<br><h3> Chapter One </h3> <b>Molecular Orbital Theory <p> <p> 1.1 The Atomic Orbitals of a Hydrogen Atom</b> <p> To understand the nature of the simplest chemical bond, that between two hydrogen atoms, we look at the effect on the electron distribution when two atoms are held within bonding distance, but first we need a picture of the hydrogen atoms themselves. Since a hydrogen atom consists of a proton and a single electron, we only need a description of the spatial distribution of that electron. This is usually expressed as a <i>wave</i> function [phi], where [[theta].sup.2]<i>d]<i>tau] is the probability of finding the electron in the volume <i>d]<i>tau], and the integral of [[phi].sup.2]<i>d]<i>tau] over the whole of space is 1. The wave function is the underlying mathematical description, and it may be positive or negative; it can even be complex with a real and an imaginary part, but this will not be needed in any of the discussion in this book. Only when squared does it correspond to anything with physical reality-the probability of finding an electron in any given space. Quantum theory gives us a number of permitted wave equations, but the only one that matters here is the lowest in energy, in which the distribution of the electron is described as being in a 1s orbital. This is spherically symmetrical about the nucleus, with a maximum at the centre, and falling off rapidly, so that the probability of finding the electron within a sphere of radius 1.4 is 90 % and within 2 better than 99%. This orbital is calculated to be 13.60 eV lower in energy than a completely separated electron and proton. <p> We need pictures to illustrate the electron distribution, and the most common is simply to draw a circle, Fig. 1.1a, which can be thought of as a section through a spherical contour, within which the electron would be found, say, 90%of the time. This picture will suffice for most of what we need in this book, but it might be worth looking at some others, because the circle alone disguises some features that are worth appreciating. Thus a section showing more contours, Fig. 1.1b, has more detail. Another picture, even less amenable to a quick drawing, is to plot the electron distribution as a section through a cloud, Fig. 1.1c, where one imagines blinking one's eyes a very large number of times, and plotting the points at which the electron was at each blink. This picture contributes to the language often used, in which the electron population in a given volume of space is referred to as the electron density. <p> Taking advantage of the spherical symmetry, we can also plot the fraction of the electron population outside a radius <i>r</i> against <i>r</i>, as in Fig. 1.2a, showing the rapid fall off of electron population with distance. The van der Waals radius at 1.2 has no theoretical significance-it is an empirical measurement from solid-state structures, being one-half of the distance apart of the hydrogen atom in a C-H bond and the hydrogen atom in the C-H bond of an adjacent molecule. It does not even have a fixed value, but is an average of several measurements. Yet another way to appreciate the electron distribution is to look at the radial density, where we plot the probability of finding the electron between one sphere of radius <i>r</i> and another of radius <i>r + dr</i>. This has a revealing form, Fig. 1.2b, with a maximum 0.529 from the nucleus, showing that, in spite of the wave function being at a maximum at the nucleus, the chance of finding an electron precisely there is very small. The distance 0.529 proves to be the same as the radius calculated for the orbit of an electron in the early but untenable planetary model of a hydrogen atom. It is called the Bohr radius [<i>a</i>.sub.0], and is often used as a unit of length in molecular orbital calculations. <p> <p> <b>1.2 Molecules Made from Hydrogen Atoms <p> <i>1.2.1 The [H.sub.2] Molecule</i></b> <p> To understand the bonding in a hydrogenmolecule, we have to see what happens when two hydrogen atoms are close enough for their atomic orbitals to interact. We now have two protons and two nuclei, and even with this small a molecule we cannot expect theory to give us complete solutions. We need a description of the electron distribution over the whole molecule-a molecular orbital. The way the problem is handled is to accept that a first approximation has the two atoms remaining more or less unchanged, so that the description of the molecule will resemble the sum of the two isolated atoms. Thus we combine the two atomic orbitals in a linear combination expressed in Equation 1.1, where the function which describes the new electron distribution, the <i>molecular orbital</i>, is called [sigma] and [[phi].sub.1] and [[phi].sub.2 are the atomic 1s wave functions on atoms 1 and 2. <p> [sigma] = [<i>c</i>.sub.1] [[phi].sub.1] + [<i>c</i>.sub.2] [[phi].sub.1] (1.1) <p> The coefficients, [<i>c</i>.sub.1] and [<i>c</i>.sub.2], are a measure of the contribution which the atomic orbital is making to the molecular orbital. They are of course equal in magnitude in this case, since the two atoms are the same, but they may be positive or negative. To obtain the electron distribution, we square the function in Equation 1.1, which is written in two ways in Equation 1.2. <p> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2) <p> Taking the expanded version, we can see that the molecular orbital [[sigma].sup.2] differs from the superposition of the two atomic orbitals [([<i>c</i>.sub.1] [[phi].sub.1]).sup.2] + [([<i>c</i>.sub.2] [[phi].sub.2]).sup.2] by the term 2[<i>c</i>.sub.1] [[phi].sub.1] [<i>c</i>.sub.2] [[phi].sub.2]. Thus we have two solutions (Fig. 1.3). In the first, both [<i>c</i>.sub.1] and [<i>c</i>.sub.2] are positive, with orbitals of the same sign placed next to each other; the electron population <i>between</i> the two atoms is increased (shaded area), and hence the negative charge which these electrons carry <i>attracts</i> the two positively charged nuclei. This results in a lowering in energy and is illustrated in Fig. 1.3, where the horizontal line next to the drawing of this orbital is placed low on the diagram. In the second way in which the orbitals can combine, [<i>c</i>.sub.1] and [<i>c</i>.sub.2] are of opposite sign, and, if there were any electrons in this orbital, there would be a low electron population in the space between the nuclei, since the function is changing sign. We represent the sign change by shading one of the orbitals, and we call the plane which divides the function at the sign change a node. If there were any electrons in this orbital, the reduced electron population between the nuclei would lead to repulsion between them; thus, if we wanted to have electrons in this orbital and still keep the nuclei reasonably close, energy would have to be put into the system. In summary, by making a bond between two hydrogen atoms, we create two new orbitals, [sigma] and [[sigma].sup.*], which we call the molecular orbitals; the former is <i>bonding</i> and the latter <i>antibonding</i> (an asterisk generally signifies an antibonding orbital). In the ground state of the molecule, the two electrons will be in the orbital labelled [sigma]. There is, therefore, when we make a bond, a lowering of energy equal to twice the value of [E.sub.[sigma]] in Fig. 1.3 (<i>twice</i> the value, because there are two electrons in the bonding orbital). <p> The force holding the two atoms together is obviously dependent upon the extent of the overlap in the bonding orbital. If we bring the two 1s orbitals from a position where there is essentially no overlap at 3 through the bonding arrangement to superimposition, the extent of overlap steadily increases. The mathematical description of the overlap is an integral [<i>S</i>.sub.12] (Equation 1.3) called the <i>overlap integral</i>, which, for a pair of 1s orbitals, rises from 0 at infinite separation to 1 at superimposition (Fig. 1.4). <p> [<i>S</i>.sub.12] = [integral] [[phi].sup.1] [[phi].sup.2]<i>d]<i>tau] 1.3 <p> The mathematical description of the effect of overlap on the electronic energy is complex, but some of the terminology is worth recognising, and will be used from time to time in the rest of this book. The energy <i>E</i> of an electron in a bonding molecular orbital is given by Equation 1.4 and for the antibonding molecular orbital is given by Equation 1.5: <p> <i>E</i> = [alpha] + [beta]/1 + <i>S</i> 1.4 <p> <i>E</i> = [alpha] - [beta]/1 - <i>S</i> 1.5 <p> in which the symbol [alpha] represents the energy of an electron in an isolated atomic orbital, and is called a <i>Coulomb integral</i>. The function represented by the symbol contributes to the energy of an electron in the field of both nuclei, and is called the <i>resonance</i> integral. It is roughly proportional to S, and so the overlap integral appears in the equations twice. It is important to realise that the use of the word resonance does not imply an oscillation, nor is it exactly the same as the 'resonance' of valence bond theory. In both cases the word is used because the mathematical form of the function is similar to that for the mechanical coupling of oscillators. We also use the words <i>delocalised</i> and <i>delocalisation</i> to describe the electron distribution enshrined in the function-unlike the words resonating and resonance, these are not misleading, and are the better words to use. <p> The function is a negative number, lowering the value of <i>E</i> in Equation 1.4 and raising it in Equation 1.5. In this book, will not be given a sign on the diagrams on which it is used, because the sign can be misleading. The symbol should be interpreted as [absolute value of ], the positive absolute value of . Since the diagrams are always plotted with energy upwards and almost always with the [alpha] value visible, it should be obvious which values refer to a lowering of the energy below the [alpha] level, and which to raising the energy above it. <p> The overall effect on the energy of the hydrogen molecule relative to that of two separate hydrogen atoms as a function of the internuclear distance is given in Fig. 1.5. If the bonding orbital is filled (Fig. 1.5a), the energy derived from the electronic contribution (Equation 1.4) steadily falls as the two hydrogen atoms are moved from infinity towards one another (curve A). At the same time the nuclei repel each other ever more strongly, and the nuclear contribution to the energy goes steadily up (curve B). The sum of these two is the familiar Morse plot (curve C) for the relationship between internuclear distance and energy, with a minimum at the bond length. If we had filled the antibonding orbital instead (Fig. 1.5b), there would have been no change to curve B. The electronic energy would be given by Equation 1.5 which provides only a little shielding between the separated nuclei giving at first a small curve down for curve A, and even that would change to a repulsion earlier than in the Morse curve. The resultant curve, C, is a steady increase in energy as the nuclei are pushed together. The characteristic of a bonding orbital is that the nuclei are held together, whereas the characteristic of an antibonding orbital, if it were to be filled, is that the nuclei would fly apart unless there are enough compensating filled bonding orbitals. In hydrogen, having both orbitals occupied is overall antibonding, and there is no possibility of compensating for a filled antibonding orbital. <p> We can see from the form of Equations 1.4 and 1.5 that the term [alpha] relates to the energy levels of the isolated atoms labelled 1[s.sub.H] in Fig. 1.3, and the term to the drop in energy labelled [<i>E</i>.sub.[sigma]] (and the rise labeled [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Equations 1.4 and 1.5 show that, since the denominator in the bonding combination is 1 + <i>S</i> and the denominator in the antibonding combination is 1 - <i>S</i>, the bonding orbital is not as much lowered in energy as the antibonding is raised. In addition, putting <i>two</i> electrons into a bonding orbital does not achieve exactly twice the energy-lowering of putting <i>one</i> electron into it. We are <i>allowed</i> to put two electrons into the one orbital if they have opposite spins, but they still repel each other, because they have to share the same space; consequently, in forcing a second electron into the [sigma] orbital, we lose some of the bonding we might otherwise have gained. For this reason too, the value of [<i>E</i>.sub.[sigma]] in Fig. 1.3 is smaller than that of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is why two helium atoms do not combine to form an [He.sub.2] molecule. There are four electrons in two helium atoms, two of which would go into the [sigma]-bonding orbital in an He2 molecule and two into the [[sigma].sup.*]-antibonding orbital. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is greater than 2[E.sub.[sigma]], we would need extra energy to keep the two helium atoms together. <p> Two electrons in the same orbital can keep out of each other's way, with one electron on one side of the orbital, while the other is on the other side most of the time, and so the energetic penalty for having a second electron in the orbital is not large. This synchronisation of the electrons' movements is referred to as <i>electron correlation</i>. The energy-raising effect of the repulsion of one electron by the other is automatically included in calculations based on Equations 1.4 and 1.5, but each electron is treated as having an average distribution with respect to the other. The effect of electron correlation is often not included, without much penalty in accuracy, but when it is included the calculation is described as being with <i>configuration interaction</i>, a bit of fine tuning sometimes added to a careful calculation. <p> The detailed form that [alpha] and take is where the mathematical complexity appears. They come from the Schrdinger equation, and they are integrals over all coordinates, represented here simply by <i>d]<i>tau], in the form of Equations 1.6 and 1.7: <p> [alpha] = [integral] [[phi].sub.1]<i>H]<i>[phi].sub.1]<i>d]<i>tau] 1.6 <p> [beta] = [integral] [[phi].sub.1]<i>H]<i>[phi].sub.2]<i>d]<i>tau] 1.7 <p> where <i>H</i> is the energy operator known as a Hamiltonian. Even without going into this in more detail, it is clear how the term [alpha] relates to the atom, and the term to the interaction of one atom with another. <p> As with atomic orbitals, we need pictures to illustrate the electron distribution in the molecular orbitals. For most purposes, the conventional drawings of the bonding and antibonding orbitals in Fig. 1.3 are clear enough-we simply make mental reservations about what they represent. In order to be sure that we do understand enough detail, we can look at a slice through the two atoms showing the contours (Fig. 1.6). Here we see in the bonding orbital that the electron population close in to the nucleus is pulled in to the midpoint between the nuclei (Fig. 1.6a), but that further out the contours are an elliptical envelope with the nuclei as the foci. The antibonding orbital, however, still has some dense contours between the nuclei, but further out the electron population is pushed out on the back side of each nucleus. The node is half way between the nuclei, with the change of sign in the wave function symbolised by the shaded contours on the one side. If there were electrons in this orbital, their distribution on the outside would pull the nuclei apart-the closer the atoms get, the more the electrons are pushed to the outside, explaining the rise in energy of curve A in Fig. 1.5b. <p> We can take away the sign changes in the wave function by plotting [[sigma].sup.2] along the internuclear axis, as in Fig. 1.7. The solid lines are the plots for the molecular orbitals, and the dashed lines are plots, for comparison, of the undisturbed atomic orbitals [[phi].sup.2]. The electron population in the bonding orbital (Fig. 1.7a) can be seen to be slightly contracted relative to the sum of the squares of the atomic orbitals, and the electron population between the nuclei is increased relative to that sum, as we saw when we considered Equation 1.2. In the antibonding orbital (Fig. 1.7b) it is the other way round, if there were electrons in the molecular orbital, the electron population would be slightly expanded relative to a simple addition of the squares of the atomic orbitals, and the electron population between the nuclei is correspondingly decreased. <p> <i>(Continues...)</i> <p> <!-- copyright notice --> <br></pre> <blockquote><hr noshade size='1'><font size='-2'> Excerpted from <b>Molecular Orbitals and Organic Chemical Reactions</b> by <b>Ian Fleming</b> Copyright © 2010 by John Wiley & Sons, Ltd. Excerpted by permission.<br> All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.<br>Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.