The following exercises are to be completed in the computational
laboratory and reports handed in to Miss Lisa Benbow in 442
on the dates indicated. Work submitted late will be penalised.

The maximum number of figures allowed in your report is 20.

Note that during scheduled sessions the computational laboratory is
reserved solely for your use.

Group B: 19^{th} January to 23^{rd} January;

deadline 13:00 on Wed 28^{th} January.

Group A: 23^{rd} February to 27^{th} February;

In each exercise there are a number of explicit *Questions*
and
often also an opportunity to *Speculate* on the results obtained.
The final report will be expected to contain answers to these questions
and, where you feel able, further speculation.

Questions related to this computational experiment can be directed
to
Prof. Nicholas Harrison
and Dr Giuseppe Mallia.

The aims of this lab are

- to show that a solid can be considered as an infinite molecule and
- to understand
how the energy level diagram of a molecule (a finite system)

becomes a band structure for a periodic system, like a polymer/chain, a slab/layer and a crystal.

An exciting journey is about to begin. The hydrogen atom is the departure station and the hypothetical hydrogen chain or polymer (an infinite one-dimentional period system) is the destination one. There are many stops in between: the hydrogen molecule is the first one. Then, the route will split in two and both will travel parallel each other. At each stop, the number of hydrogen will be increased in two different ways:

- by adding an atom to the linear chain and obtaining
the H
_{3}linear cluster, H_{4}, H_{5}, ... - by forming a ring, the H
_{3}cyclic cluster, and by inserting a new hydrogen atom at each next stop.

The hydrogen chain is hypothetical, since this system is unstable and will dissociate forming hydrogen molecules. This is also the reason why the word "cluster" has been adopted for the systems with more than 2 atoms. Nonetheless, the hydrogen chain and the considered clusters result to be a good set of models for understanding band structure.

Once the hydrogen chain is reached, there will be an extention of the periodicity, from 1 to 2 dimensions, the slab/layer, and from 2 to 3 dimensions, the hydrogen crystal.

The concept of **density-of-states (DOS)** will be also introduced.
A DOS is a way to group levels according to their energy value.
Then, the DOS curve counts the levels and
is a function of the energy.
The integral of the DOS from the energy of the lowest occupied orbital
to the highest occupied orbital gives the total
number of occupied molecular orbitals (MOs) for a molecule ( or a
finite system) and the number of occupied crystalline orbitals
for a solid (or a infinite periodic system).

Linux provides an excellent environment for numerical simulations so the first step is to reboot your computer into RedHat Linux.

The environment is not hugely dissimilar to that provided by Microsoft Windows. You will find a web browser (Mozilla) on the tool bar and under the Start Menu you will find some office tools (Writer, Math etc.) which are similar to those in Microsoft Office (Word, Excel etc.) and you may find them useful in plotting your data and writing your report.

DLVisualize is a general purpose graphical user interface for modelling. It will give you relatively easy access to a number of quantum mechanical and empirical simulation codes. In this case the interface to the code CRYSTAL.

The CRYSTAL program computes the ground state energy, electronic wave function and properties of periodic systems within Hartree Fock, density functional or various hybrid approximations, will be used.

There are web sites devoted to both DLV and CRYSTAL where you can find some additional information.

Further information about DLV

Further information about CRYSTAL

** Contents: **

- 0D system: Molecule and Cluster

- The H atom

- The H_{2}molecule

- The H_{3}cluster: - The H_{4}cluster: - The H_{6}cluster: Linear and Cyclic

- The H_{8}cluster: Linear and Cyclic

- The H_{9}cluster: Linear and Cyclic

- The H_{10}cluster: Linear and Cyclic

- The H_{20}cluster: Linear and Cyclic**(Optional)**

- The H_{30}cluster: Linear and Cyclic**(Optional)**

- The H_{40}cluster: Linear and Cyclic**(Optional)**

- The H_{50}cluster: Linear and Cyclic

- The H_{100}cluster: Linear and Cyclic

- The H_{100}cluster II: Linear and Cyclic

- 1D system: Polymer/Chain

- The H cell

- The H_{2}cell

- The H_{3}cell

- The H_{4}cell

- The H_{6}cell - 2D system: Slab/Layer

- The H cell - 3D system: Crystal

- The H cell

How to save a picture for your report

**Additional material.**

- "How Chemistry and Physics Meet in the Solid State" By Roald Hoffmann, Angew Chem Inr. Ed Engl 26 (1987) 846-878
- "Bonding and structure of molecules and solids" by D. Pettifor, Oxford Science Publications
- "Electronic Structure of Materials" by Adrian P. Sutton, Oxford Science Publications
- Introduction