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Molecular Orbital Theory Unlike hybrid orbitals that are constructed from the orbital set of a single atom, molecular orbitals are Linear Combinations of Atomic Orbitals that are distributed over the entire molecule. Molecular orbitals have associated with them a set of energy levels that follow the Aufbau Principle and also satisfy the Pauli Exclusion Principle. Remember that for the construction of hybrid orbitals, the number of hybrid orbitals equals the number of atomic orbitals from which they are constructed. The same is true for MO’s. Number of molecular orbitals = number of atomic orbitals from which they are constructed

MO’s based on 1s orbitals We start by taking a linear combination of 1s atomic orbitals on two distinct hydrogen atoms. Remember that the increased electron density between bonding atoms results in a lowering of energy. We also use the language of waves to think of this increase in electron density as complete constructive interference. We call the bonding molecular orbital the σ1s MO. Following the Pauli Exclusion Principle, we place two electrons in the MO with electron spins paired, as shown on the left. Mathematically, we would say: ΨMO, bonding ∝ ψ(1sA) + ψ(1sB)

Since we used two 1s wavefunctions, one on each atom, we must also have a second molecular orbital. The second MO is equivalent to complete destructive interference of the matter wave amplitude between the atoms. The energy of this antibonding MO lies above that of the separated atoms and is denoted σ*1s. Matematically, we write ΨMO, antibonding ∝ ψ(1sA) - ψ(1sB)

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We can visualize the electron density distributions for the bonding and antibonding MO’s from the following diagrams:

Note that the antibonding MO changes sign between the lobes, so there is a nodal plane perpendicular to the bond axis. Remember that the orbitals are probability amplitudes, and can be positive or negative. The square of the amplitude is always positive, and represents the true probability of finding a particle in a region of space. We can place the number of electrons from the constituent atoms in the orbitals to understand electron configurations in molecules. For H2, we write the electron figuration as (σ1s)2. If we remove a single electron, we make the H2+ molecular ion, and its electron configuration is (σ1s)1. The H2+ molecular ion is well known in mass spectrometry. Bond energy for H2 = 432 kJ/mol Bond energy for H2+ = 255 kJ/mol We can construct the electron configurations for two helium atoms in like manner, using the fact that each He atom contributes two electrons to the bonding scheme. Note that the third and fourth electrons go into the antibonding orbital.

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We define the bond order (BO) as ½(number of bonding electrons – number of antibonding electrons) According to this definition, the BO for H2 is 1, that for H2+ is 0.5, and for He2, the BO is zero. The bond energies for H2 and H2+ indicated above are in semiquantitative agreement with the concept of BO. A bond order of zero means that there is no covalent bonding (which involves the sharing of electrons), but only the weak van der Waals force. The binding energy of He2, which comes entirely from vdW forces, is about 0.1 kJ/mol. We can also add and remove electrons to examine electron configurations of various molecular ions. If we remove a single antibonding electron to make He2+, the BO is calculated to be 0.5, and the bond energy of this species has been measured to be ~230 kJ/mol. So, its bond strength is about half that of H2. In contrast, H2-, which also has a BO of 0.5 is known to be unstable. Negative ions in general are “peculiar” entities, and it is difficult to predict a priori whether a given species is stable or not. MO’s of molecules comprised of second row atoms The first species we encounter in the second row is Li2. When we look at the electron distributions in the atomic orbitals of Li, we see that there is negligible overlap of the electrons in the 1s orbitals. MO’s are formed from the valence electrons. There is significant overlap of the electrons in the 2s orbitals. We can construct bonding and antibonding MO’s form the 2s orbital set on each Li atom, and the MO energies look qualitatively the same as those made from 1s orbitals. The electron configuration for Li2 is (σ2s)2.

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We can also make the molecular ions Li2+ and Li2-. Both of these species exist (unlike the corresponding cases with hydrogen). For your cultural enrichment, the bond strength in Li2 is ~101 kJ/mol, while in Li2+ (with one fewer bonding electron), the bond energy is 139 kJ/mol. Devotees of simple MO theory hate this example. There is no simple explanation. This won’t be on the test.

MO’s involving 2p-orbitals When the 2p orbitals on each atom interact, two distinct kinds of interactions can occur. The first homonuclear diatomic molecule in which the 2p orbitals can interact is constructed from two boron atoms. Figure (a) below shows the orbitals orientations that can arise from the interaction of two boron atoms.

Panel (d) in the above figure shows that only one of the 2p orbitals on each atom lies along the bond axis. Those two orbitals overlap to give an electron density distribution that is cylindrically symmetric about the bond axis (See the figure). The remaining four orbitals, two per atom, generate MO’s that are fundamentally different in shape. The axes defining the directions (orientations) of the p orbitals are parallel to one another when electron overlap occurs. The electron density distribution is no longer cylindrically symmetric. Panels (b) and (c) show how the orbitals interact to form π-type MO’s. Note also that the two π-type interactions differ only by a rotation of 90° about the bond axis, i.e., they are degenerate.

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When we construct MO’s we always have a symmetric and an antisymmetric combination of orbitals, corresponding to constructive and destructive interference respectively, of amplitude between the bonding atoms. The figure below shows how these combinations occur. We have one σ2p (bonding) and one σ*2p (antibonding) MO. Note that the electron density is enhanced along the bond axis for the σ2p (bonding) orbital, and there is a node for the σ*2p (antibonding) orbital. The upper two orbitals in the figure below show those orbitals. We can also construct bonding and antibonding MO’s from the π-type orbital set. These MO’s are not cylindrically symmetric about the bond axis. The π2p MO’s are bonding,

with increased electron density (remember that the electron density is proportional to the square of the wavefunction) between the atoms. In contrast, the π*2p antibonding orbitals have a nodal plane between the atoms. Energy Ordering the MO’s Of course, the bonding MO’s are lower in energy than the energies of the isolated atoms. But, we have no a priori way to determine whether the σ or π orbitals are lower. The next two diagrams illustrate the possibilities.

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One possibility: The figure shows that if the σ MO is the lowest in energy, then the two 2p electrons from the boron atoms should have paired electron spins in B2. This would make B2 diamagnetic, i.e., weakly repelled by an external magnetic field. In fact, B2 is strongly attracted by an external magnetic field.

This indicates that B2 has unpaired electrons, and the correct picture of the orbitals is that shown in the lower figure. Hund’s Rule states that we must place electrons in distinct, degenerate orbitals (the two π2p orbitals) with the same spin.

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For the first three elements in the 2p series (B, C, N), the π2p orbitals lie below the σ2p orbitals. For the atoms in the last half of the series (O, F, Ne), the order is reversed. The figure below shows the electron configurations for these species. One important consequence of this ordering is the fact that molecular oxygen, O2,is paramagnetic, a fact that has important consequences in many chemical applications, especially in photochemistry and atmospheric chemistry.

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Heteronuclear diatomics There are many examples of such species. As a simple example, consider HF. The MO diagram is shown below. Note that the positions of the atomic orbitals are asymmetric, with the ionization potential of the atom determining its position:

Ionization limit

Hydrogen ionization energy = 1312 kJ/mol

Fluorine ionization energy = 1681 kJ/mol

The larger ionization energy of fluorine makes its valence orbital lie below that of the 1s orbital in hydrogen.

Comment on the NO molecule. Although NO has one atom with a p2 configuration and one with a p4 configuration, the correct MO diagram for NO has the σ orbitals below the π orbitals, just like in O2.