# The electronic structure of molecules, polymers, surfaces and solids

## Instructions: Spring 2010

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: 18th January to 22nd January;

deadline 13:00 on Wed 27th January. (Postponed due to network problem)

### deadline 13:00 on Fri 29th January.
(Please note you need to hand in your reports to Ms Hartley-Forbes in room 861, level 8, Chemistry)

Group A: 22nd February to 26th February;

### deadline 13:00 on Wed 03rd March.

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.

## Introduction

The aims of this lab are

1. to show that a solid can be considered as an infinite molecule and
2. 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.

This knowledge will allow analyse the band structure of a generic periodic system under investigation and to predict its electronic properties, depending on the band structure.
A 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:

1. by adding an atom to the linear chain and obtaining the H3 linear cluster, H4, H5, ...
2. by forming a ring, the H3 cyclic cluster, and by inserting a new hydrogen atom at each next stop.

In the limit of an infinite cluster, in both the cases (linear and cyclic), the hydrogen chain is reached. This approach has the advantage to show that a translation can be represented as a rotation on of a circle of infinite radius.

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).

## The Software: RedHat Linux, DLVisualize and CRYSTAL

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:

1. 0D system: Molecule and Cluster
   - The H atom
   - The H2 molecule
   - The H3 cluster:
   Linear
   Cyclic
   - The H4 cluster:
   Linear
   Cyclic
   - The H6 cluster: Linear and Cyclic
   - The H8 cluster: Linear and Cyclic
   - The H9 cluster: Linear and Cyclic
   - The H10 cluster: Linear and Cyclic
   - The H20 cluster: Linear and Cyclic (Optional)
   - The H30 cluster: Linear and Cyclic (Optional)
   - The H40 cluster: Linear and Cyclic (Optional)
   - The H50 cluster: Linear and Cyclic
   - The H100 cluster: Linear and Cyclic
   - The H100 cluster II: Linear and Cyclic

2. 1D system: Polymer/Chain
   - The H cell
   - The H2 cell
   - The H3 cell
   - The H4 cell
   - The H6 cell

3. 2D system: Slab/Layer
   - The H cell

4. 3D system: Crystal
   - The H cell

How to start DLVisualize

How to save a picture for your report

Additional material.

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