Orbital Interactions in Chemistry

By Thomas A. Albright, Jeremy K. Burdett and Myung-Hwan Whangbo

Explains the underlying structure that unites all disciplinesin chemistry

Now in its second edition, this book explores organic,organometallic, inorganic, solid state, and materials chemistry,demonstrating how common molecular orbital situations arisethroughout the whole chemical spectrum. The authors explore therelationships that enable readers to grasp the theory thatunderlies and connects traditional fields of study withinchemistry, thereby providing a conceptual framework with which tothink about chemical structure and reactivity problems.

Orbital Interactions in Chemistry begins by developingmodels and reviewing molecular orbital theory. Next, the bookexplores orbitals in the organic-main group as well as in solids.Lastly, the book examines orbital interaction patterns that occurin inorganic-organometallic fields as well as clusterchemistry, surface chemistry, and magnetism in solids.

This Second Edition has been thoroughly revised andupdated with new discoveries and computational tools since thepublication of the first edition more than twenty-five years ago.Among the new content, readers will find:
* Two new chapters dedicated to surface science and magneticproperties
* Additional examples of quantum calculations, focusing oninorganic and organometallic chemistry
* Expanded treatment of group theory
* New results from photoelectron spectroscopy

Each section ends with a set of problems, enabling readers totest their grasp of new concepts as they progress through the text.Solutions are available on the book's ftp site.

Orbital Interactions in Chemistry is written for bothresearchers and students in organic, inorganic, solid state,materials, and computational chemistry. All readers will discoverthe underlying structure that unites all disciplines inchemistry.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Mar 28, 2013
• ISBN: 9781118558256

Book preview

Preface

Use of molecular orbital theory facilitates an understanding of physical properties associated with molecules and the pathways taken by chemical reactions. The gigantic strides in computational resources as well as a plethora of standardized quantum chemistry packages have created a working environment for theoreticians and experimentalists to explore the structures and energy relationships associated with virtually any molecule or solid. There are many books that cover the fundamentals of quantum mechanics and offer summaries of how to tackle computational problems. It is normally a straightforward procedure to validate a computational procedure for a specific problem and then compute geometries and associated energies. There are also prescriptions for handling solvation. So, does it mean that all a chemist needs to do is to plug the problem into the black-box and he or she will receive understanding in terms of a pile of numbers? We certainly think not.

This book takes the problem one step further. We shall study in some detail the mechanics behind the molecular orbital level structures of molecules. We shall ask why these orbitals have a particular form and are energetically ordered in the way that they are, and whether they are generated by a Hartree–Fock (HF), density functional, or semiempirical technique. Furthermore, we want to understand in a qualitative or semiquantitative sense what happens to the shape and energy of orbitals when the molecule distorts or undergoes a chemical reaction. These models are useful to the chemical community. They collect data to generate patterns and ideally offer predictions about the directions of future research. An experimentalist must have an understanding of why molecules of concern react the way they do, as well as what determines their molecular structure and how this influences reactivity. So too, it is the duty and obligation of a theorist (or an experimentalist doing calculations on the side) to understand why the numbers from a calculation come out the way they do. Models in this vein must be simple. The ones we use here are based on concepts such as symmetry, overlap, and electronegativity. The numerical and computational aspects of the subject in this book are deliberately de-emphasized. In fact there were only a couple of computational numbers cited in the first edition. People sometimes expressed the opinion that the book was based on extended Hückel theory. It, in fact, was and is not. An even more parochial attitude (and unfortunately common one) was expressed recently I imagine that there are still people that do HF calculations too. But these days they cannot be taken too seriously. In this edition, computational results from a wide variety of levels have been cited. This is certainly not to say that computations at a specific level of theory will accurately reproduce experimental data. It is reassuring to chemists that, say, a geometry optimization replicates the experimental structure for a molecule. But that does not mean that the calculation tells the user why the molecule does have the geometry that it does or what other molecules have a similar bonding scheme. The goal of our approach is the generation of global ideas that will lead to a qualitative understanding of electronic structure no matter what computational levels have been used.

An important aim of this book is then to show how common orbital situations arise throughout the whole chemical spectrum. For example, there are isomorphisms between the electronic structure of CH2, Fe(CO)4, and Ni(PR3)2 and between the Jahn–Teller instability in cyclobutadiene and the Peierls distortion in solids. These relationships will be highlighted, and to a certain extent, we have chosen problems that allow us to make such theoretical connections across the traditional boundaries between the subdisciplines of chemistry.

Qualitative methods of understanding molecular electronic structures are based on either valence bond theory promoted largely by Linus Pauling or delocalized molecular orbital theory following the philosophy suggested by Robert Mulliken. The orbital interaction model that we use in our book, which is based on delocalized molecular orbital theory, was largely pioneered by Roald Hoffmann and Kenichi Fukui. This is one of several models that can be employed to analyze the results of computations. This model is simple and yet very powerful. Although chemists are more familiar with valence bond and resonance concepts, the delocalized orbital interaction model has many advantages. In our book, we often point out links between the two viewpoints.

There are roughly three sections in this book. The first develops the models we use in a formal way and serves as a review of molecular orbital theory. The second covers the organic main group areas with a diversion into solids. Typical concerns in the inorganic–organometallic fields are covered in the third section along with cluster chemistry, chemistry on the surface, and magnetism in solids. Each section is essentially self-contained, but we hope that the organic chemist will read on further into the inorganic–organometallic chapters and vice versa. For space considerations, many interesting problems were not included. We have attempted to treat those areas of chemistry that can be appreciated by a general audience. Nevertheless, the strategies and arguments employed should cover many of the structure and reactivity problems that one is likely to encounter. We hope that readers will come away from this work with the idea that there is an underlying structure to all of chemistry and that the conventional divisions into organic, inorganic, organometallic, and solid state are largely artificial. Introductory material in quantum mechanics along with undergraduate organic and inorganic chemistry constitutes the necessary background information for this book.

The coverage in the second edition of this book has been considerably expanded. The number of papers that contain quantum calculations has exploded since the first edition 28 years ago and, therefore, more examples have been given especially in the inorganic–organometallic areas. We have emphasized trends more than before across the Periodic Table or varying substituents. A much fuller treatment of group theory is given and the results from photoelectron spectroscopy have been highlighted. Each self-contained chapter comes with problems at the end, the solutions to which are located at ftp://ftp.wiley.com/public/sci_tech_med/orbital_interactions_2e. Finally, two new chapters, one on surface science and the other on magnetism, have been added.

It is impossible to list all the people whose ideas we have borrowed or adapted in this book. We do, however, owe a great debt to a diverse collection of chemists who have gone before us and have left their mark on particular chemical problems. Dennis Lichtenberger graciously provided us with many of the photoelectron spectra displayed here. The genesis of this book came about when the three of us worked at Cornell University with Roald Hoffmann. This book is dedicated to the memory of our old friend and colleague, Jeremy Burdett, who passed away on June 23, 1997. We would like to thank our wives, Janice and Jin-Ok, as well as our children Alex, Holly, Robby, Jonathan, Rufus, Harry, Jennifer and Albert, for their patience and moral support.

Thomas A. Albright

Jeremy K. Burdett∗

Myung-Hwan Whangbo

April 2012

Note

∗. Deceased

About the Authors

Thomas Albright is currently Professor Emeritus at the Department of Chemistry, University of Houston. He has been awarded the Camille and Henry Dreyfus Teacher Scholar and Alfred P. Sloan Research fellowships. He is the author and coauthor of 118 publications. He has been elected to serve on the Editorial Advisory Board of Organometallics and the US National Representative to IUPAC. His current research is directed toward reaction dynamics in organometallic chemistry. He received his PhD degree at the University of Delaware and did postdoctoral research with Roald Hoffmann at Cornell.

Jeremy Burdett¹ was a Professor and Chair in the Chemistry Department, University of Chicago. He was awarded the Tilden Medal and Meldola Medal by the Royal Chemical Society. He was a fellow of the Camille and Henry Dreyfus Teacher Scholar, the John Gugenheim Memorial, and the Alfred P. Sloan Research foundations. He has published over 220 publications. He received his PhD degree at the University of Cambridge and did postdoctoral research with Roald Hoffmann at Cornell.

Myung-Hwan Whangbo is a Distinguished Professor in the Chemistry Department at North Carolina State University. He has been awarded the Camille and Henry Dreyfus Fellowship, the Alexander von Humboldt Research Award to Senior US Scientists, and the Ho-Am Prize for Basic Science. He is the author and coauthor of over 600 journal articles and monographs. His current research interests lie in the areas of solid-state theory and magnetism. He has been elected to the editorial advisory board of Inorganic Chemistry, Solid State Sciences, Materials Research Bulletin, and Theoretical Chemistry Accounts. He received his PhD degree at Queen's University and did postdoctoral research with Roald Hoffmann at Cornell.

1. Deceased June 23, 1997.

Chapter 1

Atomic and Molecular Orbitals

1.1 Introduction

The goal of this book is to show the reader how to work with and understand the electronic structures of molecules and solids. It is not our intention to present a formal discussion on the tenets of quantum mechanics or to discuss the methods and approximations used to solve the molecular Schrödinger equation. There are several excellent books [1–6], which do this, and many canned computer programs that are readily available to carry out the numerical calculations at different levels of sophistication with associated user manuals [7–9]. The real challenge, and the motivation behind this volume, is to be able to understand where the numbers generated by such computations actually come from. The first part of the book contains some mathematical material using which we have built a largely qualitative discussion of molecular orbital (MO) structure. Let us see how the molecular orbitals of complex molecules or solids may be constructed from smaller portions using concepts from perturbation theory and symmetry. Furthermore, we show how these orbitals change as a function of a geometrical perturbation, the substitution of one atom for another, or as a result of the presence of a second molecule as in a chemical reaction. Many concepts and results together form a common thread, which enables different fields of chemistry to be linked in a satisfying way. The emphasis of this book is on qualitative features and not on quantitative details. Our feeling is that just this perspective leads to predictive capabilities and insight.

1.2 Atomic Orbitals

The molecular orbitals of a molecule are usually expressed as a linear combination of the atomic orbitals (LCAOs) centered on its constituent atoms, which is discussed in Section 1.3. These atomic orbitals (AOs) using polar coordinates have the form shown in equation 1.1. This is a simple product of a function, R(r),

(1.1) equation

which only depends on the distance, r, of the electron from the nucleus, and a function Y(θ, ϕ), which contains all the angular information needed to describe the wavefunction. The Schrödinger wave equation may only be solved exactly for one-electron (hydrogenic) atoms (e.g., H, Li²+) where analytical expressions for R and Y are found. For many-electron atoms, the angular form of the atomic orbitals is the same as for the one-electron atom (Table 1.1), but now, the radial function R(r) is approximated in some way as shown later. The center column in Table 1.1 gives the form of Y in Cartesian coordinate space while that in the far right-hand side uses polar coordinates.

Table 1.1 Angular Components of Some Common Wavefunctions.

Figure 1.1a shows a plot of the amplitude of the wavefunction χ for an electron in a ls orbital as a function of distance from the nucleus. This has been chosen to be the x-axis of an arbitrary coordinate system. With increasing x, the amplitude of χ sharply decreases in an exponential fashion and becomes negligible outside a certain region indicated by the dashed lines. The boundary surface of the s orbital, outside of which the wavefunction has some critical (small) value, is shown in Figure 1.1c. The corresponding diagrams for a 2px orbital are shown in Figure 1.1b, d. Note that the wavefunction for this p orbital changes sign when x → −x. It is often more convenient to represent the sign of the wavefunction by the presence or absence of shading of the orbital lobes as in 1.1 and 1.2. The characteristic features of s, p, and d orbitals

using this convention are shown in Figure 1.2 where the positive lobes have been shaded. Each representation in Figure 1.2 then represents an atomic orbital with a positive coefficient. Squaring the wavefunction and integrating over a volume element gives the probability of finding an electron within that element. So, there is a correspondence between the pictorial representations in Figure 1.2 and the electron density distribution in that orbital. In particular, the probability function or electron density is exactly zero for the px orbital at the nucleus (x = 0). In fact, the wavefunction is zero at all points on the yz plane at the nucleus. This is the definition of a nodal plane. In general, an s orbital has no such angular nodes, a p orbital one node, and a d orbital two. The exact form of R(r) for a 1s hydrogenic atomic orbital is

equation

while that for a 2s atomic orbital is

equation

where a0 is 1 bohr (i.e., 0.5292 Å) and Z is the nuclear charge. When r = 2a0/Z, the wavefunction is zero for the 2s function. This defines a radial node. In general, an atomic wavefunction with quantum numbers n, l, m will have n − 1 nodes altogether, of which the ls are angular nodes, and, therefore, n − l − 1 are radial nodes. (Sometimes it is stated that there are n nodes altogether. In this case, the node that always occurs as r → ∞ is included in the count.) Contour plots of some common atomic orbitals are illustrated in Figure 1.3. The solid lines represent positive values of the wavefunction and the dashed lines negative ones. Dotted lines show the angular nodes. One important feature to notice is that the 2s atomic orbital is more diffuse than the 1s one. A probability density, defined as the probability of finding an electron within a finite volume element, is given for a hydrogenic atom by . The maximum for the plot of this function when Z = 1 occurs at 1a0 for the 1s orbital and 5.2a0 for the 2s orbital. The maximum for the 2p function (Z = 1) occurs at 4a0. These maxima correspond to the most probable distance of finding the electron from the nucleus, in a sense the radius associated with the electron. The value of the radius changes a little when the angular quantum number l varies. However, as one proceeds down a column in the periodic table (i.e., the principal quantum number n is larger), the valence orbitals become progressively more diffuse. The two nodal planes for the dyz (as well as for dxy, dxz, and ) are at right angles to each other. There are two nodal cones associated with the atomic orbital (see the representation in Figure 1.2). In the contour plot of Figure 1.3, the angle made between the node and the z-axis is 54.73°.

Figure 1.1 Radial part of the wavefunction for a ls (a) and 2p (b) orbitals showing an arbitrary cutoff beyond which R(r) is less than some small value. The surface in three dimensions defined by this radial cutoff is shown in (c) for the ls orbital and in (d) for the 2p orbital.

Figure 1.2 Atomic s, p, and d orbitals drawn using the shading convention described in the text.

Figure 1.3 Contour plots for some common atomic orbitals. The solid lines are positive values of the wavefunction and dashed lines correspond to negative ones. The dotted lines plot angular nodes. The distance marker in each plot represents 1 bohr and the value of the smallest contour is 1 bohr³/² for the s and p atomic orbitals and 2 bohr³/² for the two d atomic orbitals. The value of each successive contour is 1/2 of the value of the inner one. These orbitals are STO-3G functions; therefore, there is no radial node associated with the C 2s orbital.

As mentioned earlier, the radial function R(r) for many-electron atoms needs to be approximated in some way. The atomic orbitals most frequently employed in molecular calculations are Slater type orbitals (STOs) and Gaussian type orbitals (GTOs). Their mathematical form makes them relatively easy, especially the latter, to handle in computer calculations. An STO with principal quantum number n is written as

(1.2) equation

where ζ is the orbital exponent. The value of ζ can be obtained by applying the variational theorem to the atomic energy evaluated using the wavefunction of equation 1.2. This theorem tells us that an approximate wavefunction will always overestimate the energy of a given system. So, minimization of the energy with respect to the variational parameter ζ will lead to determination of the best wavefunction of this type. A listing of the energy optimized ζ values for the neutral main group atoms in the periodic table [10] is shown in 1.3. The value of

ζ for the valence s orbital is directly below the atomic symbol and that for the valence p below it. There are no entries for the valence p atomic orbitals in groups 1 and 2 since there are no p electrons for the neutral atoms in their ground state; however, one would certainly want to include these orbitals in a molecular calculation. Note from the functional form of equation 1.2 that when ζ becomes larger, the atomic orbital is more contracted. Therefore, in 1.3, ζ is larger going from left to right across a column in the periodic table; it scales similar to the electronegativity of the atom. The rn−1 factor in the radial portion of the STO ensures that the orbital will become more diffuse and have a maximal probability at a farther distance from the nucleus as one goes down a column. In fact, that distance, rmax, is given by

(1.3) equation

where the effective nuclear charge Z∗ is given by the nuclear charge Z minus the screening constant S, which commonly is determined by a set of empirical rules [4, 5] devised by Slater or more realistic ones from Clementi and Raimondi [11]. Notice also from 1.3 that the values of ζ for the s and p atomic orbitals of an atom are increasingly more dissimilar as one goes down from the second row. This also occurs with Z∗ using the Clementi and Raimondi values. In particular, the valence s orbital becomes more contracted than the p. We have used this result in Chapter 7. A simplified rationale for this behavior can be constructed [12] along the following lines. There is a Pauli repulsion experienced by valence electrons which prevents them from penetrating into the core, since atomic orbitals with the same angular quantum number must be orthogonal. There is, however, a special situation for the first row elements. The 2p atomic orbitals have no corresponding core electrons, so they do not experience the Pauli repulsion that the 2s electrons do from the 1s core. The sizes for the 2s and 2p atomic orbitals in the first row are then similar, whereas in the remaining portion of the periodic table, both s and p core electrons exist and the valence p functions are more diffuse than the s. The STOs in equation 1.2 have no radial nodes, unlike their hydrogenic counterparts. This does not cause any particular problem in a calculation for an atom with, say, 1s and 2s atomic orbitals because of the orthogonality constraint, which is presented in Section 1.3. Sometimes, one may wish to be more exact and choose a double zeta basis set for our molecular calculation made up of wavefunctions of the type

(1.4) equation

where now the atomic energy has been minimized with respect to ζ1 and ζ2. This gives the wavefunction greater flexibility to expand or contract when more or less electron density, respectively, becomes concentrated on the atomic center in a molecule. For example, the valence atomic orbitals of carbon for CH3− should be more diffuse than those for CH3+ because of the presence of two additional electrons. A double or triple zeta basis set allows for this. Furthermore, the STOs for the d orbitals in the transition metals yield radial distributions, which mimic full atomic calculations only when a double zeta formulation is used. Often it is found that observables such as molecular geometry or electron correlation calculations are best carried out by ab initio calculations if polarization functions are added to the basis set. For example, for carbon, nitrogen, and oxygen atoms (n = 2), we might add 3d functions that have the angular function, Y, corresponding to a d orbital and the radial part of equation 1.2 for n = 3. Commonly, p functions are added to the basis set for hydrogen atoms. These polarization functions will lower the total energy calculated for the molecule according to the variation principle, and their inclusion may lead to a better matching of observed and calculated geometries. However, these polarization functions do not generally mix strongly into the occupied molecular orbitals and are not chemically significant. The increased angular nodes of polarization functions tailor the electron density.

A general expression for a Gaussian type orbital is

(1.5) equation

where i, j, k are positive integers or zero and represent the angular portion using Cartesian coordinates. Here α is the orbital exponent. Orbitals of s, p, and d type result when i + j + k = 0, 1, 2, respectively. For example, a px orbital results for i = 1 and j = k = 0. The one major difference between STOs and GTOs is shown in 1.4 and 1.5. Unlike GTOs, STOs are not smooth functions at the origin like their

hydrogenic counterparts. The great convenience of GTOs, however, lies in the fact that evaluation of the molecular integrals needed in ab initio calculations is performed much more efficiently if GTOs are used. In practice, the functional behavior of an STO is simulated by a number of GTOs with different orbital exponents (equation 1.6)

(1.6)

equation

where GTOs with large and small exponents are designed to fit the center and tail portions, respectively, of an STO. If n GTOs are used to fit each STO, then the atomic wavefunctions are of STO-nG quality, using terminology in current usage. The contour plots in Figure 1.3 are in fact STO-3G orbitals. A very common basis set for the main group elements is designated as 3-21G. Here, all orbitals corresponding to the core electrons consist of three primitive Gaussian functions contracted as in equation 1.6 while the valence atomic orbitals are constructed by two primitive Gaussians contracted together and a single Gaussian function which is more diffuse. Thus, they are of the double zeta quality for the valence region. A much more accurate basis, normally restricted to atoms of the first and second rows in the periodic table, is 6-311G∗∗. Now there are six primitive Gaussians contracted to one for the core, a triple zeta formulation for the valence where three, one, and one Gaussians are used, and d polarization functions are added for all atoms except hydrogen, which uses p functions. There is considerable choice as to the basis set (equations 1.2–1.6) and indeed of the exponents, ζ, themselves. In practice, the details of the basis set chosen for a given problem rely heavily on previous experience [6, 7, 13, 14].

1.3 Molecular Orbitals

For a molecule with a total of m atomic basis functions {χl, χ2, ..., χm}, there will be a total of m resultant molecular orbitals constructed from them. For most purposes, these atomic orbitals can be assumed to be real functions and normalized (equation 1.7) such that the probability of finding an electron in χμ when integrated over all space is unity. Here χμ∗ is the complex conjugate of χμ. In equation 1.8, we show an alternative, useful way of writing such integrals.

(1.7) equation

(1.8) equation

The molecular orbitals of a molecule are usually approximated by writing them as a linear combination of atomic orbitals such that

(1.9)

equation

where i = 1, 2, ..., m. These MOs are normalized and orthogonal (i.e., orthonormal), namely,

(1.10) equation

where δij = 1 if i = j and δij = 0 if i ≠ j. Note that the sum in equation 1.9 runs over all the atomic orbitals of the basis set. The cμis are called the molecular orbital coefficients. They may be either positive or negative, and the magnitude of the coefficient is related to the weight of that atomic orbital in the molecular orbital. An organizational note is in order here. We shall use Greek characters to represent atomic orbitals and the Roman alphabet in italics for molecular orbitals in generalized situations. For the mixing coefficients, the former will always be indexed before the latter. Thus, cμi stands for the mixing coefficient of the μth atomic orbital in the molecular orbital i for a general situation and c12 represents that for atomic orbital 1 in molecular orbital 2 in a specific situation. Equation 1.9 is perhaps at first sight the most frightening aspect of delocalized molecular orbital theory. For a molecule of any reasonable size, this obviously represents quite a large sum. In fact, not all of the cμi will be significant in a given molecular orbital ψi. We shall learn how to gauge this using perturbation theory in Chapter 3. Some will be exactly zero, forced to be so by the symmetry of the molecule. In general, the more symmetric the molecule, the larger the number of cμis which are zero. Furthermore, symmetry requirements often dictate relationships (sign and magnitude) between orbitals on different atoms. This is covered in Chapter 4. We devote a considerable amount of effort to provide simple ways to understand how and why the orbital coefficients in the molecular orbitals of molecules and solids turn out the way they do.

The molecular orbital coefficients cμi (μ, i = 1, 2, ..., m) which specify the nature, and hence, energy of the orbital ψi, are determined by solving the eigenvalue equation of the effective one-electron Hamiltonian, Heff, associated with the molecule (equation 1.11):

(1.11) equation

We shall leave for the moment what Heff exactly is and discuss this more fully in Chapters 2 and 8. The resultant energy ei measures the effective potential exerted on an electron located in ψi. This molecular orbital energy is the expectation value of Heff, that is,

(1.12) equation

(1.13) equation

Given two atomic orbitals χμ and χν centered on two different atoms, the overlap integral Sμν is defined as

(1.14) equation

Its origin is clear from the spatial overlap of the two wavefunctions in 1.6, where we have chosen two ls orbitals from Figure 1.1 as examples. An alternative representation 1.7 shows this in terms of two orbital lobes. For the purposes of graphical clarity, this is better written as in 1.8. According to the sign convention of 1.1, the overlap integrals in 1.9 and 1.10 are given by equations 1.15 and 1.16, respectively. This simply shows

(1.15) equation

(1.16) equation

that the overlap integral between two orbitals is positive when lobes have the same sign within the internuclear region of overlap and negative when the two lobes have opposite signs within this region.

The qualitative magnitude of the overlap integral is a principal topic of concern throughout this book. When two orbitals interact with each other, the extent of the interaction is determined by their overlap. There are several ways to gauge this without recourse to numerical calculation. As indicated earlier, symmetry often will dictate whether the overlap integral is precisely zero (or not). This is covered in Chapter 4. Second, the type of overlap will frequently determine its magnitude in a qualitative sense. Figure 1.4 shows pictorially some of the various types of overlap integrals that are encountered in practice. The σ type overlaps shown in Figure 1.4a–d contain no nodes along the internuclear axis, the π type overlaps (Figure 1.4e–g) are between orbitals with one nodal plane containing this axis, and those of δ type (Figure 1.4h, i) contain two such nodal planes. Nodes along the internuclear axis decrease the mutual overlap between orbitals and, therefore, the important general result is that the overlap integral varies in the order σ > π > δ. There are, of course, many exceptions to this rule of thumb that can be presented, that is, the overlap between two uranium 1s atomic orbitals will be smaller than the π overlap between two carbon 2p orbitals. However, when one considers valence orbitals from atoms in the same row of the periodic table, then this order is universal. Third, overlap depends on the n quantum number of the atomic orbitals involved. From Section 1.2 recall that the atomic orbital becomes more diffuse as n increases; this in turn normally creates a smaller overlap. Thus, the overlap between two 3p atomic orbitals will be less than that between two 2p orbitals. A cautionary note needs to be added here. Overlap, as we shall see, is very sensitive to the internuclear distance between the two atoms. It does not immediately follow that, for example, the π overlap between two boron 2p atomic orbitals is less than that between two fluorine atoms because boron is much less electronegative than fluorine and, consequently, its orbitals are more diffuse. The two distances are certainly going to be quite different and each will have a maximal overlap at a different distance. In transition metal complexes, one also has a situation that runs counter to the generalization just given. The metal 3d orbitals are actually so contracted that at reasonable metal– ligand distances, 4d and 5d valence atomic orbitals actually overlap with the ligand orbitals to a greater extent than the 3d valence orbitals do. The contracted 3d atomic orbitals compared to 4d and 5d counterparts will also play an important role in determining spin states (Chapters 15, 16 and 24). Last, overlap is very sensitive to the geometry present in a molecule or solid. The variation of the overlap integral with the distance between the two atomic centers depends in detail on the form of R(r) chosen in equation 1.1, but clearly will approach zero at large internuclear distances. When the two interacting orbitals are identical, the overlap integral will be unity when the separation is zero as shown by equation 1.7 for this hypothetical example. A complete S versus r curve for the case of two ls orbitals is shown in 1.11. It may be readily seen from Figure 1.1 that the overlap between an s orbital and a p orbital at r = 0 is identically zero, as shown in 1.12.

Figure 1.4 Types of overlap integrals between atomic orbitals, (a)–(d) correspond to σ overlap, (e)–(g) correspond to π overlap, and (h), (i) correspond to δ overlap.

Maximal overlap will occur at some finite value of r which depends on the magnitude of the orbital exponents for the two atoms. The angular dependence of the overlap integral follows immediately from the analytic form of Y(θ, ϕ) in equation 1.1 and expressed in Table 1.1. We can often write the overlap integral as in equation 1.17:

(1.17) equation

depends on the distance between the two orbitals and the nature (λ = σ, π, or δ) of the overlap between them. It is also, of course, strongly dependent upon the identity of the atoms on which the orbitals μ and ν are located. The angular geometry dependent term is independent of the nature of the atoms themselves and only depends on the description (s, p, or d) of the two orbitals [15]. The angular variations of some of the more common types of overlap integral are shown in Figure 1.5. In the first three examples, the overlap is precisely zero when the probe s atomic orbital enters the nodal plane of the other orbital. Also notice that the overlap with a (in terms of absolute magnitude) is considerably less at the torus than along the z-axis. The angular variations displayed in Figure 1.5 will be used many times in this book.

Figure 1.5 Angular dependence of the overlap integral for some commonly encountered pairs of atomic orbitals.

The energy of interaction associated with two overlapping atomic orbitals χμ and χν is given by

(1.18) equation

The diagonal element Hμμ (when ν = μ in equation 1.18) refers to the effective potential of an electron in the atomic orbital χμ. It then has some relationship to the ionization potential of an electron in χμ, which will be modified by the effective field of the other electrons and nuclei in the molecule. The off-diagonal element Hμν is often called the resonance or hopping integral. It measures the potential of an electron when it is associated with χμ and χν. The magnitude of Hμν will then determine how much a bonding molecular orbital is stabilized and an antibonding one destabilized. It can be approximated by the equation

(1.19) equation

which is known as the Wolfsberg–Helmholtz formula. (K is a proportionality constant.) Since the Hμνs are negative quantities, Hμν ∝ −Sμν, which implies that the interaction energy between two orbitals is negative (i.e., stabilizing) when their overlap integral is positive. There are a number of ways to compute Hμν depending upon the level of approximation. The important result, however, is that, whatever the exact functional form, there is a direct relationship between Hμν and Sμν. Furthermore, as indicated earlier, there are a number of ways to gauge the magnitude of Sμν (and, hence Hμν) in a qualitative sense.

The overlap integral, Sμν, and the interaction integral Hμν are symmetric such that Sμν = Sνμ and Hμν = Hνμ. (This second equality arises because of the Hermitian properties of the Hamiltonian.) For an arbitrary function ψi (equation 1.9), the integrals needed in equation 1.12 may be written as

(1.20) equation

and

(1.21) equation

If ψi is an eigenfunction of Heff, then it will be normalized to unity and equation 1.12 will result. However, for an arbitrary ψi, A will not be equal to unity. From equation 1.12, the energy ei is given by

(1.22) equation

According to the variational theorem, the coefficients cμi (recall that μ, i = 1, 2, ..., m for m atomic orbitals) are chosen such that the energy is minimized, that is,

(1.23) equation

For any arbitrary coefficient cκi (κ = 1, 2, ..., m)

(1.24) equation

Therefore,

(1.25) equation

Since the indices μ and ν in equations 1.20 and 1.21 are only used for summation,

(1.26) equation

Similarly,

(1.27) equation

Combining equations 1.25–1.27,

(1.28)

equation

Here as a reminder, i indexes the molecular orbital level while μ, ν, and κ index the μth, νth, and κth atomic orbitals, respectively, in the LCAO expansion of equation 1.9. Equation 1.28 is satisfied for κ = 1, 2,..., m, and the explicit form of these m equations called the secular equations, is

(1.29)

equation

A well-known mathematical result from the theory of such simultaneous equations requires the following determinant, called the secular determinant, to vanish.

(1.30)

equation

Solution of the polynomial equation that results from expansion of the secular determinant equation 1.30 provides m orbital energies ei (i = 1, 2,..., m) which, according to the variational theorem, are a set of upper bounds to the true orbital energies. Written in matrix notation, equation 1.30 becomes

(1.31) equation

As seen in Chapter 2, the coefficients cμi are determined from the secular equations (equation 1.29) and the normalization condition

(1.32) equation

The reader should not despair at the complexity introduced by equations 1.29 and 1.30. Symmetry and perturbation theory will allow us to treat any problem as an example of two or three orbitals interacting with each other. The former will be explicitly treated in Chapter 2 using equations 1.29 and 1.30.

Problems

1.1. Consider the H3 molecule composed of one atomic s AO on each atom with the linear geometry shown below:

a. Write down the secular determinant and equations for the general case (not specifying anything about r12 and r23.

b. Let r12 = r23; write the new secular determinant.

c. Now, suppose that the r13 distance is long enough so that S13 ≈ 0. Simplify the secular determinant further.

d. Using the results from (c) let H11 = −13.60 eV, H12 = −14.18 eV and S12 = 0.596. Solve for e1 − e3 and determine the orbital coefficients for ψ1 − ψ3.

e. For H3 in a geometry given by and equilateral triangle write down the secular determinant and equations. Using the parameters in part (d) compute the eigenvalues and eigenvectors associated with each MO.

1.2. Draw a qualitative sketch of Sμν for each of the situations shown below.

1.3. Given a set of AOs {χ1, χ2}, the LCAO-MOs ψi (i = 1, 2) are obtained by solving the Schrödinger equation

equation

This equation gives rise to the following matrices defined in terms of the AOs:

equation

a. What is the relationship between the above four matrices?

b. The elements of the matrix are defined in terms of the MOs as (i, j = 1, 2). Likewise, the elements of the matrix are defined in terms of the MOs as (i, j = 1, 2). Show the elements of the matrices and .

c. What is the terminology describing the transformation from H to ?

Note

1. Solutions to chapter problems are located at ftp://ftp.wiley.com/public/sci_tech_med/orbital_interactions_2e.

References

1. T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic-Structure Theory, John Wiley & Sons, Chichester(2000).

2. J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw-Hill, New York(1970).

3. M. J. S. Dewar, The Molecular Orbital Theory of Organic Chemistry, McGraw-Hill, New York(1969).

4. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam(2007).

5. W. Kutzelnigg, Einführung in die Theoretische Chemie, Band 2, Verlag Chemie, Weinheim(1978).

6. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, McGraw-Hill, New York(1989).

7. W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molecular Orbital Theory, John Wiley & Sons, New York(1986).

8. J. B. Foresman andÆ. Frisch, Exploring Chemistry with Electronic Structure Methods,2nd edition, Gaussian Inc., Pittsburgh(1996).

9. F. Jensen, Introduction to Computational Chemistry,2nd edition, John Wiley & Sons, Chichester(2007);

C. J. Cramer, Essentials of Computational Chemistry,2nd edition, John Wiley & Sons, Chichester(2004).

10. E. Clementi and C. Roetti, At. Nucl. Data Tables, 14, 177(1974).

11. E. Clementi and D. L. Raimondi, J. Chem. Phys., 38, 2868(1963).

12. W. Kutzelnigg, Angew. Chem. Int. Ed., 23, 272(1984).

13. R. Poirier, R. Kari, and I. G. Csizmadia, Handbook of Gaussian Basis Sets, Elsevier, Amsterdam(1985);

S. Huzinaga, Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam(1984).

14. K. L. Schuchardt, B. T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J. Chase, J. Li, and T. L. Windus, J. Chem. Inf. Model., 47, 1045(2007). A rather complete collection of basis sets may be found at https://bse.pnl.gov/bse/portal.

15. J. K. Burdett, Molecular Shapes, John Wiley & Sons, New York(1980).

Chapter 2

Concepts of Bonding and Orbital Interaction

2.1 Orbital Interaction Energy

The derivations of Chapter 1 were very general ones. Here we look in some detail at the illustrative case of a two-center two-orbital problem. Two atomic orbitals, χ1 and χ2, are centered on the two atoms A and B (2.1). (In Chapter 3, we show how

the results can be generalized to the case of two orbitals located on molecular fragments A and B.) The molecular orbitals (MOs) resulting from the interaction between χl and χ2 can be written as:

(2.1) equation

For the mixing coefficients, cμi, we use the convention that the first subscript refers to the atomic orbital and the second to the molecular orbital. The overlap and interaction integrals to consider are as follows:

(2.2) equation

and

(2.3) equation

Recall from Section 1.3 that S12 = S2l so that H12 = H21. If the phases of χ1 and χ2 are arranged so that S12 is positive, then from equation 1.19,

(2.4) equation

The molecular orbital energies in this two-orbital case, ei (i = 1, 2), are obtained by solving the secular determinant (equation 1.30) shown in equation 2.5 for this particular example

(2.5) equation

Expansion of equation 2.5 leads to

(2.6) equation

The solutions for ei in this equation will be examined for a degenerate case and for the general nondegenerate case .

2.1.1 Degenerate Interaction

For , solution of equation 2.6 leads to two values for the ei (i = 1, 2)

(2.7) equation

When the interaction between χ1 and χ2 is not strong (Sl2 is small), some very useful mathematical approximations may be used to simplify equation 2.7. Using the first two expressions in Table 2.1, results in equations 2.8 and 2.9;

(2.8)

equation

(2.9)

equation

For any realistic case, is negative and normally is negative too (i.e., ). Hence, ψ1 is stabilized by the presence of the second term in equation 2.8, but ψ2 is destabilized by the second term in equation 2.9. Both levels are destabilized by the third term in equations 2.8 and (2.9). These results are shown pictorially in 2.2. The important result is that with respect to the atomic orbital at an

energy the raising (destabilization) of the level is greater than the lowering (stabilization) of the e1 level. The origin of this effect is easy to see. It arises because the orbitals χ1 and χ2 are not orthogonal (i.e., Sl2 ≠ 0). Putting Sl2 = 0 in equation 2.7 leads to and this asymmetry disappears.

Table 2.1 Some Mathematical Simplifications.

Putting electrons into these resultant molecular orbitals allows calculation of the total interaction energy, ΔE, on bringing together the two atomic orbitals χ1 and χ2. Two important cases are shown in 2.3 and 2.4, the two-orbital two-electron case and the two-orbital four-electron case, respectively. These orbital interaction diagrams

indicate the relative energy of the starting and resultant orbitals by use of heavy bars drawn in the horizontal direction. Then, the vertical axis is a scale of the energy associated with each orbital and the tie-lines show which orbitals interact with each other. The small vertical lines represent electrons. Using the results of equations 2.8 and (2.9), and weighting each orbital energy by the number of electrons in that orbital leads to

(2.10)

equation

(2.11)

equation

Since for Sl2 > 0 the term is negative, the two-orbital–two-electron interaction is stabilizing (i.e., ΔE(2) < 0) but the two-orbital–four electron interaction is destabilizing (i.e., ΔE(4) > 0).

The arrangement shown in 2.3 is not the only way to put two electrons into these two molecular orbitals. An alternative pattern is shown in 2.5 with a total interaction energy of ΔE(4)/2. The pattern in 2.5 is called the high-spin case, to be contrasted with the low-spin arrangement of 2.3, where they are paired. We shall

consistently employ arrows throughout the book to indicate the electron spin when it is important, as in 2.5. The stability of the high-spin state compared to the low-spin state will be examined in detail in Chapter 8. As a general rule of thumb, when the interaction between the atomic orbitals is strong, the resultant molecular orbitals are split by a moderate to large amount, and the low-spin situation is favored. When the two molecular orbitals are degenerate or close together in energy then the high-spin arrangement is more stable. This is the molecular analog of Hund's rule.

2.1.2 Nondegenerate Interaction

When without loss of generality may be assumed to be lower in energy than , that is, . Rearrangement of equation 2.6 leads to

(2.12)

equation

and the solutions of this quadratic equation are given by

(2.13) equation

where

(2.14) equation

with

equation

Approximate expressions for el and e2 are found as follows. First D can be expanded as

(2.15)

equation

From Table 2.1,

(2.16)

equation

assuming a small interaction between χ1 and χ2 as before. We have a negative sign in front of (<0) to ensure that . By manipulation of equations 2.13, 2.14 and (2.16),

(2.17)

equation

and so

(2.18) equation

A similar expression is found analogously for e2:

(2.19) equation

The orbital energies are shown pictorially in 2.6. As a result of the interaction, the lower level is depressed in energy, and the higher level is raised in energy.

Notice that since . In other words, the higher energy orbital is destabilized more than the lower energy orbital is stabilized, just as found for the degenerate case above. The total interaction energies for the analogous two-orbital two-electron and the two-orbital four-electron cases of 2.7 and 2.8 are simply obtained. Since , ΔE(2) is negative, that is, the

equation

(2.20) equation

equation

(2.21) equation

two-orbital–two-electron interaction is stabilizing. We have already noted that and are negative if Sl2 > 0 and thus ΔE(4) is positive, that is, the two-orbital–four-electron interaction is destabilizing.

2.2 Molecular Orbital Coefficients

The MO coefficients cli and c2i of equation 2.1 are determined from the simultaneous equation 1.29 (shown for the present case in equation 2.22) and the normalization condition, equation 2.23.

equation

(2.22) equation

(2.23) equation

The coefficients cli and c2i for i = 1, 2 will be obtained for the degenerate and nondegenerate cases described earlier.

2.2.1 Degenerate Interaction

Since either of the equations 2.22 leads to

(2.24) equation

and so from equation 2.23,

(2.25) equation

which leads to

(2.26) equation

The coefficients of the ψ2 molecular orbital are obtained in a similar manner

(2.27) equation

Use of the normalization condition leads to

(2.28) equation

The nodal properties of the MOs ψ1 and ψ2 are shown in the orbital interaction diagram, Figure 2.1, where the positive signs from equations 2.26 and 2.28 are arbitrarily chosen. Equation 1.11 shows that if ψi is an eigenfunction of Heff, so is −ψi. What is important, therefore, is not the overall sign of the MO ψi, but the relative signs of its MO coefficients. Irrespective of the overall sign chosen for ψi, the important point is that χ1 and χ2 are combined in-phase for the lower lying orbital ψ1 and out-of-phase in the higher lying orbital ψ2. Henceforth, we only show one sign for our MOs. Contour plots for ψ1 and ψ2 (σ and σ∗ orbitals, respectively) in H2 using an Slater type orbital (STO)-3G basis set are shown in Figure 2.2. The solid contours plot the positive values of the wavefunction and the dotted lines negative ones. The dashed line indicates the nodal plane in σ∗, which bisects the H–H internuclear axis.

Figure 2.1 Molecular orbital diagram showing details of the degenerate interaction between the two atomic s orbitals, χl, and χ2.

Figure 2.2 Contour plots of the σ and σ∗ molecular orbitals of H2. The positive and negative values of the wavefunction are represented by solid and dotted lines, respectively.

While , it is clear from equations 2.26 and 2.28 that c11 ≠ cl2. This is a consequence of the relationship 1 > Sl2 > 0. The general result is that the atomic coefficients for the higher lying level in Figure 2.1 are larger than those for the lower lying level. This is also evident from the contour plots of the σ and σ∗ molecular orbitals for H2 in Figure 2.2.

2.2.2 Nondegenerate Interaction

From equations 2.18 and (2.22),

(2.29) equation

where . From Table 2.1, this ratio may be rewritten as:

(2.30) equation

by neglecting terms greater than second order in t and Sl2. Using the normalization condition (equation 2.23) and this result

(2.31)

equation

From Table 2.1, this may be rearranged and approximated as

(2.32) equation

Combined with equation 2.30,

(2.33) equation

where, as before, terms greater than second order in t and Sl2 have been neglected. The final form of the MO ψ1 is then

(2.34) equation

with a similar expression for ψ2

(2.35) equation

where . The two functions t and t′ are often called the mixing coefficients because t, for example, describes how orbital χ2 mixes into χ1 to give an orbital still largely χ1 in character.

Since invariably ,

(2.36) equation

and

(2.37) equation

where the symbols (+) and (−) indicate that the mathematical quantities represented by the parentheses have the positive and negative signs, respectively. In a normal case of orbital interaction, therefore, the higher energy orbital χ2 mixes in-phase into the lower level χ1 to give the lower lying MO ψ2, whereas the lower level χ1 mixes out-of-phase into the higher level χ2 to give the higher lying MO ψ2. The magnitudes of the mixing coefficients t and are small when Sl2 and H12 are small. Hence, the major orbital character of the lower lying MO, ψ1, is given by the lower atomic orbital χ1. Conversely, the major orbital character in ψ2 is contributed by χ2. As the two levels and become closer in energy, the weight of the higher atomic level χ2 in the lower lying MO ψ1 increases, as does the weight of χ1 in ψ2. As examined in the degenerate case, when , χ1 and χ2 have equal weights in ψ1 and ψ2. The nodal properties of the MOs ψ1 and ψ2 are illustrated in Figure 2.3. One last graphical convention is needed here to represent the qualitative features of molecular orbitals. The relative magnitudes of the coefficients cli and c2i (i = 1, 2) are represented by the relative sizes of the orbital lobes χ1 and χ2, respectively. Here again it may easily be shown that , or in other words, that the atomic coefficients for the high-lying level ψ2 in Figure 2.2 will be larger than those for the low-energy combination ψ1.

Figure 2.3 Molecular orbital diagram showing details of the nondegenerate interaction between two atomic s orbitals, χl and χ2.

2.3 The Two-Orbital Problem—Summary

The two-orbital problem is extremely important in that many of the bonding situations in chemistry can be distilled into just this form. We have waded through a laborious mathematical derivation. Let us review what we have uncovered thus far.

The qualitative aspects of the energy associated with orbital interactions are summarized in Table 2.2, which shows that:

1. In both degenerate and nondegenerate cases, the resultant upper molecular level is destabilized more than the lower molecular level is stabilized.

2. Regardless of whether the orbital picture contains two or four electrons, the magnitude of the total interaction energy increases with increasing overlap.

3. In a nondegenerate orbital interaction, the magnitude of the interaction energy is inversely proportional to the energy difference between the interacting orbitals.

4. In both degenerate and nondegenerate orbital interaction cases, a two-orbital–two-electron interaction is stabilizing, while a two-orbital–four-electron interaction is destabilizing.

Table 2.2 Summary of Orbital Interactions.

It is worth mentioning that the destabilization associated with the two-orbital–four-electron situation is behind the nonexistence of a bound molecule for He2 or Ne2, which have this orbital situation. The situation is complicated for three electrons. Using equations 2.8 and 2.9 for the degenerate case, along with equations 2.18 and 2.19 for the nondegenerate one, we find that there is a net stabilization still present as long as S12 remains small. However, when the overlap becomes large there is a critical point (S12 = 1/3 for the degenerate situation) when the net interaction becomes repulsive.

In any two-orbital interaction, the resultant molecular orbitals display the following patterns:

1. The lower (more stable) molecular orbital is always mixed in-phase (bonding), and the upper molecular orbital is out-of-phase (antibonding) for the degenerate and nondegenerate cases. Thus, the lower molecular orbital contains no nodes perpendicular to and contained within the internuclear axis, and the upper level contains one such node.

2. In the degenerate and nondegenerate cases, the mixing coefficients for the antibonding orbitals are larger than their bonding counterparts.

3. For the nondegenerate situation, the molecular orbital most strongly resembles that starting atomic orbital closest to it in energy. The reader is referred to Figure 2.3.

The results here are very general and will be used throughout the course of this book. They will also apply to situations wherein one or both of the starting orbitals are not atomic orbitals but molecular orbitals from a fragment, which is covered in Chapter 3.

One point that frequently causes concern is the placement of starting and resultant orbitals in an orbital interaction diagram, for example, that shown in Figure 2.3 for a nondegenerate case. There are two qualitative aspects that must be considered. First, the amount that χ1 is stabilized and χ2 is destabilized (relative to the starting energies, and , respectively) after interaction is directly proportional to . From equation 1.19 recall that this is proportional to ; a detailed discussion of the factors that influence S12 has been given in Section 1.3. The stabilization and destabilization of the resultant molecular orbitals are also inversely dependent on the energy gap between χ1 and χ2, . Second, we must have some idea about where to position the energy of χ1 relative to that of χ2. The experimental state averaged ionization potentials, in electron volts, for the main group atoms are shown in 2.9 [1]. However, those p atomic orbitals for groups 1 and 2 and for the s and p

orbitals of the sixth row are not experimentally known and hence, calculated values (indicated by an asterisk) have been used [2, 3]. The latter include relativistic corrections. The general trends are easy to see. As we proceed from the left to the right in any row, the s and p orbitals go down in energy. This is a consequence of the fact that the valence electrons do not screen each other effectively. Thus, the addition of one proton to the nucleus and one electron does not cancel; instead, the valence electrons feel an increased nuclear charge. This is especially true for the s electrons because they penetrate closer to the nucleus than the p electrons do. Therefore, the s–p energy gap increases on going from left to right in the periodic table. The valence orbitals become more diffuse and the most probable distance of the electron to the nucleus increases as one descends a column in the periodic table. The energies of the valence electrons consequently increase. There are, however, two exceptions. The filled 3d shell of electrons does not completely screen the 4s and 4p electrons. This effect is more important for the 4s electrons so that the 4s orbitals of Ga, Ge, and As are actually lower in energy than the 3s orbitals of Al, Si, and P, respectively. Second, there is an important relativistic effect at work for the sixth row. The heavy mass of the nucleus for Tl through Bi causes a contraction of the inner s and p shells, which is transmitted out to the valence region. Again, this is more important for the 6s electrons than the 6p because of the greater penetration of the former. We might think that the s–p energy gap will decrease as one goes down a column in the periodic table since the valence orbitals become more diffuse. This does, indeed, happen in comparing the second and third rows. However, this is not a general phenomenon because the two factors, as just discussed, operate in the opposite direction. It is these considerations that yield screening constants and effective nuclear charges, discussed in Section 1.2. The values of the valence orbital energies in 2.9 should not be taken in a quantitative fashion when constructing an orbital interaction diagram. They merely are a guide. The values of and will also be sensitive to charging effects in the molecular environment. The most common way [4] to incorporate charging effects is to scale the orbital energies by

equation

where q is the charge computed for the atom in the molecule, A and B are constants that depend on the atom type, and C is the orbital energy given in 2.9. A more useful guide to qualitative placement of orbital energies is the electronegativity of the atom. Electronegativity has been defined in many ways, perhaps the most common being the Pauling, Mulliken, and Allred–Rochow scales. The particular formulation, which we use, was developed by Allen [1]. Here, the electronegativity, χspec, is defined as

equation

where and are the valence p and s energies, respectively, taken from 2.9, m and n are the number of valence p and s electrons, respectively, and K is a single scale factor, which sets the electronegativity values in the Allen scale on par to those from the Pauling and Allred–Rochow scales. A plot of χspec is shown in Figure 2.4. This very conveniently encompasses all of the trends that we have just discussed. Namely, the electronegativity increases going from the left to the right along a row (the vertical direction in Figure 2.4). Along a column, it always decreases from the second to third rows and then is relatively constant with only minor decreases or in some cases even increases.

Figure 2.4 Plot of the electronegativity versus row for the main group atoms in the periodic table.

2.4 Electron Density Distribution

One way that provides further insight into the energy changes that occur when χ1 and χ2 are allowed to interact is to use equation 1.13 along with the form of the ψi to calculate the new orbital energies

(2.38)

equation

Here, terms greater than second order in t and S12 have been omitted. It is easy to show that equations 2.38 and 2.18 are identical. An analogous equation holds for e2:

(2.39) equation

The origin of the various terms