Chapter 2 – Molecular Orbital Theory Big-picture: Now that we understand aspects of molecular structure, we can look in more detail at bonding – how atoms are bonded together, energetics of bonding, where electrons reside in bonds, etc. These characteristics of a molecule allow us to explain and predict even more of its properties and applications. Learning goals: Be able to construct molecular orbital diagrams for homonuclear diatomic, heteronuclear diatomic, homonuclear triatomic, and heteronuclear triatomic molecules. Understand and be able to articulate how molecular orbitals form – conceptually, visually, graphically, and (semi)mathematically. Interrelate bond order, bond length, and bond strength for diatomic and triatomic molecules, including neutral and ionized forms. Use molecular orbital theory to predict molecular geometry for simple triatomic systems Rationalize molecular structure for several specific systems in terms of orbital overlap and bonding. Textbook Chapter 2

O2 Draw the Lewis dot structure for O2. Compare to the experimentally observed chemistry of O2. What does the Lewis dot structure get right, and what does it get wrong?

O2

Introduction to MO theory Valence bond theory does a reasonable job of explaining the bonding in some simple molecules, and therefore it is an appropriate first approximation. However, it fails to accurately describe important features of even simple molecules such as O2, and (as you can imagine) it does not do a good job with even more complex polyatomic molecules. Parallel to what we have experienced with the development of models that describe the structure of an atom, simple theories are a great starting point, but more sophisticated models are needed to more accurately account for all observations. Molecular orbital (MO) theory is a more sophisticated quantum mechanical model of bonding in molecules that can be applied successfully to both simple and complex molecules.

Introduction to MO theory Atomic orbitals are regions of space in which electrons have a high probability of residing – electrons are “spread out” over the orbitals that “surround” (or “comprise”) an atom. Molecular orbitals can be thought of as a natural extension of this concept: they are orbitals that are spread over all of the atoms in a molecule and therefore they describe where electrons reside within the molecule. We will consider MO theory fairly qualitatively, but as with atomic orbitals, it can be treated much more quantitatively. High-level calculations using MO theory can be quite accurate, doing a good job of predicting bond lengths, vibration frequencies, vibrational spectra, electronic spectra, NMR spectra, …

Brief review of atomic orbitals (AOs) De Broglie, 1924: Wave-particle duality describes electromagnetic radiation and matter (such as electrons, protons...) Schrodinger equation: Accounts for wave-particle duality, the motion of electrons in an atom, and quantized nature of atomic structure Total energy = KE + PE Eψ = KEψ + PEψ

Wavefunctions (ψ) and Atomic Orbitals The Schrodinger equation is a 2nd order differential equation that can be solved exactly for hydrogen, but only numerically for many-electron atoms and molecules. Only certain solutions produce physically acceptable results – quantization. Wavefunction: Solution to the Schrodinger equation, describes the behavior of an electron moving in x, y, and z directions – where it is and what it is doing Each wavefunction that is a solution to the Schrodinger equation is an atomic orbital Quantum numbers uniquely label each orbital

vs. Probability density The probability of finding an electron at a particular point in space is proportional to the square of the wavefunction at that point (x, y, z) vs.

Nodes Nodes: Regions where wavefunctions pass through zero 1s 2s 3s Radial nodes: Radial component of the wavefunction passes through zero Angular nodes: Angular component of the wavefunction passes through zero

There are always n-1 nodes, where n is the principal quantum number Counting nodes There are always n-1 nodes, where n is the principal quantum number The number of angular nodes is ℓ Which orbitals are these? How many radial and angular nodes does each one have? How many radial and angular nodes does a 5d orbital have?

Making MOs from AOs As with atomic orbitals, Schrodinger equations can be written for electrons in molecules. Approximate solutions to these equations can be constructed from linear combinations of atomic orbitals (LCAO), which are the sums and differences of the atomic wave functions. The coefficients can be equal or unequal (when?), positive or negative (when?), depending on the individual orbitals and their energies.

Making MOs from AOs As two atoms approach and the distance between them decreases, their orbitals overlap, and in the overlap region there is a significant probability for electrons to be found – why (qualitatively and quantitatively)? Molecular orbitals form from this overlap, and electrons in bonding molecular orbitals have a high probability of occupying the space between nuclei – the electrostatic forces between the electrons and the two positive nuclei hold the atoms together.

LCAO’s: Bonding for H 1s

LCAO’s: Antibonding for H 1s

What do these MOs look like?

How are their energies related?

MOs from p-orbital overlap How can the p-orbitals overlap, and what are their shapes and relative energies?

MOs from p-orbital overlap

What all does this predict? MO diagram for O2 What all does this predict?

Three conditions are essential for overlap to lead to bonding: Considerations Three conditions are essential for overlap to lead to bonding: Symmetries Energies Distance

Considerations What is the relationship between the number of atomic orbitals and the number of molecular orbitals?

Bond order, bond length, bond strength How are these correlated? Use O2 and its ions as an example.

Are there any orbitals you would predict to overlap in this way? Orbital mixing The energies of the molecular orbitals depend on the degree of overlap, which depends on symmetries and the energies of the constituent atomic orbitals. So far we considered only interactions of orbitals that have identical energies. However, orbitals with similar, but not equal, energies can also interact if they have appropriate symmetries. Are there any orbitals you would predict to overlap in this way?

Based on symmetry, which can interact?

Based on energy, which can interact?

2p 2s

MO diagram for N2

N2, O2, F2, Ne2?

Heteronuclear diatomic molecules How do heteronuclear MOs differ from homonuclear MOs? (qualitatively and quantitatively)

How do we predict relative orbital energies? This can be approximated by electronegativity, which tracks with relative orbital energies (bonding electrons held close to atom).

HCl What would you predict the MO diagram for HCl to look like? (Account for all orbitals)

Bonding continuum Progression from covalent to ionic bonding shown nicely with MO theory (not a hard, definitive cut-off)

HF What would you predict the MO diagram for HF to look like? (Account for all orbitals) How does this compare to the Lewis dot structure you would draw? What additional information does it provide?

Construct the MO diagram for FHF– Polyatomic molecules Recall: N atomic orbitals yields N molecular orbitals. We are not limited to diatomic molecules. Consider, for example, a simple triatomic. We can treat this much more quantitatively, integrating symmetry considerations, but we will approach this more intuitively and qualitatively. The FHF– ion is linear, with hydrogen in the middle. Construct the MO diagram for FHF– How do we begin?

FHF– As with HF, only H-1s and F-2s,2p are energetically similar enough to interact. Begin by putting H-1s (spherically symmetric) in the middle and looking at how it overlaps – symmetry, phase, energy – with all permutations of the F-2s and F-2p orbitals on each side.

FHF– How can H-1s interact with F-2s? How significant will this interaction be?

FHF– How can H-1s interact with F-2pz? How significant will this interaction be?

FHF– How can H-1s interact with F-2px,y? How significant will this interaction be?

FHF– molecular orbital diagram

Using MO theory to predict molecular geometry How can we predict the shape of a molecule using MO theory? Consider the H3 molecule. Pretend it is linear. What H-1s orbital configurations are possible?

H3+ molecular orbital diagram

What happens if we begin with H3 linear, and then bend it? H3: From linear to bent What happens if we begin with H3 linear, and then bend it?

What happens if we begin with H3 linear, and then bend it? H3: From linear to bent What happens if we begin with H3 linear, and then bend it?

H3: From linear to bent

H3: From linear to bent

Walsh diagram

What does this suggest about the molecular geometry of H3+ vs. H3–?

π-bonding: 2nd row vs. 3rd (4th, 5th, 6th) rows Ethylene: Stable molecule, doesn't polymerize without a catalyst. Silylene: Never isolated, spontaneously polymerizes. The large Ne core of Si atoms inhibits sideways overlap of 3p orbitals → weak π-bond N can make π-bonds, so N2 has a very strong triple bond and is a relatively inert diatomic gas “RTV” silicone polymer (4 single bonds to Si) vs. acetone (C=O double bond)

p-d orbital interactions Other orbitals that are similar in energy and symmetry can also interact

d-d orbital interactions