Autores: A partir do fascículo 39/9 a revista Química Nova adotou a licença CC-BY. Mais informações a respeito dessa licença podem ser obtidas aqui.

Explaining the geometry of simple molecules using molecular orbital energy-level diagrams built by using symmetry principles |

Sérgio P. Machado; Roberto B. Faria Departamento de Química Inorgânica, Instituto de Química, Universidade Federal do Rio de Janeiro, 21941-909 Rio de Janeiro - RJ, Brasil Recebido em 27/08/2017 Endereço para correspondência
*e-mail: faria@iq.ufrj.br RESUMO
The built of qualitative energy-level molecular diagrams for different geometries of simple molecules allow to explain the preferred geometry. The diagrams are built using simple symmetry principles and explain, on basis of the number of nonbonding electrons, for example, why the molecule of water is bent and not linear and ammonia is pyramidal and not planar. This simple energy principle does not need to consider the Valence Shell Electron-Pair Repulsion theory (VSEPR theory) neither hybrid orbitals to explain the geometry of simple molecules. This discussion is more appropriate to inorganic chemistry courses where symmetry is a common topic. Palavras-chave: symmetry; group theory; molecular orbitals; molecular geometry
The geometries of molecules like H When using hybrid orbitals, the central atom in all these molecules is considered to use sp On the other hand, in the inorganic chemistry courses, when part of the time is dedicated to the application of symmetry to build the symmetry-adapted linear combinations of atomic orbitals, the stability of these molecules can be discussed more properly based on the molecular orbital energy-level diagrams for these molecules. Several inorganic chemistry textbooks present a very detailed molecular orbital energy-level diagrams for several simple molecules, for example, BeH Other works deal with the same subject we present in this article. Mulliken In this work we present the use of qualitative molecular orbital energy-level diagrams built from simple group theory principles of symmetry. However, differently from the articles cited above, our approach is based on the different occupancy of nonbonding molecular orbitals for each geometry of the molecule considered. This allows explain the preferred geometry of simple molecules without requiring the building of complete Walsh diagram. Our approach is simpler and is based mainly in the number of electrons on nonbonding molecular orbitals. Symmetry is a common subject in inorganic chemistry courses (most commonly third or fourth semester or even later) and undergraduate students are very familiar with it. In this way, we have used the approach described below in the undergraduate inorganic chemistry course. We observed that the comparison of the energy-level molecular diagrams for different geometries of an specific molecule is a very significant application of symmetry and molecular orbital theory which gives a very strong support to more deep discussions of chemical bond based on molecular orbitals. The inclusion of this subject allowed show for the students a simple and direct application of symmetry and group theory on the very basic topic of molecular geometry together with the presentation of fundamental aspects of symmetry. This was a very significant improvement because we start to show an application of these principles earlier than before. Students, usually, are very anxious about where they will apply this knowledge, which is abstract, mathematical and spatial reasoning, being difficult for most part of the class. Before we have included this subject, the students need wait for the first application of symmetry until the discussing the electronic spectroscopy of transition metal complexes. As a consequence, we observed a very clear improvement in the number of the students which obtained higher grades in the questions related with the basic principles of symmetry and group theory. We did not apply a tool to measure this improvement but we estimated that 70 to 80% of the class have increased their comprehension level. The starting point of this approach is the identification of the irreducible representation of the atomic orbitals. After that, the energy level diagram of the molecular orbitals is built, combining the atomic orbitals of the same irreducible representation. Then, the molecular orbitals are filled with the electrons using the aufbau principle. Comparison of the number of electrons which occupy bonding, antibonding, and nonbonding orbitals, at each different geometry of the same molecule, allows explain its preferred molecular geometry. In the following we consider that the reader knows the basic principles of Group Theory applied to chemistry including symmetry operations and the use of the Character Tables.
This molecule has only two possible geometries indicated in Figure 1: linear (D
Figure 1. The two possible geometries for the water molecule: linear (D_{∞h}) and bent (C_{2v})
In the case of the bent geometry the molecule belongs to the point group C
In the case of the
Figure 2. Results for the application of the Ĉ_{2} operator (on the z axis), of the (C_{2v}) point group, over the 2p_{x}, 2p_{y}, and 2p_{z} oxygen orbitals of the water molecule
Figure 3. Results for the application of the operator _{v}^{(xz)} , of the (C_{2v}) point group, over the 2p_{x}, 2p_{y}, and 2p_{z} oxygen orbitals of the water molecule
Figure 4. Results for the application of the operator _{v}^{(xz)}, of the (C_{2v}) point group, over the 2p_{x}, 2p_{y}, and 2p_{z} oxygen orbitals of the water molecule
To identify the irreducible representation for each orbital the result of the application of the symmetry operator is taken as equal to 1 if the orbital does not change and the result is -1 if the orbital changes its signal. In this way, Table 1 shows the results and the attribution of an irreducible representation for each oxygen atomic orbital 2p For the 1s atomic orbitals on each hydrogen (peripheral atoms), indicated by s This notation can be simplified considering that the first column and first line of the matrices refer to the sH1 atomic orbital and the second column and second line refer to the sH2 atomic orbital. In addition, the character or the trace of the matrix (χ, the sum of the main diagonal elements) is also indicated. These results show that 1s atomic orbitals of the hydrogens produce a reducible representation which can be shown to be formed by the sum A
With this information we can set up the energy level diagram for the molecular orbitals of water (Figure 5), considering that only atomic orbitals of the same symmetry species can combine. The symmetry labels are written in lower case when indicating atomic and molecular orbitals. Note also that the 2p
Figure 5. Qualitative molecular orbital energy-level diagram for water in the bent geometry (C_{2v})
Now, let us consider the possibility that the water molecule is linear. In this case the molecule will belongs to the point group D
In the case of the hydrogen 1s atomic orbitals, any These results show that 1s hydrogen orbitals (s Using these results we can build the molecular orbital energy-level diagram for the water molecule in the linear geometry (D
Figure 6. Qualitative molecular orbital energy-level diagram for water in the linear geometry (D_{∞h})
Comparison between Figures 5 and 6 shows that in the linear geometry the water molecule has four electrons in two π Clearly the reason for the bent geometry of the water molecule is that one of the π Some authors It is worth to say that the conclusion that water is bent was obtained without consider the repulsion between the electrons pairs in the VSEPR theory or by the use of sp
The same approach used with the water molecule can be applied to explain the most stable geometry of other simple molecules. Let us now consider the NH
Figure 7. The two possible geometries for the ammonia molecule: pyramidal (C_{3v}) and planar (D_{3h})
From these results we can say that both nitrogen 2s and 2p
Figure 8. Qualitative molecular orbital energy-level diagram for NH_{3} in the pyramidal geometry (C_{3v})
Let us now consider the NH
The three hydrogen 1s orbitals produce a reducible representation which is formed by A
Figure 9. Qualitative molecular orbital energy-level diagram for NH_{3} in the planar geometry (D_{3h})
Comparison between Figures 8 and 9 shows that NH
We can now consider the tetrahedral and planar geometry of methane, which belongs to the point groups T
Figure 10. The two possible geometries for the methane molecule: square planar (D_{4h}) and tetrahedral (T_{d})
Figure 11. Qualitative molecular orbital energy-level diagram for CH_{4} in the tetrahedral geometry (T_{d})
Figure 12. Qualitative molecular orbital energy-level diagram for CH_{4} in the planar geometry (D_{4h})
As can be seen from Figures 11 and 12 the methane in the planar geometry (D
In addition to these simple cases above, it is intriguing, for example, why is the ozone molecule bent and CO
Figure 13. Simplified molecular orbital energy-level diagram for CO_{2} (16 valence electrons) showing only the frontier π orbitals
As can be seen, the HOMO is a nonbonding π
The use of qualitative molecular orbital energy-level diagrams build from simple principles of symmetry and group theory allows forecast the correct geometry between two possibilities for some simple molecules. The choice of the most stable geometry depends on the energy of all electrons in the molecule and their interactions. However, for the molecules H This approach reinforces the importance of the knowledge of symmetry and molecular orbital principles and gives additional support to discussions of bonding and molecular geometry based in other principles as, for example, electron pair repulsion based in VSEPR theory or hybrid orbitals, which are more commonly teach in the beginning levels. Molecules of the AH
The authors thank the financial support by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Grants no. 306.050/2016-1 and 304.423/2014-9).
1. Brady, J. E.; Russell, J. W.; Holum, J. A.; Chemistry: The Study of Matter and Its Changes, 3 2. Kotz, J. C.; Treichel, P. Jr.; Chemistry and Chemical Reactivity, 9 3. Brown, T. E.; LeMay, H. E., Jr.; Bursten, B. E.; Murphy, C.; Woodward, P.; Stoltzfus, M. E.; Chemistry: The Central Science, 13 4. Weller, M.; Overton, T.; Rourke, J.; Armstrong, F.; Inorganic Chemistry, 6 5. Miessler, G. L.; Fischer, P. J.; Tarr, D. A.; Inorganic Chemistry, 5 6. Housecroft, C.; Sharpe, A. G.; Inorganic Chemistry, 4 7. Bernath, P. F.; Spectra of Atoms and Molecules, Oxford University Press: Oxford, 1995. 8. Walsh, A. D.; 9. Walsh, A. D.; 10. Walsh, A. D.; 11. Walsh, A. D.; 12. Walsh, A. D.; 13. Walsh, A. D.; 14. Walsh, A. D.; 15. Mulliken, R. S.; 16. Baird, N. C.; 17. Miller, C. S.; Ellison, M.; 18. Orchin, M. M.; Jaffé, H. H.; 19. Harris, D. C.; Bertolucci, M. D.; Symmetry and Spectroscopy, Oxford University Press: Oxford, 1978. 20. Cotton, F.A.; Chemical Applications of Group Theory, 3 21. Carter, R; L.; Molecular Symmetry and Group Theory, John Wiley & Sons: New York, 1998. 22. Kettle, S. F. A.; Symmetry and Structure - Readable Group Theory for Chemists, 3 23. Laing, M.; |

On-line version ISSN 1678-7064 Printed version ISSN 0100-4042

Química Nova

Publicações da Sociedade Brasileira de Química

Caixa Postal: 26037
05513-970 São Paulo - SP

Tel/Fax: +55.11.3032.2299/+55.11.3814.3602

Free access