Computational Chemistry Practical exercise 1 Electronic and geometric structure of carbon nanotubes Introduction Single-walled carbon nanotubes (SWNTs) are allotropes of carbon with cylindrical structures. They may exhibit a wide range of mechanical, thermal and electrical properties depending on their specific structures. Conceptually, a SWNT) an be constructed from the rolling of a graphene sheet, as illustrated in Figure 1. The properties of the tubes depend upon their geometry. SWNTs can be classified by a pair of numbers which denote the carbon atom which meets up with the origin upon rolling, as also shown in Figure 1. In so-called â€œarmchair tubesâ€, the two numbers (n, n) are the same, and the armchair edge is shown in Figure 2. So-called â€œzigzag tubesâ€ are characterized by (n, 0) and their edges have a zigzag motif, also shown in Figure 2. The last class of tubes are helical in nature and have (n, m) where n is not equal to m. These represent the three different classes of SWNT. Figure 1: The construction of a single-walled carbon nanotube from a graphene sheet. Rolling the sheet so that the origin meets with a black/gray/white circle yields a zigzag/armchair/helical nanotube. The arrows indicate how a (4,0) and a (4,4) nanotube would be made from rolling this sheet. Figure 2: A finite fragment of a (5,5) armchair (left) and a (9,0) zigzag (right) SWNT. The (n, m) values not only determine the diameter â€“ they also determine the properties of the SWNTs. For example, armchair tubes are typically metallic (electricity can flow through them), so in principle they can be used to make molecular wires. Other tubes may be semiconductors or even insulators. The SWNTs made in experiments are typically very long; in fact, their length can be a million times greater than their diameter. One way to study these systems computationally is to treat them as being infinitely long, and employ computational programs used to study solids. However, short nanotubesComputational Chemistry Practical exercise 2 can be made too, and another way to study these systems computationally is by using molecular calculations to see how the properties change as a short SWNT is lengthened The aim of this experiment is to use computational methods to gain an understanding of the electronic structure and properties of short SWNTs. Methods Electronic structure In this section the aim is to calculate the electronic properties of (3,3) finite nanotubes with varying lengths. (3,3) nanotubes exhibit the armchair structure shown in Figure 2, and can be considered to comprise the repeat units shown in Figure 3, three of which make a ring. This ring is the shortest possible SWNT with an armchair structure, and the structure is shown in Figure 4. Construct such a ring, and add sufficient hydrogen atoms such that the C atoms are sp2-hybridised. The easiest way is to first make the ring of carbon atoms, then add the H atoms at the end, and you will probably also find it easier if you select â€œBalls and Cylindersâ€ under the Display -> Rendering menu. When constructing this and the other structures in this work, do not use the Model Builder option to cleanup/pre-optimise your structure because it will add too many hydrogens to the structures; instead you may want to pre-optimise the structures using the MM+ method (with the exception of this first structure, for which you should not pre-optimise the structure at all). Hyperchem can (and does) crash, particularly when performing calculations on structures containing large numbers of atoms such as those performed later in this exercise. Save your structures regularly to avoid wasted time should it crash, and ensure each of your optimised structures are saved for the second part of the exercise. Figure 3 Repeat unit of a SWNT with an armchair structure Figure 4 The structure of a (3,3) SWNT of length 1 Optimise the structure using the AM1 semi-empirical method, and record the HOMO and LUMO energies (using Compute -> Orbitals), and the approximate length of the tube noting how you have chosen to define it. Modify your structure by adding another set of 12 carbon atoms to give the structure shown in Figure 5, optimise it using the AM1 method, and record the HOMO and LUMO energies and length again.Computational Chemistry Practical exercise 3 Figure 5 The structure of a (3,3) SWNT of length 2 Continue adding sets of 12 carbon atoms in the same way, each time extending the nanotube by the length of one repeat unit (shown in Figure 3), remembering to save your structures. The spacebar is a useful shortcut to rescale the view as you are enlarging the molecule. For each structure, optimise it in the same way, and record the HOMO and LUMO energies. Continue this process until you have 7 HOMO and LUMO energies and 7 measured lengths; your final structure should contain 84 carbon atoms. For one of your structures (with at least 24 C atoms) visualise the HOMO and LUMO orbitals, and create figures for your report. Determine the HOMO-LUMO gap for each length of tube, and plot these values against the number of carbon atoms in each tube. Add a linear regression line to your plot, ensuring you show its equation. Consider excluding data point(s) from your line if this is justifiable. Using this line, predict the number of carbon atoms needed for a (3,3) SWNT to be electrically conductive. Remember that insulators have a gap between the HOMO and LUMO levels whereas conductors do not. Curvature induced strain In the previous section the influence of the SWNT length was assessed, but their width (radius) is also important to their properties. In a flat graphite sheet all C-C bond lengths are 1.44 Ã…, but nanotubes are strained, and the bond lengths may vary depending on the bond orientation relative to the orientation of the tube axis. In armchair SWNTs the bonds may be characterized as either perpendicular to the tube axis, or as diagonal. From Blackboard, download the (3,3)nanotube structure. This is a (3,3)-nanotube optimised using the HF method and the STO-3G basis set, which would take too long to optimise for the time available in this session. Measure the bond lengths of perpendicular and diagonal C-C bonds within the structure, but avoid using bonds containing C atoms to which H atoms are attached. For an optimised structure, lengths of equivalent bonds should be almost identical. Record these values as well as the total number of carbon atoms in the structure. Then carry out a single-point energy calculation on the structure using the AM1 method and record the energy of the structure. Download the equivalent structures of (4,4), (5,5), (6,6), and (7,7) SWNTs, and optimise them using the same method as used in the previous section. Record and tabulate the energy of each structure, the lengths of the perpendicular and diagonal bonds, and the number of carbon atoms in each structure.Computational Chemistry Practical exercise 4 Once all your energies, numbers of carbon atoms and bond lengths are tabulated, for each structure determine the energy per carbon atom and plot a graph of these values against the number of carbon atoms. Also plot the lengths of the perpendicular and diagonal carbon-carbon bonds in the tube versus the total number of carbon atoms in the tube, and using this plot estimate the number of carbon atoms that the same length tube would need for the two C-C bond lengths to be equal. Write-up The report should be less than 2500 words in total (but may well be much shorted). Include a brief introduction outlining the background and relevance of the exercise. A short methods section should include sufficient information for a reader to replicate the experiment. A results and discussion section should set out your main results, including tables of the values you have determined. In the electronic structure section, include a plot of HOMO and LUMO energy against the number of carbon atoms, a plot of the HOMO â€“ LUMO energy gap against the number of carbon atoms, and a figure showing your HOMO and LUMO visualisations. In the curvature induced strain
section include your tabulated values as well as plots of energy-per-carbon-atom against the number of carbon atoms and bond-lengths against the number of carbon atoms. Make sure that all tables and figures have a number and caption. Finish with a short conclusions section outlining the significance and implications of the results, along with areas for potential future work. Questions to consider: – Why are semi-empirical methods well suited to this investigation? What would be the advantages/disadvantages of using a more thorough method? – What are the characters of the HOMO and LUMO orbitals you have prepared visualisations of? Are they localised on the carbon or hydrogen atoms, or delocalised? What is the apparent hybridisation of these orbitals? – In the context of SWNT diameter, do you think narrow or wide diameter tubes are more easily synthesised and why? Which would be likely to be more stable?