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The Quantum Chemistry of Loosely-Bound Electrons

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The Quantum Chemistry of Loosely-Bound Electrons CHAPTER 8 The Quantum Chemistry of Loosely-Bound Electrons John M. Herbert Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA INTRODUCTION AND OVERVIEW What Is a Loosely-Bound Electron? By some measure, the title of this chapter could have been “The quantum chemistry of weakly-bound anions,” because much of it will focus on how to describe the weak binding of an “extra” electron to a stable, neutral molecule using electronic structure theory. Electron binding energies in such cases may be quite small, typically less than the largest atomic electron affinities (EAs)1 (3.4 eV for F and 3.6 eV for Cl), and even less than 0.1 eV in some cases. Unlike the case where a neutral molecule, M, is ionized, an electron separated from the anion M− does not experience an attractive −1∕r potential at large separations,2,3 but rather only charge–dipole and or higher order charge–multipole interactions. Cases where the electron affinity of M is ≲ 0.5 eV are the signature of a short-range valence potential that is weakly attractive at best, such that electron binding in M− results primarily from long-range electron–molecule, charge–multipole interactions. In such cases, one expects to find an unpaired electron− inM that is radially diffuse, much more so than in F− or Cl−, for example. It is in this sense that the odd electron in M− is “loosely-bound.” Moreover, the preceding discussion assumes that M− is bound at all, but in fact we will also consider cases in which the electron Reviews in Computational Chemistry, Volume 28, First Edition. Edited by Abby L. Parrill and Kenny B. Lipkowitz. © 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc. 391 392 The Quantum Chemistry of Loosely-Bound Electrons is adiabatically unbound (higher in energy than M + e−), but where the species M− can exist as a temporary anion resonance, trapped behind some energy barrier that often originates from the centrifugal potential required to conserve angular moment when removing an electron from an orbital with > 0. We will further broaden our definition of what constitutes a loosely-bound electron to include a discussion of solvated electrons, which one − might define as cluster anionsN M where the odd electron is bound collectively by the solvent molecules, insofar as the anion M− of a single solvent molecule − is not a bound species. In a small cluster, MN might be weakly-bound, but depending on the number (N) and nature of the solvent molecules, the electron − binding energy of MN can sometimes be quite large, up to a few eV in some cases. This is still much smaller than the ionization energy of the neutral cluster MN, and in this sense the “extra” electron may still be considered to be weakly-bound. As a definite example, consider the case M = H2O. Vertical − electron binding energies in (H2O)N clusters, as measured by photoelectron spectroscopy, can exceed 2 eV for N ≳ 100,4,5 and the best estimates in the → 5–10 − bulk limit (N ∞) lie in the range of 3.3–4.0 eV. Insofar as H2O is not bound,11 however, the unpaired electron cannot be said to be associated to any one particular water molecule and is thus “loosely-bound.” As another example of what we have categorized, for the purpose of this chapter, as loosely-bound electrons, we will consider excited electronic states of anions that possess enough energy to access an electronic contin- uum, or in other words, excited states where the excitation energy is greater than the electron detachment energy. Such states exist, if at all, only as tem- porary, “auto-ionizing” resonances. Other temporary anion resonances (viz, shape resonances and Feshbach resonances) will be explained and discussed as well. These temporary anion resonances are chemically important in the context of dissociative electron attachment (DEA),12 in which resonant attach- ment of low-energy electrons is followed by internal conversion to a state where covalent bond dissociation is energetically feasible. DEA provides a mechanism wherein energy barriers that would be thermodynamically insurmountable on the Born–Oppenheimer potential surface for M + e− are bypassed by means of nonadiabatic transitions, leading to molecular fragmentation in the presence of electrons whose kinetic energies are “just right.” Scope of This Review Each of the aforementioned phenomena presents special challenges for quantum chemistry, and the methods required to meet these challenges are the primary topic of this chapter. In addition, this chapter provides a discussion of the basic quantum mechanical concepts that underlie the collection of phenomena that this author has categorized as “loosely-bound electrons.” A limited discussion of some of the interesting chemical systems that fall under this moniker is provided as well, although this chapter is not intended Introduction and Overview 393 to be a comprehensive review of the field of weakly-bound anions, solvated electrons, or anything else. Several excellent topical reviews have appeared in the past few years, to which the reader is referred for a thorough discussion of the chemistry. These include a comprehensive 2008 review of the whole field of molecular anions (experiment as well as theory),2 and a more recent review focused on theoretical calculations.3 Reactions induced by low-energy electrons have also been thoroughly reviewed in the past few years, including several general overviews of electron-induced reactions,12–14 a review of biological radiation chemistry and the role of “presolvated” electrons in biological radiation damage,13 and a review of electron attachment (EA) to DNA, focusing on electronic structure calculations.14 Solvated electrons, both in clusters and in bulk liquids, have been reviewed recently from both a theoretical perspective15,16 and an experimental perspective.17–19 This work is intended as an introduction to these topics and a tutorial guide to performing quantum chemistry calculations intended to model these types of molecular systems and phenomena. The focus here is on methods that are readily available in standard quantum chemistry software packages and thus, for example, we will discuss the calculation of resonance states using modifications of bound-state methodology,20 since bound states are what one computes in traditional quantum chemistry. Alternative formalisms such as scattering theory21–26 or the explicit treatment of the interaction of a discrete state with a continuum state27,28 will not be discussed here. The use of complex absorbing potentials29–31 is discussed only briefly. Our discussion of electronic structure calculations is further limited to methods based on Gaussian basis sets rather than plane waves. This restric- tion is partly a matter of taste, and in fact one can make a good case that a plane-wave basis is better suited for representing the most diffuse parts of a weakly-bound anion’s electron density, as compared to a basis comprised of localized, atom-centered functions. However, a more important concern (in this author’s view) is the fact that Hartree–Fock exchange is prohibitively expensive to compute in a plane-wave basis. Although significant progress has been made in this respect,32 the cost remains prohibitive unless the nonlocal exchange interaction is screened at long range.33,34 As a result, the hybrid functionals that provide the best performance for many molecular properties are ordinar- ily not available in plane-wave density functional theory (DFT) calculations, which proves particularly egregious in the case of weakly-bound anions. Cor- related, post-Hartree–Fock wave functions are similarly unavailable in most plane-wave codes, which is another reason to dismiss plane waves in the present context. As the title of this chapter suggests, the discussion here is limited, for the most part, to loosely bound electrons, meaning that valence ionization is not considered to any significant extent. It is important to keep in mind, however, that “loosely-bound” need not mean “weakly-bound.” As mentioned earlier, vertical electron detachment energies in excess of 1–2 eV are possible even 394 The Quantum Chemistry of Loosely-Bound Electrons in solvated-electron clusters where the “extra” electron is not strongly asso- ciated with any particular molecular unit or chemical moiety. That said, the quantum chemistry of weakly-bound, gas-phase anions is certainly discussed herein, at a somewhat more pedagogical level as compared to previous reviews on that subject.2,11,35 An exception is that Refs. 2 and 37 describe the under- lying theory behind various post-Hartree–Fock quantum chemistry models, material that is not covered here at all (it has been covered in previous chapters in this series36,37). This review assumes a basic familiarity with the nomencla- ture of quantum chemical models and basis sets, at the level of introductory textbooks35,38,39 or previous chapters in this series.36,40,41 The performance of a variety of quantum chemical models, as applied to weakly-bound anions, is addressed in detail, but the inner workings of these models is not discussed here. A unique feature of this chapter, as compared to other reviews of anion quantum chemistry that have appeared in the past decade,2,11,42 is a greater emphasis on treating larger systems, including anions in the condensed phase. In large systems, compromises must inevitably be made in terms of the theoreti- cal methods that can be deployed. As such, density functional methods – which have hardly been discussed at all in previous reviews of anion quantum chem- istry, except briefly in Ref. 2 and in a benchmarking capacity in Ref. 43–are discussed at length here. Previous chapters in this series provide an introduction to DFT itself,40,41 but the focus here is on the performance of the models, not their intimate details.
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