*CHEM2020 *Intermediate Physical Chemistry I

## Module Overview

The aim of this module is to provide a basis for future studies in chemistry and allied subjects. Students select two areas of Chemistry from Inorganic, Organic, and Physical Chemistry according to the needs of their programme of study. Please consult with the leader of your programme or your personal academic tutor in deciding which two modules to follow. Note that this module is not available for students enrolled in any of the Chemistry degree programmes.

### Aims and Objectives

#### Module Aims

The aims of the module are to provide: • an introduction to the quantum mechanical description of atoms and molecules; • the form and solutions to the Schrödinger equation; • a working understanding of operators and wavefunctions. • a qualitative understanding of the physical origin of intermolecular interactions and their influence on the structure of molecular assemblies; • quantitative descriptions of the physical contributions to interactions between molecules; • an introduction to common model potentials for describing intermolecular interactions. • an understanding of: operators; eigensystems; matrix functions and determinants; complex numbers and functions;

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

- apply the basic principles of quantum mechanics that underpin the interactions between electrons and nuclei to obtain the energy levels and electronic structure of atoms and molecules;
- form molecular wavefunctions as a linear combination of atomic orbital, including the application of Huckel theory to conjugated molecules;
- calculate atomic and molecular properties using quantum mechanical techniques.
- recognise the types of forces acting within individual molecules and in molecular assemblies and classify these in terms of their relative strength and origin;
- provide and apply mathematical descriptions of electrostatic, polarisation, dispersion and repulsion contributions to intermolecular interactions;
- calculate intermolecular interactions using model potentials.

### Syllabus

• The Schrödinger eigenvalue equation is introduced as the central principle which governs the behaviour of electrons, atoms and molecules. The solution of the equation produces the wavefunction from which all experimentally observable properties of matter can be computed. The conditions that must be satisfied by chemically acceptable wavefunctions are discussed. • Differentiation of functions is revisited and operator notation is introduced. Quantum mechanical operators are defined. The Hamiltonian operator is built from the kinetic energy and potential energy operators. • Integration of functions is revisited and the calculation of experimentally observable properties (e.g. energy, dipole moment, etc.) from the wavefunction is presented. Normalisation and orthogonality of wavefunctions are discussed. • Vectors are reviewed. The Coulomb potential for interaction between point charges is introduced and with the help of vectors it is defined in 1, 2, and 3 dimensions. The importance of this potential in all sorts of chemical situations is discussed and demonstration of how the same potential describes interactions between electrons, nuclei and either is shown. • The problem of the particle in a 1-dimensional box is solved and its solutions are used to construct a simple model for the energy levels of conjugated polyenes. The particle in a 2-dimensional box is solved as a model quantum system where there is degeneracy between its energy levels. • The Hamiltonian operator is constructed, first for the hydrogen atom and then for any atom. • The wavefunctions of the Hydrogen atom which are built from Spherical Harmonics and radial functions are introduced and various properties such as their energy levels are examined. • The Pauli exclusion principle is introduced and it is shown how the Hydrogen wavefunctions can be extended to describe polyelectronic atoms and hence the periodic table. • The Born-Oppenheimer approximation to separate electronic from nuclear motion is described and the Hamiltonian operator (and Schrödinger equation) for molecules is constructed. • Molecular wavefunctions are approximated as products of molecular orbitals. The molecular orbitals are constructed from atomic orbitals via the Linear Combination of Atomic Orbitals (LCAO) approach. • Matrices, determinants and operations such as matrix multiplication are reviewed. The variational principle that allows LCAO calculations to be performed is described. Examples of LCAO calculations are presented and discussed in terms of the Huckel theory of conjugated molecules and other approaches of more general applicability. • The role intermolecular forces play in chemistry is described. • The Coulomb potential is revisited as the principle for describing permanent inter-molecular electrostatic interactions, using a partial charge approach. • The concept of multipole moments is described, and how the lowest multipole moment of a molecule may be determined. The role of electrostatic potential in understanding permanent electrostatic interaction is discussed, along with how a combination of multipole moments may be used to reproduce molecular electrostatic potentials. • The direct calculation of dipole-dipole interaction energies is given, and the distance dependence of all other multipole-multipole interactions is outlined. The link between the partial charge model and multipole descriptions is presented. • Induced interactions are described, together with how they are related to the electric field. The energy expression for the induction energy arising from a dipole is given. The distance dependency and non-pairwise additivity of induction is emphasised. • Dispersion interactions are introduced as a quantum mechanical effect due to electronic correlation and an approximate classical model for describing them is presented. • The relative strengths of the long-range intermolecular forces are discussed for a range of atomic and molecular systems. • The physical basis of hydrogen bonds and their role in affecting the properties of matter is given. • Repulsive forces are described in terms of a simple physical picture. An exponential dependence on distance is outlined. • Model potentials are introduced as an approximate method for computing molecular structure and interactions, starting from the Lennard-Jones potential as an example. The concepts of effective pair potentials and combining rules are outlined. • The advantages and disadvantages of model potentials as compared to the quantum mechanical calculations for molecules are discussed and demonstrated by examples. • The mathematics part of the module enables the students to understand the basic algebraic principles behind the quantum mechanical derivations and approximations used in the chemistry part of the module. At the end of the module the students will be comfortable with the mathematical foundations of quantum theory and well prepared for future work in the area of quantum chemistry.

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem-solving tutorials with group working and tutor support. 12 hours workshops is maths workshops 13 hours of preparation for scheduled sessions includes other independent study

Type | Hours |
---|---|

Workshops | 12 |

Preparation for scheduled sessions | 13 |

Revision | 10 |

Workshops | 10 |

Lecture | 30 |

Total study time | 75 |

#### Resources & Reading list

Monk and Munro (2010). Maths for Chemistry.

D. O. Hayward (2002). Quantum Mechanics for Chemists.

Ken A. Dill & Sarina Bromberg (2010). Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience.

Atkins & de Paula (2014). Physical Chemistry: Thermodynamics, Structure, and Change.

Steiner (2008). The Chemistry Maths Book.

### Assessment

#### Formative

In-class Test

#### Summative

Method | Percentage contribution |
---|---|

Examination (2 hours) | 100% |

#### Referral

Method | Percentage contribution |
---|---|

Examination (2 hours) | 100% |