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Inorganic Chemistry, Gary L. Miessler, Donald A. Tarr Textbook Chapter 5

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McGill University
CHEM 212
Richard Oakley

Molecular orbital theory uses the methods of group theory to describe the bonding in inolecules and complements and extends the simple pictures of bonding introduced in 3 The symmetry properties and relative energies of atomic orbitals determine Chapter how they interact to form molecular orbitals These molecular orbitals are then filled with the available electrons according to the same rules used for atomic orbitals and the total energy of the electrons in the molecular orbitals is compared with the initial total energy of electrons in the atomic orbitals If the total energy of the electrons in the mol ecular orbitals is less than in the atomic orbitals the molecule is stable compared with form We will the atoms if not the molecule is unstable and the compound does not first describe the bonding or lack of it in the first ten homonuclear diatomic molecules H2 through Ne2 and then expand the treatment to heteronuclear diatomic molecules and to molecules having more than two atoms A simple pictorial approach is adequate to describe bonding in many cases and can provide clues to more complete descriptions in more difficult cases On the other hand it is helpful to know how a more elaborate group theoretical approach can be used both to provide background for the simpler approach and to have it available in cases in which it is needed In this chapter we will describe both approaches showing the simpler pictorial approach and developing the symmetry arguments required for some of the more complex cases 51 As in the case of atomic orbitals Schrodinger equations can he written for electrons in molecules Approximate solutions to these molecular Schrodinger equations can be FORMATION OF constructed from linear combinations of the atomic orbitals LCAO the sums and MOLECULAR differences of the atomic wave functions For diatomic molecules such as Hz such ORBITALS FROM wave functions have the form ATOMIC ORBITALS T is the molecular wave functionand h are atomic wave functions and c where 1 17 51 Formation of Molecular Orbitals from Atomic Orbitals and cb are adjustable coefficients The coefficients can be equal or unequal positive or negative depending on the individual orbitals and their energies As the distance be tween two atoms is decreased their orbitals overlap wilh significant probability for electrons from both atoms in the region of overlap As a result molecular orbitals form Electrons in bonding molecular orbitals occupy the space between the nuclei and the electrostatic forces between the electrons and the two positive nuclei hold the atoms together Three conditions are essential for overlap to lead to bonding First the symmetry of the orbitals must be such that regions with the same sign ofoverlap Second the en ergies of the atomic orbitals must be similar When the energies differ by a large amount the change in energy on formation of the molecular orbitals is small and the net reduc tion in energy of the electrons is too small for significant bonding Third the distance between the atoms must be short enough to provide good overlap of the orbitals but not so short that repulsive forces of other electrons or the nuclei interfere When these con ditions are met the overall energy of the electrons in the occupied molecular orbitals will be lower in energy than the overall energy of the electrons in the original atomic orbitals and the resulting molecule has a lower total energy than the separated atoms 51 1 MOLECULAR ORBITALS FROM s ORBITALS We will consider first the combination of two s orbitals as in Hz For convenience we a and b so the atomic orbital wave functions are label the atoms of a diatomic molecule lsa and lsb We can visualize the two atoms moving closer to each other until the electron clouds overlap and merge into larger molecular electron clouds The result ing molecular orbitals are linear combinations of the atomic orbitals the sum of the two orbitals and the difference between them In general terms For H2 and Tlr dI and c and cb are adjustable co N is the normalizing factor so efficients In this case the two atomic orbitals are identical and the coefficients are nearly identical as well These orbitals are depicted in Figure 5 1 In this diagram as in all the orbital diagrams in this book such as Table 23 and Figure 26 the signs of orbital lobes are indicated by shading Light and dark lobes indicate opposite signs of The choice of positive and negative for specific atomic orbitals is arbitrary what is im portant is how they fit together to form molecular orbitals In the diagrams on the right side in the figure light and dark shading show opposite signs of the wave functionore precise calculations show that the coefficients of the a orbital are slightly larger than for the a orbital but this difference is usually ignored in the simple approach we use For identical atoms we will use ccb1 and NI d The difference in coefficients for the a and a orbitals also results in a larger a molecular orbitals than for the a orbitals decrease energy change increase from atomic to the Chapter 5 Molecular Orbitals 1 18 Because the u molecular orbital is the sum of the two atomic orbitals 1Isl s and results in an increased concentration of electrons between 6 the two nuclei where both atomic wave functions contribute it is a bonding molecular and has a lower energy than the starting atomic orbilals The u molecular orbital 1lsb It has a orbital is the difference of the two atomic orbitals ls di node with zero electron density between the nuclei caused by cancellation of the two wave functions and has a higher energy it is therefore called an antibonding orbital Electrons in bonding orbitals are concentrated between the nuclei and attract the nuclei and hold them together Antibonding orbitals have one or more nodes between the nu clei electrons in these orbitals cause a mutual repulsion between the atoms The differ ence in energy between an antibonding orbital and the initial atomic orbitals is slightly larger than the same difference between a bonding orbital and the initial atomic orbitals Nonbonding orbitals are also possible The energy of a nonbonding orbital is essen tially that of an atomic orbital either because the orbital on one atom has a symmetry that does not match any orbitals on the other atom or the energy of the molecular orbital matches that of the atomic orbital by coincidence The u sigma notation indicates orbitals that are symmetric to rotation about the line connecting the nuclei o from s orbital o from p orbital An asterisk is frequently used to indicate antibonding orbitals the orbitals of higherenergy Because the bonding nonbonding or antibonding nature of a molecular orbital is sometimes uncertain the asterisk notation will be used only in the simpler cases in which the bonding and antibonding characters are clear The pattern described for H2 is the usual model for combining two orbitals two atomic orbitals combine to form two molecular orbitals one bonding orbital with a lower energy and one antibonding orbital with a higher energy Regardless of the num i ber of orbitals the unvarying rule is that the number of resulting molecular orbitals is i the same as the initial number of atomic orbitals in the atoms
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