Reactive Intermediate Chemistry

4.2.3.Why Are the Cubyl Hydrogens of Methylcubane More Reactive than the Methyl Hydrogens Toward Abstraction by tert-Butoxyl

4.3.Can Hyperconjugation in a 1,3-Diradical Control the Stereochemistry

of Cyclopropane Ring Opening and Make a Singlet the Electronic Ground

1. INTRODUCTION

Because reactive intermediates (RIs) undergo facile intraand/or intermolecular reactions, they are, by definition, short lived. As described in the other chapters in this volume, the transient nature of RIs provides a challenge to their direct spectroscopic observation. When the spectra of a putative RI can be obtained, it may not be certain that the observed spectra actually belong to the RI.

What makes RIs short lived is, of course, their unusual electronic structures. The effect of their unusual electronic structures on the geometries and relative energies of RIs is often of tremendous interest. Unfortunately, the short lifetimes of RIs usually precludes their study by X-ray crystallography; and measuring the energies of RIs, especially relative to those of stable molecules, can be very challenging.

The unusual electronic structures of RIs frequently also make their chemistries very unusual. Understanding the electronic structures of RIs well enough to make both qualitative and quantitative predictions about their reactivities is an important goal of research in this area of chemistry.

It is easy to appreciate why electronic structure calculations that are accurate enough to compute reliably the geometries, spectra, and relative energies of RIs and to predict their reactivities, would be a very important adjunct to experiments. Fortunately, during the last three decades of the twentieth century, advances in both computer hardware and software began to make it possible to perform electronic structure calculations on a wide variety of RIs with the required accuracy.

INTRODUCTION 963

This fact was first brought to the attention of experimentalists, interested in RIs, by two different computational predictions, made in the 1970s, about methylene.1 Triplet methylene was computed to have a bent geometry, with an H C H bond angle of 135 ; and the best calculations predicted the triplet to lie 10 kcal/mol below the lowest singlet state. Both of these computational results, when published, were in apparent conflict with experiments, which had been interpreted as indicating that triplet methylene is linear and that it lies >20 kcal/mol below the lowest singlet state. However, subsequent experiments proved the calculations to be correct.

At the same time that experimentalists were beginning to realize that electronic structure calculations were capable of making predictions, accurate enough to be useful, the development by John Pople (who shared the Nobel Prize in Chemistry in 1998) and his co-workers of the Gaussian series of programs made performing such calculations relatively easy, even for experimentalists. Experimentalists began to stop relying on their theoretician friends to perform electronic structure calculations for them and started carrying out calculations themselves.

In many, if not most, groups currently engaged in experimental studies of RIs, electronic structure calculations play an important role; and theoretical chemists continue to find RIs fascinating subjects for computational studies. The partnership between electronic structure calculations and experiments in the study of RIs is the subject of this chapter.

Section 2 describes in general terms the types of information, useful to experimentalists, about RIs that can be obtained from electronic structure calculations. References are given to various applications of electronic structure calculations that are discussed in other chapters of this book.

In both Sections 2 and 3, terms that are frequently used in the literature in discussing the results of electronic structure calculations, are given in italics. It is hoped that, by having their attention directed to these terms, the reader will become familiar with them and will learn what each term means.

Section 3 provides a brief introduction to the methods used in performing electronic structure calculations. Methods based on both wave functions and on density functional theory (DFT) are discussed. The assumptions that underlie each of these two different types of calculation are described, and a comparison between these methods is provided.

Section 4 discusses the application of electronic structure calculations to understanding and predicting the outcome of experiments on three different types of RI. The examples in this section are taken from research, performed in collaborations between the author’s group and the groups of experimentalists. The three RIs discussed in Section 4 are phenylnitrene, cubyl cation, and propane- 1,3-diyls.

Section 5 provides some final observations about the uses of electronic structure calculations in the experimental studies of RIs; and Section 6 gives suggestions for further reading, both about electronic structure theory and about its application to specific experimental studies of RIs. A brief description of each of the suggestions for further reading is provided.

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2. WHAT ROLES CAN ELECTRONIC STRUCTURE CALCULATIONS PLAY IN EXPERIMENTAL STUDIES OF REACTIVE INTERMEDIATES?

2.1. Predictions of Spectra

Because of the fleeting existence of RIs under ordinary conditions, if RIs are to be observed directly, they must be observed either by detection during their very short lifetimes or by extending their lifetimes. The latter strategy is sometimes pursued by studying RIs in the gas phase, under conditions where bimolecular reactions can be minimized. However, immobilizing RIs in glasses or rare gas matrices at low temperatures has increasingly become the technique of choice for obtaining detailed spectroscopic information about RIs.

Whether laser flash photolysis (LFP) is used to detect RIs before they react, or matrix isolation at very low temperatures is employed to slow down or quench these reactions, spectroscopic characterization of RIs is frequently limited to infrared (IR) and/or ultraviolet–visible (UV–vis) spectroscopy. Nuclear magnetic resonance (NMR) spectroscopy, which is generally the most useful spectroscopic technique for unequivocally assigning structures to stable organic molecules, is inapplicable to many types of RI.

It is often the case that the putative structure assigned to an RI is so unusual that vibrational and electronic spectra of analogous molecules are unavailable. How then can an experimentalist know whether an observed IR and/or UV–vis spectrum actually does belong to the putative structure assigned to the RI? As exemplified in the sections on matrix-isolated benzynes in Chapter 16 in this volume and also discussed in Chapter 17 in this volume, suitable electronic structure calculations can provide accurate predictions of the IR spectra and UV–vis excitation energies for RIs. The IR and UV–vis spectra, predicted for a particular structure, can then be compared with those obtained experimentally.

The ability of calculations, especially those based on DFT, to predict IR spectra accurately is generally very good. Currently, most, if not all of the papers that report the IR spectrum of a matrix-isolated RI, contain a figure that shows graphically how well the observed spectrum and the spectrum computed for the structure assigned to the RI correspond.

As described in Chapter 1 in this volume, electronic structure theory calculations are also very useful for computing 13C chemical shifts in carbocations. The predicted 13C NMR spectra and the experimental spectra, obtained in superacids under conditions where the carbocations are stable, can be compared. These comparisons are particularly helpful in differentiating between classical and nonclassical structures for these electron-deficient species.

2.2. Predictions of Geometries

One of the most desirable and experimentally least accessible pieces of information about an RI is its molecular structure. Unusual features of the electronic structures

of RIs (e.g., nonclassical bonding in carbocations) often result in unusual bond lengths and bond angles. Unfortunately, although the geometries of a few carbocations have been obtained by X-ray crystallography, most RIs are too reactive to allow their structures to be determined experimentally.

However, except in pathological cases (e.g., that of m-benzyne, which is discussed in Chapter 16 in this volume), even fairly low-level calculations can produce rather accurate structural parameters. Therefore, it is almost always much easier to compute the structural parameters for an RI than to measure them. If there is good agreement between the spectra observed for an RI and those computed for the structure assigned to it, then it is usually assumed that the bond lengths and bond angles in the calculated structure are very close to those in the RI.

2.3. Predictions of Enthalpies

As discussed in Section 3, it is usually more challenging to compute accurately the energies than the geometries of RIs. In contrast, although geometries of most RIs are hard to obtain experimentally, it is often possible to measure the enthalpy of activation for the formation and/or disappearance of an RI accurately. This experimental value can then be compared with the values predicted by different levels of electronic structure calculations, in order to test the ability of a particular level of theory to mirror accurately the experimental energetics.

In computing an activation enthalpy for a reaction, it is first necessary to optimize the geometry of the reactant and to locate the transition structure (TS). Both geometries are stationary points (i.e., points at which all the first derivatives of the energy are zero) on the potential energy surface (PES); but a TS is a saddle point (i.e., a stationary point with one and only one negative second derivative). Finding a TS is computationally more demanding than optimizing the geometry of an energy minimum. Nevertheless, in many cases, a level of electronic structure theory that is sufficient to provide an accurate geometry for an RI also suffices to find with comparable accuracy the geometry of the TS by which the RI is formed and/or reacts.

The adequacy of calculations, performed at a particular level of electronic structure theory, for finding equilibrium and TS geometries, can be tested, even if comparisons with experimental enthalpy differences cannot be made. The relative energies of two geometries, calculated at the level of theory that is used to find these stationary points, can be recomputed by single-point calculations, performed at the same two geometries, but with a higher level of electronic structure theory (usually one that is too demanding of computational resources to be used routinely for finding geometries). If the two different levels of theory give similar energy differences, this finding provides evidence that the lower level of theory is probably as good as the higher level would be for finding the geometries of these two stationary points on the PES.

In order to compute the enthalpy difference between two stationary points, the difference between their electronic energies must be corrected for the difference between their vibrational energies at 0 K and also for the difference between the amounts of thermal energy that each absorbs between 0 K and the temperature

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of interest. These zero-point and heat capacity corrections are available from vibrational analyses, which require calculation of the second derivatives of the energy with respect to all internal coordinates at each stationary point. Vibrational analyses must also be performed in order to confirm that the geometry of a putative energy minimum has only real vibrational frequencies and that the geometry of a stationary point, which is thought to be a TS, has one and only one vibrational frequency that is imaginary.

Good agreement between a measured enthalpy of activation and that computed at a particular level of theory, furnishes evidence that calculations at this level of theory are accurate enough to provide reliable information about the enthalpy differences between the reactant, the TS, and other stationary points on the PES for an RI. As already noted, differences between the heats of formation of an RI and other energy minima on a PES (i.e., stable molecules and other RIs, which are either formed from an RI or from which an RI is formed) are usually harder to measure experimentally than the activation enthalpy for appearance or disappearance of an RI. Therefore, being able to compute accurately the enthalpy differences between an RI and other energy minima on a PES can provide very valuable information that is usually not easy to obtain experimentally.

2.4. Predictions of Free Energies and Isotope Effects

Rotational and vibrational partition functions can be computed from the geometry and vibrational frequencies that are calculated for a molecule or TS. The entropy can then be obtained from these partition functions. Thus, electronic structure calculations can be used to compute not only the enthalpy difference between two stationary points but also the entropy and free energy differences.

The ability to compute the free energy difference between two minima on a PES allows the effect of isotopic substitution on the equilibrium constant between them to be calculated. Similarly, a kinetic isotope effect (KIE) can be predicted from the effect of isotopic substitution on the free energy difference between a minimum and a TS (i.e., a free energy of activation).

KIEs are often very sensitive to the lengths of the forming and breaking bonds in the TS especially when rehybridization occurs at the site of isotopic substitution in a TS. By comparing measured KIEs with those computed for different TS geometries, Houk co-workers2 obtained valuable information about the geometries and lengths of the forming and breaking bonds in the TSs for a wide variety of reactions. The use of KIE effect calculations in an attempt to reconcile the results of two experiments on an RI, propane-1,3-diyl, is described in Section 4.3.1.

2.5. Development of Qualitative Models

Perhaps more valuable over time than the quantitative predictions of spectra, structural parameters, and relative enthalpies and entropies of RIs, which can be obtained from electronic structure calculations, are the qualitative models of the electronic structures and reactivities of RIs that emerge from the computational results. Any model, to be successful, must do two things.

First, a successful model must provide explanations of all the experimental and computational results that are known at the time that the model is formulated. Since the electronic structures of RIs are usually rather different from those of stable molecules, a new model, or a modification of an old one, is frequently necessary in order to explain the puzzling chemistry and/or spectroscopy of an RI.

Second, a successful model makes predictions that can be tested experimentally. The formulation of a model that stimulates experiments, whose results may lead to more calculations, is probably the most important type of synergism between calculations and experiments. Section 4 provides some specific examples of this type of partnership in the study of RIs and of the reactions that they undergo.

3. METHODS FOR PERFORMING ELECTRONIC STRUCTURE CALCULATIONS

This section provides a ‘‘beginners’ guide’’ to electronic structure calculations. A much more detailed introduction than is possible here can be found in the authoritative and very readable monograph by Cramer.3a

Until the last decade, for all but a few theoreticians, electronic structure calculations meant calculations based on approximate solution of the time-independent Schro¨dinger equation,

In Eq. 1, H is the Hamiltonian operator, is one of the possible wave functions for all of the electrons in a molecule, and E is the energy associated with each of these possible many-electron wave functions.

However, within the last 10 years, calculations based on density functional theory (DFT) have come into wide use. The DFT calculations are usually much less computationally demanding than wave function based calculations that provide results of comparable accuracy. Hence, DFT calculations are particularly useful for very large molecules, for which wave function based calculations of sufficient accuracy may be too computer intensive to perform.

3.1. The Born–Oppenheimer Approximation

Since a molecule consists of both nuclei and electrons, it may not be at all obvious why calculations can be performed on just the electrons. However, the huge differences in masses between nuclei and electrons justify the Born–Oppenheimer approximation, which allows the motions of the electrons to be separated from the motions of the nuclei.

Using the Born–Oppenheimer approximation, electronic structure calculations are performed at a fixed set of nuclear coordinates, from which the electronic wave functions and energies at that geometry can be obtained. The first and second derivatives of the electronic energies at a series of molecular geometries can be computed and used to find energy minima and to locate TSs on a PES.

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The wave functions for the periodic motions (i.e., vibrations) of the nuclei and the associated vibrational energy levels can then be obtained from the geometries and the second derivatives of the energy at the stationary points. As described in Sections 2.3 and 2.4, the vibrational energies are necessary to correct the difference between the electronic energies of two stationary points into differences between their enthalpies and free energies.

3.2. Wave Function Based Calculations

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in H gives the kinetic energy of each electron in , by computing the second derivative of the electron’s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb’s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each pair of electrons, and (c) the repulsion between each pair of nuclei.

The last set of terms is independent of the electrons and depends only on the coordinates of the nuclei. At a particular geometry, the sum of these terms provides the total nuclear repulsion energy, which is just an additive constant to the electronic energy. Consequently, the terms for the nuclear–nuclear repulsion energy in H can be neglected when attempting to solve Eq. 1 for the electronic wave functions and their associated energies.

Since in Eq. 1 is the wave function for all of the electrons in the molecule, it is simplest to begin trying to find by assuming that it can be approximated as the product of one-electron wave functions, one wave function for each of the electrons in a molecule. These one-electron wave functions are called orbitals, and they are distinguished from the many-electron wave function by using a lower case psi (c) for the former and an upper case psi ( ) for the latter.

In order for to embody the Pauli exclusion principle, it must be an antisymmetrized wave function. Antisymmetrization requires that exchange of any two electrons between orbitals or exchange of the spins between electrons in the same orbital causes to change sign.

Antisymmetrization results in vanishing, not only if two electrons with the same spin occupy the same orbital, but also if electrons with the same spin have the same set of spatial coordinates. Thus, antisymmetrization of results not only in the Pauli exclusion principle but also in the correlation of electrons of the same spin.

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hund’s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins.

The simplest expression for consists of a single configuration, in which each electron is assigned to an orbital with spin up (a) or spin down (b). If two electrons with opposite spin occupy the same orbital, ci, it is notationally more convenient to denote this orbital occupancy by replacing cai cib with 2i .

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, , that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin.

The major difficulty in wave function based calculations is that, starting from an independent-particle model, correlation between electrons of opposite spin must somehow be introduced into . Inclusion of this type of electron correlation is essential if energies are to be computed with any degree of accuracy. How, through the use of multiconfigurational wave functions, correlation between electrons of opposite spin is incorporated into , is the subject of Section 3.2.3.

3.2.1. Hartree–Fock Theory. In the lowest electronic state of most stable molecules the n orbitals of lowest energy are all doubly occupied, thus forming a closed shell. If, at least as a first approximation, such an electronic state is described by a single configuration, the wave function for this state can be written as

where the Dirac ‘‘ket’’ (j i) is used to symbolize that has been antisymmetrized. Minimizing the energy, E, in Eq. 1 with respect to variations in the filled orbitals, c1; c2; . . . ; cm; cn, in Eq. 2, leads to the finding that the optimal orbitals must

satisfy

The operator F in this equation is called the Fock operator, and ei is the energy of orbital ci. According to Koopmans’ theorem, ei is approximately equal to the energy required to ionize a molecule by removing an electron from ci.

Two of the terms in F are the same one-electron operators that appear in H. One of these operators gives the kinetic energy of an electron in ci, and the other computes the attractive Coulombic energy between an electron in ci and each of the nuclei in the molecule.

However, there are two more types of one-electron operators in F. One of them, J j, is called the Coulomb operator. The other, Kj, is called the exchange operator. Together, they replace the two-electron operators, e2=rkl, in H, which give the Coulombic repulsion energy between each pair of electrons, k and l.

2J j Kj operating on ci gives the expression in Hartree–Fock (HF) theory for the effective Coulombic repulsion energy between an electron ci and the pair of

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electrons in cj. In the Fock operator, 2J j Kj is summed over the n doubly occupied orbitals; so the sum, ð2J j KjÞ in F, gives the HF expression for the effective Coulombic repulsion energy between an electron in ci and all the other electrons in a molecule.

The operator 2J j computes the Coulombic repulsion energy between an electron in ci and the pair of electrons in cj, assuming that the electrons in ci and in cj move independently of each other. The operator Kj corrects the Coulombic repulsion energy, computed from 2J j, for the fact that antisymmetrization of results in correlation between an electron in ci and the electron of the same spin in cj.

The operators J i and Ki give the same result when they operate on ci. Thus, 2J i Ki operating on ci has the same effect as just J i operating on ci. This cancellation is physically correct, because an electron in ci only feels the field from the one other electron in ci, not from itself. Thus, the exchange operator, Ki, in F also has the effect of canceling the fictional repulsion between an electron in ci and itself, which is included in 2J i.

The n doubly occupied orbitals in the closed-shell wave function in Eq. 2 are required in order to construct (i.e., compute) the Coulomb and exchange operators in F. Thus, it might seem impossible to use Eq. 3 to find the optimal orbitals for a molecule, because these orbitals must already be known in order to construct the Fock operator in Eq. 3. However, this seemingly impossible task is made possible by using an iterative process.

Starting with a guess as to what the optimal orbitals are, this guess is used to construct the Coulomb and exchange operators in the Fock operator, and Eq. 3 is then used to find an improved set of orbitals. This set of orbitals is used to construct the Coulomb and exchange operators for a new Fock operator; and, by using this new F operator in Eq. 3, another set of ci is obtained. The process is repeated until the set of ci that are used to construct F is essentially the same as the set of ci obtained by using this Fock operator in Eq. 3. These converged orbitals are said to have reached self-consistency, and they are the orbitals that give the lowest possible energy for the HF wave function in Eq. 2.

Since the electric field, computed from the filled ci, is used to construct the Coulomb operator in F, the electric field that is used to construct F from the converged orbitals is the same as the electric field that is computed from the orbitals that solve Eq. 3 for this Fock operator. Therefore, at convergence, both the orbitals and the electric field computed from them are self-consistent. Consequently, HF theory is also known as self-consistent field (SCF) theory.

3.2.2. Basis Sets of Atomic Orbitals and the LCAO–MO Approximation. It seems reasonable to assume that, like isolated atoms, atoms in molecules have orbitals. Therefore, it makes physical sense to try to write the orbitals, ci, for a molecule as linear combinations of the atomic orbitals (LCAOs), fr , of the atoms in the molecule.