QMMS Lecture #1 (Venables/Heggie)

Notes for QMMS Lecture 1 (Venables/Heggie)

Lecture notes by John A. Venables and Edward Hernández. Latest version 26 Feb 08.

The references for this lecture are here. Note that this lecture needs the Symbol font enabled on your browser.

1. Bonds and Bands: chemical and physical approaches

1.1 A brief revision of Quantum Mechanics

1.1.1 Hamiltonians and Eigenvalue equations

The object of this first lecture is to revise some Quantum Mechanics ideas which are going to be needed at later stages of the course, and introduce some relevant terminology. We also apply Quantum Mechanics to diatomic molecules, as this helps clarify concepts and understanding that are later applied to solids and clusters. Since you already have had a Quantum Mechanics course, we are not going into any great depth, but you may want to go back to your preferred Quantum Mechanics textbook and look again at some of the ideas discussed.

Quantum Mechanics (QM) was developed after it was realized that what we now call Classical Mechanics was not applicable to the microscopic world. In particular, Classical Mechanics failed to account for the spectroscopic observation of the discrete energy levels in atoms and molecules, the photoelectric effect, and black body radiation, among other observations. Efforts of the scientific community to explain these observations eventually resulted in today's formulation of QM.

According to QM, a given system (an atom or molecule, say) is characterised by a Hamiltonian H, and by a wave function y(r) (generally a complex function), the two being related through the Schrödinger equation: Hy = Ey, where E is the energy associated with y. The Schrödinger equation is of fundamental importance; we will be referring to it all the time. The wave function is also crucial, as, according to the principles of QM, it contains all the information that we can possibly extract from the system. In mathematical language, the Hamiltonian is an operator, an entity which transforms a function into another function. An operator acting on a function can (for example) scale it by a constant factor, multiply by another function, take the derivative or integrate it.

The Schrödinger equation is called an eigen-value equation, which means that, given H, there are certain functions un, called the eigen-functions of H, which when transformed by the Hamiltonian result in the same function un, multiplied by a constant number En, the eigenvalue associated with un. In the particular case of the Hamiltonian, the eigenvalues are real numbers, because the Hamiltonian is a Hermitian operator (you may want to revise at this stage the definition of Hermitian operators). Some operator and matrix pages prepared for John's Quantum Physics Course may be relevant here.

We say that un characterizes a state of the system, and that En is the energy of that state. Depending on the system, and hence on its Hamiltonian, the eigenvalues En may vary discretely and/or continuously. Discrete eigenvalues correspond to bound states, and continuously varying eigenvalues correspond to unbound states. Some systems only have discrete eigenvalues such as the harmonic oscillator or the particle in a box with infinite potential bounding the box (see Lecture 2). Others have continously varying eigenvalues (like the free electron gas), and others have a mixture of discrete and continuum spectra of eigenvalues (atoms, molecules, etc).

Solving a problem in QM means solving the corresponding Schrödinger equation. However, there are only a handful of cases for which this can be done analytically (i.e. with pencil and paper). Two such cases are the harmonic oscillator and the hydrogen atom, but in general for real situations one has to use numerical approximations. An appreciation of these numerical methods, and associated practice with computing, form an important part of this course.

1.1.2 Wave functions, states and representations

Up to now we have spoken of wave functions as being functions of position r, also called real space. But in fact there is a great deal of flexibility. We can represent wave functions in terms of other variables, such as momentum. Frequently, problems relating to solids are easier to work out in momentum or reciprocal space rather than in real space, as will become apparent in later lectures. This is the issue of representation; we can work in whatever representation turns out to be most convenient for our particular problem, because the laws of QM do not depend on the particular representation used.

Therefore, it makes sense to speak of the states of a given system without making any reference to the way in which we choose to represent them. To do this we make use of Dirac's notation, by which a state characterised by the wave function un(r) (in real space) is denoted by |un> or simply by |n>. |n> is the ket notation of state n. We also need to manipulate the complex conjugate of kets, for example when calculating expectation values, and these are denoted by <n|, and are called bra. Therefore in terms of bras and kets, the Schrödinger equation is written:

H|n> = E|n>.         (1.1)

The abstract representation of wave functions (states) in terms of bras and kets brings out a parallelism with vectors. In vector algebra we can manipulate vectors in an abstract way, without making use of a specific frame of reference; for specific problems certain frames of reference may be more useful than others. In QM it is the same: we can manipulate bras and kets, and only use a specific representation when it happens to be more useful for the problem at hand.

The parallelism between kets (and bras) and vectors becomes even more apparent after the introduction of orthonormal basis sets. An orthonormal basis set is a collection of kets {|n>} which are all normalised, i.e. <n|n> = 1, and which fulfil the condition of being linearly independent. Linear independence means that no ket from the set can be expressed as a linear combination of the other kets in the set. This implies that <n|m> = 0 for n not equal to m. You will sometimes see this pair of equations written in terms of the Kronecker d as

<n|m> = dnm.         (1.2)

Orthonormal basis sets are useful because in the world of bras and kets they play the same role as frames of reference play in vector algebra. To give a specific example let us assume that we have an orthonormal basis set {|n>} which spans a given space. Contained within this space we have a state |f>. We can then express |f> in terms of the set {|n>} as

|f> = Σn <n|f>|n>,         (1.3)

where the sum extends over all elements of the basis set. Remember that in vector algebra a vector v is expressed in terms of the set of unit vectors {en} forming a frame of reference as

v = Σn (e n · v) en.         (1.4)

The term contained within brackets represents the scalar product of vector en with vector v. The parallelism between kets and vectors is thus obvious, as well described by Sutton. Note that we need you to read the relevant parts of chapters 1 & 2 following this lecture, as explained in the reference list.

1.1.3 Practical computational schemes: matrices

But what about operators? Are operators also isomorphic with vectors? No, operators are represented as matrices, which is not so surprising if one bears in mind that a matrix transforms one vector into another vector (remember that an operator transforms a ket into another ket!). This matrix/vector formulation of QM may sound abstract and convoluted, but it is in fact extremely useful, and constitutes the starting point by which most practical problems are solved, as we shall see.

Let us consider again the Schrödinger equation H|n> = En|n>. When we want to solve this equation for a specific system, we usually know the Hamiltonian, but the eigenvectors |n> and eigenvalues En are unknown and our task is to find them. We then proceed as follows:

  1. We choose an appropriate basis set of functions. This choice is generally physically motivated. For example, if we want to solve the Schrödinger equation for a molecule, an appropriate basis set may be the set of atomic eigenstates of the atoms constituting the molecule.
  2. We evaluate the Hamiltonian matrix elements <n|H|m> for every n, m in the basis set. Note that we are not making reference to any representation here; it can be done in any representation that we choose.
  3. Then, with the Hamiltonian in matrix form, solving the Schrödinger equation is equivalent to the matrix operation called diagonalisation, which is a standard matrix problem for which many practical algorithms exist. This process returns the eigenvalues and eigenvectors (expressed in the basis set chosen) of the Hamiltonian. Computers are very good at solving problems involving vectors and matrices, and this is why the matrix/vector representation of QM is so useful.

In the next section we will illustrate all this machinery with a specific example, and then pose some problems for you to do.

1.2 Homonuclear diatomic molecules: bonding and anti-bonding states

1.2.1 The molecular hydogen ion, H2+

Let us now consider a specific example. We shall solve the Schrödinger equation in matrix form for the H2+ molecule, i.e. the ionized hydrogen molecule, the simplest possible molecule, consisting of two protons and one electron. Since we are interested in the electronic structure, we are going to assume that the protons remain fixed, and that they contribute only an external Coulomb field in which the electron moves. We can do this by virtue of the so-called Born-Oppenheimer approximation, which states that given the low ratio of the mass of the electron to the mass of the proton, the motion of the latter is much slower. As a consequence, the motion of the electron and that of the nuclei are largely decoupled and can be considered separately. This is of course an approximation, but it is so often made that it is often not mentioned at all; this is a warning ahead for when we want to compare theory with experiment: look at the fine print!

The following treatment is of course quite simplified, but it serves as an illustration of the process. Initially, consider an H atom (proton+electron) in its ground state (i.e. the electron in the 1s atomic state) and a proton, very far away from each other. As we bring the H atom and the proton closer together, the electron will start feeling the presence of the extra proton, and the total energy of the system will be lower if the electron is located in those areas of space in which it is closest to both protons simultaneously. This is the essence of covalent bonding; now we see how this intuitive picture arises in QM.

It is reasonable, as a first approximation, to adopt a basis set which consists only of two elements (two kets), namely two 1s atomic orbitals, one centred on the H atom, the other centred on the proton (at this stage you might like to revise the form of 1s functions, which is discussed for example in Chapter 1 of Sutton's book, as well as in any Quantum Mechanics or Physical Chemistry textbook). So let us now construct the matrix form of the Hamiltonian in this basis. For the H+2 system the Hamiltonian has the following form:

H = T + V1(r) + V2(r),         (1.5)

where T is the kinetic energy operator for the electron, V1(r) is the Coulomb potential energy operator describing the interaction of the electron with proton 1, and the last term, V2(r) is just the same thing but with proton 2.

1.2.2 Matrix elements and determinental solutions

We will label our basis functions simply as |1> (1s function centred on proton 1) and |2> (1s function centred on proton 2). Then we have:

<1|H|1> = <1|T + V1(r)|1> + <1|V2(r)|1> = E1s + V.         (1.6)

Here, E1s is the energy of the ground state of the isolated hydrogen atom, and V is the energy of interaction of the electron with the second proton.

The second matrix element, <1|H|2>, will look like this: <1|H|2> = <1|T + V2(r)|2> + <1|V1(r)|2>. But notice that the first term in the right hand side is zero, because we are assuming that <1| and |2> form an orthonormal set, and thus <1|T + V2(r)|2> = <1|E1s |2> = E1s <1|2> = 0. Therefore we have

<1|H|2> = <1|V1(r)|2> = W.         (1.7)

Likewise, it is easy to see that the other remaining integrals are

       <2|H|1> = <1|H|2> = W,            (1.8)

and <2|H|2> = <1|H|1> = E1s + V.    (1.9)

So we now have the matrix form of the Hamiltonian for the H+2 molecule. Now, let's turn to the wave function; we still don't know what this is, but we do know that it will be expressed in terms of our basis set as |y> = C1 |1> + C2 |2>, i.e. as a linear combination of our chosen basis set, the two 1s functions centred on either proton. And we also know that the wave function will be the solution of the Schrödinger equation. So let us write down the Schrödinger equation in matrix form:

which, after some rearrangement is seen to be equivalent to
This system of equations can only have non-trivial solutions if the following condition holds
which gives us a quadratic equation for the eigenvalue E, which once solved has two solutions:

Eb = E1s + V + W,             (1.10a)

and Ea = E1s + V - W.         (1.10b)

Thus we have obtained, in equations (1.10), the two eigenvalues of the system.

1.2.3 Wave functions and bond charge

To obtain the corresponding eigenstates, we take each of the found eigenvalues in turn and, substituting them in the Schrödinger equation, we solve for the values of C1 and C2. When we do this we find

|yb> = N ( |1> + |2>),              (1.11a)

and |ya> = N ( |1> - |2>),         (1.11b)

where N is a normalisation constant (equal to 2). Because W is negative, Eb is the lowest of the two eigenvalues, i.e. it is the energy of the ground state. |yb>, the ground state wave function, looks pictorially like this:

The charge density, |yb |2, looks more or less similar. As you can see, there is a build up of electronic charge in the region between the two protons, and this build up of charge helps to hold the molecule together, constituting a chemical bond. Because the molecule is actually held together by the sharing of an electron between two nuclei, this is an example of a covalent chemical bond.

However, for the excited state, the wave function looks like this:

and the charge density (the squared norm of the wave function) looks like this:
You can see that the wave function has a node at the mid point between the two protons. When we look at the charge density associated with this state, one finds that it does not accumulate between the two protons, but rather, it is most probable to find the electron on either end of the molecule. So in this case the electron is not helping to hold the molecule together, and a bond is not formed. In fact, the energy of this state is higher than the energy of an isolated hydrogen atom and a proton far away from each other, and we thus say that this is an anti-bonding state.

There are some differences between the above treatment and that presented by Sutton on pages 25-31. Make sure that you are clear about these differences via problem 1.4.2.

1.3 Heteronuclear diatomic molecules: covalency and ionicity

The worked-out example for the H2+ molecule can serve as a template for the heteronuclear diatomic molecule. The process of solution is essentially identical, but the atomic levels are now different. Nevertheless, the fact that the nuclei are not the same has some profound consequences, which can result, in the extreme case, in ionic bonding in the molecule.

We suggest that you try to work out problem 1.4.3 for the heteronuclear diatomic molecule for yourself, and then consider the consequences of your findings on the nature of the chemical bond.

1.4 Problems relating to this topic

1.4.1: The unit matrix

Use equation (1.3) to show that the unit matrix I is given by Σn|n><n|, and explain the corresponding situation with ordinary vectors in real (3D) space.

1.4.2: Bond charges in the hydrogen molecule and molecular ion

As spelt out by Sutton on page 29, the charge density in the bond is rbond(r) = 2y1(r) y2(r). Evaluate this difference between the the total charge density and the charge density of the constituent parts in the case of the hydrogen molecular ion.

There seems to be some shifty footwork going on: we discuss the molecular ion, while Sutton seems to be discussing the molecule. See if you can get clear what is going on, and if not pose a question in class.

1.4.3: The heterodiatomic molecule

As set out in section 1.3 above, the methods demonstrated in section 1.2 will work for the heterodiatomic molecular ion, where now E2 is not equal to E1. Work through this problem, and plot a graph showing how the bonding and antibonding molecular levels depend on (W/DE), where DE = Ea - Eb. Find out, or discuss in class, how the matrix solution is related to the result obtained by using a perturbation expansion.

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