Molecular Anions

Jack Simons

Chemistry Department

Henry Eyring Center for Theoretical Chemistry

University of Utah

Opening remarks:

This web-based text book offers many web links to researchers who have contributed much to the study of molecular anions. It also offers many literature references pertaining to the examples I use to illustrate the families of molecular anions discussed. It is my hope that the reader will find this text to be a useful resource for learning why the experimental and theoretical study of anions is such an exciting endeavor for so many chemists. I also hope it contains some surprises that offer even the most knowledgeable reader new insight into the behavior of negative molecular ions. If I have been successful, I am confident that wonderful new knowledge about molecular anions will be produced by readers of this text and that new workers will be drawn to this exciting field of study.

If you wish to download a .pdf version of this text, please click here. I am still working out the bugs encountered in posting the html version on the web (e.g., the Greek and math symbols). If you wish to access the most recent version, click here (I found that Internet Explorer does a better job than other browsers at displaying the symbols). If you want to offer input for future improvements or additions, please click here to send me email. I hope you enjoy browing through this text and connecting to its many web links.
Table of Contents (click on a Chapter or Section header label to go to it)

Introduction

Chapter 1. Introduction to Molecular Anions

I. Anions Have Very Different Valence-Region Electronic Potential Energies Than Neutrals and Cations

II. Anions Are Difficult to Prepare and Study as Isolated Species

A. Making Anions

B. Selecting Specific Anions

C. How Fields Are Used to Focus and Select Anions

D. Problems That Can Occur

III. Anions Experience Strong Environmental Effects

IV. Spectroscopic Probes

V. Reaction Dynamics Probes

Chapter 2. Anions Also Present Special Challenges to Theoretical Study

I. Special Atomic Basis Sets Must be Used

II. The Hartree-Fock SCF Process is Usually the Starting Point

III. Koopmans’ Theorem Gives the First Approximation to the Electron Affinity

IV. Electron Correlation Involving the Excess Electron Usually Must be Treated

V. Various Methods Can be Used to Treat Correlation

A. The multiconfigurational self-consistent field (MCSCF) method

B. The configuration interaction (CI) method

C. The Møller-Plesset perturbation (MPPT) method

D. The Coupled-Cluster (CC) method

E. Density Functional Theory (DFT)

VI. Computational Requirements, Strengths, and Weaknesses of Various Methods

VII. Direct Calculation of Intensive Energy Methods

VIII. Complete Basis Extrapolations

IX. Why is Electron Correlation So Important for EAs?

X. Reaction Paths

XI. Summary

Chapter 3. Chemically Conventional Anions

I. What Makes these Anions Conventional?

II. They May be Conventional, but They Involve Complexities

III. Common Multiply Charged Anions are Not Conventional

Chapter 4. Multipole-Bound Anions

I. Electrostatic Attractions

A. The Point and Fixed Finite Dipole Models

B. Binding to Real Molecules

C. Summary

II. Binding an Electron to Quadrupolar Molecules

A. Is There a Critical Value for the Quadrupole Moment?

B. Real Molecules that Quadrupole Bind

III. Binding Through Higher Moments

IV. Double-Rydberg Anions

V. Zwitterion-Bound Anions

Chapter 5. Multiply Charged Anions

I. Binding to Polar Molecules

A. What the PD and FFD Models Suggest

B. Real Cases

II. Binding to Two Distant Sites in a Single Molecule

III. Binding to Proximate Sites

IV. Special Techniques are Needed to Handle Metastable Anions

Chapter 6. Cluster Anions

I. Anions that are Solvated

II. Clusters With an Electron Attached

III. Clusters Can be Used to Probe Chemical Reactions

IV. Sometimes the Isomers are New Anions

V. Covalent and Metallic Cluster Anions

Chapter 7. Anions of Biological Molecules

I. Dipole-bound and Valence Anions

II. Virtual Orbitals May Not be Electronically Stable

III. Electrons Attached to DNA

IV. Electrons Fragmenting Peptides

Introduction

Within the pages of this book, my personal perspectives are offered on the chemical study of negative molecular ions. Not much emphasis will be placed on discussing atomic anions as isolated species because it is my view that chemistry deals primarily with molecules and materials and with their reactions and properties, and I think the world of molecules begins with two or more atoms held together by chemical bonds. Therefore, I view the study of isolated atomic anions as primarily the domain of the atomic physics community although, of course, I do think it useful to discuss atoms as building blocks that form molecules. A recent review by Professor David J. Pegg from the point of view of a physicist with emphasis on atomic anions can be found at this online link.

For insight into the experimental study of negative molecular ions from a chemist’s point of view beyond what is presented in this text, I refer the reader to the web sites of Professor W. C. Lineberger and Professor John I. Brauman.

Carl Lineberger, University of Colorado John Brauman, Stanford University

These two scholars have done as much if not more than anyone else over the past forty years to contribute to chemists’ knowledge about electron affinities and the chemical structures, reactions, and spectroscopy of molecular anions. Their groups have also pioneered many of the most useful experimental tools for studying molecular anions and have generated many scientific offspring who became major figures in this field. Of course, even they stood on the shoulders of earlier masters such as Louis Branscomb (Atomic and Molecular Processes, edited by D. R. Bates, Academic Press, New York, N.Y. (1962)), George Schulz (Rev. Mod. Phys. 45, 373, 423 (1973)), and Sir H. S. W. Massey (Negative Ions, Cambridge Univ. Press (1976), Cambridge, England).

I will make use of many examples of chemical studies carried out by Professors Brauman and Lineberger as well as results from the laboratories of many others I mention throughout this text. In so doing, I do not mean to suggest that only the groups I mention in each example have contributed to such studies; in fact, most of the groups I highlight pursue work on many if not most of the molecular anions treated in this book. However, for brevity, I have had to select but a few examples for each of the classes of anions treated from among the many studies these workers have undertaken.

Many other senior scholars have contributed much to the advancement of experimental studies of molecular anions in recent decades and continue to do so. Several of them are shown below. They and many of their scientific offspring continue to expand the horizons of this field of study.

Kit Bowen, Johns Hopkins Jim Coe, Ohio State Dan Neumark, Berkeley

Bob Compton, Tennessee Mark Johnson, Yale Torkild Andersen, Aarhus

Lars Andersen, Aarhus Paul Burrow, Nebraska Lai-Sheng Wang, Washington State

Jack Beauchamp, Caltech Kent Ervin, Nevada, Reno Michael Allan, Fribourg

Veronica Bierbaum, Colorado; Barney Ellison, Colorado; Eugen Illenberger, Berlin

Leon Sanche, Sherbrooke Ron Naaman, Weizmann Tilmann Märk, Innsbruck

Paul Kebarle, Alberta Will Castleman, Penn State

The University of Colorado’s Joint Institute for Laboratory Astrophysics (JILA) has a very long tradition of experimental advances and studies of molecular anions. Below we see several JILA scientists whose scientific careers are linked strongly to this field of study; can you identify them?

The University of Colorado, JILA, Ion Gang in 1980.

Throughout this text, I will show many examples from labs of the people shown in the above figures of experimental data on a wide range of molecular anions.

Of course, there have been theoretical chemists who have also advanced our knowledge of molecular anions during the same timeframe. Professor R. S. Berry was among the earliest pioneers (R. S. Berry, Chem. Rev. 69, 533 (1969)) of such studies.

Steve Berry, Chicago

Several other senior chemistry scholars who have contributed much to the advancement of the theoretical study of molecular ions are shown below. They and their scientific offspring continue to advance this field of study.

Lenz Cederbaum, Heidelberg Alex Boldyrev, Utah State Ken Jordan, Pittsburgh

Josef Kalcher, Gratz Piotr Skurski, Gdansk Maciej Gutowski, Edinburgh

Vince Ortiz, Auburn Ludwik Adamowicz, Arizona Howard Taylor, USC

Fritz Schaefer, Georgia Ernest Davidson, Washington

Kwang Kim, Pohang Peter Rossky, Texas Pavel Rosmus, de Marne la Vallée

I will also show results from these scientists’ research efforts throughout this text, but again only a small fraction of what they have contributed can be covered.

Prior to the time most of the people shown above began to study molecular anion chemistry, the electron affinities of most atoms were not known and very little was known about the electron affinities of molecules and radicals. It was largely because of experimental advances in ion sources and spectroscopic probes that the determination of molecular electron affinities and the study of molecular anions began to advance rapidly in the 1960s and 1970s. Once experimental chemists began to be able to make and study negative molecular ions, it was natural for theoretical chemists to become involved in this field. However, they too faced significant challenges and had to develop new models and new computational tools to study anions as I will show later in this text.

My own history in the field dates to 1973, when our first paper (Theory of Electron Affinities of Small Molecules, J. Simons, and W. D. Smith, J. Chem. Phys., 50, 4899-4907 (1973)) dealing with the ab initio calculation of electron affinities (EAs) using what we termed the equations of motion (EOM) method was published. At about this same time, Professor Lenz Cederbaum was developing what turned out to be an equivalent method [[1]] for directly calculating ion-molecule energy differences, as were other groups [[2]]. Prior to this time, quantum chemical calculations of molecular EAs [[3]], including many from Professors Enrico Clementi, Ernest Davidson and Fritz Schaefer were most commonly carried out using approximate solutions to the Schrödinger equations to obtain the total electronic energies of the neutral (E_neu) and anionic (E_an) species and subtracting these two quantities to compute the EA as

EA = E_neu – E_an.

However, because the EA is a very small fraction of the total electronic energies of the neutral or the anion, this process is fraught with danger because one must obtain each of the two total energies to very high percent accuracies to obtain the EA to a chemically useful accuracy. To illustrate, we note that EAs typically lie in the 0.01-5 eV range, but the total electronic energy of even a small molecule is usually several orders of magnitude larger. For example, the EA of the ⁴S_3/2 state of the carbon atom is [[4]] 1.262119 ± 0.000020 eV, whereas the total electronic energy of this state of C is

–1030.080 eV (this total energy is defined relative to a C⁶⁺ nucleus and six electrons infinitely distant and not moving). Since the EA is ca. 0.1 % of the total energy of C, one needs to compute the C and C^- electronic energies to accuracies of 0.01 % or better to calculate the EA to within 10%.

Moreover, because the EA is an intensive quantity but the total energy is an extensive quantity, the difficulty in evaluating EAs to within a fixed specified (e.g., ± 0.1 eV) accuracy based on subtracting total energies becomes more and more difficult as the size and number of electrons in the molecule grows. For example, the EA of C₂ in its X ground electronic state [4] is 3.269 ± 0.006 eV near the equilibrium bond length R_e but only 1.2621 eV at R ¥ (i.e., the same as the EA of a carbon atom). However, the total electronic energy of C₂ is –2060.160 eV at R ¥ and lower by ca. 3.6 eV (the dissociation energy [[5]] of C₂) at R_e, so again the EA is a very small fraction of the total energies. For buckyball C₆₀, the EA is [[6]] 2.683 ± 0.008 eV, but the total electronic energy is sixty times –1030.080 eV minus the atomization energy (i.e., the energy change for C₆₀ 60 C) of this compound. This situation becomes especially problematic when studying extended systems such as solids, polymers, or surfaces for which the EA is an infinitesimal fraction of the total energy. I should note that this same difficulty plagues the theoretical evaluation of other intensive properties such as ionization potentials, electronic excitation energies, bond energies, heats of formation, etc.

These examples show that computing the EA of a molecule by using the total energies of its neutral and anion may not be a wise approach. How do most experiments determine molecular EAs? The most direct technique involves using a tunable light source of frequency n to photodetach an electron from a molecular anion A^-. By determining the minimum photon energy hn needed to detach an electron to form the neutral molecule A, one determines the EA. This offers an example of how the EA is determined directly. Nowhere in this experiment is the extensive total energy of either the anion or the neutral measured. So, it would appear natural to seek a theoretical approach to determining EAs that follows the experimental example.

In the 1973 paper mentioned above, we did so by developing the equations of motion (EOM) method as a route to calculating the intensive EAs directly as eigenvalues of a set of working equations. In this theoretical development, one avoids (approximately) solving the Schrödinger equation for the extensive energies of the neutral and anion and then subtracting the two extensive energies to obtain the desired intensive EA. In numerous of our subsequent publications, the EOM method was refined and applied to a variety of molecular anions. In the intervening years, our group and others [[7]] have greatly extended the EOM method beyond the Møller-Plesset framework that we initially used to allow more powerful coupled-cluster, multi-configurational, and other wave function classes to be employed. Most of the subsequent developments of these theoretical tools have been cast within the language of so-called Greens function or propagators, but they could just as well have been written in our EOM language. As a result of such advances by many different groups, several direct-calculation techniques are now routinely used to compute EAs; that of Professor Vince Ortiz [7] is even contained within the widely used Gaussian suite of programs [[8]] that many chemists use routinely.

In the early studies of anions carried out in the 1970s and 80, emphasis was placed on simply determining electron affinities (EAs) rather than on probing the potential energy surfaces of chemical reactions involving anions, determining their spectrocsopic and structural properties, or attempting to design anions with novel structural or bonding characters. This was true both of the theoretical and experimental investigation of anions primarily because

a. prior to 1970, even these most fundamental thermodynamic data (i.e., EAs) had not been directly (i.e., by laser photodetachment) determined for most atoms, molecules, and radicals, and

b. the experimental and theoretical tools available to determine EAs were in their formative stages and needed to be tested on species whose EAs were reasonably well known from other sources.

In the subsequent thirty years, the field has broadened considerably to where the study of molecular anions is now motivated by a variety of reasons including designing new anions having specific bonding behavior or energy content and probing the influence of electrons attached to biological molecules, water clusters, interfaces, and within nanoscopic materials. Over this same period, the number of research groups focusing on anion chemistry has grown tremendously. In the 1970s, issues of J. Chem. Phys. or J. Phys. Chem. contained very few papers on anions, but now essentially any issue of either of these journals contains more than one anion paper and the number and range of such papers in increasing rapidly.

Because our knowledge of molecular anions has reached a stage in which the field has very broad interest and impact, I felt it was time to offer a source from which one could gain perspective about these species. By no means does this book intend to thoroughly review the vast body of knowledge that has been established on molecular anions’ properties or to tabulate molecular EAs. Rather, it focuses on providing references to many practicing scientists and other valuable sources of information and on introducing the reader to

a. the fundamental properties that make anions qualitatively different from neutrals and cations,

b. introducing several classes of anions whose study has substantially expanded in recent years but which still offer promise of many more discoveries.

c. illustrating the special challenges that the study of molecular anions present.

In my mind, this book is a text from which one can learn rather than a reference book where one can look up all that is known.

If one is searching for tabulated values of atomic or molecular electron affinities (EAs), the best places to search for such information are:

1. For atoms, the early reviews of Hotop and Lineberger [[9]], and the more recent review by Andersen, Haugen, and Hotop [[10]] remain excellent sources.

2. For molecules, there are several sources [[11], [12], [13], [14], [15]] that span many years, some of which are accessible on the web.

The primary focus of the present work is to first (Chapters 1 and 2) give an introduction to some of the special challenges that negative molecular ions present both in terms of experimental study and theoretical investigation. I begin by considering some of the characteristics of negative ions that make them qualitatively different from neutrals or cations. Also, I offer a brief introduction to some of the challenges that one must face when studying anions in the laboratory. Although I am a theoretical chemist and is not familiar with all of the details involved in carrying out experiments on anions, I believe it is essential for me to discuss such matters so readers will appreciate how difficult it is to make anions in appreciable numbers and to confine them so they can be probed, and how their low electron binding energies further complicate matters.

Subsequent focus (in Chapters 3-7) is aimed at introducing the reader to the wide variety of negative ions that one encounters in chemical science and giving a few examples of several such classes. As a result, these Chapters provide an introduction to various kinds of molecular anions and the special characteristics that they possess, but by no means do they offer exhaustive reviews of the extensive literature on these anions.

Now, let us begin the journey through the world of negative molecular ions by examining in Chapters 1 and 2 what makes anions significantly different from neutrals and cations, why these differences are important, and what makes their experimental and theoretical study challenging.

Chapter 1. Introduction to Molecular Anions

I. The Valence Electrons in Anions Experience Very Different Potential Energies Than in Neutrals and Cations

The physical and chemical properties of anions are very different from those of neutral molecules or of cations. Obviously, their negative charge causes them to interact with surrounding molecules and ions differently than do cations or neutrals. For example, when hydrated, anions are surrounded by H₂O molecules whose dipoles tend to have their positive ends directed toward the anion. For cations, the dipoles are directed oppositely and for neutrals, the local solvent’s orientation depends upon the polarity of the functional group on the solute nearest the solvent. Moreover, anions polarize the electron clouds of nearby molecules in the opposite sense that cations do. Because of their weakly bound valence electron densities, anions have large polarizabilities and thus tend to have stronger van der Waals interactions with surrounding molecules than do more compact, less polarizable neutrals and cations. The valence electron binding energies in anions tend to be smaller than in neutrals or cations, and anions seldom have bound excited electronic states whereas neutrals and cations have many bound excited states including Rydberg progressions. The source of all of these differences lies in the potentials that govern the movements of the valence electrons of the anions, cations, and neutrals.

As chemists know well, it is an atom or molecule’s outermost (i.e., valence) orbitals that govern the size, electron binding energy, and much of the chemical reactivity of that species. When an anion’s electrons move to the regions of space occupied by its valence orbitals, they experience an attractive potential that is qualitatively different from in neutrals and cations. It is these differences that we need to now spend time discussing because these differences are of fundamental importance in determining many of the physical and chemical properties of anions that make them different.

Specifically, an electron in the valence regions of an anion experiences no net Coulomb attraction in its asymptotic (i.e., large-r) regions, but corresponding electrons in neutrals and cations do experience such –Ze²/r attractions (e.g., Z = 1 for a neutral and 2 for a singly-charged cation, etc.). In fact, the longest-range attractive potentials appropriate to an electron in singly charged anions are the charge-dipole (-mr e/r³), the charge-quadrupole (-Q(3rr –r²1) e/3r⁵), and the charge-induced-dipole

(-arre²/2r⁶) potentials. Here, m, Q, and a are the corresponding neutral molecule’s dipole moment vector, quadrupole moment tensor, and polarizability tensor, respectively; and r is the position vector of the electron. The symbols indicate dot products with the vectors or tensors, and 1 is the unit tensor. The most important thing to note is that for cations and neutrals, the large-r attractive potential falls of as –Ze²/r, whereas for anions, it falls off as a higher power of 1/r.

These differences are what produce major differences in the radial size, electron binding energy, and pattern of bound electronic states of anions compared to neutrals and cations. For example, recall that it is the 1/r dependence of the Coulomb attraction combined with the 1/r² scaling of the radial kinetic energy operator (-h²/(2mr²) /r(rr)) that produces the well known E = -13.6 eV(Z²/n²) Bohr formula for the infinite series of energies of hydrogen-like atoms having one electron moving about a nucleus of charge Z in an orbital of principal quantum number n. Anions do not have series of bound electronic states that obey this equation because their potentials do not vary as 1/r at large-r. In contrast, molecules and cations do possess excited states (i.e., so-called Rydberg states) whose energies can be fit to a formula like E = -13.6 eV(Z_eff²/(n-d)²). Here Z_eff is an effective nuclear charge and d is called a quantum defect; both are designed to embody the effects of the inner-shell electrons in screening the outermost Rydberg electron.

For multiply charged anions, the asymptotic potential an electron experiences is repulsive and has the Coulomb form (Z-1) e²/r, where Z is the magnitude of the (negative) charge of the anion. For example, for SO₄^2-, Z = 2. It is only in the inner valence regions that the net potential in multiply charged anions may become attractive enough to permit electron binding (e.g., in SO₄^2-, as an electron approaches one of the very electronegative oxygen centers, the potential is attractive meaning the radial force -dV/dr is directed inward). The kind of potentials discussed above are illustrated in Fig. 1.1 where it is also suggested how the shorter range repulsive potentials (due to the remaining electrons’ Coulomb and exchange interactions) eventually cut off these long-range behaviors at smaller values of r (it is assumed that the magnitudes of m, a, and Q as well as the range of the short-range repulsive potentials are within ranges that are commonly encountered).

Figure 1.1. Plots of potentials experienced by valence electron in neutrals and cations, in singly charged anions, and in multiply charged anions.

In addition to these differences in long-range potentials, there are also qualitative differences in the inner valence-range potentials appropriate to anions, neutrals, and cations. Specifically, an electron in any molecule or ion containing N total electrons experiences Coulomb attractions (-S_a Z_ae²/|r-R_a|) to each nucleus (having charge Z_a); the total of such attractive charges is Z_tot = S_a Z_a. This same electron experiences repulsive Coulomb and exchange interactions (e.g., given as a sum of Coulomb J_j and exchange K_j operators S_j=1,N (J_j – K_j) within the Hartree-Fock approximation that I will discuss in Chapter 2) with a total of N-1 other electrons. However, as Fig. 1.1 suggests, the balance between these Z_tot attractions and N-1 repulsions is very different among neutrals, singly charged anions, multiply charged anions, and cations.

For a singly charged anion, Z_tot = N-1, so, as noted above, there is no net Coulomb attraction or repulsion in regions of space where the electron-electron Coulomb and exchange terms cancel the nuclear attraction terms (e.g., for large r). However, in regions of space where this cancellation is not fully realized (e.g., within an oxygen p orbital of SO₄^2- or inside the lone-pair orbital of the H₃C^- anion shown in Fig. 1.2 where the extra electron is not entirely shielded from the carbon nucleus by the other electron occupying this same orbital), a net attraction occurs. It is this net attractive potential and the fact that it has no long-range Coulomb character that ultimately determines the orbital shape and radial extent as well as the binding energy for the singly charged anion’s extra electron.

Figure 1.2. Doubly occupied orbital in the H₃C^- anion (left) and singly occupied orbital of the CH₃ radical (right).

Note that the singly occupied orbital of the CH₃ radical is drawn in Fig. 1.2 as being radially more compact than the corresponding doubly occupied orbital of the anion. This difference is size is due to the fact that the electron occupying the orbital in the neutral does not experience a Coulomb repulsion from a second electron in this same orbital (as occurs in the anion) and, as a result, this electron experiences a more attractive potential within the valence-orbital range and at large-r where the potential is of the attractive Coulomb form.

In contrast, for a neutral molecule or cation, Z_tot is larger than N-1, so there exists a net Coulomb attraction at long range, as well as valence-range net attractive potentials, and repulsive potentials at even shorter range (due to repulsion from inner-shell electrons). The fact that the same kind of valence attractive potentials as in the anion are augmented by a long-range Coulomb attractive potential gives rise to stronger electron binding and smaller radial extent in such cases. For this reason, the minima in the potentials shown in Fig. 1.1 for the neutral and cation, are usually (i.e., for commonly realized values of the dipole moment, polarizability, etc.) deeper than for the anion cases. As a result, ionization potentials (IPs) of neutrals or of cations [[16]] usually exceed electron binding energies in anions (alternatively, electron affinities EAs of the corresponding neutrals [[17]]). This difference in the range of magnitudes of EAs and IPs is one of the most problematic facts for the theoretical study of anions. Specifically, any theoretical method that is able to produce electronic energy differences to an accuracy of 0.5 eV can prove valuable for studying IPs, which are usually significantly greater than 0.5 eV. However, many EAs are comparable to or less than 0.5 eV, so such theoretical predictions are of much less value for anions.

It is important to stress another attribute of the -1/r character of the attractive Coulomb potential appearing in neutrals and cations. As noted briefly earlier, when combined with the 1/r² dependence of the electronic kinetic energy operator (-h²/2m²), the –1/r Coulomb potential gives rise to the well known Rydberg series of bound electronic states whose energies vary with quantum number n as –R/(n-d)² (R being the Rydberg constant equal to 13.6 eV and d the quantum defect parameter embodying the effects of inner-shell electrons’ repulsions). Anions do not possess the same kind of infinite series of bound electronic states because their long-range potentials vary as higher powers of 1/r. In fact, anions in which the excess electron is bound in a valence orbital (e.g., H₃C^- or HO^-) have only one bound state rather than the infinite progressions of bound states that arise in neutrals and cations. On the other hand, anions with large dipole moments (> ca. 2 Debyes) can also bind an electron via their charge-dipole potential, but, unless the species has an extremely large dipole moment, only one electronic state is significantly bound (i.e., bound by > 100 cm^-1). So, again, one does not observe an infinite progression of substantially bound states when dipole binding is operative; usually only one weakly bound state is seen. The bottom line is that one should not expect a molecular anion to possess any significantly bound excited electronic states; some anions have a few bound excited states, but most do not. The lack of bound excited electronic states presents significant difficulties for using spectroscopic tools to probe anions because such tools (e.g. laser induced fluorescence (LIF), resonance-enhanced multi-photon detachment (REMPD)) often rely on access to a bound excited state.

It is very important to be aware of the patterns of bound electronic states that occur in cations and neutrals and the paucity of bound states that characterize anions because one can not depend on the existence of an experimentally accessible progression of bound excited states to probe the electron detachment thresholds of anions as one often uses Rydberg series to approach ionization thresholds of neutral species. Moreover, the kind of vibration-induced electron detachment [[18]] processes that take place in Rydberg states of neutrals (in which a sequence of vibrational energy losses is accompanied by a series of Rydberg-state electronic energy gains) cannot occur in molecular anions. The anions do not have such a series of electronic states that can accept the sequence of vibrational energy losses and thus effect electron detachment, so one must use a different theory [[19]] to model such processes. In particular, the theory must allow for the anion to accept, in a single transition, enough vibration-rotation energy to undergo a bound-to-free transition in a single step.

For multiply charged anions, yet another situation arises since Z_tot is smaller than N-1. As a result, at long range a repulsive Coulomb potential is operative. However, in the valence regions near the nuclei of the constituent atoms, the nuclear attractive potentials may be strong enough to overcome electron-electron Coulomb repulsions in certain regions of space (e.g., near the oxygen centers in TeF₈^2-). In those regions, an electron may experience a net attractive potential that may be strong enough to bind the electron in which case the net potential will have the form shown in the upper part of Fig. 1.3 and one can have an electronically stable di-anion. Alternatively, if the valence-range attractive potentials are not strong enough to overcome the Coulomb repulsion, a potential such as in the bottom of Fig. 1.3 can result. In this latter case, a metastable state of the multiply charged anion may result (as in SO₄^2-). In such a state, an extra electron can undergo autodetachment by tunneling through what is called the repulsive Coulomb barrier (RCB).

In both cases, one observes the RCB within which a bound or metastable state may exist. It should be noted that the long-range Coulomb repulsion that is operative in multiply charged anions has both destabilizing and stabilizing effects. The internal Coulomb repulsions certainly reduce the intrinsic electron binding potential of the shorter-range attractive potentials. However, the Coulomb repulsion also produces the long-range barrier that acts to confine or trap the electron; this confinement is especially important to consider when the multiply charged anion is metastable rather than electronically stable and thus susceptible to autodetachment. As we will see in Chapter 5, the RCB can cause multiply charged anions that are adiabatically quite unstable to have lifetimes exceeding seconds or minutes thus rendering them very amenable to detection and characterization.

Figure 1.3. Effective radial potentials experienced by the outermost electron in a doubly charged stable (top) and metastable (bottom) anion.

Of course, in a real multiply charged molecular anion the poential is not angularly isotropic; that is, it depends on the direction an electron moves as it attempts to escape. Two more quantitative representations of such potentials for doubly charged anions are shown in Figs. 1.4a and 1.4b. The former describes the potential experienced by the second excess electron in a linear structure of C₆^2- [[20]] while the latter shows the corresponding potential in the tetrahedral species N(BF₃)₄^2- [[21]] with the potential plotted for a direction along one of the N-B bonds. In both cases, the potential is defined as zero when the second excess electron is infinitely far from the corresponding mono-anion.

Figure 1.4a. Potential experienced by second excess electron in C₆^2- with the molecule’s center being located at (0,0) in the figure [20].

Figure 1.4b. Potential experienced by second excess electron in N(BF₃)₄^2- with the nitrogen atom being located at (0,0) and the molecule oriented as shown [21].

The examples shown in Figs. 1.4 illustrate that, although the Coulomb interactions between the two excess electrons produce a repulsive Coulomb barrier, the height of this barrier is not equal in all directions. This means that the second excess electron will escape by tunneling (when it is metastable with respect to a free electron and a mono-anion) with a highly anisotropic angular distribution. That is, the electron will be ejected preferentially in directions where the barrier is low and/or narrow and thus where tunneling is most favorable. Another implication of this observation is that, when carrying out theoretical studies on such dianions, one can obtain a reasonable estimate of the tunneling lifetime if one identifies regions where the Coulomb barrier is small and thin and computes (as an approximation) tunneling rates in such regions.

The differences in long-range and valence-range potentials experienced by the electrons produce some of the most profound differences in the physical and chemical properties of singly charged anions, multiply charged anions, and neutrals or cations. On a qualitative level, the fact that a Coulomb attractive potential, even with Z=1, is longer range (i.e., falls off as a lower power of 1/r) than charge-dipole, charge-quadrupole, or charge-induced-dipole potentials and produces a deeper well (i.e., for typical values of m, Q, and a found in typical molecules) than do the other potentials causes IPs to usually be larger than EAs. This in turn causes the sizes (i.e., radial extent of the outermost valence orbitals) and polarizabilities of anions to be larger than those of neutrals or cations of the same parent species. Moreover, these differences in potentials make the pattern of bound states very different for anions (i.e., few if any significantly bound excited states) than neutrals or cations (i.e., the infinite Rydberg progression of states as well as bound valence-excited states).

Also, as noted above, the Coulomb repulsive potential that occurs in multiply charged anions causes many such species to be metastable with respect to electron detachment or with respect to bond rupture (which subsequently produces Coulomb explosion as we will discuss in Chapter 5). For example, gas-phase (i.e., isolated) SO₄², CO₃^2-, and PO₄^3- are not stable with respect to loss of an electron; these multiply charged anions undergo rapid autodetachment [[22]] in the gas phase. Only when strongly solvated (e.g., in aqueous solution or in the presence of several solvent molecules) are such multiply charged anions stable with respect to electron loss. In contrast, dicarboxylate dianions ^–O₂C-(CH₂)_n-CO₂^- in which three or more methylene units separate the two anion centers are both electronically stable (i.e., neither excess charge spontaneously departs) and geometrically stable with respect to bond rupture and Coulomb explosion [[23]]. The primary difference between the SO₄², CO₃^2-, and PO₄^3- and ^–O₂C-(CH₂)_n-CO₂^- cases is the distance between the two or three excess charges, which, in turn, governs the strength of the repulsive Coulomb barrier. In the former three cases, the charges are too close to produce electronic stability (as in the bottom of Fig. 1.3). In the latter (at least for n ³ 3)), the distance between the two negatively charged sites is large enough to not render the dianion metastable.

Finally, it is worth mentioning how the differences in large-r potentials and subsequent differences in radial extent and electron binding energies can provide special challenges to the theoretical study of singly and multiply charged anions. In particular, when studying anions, it is important to utilize a theoretical approach that

a. properly describes the large-r functional form of the potential (as we discuss in Chapter 2, not all commonly used quantum chemistry tools meet this criterion), especially for anions with very small EAs for which significant electron density exists at large r;

b. is accurate enough to produce EAs of sufficient accuracy (this usually means that electron correlation effects must be included as we discuss in Chapter 2); and

c. is capable of treating electronic metastability when the anion is not electronically stable (this is very difficult to do and is not a feature of most commonly used quantum chemistry software; we treat the special tools needed in such cases later in Chapter 5).

For singly charged anions in which the excess electron is bound tightly in a valence orbital (e.g., in F^-, and organic RO^-, RNH^-, RCOO^- anions), special atomic orbital basis sets are often not essential because the large-r amplitude of the anion’s wave function is small. That is, most of the excess electron’s density exists in the valence-orbital region. Such anions can be handled with the same kind of theoretical tools that have proven most useful in treating neutrals and cations.

However, when treating anions with very small electron binding energies, and thus large radial extent (e.g., dipole-bound anions such as NC-CH₃^-, (HF)_n^-, and NCH^-), using an accurate method (because the EA is so small) and one that is proper at large r (i.e., contains charge-dipole interaction of a correct magnitude and no net Coulomb attraction) is important. In addition, using accurate methods that are correct at large r and which can handle metastable states is important when dealing with multiply charged anions. As discussed in Chapter 2, not all theoretical methods fulfill the criteria detailed above for use on weakly bound anions or multiply charged anions. In particular, most commonly used density functional theories (DFTs) contain potentials that do not behave properly at large r; they contain an attractive –c/r Coulomb-type potential at large r, which clearly is not appropriate when treating such anions. However, efforts are being made to remedy this [[24]]. For example, Professor Don Truhlar’s group has designed a new functional [[25]] that is asymptotically correct and seems to yield accurate electron binding energies.

II. Anions Are Difficult to Prepare, Control, and Study as Isolated Species

A. Making Anions

The fact that most anions (and multiply charged anions) bind their outermost electrons less tightly than do most neutrals or cations contributes to the significant experimental difficulty one has in making substantial quantities of anions. Simple collisional attachment of an electron to a molecule M having a positive EA to form the anion M^- is often not a fruitful means for preparing M^- in gas-phase environments. Because the electron-attachment process is exothermic, and because total energy must be conserved in any binary collision, it may be impossible to form the stable ground state of M^- directly in such gas-phase experiments. One needs to have some way to remove the excess energy (i.e., the exothermicity) released in forming M^-. Moreover, as was emphasized earlier, because most anions do not have progressions of significantly bound excited electronic states, electron capture into an excited state, followed by radiationless relaxation to the ground state of M^- is also not feasible. Thus, unlike cations C⁺, for which electron capture into a Rydberg state C**

C⁺ + e C** C,

followed by radiationless relaxation to lower states, is often an effective means of production of the neutral C, the analogous avenue is infrequently available to generate anions.

In contrast, dissociative electron attachment, followed by fragmentation to yield a fragment anion, is a commonly employed tool for forming molecular anions. In this process, an electron initially attaches by entering (usually) an antibonding orbital of the parent neutral molecule M-X:

M-X + e (M-X)^-*

to form a metastable state of the anion (M-X)^-*. This state is metastable because the reverse process, autodetachment to return to M-X plus a free electron, remains possible so the (M-X)^-* anion has a finite lifetime. There is a long and rich history of the experimental [[26]] and theoretical [[27]] study of such electronically metastable anions. These approaches provide some of the most direct data on antibonding molecular orbitals (e.g., in the e^- + H₃CS-SCH₃ (H₃CS-SCH₃^-)* H₃CS^- + SCH₃ process, the electron enters an S-S antibonding s* orbital) and, as we emphasize here, they offer a good way to create a negative ion. Subsequent to such attachment, a fraction of the (M-X)^-* species undergo bond rupture to form fragments M and X ^– before electron detachment from M-X^-* occurs:

M-X^-* M + X^-.

Of course, the amount of X^- formed depends on the rates of fragmentation and of autodetachment. Often the latter rate is very fast (e.g., 10^13--10¹⁴ s^-1 or faster) and thus severely limits the yield of X^- because fragmentation must take place on a timescale over which the M-X bond can appreciably elongate. The fragmentation is driven by the fact that the extra electron entered an antibonding orbital of M-X. Many anions have been generated by this kind of dissociative electron attachment processes in the gas phase using electron beams or electric discharges.

There are other avenues for forming molecular anions that are also commonly used. Once one has a source of one anion (say X^-), one can generate other anions Y^- by chemical reaction. For example, reactions of the type (R represents an organic functional group)

X^- + R-Y Y^- + R-X

X^- + H-Y X-H + Y^-

can be used when they are exothermic and proceed with no barrier. For example, the anion of a strong acid H-Y can be formed by reacting H-Y with the anion X^- of a weaker acid H-X. Such reactions can also be used to, for example, rank order acid strengths; if X^- does not abstract a proton from H-Y to form H-X + Y^-, then HX must be a stronger acid than HY.

Another technique for generating anions is to collide the parent neutral M with a highly excited (often Rydberg) atom or molecule R**. A key to the success of such an approach is to find an excited state R** for which the energy required to remove its outermost electron matches the electron affinity of M, so that the process

M + R** M^- + R⁺

is thermo-neutral (or nearly so), in which case we say the electron transfer event is in resonance. Such resonance electron transfer collisions have especially high cross-sections (both because the Rydberg orbitals usually employed are spatially large and because of their energy resonance) and thus offer a good means for generating significant amounts of the desired anion. The groups of Professors Jean Pierre Schermann and Charles Desfrançois and Bob Compton have made [[28]] much use of this technique for creating a variety of anions including dipole bound anions (we discuss them in Chapter 4) by colliding a Rydberg atom with a highly polar molecule. In fact, the former workers have even been able to determine [[29]] the electron binding energy of the dipole-bound state thus formed by measuring the dependence of the cross-section for anion formation upon the principal quantum number of the Rydberg atom used to effect the electron transfer.

To form certain anions, one can use so-called laser ablation techniques. Here, one impinges a laser, whose photon energy hn and intensity can be controlled, onto a sample (usually a solid) of the material to be ablated. The ablation process causes fragments of the material to enter the gas phase with some of these fragments also undergoing ionization to form anions and cations. For example, a piece of solid aluminum subjected to laser ablation can generate Al, Al^-, Al⁺, and various Al_n^-, Al_n, and Al_n⁺ cluster species. The size- distribution of the fragments will depend on the laser characteristics (fluence and energy), which are usually tuned to optimize production of the most desired species. In any event, the output of such a laser ablation ion source contains neutrals, cations, and anions of various cluster sizes. Because the anions are charged, mass spectrometric methods can then be used to select the species of the desired charge-to-mass ratio and to guide the anions into a reaction or spectroscopic-observation region.

There are several other approaches that one can use to form gas-phase samples of molecular anions. Because the intention here is to offer a brief introduction to some of the difficulties that arise in experimental studies of anions rather than to review all possible means of forming anions, we will not go further into this subject now. Instead, let us turn to focus on other aspects of the experimental studies.

When one is faced with forming multiply charged anions, special challenges arise, and another type of ion source is often used to overcome these difficulties. The so-called spray techniques are often used to form gas-phase samples of multiply charged anions (n.b., these sources can also be used to form singly-charged anions). Professor Lai-Sheng Wang‘s group has many papers in which these techniques are described in detail. In these methods, one typically begins with a liquid-phase sample containing the desired anion (usually existing in a strongly-solvated and hence highly stabilized state). One then injects a burst of the liquid sample into the gas-phase within the source region of, for example, a mass-selection device that we will discuss in the following Section. Injection is effected by using one of several spray techniques (e.g., electrospray, thermospray, etc.). As a result, one forms a gaseous sample containing

1. solvent molecules S and perhaps solvent ions;

2. the anion or multiply charged anion of interest M^-n;

3. the M^-n species clustered with various numbers of solvent molecules MS_K^-n.

4. other ions and solvated ions.

Many of the ion-solvent clusters formed in the initial spray event subsequently eject one or more solvent molecules losing mass and undergoing cooling in the process. This solvent evaporation process assists in producing internally (i.e., vibrationally and electronically) cold ion samples that can be mass-selected and subjected to subsequent reactions or spectroscopic examination.

As is the case with most techniques used to create molecular anions, the initial source preparation usually produces a complex mixture of ions that must be identified and selected to choose the particular ion whose behavior is to be examined. A mass spectrum of a sample derived from NH₃ and H₂ is shown in Fig. 1.5.

Figure 1.5. Intensity (vertical axis) of anions having various masses (horizontal axis) produced in a mixture of H₂ and NH₃ illustrating how various anions are usually identified and mass selected [258].

Clearly, there are many different negative ions in the gaseous sample whose mass spectrum is shown. If, for example, one were interested in studying H^-(NH₃) in a subsequent spectroscopic or reaction event, one must subsequently subject this sample to a mass-selection process to extract and control the desired anion.

For those readers who wish more up-to-date overviews of how molecular anions are formed in laboratory settings, there is a very recent review of electron affinities by Professors Barney Ellison and Fritz Schaefer [15]. Professor Ellison is one of the leading experimental figures in this rapidly expanding field; his contribution to that review provides the reader with much insight into how experiments on anions are carried out. That review also offers a wonderful avenue to much of the earlier experimental and theoretical studies of atomic and molecular electron affinities.

B. Selecting Specific Anions

Once an anion has been formed, it can be selectively removed from the source chamber using mass spectrometric, including ion cyclotron resonance (ICR), tools which rely on bending the trajectories of the various ions into arcs whose radii depend on the ion’s charge-to-mass ratio. Let us discuss some of the basic physics involved to illustrate.

An ion of charge q and mass m moving with velocity v interacting with a magnetic field B experiences a Lorentz force directed perpendicular to the ion’s velocity and perpendicular to the magnetic field

F_L = q v x B.

This force has no component along the magnetic field direction (z), so the ion’s motion along z is unperturbed. Within the x, y plane, any radial component of the ion’s velocity experiences a torque and any angular component of the ion’s velocity experiences a radial force. The outward-directed (i.e., radial in the x, y plane) centrifugal force F_C generated by having the trajectory bent is

F_C = m v²/r

where r is the instantaneous radius of curvature of the trajectory and v is the magnitude of the ion’s velocity in the x, y plane. When the radial Lorentz and centrifugal forces come into balance, a stable circular orbit of radius

r_stable = m v/(qB)

is formed. Once such a stable orbit is formed, the ions will move along the direction z of the magnetic field with unchanged speed v_Z and will undergo periodic circular motions in the x,y plane perpendicular to B with a speed v that is unchanged. The force no longer change the speed (v = |v|) of the ion because the work

W = Fdr

done on the ion by this force is zero because F is always perpendicular to the trajectory’s (angular) motion and hence no energy is imparted to the ion.

This shows that ions having mass-to-charge ratios m/q will, in the plane perpendicular to B, evolve into circular trajectories of different radii; the frequency n with which ions of a given q/m ratio move around these circular orbits is given by

n = (2p r_stable/v) = (2p/B) (q/m).

This analysis suggests that, if one has a mixture of ions having different q/m ratios and a distribution of velocities (both v in the x,y plane and v_Z along B), the ions will move unperturbed along the direction of B (i.e., with whatever speeds v_z they initially possessed) but will be distributed in a series of circular orbits about the magnetic field direction. Although all ions of a given q/m ratio will have identical circular orbit frequencies, the radius of each ion’s orbit will depend on the speed v in the x, y plane with which the ion began its motion (n.b., as mentioned above, this speed is conserved). If one wants to separate such a mixture of ions according to their q/m ratios, one could achieve spatial (i.e., radial) separation if one could force all of the ions to have the same speed.

To illustrate how one might achieve velocity selection, consider (it is illustrated for positive ions but the analysis also holds for anions) the crossed magnetic B and electric E field setup shown in Fig. 1.6.

Figure 1.6. Illustration of a velocity-selection experimental setup.

Ions entering the magnetic and electric field region and having the dominant component v of their velocities lying along the direction perpendicular to both B and E will experience two forces of magnitude:

F_B = q v B

and

F_E = q E.

These two forces will oppose one another (in the direction of the electric field) and will thus deflect the ions’ trajectories except for those ions whose velocities v happens to match

v= E/B.

These ions will not be deflected and thus will pass through this so-called velocity (or Wien) filter. By adjusting the E/B field ratio, one can then tune the ions’ speeds if needed and one can guarantee that only ions of the same speed exit the filter.

The above analysis shows how one can bend trajectories of ions and make them undergo periodic orbiting motions whose frequencies depend on the q/m ratios and how one can velocity select ions. Now, let us explain the basics of how many mass spectrometers function. Most instruments, after ion formation, first subject all ions exiting the source to an accelerating electric field through which the ions undergo a potential change V. This causes them to gain kinetic energy by an amount

½ m v² = q V.

If the accelerating potential is high enough, this kinetic energy will vastly outweigh any (e.g., thermal) kinetic energy the ions may have had prior to being accelerated. So, it is safe to assume that v given above is the total speed of the ions of mass m and charge q after they have been subjected to this acceleration. Solving for the speed v in terms of the potential V and substituting into the expression for the stable periodic orbit of radius r_sstable, we obtain

m/q = B² r_stable² /(2V).

Thus, if one accelerates all ions in a sample through an electric field of potential V and then subjects them to a magnetic field B, the different (i.e., having different m/q) ions’ trajectories will be bent into orbits of different radii. So, if one has a way to sample the ions that are moving in the magnetic sector at a radius r_stable, one will sample ions of a fixed m/q ratio.

Now, consider what happens if one has an instrument that passes a sample of ions through an accelerating electric potential V into a magnetic field region that has entry and exit slits that happen to be connected by a circular path of radius r₀ as sown in Fig. 1.7.

Figure 1.7 Schematic diagram showing ions of different m/q ratios bending with different curved trajectory with ions having radius r₀ being selected to enter the detection chamber.

Then only those ions having m/q values given by

m/q = B² r₀²/(2V)

will strike the exit slit and thus exit the magnetic sector to be detected in the next region of the instrument. However, by scanning the magnetic field strength B, one can cause ions of various m/q ratios to strike the exit slit and to thus be subject to detection. If one were to use a velocity filter containing electric and magnetic fields of strengths E_S and B_S, respectively, prior to injecting the ions into the mass-selection magnetic sector (of filed strength B), an ion having a given m/q ratio would be bent into an arc of radius

r = m v/(qB)

and its velocity would be given by

v = E_S/B_S.

So, the ions would move as shown in Fig. 1.8, and a detector placed on the outside of a slit at a distance r* could be used to detect ions of a selected m/q ratio by scanning the magnetic sector’s field strength B until those ion’s r value matched r*.

Figure 1.8. Schematic of mass spectrometer set up with velocity selection prior to injection into the magnetic sector.

Another example of a mass spectrometric ion-selection and detection apparatus that performs such tasks is shown in schematic form in Fig. 1.9. Such devices usually have several components including:

1. A source region (on the left) in which the anions are formed.

2. A region proximal to the source where electric fields are used to separate neutrals, positive ions, and anions and to accelerate and focus the anions (to the right in the Fig. 1.9);

3. In the region where the anions are accelerated from left to right, a series of so-called electrostatic lenses are used to focus the anion beam onto a slit or pinhole opening in the next sector of the instrument.

4. A magnetic sector within which the an electric field and a perpendicular magnetic field act to bend the anions’ trajectories into circular arcs whose radii depend on their charge-to-mass ratio and to thus mass-select the ions;

5. A region, subsequent to the mass-selection sector, where the anions whose radii of motion cause them to strike the entrance slit or hole of this region are collected (other anions strike the walls and are thus eliminated).

Figure 1.9. Schematic drawing of a mass-selection and detection device.

In the latter region, the mass-selected anion beam can, for example, be crossed with a photon beam to carry out photoelectron spectroscopy experiments. Alternatively, the beam can impact another beam (containing neutrals or other ions), a chamber containing a gaseous sample of other species, or a surface on which other species reside. Such impacts may then result in reactions whose products may be monitored using photon absorption, fluorescence, or mass spectroscopic techniques. More about these spectroscopic and reaction probes will be said in Sections IV and V of this Chapter.

An example of data obtained in a photoelectron spectroscopy experiment carried out on mass-selected ions is offered in Fig. 1.10 where the mass selection has allowed the workers to focus on the copper dimer anion.

Figure 1.10. Photoelectron spectrum of Cu₂^- in which the number of electrons ejected as a function of the kinetic energy of the ejected electrons is plotted.

In such photoelectron experiments, a fixed-frequency light source shines on the mass-selected anion sample and the number of ejected electrons per unit time is monitored as a function of the kinetic energy of the ejected electrons. As Fig. 1.11 illustrates, the spacings between the peaks in Fig. 1.10 relate to the vibrational spacings of the neutral molecule produced when the electron is detached. Also, the photon energy hn minus the kinetic energy of the electrons ejected in the v=0 v=0 peak (if one can properly identify this) gives the adiabatic electron binding energy:

BE = hn - KE(0,0).

Figure 1.11. Anion (lower) and neutral (upper) potential surfaces with transition induced by absorbed photon and kinetic energies of ejected electrons.

In addition, if hot bands (i.e., transitions originating from excited vibrational levels of the anion) are observed, their energy spacings can be used to determine the vibrational level spacings of the anion.

If the neutral molecule accessed by detaching an electron from a molecular anion has low-lying electronic states, it is also possible to also observe peaks (at lower kinetic energy of the ejected electron or, alternatively, higher binding energy) corresponding to such excited states. An example of such a case is the iron dimer anion Fe₂^- for which there are two low-energy electronic states of Fe₂ spaced by ca. 0.6 eV; the spectrum of this anion is shown in Fig. 1.12.

Figure 1.12. Photoelectron spectrum of Fe₂^- showing two sets of vibrational progressions, one for each of two electronic states.

The research group of Professor Carl Lineberger has, for many years, carried out such photoelectron experiments on atomic and molecular anions from which they have extracted many of the most up-to-date EA data on atoms, molecules and radicals, as well as vibrational level-spacing data on neutrals, radicals, and anions. The Lineberger group has also pioneered many of the experimental techniques used to form, select, and spectroscopically probe such anions. The group of Professor Kit Bowen at Johns Hopkins University has also carried out a large number of spectroscopic measurements on molecular anions to obtain the kind of information discussed above.

Another technique that relies on electric and magnetic fields to select and study ions according to their m/q ratios involves the ion cyclotron resonance (ICR) cell, which is illustrated in Fig. 1.13.

Figure 1.13. Schematic drawing of an ion cyclotron resonance cell.

A strong magnetic field B along the z-direction causes the ions to undergo periodic circular motions within the x, y plane as discussed earlier. The radius of such motion

r = (m/q) (v/B)

depends on the ion’s m/q ratio and its speed v as well as the magnetic field strength B. For an ion with m/q near 100 Daltons moving at room-temperature thermal speeds and a magnetic field of 7 Tesla, the radius is ca. 4 x10^-2 mm. The frequency of this orbiting motion

n = 2 p B (q/m)

will be ca. 1 MHz under these conditions and thus be in the radio frequency (RF) range. In addition to the magnetic field along the z- axis, the ICR cell has two so-called trapping plates located at z = -L and z = L (z = 0 corresponding to the center of the cell) between which an electrostatic potential with spatial dependence of the form

V = ½ V_T + ½ k z²

is applied. This potential exerts a force

F_Z = - k q z

on the ions along the z-direction that acts to constrain the ions near the center of the cell (z = 0). In fact, this trapping electrostatic potential causes the ions to undergo harmonic motion along the z-axis of frequency

n_z = 2p(kq/m)^1/2

that depends on the m/q ratio of the ions.

So, in an ICR cell, the ions undergo periodic motions in the x, y plane of frequency 2pB (q/m) and along the z-axis of frequency 2p(kq/m)^1/2. In Fig. 1.13 we also see two faces of the cell that are called excitation plates. If an RF field with a frequency matching the cyclotron frequency 2pB (q/m) of a group of ions were applied to these plates, energy would flow from this RF field and cause these ions to gain angular kinetic energy and to move into circular orbits of larger and larger radius. The coherence of this RF field would also cause the ions in resonance with it to move together coherently; prior to application of this field, all these ions moved with the same frequency 2pB(q/m) but their angular movements were not coherently coordinated. This group of ions moving coherently together will then induce a time-dependent image current in the detector plates shown in Fig. 1.13. If this current is measured and digitized, the signal can be Fourier transformed and, not surprisingly, will produce a frequency spectrum with one component n= 2pB (q/m).

If, instead of applying an RF field of one chosen frequency, one applied a broad-band RF pulse, one could resonantly and coherently excite the cyclotron motions of all (or at least for a wide distribution of q/m values) of the ions in the ICR cell. The motions of these ions would, in turn, generate a time-dependent image current in the detector plates. Upon digitizing this time-dependent current and Fourier transforming the resulting signal, one obtains a frequency spectrum with peaks at each of the 2pB(q/m) frequencies belonging to each group of ions in the cell. The intensity of each peak is proportional to the concentration of ions having the corresponding q/m values in the cell. In this manner, the ICR experiment can identify a wide range of q/m values using a single RF excitation pulse strategy in contrast to scanning the excitation frequency.

Before closing this Section dealing with mass (actually q/m) selection, it is important to mention a selection device that does not use magnetic fields at all. A so-called time-of-flight (TOF) mass spectrometer accelerates a mixture of ions through a potential drop V using an electric field. After exiting this acceleration stage, an ion will have a kinetic energy

½ m v² = q V,

so its speed along the direction of the electric field (z) will be

v = (2 V q/m)^1/2 .

These ions are allowed to undergo undisturbed (by collisions or fields) movement along the z-direction for a distance D at which position they are detected. The time t it takes an ion to reach the detection position is

t = D/v = D (m/q)^1/2 (1/2V)^1/2.

So, ions with small m/q values will reach the detector before ions with higher m/q values. By determining the times at which various ions reach the detector (so-called arrival times), one can thus determine their m/q values.

Professor John Brauman‘s group has pioneered the use of ICR methods to both separate and trap (i.e., contain for long times) ions of a chosen mass to charge ratio. As we will discuss in Sec. V of this Chapter, chemical reactions can also be carried out within the ICR cell and the appearance of product ions, having different q/m values, can be monitored using the above ICR methods.

By injecting radiation into the ICR chamber that is resonant with ions having q_J/M_J, one causes such ions to be ejected from the chamber. One can thus eject all ions but those whose q/M ratio corresponds to the desired ion. These mass-selected anions then undergo circular motion in the ICR source until collisions or radiation causes them to change trajectory and thus be eliminated. Trapping times in the seconds or minutes range are not uncommon in such experiments. One of the main advantages of an ICR source is the long time that one can trap ions for subsequent study. For example, if one wishes to probe the infrared (IR) absorption or emission of anions, it is useful to have the ions within the spectral regions for long times because IR absorption and emission rates are quite low. The ICR chamber can also be used as a region where photodetachment or chemical reactions of the selected anions occur. That is, as the anions circulate throughout the ICR chamber, they can be subjected to radiation or to collisions with other reagents and the outcomes (i.e., ejected electrons or production of reaction product ions) of such processes can be examined.

It is possible to use the same kind of physics just discussed to bend and accelerate ions not into small circular orbits but into large paths (i.e., several meters in diameter) that constitute a storage ring. The groups of Professors Torkild Andersen and Lars Andersen in Aarhus have used such instruments to create, store, and study spectrocopically a wide variety of molecular anions.

C. How Fields Are Used to Focus and Select Anions

As shown in Fig. 1.9, when ions leave the source region, it is often found useful to arrange for them to be spatially directed before they enter, for example, a velocity selection or mass separation region. This step allows one to cause a larger fraction of the ions produced in the source region to be used in the experiment. It is theefore instructive to discuss how the collimating and collection sectors shown in Fig. 1.9 operate. In the latter, the (usually cylindrical) tube that forms this sector has several rods arranged symmetrically about its outer edge. For example, if one were to view this sector looking down the length of the tube, one would see what is qualitatively depicted in Figure 1.14 where octopole (left) and quadrupole (right) arrangements appear.

Figure 1.14. Time varying positively and negatively charged octopole rods (left) and rods in a quadrupole arrangement (right).

Various numbers of rods can be employed; many rods as in the left of Fig. 1.14 produce a field near the center of the circle that is flat bottomed as that shown in the left of Fig. 1.15, while few rods as in the quadrupole on the right of Fig. 1.14 give a field like that shown in the right.

Figure 1.15. Radial constraining field for multipoles of high and low orders (top) and trajectories of light ions (left), heavy ions (middle), and selected ions (right) under combined RF and DC potentials.

An instrument having 2N rods produces a radial constraining field that varies as 1/R^2N-2, where R is the distance from the center of the circle. So, the octopole arrangement shown in Fig. 1.14 would produce a field varying with R as 1/R⁶ and a quadrupole produces a potential that is quadratic and varies as 1/R² as in Fig. 1.15 on the right.

An anion located in the center of this circle and moving axially down the tube will experience no net force from the positive and negative charges of the rods. However, an anion located away from the center of this circle will experience a net force; it will be attracted to the positive rods and repelled from the negative rods. If the rods retained fixed charges, the anions would eventually strike a positive rod and be removed from the beam. However, when operated in a mode to guide an ion beam but not separate ions by q/m values, the rods’ charges are alternated by application of an external alternating potential (usually in the RF range). If the period of this oscillation is short enough, an anion initially attracted to a positive rod will soon be repelled from this same rod (as it becomes negatively charged) and attracted to the oppositely charged rods. The net result is that an anion will experience a time-averaged field that varies with distance R away from the center of the circle shown in Figure 1.15. This potential acts to trap the ions radially. For a low-order multipole, it also acts to focus the ions toward the center of the cylinder, whereas the flat-bottomed nature of the higher order multipole’s potential does not focus to such an extent but it still guides the ions down the cylinder.

Now, let’s consider what happens if one applies both an AC and a DC field (V_DC + V_AC cos(wt)) to two opposite poles of a quadrupole while applying (-V_DC - V_AC cos(wt)) to the other two poles. The alternating electric field causes the ions move in spiral paths of larger and larger radial size as they pass down the quadrupole’s long axis. The DC voltage acts to drag them in one direction, towards one pair of electrodes. A light ion will be dragged a large distance by the alternating field, and will quickly collide with an electrode and disappear as shown on the bottom left of Fig. 1.15. A heavy ion will not be deflected radially as much by the alternating field, but will be gradually pulled by the DC field as shown in the middle of Fig. 1.15 so it will also collide with an electrode, and be lost. In contrast, an ion that has just the right q/m will drifs slightly due to the DC field, but will be pulled back toward the center of the quadrpole by the AC field as long as the amplitude of the AC field is not large enough to make this ion spiral out of control into an electrode. Thus an ion just the right size is stable in this quadrupole field and reaches the end, where it can be measured. By scanning the magnitudes of the AC and DC fields, one can arrange for ions of a desired q/m value to be stable within the quadrupole filter. In this way, a quadrupole can act as a mass-selection device. In addition, by choosing the strength of the DC field to be stronger than that of the AC field, heavy ions will be pulled out of the center while the lighter ions will be stabilized by the DC field, so one can create a so-called high-mass filter. In reverse, by choosing the AC field to be stronger than the DC field, the light ions will be destabilized and thus ejected while the heavier ions will respond mainly to the DC field and have a better chance of passing down the quadrupole, thus creatng a low-mass filter. Finally, the use of multipole fields can also allow one to confine ions within a three dimensional region of space for long periods of time as in a so-called quadrupole (or Paul) trap. The American Association of Mass Spectrometry has a nice web site that explains how various mass-selection and ion-trapping devices work.

Let us now turn to our discussion of the collimating sector where a series of electrostatic lenses is used to both accelerate the anions (from left to right in Fig. 1.9) and to focus the ion beam toward the center so it can pass through an entrance hole or slit of the magnetic sector. The lenses often consist of a series of circular plates, such as shown in Fig. 1.16, with successive plates held at a higher voltage than the preceding plate.

Figure 1.16. Series of plates constituting an electrostatic lens.

An anion moving from left to right down the z-axis of such circular plates will be accelerated because it experiences an electric field gradient. Such a gradient V(z)/z produces a force F_z = - qV(z)/z along the z-axis (here q is the anion’s charge) that acts to accelerate the anions.

A closer look at the electric fields within and between successive plate regions is shown in Fig. 1.17 for a pair of plates.

Figure 1.17. Electric field contour lines within and between two plates.

Because the force acting on an anion is proportional to the gradient of the electric field, this force is directed perpendicular to the contour lines of the electric field. At regions deep within a plate, the force is directed along the z-axis. However, in the regions between plates and extending somewhat inside each plate, the contours have the curved shapes shown in Fig. 1.17. The resultant field gradients produce forces that cause an anion to be moved inward toward the center of the circular plates. In this manner, the anions’ trajectories are focused along the z-axis as they are accelerated along this direction.

D. Problems That Can Occur

A significant problem that arises in ion beam experiments relates to what is called space charge effects. When an ion beam is collimated and focused, the ions are forced close together and thus repel one another strongly. The Coulomb repulsion among the ions tends to resist the forces applied by external fields designed to collimate and/or focus the beam. As a result, it is very difficult to retain a tightly collimated and intense beam of ions even using the devices discussed above.

Another difficulty arises from the ability of any applied electric field to pull the excess electron(s) off the anion of interest. To understand how this field-detachment process works, consider the radial potential that an excess electron experiences when an external electric field of strength E is applied. In Fig. 1.18, we illustrate the two potentials that such an excess electron experiences- the potential intrinsic in the electron-molecule interaction and the potential due to the electron-field interaction.

Figure 1.18. Long- and short- range potential experienced by excess electron (smooth curve) and potential E r cosq due to applied electric field.

The smooth curve is meant to describe the kind of potentials discussed in detail earlier in Section I of this Chapter (see Fig. 1.1), while the straight line describes the r-dependence of the charge-field potential - E r cosq due to the external electric field of strength E (q being the angle between the field direction and the spatial location vector r of the electron). Also shown in Fig. 1.18 is the energy of the bound state of the anion in the absence of the external electric field.

Of course, the excess electron moves under the influence of a total potential that is a sum of the charge-field potential and the potential operative in the absence of the field. In Fig. 1.19, this total potential is depicted (for two values of the field strength E and for q such that cosq is positive).

Fig. 1.19. Total radial potential experienced by excess electron in an anion in the presence of an external electric field.

The important thing to notice in Fig. 1.19 is that the total effective potential has a barrier beyond which the potential decreases as r increases. The form of this potential allows the attached electron to tunnel through it and thus undergo detachment. Of course, the lifetime of such an anion with respect to tunneling will depend on the binding energy and the strength of the applied field. The stronger the field, the lower is the barrier in the potential as shown in Fig. 1.19 and the shorter is the lifetime. In fact, if the field is strong enough, the barrier will occur at the energy of the electronic state and tunneling will not be required for the electron to escape. At this critical field strength, the electron can simply fall off the barrier.

To shed further light on the matter of field-induced detachment let us examine the case in which no (or little) tunneling is required for electron detachment. We begin by assuming that the long-range part of the potential shown above (in the absence of the external field) is of the form

V_long-range = - A r^-n.

Such an expression is consistent with the prototypical dipole, quadrupole, or polarization potentials discussed earlier as well as with the n = 1 Coulomb potential appropriate to neutrals and cations. Adding to this long-range attractive potential the electron-field interaction potential - E r cos q, we obtain the following total potential at large-r:

V_total = - A r^-n - E r cosq.

Taking the derivative of this with respect to r and setting the derivative equal to zero (to determine the location and the energy of the barrier), we find:

r_barrier = (nA/Ecosq)^1/(n+1).

At this value of r, the total potential is

V_total = - A[Ecosq/nA]^n/(n+1) – Ecosq [nA/Ecosq]^1/(n+1)

= - [nA/Ecosq]^1/(n+1) Ecosq {1 + 1/n}.

Along the direction where cosq is largest (i.e., along which the field effect is strongest), the value of V_total at the barrier reduces to - [nA/E]^1/(n+1) E {1 + 1/n}. For example, in the case of dipole binding (n = 2) V_total = - 3/2 (2A)^1/3 E^2/3. In comparison, for states of neutrals or cations for which the longest-range potential is the Coulomb potential (n = 1), V_total = -2A^1/2 E^1/2. The point of this analysis is to show that the barrier in the total potential lies below zero (i.e., below the detachment threshold) by an amount that varies as the 2/3 power of the applied electric field for dipole binding but as the 1/2 power of the electric field for the Coulomb potential, which relates to neutrals and cations, not anions. So, again, we see a qualitative difference in the behavior of anions and other species.

As noted above, as the applied electric field in increased, the barrier eventually reaches a level at which the bound anionic state (e.g., the level of the horizontal line in Fig. 1.19) becomes unstable. At such field strengths, the quasi-bound level no longer requires tunneling to effect electron detachment; the electron can simply detach by falling over the barrier. So, electric field detachment can cause problems if the applied field is strong enough to cause the anion to lose its electron(s) before its properties are studied.

Alternatively, the strength of binding of the excess electron(s) in an anion can be studied by subjecting the species to external electric fields of increasing strength, and determining at what value E of the field electron loss becomes very facile (i.e., when the barrier drops below the energy of the quasi-bound state). Indeed, this effect has become a powerful and widely used tool for determining electron binding energies, especially in species with quite small EAs.

Before closing this Section dealing with the problems that can arise when studying anions in the laboratory, we need to offer another note of caution. Even when utmost care is taken to optimize source conditions and carry out mass selection and extraction, one must keep in mind that the final anion sample may not contain only the anions that one has in mind. For example, imagine that one were to prepare a sample containing H^- anions clustered with various numbers of NH₃ molecules. After one extracts the anions from this sample and subjects them to mass selection (being careful to select only ions of mass-to-charge ratio of 18), one expects to have a beam of H^- (NH₃) anions. However, there may be other species in this beam!

For example, if the source contained any oxygen atoms, one may have OD^- ions also present. This problem can be dealt with by realizing that OD^- does not weigh exactly the same as H^-(NH₃) and that OD^- has an electron detachment energy of ca. 1.8 eV, whereas H^-(NH₃) binds its electron by only 1.4 eV. For these reasons, one could either increase the resolution of the mass selection to permit only the desired H^-(NH₃) ions to enter the detection region, or spectroscopically probe for anions that detach below 1.8 eV.

However, another kind of problem is more difficult to handle- that of structural isomers. For the H^-NH₃ example at hand, one may have an appreciable amount of the double-Rydberg anion (we discuss such anions in detail later) NH₄^- that is an isomer of H^-(NH₃). NH₄^‑ consists of an NH₄⁺ cation to which a pair of electrons is bound in a diffuse Rydberg orbital such as shown in Fig. 1.20. All molecular cations have such Rydberg orbitals because they possess a positive charge whose Coulomb potential can attract and bind at least one electron.

Figure 1.20. Rydberg orbitals of NH₄⁺, H₃O⁺, and H₃C-NH₃⁺

One way to think of the NH₄^- anion is in terms of its isoelectronic analog, the Na^- anion. If one thinks of taking the +11 nucleus of Na and splitting it into five parts- a +7 nucleus at the origin and four +1 nuclei placed tetrahedrally about the origin, one forms the nuclear geometry of NH₄^-. By then allowing the eleven electrons of Na^- to move not in the spherical potential of a single +11 nucleus but in the presence of the five positive nuclei, one forms NH₄^-. So, Rydberg species can be viewed in terms of their united-atom isoelectronic analogs. It turns out that all molecular cations have Rydberg orbitals that can be thought of as arising from the attractive long-range Coulomb potential -e²/r combined with valence-range repulsive potentials from the Coulomb and exchange interactions with the cation’s other electrons (e.g., the N 1s and four N-H bond pairs in the NH₄⁺ example). It is these inner-shell electrons that cause the energy level patterns of Rydberg states to fit the –R/(n-d)² pattern with a non-zero quantum defect d. Placing one electron into a Rydberg orbital of a cation generates a neutral Rydberg species such as NH₄ whose electronic states can indeed be fit to such a formula. Placing two electrons into such Rydberg orbitals generates what is called a double-Rydberg anion such as NH₄^- about which I will have more to say in Chapter 4. Let us return now to discuss how such geometrical isomers can plague experiments.

If one were to assume that only H^-(NH₃) were present and, for example, carry out endothermic reactions of the mass-selected anion beam with reagents X for which formation of HX^- + NH₃ from X + H^-NH₃ were say 1.1 eV endothermic, one would obtain surprising results if an appreciable amount of the NH₄^- isomer were present. Because the double-Rydberg isomer lies 0.8 eV above H^-(NH₃) on this anion’s ground-state energy surface, its reaction with X to give HX^- + NH₃ is exothermic by only 1.1-0.8 = 0.3 eV. As a result, when the anion beam is accelerated to a kinetic energy of 0.3 eV and allowed to undergo collisions with X, formation of HX^- product ions would be observed. If one were to interpret this threshold for HX^- production in terms of the DE for the reaction

H^-(NH₃) + X HX^- + NH₃,

one would be incorrect. This threshold really related to the DE of the following reaction:

NH₄^- + X HX^- + NH₃.

So, in addition to using mass selection, it may help to also carry out, for example, photoelectron spectroscopic probes of the reactant-ion sample to determine whether more than one isomer is present. In fact, it was precisely through such a careful examination of the photoelectron spectrum of H^-(NH₃) that Professor Kit Bowen discovered the double-Rydberg species (NH₄)^- that we have been discussing. The Bowen group has been active for many years studying such anion clusters as well as more exotic species (e.g., dipole-bound anions) discussed in Chapter 4 and biological-molecule anions treated in Chapter 7.

Finally, I want to mention one more difficulty that can occur in the kind of experiments outlined above. Again using the NH₄^‑ example, if one mass selects on M/q = 18, one may have some [(NH₄)(NH₄)]^2- dianions present in addition to the desired H^-(NH₃). So, species that are dimers or higher oligomers of the species one is attempting to select can also be present in the mass-selected beam. All of the complications discussed above present serious challenges to the experimental chemist and, at times, make interpretation of experimental data ambiguous or, at least, very challenging.

The main point of the above examples is to illustrate that mass selection alone does not guarantee one has only one kind of anion in the beam that is thereby produced. One must always be aware of the possibilities of species with nearly identical masses and isomers and dimers (and other oligomers) of the species one wishes to study.

This concludes this overview of how anions can be made and subsequently selected according to their mass-to-charge ratio. Such mass-selected ion samples can then be subjected to, for example, spectroscopic probes or collisions with other species with which they might react. In the latter case, other product ions may be formed, so this reaction region would have to be equipped with an ion extraction and second-stage m/q detection region to probe the identities and abundances of these product ions.

E. Anions Experience Strong Environmental Effects

When an anion is surrounded by other molecules as in solution, in a cluster, at an interface, or in a solid matrix, it experiences very strong intermolecular potentials. Moreover, because anions’ valence electron densities are typically more diffuse and less tightly bound than those of cations, the anions’ outmost orbitals can interact more strongly with surrounding molecules. This does not mean that the solvation energies of anions exceed those of cations but that the valence orbitals of anions can be more strongly affected (e.g., their electron binding energies can be altered by a larger percent) by solvation.

To illustrate, a typical energy with which a single H₂O or NH₃ molecule is bound to a small anion such as F^-, Cl^-, or OH^- is in the 20-30 kcal mol^-1 range as are the corresponding ion-molecule interaction energies for K⁺ and Li⁺ (19 and 32 kcal mol^-1, respectively). In H₂O or NH₃, the total solvation energies of each of these ions is larger than the energies [[30]] just quoted because solvation involves many ion-solvent and solvent-solvent interactions. In contrast to the strength of the ion-solvent interaction energies, the energy of attraction between a pair of H₂O or of NH₃ molecules is in the 3-7 kcal mol^-1 range, and these hydrogen bonding solvents contain among the stronger attractive potentials that act between pairs of neutrals.

One of the most important influences of the large solvation energies that anions experience occurs in the electron binding energies of such solvated anions. Because the anion M^- is more strongly than its parent neutral M, the M^- M energy gap is significantly larger for the solvated species than for the gas-phase counterparts. As a result, the photoelectron spectra of solvated anions have their peaks blue-shifted (i.e., moved to higher detachment energies) compared to their gas-phase counterparts. Of course, analogous but smaller spectral shifts are observed when M^- is partially solvated (e.g., as in M^-(H₂O)_n cluster ions).

Differential solvation effects can also affect reaction rates and the energy profiles along reaction paths. For example, in the widely studied S_N2 reactions such as Cl^- + H₃C-Br Cl-CH₃ + Br^-, the energy of the transition state [Cl—CH₃—Br]^- relative to that of the reactants and of the products is not the same in solution as in the gas phase. Hence, the activation energy and the forward and backward reaction rates are affected by solvation. Professor John Brauman’s group has done more than any other to elucidate such solvation effects on reaction energy profiles for a wide range of organic reactions. The physical origin of solvent effects on such reaction profiles lies in the different solvation energies of the Cl^- and Br^- anions as well as of the transition-state anion. In particular, because the negative charge is delocalized over both halogen sites in the transition state, the solvation energy of this species is smaller than are the solvation energies of the more charge-localized Cl^- and Br^- ions. In Fig. 1.21 we illustrate these effects.

Figure 1.21. Energy profile for typical S_N2 reaction in the gas phase (black) and in a strong solvent (red).

Because of the very large differential solvation of the charge-localized reactant and product ions, the energy of the transition state changes from lying below that of the reactants (i.e., in the gas-phase) to lying above the reactants (i.e. in solution). Hence, solvation can even qualitatively alter an ionic reaction’s energy landscape.

The Brauman group has used ion-cyclotron resonance techniques coupled with gas-phase photoelectron spectroscopy to probe such reaction-path energy profiles for a very wide variety of important organic ionic reactions. By comparing their results to what was known earlier about the same reactions in solution, they have been able to clearly identify the roles of intrinsic (i.e., in the absence of solvation) electronic effects and those of solvation. Such studies have essentially revolutionized how we view many organic reaction mechanisms.

The substantial differential stabilization of the anion relative to the corresponding species with one fewer electron can even cause ions that are unstable with respect to spontaneous electron loss in the gas phase to become stable when solvated. For example, isolated (i.e., gas phase) SO₄^-2 and PO₄^-3 are not electronically stable; they spontaneously eject an electron to produce SO₄^-1 and PO₄^-2, respectively (PO₄^2- even ejects another electron to give PO₄^-). However, when solvated by a few H₂O molecules to form SO₄(H₂O)_n^-2 and PO₄(H₂O)_n^-3, these ubiquitous anions become electronically stable. Professor Lai-Sheng Wang has examined several such metastable anions and has focused proper attention on the role of solvation in stabilizing these species. The Wang group uses electrospray-type sources and so-called magnetic bottle methods to contain the selected ions for long times. They then carry out photoelectron spectroscopy experiments on these anions. Using such methods, they have been successful in studying a wide range of multiply charged anions and cluster anions including many metastable anions.

When anions are stabilized by surrounding solvent molecules, not only do their electron binding energies increase, but the radial extent of the outermost orbitals containing the excess electron(s) is also reduced. Of course, the two effects- an increase in binding energy and shrinkage in orbital size- go hand-in-hand, as they do for neutrals and cations. In fact, the functional form of the exponential decay that governs the radial extent of any orbital is related to the electron binding energy of that orbital as

exp(-r(2mDE/h²)^1/2), where the electron detachment energy (DE) is the energy needed to (vertically) remove an electron (with mass m) from that orbital. From this relationship, it is clear that species such as anions with small electron binding energies must have more diffuse (i.e., radially extended) electron distributions. The average value of the radius of such an orbital is given by

<r> = 3h/(2(2mDE)^1/2),

which suggests how the radial size (of the outermost orbitals) depends upon DE.

IV. Spectroscopic Probes

Once the anions have been mass selected and extracted, they can be subjected to various probes. The two most common interrogations involve carrying out some kind of spectroscopy on the ions or allowing the ions to undergo reactive collisions with another gas (or with a solid surface) after which product ions’ identities and abundances are determined. In this Section, we will discuss some spectroscopic studies that can be carried out. However, in Chapters 3-7 many more examples of spectroscopic and chemical reaction studies of molecular anions will be given.

The number density of anions that are available (after formation and mass selection) for spectroscopic study is often low because of space charge effects or because the anions may be difficult to make in significant abundance. Thus straightforward absorption-type spectroscopies in which one monitors the attenuation of the intensity of a photon beam and uses the Beer-Lambert law

log (I₀/I) = e C L

to determine the ion’s concentration C or to monitor the wavelength dependence of its extinction coefficient e are not feasible. There is just too little absorption to measure it accurately. Also, because molecular anions seldom have bound excited electronic states, methods such as laser-induced fluorescence (LIF) cannot be employed.

For these reasons, one must resort to so-called action-based spectroscopic methods to study molecular anions. In these approaches, one measures an outcome of light absorption that can be quantified with high sensitivity and resolution against a near-zero background signal. The simplest example, and one of the most widely used techniques involves photoelectron and photodetachment spectroscopies. In the former, one uses a fixed-frequency (n) light source (these days, most likely a laser) to eject electrons from a molecular anion source and then one measures (as the action) the appearance of emitted electrons whose kinetic energies KE one also measures. To increase the path length over which the anions are irradiated, it is useful to align the light beam coaxially to the ion beam.

For a transition in which an anion M^- (e,v,J) in electronic state e, vibrational level v, and rotational level J ejects an electron to generate a neutral M (e’,v’,J’) in its corresponding states, electrons will be detected with kinetic energies given by

KE = hn - {E[M (e’,v’,J’)] – E[M^- (e,v,J)]}.

If one is able to identify the peak (i.e., grouping of ejected electrons) corresponding to a transition from the ground e, v, J state of the anion into the ground e’,v’,J’ state of the neutral, the adiabatic electron affinity (EA) can be determined from

EA = hn - KE⁰,

where KE⁰ is the kinetic energy of the electrons ejected for this peak. An example of a photoelectron spectrum for the DOO^- anion [[31]] from the laboratories of Professors Veronica Bierbaum, Barney Ellison, and Carl Lineberger is shown in Fig. 1.22 where the intensity of ejected electrons is plotted as a function of the electron binding energy (BE) for each peak determined in terms of the kinetic energy of the electrons ejected in that peak as

BE = hn - KE.

Figure 1.22. Number of photoelectrons as a function of electron binding energy for various vibrational transitions in DOO^- DOO + e^-.

In ref. 31 the peaks labeled a0 through a6 were assigned to transitions in which the neutral HOO radical is formed in its ground X ²A’’ electronic state with zero (a0) through six (a6) quanta of vibrational energy in the O-O stretching mode. Peaks b0 through b4 were determined to correspond to transitions in which HOO is formed in it’s excited A 2A’ electronic state with zero (b0) through four (b4) quanta of vibrational energy in the O-O stretching mode. Thus, the adiabatic EA can be determined from the energy of the a0 peak to be 1.08 eV and the electronic X A excitation energy in HOO is given by the spacing between the b0 and a0 peaks to be 0.87 eV.

As in all spectroscopic work, determining what transition each peak corresponds to is difficult work. Of course, one looks for series of peaks whose spacings seem to fit a progression (i.e., become closer spaced due to anhramonicity as one moves to higher levels). However, if the geometry change accompanying electron ejection is large, there may be very low Franck-Condon factors connecting to the neutral’s lowest vibrational level, so identifying the peak corresponding to transitions to this level. This limitation can make it difficult to evaluate the adiabatic EA. In addition, hot bands (i.e., transitions originating from excited vibrational levels of the parent anion) will produce peaks having low electron bininding energies and may further complicate the peak-assignment challenge.

The workers in ref. 31 used the EA data determined from the photoelectron spectrum to determine the enthalpy of formation of HOO (and DOO) as follows:

1. In a separate reaction dynamics experiment, they determined DG₂₉₈ for the reaction

.

2. Knowing the gas-phase acidity DG^acid of acetylene, they could determine the gas-phase acidity of HOOH from

DG^rxn₂₉₈ = DG^acid (HCCH) - DG^acid (HOOH).

3. The gas-phase acidity of HOOH relates to the reaction

HOO-H H⁺ + HOO^-,

so one can obtain the DH_bond for the homolytic bond cleavage

HOO-H H + HOO

if one knows the IP of the hydrogen atom (which is very accurately known) and the EA of the HOO radical, which they determined in the photoelectron experiment. In this way, they were able to determine the enthalpy of formation of HOO by

DH_f,298(HOO) = DH_bond - DH_f,298 (H) + DH_f,298(HOOH),

using the known enthalpy of formation of HOOH, and obtained DH_f,298(HOO) = 3.2 ± 0.5 kcal mol^-1.

In Fig. 1.23 we show another example of a photoelectron spectrum [[32]] for the H₃COO^- anion also obtained by the Boulder team mentioned above.

Figure 1.23. Photoelectron spectrum (a) of H₃C-OO^- showing several vibrational progressions and (b) a similar plot focused on the region of peaks c1-c6.

By determining that peak a1 corresponds to producing H₃COO radical in its X ²A’’ electronic state with no excess vibrational energy (from the corresponding anion in its lowest electronic and vibrational level), the workers of ref. 32 determined the EA to be 1.16 eV. Be determining that peak c1 corresponds to producing H₃COO in it’s A ²A’ excited electronic state with no excess vibrational energy, they determined the X A energy gap to be 0.91 eV. From these examples, we see how EA data and photoelectron spectroscopy provide data that is useful in determining energies of excited electronic states of radicals (formed by removing an electron from the anion) and in determining a wide range of thermodynamic data on such species.

In photoelectron spectroscopy (of anions and of neutrals), a limiting factor in how accurately electron binding energies can be measured is the ability to measure the kinetic energies of the ejected electrons. Typically, one can not measure electron energies better than ca. 0.03 eV, which, of course, means EA values can not be determined to better than this limit. By using the fact that the photon energy hn can be specified to much higher accuracy than the ejected electrons’ kinetic energies can be determined, one can do better as we now discuss.

In threshold photodetachment spectroscopy one does not use a fixed-frequency laser. Instead one uses a tunable laser. By increasing the energy of the light source until one observes ejected electrons, one can determine the energy gap between the anion and the neutral. At higher photon energies, neutrals in excited (vibration-rotation or electronic) states can be produced, which one quantitates by looking for an increase in electron yield as hn increases (i.e., each such channel opening generates an increase in the electron ejection rate). A very important advantage of this approach is that one does not need to measure the ejected electrons’ kinetic energy; all one needs to do is to detect the electrons that have been ejected. In this way, one can achieve spectral peak resolutions in the range of 0.003 eV, thus increasing the resolution compared to photoelectron spectroscopy by approximately an order of magnitude. An example [[33]] from Professor Dan Neumark‘s lab comparing photoelectron and threshold photodetachment spectra on IHI^- is given in Figs. 1.24 and 1.25. In the former, the photoelectron spectrum is shown, while in the latter, the threshold photodetachment spectrum is shown near the v₃’ = 2 peak of the former.

Figure 1.24. Photoelectron spectrum of IHI^- IHI + e^- producing the neutral IHI in the v₃’ = 0, 2, or 4 level of its I^…H^…I asymmetric stretching mode.

Figure 1.25. Threshold photodetachment spectrum of IHI^- IHI + e^- producing the neutral IHI in the v₃’ = 2 level of its I^…H^…I asymmetric stretching mode.

The spacings among the v₃’ = 0, 2, and 4 peaks in Fig. 1.24 are ca. 1400 cm^-1; this progression corresponds to the asymmetric I^…H^…I vibrational mode in the neutral. The spacings among peaks A, B, and C in Fig. 1.25 are ca 100 cm^-1 and correspond to the symmetric IHI stretching mode. So, from Fig. 1.24 we can do not see a resolved symmetric stretching progression within the v₃’ = 2 band (although there may be signs on the left side of this peak). However, in the higher-resolution photodetachment spectrum of Fig. 1.25, the symmetric stretching progression is clear.

Another kind of action spectrum has proven very useful in studying molecular anions arises when one wishes to probe the vibrational spectra of anions using infrared (IR) light. Again, it is essentially impossible to carry out a straightforward IR absorption (or emission) spectrum on a molecular anion sample; the number density of ions and the weak IR oscillator strength do not allow this. However, if one attaches to the molecular anion of interest M^- a passive species such as one or more noble gas atoms (e.g., Ar_n) and subjects the M^-…Ar_n mass-selected ion to IR radiation, one can succeed. In Fig. 1.26 we show an example [[34]] from Professor Mark Johnson‘s group of an IR spectrum of OH^-(H₂O)_n (n = 1, 2, … 5) obtained by monitoring for the appearance (this is the action) of OH^-(H₂O)_n ions when mass-selected OH^-(H₂O)_nAr ions are irradiated with IR light. The OH^-(H₂O)_n ions are formed when an IR photon is absorbed by an OH stretching mode in OH^-(H₂O)_nAr and the vibrational energy is converted into the motion of the Ar, thus inducing ejection of the Ar atom. Thus, the action signal, the appearance of OH^-(H₂O)_n, is a direct probe of the absorption of IR energy. Because the action signal involves detecting the mass and quantity of ions, it can be measured with high sensitivity and accuracy.

Figure 1.26. Yield of OH^-(H₂O)_n ions from OH^-(H₂O)_nAr as a function of IR photon energy for n = 1 (A) through n = 5 (E).

The features seen in Fig. 1.26 correspond to various vibrations of the OH^-(H₂O)_n complex. For example, F labels a vibrational excitation of a free OH bond (one not involved in hydrogen bonding to the OH^- or to other water molecules) and IHB labels a stretching mode for an OH bond that is hydrogen bonded to the OH^- ion.

V. Reaction Dynamics Probes

As with spectroscopic probes, there are a variety of different reaction experiments that mass-selected anions are commonly subjected to. In guided-ion beam collision induced dissociation (CID) and collisonal reaction experiments, one accelerates a mass-selected and collimated ion beam to a specificed kinetic energy E (in the laboratory frame) and allows these ions to collide either with an inert gas (in CID) or with a reactant gas (in collisional reaction). The inert or reactant gas usually exists in a collision chamber held at some temperature T, so these gas molecules are moving randomly with a Maxwell-Boltzmann distribution of kinetic energies and with a thermal distribution of internal energies. Upon collisions between the guided-ion beam and the gas in the collision chamber, dissociation or chemical reaction can occur. The products of these events are then subjected to mass analysis and quantitation, again using mass spectrometric means. In carrying out such experiments, one must usually vary the concentration (i.e., the pressure) of the gas molecules to make sure that one is observing single-collision events. One usually does this by extrapolating results to the low-pressure limit.

In a collision between an anion having mass M and an inert or reactive gas molecule of mass m, not all of the laboratory-frame kinetic energy E of the ion is available to induce dissociation or reaction. Only the so-called center-of-mass kinetic energy

E_CoM = ½ (mM/(m+M)) v_rel²

is available. Here, v_rel is the relative velocity with which the ion and the gas molecule collide. If the anion beam’s laboratory velocity v_Lab exceeds the average thermal velocities of the gas molecules, then v_rel can be approximated as v_Lab, and the E_CoM can be related to the ion’s laboratory kinetic energy E as

E_CoM = ½ (mM/(m+M)) v_Lab² = (m/(m+M)) E.

This shows that only the fraction m/(m+M) of the laboratory collision energy E is available to induce dissociation or reaction. This fraction can be very low when the ion mass M exceeds that of the collision gas m by a large amount. For this reason, in CID experiments, heavy inert gases such as Xe are often employed.

In Fig. 1.27 we show guided-ion beam measured reaction cross-sections [[35]] for S^- anion abstracting a hydrogen atom from H₂, methane, or ethane taken from Professor Kent Ervin‘s laboratory. In these cases, it is the HS^- product ions that are detected as a function of the S^- ion beam’s kinetic energy E_CoM.

Figure 1.27. Cross-sections for S^- abstracting a hydrogen atom from H₂ (a), CH₄(b) and C₂H₆(c) as functions of the center-of-mass collision energy in eV.

For the reaction with H₂, the threshold was determined to be 59.0 kJ mol^-1, which is very close to the endothermicity (59.4 kJ mol^-1) of the reaction

S^- + H₂ HS^- + H.

This suggests that the hydrogen abstraction proceeds with no energy barrier above the endothermicity. In contrast, for reactions with CH₄ and C₂H₆, the thresholds appear at 124 and 107 kJ mol^-1, respectively. However, the corresponding endothermicities are 60 and 44 kJ mol^-1, suggesting that reaction barriers do exist for these reactions. In addition, the magnitudes of the latter two reactions’ cross-sections are much smaller than for the H₂ reaction; this is consistent with barriers appearing in the latter but not in the former. Indeed, as shown in Fig. 1.28, barriers are observed in the latter two reactions’ potential energy profiles when computed using the methods detailed in Chapter 2. An analogous examination of the energy profile for the reaction with H₂ shows no barrier above the reaction endothermicity.

Figure 1.28. Energy as a function of reaction progress along intrinsic reaction paths for S^- abstracting a hydrogen atom from methane (top) and ethane (bottom).

To determine the threshold E₀ for dissociation or reaction, a model [[36]] for how the cross-section s depends upon collision energy E_CoM is employed:

.

Here s₀ is a parameter related to the maximum of the cross-section and E_i is the energy of the i^th internal (vibration-rotation-electronic) state of the reactants whose population is g_i. The variable n controls the shape of the energy dependence of s. The time variable t is the average experimental time available for the dissociation or reaction to occur (i.e., how long the ion-molecule collision complex remains in the instrument subject to detection), and k(e+E_i) is the unimolecular rate constant for dissociation of the ion-molecule collision complex having energy equal to E_i +e. If the residence time t is long enough to assure that all collision complexes that are going to dissociate or react have time to do so, the above expression reduces to

.

Clearly, the latter expression shows that s will vanish whenever the available collision energy plus the internal energy of the reactants falls below the threshold E₀. So, if one has reasonable knowledge about the populations g_i of the reactants’ internal states, one can fit this expression to the experimentally observed energy dependence of s to extract E₀. However, whenever (e.g, for large molecules) the ion-molecule collision complex survives without fragmenting longer than the experimental residence time t, one cannot use the simplified cross-section expression shown above. In such cases, although the total amount of energy contained within the collision complex may exceed the reaction threshold, it simply takes too long for this energy to end up in the critical reaction coordinate that permits fragmentation to occur.

In Fig. 1.29 we show another example [[37]] of guided-ion beam data from Kent Ervin’s laboratory. In this case, an ion complex consisting of HS^-…HCN is subjected to collisions with Xe gas and the number and masses of various product ions are determined. Three primary reactions are observed: collision induced dissociation (CID) of the complex to produce HS^- (and HCN), proton transfer to yield CN^- (and H₂S), and formation of HS^-…Xe (and HCN). In Fig. 1.30, we see the reaction energy profile computed in ref. 37.

Figure 1.29. Guided-ion beam cross-sections for dissocitation of HS^-…HCN complex by Xe atoms as functions of collision energy (eV)

Figure 1.30. Reaction energy profiles for HS^-…HCN (solid line) and HS^-…(HNC) (dashed line) to yield HS^- + HCN (right) and H₂S + CN^- (left) vs. reaction coordinate.

An interesting feature of the data shown in Fig. 1.29 is that, while the thresholds for production of HS^- + HCN and H₂S + CN^- are nearly identical (ca. 0.4 eV), the maximum magnitudes of the corresponding cross-sections differ by approximately a factor of ten. The reaction energy profiles for forming these two products seen in Fig. 1.30 suggest that the thresholds for forming these two products should indeed be similar. By using the kind of electronic structure tools discussed in Chapter 2, the workers of ref. 37 we able to determine the energies, geometries, and vibrational frequencies of the transition states connecting HS^-…HCN to HS^- + HCN and to H₂S + CN^-. They found that the transition state leading to H₂S + CN^- is very tight (i.e., a compact structure with high-frequency inter-fragment vibrations) while that leading to HS^- + HCN is loose. These structural differences in the transition states caused rates to differ by an order of magnitude (i.e., because the transition state’s partition function for forming HS^- + HCN is larger than that for forming H₂S + CN^-).

Another instrumental setup that has produced a wealth of data on reactions of molecular anions is referred to as a flow tube. An example of this instrument (used for positive-ion studies in this case) is shown schematically in Fig. 1.31.

Figure 1.31. Schematic of a flow tube instrument.

In this kind of experiment, ions are typically created (at the left in Fig. 1.31) using an electric discharge within a flowing gas containing precursors of the anions of interest and a carrier gas (He in Fig. 1.31). This discharge creates cations, radicals, and anions, many in excited electronic states that often emit an observable visible glow, so such setups often are called flowing afterglow instruments. Subsequent to creating this ion and radical mixture, a mass-selection device such as the quadrupole mass filter shown in Fig. 1.31 can be used to extract ions of a given charge and q/m ratio and to allow these ions to enter the flow tube region. Such a mass selection step within the flow tube instrument is described as using a selected-ion flow tube or SIFT step. After exiting the mass filter, the selected ions flow, under a carrier gas from left to right through the flow tube. While in this tube, the ions can be subjected to collisions with other gases that can be injected using any of the gas inlet ports as shown in Fig. 1.31.

The anions, carrier gas, and any injected reactant gas flow down the tube at a fixed velocity v (determined by the pressure of the carrier gas). So, the time t it takes for an anion (or reactant or carrier species) to move from the location of an inlet port to the end of the flow tube where it enters the detection region (right of the flow tube in Fig. 1.31) will be related to the distance d separating the inlet port and the detection region by

t = d/v.

So, by injecting reactants at various inlet ports, one can vary d; and by changing the flow rate, one can alter v. Through both of these means, one can alter the time interval t over which the selected anions are in contact with a reactant gas. Of course, the range of flow rates that can be achieved and physical limits on the length of the flow tube place limits on the time t, which, in turn, limits the rates of reactions that can be studied by such flow tube methods.

Assuming that the anions A^- and the reactant gas molecules B undergo bimolecular collisions to generate product ions P^-

A^- + B P^-,

the rate of appearance of P^- ions should be determined by the following kinetic equation

d[P^-]/dt = k [B] [A^-] = - d[A^-]/dt

If, as is usually the case, the concentration of the reactant gas greatly exceeds that of the reactant anions, the concentration [B] will remain essentially unchanged (at [B]₀) throughout the reaction, so this kinetic expression will reduce to a pseudo- first-order form

d[P^-]/dt = k [B]₀ [A^-] = - d[A^-]/dt,

which can be integrated to yield

[A^-](t=t) = [A^-](t=0) exp(-k[B] t), and

[P^-] = [A^-](t=o) (1-exp(-k[B] t)).

Here, t = 0 corresponds to the time the anions pass the inlet port at which reactant gas B is injected, and t = t is the time when this collection of gases (ions, neutrals, and carrier gas) exit the flow tube and enter the detection region. Because t is given by d/v, these solutions to the kinetic equations can be written in terms of exp(-k[B]d/v) instead.

So, one carries out a series of such experiments in which the reactant gas molecules B are injected from inlet ports at varying distances from the detector. By then monitoring the concentration of A^- ions or of product P^- ions arriving at the detector, one can, for example, plot ln[A^-] vs. the distance d to extract the pseudo first-order rate constant k[B]. An example of such a plot is shown in Fig. 1.32 from an experiment in Professor Veronica Bierbaum’s lab [[38]] where the reaction of the isoprene anion H₂C=CH-(CH₂)₂^- with D₂O to exchange one, two or three D atoms for H atoms has been studied. Knowing the flow velocity v and the concentration of D₂O, these workers could also determine the rate constants for these reactions.

Figure 1.32. Percent of allyl anions that have undergone one (d1) through three (d3) deuterium-for-hydrogen atom exchanges as a function of distance along the flow tube.

Another example from this same lab [[39]] in a study of F^- anions reacting with H₃C-OOH produces the data shown in Fig. 1.33 in which the flow tube distance variable d has already been converted to time using the known flow velocity.

Figure 1.32. Concentrations of F^- reactant and OH^-, and CH₃-OO^- product ions as functions of time.

An interesting lesson learned in this study is that the H₃COO^- anion is not formed only by proton abstraction from the O-H unit

F^- + H₃C-OO-H FH + H₃C-OO^-.

Instead, a proton can be lost from the methyl group

F^- + H₃C-OOH ^-H₂C-OOH + FH

after which the ^-H₂C-OOH anion decomposes to produce OH^- and formaldehyde

^-H₂C-OOH OH^- + H₂C=O.

The OH^- can then go on to react with another H₃C-OOH to generate the H₃C-OO^- anion in another route

OH^- + H₃C-OOH H₂O + H₃C-OO^-.

The amount of H₃C-OO^- produced by direct proton abstraction is shown by the dashed line in Fig. 1.32; the total amount is shown in the solid line.

Professor Paul Kebarle carries out a different kind of experiment using mass spectrometric tools. In particular, he studies ion reactions under equilibrium conditions, and, by determining the temperature dependence of various equilibrium constants, he determines reaction DH and DS vallues. An example of one of his studies [[40]] of the (OH)₂PO₂^- anions being sequentially hydrated

(OH)₂PO₂^-(H₂O)_n-1 + H₂O (OH)₂PO₂^-(H₂O)_n

produced the data shown in Fig. 1.33.

Figure 1.33 Observed intensity ratio for (OH)₂PO₂^- ions with one (I₁) and zero (I₀) water molecules attached as a function of water pressure at three temperatures (22, 56, and 88 C).

The equilibrium constant K_0,1 for the equilibrium

(OH)₂PO₂^-(H₂O)_n-1 + H₂O Û (OH)₂PO₂^-(H₂O)_n

is obtained as the slope of the I₁/I₀ plot vs. P(H₂O). When these equilibrium constants, determined at various temperatures are then plotted vs. 1/T to form a van’t Hoff plot, Fig. 1.34 was obtained.

Figure 1.34. Van’t Hoff plot of K_0,1 and K_1,2 vs. 1/T for sequential hydration of (OH)₂PO₂^-

From the slopes and intercepts of these two plots, the Kebarle group obtained DH_0,1 = -14.0, DH_1,2 = - 12.3 kcal mol^-1 and DS_0,1 = -21.0, DS_1,2 = -20.8 cal mol^-1 K^-1. Using this kind of equilibrium measurement, the Kebarle group has been able to determine sequential hydration energies (and energies for binding many other ligands) for a wide variety of anions and cations.

This concludes the discussion of how anions are made, controlled, and studied in the laboratory. I hope you have learned why anions are different from cations and neutrals in ways that relate to the potentials that bind their valence-level electrons. I hope you have also gained some appreciation of how spectroscopic and reaction dynamics probes can be used to generate information about molecular anions.

As I said earlier, because my background lies in theoretical chemistry, I am not able to offer as much insight into the experimental tools used to probe anions as a first-rate experimental chemist who studies them. For this reason, I encourage the reader to go to the web sites of some of the experimental chemists I mention througout this text to learn in more detail how the experiments are carried out and about their limitations and sources of error. Now, let us turn our attention to some of the challenges that molecular anions pose to their theoretical study.

Chapter 2. Anions Also Present Special Challenges to Theoretical Study

As with their experimental study, the theoretical investigation of molecular anions is fraught with difficulties, many of which do not occur for neutrals or cations. Some of the challenges that are, to a large extent, specific to molecular anions include:

1. Molecular electron affinities (EAs) are small (almost always below 4 eV, often less than 1 eV, and, at times, in the 0.01-0.1 eV range). Therefore, theoretical methods capable of high absolute accuracy must often be employed. To achieve high accuracy, one must use a method that treats the dynamical correlations among the electrons’ movements.

2. The small EAs produce radially diffuse electron densities, so special atomic orbital basis sets capable of describing such densities are needed.

3. Some molecular anions bind their excess electron in a Rydberg or dipole-bound orbital rather than in conventional valence-type orbitals. In such cases, basis sets able to describe such orbitals must be used (this often requires constructing such bases “from scratch”).

4. Electron binding energies are often of the same magnitude as vibrational energy quanta. This means that a vibrationally excited molecular anion can be iso-energetic with the corresponding neutral molecule in a lower vibrational level plus an ejected electron. Such degeneracies of anion and neutral energies require one to treat vibration-to-electronic energy coupling and to consider the resulting autodetachment processes.

5. Some molecular anions have negative EAs corresponding to metastable electronic states. The proper theoretical treatment of these states requires one to use special techniques designed to describe both the quasi-bound valence character and the free-electron continuum character of their wave functions.

This Chapter will introduce you to a variety of theoretical tools needed to address the special difficulties noted above.

There are several sources that one can access to read about how the theoretical study of anions has evolved over the past few decades. These include reviews by: Boldyrev and Gutsev [[41]], Baker, Nobes, and Radom [[42]], Jordan [[43]], Simons and Jordan [[44]], Kalcher and Sax [[45]], Kalcher [[46]] and Berry [[47]], as well as classic earlier overviews by Massey [[48]] and Branscomb [[49]], and a book [[50]] edited by Professor Josef Kalcher. Later in this text, other review articles relating to particular families of anions are also cited.

The reader may expect this Chapter will conclude with specific recommendations about the theoretical tools that are optimal for studying anions and computing EAs. Unfortunately, this is not going to be possible although there are certain aspects (e.g., the use of special diffuse basis sets and the need to treat what is called electron correlation) that are common to any good approach for theoretically studying anions. The fact is there are several reliable and accurate theoretical approaches that can be used; some are better in certain circumstances and others are better in others. Moreover, the computational effort and degree of accuracy involved in various methods varies greatly from method to method, with the most accurate approaches almost always having the highest computational demand. In addition, this demand scales in a highly non-linear (e.g., as the third or higher power of the number of electrons) manner with the size of the molecule. It is therefore not always practical to invoke the most accurate calculation, so one is often faced with balancing computational effort against needed accuracy. For such reasons, it is necessary to explain the strengths and weaknesses of several different methods so the reader will understand and thus be optimally positioned to apply the most appropriate methods in cases of her or his interest.

Before discussing many of the theoretical methods available for calculating EAs and studying the structures, energies, reactivities, and spectroscopic behavior of anions, I want to first reiterate the magnitude of the difficulty inherent in these tasks by considering how small a percent of the total electronic energy the EA is. Let us begin with the simplest case and the situation in which the task appears the most straightforward. For the anion containing the fewest electrons H^-, the EA is 0.75 eV whereas the total electronic energy is –14.35 eV (i.e., the sum of the EA and the ionization energy of H). So, for H, the EA is 5.2 % of the total energy. Therefore, if one is able to compute total electronic energies of H and H^‑ accurate to a few percent, computation of this EA to within an accuracy of say 30% appears to be quite feasible.

However, for atoms and molecules containing more electrons, the situation is much worse because the total electronic energy grows rapidly with the number of electrons, whereas the EAs of most atoms and molecules remain in the 0.01- 4 eV range.

So, if one has available a tool that promises to compute electronic energies to say 1%, one soon (i.e., for some reasonable size molecule) finds that the full magnitude of the EA is less than this percent of the total electronic energy.

To further illustrate, let us consider the EA of a carbon atom which has been measured by examining the C^-(⁴S_3/2) C(³P₀) photodetachment threshold to be 1.262119 ± 0.000020 eV. The total electronic energy of the C atom is –1030.080 eV (obtained by adding the C C⁺, C⁺ C²⁺ , C²⁺ C³⁺ , C³⁺ C⁴⁺, C⁴⁺ C⁵⁺ , and C⁵⁺ C⁶⁺ ionization energies). The total energy is a negative quantity because the ground-state energy of C is defined (as is the case for all atoms and molecules) relative to a reference in which the zero of energy corresponds to the bare nuclei and bare electrons all infinitely far from one another and all having no kinetic energy. To compute the EA of C even to an accuracy of 0.1 eV requires either

a. that one compute the total electronic energies of both C and C^- to this accuracy (which is only 9.7 x10^-3 % of the total energy) or

b. one rely on cancellation of systematic errors when these two energies are subtracted to obtain the EA.

In any event, as we discussed earlier in this text, the evaluation of EAs is complicated by the fact that total electronic state energies are extensive quantities but EAs are intensive quantities as was noted earlier in this text.

Not surprisingly, much of the total energy of a carbon atom derives from the energies of the two 1s electrons (as reflected in the fifth and sixth ionization energies being 392.077 and 489.981 eV, respectively), To the extent that the 1s inner-shell electrons are unaffected by adding an electron to C to form C^-, errors made in computing their total energy contributions should cancel when EA is calculated. As most chemists know well, inner-shell orbitals are indeed little affected by changes made to the valence-orbital occupancies. However, the inner-shell orbitals are altered to some extent, even if only to a small degree. Thus, if one is faced with the challenge of computing EAs to high accuracy, one cannot ignore changes in the core orbitals’ energies. However, one often relies on the approximate cancellation of the energies of the inner-shell electrons and computes EAs by focusing much of the computational detail and effort on the dynamics of the valence-level electrons. This approach is usually capable of yielding EAs accurate to ca. 30% if one handles what is called the correlation energy of the valence-level electrons using methods that we detail later.

Another issue to be aware of is the magnitude of the electron-electron Coulomb interaction in comparison with the total energies. This is important because it is precisely these contributions to the total energies that essentially all quantum chemistry tools have difficulty treating to high accuracy, and it is these energies that render the Schrödinger equation not analytically (or even numerically to high precision) soluble for atoms and molecules. Again using the C/C^- example, we note that the difference between the fifth and sixth ionization energies of the C atom is ca. 97 eV. This difference offers an estimate of the Coulomb repulsion between the two 1s electrons. Clearly, because these two electrons reside in close proximity to one another on average, their repulsion is quite large. In contrast, the difference between the first and second ionization energies is ca. 13 eV, which gives an estimate of the Coulomb repulsion between two electrons in two orthogonal 2p orbitals of C (e.g., 2p_x and 2p_y). Even this repulsion energy is large when compared to the accuracy (i.e., ca. 0.1 eV) to which we usually aspire to compute atomic and molecular EAs.

The relevance of these observations about the sizes of Coulomb interactions is that essentially all quantum chemistry methods begin with a so-called mean-field description of these interactions (e.g., as in the Hartree-Fock method discussed later in thi Chapter). That is, the interactions are approximated in terms of a Coulomb integral

J_i,j = |f_i(r)|² e²/|r-r’| |f_j(r’)|² dr dr’,

where |f_i(r)|² and |f_j(r’)|² give the mean-field estimates of the spatial probability densities for finding an electron in orbital f_i at location r and another electron in f_j at location r’, respectively. This estimate of the average Coulomb interaction between the pair of electrons is not fully correct because it ignores correlations in the electrons’ motions. That is, it assumes that the probability density for finding one electron at r does not depend on where the other electron is located. It turns out that such uncorrelated estimates of Coulomb interactions between pairs of electrons are accurate to 5-10 %. However, for the two 2p orbitals of C discussed above, a 5-10 % error in the interaction energy of 13 eV is still too large an error to tolerate if one is attempting to compute the EA of C to within 0.1 eV. For such reasons, one is forced to use theoretical descriptions of the electronic structures of the neutral and anion that more adequately describe the correlated movements of at least the valence-level orbitals (then assuming that systematic errors in the energies of the inner-shell orbitals cancel). Several approaches to treating inter-electron correlations are described later in this Chapter.

Having given insight into the fundamental difficulties behind accurate evaluation of atomic or molecular EAs (e.g., EAs are intensive and small fractions of total energies, and electrons undergo correlated not mean-field motions), let us now move on to discuss the variety of tools that can be applied to this challenging task.

I. Special Atomic Basis Sets Must be Used

Because of the diffuse character of the electron densities of anions, one must employ atomic orbital (AO) basis sets that decay slowly with radial distance r. As noted earlier, because electron binding energies are, by nature, small quantities, one must compute EAs with high absolute accuracy to achieve acceptable percent errors. The latter fact requires that the AO basis set be flexible enough to describe accurately the spatial distributions of the electrons as well as their so-called dynamical correlation (we discuss this later). That is, the quality of the basis influences both our ability to describe the radial and angular shapes of the orbitals that contain the electrons as well as our ability to treat the correlated motions of the electrons.

Let us now briefly review what constitutes the kinds of AO bases that are most commonly used for such studies. The basis orbitals commonly used to express the molecular orbitals (MOs) f_j as linear combinations of AOs c_m via the linear combination of AOs to form MOs (LCAO-MO) process fall into two primary classes:

1. Slater-type orbitals (STOs) have the following angular and radial form

c_n,l,m (r,q,f) = N_n,l,m,z Y_l,m (q,f) r^n-1 exp(-zr)

and are characterized by quantum numbers n, l, and m and exponents (which characterize the radial 'size' ) z. The symbol N_n,l,m,z denotes the normalization constant.

2. Cartesian Gaussian-type orbitals (GTOs) have the following angular and radial form

c_a,b,c (r,q,f) = N'_a,b,c,a x^a y^b z^c exp(-ar²),

and are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents a which govern the radial 'size'.

For both types of AOs, the coordinates r, q, and f refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. It is most common to locate such orbitals on the atomic nuclei, but, at times, additional so-called floating AOs are placed elsewhere. For example, when describing the binding of an electron to the NH₄⁺ cation discussed earlier, it would be appropriate to center diffuse s-type basis functions designed to treat this species’ Rydberg orbital on the nitrogen nucleus. When studying an electron solvated within a cavity formed by four tetrahedrally placed H₂O molecules whose dipoles are oriented inward, one most likely would place floating s- and p- type AOs at the center of the tetrahedral cavity rather than on any of the O or H nuclei of the surrounding water molecules. Of course, if any basis were mathematically complete, it would not matter where the functions were located. However, it is essentially never possible to utilize a complete basis or one that approaches completeness, so one is usually forced to choose where to place the AOs based upon knowledge or intuition about where the attractive potential is likely to accumulate electron density.

The two families of basis AOs mentioned above have their own strengths and weaknesses. Slater-type orbitals are similar to Hydrogenic orbitals in the regions close to the nuclei. Specifically, they have a non-zero slope near the nucleus on which they are located (i.e., d/dr(exp(-zr))_r=0 = -z) and the more radially compact the STO, the larger is this slope. In contrast, GTOs, have zero slope near r=0 because d/dr(exp(-ar²))_r=0 = 0. We say that STOs display a cusp at r = 0 that is characteristic of the Hydrogenic solutions, whereas GTOs do not. This characteristic favors STOs over GTOs because we know that the correct solutions to the Schrödinger equation have such cusps at each nucleus of a molecule (because near a nucleus the –Ze²/r potential is dominant).

Although STOs have the proper 'cusp' behavior near nuclei, they are used primarily for atomic and linear-molecule calculations because the multi-center integrals that arise in polyatomic-molecule calculations (we will discuss them later) cannot efficiently be evaluated when STOs are employed as basis AOs. In contrast, such integrals can routinely be computed when GTOs are used. This fundamental advantage of GTOs has lead to the dominance of these functions in molecular quantum chemistry.

To overcome the primary weakness of GTO functions (i.e., their radial derivatives vanish at the nucleus), it is common to combine two, three, or more GTOs, with combination coefficients that are fixed and not treated as LCAO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs (i.e., functions with large, intermediate, and small exponents a) are multiplied by contraction coefficients and summed to produce a CGTO that approximates the proper 'cusp' at the nuclear center. However, it is not possible to correctly produce a cusp by combining any number of Gaussian functions because every Gaussian has a zero slope at r = 0 as illustrated in Fig. 2.1.

Figure 2.1. Plots of a tight, medium and loose Gaussian radial function showing the zero slope that all Gaussians have at r = 0 as well as the function the contraction of the Gaussians attempts to approximate.

Although most calculations on molecules are now performed using Gaussian orbitals, it should be noted that other basis sets can be used as long as they span enough of the regions of space (radial and angular) where significant electron density resides. In fact, it is possible to use plane wave orbitals [[51]] of the form

c (r,q,f) = N exp[i(k_x r sinq cosf + k_y r sinq sinf + k_z r cosq)],

where N is a normalization constant and k_x , k_y , and k_z are quantum numbers detailing the momenta of the orbital along the x, y, and z Cartesian directions. The advantage to using such simple orbitals is that the integrals one must perform are much easier to handle with such functions; the disadvantage is that one must use many such functions to accurately describe sharply peaked charge distributions of, for example, inner shell core orbitals as well as the slowly-varying diffuse orbitals characteristic of anions’ valence regions.

Much effort has been devoted to developing sets of STO or GTO basis orbitals for main-group elements and the lighter transition metals. This ongoing effort is aimed at providing standard basis set libraries which:

1. Yield predictable chemical accuracy in the resultant energies (for the ground and excited and ionized states).

2. Are computationally cost effective to use in practical calculations.

3. Are relatively transferable so that a given atom's basis is flexible enough to be used for that atom in various bonding environments.

Each such basis set has several components designed to handle various aspects of the electronic structure issue. In the following sub-sections, we briefly describe these components.

A. The Fundamental Core and Valence Basis

Within this category, the following choices are common (we illustrate these choices for Gaussian type orbitals because they are the most commonly used):

1. A minimal basis in which the number of CGTO orbitals is equal to the number of core and valence atomic orbitals in the atom. For example, for a carbon atom, one would use one tight s-type CGTO, one looser s-type CGTO and a set of three (i.e., x, y, and z) looser p-type CGTOs.

2. A double-zeta (DZ) basis in which one uses twice as many CGTOs as there are core and valence atomic orbitals (e.g., two tight s, two looser s, and two sets of three looser p CGTOs for carbon). The use of more basis functions is motivated by a desire to provide additional variational flexibility so the LCAO process can generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies. For example, the 2p molecular orbital in a neutral carbon atom does not have the same radial extent as the lone pair 2p orbital in the H₃C^‑ anion, so one needs to have a basis set that can describe either 2p orbital. In the DZ basis, the neutral carbon’s 2p orbital would have a larger LCAO-MO coefficient multiplying the tighter p-type basis function; the CH₃^- lone pair orbital would have a larger coefficient multiplying the looser basis function.

3. A triple-zeta (TZ) basis in which three times as many CGTOs are used as the number of core and valence atomic orbitals (extensions of this sequence of NZ bases to quadruple-zeta and higher-zeta bases also exist).

Optimization of the orbital exponents (z’s or a's) and the GTO-to-CGTO contraction coefficients for the kind of bases described above has undergone explosive growth in recent years. The theory group at the Pacific Northwest National Labs (PNNL) offer a world wide web site (http://www.emsl.pnl.gov:2080/forms/basisform.html) from which one can find (and even download in a form prepared for input to any of several commonly used electronic structure codes) a wide variety of Gaussian atomic basis sets.

B. Polarization Functions

One usually enhances any core and valence functions by adding a set of so-called polarization functions. These are functions of one higher angular momentum than appears in the atom's valence orbital space (e.g, d-functions for C, N, and O and p-functions for H), but they have exponents (z or a) that cause their radial sizes to be similar to the sizes of the valence orbitals (i.e., the polarization p orbitals of the H atom are similar in size to the 1s orbital and the polarization d orbitals of C are similar in size to the 2s and 2p orbitals). Thus, polarization functions are not orbitals that describe the atom's valence orbital with one higher l-value; such higher-l valence orbitals would be radially more diffuse. For example, a carbon atom’s 3d orbital in, for example, its 1s² 2s² 2p¹ 3d¹ configuration is quite different in radial character from the polarization d-orbital, which has a radial extent similar to the 2p orbital.

One primary purpose of polarization functions is to give additional angular flexibility to the LCAO process in forming bonding molecular orbitals between pairs of valence atomic orbitals. This is illustrated in Fig. 2.2 where polarization d_p orbitals are seen to contribute to formation of the bonding p orbital of a carbonyl group by allowing polarization of the carbon atom's p_p orbital toward the right and of the oxygen atom's p_p orbital toward the left.

Figure 2.2. Illustrations of the use of polarization functions of p symmetry in CO.

Polarization functions are essential in strained ring compounds such as cyclopropane and cyclopropene because they provide the angular flexibility needed to direct the electron density into regions between bonded atoms. It is just not possible to hybridize the carbon 2s and 2p orbitals to construct bonds that have ca. 60 bond angles as in the above cyclic compounds.

Polarization functions are not only important for allowing charge density to flow into regions between atoms and to thus form directed bonds. It turns out that polarization functions are also important for describing the correlated motions of electrons (this is described in greater detail later in this Chapter), especially their angular correlations. Simply put, electrons avoid one another because they have identical charges. They can do this by moving in different radial regions; that is, one electron can be at small-r when another is at large-r. Alternatively, they can undergo angular correlations; one electron can be on the left while another is on the right. For treating such angular correlations, basis functions having angular momentum quantum numbers higher than those of the valence orbitals are useful to include. In fact, to achieve highly accurate descriptions of angular correlation, one must employ basis functions having quite high L-values. Substantial effort has been devoted to analyzing the L-dependence of the electronic energy so that results obtained from bases obtaining modest L-values can be extrapolated to high-L thus obviating the need to perform calculations with the high-L basis. We will have more to say about such complete-basis extrapolation strategies later.

C. Diffuse Functions

When dealing with anions or Rydberg species, one must further augment the AO basis set by adding so-called diffuse basis orbitals. The valence and polarization functions described above may not provide enough radial flexibility to adequately describe either of these cases. Once again, the PNNL web site database offers a good source for obtaining diffuse functions appropriate to a variety of atoms but not for situations in which very weakly bound anions (e.g., having EAs of 0.1 eV or less) occur.

These tabulated diffuse functions are appropriate if the anion under study has its excess electron in a valence-type orbital (e.g., as in F^-, OH^-, carboxylates, MgF₄^2-, etc.). However, if the excess electron resides in a Rydberg orbital, in an orbital centered on the positive site of a zwitterion species (we will discuss these cases in Chapter 4), or in a so-called dipole-bound orbital (we will also treat them in Chapter 4), one must add to the bases containing valence, polarization, and conventional diffuse functions yet another set of functions that are extra diffuse. The exponents of these extra diffuse basis sets should be small enough to describe the diffuse charge distribution of the excess electron. In dipole-bound anions, not only s but also p and sometimes d symmetry functions are required to describe a dipole-bound orbital localized on the positive side of the molecular dipole. For example, in the NCH^- dipole-bound anion, the extra diffuse functions would likely be centered on the H atom because this nucleus is near the positive end of HCN’s dipole as shown in Fig. 2.3.

Figure 2.3. Orbital holding the excess electron in HCN^-; the hydrogen nucleus is on the left side of the molecular framework which can barely be seen because of the large size of the orbital.

It would be essentially impossible to describe such a diffuse orbital using conventional AO basis sets even when conventional diffuse functions are included.

Moreover, the extra-diffuse set of AOs needs to be flexible enough to describe dispersion stabilization between the excess electron and the electrons of the neutral species. Dispersion is also an electron correlation effect, so the remarks made in the preceeding subsection about using basis functions of higher L-values also apply to it. In the NCH^- example discussed above, the dispersion interaction we refer to is that between the electron in the large, diffuse, highly polarizable dipole-bound orbital shown in Fig. 2.3 and the fourteen electrons of the HCN molecule. We note that the kind of extra diffuse basis sets discussed here have been developed [[52]] by the author and Professors Maciej Gutowski and Piotr Skurski and are currently experiencing wide use in the electronic structure community.

D. Basis Notations

It has become common to describe valence, polarization, and (conventional) diffuse AO basis sets using one of several short hand notations. The various notations derive from the rich history of the several research groups that developed these now widely used basis sets. Rather than review this history, we will simply summarize the notations that are most commonly used. First, it is common in all of the basis sets we will discuss to treat the core and valence atomic orbitals (e.g., for carbon the 1s orbital is a core orbital and the 2s and 2p are valence; for transition metals such as Ti, the 3d and 4s orbitals constitute the valence and the 1s, 2s, 2p, 3s, and 3p are the core) differently. In particular, a more flexible set of contracted orbitals is usually employed for the valence orbitals because it is the valence orbitals that change most (i.e., in their radial extent) depending on what other atoms are involved in the bonding.

The AO basis sets developed largely by Thom Dunning and co-workers use notation of the following form: aug-cc-pVTZ or cc-pVQZ or pVDZ. The VDZ, VTZ, VQZ or V5Z component of the notation is used to specify at what level (double-zeta DZ, triple-zeta TZ, quadruple zeta QZ, or quintuple zeta 5Z) the valence (V) AOs are described. Nothing is said about the core orbitals because each of them is described by a single contracted Gaussian type basis orbital. The term cc is used to specify that the orbital exponents and contraction coefficients in each of the contracted AOs were determined by requiring the atomic energies (usually of all term symbols arising from the conventional lowest-energy electronic configuration), when computed using a correlated rather than Hartree-Fock method, agree to within some tolerance with experimental data. If cc is missing from the notation, the AO exponents and contraction coefficients were determined to make the Hartree-Fock atomic state energies agree with experiment to some precision. The notation p is used to specify that polarization basis orbitals have been included in the basis (if the p is absent, no polarization functions have been added). However, the number and kind of polarization functions differs depending on what level (i.e., VDZ through V5Z) the valence orbitals are treated. At the VDZ level, only one set of d polarization functions is added; at the VTZ, two sets of d and one set of (7) f polarization functions are included. At the VTZ level, three d, two f, and one g set of polarization functions are present, and at the V5Z, four d, three f, two g and one h sets of polarization functions are included. The notation aug is used to specify that (conventional) diffuse basis functions have been added to augment the basis, but again the number and kind depend on how the valence basis is described. At the pVDZ level, one s, one p, and one d diffuse function appear; at pVTZ a diffuse f function also is present; at pVQZ a diffuse g set is also added; and at pV5Z a diffuse h set is present.

For example, a carbon atom basis of aug-cc-p-VTZ quality has a single 1s contracted function, three s-type (i.e., of three distinct radial extent because the valence basis is of triple-zeta quality) valence functions, three sets (i.e., x, y, z) of p-type valence functions of various radial size, two sets (xy, xy, yz, x²-y², z²) of d-type polarization functions (having radial size similar to the valence s and p functions), one set (containing seven) of f-type polarization functions, a more diffuse s and sets (x, y, z) of more diffuse p, d, and f functions. This basis thus contains a total of 46 contracted AOs. An aug-cc-p-VDZ basis would contain 23 AOs, an aug-cc-p-VQZ basis 80, and an aug-cc-p-V5Z basis 127 AOs. A full listing of the aug-cc-p-V5Z basis is shown below with the left column telling the kind of CGTO (i.e., s, p, d, etc.) as well as the Gaussian orbital exponents (a_J) of each primitive Gaussian and the right column giving the contraction coefficient telling how to combine the primitive Gaussians to form the contracted Gaussian AO.

CARBON, Aug-cc-pV5Z: (127 basis functions, 209 primitive functions)

Standard basis: Aug-CC-pV5Z (5D, 7F)

Basis set in the form of general basis input:

1 0

S 10 1.00

0.9677000000D+05 0.2500000000D-04

0.1450000000D+05 0.1900000000D-03

0.3300000000D+04 0.1000000000D-02

0.9358000000D+03 0.4183000000D-02

0.3062000000D+03 0.1485900000D-01

0.1113000000D+03 0.4530100000D-01

0.4390000000D+02 0.1165040000D+00

0.1840000000D+02 0.2402490000D+00

0.8054000000D+01 0.3587990000D+00

0.3637000000D+01 0.2939410000D+00

S 10 1.00

0.9677000000D+05 -0.5000000000D-05

0.1450000000D+05 -0.4100000000D-04

0.3300000000D+04 -0.2130000000D-03

0.9358000000D+03 -0.8970000000D-03

0.3062000000D+03 -0.3187000000D-02

0.1113000000D+03 -0.9961000000D-02

0.4390000000D+02 -0.2637500000D-01

0.1840000000D+02 -0.6000100000D-01

0.8054000000D+01 -0.1068250000D+00

0.3637000000D+01 -0.1441660000D+00

S 1 1.00

0.1656000000D+01 0.1000000000D+01

S 1 1.00

0.6333000000D+00 0.1000000000D+01

S 1 1.00

0.2545000000D+00 0.1000000000D+01

S 1 1.00

0.1019000000D+00 0.1000000000D+01

P 4 1.00

0.1018000000D+03 0.8910000000D-03

0.2404000000D+02 0.6976000000D-02

0.7571000000D+01 0.3166900000D-01

0.2732000000D+01 0.1040060000D+00

P 1 1.00

0.1085000000D+01 0.1000000000D+01

P 1 1.00

0.4496000000D+00 0.1000000000D+01

P 1 1.00

0.1876000000D+00 0.1000000000D+01

P 1 1.00

0.7606000000D-01 0.1000000000D+01

D 1 1.00

0.3134000000D+01 0.1000000000D+01

D 1 1.00

0.1233000000D+01 0.1000000000D+01

D 1 1.00

0.4850000000D+00 0.1000000000D+01

D 1 1.00

0.1910000000D+00 0.1000000000D+01

F 1 1.00

0.2006000000D+01 0.1000000000D+01

F 1 1.00

0.8380000000D+00 0.1000000000D+01

F 1 1.00

0.3500000000D+00 0.1000000000D+01

G 1 1.00

0.1753000000D+01 0.1000000000D+01

G 1 1.00

0.6780000000D+00 0.1000000000D+01

H 1 1.00

0.1259000000D+01 0.1000000000D+01

S 1 1.00

0.3940000000D-01 0.1000000000D+01

P 1 1.00

0.2720000000D-01 0.1000000000D+01

D 1 1.00

0.7010000000D-01 0.1000000000D+01

F 1 1.00

0.1380000000D+00 0.1000000000D+01

G 1 1.00

0.3190000000D+00 0.1000000000D+01

H 1 1.00

0.5860000000D+00 0.1000000000D+01

****

127 basis functions 209 primitive gaussians

The AO basis sets developed largely by the late Professor John Pople and co-workers use a different notation to specify essentially the same information. For example, they use notation of the form 6-31+G** or 3-21G*, 6-311+G*, or 6-31++G. The 3- or 6- component of the notation is used to specify that the core orbitals are each described in terms of a single contracted Gaussian orbital having 3 or 6 terms in its contraction. The –21 or –31 is used to specify that there are two valence basis functions of each type (i.e., the valence basis is of double-zeta quality), one being a contraction of 2 or 3 Gaussian orbitals and the other (the more diffuse of the two) being a contraction of a single Gaussian orbital. When –311 is used, it specifies that the valence orbitals are treated at the triple-zeta level with the tightest contracted function being a combination of 3 Gaussian orbitals and the two looser functions being a single Gaussian function. The * symbol is used to specify that polarization functions have been included on the atoms other than hydrogen; the ** specifies that polarization functions are included on all atoms, including the hydrogen atoms. Finally, the + is used to denote that a single set of (conventional) diffuse valence basis AOs have been included; ++ means that two such sets of diffuse valence basis AOs are present.

Finally, we should say that no common notation exists to specify that extra diffuse basis AOs such as those needed for very weakly bound anions are included. One must simply state this explicitly when detailing the AO basis sets that are employed. It is useful to point out that the kind of extra diffuse basis sets that have been designed for such purposes were constructed by taking each conventional diffuse valence AO and adding a series of successively more diffuse orbitals of identical angular character but with orbital exponents a_J that are related in an even tempered manner to the exponent a_O of the conventional diffuse AO. For example, to add three extra diffuse p-type AOs to a carbon atom whose conventional diffuse p-type AO has an exponent of 0.25, we would add p-functions with exponents 0.25/3, 0.25/9 and 0.25/27 (i.e., with an even tempering

scale factor of 1/3).

E. Computational Cost Depends on the Basis Size

It is essential to be aware of the total number of AO basis functions used in a calculation because the computational cost (i.e., CPU time) and data storage (in main memory or on disk) involved in various calculations depends in a highly non-linear manner on the basis size which I will denote M. The non-linear scaling arises primarily from two sources:

i. In essentially all calculations of electronic energies and wave functions, certain integrals involving products of two or four Gaussian AOs also involving components of the Hamiltonian operator (i.e., the kinetic energy operator, the electron-nuclei Coulomb potential, or the electron-electron Coulomb potential) need to be computed. The number of such integrals is proportional to the square or the fourth power of the number of AOs, respectively. This results in an M⁴ scaling in the CPU time needed for the evaluation and an M⁴ scaling for the data storage.

ii. To compute the electronic energy and wave function, one ends up either (a) having to solve for an eigenvalue of a large (sparse) matrix whose dimension varies at least linearly (more likely at least quadratically) with the basis size M or (b) having to sum a number of terms whose number varies as M⁴ or higher. The former case arises in Hartree-Fock (HF) and configuration interaction (CI) calculations and the latter in Møller-Plesset perturbation theory (MPPT) calculations, both of which we discuss later. To solve for a single eigenvalue of a matrix (A) of dimension D requires CPU time proportional to D² because one must compute elements of a vector v equal to the product of the matrix A with another (so-called trial) vector u: v_J = S_K A_J,K u_K which clearly involves M² operations. The dimension of the HF Hamiltonian matrix is M, so the CPU time needed to evaluate a single HF molecular orbital and its energy will scale as M². Because the dimension D of the CI Hamiltonian matrix most likely scales as M² or higher, the CPU time will vary as M⁴ or more strongly in CI calculations.

Although we illustrate the strong dependence of CPU time and data storage on the size of the AO basis for only three (HF, CI and MPPT) methods, suffice it to say that the cost of all electronic structure methods can rapidly get out of hand if the AO basis grows too large. For example, an aug-cc-p-VDZ calculation on buckyball C₆₀ would involve 23 contracted Gaussian orbitals per carbon atom or 1380 total AOs. The number of so-called two-electron integrals (we define these later in this Chapter) is M⁴/8; for this basis there would be 4.5 x10¹¹ such integrals that need to be calculated and (perhaps) stored. For aug-cc-p-VTZ or aug-cc-p-VQZ bases, the total number of AOs would be 2760 or 4800, respectively, and the number of two-electron integrals would be 7.3 x10¹² or 6.6 x10¹³. It would thus appear that, with a modern computer capable of carrying out ca. 10⁹ CPU operations per second (and knowing that each two-electron integral requires several floating-point arithmetic operations to evaluate) that calculations on C₆₀ using any of these bases would be feasible in a few thousand to a million seconds. However if a calculation whose CPU cost scales as M⁵ were to be employed, these three example calculations would require 1380-4800 times as much effort, which may prove prohibitive even with a 10⁹ operations-per-second computer. It is therefore very important to use bases that are adequate to the task at hand but not unnecessarily large and to anticipate the CPU and storage needs that will arise as a basis is expanded to include additional functions.

II. The Hartree-Fock SCF Process is Usually the Starting Point

Once one has specified an AO basis for each atom in the molecule or anion, the LCAO-MO procedure can be used to determine the C_i,m coefficients that describe the occupied and virtual orbitals. It is important to keep in mind that the basis AOs are not themselves the SCF orbitals of the isolated atoms; even the SCF orbitals are combinations (with atomic values for the C_i,m coefficients) of the basis functions. The LCAO-MO-SCF process itself determines the magnitudes and signs of the C_i,m coefficients, and alternations in the signs of these coefficients allow radial nodes to form.

Because the full electronic Hamiltonian

H = S_j {- h²/2m ²_j - Ze²/r_j} + 1/2 S_j,k e²/|r_j-r_k|

is invariant under the operation P_i,j in which any pair of electrons have their labels (i, j) permuted, we say that H commutes with the permutation operator P_i,j. This fact implies that any solution Y to HY = EY must also be an eigenfunction of P_i,j As a result of H commuting with electron permutation operators, the eigenfunctions Y must either be odd or even under the application of any such permutation. Because electrons are Fermions, their Y functions must be odd under such permutations.

The simple spin-orbital product function

Y = P_k=1,N f_k,

which is what one imagines when one specifies, for example, that carbon is in its 1sa 1sb 2sa 2sb 2p_xa 2p_ya configuration, does not have the correct permutational symmetry. Likewise, the Be atom spin-orbital product wave function

Y = 1sa(1) 1sb(2) 2sa(3) 2sb(4)

is not odd under the interchange of the labels of electrons’ 3 and 4 (or of any pair of electrons); instead one obtains 1sa(1) 1sb(2) 2sa(4) 2sb(3) when the permutation is carried out. However, such products of spin-orbitals (i.e., orbitals multiplied by a or b spin functions) can be made into properly antisymmetric functions by forming the determinant of an NxN matrix whose row index K labels the spin orbital and whose column index J labels the electrons. For example, the making Be atom function 1sa(1) 1sb(2) 2sa(3) 2sb(4) antisymmetric produces the 4x4 matrix whose determinant is shown below

Clearly, if one were to interchange any columns of this determinant, one changes the sign of the function. Moreover, if a determinant contains two or more rows that are identical (i.e., if one attempts to form such a function having two or more spin-orbitals equal), it vanishes. This is how such antisymmetric wave functions embody the Pauli exclusion principle.

A convenient way to write such a determinant is as follows:

S_P (-1)^p f_P1 (1) f_P2(2) … f_PN(N),

where the sum is over all N! permutations of the N spin-orbitals and the notation (-1)^p means that a (–1) is affixed to any permutation that involves an odd number of pairwise interchanges of spin-orbitals and a +1 sign is given to any that involves an even number. To properly normalize such a determinental wave function, one must multiply it by

(N!)^-1/2. So, the final result is that wave functions of the form

Y = (N!)^-1/2 S_P (-1)^p f_P1 (1) f_P2(2) … f_PN(N)

have the proper permutational antisymmetry.

If one uses a single-determinental wave function to form the expectation value of the Hamiltonian <Y|H|Y> and subsequently minimizes this energy by varying the LCAO-MO coefficients in the spin-orbitals, one arrives at a Schrödinger equation appropriate for determining the optimal spin-orbitals

h_e f_J = {– h²/2m ² -Ze²/r + S_K <f_K(r’) |(e²/|r-r’|) | f_K(r’)>} f_J(r)

- S_K <f_K(r’) |(e²/|r-r’|) | f_J(r’)>} f_K(r) = e_J f_J(r).

In this expression, which is known as the Hartree-Fock equation, the kinetic and nuclear attraction potentials

– h²/2m ² - Ze²/r

occur, as does the Coulomb potential

S_K f_K(r’) e²/|r-r’| f_K(r’) dr’ = S_K <f_K(r’)|e²/|r-r’| |f_k(r’)> = S_K J_K,K (r).

One also sees an exchange contribution to the Hartree-Fock potential that, when acting on the spin-orbital f_J is equal to

S_L <f_L(r’) |(e²/|r-r’|) | f_J(r’)>} f_L(r)

often written in short-hand notation as S_L K_L,L f_J(r). Notice that the Coulomb and exchange terms cancel for the L=J case so there is no artificial self-interaction term J_L,L f_L(r) in which spin-orbital f_L interacts with itself. The sum of all the kinetic, electron-nuclear Coulomb, electron-electron Coulomb and exchange operators add up to a one-electron Hamiltonian operator that I will denote h_e; this is called the Fock operator and sometimes written as F instead of h_e.

When the LCAO expansion of each Hartree-Fock (HF) spin-orbital is substituted into the above HF Schrödinger equation, a matrix equation is obtained:

S_m <c_n |h_e| c_m> C_J,m = e_J S_m <c_n|c_m> C_J,m

where the overlap integral is <c_n|c_m> and the h_e matrix element is

<c_n| h_e| c_m> = <c_n| – h²/2m ² |c_m> + <c_n| -Ze²/|r |c_m> + S_K C_K,h C_K,g

[<c_n(r) c_h(r’) |(e²/|r-r’|) | c_m(r) c_g(r’)> - <c_n(r) c_h(r’) |(e²/|r-r’|) | c_g(r) c_m(r’)>].

In the version of the Hartree-Fock self-consistent field (SCF) method outlined above, each spin-orbital is assigned an independent set of LCAO-MO coefficients C_j,m. This has important consequences including making the resultant single-determinant wave function not an eigenfunction of the total electron spin operator S² = (S_K=1,N S_K,x)² +(S_K=1,N S_K,y)² + (S_K=1,N S_K,z)² although such a determinant is an eigenfunction of S_Z = S_K=1,N S_K,z. For example, when carrying out such an SCF calculation on a carbon atom using |1sa 1sb 2sa 2sb 2p_za 2p_ya| as the Slater determinant, the 1sa and 1sb spin-orbitals are not restricted to have identical C_K,n coefficients; nor are the 2sa and 2sb spin-orbitals. This kind of SCF wave function is called an unrestricted Hartree-Fock (UHF) function because it allows all spin-orbitals to have independent C_K,n coefficients.

Why do the 1sa and 1sb spin orbitals turn out to have unequal LCAO-MO coefficients? Because, the matrix elements of the Fock operator shown above <c_n| h_e |c_m> are different for an a and a b spin-orbital. Why? Because the sum S_K C_K,h C_K,g appearing in these matrix elements runs over all N of the occupied spin-orbitals. Thus, for the carbon atom example at hand, K runs over 1sa, 1sb, 2sa, 2sb, 2p_za, and 2p_ya. If, for example, the spin-orbitals whose LCAO-MO coefficients and orbital energies are being solved for is of a type, there will be Coulomb integrals S_K C_K,h C_K,g <c_n(r) c_h(r’) |(e²/|r-r’|) | c_m(r) c_g(r’)> for all N spin-orbitals (i.e., for K = 1sa, 1sb, 2sa, 2sb, 2p_za, and 2p_ya.). However, there will be exchange contributions -S_K C_K,h C_K,g <c_n(r) c_h(r’) |(e²/|r-r’|) | c_g(r) c_m(r’)> only for K = 1sa, 2sa, 2p_za, and 2p_ya., because the exchange integrals involving 1sb and 2sb vanish. On the other hand, when solving for LCAO-MO coefficients and orbital energies of spin-orbitals of b type, there will be Coulomb integrals S_K C_K,h C_K,g <c_n(r) c_h(r’) |(e²/|r-r’|) | c_m(r) c_g(r’)> for K = 1sa, 1sb, 2sa, 2sb, 2p_za, and 2p_ya. but exchange contributions -S_K C_K,h C_K,g <c_n(r) c_h(r’) |(e²/|r-r’|) | c_g(r) c_m(r’)> only for K =1sb and 2sb. Notice that the number of such exchange contributions is not the same as for a-type spin-orbitals. The bottom line is that the Fock matrix elements for a and for b spin-orbitals are different for such open-shell cases in which not each orbital is doubly occupied. In physical terms this means that the 1sa and 1sb spin-orbitals experience different potentials (as do the 2sa and 2sb) so they turn out to have different orbital energies and different LCAO-MO coefficients. This spin polarization is not wrong in that experiments do indeed find, for example, that the energies of the 1s a and 1s b spin-orbitals of carbon (as measured by X-ray photoionization) are not equal. However, the degee of spin polarization (i.e., the differences between a- and b- C_K,n coefficients) obtained in a UHF calculation is found to not be very accurate. Moreover, the fact that the antisymmetrized product of spin-polarized spin-orbitals is not an S² eigenfunction plagues this approach.

So, it is important to keep in mind that UHF wave functions are not eigenfunctions of S²; the degree of so-called spin contamination (i.e., the extent to which they are not spin eigenfunctions) depends on the degree to which the a and b spin-orbitals’ LCAO-MO coefficients differ. Often, the expectation value of S² is computed for such a UHF wave function and reported as part of the computer output; in this way, one can judge to what extent this value differs from the nominal value of S² (e.g., S(S+1) = 1(2) for the |1sa 1sb 2sa 2sb 2p_za 2p_ya| carbon atom example).

It is, of course, possible to develop the working Fock-type equations appropriate to a single Slater determinant in which the LCAO-MO coefficients of nominally equivalent orbitals (e.g., 1sa and1sb or 2sa and 2sb) are restricted to be equal; in this way, the spin contamination property of UHF theory can be overcome. However, the working equations of such a so-called restricted Hatree-Fock (RHF) calculation are more complicated than shown above.

Regardless of whether one uses an RHF or a UHF wave function, it is important to keep in mind that it is sometimes not possible to, even qualitatively, represent the wave function in terms of a single determinant. Later we will see cases in which so-called electron correlation (i.e. the tendency of electrons to avoid one another because of their mutual Coulomb repulsion) needs to be taken into account and where a single-determinant wave function is inadequate. However, there are other cases where one attempts to describe an electronic state even in an uncorrelated manner where more than one determinant must also be used. For example, although the determinant |1sa 1sb 2sa 2sb 2p_za 2p_ya| is an acceptable approximation to the carbon ³P state if the 1s and 2s spin-orbitals are restricted to be equal for a and b spins, the ¹S state arising in this same 1s²2s²2p² configuration can not be represented as a single determinant. In fact, the ¹S state requires a minimum of the following three-determinant wave function:

Y = 3^-1/2 [1sa 1sb 2sa 2sb 2p_za 2p_zb| - 1sa 1sb 2sa 2sb 2p_xa 2p_xb|

- 1sa 1sb 2sa 2sb 2p_ya 2p_yb|].

The implications of whether a state of an atom or molecule (or anion) can be qualitatively represented in terms of a single Slater determinant are important to know about. If the state cannot be so represented, one simply should not use theoretical methods that are predicated on the existence of a dominant single determinant in the expansion of the full wave function. As we will see later, several of the most commonly employed theories are derived assuming that a single determinant wave function is a good starting point. When this is not the case (e.g., for ¹S carbon), one should avoid using such theories. This warning is difficult to over emphasize, but many workers do not seem to be aware of it or ignore it when carrying out calculations on such open-shell systems for which some of the term symbols’ wave functions just can not be written an a single-determinant fashion.

Before closing this discussion, it is useful to reflect on the physical meaning of the Coulomb and exchange interactions between pairs of orbitals. For example, the Coulomb integral J_1,2= |f₁(r)|² e²/|r-r’|f₂(r’)|² dr dr’ appropriate to the two orbitals shown in Fig. 2.4 represents the Coulomb repulsion energy of two charge densities, |f₁|² and |f₂|², integrated over all locations r and r’ of the two electrons.

Figure 2.4. Two orbitals’ overlap region.

In contrast, the exchange integral

K_1,2= f₁(r) f₂(r’) e²/|r-r’|f₂(r) f₁(r’)dr dr’

can be thought of as a Coulomb repulsion between two electrons whose coordinates r and r’ are both distributed throughout the overlap region f₁ f₂. This overlap region is where both f₁ and f₂ have appreciable magnitude. This interpretation of exchange integrals as Coulomb interactions between two electrons confined to the overlap region helps us understand why exchange integrals tend to be smaller in magnitude than Coulomb integrals involving the same functions (because the overlap region is a fraction of the total volume of either orbital). It also helps to understand why exchange integrals decay rapidly (and exponentially) with the distance R between the two centers on which the two orbitals are located but Coulomb integrals decay less rapidly (and as 1/R).

As noted above, the Coulomb and exchange integrals in which the two spin-orbital indices are identical cancel (i.e., J_I,I = K_I,I). This cancellation is important because it assures that the SCF equations do not contain a so-called Coulomb self-interaction of an electron with itself. It may appear obvious that this should be an attribute of any acceptable model for the interactions among electrons, but it turns out that not all commonly used theoretical methods possess it. In particular, most density functional theory (DFT) potentials [[53]] do not guarantee such cancellation in the large-r regions (i.e., where both electrons are far from nuclear centers). As a result, such potentials do not properly yield asymptotic behavior that contains no Coulomb interaction, as should be the case for a singly charged anion. Recall from our earlier discussion about the nature of the long-range attractive and repulsive potentials in neutrals and cations and in anions and multiply charged anions that such potentials differ qualitatively. For these reasons, it is essential to properly describe the large-r behavior of the anion’s electron-attracting potential, and thus it is important to use methods that offer this possibility.

III. Koopmans’ Theorem Gives the First Approximation to the Electron Affinity

The HF-SCF equations

h_e f_i = e_i f_i

imply that the SCF orbital enegies e_i can be written as:

e_i = < f_i | h_e | f_i > = < f_i | T + V | f_i > + S_j(occupied) < f_i | J_j - K_j | f_i >

= < f_i | T + V | f_i > + S_j(occupied) [J_i,j - K_i,j ],

where T + V represents the kinetic (T) and nuclear attraction (V) energies, respectively. Thus, ei is the average value of the kinetic energy plus Coulomb attraction to the nuclei for an electron in f_i plus the sum over all of the spin-orbitals occupied in Y of Coulomb minus exchange interactions of spin-orbital f_i with the other occupied spin-orbitals.

If fi is an occupied spin-orbital, the term [J_i,i - K_i,i] disappears and the remaining terms in the sum represents the Coulomb minus exchange interaction of f_i with all of the N-1 other occupied spin-orbitals. If f_i is a virtual spin-orbital, this cancellation does not occur (because j cannot equal i), and one obtains the Coulomb minus exchange interaction of f_i with all N of the occupied spin-orbitals in Y. Hence the energies of occupied orbitals pertain to interactions appropriate to a total of N electrons, while the energies of virtual orbitals pertain to a system with N+1 electrons. For this reason, virtual orbitals also tend to be radially more diffuse than occupied orbitals.

Let us consider the following model of the detachment or attachment of an electron in an N-electron system.

1. In this model, both the parent molecule and the species generated by adding or removing an electron are treated at the HF level.

2. The Hartree-Fock orbitals of the parent molecule are used to describe both species. It is said that such a model neglects ‘orbital relaxation' (i.e., the re-optimization of the spin-orbitals to allow them to become appropriate to the daughter anion or cation species).

Within this model, the energy difference between the daughter and the parent can be written as follows (f_k represents the particular spin-orbital that is added or removed):

For electron detachment:

E^N-1 - E^N = - e_k ;

For electron attachment:

E^N - E^N+1 = - e_k .

So, within the limitations of the HF, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negatives of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans' theorem; it is used extensively in quantum chemical calculations as a means of estimating IPs and EAs and often yields results that are qualitatively correct (i.e., ± 0.5 eV) but not usually more accurate than this.

It is useful at this time to reflect a bit more on the physical meaning of the Hartree-Fock orbitals and their energies; let us do so with a concrete example of the N₂ molecule. A HF calculation using a reasonable AO basis set on N₂ will generate a set of occupied MOs of s_g, s_u, and p_u symmetry as well as a set of unoccupied (called virtual) orbitals of s_g, s_u, p_g, and p_u symmetries. The occupied orbital of p_u symmetry corresponds to the bonding p orbital and will have a negative HF orbital energy that, via. Koopmans’ theorem, gives an approximation to the energy needed to remove an electron from this orbital:

.

However, the lowest-energy unoccupied orbital will not necessarily appear very much like one might expect; that is, it will not necessarily correspond to an anti-bonding p_gorbital. Why not? Because the Coulomb and exchange potentials that act on this virtual orbital correpsond to a total of fourteen electrons. This same combination of Coulomb minus exchange potentials acting on, for example, the bonding p_u orbital describe only thirteen electrons acting on this orbital. That is, the term

S_j(occupied) [J_i,j - K_i,j ]

in the HF Hamiltonian, when acting on one of the occupied orbitals f_o, is of the form

S_j(occupied) [J_o,j - K_o,j ].

In the sum over j, the terms j = o cancel. In contrast, when acting on an unoccupied orbital f_u, one obtains

S_j(occupied) [J_u,j - K_u,j ].

Nowhere in the sum over j does the term j = u occur, so one does not obtain the kind of cancellation that occurred in the occupied-orbital case. This means that in and N-electron species, the occupied orbitals feel only the N-1 other electrons’ Coulomb and exchange potentials as one expects (i.e., an electron does not experience such an interaction with itself). So, the occupied p_u orbital of N₂ is an eigenfunction of a Hamiltonian that contains the effects of the remaining thirteen electrons. In contrast, the unoccupied orbitals feel Coulomb and exchange potentials from all N of the occupied orbitals. Hence, the unoccupied p_g orbital is an eigenfunction of a Hamiltonian that contains the effects of the fourteen occupied orbitals’ electrons. It is for this reason that we often say that virtual HF orbitals are more appropriate for describing the anion (which, of course, is what Koopmans’ theorem suggests) while the occupied orbitals more properly relate to the neutral parent.

This example shows that one must be very careful when interpreting virtual orbitals’ energies and radial characters. However, it does not answer the question about what one should do if one wants to obtain an orbital of p_g symmetry that can be used, for example, to describe a p_u¹ p_g¹ state of N₂. To approach this problem, one can carry out a separate HF calculation in which the orbitals defined as occupied (i.e., those indexed j in the sum S_j(occupied) [J_i,j - K_i,j]) include one p_u and one p_g orbital. So, one should use HF orbitals that have been obtained using an occupancy definition that fits the electronic state one wishes to study.

Above, we noted that the virtual orbitals of N₂ are more appropriate to the N₂^- anion than to neutral N₂. However, such virtual orbitals cannot be trusted (e.g., their Koompmans’ estimate for the EA should not be used) if, as is the case for N₂^-, the anion is electronically metastable with respect to electron autodetachment. In such cases, the specialized techniques detailed in Section IV of Chapter 5 need to be employed to gain even a qualitatively correct description of the metastable anion’s orbitals.

IV. Electron Correlation Involving the Excess Electron Usually Must be Treated

To achieve reasonable chemical accuracy (e.g., ± 5 kcal/mole) in electronic structure calculations, one cannot describe the wave function Y in terms of a single determinant as is done in the Hartree-Fock approach. Above and beyond cases such as the ¹S state of the carbon atom discussed earlier in which three determinants are needed just to form a function of proper spin and spatial symmetry, there are other situations in which more than one determinant should be used. The reason a single-determinant wave function is inadequate for all species when one desires high accuracy is because the spatial probability density functions are not correlated when such a function is used. In other words, the probability P(r,r’) of finding one electron at point r and a second electron at r’ for a single-determinant wave function turns out as we show later to be the product p(r) of finding one electron at r times the probability p(r’) of finding an electron at r’ P(r,r’) = p(r) p(r’). This means the probability of finding one electron at position r is independent of where the other electrons are, which is absurd because the electrons’ mutual Coulomb repulsion causes them to avoid one another.

This mutual avoidance is what we call electron correlation because the electrons’ motions as reflected in their spatial probability densities are correlated (i.e., inter-related). It turns out that the differences between predictions of non-correlated (e.g., Hartree-Fock) and correlated theories of EAs are quite substantial and often amount to ca. 0.5 eV, so it is essential that one use correlated methods to achieve reliable EAs.

Let us consider a simple example to illustrate this problem with single determinant functions. The |1sa(r) 1sb(r’)| determinant appropriate, for example to a He atom, when written as

|1sa(r) 1sb(r’)| = 2^-1/2{1sa(r) 1sb(r’) - 1sa(r’) 1sb(r)}

can be multiplied by itself to produce the 2-electron probability density:

P(r, r’) = 1/2{[1sa(r) 1sb(r’)]² + [1sa(r’) 1sb(r)]²

-1sa(r) 1sb(r’) 1sa(r’) 1sb(r) - 1sa(r’) 1sb(r) 1sa(r) 1sb(r’)}.

If we now integrate over the spins of the two electrons and make use of the orthonormality of the a and b spin functions

<a|a> = <b|b> = 1, and <a}b> = <b}a> = 0,

we obtain the following spatial (i.e., with spin absent) probability density:

P(r,r’) = |1s(r)|² |1s(r’)|².

This probability, being a product of the probability density for finding one electron at r times the density of finding another electron at r’, clearly has no correlation in it. That is, the probability of finding one electron at r does not depend on where (r’) the other electron is. This product form for P(r,r’) is a direct result of the single-determinant form for Y, so this form must be wrong if electron correlation is to be accounted for. In fact, the Coulomb repulsions between pairs of electrons causes the true pair distribution function P(r,r’) to vanish as r approaches r’. In fact, P(r,r’) has a cusp whenever |r – r’| 0; unlike the cusp in the electronic wave functions at geometries where r approaches a nuclear center (n.b., this probability density is non-vanishing at the nuclear center where it has a non-zero slope), the electron-pair probability vanishes at |r – r’| = 0 and has a non-zero slope.

Now, we need to ask how Y should be written if such correlation effects are to be taken into account. One approach is to introduce into the form of the ansatz wave function terms that depend explicitly on the inter-electron coordinates r_i,j and to cast these terms in a way that causes the wave function to decay whenever any r_i,j becomes small. Such so-called explicitly correlated approaches are indeed possible to use, but, because of the added complexity that such functional forms introduce into their practical implementations, their use currently is limited to rather small molecules and ions. Therefore, it is more common (and practical) to use other means to allow the electrons to avoid one another and it is such approaches that we shall focus on here.

It turns out that one can account for electron avoidance (for each pair of electrons) by taking Y to be a combination of two or more determinants that differ by the promotion of two electrons from one orbital to another orbital (i.e., from an occupied orbital to a so-called virtual orbital-one that is not occupied in the more approximate wave function). In such a wave function, we say that doubly excited determinants have been included, and the approach of combining two or more determinants to form a wave function is called configuration interaction (CI).

For example, in describing the p² bonding electron pair of an olefin or the ns² electron pair in alkaline earth atoms, we find it important to mix in doubly excited determinants of the form (p*)² or np², respectively. It turns out that the avoidance of electrons that occupy the same orbital are the most important to include (i.e, have the largest energy contribution), but correlations between electrons occupying different orbitals (e.g., 1s 2s electron correlation in Be) are also important if one wishes to achieve high accuracy.

Briefly, the physical importance of such doubly-excited determinants can be made clear by using the following identity involving the combination of any pair of determinants denoted | ..fa fb..| and | ..f’a f’b..| that differ by only two spin-orbitals (fa and fb vs. f’a and f’b) :

Y = C₁ | ..fa fb..| - C₂ | ..f'a f'b..|

= C₁/2 { | ..( f - xf')a ( f + xf')b..| - | ..( f - xf')b ( f + xf')a..| },

where

x = (C₂/C₁)^1/2 .

This identity allows one to interpret the combination (C₁ | ..fa fb..| - C₂ | ..f'a f'b..|) of two determinants that differ from one another by a double promotion from one orbital (f) to another (f') as equivalent to a singlet coupling (i.e., having 2^-1/2 (ab-ba) spin function) of two different orbitals (f - xf') and (f + xf'). Let us now see how this identity helps us understand how such CI wave functions can incorporate correlations among the electrons’ motions.

In the olefin example mentioned above, the two non-orthogonal polarized orbital pairs (f - xf') and (f + xf') involve mixing the f = p and f’ = p* orbitals to produce two left-right polarized orbitals as depicted in Fig. 2.5.

Figure 2.5. Polarized p orbital pairs.

In this case, one says that the p² electron pair undergoes left-right correlation when the (p*)² determinant is mixed into the CI wave function. One electron is in p+xp* on the left while the other is in p-xp* on the right.

In the alkaline earth atom case, the polarized orbital pairs are formed by mixing the ns and np orbitals (actually, one must mix in equal amounts of px, py , and pz orbitals to preserve overall ¹S symmetry in this case), which gives rise to angular correlation of the electron pair. Such a pair of polarized orbitals is shown in Fig. 2.6.

Figure 2.6. Polarized s and p orbital pairs.

More specifically, the following four determinants are included to form polarized orbital pairs and to preserve the ¹S spin and orbital angular momentum character in Y:

Y = C₁ |1s²2s² | - C₂ [|1s²2p_x² | +|1s²2p_y² | +|1s²2p_z² |].

The fact that the latter three terms possess the same amplitude C₂ is a result of the requirement that a state of ¹S symmetry is involved. It can be shown that this four-determinant function is equivalent to:

Y = 1/6 C₁ |1sa1sb{[(2s-a2p_x)a (2s+a2p_x)b - (2s-a2p_x)b(2s+a2p_x)a]

+[(2s-a2p_y)a(2s+a2p_y)b - (2s-a2p_y)b(2s+a2p_y)a]

+[(2s-a2p_z)a(2s+a2p_z)b - (2s-a2p_z)b(2s+a2p_z)a] |,

where a = (3C₂/C₁)^1/2. Here two electrons occupy the 1s orbital (with opposite, a and b spins) and are thus treated in a non-correlated manner while the other pair resides in 2s-2p polarized orbitals in a manner that instantaneously correlates their motions. This illustrates that wave functions can be created which correlate selected subsets of the occupied spin-orbitals while treating others at the non-correlated level.

The polarized orbital pairs (2s ± a 2p_{x,y, or z}) are formed by combining the 2s orbital with the 2p_{x,y, or z} orbital in a ratio determined by C₂/C₁. As we will see later when we deal with how to evaluate Hamiltonian matrix elements between pairs of determinental wave functions, this ratio C₂/C₁ can be shown to be proportional to the magnitude of the coupling matrix element

<1s²2s² |H|1s²2p² >

between the two determinants involved and inversely proportional to the energy difference

[<1s²2s²H|1s²2s²> - <1s²2p²|H|1s²2p²>]

between these configurations. In general, configurations that have similar Hamiltonian expectation values and that are coupled strongly give rise to strongly mixed (i.e., with large |C₂/C₁| ratios) polarized orbital pairs.

In each of the three equivalent terms in the above alkaline earth wave function, one of the valence electrons moves in a 2s+a2p orbital polarized in one direction (x, y, or z) while the other valence electron moves in the 2s-a2p orbital polarized in the opposite x, y, or z direction. For example, the first term

[(2s-a2p_x)a(2s+a2p_x)b] - (2s-a2p_x)b(2s+a2p_x)a]

describes one electron occupying a 2s-a2p_x polarized orbital while the other electron occupies the 2s+a2p_x orbital. The electrons thus reduce their Coulomb repulsion by occupying different regions of space; in the SCF picture 1s²2s², both valence electrons reside in the same 2s region of space. In this particular example, the electrons undergo angular correlation to 'avoid' one another. To achieve an even higher-level description of angular correlation, one can also employ functions with angular momentum higher than the p-functions described above. That is, L=2, 3, … basis functions can be used to gain an even more accurate description of angular correlation.

The use of doubly excited determinants is thus seen to be a mechanism by which Y can place electron pairs, which in the single-configuration picture occupy the same orbital, into different regions of space (i.e., each one into a different member of the polarized orbital pair) thereby lowering their mutual Coulomb repulsion. Such electron correlation effects are referred to as dynamical electron correlation; they are extremely important to include if one expects to achieve chemically meaningful accuracy (i.e., ± 5 kcal/mole). As mentioned earlier, because molecular EAs can be of this order of magnitude, their accurate calculation usually requires one to include such dynamical electron correlation effects.

In practical quantum chemistry calculations on anions, one must compute the C₂/C₁ ratios (more appropriately, all of the C_I coefficients in the CI expansion of Y

Y = S_J C_J F_J,

where the F_J are spin and spatial symmetry adapted configuration state functions (i.e., combinations of Slater determinants that produce the proper spin and space symmetry).There are a variety of approaches that can be used to compute the C_I coefficients as well as the energy E. The most commonly employed methods are the configuration interaction (CI), Møller-Plessset (MP) perturbation, and coupled-cluster (CC) methods. Each has strengths and weaknesses that we briefly review in the following subsection.

V. Various Methods Can be Used to Treat Correlation

There are numerous procedures currently in use for determining the 'best' wave function of the form:

Y = S_I C_I F_I,

where F_I is a spin-and space- symmetry-adapted configuration state function (CSF) consisting of linear combinations of determinants each of which we denote | f_I1 f_I2 f_I3 ... f_IN | in terms of their N occupied spin-orbitals. In all such wave functions, there are two kinds of parameters that need to be determined- the C_I coefficients and the LCAO-MO coefficients describing the f_Ik . Because treatment of electron correlation is an integral part of nearly all studies of molecular anions, it is important to survey the variety of approaches that are used for this purpose.

The most commonly employed methods used to determine these parameters include:

A. The multiconfigurational self-consistent field (MCSCF) method

In this approach, the expectation value < Y | H | Y > / < Y | Y > is treated variationally and made stationary with respect to variations in both the C_I and C_i,n coefficients. The energy functional is a quadratic function of the CI coefficients, and so one can express the stationary conditions for these variables in terms of a matrix eigenvalue problem:

S_J H_I,J C_J = E C_I .

However, E is a quartic function of the C_i,n 's because, as we will show later, each H_I,J matrix element can be expressed in terms of two-electron integrals <f_if_j | g |f_kf_l> each of which depend quartically on the C_n,i coefficients because each f_j is a linear combination of bais orbitals c_n multiplied by C_i,n.

In the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g., a few to several thousand) chosen to describe essential correlations (i.e., configuration crossings, near degeneracies, proper dissociation, all of which are often termed non-dynamical correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc. nature that are important to the anion-neutral energy difference). For the alkaline earth example used earlier, the four determinants in the two-CSF wave function Y @ C₁ |1s²2s² | - C₂ [|1s²2p_x² | +|1s²2p_y² | +|1s²2p_z² |] could be used to carry out an MCSCF calculation in which case one would speak of including angular dynamical correlations of the electrons occupying the two 2s spin-orbitals.

B. The configuration interaction (CI) method

In this approach, the LCAO-MO coefficients of all the spin-orbitals are determined first via a single-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The C_I coefficients are subsequently determined by making the energy expectation value < Y | H | Y > / < Y | Y > stationary. The CI wave function is most commonly constructed from CSFs F_J that include:

1. All of the CSFs in the SCF or MCSCF wave function that was used to generate the molecular spin-orbitals f_i . These are referred to as the 'reference' CSFs. For the alkaline earth example, the four determinants in Y @ C₁ |1s²2s² | - C₂ [|1s²2p_x² | +|1s²2p_y² | +|1s²2p_z² |] might constitute such a list of two reference CSFs.

2. CSFs generated by carrying out single, double, triple, etc. level 'excitations' (i.e., orbital replacements) relative to reference CSFs. For the alkaline earth example, determinants of the form 1s²ns² |, ms²2s² |, 1s²np² |, and 1s²23d² | (with n, m ³ 3) might be included. CI wave functions limited to include contributions through various levels of excitation are denoted S (singly), D (doubly), SD (singly and doubly), SDT (singly, doubly, and triply) excited.

As we already introduced, the orbitals from which electrons are removed can be restricted to focus attention on correlations among certain orbitals. For example, if excitations out of core electrons are excluded, one computes a total energy that contains no core correlation energy. The number of CSFs included in the CI calculation is often far in excess of the number considered in an equivalent-quality MCSCF wave function (because in the latter, one also optimizes the LCAO-MO coefficients). CI wave functions including 5,000 to 50,000 CSFs are routine, and functions with one to several billion CSFs are within the realm of practicality.

The need for such large CSF expansions in a CI wave function might be surprising but is relatively easy to understand. Consider (i) that each electron pair requires at least two CSFs to form polarized orbital pairs that can correlate their motions, (ii) there are of the order of N(N-1)/2 electron pairs for N electrons, hence (iii) the number of terms in the CI wave function scales as 2^N(N-1)/2. For a molecule containing ten electrons, there thus could be 2⁴⁵ = 3.5 x10¹³ terms in the CI expansion. This may be an over estimate of the number of CSFs needed, but it demonstrates how rapidly the number of CSFs can grow with the number of electron pairs that one wishes to correlate.

i. The Slater-Condon Rules

In all of the methods used to treat electron correlation, the H_I,J matrices are, in practice, evaluated in terms of one- and two- electron integrals over the molecular orbitals using the so-called Slater-Condon rules or their equivalent. These rules express all non-vanishing matrix elements involving either one- or two- electron operators between any pair of determinental wave functions in which the constituent spin-orbitals are orthonormal. One-electron operators (e.g., the kinetic energy and the electron-nuclei Coulomb attraction potentials) are one-electron additive and appear in any quantum mechanical operator, including the Hamiltonian, as

F = S_i f(i).

Two-electron operators (e.g., the electron-electron Coulomb repulsions) are pair wise additive and always appear as

G = S_ij g(i,j)).

The Slater-Condon rules give the matrix elements between any two determinants

| > = |f₁f₂f₃... f_N|

and

| ' > = |f'₁f'₂f'₃...f'_N|

for any quantum mechanical operator that is a sum of one- and two- electron operators (F + G). It expresses these matrix elements in terms of one-and two-electron integrals involving the spin-orbitals that appear in | > and | ' > and the operators f and g.

As a first step in applying these rules, one must examine | > and | ' > and determine by how many (if any) spin-orbitals | > and | ' > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to achieve maximal coincidence with those in the other determinant; it is essential to keep track of the number of permutations (Np) that one makes in achieving maximal coincidence. The results of the Slater-Condon rules given below are then multiplied by (-1)Np to obtain the matrix elements between the original | > and | ' >. The final result does not depend on whether one chooses to permute | > or | ' >.

The Hamiltonian is, of course, a specific example of such an operator; the electric dipole operator S_i eri and the electronic kinetic energy - h2/2meSiÑi2 are examples of one-electron operators (for which one takes g = 0); the electron-electron coulomb interaction S_i>j e2/rij is a two-electron operator (for which one takes f = 0). Once maximal coincidence has been achieved, the Slater-Condon (SC) rules provide the following prescriptions for evaluating the matrix elements of any operator F + G containing a one-electron part F = S_i f(i) and a two-electron part G = S_ij g(i,j).:

(i) If | > and | ' > are identical, then

< | F + G | > = S_i < f_i | f | f_i > +S_i>j [< f_if_j | g | f_if_j > - < f_if_j | g | f_jf_j >],

where the sums over i and j run over all spin-orbitals in | >;

(ii) If | > and | ' > differ by a single spin-orbital mismatch ( f_p f'_p ),

< | F + G | ' > = < f_p | f | f'_p > +S_j [< f_pf_j | g | f'_pf_j > - < f_pf_j | g | f_jf'_p >],

where the sum over j runs over all spin-orbitals in | > except f_p ;

(iii) If | > and | ' > differ by two spin-orbitals ( f_p f'_p and f_q f'_q):

< | F + G | ' > = < f_p f_q | g | f'_p f'_q > - < f_p f_q | g | f'_q f'_p >

(note that the F contribution vanishes in this case);

(iv) If | > and | ' > differ by three or more spin orbitals, then