions with outgoing waves 3.3 Properties of the stationary resonance state Unified Picture of Resonance States 4.1 Expansion of the stationary resonance state in time 4.2 The “death” of a resonance state The Origin of Resonances 5.1 Shape-type resonances 5.2 Feshbach-type resonances Conclusions Acknowledgment References 4 13 14 15 17 18 21 22 23 24 24 26 28 29 29 Dynamical processes in nature often involve unstable states Analyzing systems with a finite lifetime can be challenging for a practitioner of quantum mechanics To study such processes in a quantum system, one must venture Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: shachark@technion.ac.il Advances in Quantum Chemistry, Volume 63 ISSN 0065-3276, DOI: 10.1016/B978-0-12-397009-1.00001-1 c 2012 Elsevier Inc All rights reserved Shachar Klaiman and Ido Gilary into the continuum where the use of a continuous superposition of states, i.e., a wave packet, is required Most of our quantum education focuses on quantized bound states rather than on the behavior of wave packets Here, we aim to give a pedagogic introduction to the behavior and analysis of unstable states To achieve this, we introduce two complementary viewpoints by which such states can be analyzed We further discuss the physical mechanisms through which quantum unstable states are formed INTRODUCTION The word resonance is a very widespread term in the scientific world Common uses range from being in a or on resonance to resonance poles and peaks As with many such ubiquitous terms, they evolve with time and tend to take a life of their own acquiring new meaning and connotations as time goes by This can lead to some confusion and ambiguity when different definitions are evoked Here, we wish to explore the meaning of this term attributed to unstable states in quantum mechanics Given a quantum mechanical system, i.e., a Hamiltonian, one can generally separate the spectrum into two types of solutions: bound states and continuum states Regularly, introductory courses and texts in quantum mechanics focus on bound states These are found by searching for solutions ă of the time-independent Schrodinger equation (TISE) with the appropriate boundary conditions (BCs), which for bound states are such that the wavefunction vanishes at all the boundaries The imposed BCs lead to the quantization of the bound spectrum This quantization facilitates the understanding of quantum phenomena related to bound states since one can often relate the desired phenomenon with the occupation of only a few welldefined states Unfortunately, the continuum part of the spectrum is not as gratifying In the continuum, we are forced to use wave packets rather than a single eigenstate to describe quantum particles Single eigenstates in the continuum are not amenable to the usual probabilistic interpretation, which requires the normalization of the particle wavefunction Wave packets are built by integrating over a continuous range of energy eigenstates to create localized wavefunctions Therefore, when describing phenomena that require the continuous part of the spectrum, it becomes increasingly difficult to correlate an observed effect with a single eigenstate of the TISE The necessity of working with wave packets presents an intrinsic difficulty in the treatment of the system One can no longer be content with the soluă tions of the time-independent Schrodinger equation, and a solution to the ă time-dependent Schrodinger equation (TDSE) is required Although analysis based on the TDSE is certainly possible, one is often not well accustomed to it This is mainly because most of quantum mechanical textbooks build our intuition and understanding with examples of solution of the TISE, and the TDSE is mostly disregarded We should mention here a recent textbook by Tannor [1], which recognized this void and aims to fill it On Resonance: A First Glance into the Behavior of Unstable States Notwithstanding the above mentioned difficulties in treating processes in the continuum, many physical situations allow for a simpler approach based on resonance states Resonance states are solutions of the TISE, which correspond to unstable quantum states, i.e., states with a finite lifetime Although, by definition, these processes occur solely in the continuum, i.e., bound states have an infinite lifetime, resonance states are quantized solutions of the TISE Therefore, describing a continuum wave packet using such resonance states would circumvent one of the biggest difficulties in the continuum – the inability to associate the physical phenomenon with a finite number of physical states An extensive account of the theoretical framework of resonance phenomena as well as the various methods used to treat it can be found in Refs [2, 3] Quite generally, one can classify processes in the continuum into two types: a full-collision process and a half-collision process, according to the initial preparation of the system In a full-collision process, particles are scattered from a potential and are then measured in the asymptotic region, i.e., the particles start and finish in the asymptotes In a half collision, however, the system is prepared in an excited state and one measures the breakup into products of this excited state, i.e., particles that are initially located in the interaction region are measured at the asymptotic region In this chapter, we focus on half-collision processes demonstrating the connection between a wave packet solution of the TDSE and a resonance solution of the TISE This connection between the solutions of the TDSE and the TISE puts one on solid ground even when the continuum is involved Since we hope to give here an introductory account of resonances, we shall focus on systems where the dynamics is controlled by a single metastable state, i.e., an unstable state with an appreciable lifetime This might seem at first rather limiting, but in fact it accounts for many physical situations The conditions for single resonance dynamics will be expanded on further in the following In addition in order to maintain a simple picture we will illustrate everything for a single particle in one dimension although most of the arguments that will be made in this chapter could be readily generalized to many-body problems in higher dimensions One of the most famous examples as well as one of the first applications of resonance theory in quantum mechanics was given by Gamow in 1928 [4] in his study of α decay Since this phenomenon is extremely robust, many other examples can be found in various fields of physics [5] Just to name a few disciplines, these include, for example: particle [6–8], atomic [9–13], molecular [14–16], and mesoscopic [17–20] physics, as well as the interaction of such systems with electromagnetic radiation [21, 22] or their implementation in electronic applications [23–25] The simplest way to construct a system that supports a metastable state is to first consider a system with at least a single bound state If we wish to probe a particle in this bound state, we must couple it to the “outside” world where our measurement devices are There are, in general, many ways by which such a coupling can occur One common possibility is to scatter Shachar Klaiman and Ido Gilary particles of the bound target and measure the resulting cross section, i.e., a full-collision experiment This is the situation, for example, in experiments probing quantum dots via a conductance measurement [17, 26] Yet another possibility is to manipulate the bounding potential using external forces such that the bound state is pushed into the continuum, i.e., a half-collision experiment A well-known example are Stark resonances, which are the result of placing atoms inside a dc electric field [27] For pedagogic as well as illustrative reasons, we begin our discussion by presenting a solution of the TDSE for a model problem On this model problem, we demonstrate in Section that a wave packet solution of the TDSE can be of a dual nature and possess both the characteristics of a bound and a continuum state Most importantly we show a stationary nature of the time-dependent dynamics This oxymoron is at the heart of resonance theory Following the time-dependent analysis, we proceed to discuss a stationary analysis using resonances This is done in Section Ensuing from the complementary pictures of both the stationary and the time-dependent strategies, Section aims to clarify and unite the two approaches, settling the seemingly disturbing dissonance In Section 5, we present the possible quantum mechanical sources for the formation of metastable states We then present our conclusions along with possible other features that could be the subject of further study A QUANTUM MECHANICAL RESONANCE STATE FROM A TIME-DEPENDENT PERSPECTIVE At first glance, one can not hope to find general features that are common to different solutions of the TDSE There are simply to many variables, one might rightfully assume that the solution depends greatly on the initial condition and that every potential displays completely different features Unlike a solution of the TISE, where we can define a state by its energy and write its time dependence explicitly, a general wave packet solution of the TDSE cannot be so characterized in a similar manner In this section, however, we will show that under certain conditions even a wave packet, a dynamic time-dependent entity by definition, has many of the common attributes of a stationary state 2.1 From bound state to metastable state Consider, as an example, the following variation of the often-used onedimensional potential [28]: V(x) = − V0 cosh βx 2 e−αx (1) 334 Vitali Averbukh and Pˇremysl Kolorenˇc ϕl ϕm ϕj ϕk Kr+ Kr+ ϕn e− ϕa X Figure 6.6 Schematic representation of collective interatomic decay of two inner-shell vacancies, see Eq (20) in general, the collective decay occurs without facing a competition from the ICD if 1.5 < (Eiv − Ec )/Eion < 2, where Eiv is the inner valence ionization energy of the given species, Ec is the energy of Coulombic repulsion between two singly ionized atoms or molecules (typically 3–4 eV at the equilibrium distances of neutral van der Waals clusters) and Eion is the single ionization energy Some qualitative understanding of the CICD can be gained by means of Wentzel-type theory that treats the initial and final states of the decay as single Slater determinants taking electronic repulsion responsible for the transitions as a perturbation The collective decay of two inner-shell vacancies (see Figure 6.6) is a three-electron transition mediated by two-electron interaction Thus, the process is forbidden in the first-order perturbation theory, and its rate cannot be calculated by the first-order expressions, such as (1) Going to the second-order perturbation theory, the expression for the collective decay width can be written as = 2π i ˆ ˆ f |V|i i|V|0 E0 − Ei ˆ = δ(Ef − E0 ), V p