CHEM0019: PHYSICAL CHEMISTRY Quantum Mechanics

Section   Topics covered

1            Overview

2             Postulates of quantum mechanics: probability interpretation of Ψ, operators and Hamiltonians, eigenvalue equations, Schrödinger equation, expectation values, Variation principle

3             Exact solutions to the Schrödinger equation: harmonic oscillator (Peer Study Group 1, week 16), particle on a ring, particle on a sphere (rigid rotor), hydrogen atom (Active Learning Workshop 1, week 21)  Xmas break!!!

4             Beyond exact solutions: helium atom

5             Molecular orbitals: Born-Oppenheimer approximation, linear combination of atomic orbitals, secular equations, two-orbital systems (Peer Study Group 2, week 22)

6             Hückel theory for τ-electron systems; allyl radical (Active Learning Workshop 2, week 23)

7             Concluding remarks

Recommended textbooks:  Atkins Physical Chemistry – online link on Moodle (Parts of) Chapters 7-10.

Alternatively: “Quantum Mechanics for Chemists” by D. O. Hayward, RSC.

“Molecular Quantum Mechanics” by P.W Atkins and R.S. Friedman, 5th  edition, OUP, available electronically “Physical Chemistry” by P.W Atkins and J. de Paula, 8th  edition, OUP.

1. Overview

The principal aim of this course is to understand wavefunctions, how to calculate them and how to use them in chemistry. To do this we will first define some fundamental principles of quantum mechanics. To help explain these principles we will use test problems, such as the harmonic oscillator and the rigid rotor. Understanding these cases allows us to understand the electronic structure of the hydrogen atom. We will then move on to consider the electronic structure of helium atom, and an introduction to molecular orbital theory. This helps  illustrates why approximate wavefunctions can be used both qualitatively and quantitatively throughout chemistry.

“The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known,

and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved”

Dirac 1929

2. Postulates of quantum mechanics

Postulate = a stipulation, an assumption, a fundamental principle

2.1 Probability interpretation of Ψ

Postulate 1: the state of a system is fully described by a mathematical function Ψ, called the wavefunction. As an example, consider a particle which can move only in the x direction

This particle is described by a wavefunction Ψ(x).

Question: what is the probability that the particle will be found between points x and x + dx?

Answer:

This probabilistic interpretation of the wavefunction is commonly known as the Born interpretation.

Notes: 1. In this treatment we assume that the wavefunction is normalized, i.e. that the probability of finding the particle somewhere along the x direction is 1. Mathematically we write: the “bra-ket” notation first introduced by Paul Dirac.

2. The wavefunction itself has no physical meaning. It may, at any given point in space, be positive or negative, real or complex.

Ψ (x )2  = Ψ * (x )Ψ (x ) is always non-negative and real, and has the probabilistic meaning described above.

2.2 Operators and Hamiltonians

An observable is a measurable property such as bond length, dipole moment, kinetic energy etc … .

Postulate 2: every observable B is represented by an operator, and all operators can be built from the operators for position and momentum.

The operator for position in the x direction is defined as B(^) =x(^)

and for momentum in the x direction,

A very important operator is the total energy operator, which is called the Hamiltonian

H(^) = T(^)+ V(^)

where H(^) = total energy operator, T(^) = kinetic energy operator and V(^) = potential energy operator.

For motion in the x direction px = mvx and hence

The role of the operator is to operate on a wavefunction to yield information associated with the observable that the operator represents.

2.3 Eigenvalue equations

An eigenvalue equation has the general form.: Bf(^) = bf where the operator B(^) acts on the eigenfunction to

regenerate f multiplied by the eigenvalue b (a constant).

Example 1: If and

and hence eax is an eigenfunction of with eigenvalue a .

Example 2: If and f = sin ax , Hence sin(ax) is not an eigenfunction of dx/d.

2.4 The Schrödinger equation

Let us now use the operator H(^) to operate on Ψ(x):

where V(^)(x)  is the potential energy operator for motion in the x direction.

If Ψ(x) is an eigenfunction of H(^) , then it is termed an exact wavefunction, and

where the eigenvalue E is an energy (recall that H(^) is an energy operator). This equation is the Schrödinger equation for a particle moving in the x direction. (If V(^)(x) is complicated, we may not be able to find an exact Ψ (x)analytically and so may only be able to find approximate wavefunctions using a computer.)

There may very well be many more wavefunctions which are eigenfunctions of H(^) , i.e. which are solutions to H(^)Ψ (x) = EΨ (x) . Any two non-degenerate solutions (i.e. solutions of different energy) are orthogonal

Any two wavefunctions are orthonormal if

where ∂jk is the Kronecker delta.

∂jk = 1if j is the same as k i.e. j = k (the wavefunction is normalised)  and

∂ jk = 0 if j is different from k i.e. j ≠ k (the wavefunctions are orthogonal)

2.5 Expectation values

The expectation value of an operator B(^) for a wavefunction Ψ is denoted <B> and is defined as

where dτ tells us that the integration is being performed over all space. If the wavefunction Ψ  is normalised to 1, then the expectation value simplifies

If Ψ is an exact wavefunction, i.e. it is an eigenfunction of H(^) , and it is also an eigenfunction of B(^) such that BΨ(^) = bΨ , then

What would happen if we attempted to measure the   value of observable B? Because Ψ is an exact

wavefunction and an eigenfunction of B(^) , a series of identical experiments will all yield b i.e. they will give a single value.

Let us now consider the specific and real case of an atom or molecule with more than one electron.  This system

has a set of wavefunctions Ψn which satisfy the Schrödinger equation H(^)Ψ n = EΨn .  However, these

wavefunctions are unlikely to be eigenfunctions of of B(^) i.e. BΨ(^)n ≠ bnΨn . Under these circumstances a series of

identical experiments would not all yield the same answer but each individual experiment could give any one of

the eigenvalues b0, b1, b2 … . and the expectation value of B(^) is the average of all the values that would be

obtained from a large number of experiments.

This is represented by our final postulate 3 and its implication 3’:

Postulate 3: when a system is described by a wavefunction Ψ , the average value of the observable B in a series

of measurements is equal to the expectation value of the corresponding operator B(^) , i.e. B = Ψ B Ψ .

Postulate 3': when Ψ is an eigenfunction of B(^) , determination of B always yields one result, b.  When Ψ is not an eigenfunction of B(^) , a single measurement of B yields a single outcome which is one of the eigenvalues of B(^) ,

and a large number of measurements will yield an average of the eigenvalues of B(^) i.e.  .

2.6 The Variation Principle

Postulate 3 leads to a plausibility argument for the variation principle, which is used extensively for the

computational treatment of many-electron atoms and molecules, and is the foundation for the approximate methods for determining molecular orbitals in Section 5.

We want to find the energy of the ground state Ψ 0  , which has an associated energy E0 of a molecule which has a

set of wavefunctions Ψn which satisfy the Schrödinger equation H(^)Ψ n = En Ψ n .   As the exact wavefunctions for

many-electron systems are not known and we must employ an approximate wavefunction Ψtrial . As Ψtrial is not

exact, it will not be an eigenfunction of H(^) , and H(^)Ψtrial ≠ EΨtrial .

If we attempted to measure the energy of a hypothetical system described by Ψtrial , a series of identical

experiments would not all yield the same answer but each individual experiment could give any one of the

The variation theorem states that for any trial wavefunction Ψ trial , the expectation value of the energy can never be less than the true ground state energy E0 :

The expectation value of the energy is an average of the true energies of the system E0  , E1 , E2  … ., and this can never be less than E0  .