Notes for QMMS Lecture 1 (Venables/Heggie)

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1. Bonds and Bands: chemical and physical approaches

1.1 A brief revision of Quantum Mechanics

1.1.1 Hamiltonians and Eigenvalue equations

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 u_n, called the eigen-functions of H, which when transformed by the Hamiltonian result in the same function u_n, multiplied by a constant number E_n, the eigenvalue associated with u_n. 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 u_n characterizes a state of the system, and that E_n is the energy of that state. Depending on the system, and hence on its Hamiltonian, the eigenvalues E_n 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

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 u_n(r) (in real space) is denoted by |u_n> 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:

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

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

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 {e_n} forming a frame of reference as

The term contained within brackets represents the scalar product of vector e_n 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> = E_n|n>. When we want to solve this equation for a specific system, we usually know the Hamiltonian, but the eigenvectors |n> and eigenvalues E_n 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⁺

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:

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:

Here, E_1s 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|E_1s |2> = E_1s <1|2> = 0. Therefore we have

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

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:

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

1.2.3 Wave functions and bond charge

where N is a normalisation constant (equal to 2^-½). Because W is negative, E_b is the lowest of the two eigenvalues, i.e. it is the energy of the ground state. |y_b>, the ground state wave function, looks pictorially like this:

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

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

1.4.2: Bond charges in the hydrogen molecule and 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