This is an old version of the course - new version is on Blackboard

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1 Molecular Electronic Structure This is an old version of the course - new version is on Blackboard Jeremy Harvey W (internal #46991) jeremy.harvey@bris.ac.uk Recommended Books 1

2 P. W. Atkins, Physical Chemistry (Oxford University Press), Ch. 13, 14. G. H. Grant and W. G. Richards, Computational Chemistry (Oxford Primer) Frank Jensen, Introduction to Computational Chemistry (Wiley) Ira N. Levine, Quantum Chemistry (Prentice Hall) A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Dover) P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics (Oxford University Press) Links to Online Lecture Notes Lecture 1. o Why Study Electronic Structure? o The Schrödinger Equation, and the Born-Oppenheimer Approximation. Lecture 2. Molecular Orbitals, Bonding, Symmetry. Lecture 3.The Pauli Principle. Lecture 4.The Hartree-Fock Method. Part I. Lecture 5. The Hartree-Fock Method. Part II. Lecture 6. o Other Electronic Structure Methods. o Conclusions. Here you will find biographical details for some of the scientists whose work is mentioned in these pages. For those of you wishing to study these pages at home, a downloadable form (.zip file) of the whole set of webpages is available here. Course Content How the Hartree-Fock method (the most fundamental electronic structure method) works. How electronic structure relates to structure, spectroscopy, and reactivity. Application of the previous point to some typical organic and inorganic molecules and reactions. (The applications to spectroscopy will be covered in more detail in the other two units within the module.) This Unit and the Rest of the Course 2

3 This course relies a bit more heavily than many others in the course on explicitly formulated quantum mechanics, but in doing so, it builds on what you have learnt in previous years and in other units this year. But Chemists are experts on molecular electronic structure!!! And this unit addresses the same material covered in ALL the course: the Structure and Reactivity of molecules. Some strongly related course units, etc. this year: 1. Chemical Applications of Group Theory, Dr. N. C. Norman (weeks 1-6). 2. Molecular Spectroscopy, Dr. C. M. Western (weeks 11-14). 3. Photochemistry, Professor M. N. R. Ashfold (weeks 15-18). 4. Workshop - Week (Two theoretical chemistry practicals w/ Drs. N. L. Allan and A. J. Orr-Ewing.) 6. (Two physical chemistry practicals, w/ Dr. A. J. Orr-Ewing & myself.) Molecular Electronic Structure Lecture 1a Why does molecular electronic structure matter? The need for a qualitative understanding of electronic structure is obvious, because in Chemistry, we are permanently appealing to concepts relating to electronic structure, e.g.: The carbon atom of a carbonyl group has a partial positive charge, so it is electrophilic. The nitrogen atom of aniline is less nucleophilic than that of methylamine due to delocalisation of the lone pair into the aromatic ring. Pi backbonding from electron rich transition metals to carbonyl ligands weakens the C O triple bond, leading to a downfield shift in the infrared spectrum. An electron withdrawing cyano group increases the reactivity of ethene in the Diels Alder reaction, by lowering the energy of the LUMO. The electronic structure of molecules is determined by the equations of quantum mechanics. As the examples above show, understanding how these equations work out can enable one to make qualitative predictions about structure, bonding, spectroscopy, reactivity,... Solving these equations can give quantitative predictions of these properties. In the past, it was necessary to use very severe approximations when solving the equations, and the most powerful computers available were needed. The results were hard to interpret, and of qualitative value only. Nowadays, even standard desktop computers are far more powerful than state-of-the-art supercomputers of 20 years ago, and the programs used to solve the equations have become 3

4 much more powerful. Therefore, computational approaches to molecular electronic structure are much more widespread and useful, as shown by the following examples: Structure. Computational methods can be used to predict the structure and stability of novel molecules. As an example, we recently predicted that the following heterofullerene, C 12 P 8, should be a stable molecule: Ref.: R. W. Alder, J. N. Harvey, P. v. R. Schleyer & D. Moran, Organic Letters, Properties. As well as their structure, you can also predict other properties of molecules. In a recent undergraduate research project, we studied the magnetic properties of a range of dinuclear molybdenum complexes, which have been studied experimentally in the group of Professor Mike Ward. Using computation, we can predict whether these complexes have a diamagnetic or paramagnetic ground state. 4

5 Ref.: K. Neale, M. D. Ward, and J. N. Harvey, to be published. Reactivity. One of the important properties of a molecule is its reactivity. One way to predict reactivity is by examining the orbitals of the reagents. Here, for example, is what the lowest unoccupied orbital of the Tungsten complex W(CO) 5 [=CH(OCH 3 )] looks like: 5

6 Clearly, the best site for attack of a nucleophile is on the carbene carbon atom - which is exactly what is found experimentally. Reaction Mechanisms. As well as being able to characterise the structure and energy of stable molecular structures, as above, it is possible to determine the structure of transition states separating reactants and products. By comparing their energy to that of the corresponding reactants, one can also compute the activation energy of the reaction. As an example, I am currently carrying out a set of studies on the mechanism of a novel organometallic hydrogenation process, based on a ruthenium dihydride complex. This system has been examined experimentally by Prof. Bob Morris from Toronto in Canada. This is a picture of the active form of the catalyst: 6

7 And this shows the transition state for transfer of hydrogen from the catalyst to acetone: (In this preliminary case, a much smaller "model" system has been chosen for the computations on the transition state) Biochemistry. Enzymes are large molecules... so reaction mechanisms can be studied for enzymes also. In recent work conducted in collaboration with Adrian Mulholland, we have studied the mechanism of action of the Lipoxygenase enzyme. This enzyme converts unsaturated 7

8 fatty acids and molecular oxygen into hydroperoxides. The transition state of the key step in this enzyme's active cycle, abstraction of a hydrogen atom by an Fe(III)-OH moiety, is shown here: Knowing the energy and geometry of the transition state gives invaluable information as to how the enzyme works, which can be very useful in the design of pharmaceuticals. Dynamics. Modern experimental techniques are able to probe reaction mechanisms in exquisite detail. Many of these experimental studies are conducted in parallel with electronic structure studies, which can be used to predict the dynamics of a reaction, i.e. to predict how the atoms will move during the reaction. As an example, consider the contrasting behaviour in the reactions of chlorine atoms with methane and with methanol, which have been studied experimentally in Bristol by Svemir Rudic and Andrew Orr-Ewing: In the reaction with methane, HCl is produced in states with small amounts of rotational energy (mostly J=0). With methanol, the degree of rotational excitation is much larger. This does not 8

9 appear to be due to differences in the geometry of the corresponding transition states, which are rather similar. It must therefore be due to details in the dynamics (motion of the atoms) as the products part. This is shown in the following "film" (you need Chime to view this; click on any of the images to bring up the "film"; 3 snapshots are shown in case you do not have Chime): Ref.: S. Rudic, J. N. Harvey, A. J. Orr-Ewing, to be published. All of these applications are dependent to some extent on understanding the theory behind molecular electronic structure, and this course is aimed at providing you with some of that background. Some of you may one day be involved in developing newer and better programs to solve the physical equations involved. But almost all of you are able to benefit from this course, 9

10 through an enhanced understanding of how the structure and properties of molecules arise from the fundamental properties of nuclei and electrons and the laws of physics. Molecular Electronic Structure Lecture 1b The Schrödinger Equation From the standpoint of quantum mechanics, molecules and other chemical systems are physical systems, composed of smaller, charged particles, and their behaviour can be predicted from the fundamental laws of physics. In short: Molecule = Nuclei + Electrons The nuclei and electrons can be fully described by a small number of properties, such as their mass, their electrical charge, their magnetic properties,... These are the only basic properties needed to predict how molecules will behave. Such systems are described by the Time-dependent Schrödinger equation: is the Wavefunction, which describes the state of the system throughout space (position of the particles, r) and time. H is the Hamiltonian Operator, which describes how the system changes in time. For many systems, the time-dependence only contributes an unimportant phase factor: Where E is the energy of the system. Substituting this expression into the Schrödinger equation, above, yields the following: 10

11 This equation can be separated into two. The most important of the equations from our point of view deals only with the space-dependent part of the wavefunction: Even so, the wavefunction is a very complex entity: To take an example, the wavefunction for a benzene molecule depends on 162 variables: 3 Cartesian coordinates each for 6 Carbon nuclei, 6 Hydrogen nuclei, and 6 x x 1 electrons. The Hamiltonian Operator acts on all these variables, and mixes them up in a complicated way, because it depends on the inter-particle distances: Again, to take the example of benzene, there are 42 x 41 / 2 = 861 electron-electron distances! The Born-Oppenheimer Approximation This first step in simplifying the equation is based on the fact that nuclei are much more heavy than electrons. This means that on the timescale where electrons move around, the nuclei are essentially fixed. Under these conditions, it is possible to factorise the wavefunction: 11

12 In other words, the total wavefunction can be written as a product of two functions, one ( ) which depends only on the positions of the nuclei, and one ( ) which depends on the positions of the electrons, but is also parametrically dependent on the positions of the nuclei (shown symbolically in the formula by underlining R). This means that the function does not contain any terms explicitly dependent on R, but that there is a different function for each set of R. Factorising in this way leads to separating the Schrödinger equation into two separate equations, one for the electrons, and one for the nuclei. There are also two Hamiltonians. The electronic Hamiltonian is of the form: Note that the last term in this equation, which depends on the position of the nuclei, is a constant, since the nuclei are assumed to be frozen. In some texbooks, this term is not included in H elec. The corresponding equation is: For the nuclei, the Hamiltonian look like this: The first term in this expression corresponds to the kinetic energy of the nuclei, whereas the second corresponds to their potential energy, and is often referred to as the interatomic potential energy surface V(R). The motion of electrons almost always needs to be understood in a quantum-mechanical way. Nuclei, on the other hand, can often be thought to behave in a classical way. The time-dependent version of the nuclear Schrödinger equation often reduces to something very similar to Newtonian dynamics - the motion of a point on a potential energy surface V(R), which is obtained by solving the Electronic Schrödinger Equation. 12

13 Molecular Electronic Structure Lecture 2 Molecular Orbitals The problem in theoretical and computational studies of Molecular Electronic Structure is to solve the electronic Schrödinger equation for a system of N nuclei and n electrons: 13

14 The Schrödinger Equation has analytical solutions for some simple systems: Particle in a box Harmonic oscillator Morse potential (Hard!) Hydrogen atom Hydrogen molecule ion H 2 + (very hard) In most cases, one needs instead to use Numerical approaches. This involves assuming that the real wavefunction, which is too complicated to be found directly, can be approximated by a simpler function. For some types of function, it is then possible to solve the electronic Schrödinger Equation numerically. Provided the assumption one has made about the form of the function is not too drastic, one obtains a good approximation to the correct solution. Molecular electronic structure computations consist in choosing sensible approximations to the wavefunction. A variety of approximate forms can be suggested; the simpler ones will lead to an easier solution of the Schrödinger Equation, with more complex approximations being harder to solve but leading to wavefunctions which are closer to the correct solution. The most common starting point for solving the Electronic Schrödinger Equation for many electron systems is the (Molecular) Orbital Approximation. The wavefunction is taken to be a product of one electron wavefunctions: These one-electron wavefunctions are also called Orbitals, and, in the case of molecules, are generally approximately expanded in a basis set of atomic functions: Molecular orbitals are of course familiar to you already - you have discussed them in all sort of courses you have already followed. In the level 2 lecture courses of Prof. Ashfold and Dr. Mulholland, you have learned more about molecular orbitals in diatomic molecules and in conjugated ( -electron) systems. Nature of the Molecular Orbital Approximation 14

15 The molecular orbital approximation is equivalent to assuming that the electrons behave independently from each other. The probability of finding an electron in orbital i at a given place r i does not depend on where the other electrons are: As a result, the probability of finding ANY electron at a given point in space is just the sum of the probabilities of finding one there in each of the orbitals. The Orbital Approximation is also equivalent to assuming that the electronic Hamiltonian can be expressed as a sum of one-electron hamiltonians: For comparison, the full, correct electronic Hamiltonian is shown below. As can be seen, the first and second terms, which correspond to the kinetic energy of individual electrons, and to the interaction of each electron with the nuclei, respectively, are indeed sums over all the electrons. The final term is a constant, which is added separately to the sum of one-electron terms. The third term, which corresponds to the repulsion between electrons, cannot be separated exactly into one-electron terms. By imposing the separability, the Molecular Orbital Approximation thereby inevitably involves an incorrect treatment of the way in which the electrons interact with each other. For more details, see the section on the Hartree-Fock Method. Molecular Orbitals and Symmetry How can one determine the shape of the molecular orbitals? In general, one needs to solve the Schrödinger equation. One can however often predict the qualitative shape of the orbitals without computation. In this section, we will examine how molecular orbitals arise in this way from 15

16 linear combinations of atomic orbitals. This will be seen to be intimately related to the question of molecular symmetry. As already mentioned above, molecular orbitals are obtained as linear combinations of atomic orbitals: This may at first sight appear to be a significant restriction on the form molecular orbitals can take. In fact, if enough atomic orbital functions are taken, any function can be written as a linear combination of the form given. In practice, it is often found that the qualitative shape of molecular orbitals can be reproduced from a very small number of atomic orbitals. In fact, just taking the occupied valence and core orbitals of the constituent atoms is usually enough to understand the shape of molecular orbitals. There are three qualitative rules for predicting how atomic orbitals mix to give molecular orbitals: 1. Orbitals of similar energy interact most. 2. The more two orbitals overlap, the more they mix. 3. Molecular orbitals must belong to one of the irreducible representations (irreps) of the molecular Point Group. The first two rules above should be familiar to you from Prof. Ashfold's course. The third rule is in essence a consequence of the second, but may be less familiar to you so we shall now discuss it in some more detail, in the context of the molecular orbitals of the water molecule. The Molecular Orbitals of Water The water molecule belongs to the C 2V point group. This means that it has four symmetry elements: The identity operation. A two fold (C 2 ) symmetry axis (running down the bisector of the H O H angle). A plane of symmetry, running through the oxygen atom and the two fold symmetry axis, and orthogonal to the plane of the molecule. A second plane of symmetry, which is simply the plane of the molecule. 16

17 (To help you visualise these symmetry elements, you can use the following link to see a 3D representation of a water molecule. Click Here to view the water molecule (this link will open in a new window). Let us now examine the core and valence atomic orbitals of the O and H atoms. These are shown on the following picture (Note that following the usual convention, the symmetry axis of the molecule is chosen as the z axis, and the plane of the molecule is chosen as the yz plane): The individual H 1s orbitals do not belong to any of the irreps of the molecular point group. (Reminder: to belong to one of the irreducible representations, the object or function needs to be either symmetric or antisymmetric with respect to all the symmetry operations of the point group.) Because the MOs have to belong to one of the irreps, this implies that there are constraints on the values of the c ij for the H 1s orbitals in the linear combination. These coefficients have to match one of the two following conditions: An equivalent way of saying this is to say that the MOs of a molecule are not obtained directly from AOs, but from symmetry-adapted linear combinations of AOs. These are shown here, for the water molecule, and are classified by the irrep to which they belong. You should already be familiar with the names of the irreps for atoms: s, p, d, f,...; and for diatomic (or other linear) 17

18 molecules:,,,... You will learn more about point groups, irreps, symmetry labels, and other aspects of molecular symmetry in the lecture course with Dr. Norman. These symmetry-adapted AOs mix, according to the rules given above, to yield MOs for water. Here is a rough diagram showing how the mixing occurs: 18

19 Notice that the O 1s orbital does not mix with the others - this is because it is much lower in energy than them. Notice also that the O 2p x orbital does not mix with any others - this is because it is the only AO belonging to the b 2 irrep. As already discussed, the exact form of the molecular orbitals for any given molecule cannot be predicted simply from the qualitative rules given above. This is because there are typically lots of symmetry adapted AOs belonging to each irrep, and lots of MOs, and so there are hundreds of coefficients c ij. These can be obtained quantitatively by solving the Schrödinger equation. The sort of results obtained can be seen by clicking on the following links to bring up a picture of each of the five lowest MOs of water. The coefficients of each AO for each MO are also given. Note that these links will each open in a new window. Click to bring up a picture of the corresponding orbital: 1a 1 2a 1 1b 1 3a 1 1b 2 Overall, because water has ten electrons, these five lowest orbitals are all doubly occupied, giving an electronic configuration which can be written in symbolic form in either of the following ways: Molecular Orbitals in Large Molecules Building up the molecular orbitals of large molecules directly from atomic orbitals is hard to do in the same qualitative way we have used above, because there are so many orbitals to consider. In practice, of course, the orbitals are generated by solving the Schrödinger equation. If one is looking for a qualitative understanding of how the molecular orbitals arise, it can sometimes be useful to break up the process of mixing the AOs to form MOs into several steps. For example, one can predict the form of the orbitals of the fragments A and B of a molecule AB, and then concentrate on how these "fragment MOs" mix to form the final MOs. The rules for this mixing procedure between fragment MOs are the same as those we used for mixing AOs To illustrate this procedure and explain how it helps to understand structure and reactivity, we will analyze two examples chosen from organometallic chemistry. 19

20 Fischer Carbene Complexes. Various 16-electron organometallic moieties [M] form "carbene" complexes [M]CR 2. An example is the molybdenum species Mo(CO) 5 =CH 2. The structure of this complex is shown here: This complex has a total of 120 electrons, so there are 60 occupied molecular orbitals. There are 81 occupied atomic orbitals in the 14 atoms making up the molecule so a diagram showing how they all mix would be horrendously complicated. Instead, let us view this molecule as being made up of two fragments, Mo(CO) 5 and CH 2. We will assume that we already know what the MOs of these fragments look like. For CH 2, we could in fact work them out starting from the atomic orbitals, because this system is very similar to water, discussed above. In the singlet state of CH 2, the HOMO is similar to the 3a 1 orbital of water: it is carbon lone pair, roughly speaking an sp 2 hybrid, which is oriented along the Mo-C bond and is thus - loosely speaking - a orbital. The LUMO is like the 1b 2 orbital of water. It is an empty p orbital orthogonal to the Mo-C bond (a orbital). These orbitals are shown schematically here: 20

21 Click here to bring up a picture of these orbitals derived from an MO computation: HOMO LUMO The orbitals of the Mo(CO) 5 fragment are made up of many more AOs and are thus very complex. Nevertheless, they can roughly be described as being a d orbital of symmetry (HOMO), and a -symmetric orbital made up out of the d z 2 orbital, among others (LUMO): Click here to bring up a picture of these orbitals derived from an MO computation: HOMO LUMO It is now relatively easy to picture how the bond forms. The frontier orbitals of the two fragments have matching symmetries (Mo(CO) 5 is sometimes said to be isolobal to CH 2 ), so that the orbitals mix among themselves, as do the orbitals. The overlap between the orbitals is greater, so the energy splitting is larger too. This gives the following rough MO diagram: 21

22 The form of three of these orbitals, as derived from an MO computation, can be seen by clicking on the following links: sigma pi pi*. Note the form of the LUMO - this explains why carbenes react as electrophiles, with nucleophiles adding to the carbon atom. In some respects, the C=Mo double bond behaves like a C=O (carbonyl) double bond! Palladium Allyl Complexes. These species are formed by the attack of Pd(0) compounds on allyl halides and are important intermediates in the Tsuji-Trost allylic substitution reaction. The structure of the parent compound is shown here (note the plane of symmetry): 22

23 Let us break this molecule up into two fragments, Pd(PR 3 ) 2 and allyl cation. This way of breaking up the molecule corresponds to the way in which it is formed in organometallic chemistry: by reaction between some or other Pd(0) species Pd(L) 2 and an allyl-x derivative (X = acetate,...). We can now analyse the bonding between the two fragments by looking at some of their MOs. The most important mixing that occurs is between the LUMO of allyl cation, and the HOMO of the Pd fragment. To view these MOs, click here (this link will open in a new window). You can see that both orbitals are antisymmetric with respect to the plane of symmetry - they belong to the A" irrep of the C S point group. The way the mixing of these orbitals gives rise to the MOs of the complex is shown schematically in the following MO diagram: To see the HOMO of the complex, click here. Note the strong bonding between palladium and the two terminal carbon atoms. To see the LUMO of the complex, click here. (These links will open in new windows) Note the anti-bonding nature of the orbitals between the Pd and C atoms. Note also the large lobes on the terminal carbon atoms, on the side opposite to palladium. What does this suggest for the stereochemistry of nucleophilic substitution of an allyl acetate by Nu, catalysed by Pd(0)? 23

24 The analysis of reactivity in this way is referred to as Frontier Molecular Orbital (FMO) theory, developed among others by Fukui. Using FMO theory is one of the most rigorous ways to derive the Woodward-Hoffmann rules, which you have met in Organic Chemistry lectures and in Dr. Mulholland's Year 2 Lecture Course. Molecular Electronic Structure Lecture 3 The Pauli Principle and Spin Our aim now is to work out how to solve the Schrödinger equation so as determine molecular wavefunctions, within the molecular orbital (MO) approximation. Before we do this, however, we need to address an important property of electrons which we have neglected up till now: Elementary Particles Cannot be Told Apart In previous pages, we have been writing electronic wavefunctions as Products of 1-electron functions, or molecular orbitals (these products are called Hartree products): In fact, such expressions violate a very fundamental principle of quantum mechanics: like elementary particles are indistinguishable from each other; You can't tell one electron from another. Consider a two-electron wavefunction (written as in the previous pages): This is not a valid expression, because it is not the same as the indistinguishable: 24

25 To put this in a more graphical way, imagine we have a water molecule, and we "know" that such and such an electron is in the HOMO, and we then ionise the molecule as shown: We would know which electron had been removed, whereas the indistinguishability of particles tells us we cannot know that - only that we have removed one of the 10 electrons on the water molecule. So how should one write wavefunctions, preserving the spirit of them being products of molecular orbitals? The answer is to take some or other linear combination of the Hartree products obtained by switching the position of the electrons between orbitals. For the two-electron wavefunction, there are the following two possibilities: Note that for the second of these possibilities, switching the two electrons does not give the same wavefunction, because: However, all experimental observations depend on the square of the wavefunction, and one of course has: 25

26 In principle, then, either of the two types of linear combination given above could be satisfactory. It is an experimental fact, however, that for electrons, only the second type of wavefunction, which changes sign upon exchanging the two particles, is possible. This observation is at the heart of the Pauli Principle: Valid Electronic Wavefunctions must change sign upon exchanging the coordinates of any two electrons. How does this affect our picture of molecular orbitals? Consider the wavefunction for the Helium atom, which we "know" has two electrons in the 1s orbital: Electron Spin The paradoxical conclusion at the end of the previous paragraph is due to our neglect up till now of Electron Spin. We have been assuming that electrons are fully described in terms of their position in space, their speed and their charge which undergoes Coulombic interactions with other charged particles. This leads to writing electronic wavefunctions as functions of their position only: In fact, this is not correct. Experiment shows that electrons have intrinsic magnetic properties, which can be shown by passing a beam of electrons through a magnetic field. In classical physics, magnetism arises from (rotational or other) motion of charged particles. Because the electrons have intrinsic magnetic properties, i.e. they have these properties whatever they are doing, even when they are apparently at rest, the properties have been attributed to a mysterious spinning motion of the electrons, referred to simply as spin. However, this word is only used by analogy to classical physics; as a particle which has zero size, electrons cannot really "spin" so the name should not be taken too literally. Spin is a quantum-mechanical property. To fully describe an electron, you need a fourth coordinate, wavefunction should be written as:, so that a one-electron The spin coordinate is very unusual, and corresponds to nothing in classical mechanics. For our purposes, it is enough to say that all wavefunctions can be written in one of the following forms: 26

27 The function is similar to the orbitals we have been writing before, and gives all the space dependence; it is often called a Spatial Orbital. There are just two possible "functions" giving the dependence on the spin coordinate, and, which correspond classically to the electron spinning clockwise or anticlockwise, to the spin being "up" or "down". The spin "coordinate" is often omitted. The product of the spatial orbital and the spin function, here written as often called a Spin Orbital. The wavefunction which must satisfy the antisymmetry requirement of the Pauli principle is the total wavefunction, with spin. This will be based on a product of one-electron orbitals. For example, the wavefunction of Helium will be of the form:, is Unlike the expression a few pages up, this does not reduce to zero even if both electrons are placed in the same 1s spatial orbital, providing the spin part of the orbitals is different: Had we tried to put both electrons into the same spatial orbital and to give them the same spin, the function would be equal to zero (prove it!), which is impossible. This observation leads to more familiar expressions of the Pauli Principle: No more than two electrons may occupy the same orbital. Two electrons cannot have all their quantum numbers identical. Some Consequences Writing wavefunctions as an antisymmetrised sum (difference) of terms may seem to be be no big deal - all the terms look alike, after all. Actually, the consequences are rather important: 27

28 First, as discussed above, the "rule" that only two electrons may be in any given orbital is only due to spin and the Pauli Principle. Without this, all the electrons of every atom would be in 1slike orbitals and Chemistry would not happen. A more sophisticated conclusion can be drawn from the following simple example. Imagine you have two particles (e.g. electrons) in a box, and that one is in the ground state, the other in the first excited state. The square of the corresponding one-particle wavefunctions are shown here: What happens to the two-electron wavefunction? Below are shown two different 2d-plots of the square of the wavefunction. The first assumes it to be a simple product of the two one-electron wavefunctions: 28

29 But we have just shown that this expression is incorrect! The next picture shows the correct antisymmetrised difference of products: Note the differences!!! You can easily prove that the probability of finding two electrons with the same spin at the same point in space is zero - this is not the case if the electrons have opposite spin. Although the real reason is a bit more complicated, this is the gist of the reason behind Hund's rule that electrons tend to prefer to have parallel spins: Electrons with the same spin keep away from each other better than those of opposite spin, thus minimising Coulombic repulsion. This is very important in Transition Metal Chemistry, among others. Transition metals have a partly filled d shell - they have less than 10 electrons in 5 orbitals which are degenerate. If you use spectroscopy to study an isolated TM atom or ion, it always has a high-spin ground state: 29

30 In the presence of ligands, some of the orbitals are more stabilised than others, but they remain fairly close in energy. Depending on the ligands, it is more favourable for the electrons to "pair up" or to remain with parallel spin. As an example, consider haem, the cofactor responsible for carrying oxygen around the body. As a free cofactor, it has a quintet ground state (4 unpaired electrons, see above). Upon binding to CO (carbon monoxide is a poison because it binds very strongly to haem), the splitting between the orbitals increases, and a ground state singlet, with the 6 d electrons paired up, is obtained. This change of spin is important - if both systems were singlets, the reaction of CO with haem would be faster, and carbon monoxide would be an even worse poison than it already is. In fact, life as it is now would perhaps not be possible, because the human body itself produces a small amount of carbon monoxide. 30

31 Slater Determinants Wavefunctions need to be written as antisymmetrised products of spin-orbitals: 31

32 We have already seen that for a 2 electron system, the antisymmetrisation operator means that one needs to write the wavefunction as a difference of two Hartree products. For an n- electron system, the wavefunction needs to be antisymmetric with respect to interchange of any pair of electrons. To satisfy this, one will need to take a difference of 2 Hartree products for each pair of electrons! For n electrons, there are n!/2 pairs of electrons. For a three electron system (e.g. the Lithium atom), that means 6 Hartree products. With a bit of patience, you can work out that a 3-electron wavefunction will be of the form: Writing out such expressions rapidly becomes tedious; fortunately, they can be written in a completely general way as Determinants (called Slater determinants) In the common situation where there are an even number of electrons in a "closed-shell" situation, they are usually taken to occupy the spatial orbitals two by two, with paired spins: (Here for example with six electrons in three orbitals) 32

33 This can be written in even shorter form: Two properties of determinants are important: 1. The determinant changes sign when two rows (or columns) are exchanged, and therefore 2. A determinant with two identical rows (or columns) is equal to zero. From (1), you can check that exchanging the coordinates of any two electrons leads to a change in sign of the whole wavefunction, satisfying the Pauli Principle. From (2), if two electrons occupy the same Spin Orbital, i.e. two columns of the determinant are identical, the wavefunction is equal to zero - which is meaningless, so that such a wavefunction is not acceptable. Now that we know how to write many-electron wavefunctions correctly, we are almost ready to solve the Schrödinger equation for molecules. This means that we can determine the coefficients c ij which make up the wavefunction: 33

34 This will be the subject of the next lecture. Molecular Electronic Structure Lecture 4 The Hartree-Fock-Roothaan Method: Part I In the previous lecture, we learned to write a simple approximate form for molecular wavefunctions, as Slater determinants SD, i.e. as antisymmetrised products of spin-orbitals: We now want to determine the value of the coefficients c ij. To do this, we need to solve the electronic Schrödinger equation: 34

35 In fact, because we have decided to use an approximate wavefunction, there is no exact solution to this equation (no values of c ij which, inserted into the equation, make the two sides rigorously equal). Instead, we are looking for the best approximate solution. How do we define "best"? The answer to this question is given by the Variational Principle: The best wavefunction is the one with the lowest energy. The energy of an approximate wavefunction is given by: Or, more concisely, and provided the function is normalised: What we are trying to do, then, is to find the coefficients c ij (which define the molecular orbitals in terms of the atomic orbitals ) that give the lowest possible energy in the expression above. Doing this is the object of the Hartree-Fock-Roothaan method. Energy of a Slater Determinant To calculate the energy of a Slater Determinant wavefunction, we "just" have to insert the mathematical expression for it into the expression above for the energy of a wavefunction. Where: 35

36 And the Hamiltonian operator is given by: This is obviously going to be complicated!! Doing all the algebra required is, at the very least, tedious. We will more modestly try to get a qualitative idea of how the final expressions are reached. You are not expected to be able to reproduce any of the derivation! First of all, it is useful to note that the Hamiltonian is a sum of three terms: This first term does not depend on any electronic coordinates. This is a sum of one-electron operators h, so called because each depends only on the coordinates of one electron, and which are given by: And: 36

37 This is a sum of n(n-1)/2 two-electron contributions, which each depend on the coordinates of two electrons. The Hamiltonian can thus be written more concisely as a sum of zero-, one- and two-electron terms: Inserting this into the expression for the energy leads to: The first term in this sum is an integral over a constant, since the nuclear-nuclear repulsion energy does not depend on the electronic coordinates. This therefore simply gives: V NN is the potential energy due to nuclear-nuclear Coulombic repulsion. The two following terms are both integrals of sums, which can be re-written as sums of integrals, leading to: 37

38 Each one-electron operator h only acts on a very small part of the wavefunction, which leads to a considerable simplification of the first term. This term is also very easy to understand: it is a sum of the one-electron energies h ii of each orbital. Note that the one-electron operator h includes a part corresponding to the electron's kinetic energy, and another to the potential energy created by the attractive Coulombic interaction with the nuclei. Therefore, the one-electron energy of an orbital is the sum of a kinetic and a potential part: And: 38

39 T e is the electronic kinetic energy, and V Ne is the potential energy due to nuclear-electronic Coulombic attraction. The second set of integrals, the two-electron terms, are more complicated, because there are more of them, and because of the antisymmetrisation of the wavefunction!! Each term in the summation looks like: This rather daunting expression can be multiplied out, to give four terms: The first and fourth, and second and third, term are identical two by two. To understand what they both mean, it helps to rewrite them in terms of the spatial orbitals and coordinates (the spin terms are not very important here), and to rearrange them a bit. Taking the first term first (!), it can be written as: 39

40 Now, 2 (r) is the probability of finding an electron at a given point in space. So this first term is simply the energy of the Coulombic interaction between an electron in orbital i with an electron in orbital j. For this reason, this integral is often called the Coulomb Integral, written in shorthand as J ij. Because 1/r is always positive, and 2 obviously also, this term contributes a positive energy, i.e. a destabilisation - this is what you expect from a Coulombic repulsion between electrons. The other integrals become: What does this term, called an Exchange Integral, and written as K ij, mean? Unlike for the Coulomb integral, there is no immediate classical interpretation for the Exchange integral. The name Exchange Integral comes from the fact that the two electrons exchange their positions from the left to the right of the integrand. This suggests, correctly, that it has something to do with the Pauli principle. Remember that the probability density for two electrons is very different in an antisymmetrised Slater Determinant than in a simple Hartree Product. The total density is not simply a sum of the densities associated with each molecular orbital. (See the graphs for the two-electrons-in-a-box model in the section concerning the Pauli Principle.) Yet the expression for the repulsion energy between electrons given by the Coulomb integrals suggests that the total density is a sum of densities - it neglects all the effect of the antisymmetrisation on the density! However, the expressions we have been deriving are for the energy of a Slater Determinant - so the antisymmetrisation effect should be there, somewhere. In fact it is the the exchange integrals which "correct" the Coulomb integrals to take into account the antisymmetry of the wavefunction. We saw that the electrons (especially those of same spin) tend to avoid each other rather more in the Slater Determinant model than in the Hartree Product model, so the Coulomb integrals should exagerrate the Coulomb repulsion of the electrons. The Exchange integrals, which are negative, compensate for this exagerration, and are typically ca. 25% in magnitude of the Coulomb integrals. The overall contribution to the total energy of the potential energy due to electronic-electronic Coulombic repulsion, V ee is therefore given as a difference of two terms: 40

41 Overall, the energy of a Slater Determinant is given by adding up all the terms discussed above. For the general case with matrix elements expressed as spin orbitals, one reaches the following expression: A slightly different expression, which is more commonly met in books, is reached if one considers matrix elements over the spatial orbitals, for a closed-shell system (= a spin singlet where all the occupied orbitals have two electrons in them), Or: When this expression is computed for the Hartree-Fock wavefunction (i.e. for the Slater determinant obtained by minimising the energy with respect to all the coefficients c ij ), it is called the Total Energy, V tot. (In some textbooks, you will find that E SD does not include V NN, so that the total energy is given by E SD + V NN.) An Example To get a feel for how all of the above quantities vary upon forming a molecule, let us consider the bonding in the water molecule. 41

42 (A note on units. Throughout this course, we are implicitly using a set of units called Atomic Units. These are units such that many of the constants involved in the various equations have values of and are thus very convenient. The atomic unit of energy is also called a Hartree and is equal to J - rather small!! - or equivalently to 2625 kj/mol. The values in the Table are in kj/mol). Notice also that the zero of energy is the state where all the nuclei and electrons are infinitely separated and at rest. It is instructive to think about the origins of chemical bonding in terms of the four contributions shown here. In previous lecture courses, the term which has usually been used to account for bonding is V Ne - by accumulating electron density between the nuclei, you lower the overall potential energy. As you can see here, V Ne does indeed drop dramatically upon bond formation. But T e and V ee go up (the electrons are confined to a smaller region of space), and V NN does so too. The overall effect is very finely balanced! In fact, even in antibonding situations, V Ne will decrease - but not enough to compensate for the increase in the other terms. The overall energetic situation is summarized on the following picture: 42

43 (Note also that the experimental difference in energy is of ca kj/mol. The difference between this value and the HF computed number is due to the approximations involved in HF theory). The Hartree Fock Roothan Equations The previous paragraphs show how to calculate the energy of a Slater Determinant. The Hartree- Fock method is then very easy to understand: It simply involves optimizing the spin-orbitals (or in other words the coefficients c ij defining the MOs in terms of the AOs) to give the lowest energy possible. This is done using a modified version of the variational method discussed for Hückel theory in Dr. Mulholland's second year lecture course. Like in that case, a differential equation is reduced to a set of homogeneous equations which can be solved using linear algebra. It turns out that this needs to be done in an iterative way, because to solve what are called the Fock equations (which give the coefficients of the moleculars orbitals), one already needs to know the form of all the occupied orbitals (basically because one needs to be able to calculate J ij and K ij )!! In practice, a self-consistent method is used: 1. Choose initial orbitals. This is done using an approximate version of MO theory, e.g. some version of Hückel theory. 43

44 2. Solve the Fock equations, to give an improved set of orbitals. 3. Compare these orbitals to the previous set; If they are identical within some or other numerical convergence threshold, stop. 4. Repeat step 2., using the improved orbitals as input. With a reasonable set of "guess" orbitals to start off with, cycles are usually enough to converge the procedure to numerical accuracy. Molecular Electronic Structure Lecture 5 The Hartree-Fock-Roothaan Method - Part II In the previous lecture, we found out how to calculate the energy of a Slater determinant wavefunction, and thereby to optimize the wavefunction according the the variational principle. This is the so-called Hartree-Fock-Roothan method and the wavefunction obtained is the Hartree-Fock wavefunction. The Hartree-Fock energy is given by: In this lecture, we will survey the different sorts of informations that we can get from the Hartree-Fock method? 1. Energies The main property one can calculate is the total energy. This can be used as in the previous lecture to calculate bond energies. Hartree-Fock theory relies on the validity of the Molecular Orbital Approximation. In fact, wavefunctions are more complicated than single Slater Determinants, so that within the formalism of the variational principle, the Hartree-Fock total energy will inevitably be higher than the true energy. However, the difference is usually very 44

45 small, of the order of 1% or less. In absolute terms, then, Hartree-Fock energies are very accurate, which is a remarkable achievement for such a "simple" wavefunction! Nevertheless, as also discussed in the previous lecture, Hartree-Fock energies are still not accurate enough in relative terms, e.g. when computing a bond energy, which is a small number obtained as a difference between two large numbers, the total energies of the molecule and the atoms. For this reason, other methods are needed for accurate energetics (see next lecture). 2. Orbital Energies As well as the total energy, one also obtains a set of Orbital Energies,. These are given by: Orbital energies are important for constructing approximate MO diagrams, as we did in the second lecture. They can also be used to predict approximate ionisation energies. Consider a species M described by the Hartree-Fock wavefunction. What is the energy of the Slater Determinant obtained by removing an electron from occupied orbital a? This result is known as Koopmans' Theorem and provides a useful first approximation to molecular ionisation energies. Here are some examples to illustrate the level of accuracy (energies are in kj/mol): 45

46 3. Molecular Orbitals The molecular orbitals are found from solving the Fock equation. For a closed-shell spin-singlet with n elec electrons, the n elec / 2 orbitals with lowest energy are occupied, each by two electrons. The highest occupied orbital (HOMO) and the lowest unoccupied one (LUMO) are useful for understanding reactivity. 4. A Wavefunction This is written as a Slater Determinant: (here in brief form) A wavefunction allows you to calculate a huge number of properties of a system. In technical terms, this is because to each property, there is an associated Operator, and the observed value of the property is simply the value of that operator, averaged over the wavefunction (this is called the expectation value of the operator): Some examples are given below. 5. Dipole Moments 46

47 The dipole moment of a set of charged particles is given by the product of the difference between the total positive and negative charges, and the distance between the average centre-of-charge positions of the two charge clouds. This is computed very easily from a wavefunction: This is the dipole associated with a molecule of fluoromethane: The underlying reason for the dipole is that the C-F bond is polarised towards Fluorine. With some will-power, this can be seen on a plot of the corresponding MO: 47

48 Calculating dipoles is so easy that programs which perform Hartree-Fock computations will always provide you with the dipole (and the quadrupole, octapole, hexadecapole, etc.). Atomic Charges The atomic charges shown above on C and F can also be derived from the wavefunction. Basically, one takes the total electron density, and finds some way to split it up into parts "belonging" to the individual atoms. It turns out that there is no theoretically "best" way to do the splitting up. One popular procedure was suggested by Mulliken, and is called Mulliken Population Analysis. This often provides useful trends but also often produces meaningless results! 6. Electrostatic Potential With a wavefunction, it is also easy to calculate the Electrostatic Potential at each point in space. This is the energy required to bring a (hypothetical) point charge from infinity. It provides useful three- dimensional insight into the polarity of a molecule. This picture shows the Electrostatic Potential around the fluoromethane molecule (plotted onto an isodensity surface of the electron density. The reddest regions are most attractive to negatively-charged species, and the blue-est ones to positively charged ones). The electron attracting region in the centre of the methyl group can be related (with slightly less need for the eyes of faith) to the shape of the LUMO: 48

49 Plots of the electrostatic potential are often used by theoretical chemists working in biochemistry or pharmaceutical chemistry - the polarity of molecules strongly relates to how they interact with enzymes. As a (slightly) more relevant example of this, here is an Electrostatic Potential plot for the amino acid Valine, in its zwitterionic form (H 3 C) 2 CH- CH(NH 3 + )(CO 2 - ): 49

50 7. Geometries Up till now, we have considered solving the Schrödinger equation for a fixed position of the nuclei. As discussed in the first section, the electronic energy (or equivalently the Potential) is a Function of the positions of the nuclei. This defines a so-called Potential Energy Surface V(R). Stable structures of molecules correspond to R coordinates such that V(R) is low. For example, if one solves the Schrödinger 50

51 equation for the hydrogen molecule at a set of internuclear distances, one obtains the following graph: For larger molecules, it is useful to recognise that the expression shown above is a Function of the coordinates R. Like all functions, it can be differentiated. This leads to the Gradient of the Potential Energy: Calculating this derivative is not easy, but the necessary equations have been worked out, and the corresponding computation carried out once the wavefunction is known. Knowing the slope of the surface enables one to use advanced mathematical optimisation techniques so as to find optimum geometries (where the gradient is equal to zero). In this way, stable molecular structures for large molecules can be determined. Carbene 51

52 The carbene (or methylene) intermediate, CH 2 is an unstable intermediate involved in several organic reactions. Its gas-phase structure was long uncertain. In 1959, Foster and Boys predicted it to be bent on the basis of ab initio calculations, with an H-C-H angle of 129 degrees. The first experimental report was by the eminent physical chemist Gerhard Herzberg, in 1961, who analyzed the electronic spectrum and predicted the molecule to have a linear structure. In those times, computational predictions were not taken very seriously, so doubt was laid on the reliability of the calculations. In 1970, new and thorough HF calculations by Bender and Schaeffer lead to the conclusion that carbene is bent. This was followed in the next years by new experiments which found carbene to be... bent! Nowadays, computational methods are routinely used to predict molecular geometries, with an accuracy which is similar or better to that of most experimental methods. Chemical Reactivity The study of potential energy surfaces also provides insight into chemical reactivity. The key here is to find Transition States, or saddle-points, the lowest energy cols which need to be crossed on going from one valley to another: 52

53 Locating transition states gives useful information on Reaction Mechanisms, as well as on Activation Energies and Reaction Rates, combined e.g. with Transition State Theory. 8. Other Properties Many other properties can be computed: NMR chemical shift, NMR spin coupling constants, IR spectra, UV spectra,... Some of these are easy to compute - some less so. Applicability of Hartree-Fock Theory How easy is it to perform a Hartree-Fock calculation? 53

54 Some 50 years ago, the answer would have been "Not at all". Performing Hartree-Fock calculations on paper for anything but the very simplest of systems is rapidly rather tiresome... Since then, computers have made their appearance and a standard PC is now more powerful than essentially all computers were 20 years ago, and than many computers 10 years ago. General software packages for performing Hartree-Fock and other ab initio computations have been developed which make it unnecessary to delve into the murky depths of the maths. Molecules can be "built" on the screen by clicking atoms into place. How long do calculations take? The bigger the molecule, the longer. In formal terms, the most time-consuming operation is the need to calculate all of the following "two-electron" integrals over the basis functions: There are roughly N 4 /8 integrals of this form, where N is the number of basis functions. As a crude estimate, each atom needs of the order of 10 basis functions to be well described. For 2 atoms, there are therefore roughly integrals. For 10 atoms, there are This is shown on the following graph: 54

55 This poor scaling has historically made ab initio calculations difficult for large molecules. However, new, more efficient algorithms which e.g. manage to avoid calculating all of the twoelectron integrals, together with ever more powerful computers mean that computations are now possible on MUCH larger molecules than before. As an example, a HF calculation with geometry optimisation was recently reported for the (rather small, only 650 atoms) protein, Crambin: 55

56 (If you have the Chime plug-in, you can click on the image to view it in 3 dimensions.) The increasing applicability of ab initio theory means that many research papers nowadays include computational studies using the methods of electronic structure. The development of efficient ab initio software such as Gaussian, GAMESS, MOLPRO, MOLCAS, CadPac, Jaguar, ADF, Turbomole, etc., has been carried out by a large number of research groups, with British scientists among others playing a leading role. To single out one name among many others, John Pople, who was awarded the Nobel prize in 1998 for his work in the field, has done more than anyone else to make the methods more efficient and to make their application more popular among research chemists. (To illustrate these principles, Click Here to open a new window with an example (this link will open in a new window). Molecular Electronic Structure Lecture 6a 56

57 Other Ab Initio Computational Methods The computation of energies, geometries, wavefunctions, etc. by the Hartree-Fock method has several disadvantages which mean it is not suited to all problems. The main problems are: 1. Expense. Calculating then storing N 4 two electron integrals is time and memory consuming. 2. Applicability. Certain molecules or transition states have an electronic structure which cannot be described by a Slater Determinant. 3. Accuracy. The energies computed with HF methods are quite accurate in absolute terms (Total energies). However, relative energies (small differences between two large numbers!) are much less well reproduced. Semi Empirical Methods Hartree-Fock: Instead of this, an approximate expression can be used instead of H, which ultimately leads to approximate expressions being used for h ii, J ij and K ij, requiring the computation of far fewer integrals. Some of the integrals are given empirical values, adjusted to reproduce the known properties of certain simple atoms and molecules. 57

58 There are different ways of introducing the approximations, ranging from the extremely crude (Hückel theory) to the extremely sophisticated (PM3, AM1), and including models such as MNDO, CNDO,... These methods give wavefunctions, like ab initio methods, so can be used to predict a certain number of properties beyond the mere energy of the system. They can be applied, on a routine basis, even to very large molecules. Electron Correlation Problems (2) and (3) above are really two not-completely-distinct parts of the same problem: The Molecular Orbital Approximation is an approximation. A wavefunction cannot be factorised!! Equivalently, electrons do not move independently one from another: their motion is said to be Correlated. Ab Initio methods which include this effect are said to be Correlated methods; they describe electron correlation. How can one expand an unfactorisable wavefunction in terms of molecular orbitals? By taking a Linear Combination of Slater Determinants: The energy of such a wavefunction is always lower than that of the Hartree-Fock wavefunction. The difference in energy is called the correlation energy. Applicability Some wavefunctions need more than one determinant to describe them properly even in a qualitative sense. A simple example is the H 2 molecule at long bond distances: 58

59 The correct wavefunction only contains the neutral terms. This cannot be obtained by a single determinant: 59

60 Allowing the wavefunction to be given as a sum of these two determinants yields a qualitatively correct (Multi-Configuration Self-Consistent Field, MCSCF) wavefunction: Similar results can be obtained using the Valence-Bond method, but this is used more seldom in computations due to its algorithmic inconvenience. Similar Multi-Reference problems are met whenever bonds are strongly broken. For example, the Hartree-Fock method will be completely inappropriate to study the cis to trans isomerisation of cyclopropane: 60

61 This situation, known as Non-Dynamic Correlation, can also occur for excited states, or for molecules with an unusual electronic structure. Accuracy The Hartree-Fock method allows electrons (especially those of opposite spin) to approach one another because it treats the Coulombic repulsion between electrons in an average way only: We have mentioned before how any function can be expanded as a linear combination of suitable functions. In this way, the "true" wavefunction can always be expanded as a sum of Slater Determinants. It is not possible to give a simple pictorial representation of how this leads to the correct behaviour of the electron-electron relative position probability... but it does! The most straightforward way of computing correlated wavefunctions is the Configuration Interaction (CI) method, which uses the variational method to optimise the coefficients of the different Determinants. Usually, one starts from a Hartree-Fock wavefunction, and includes progressively more and more Slater Determinants which differ from the Hartree-Fock Determinant: 61

62 This effect is known as Dynamic Correlation, and plays an important role whenever quantitative relative energies are needed - e.g. between two fragments and the corresponding bonded species, or between two isomers, etc. As well as the CI method, other approaches can be used to introduce excited determinants: Perturbation Theory. Formulated by Møller & Plesset. E.g. MP2, MP4. Coupled Cluster Theory. E.g. CCSD, CCSD(T). Density Functional Theory As is so often the case with mathematical expressions in physics/chemistry, the Schrödinger equation can be recast in many different ways, which although they are all fundamentally the same, are each suggestive of different approximate methods of solution. We have been using the "traditional" method to solve the Schrödinger equation until now, but will briefly mention another approach, Density Functional Theory. Kohn and Hohenberg showed in 1964 that all the properties of a (molecular) system, including its energy, can in principle be deduced once one knows the Electronic Density function of the system. The energy is a "function" of the density function, which is often called a "Functional": 62