Molecular Orbital Theory

Model Our model of an atom is layers of atomic orbitals (AOs): 1s1s 2s2s 3s3s 2p2p 3p3p 3d3d As atoms approach each other their occupied AOs (waves) interfere Add constructively and destructively Result is molecular orbitals

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining s-orbitals =  and  * molecular orbitals: Same phase Opposite phase AO 1AO 2 +  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining s and p also =  and  * molecular orbitals: Same phase Opposite phase AO 1AO 2  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining p and p also =  and  * molecular orbitals: Same phase Opposite phase AO 1AO 2  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining p and p =  and  * molecular orbitals: Same phase Opposite phase AO 1 AO 2  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining d and d =  and  * molecular orbitals:  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining d and d =  and  * molecular orbitals: Same phase Opposite phase AO 1 AO 2  “bonding” MO  * “anti-bonding” MO

Let’s start simple Qualitative combinations of orbitals on different atoms, “add” phases: Combining d and d =  and  * molecular orbitals:  “bonding” MO  * “anti-bonding” MO

Summary of common combinations of AOs to form MOs: s and s =  ’s only p and p =  ’s and  ’s only d and d =  ’s,  ’s, and  ’s only Let’s start simple In general when building molecules, all AOs can combine with each other in all possible to varying degrees! Very quickly becomes a computationally demanding problems to predict molecular structures!

To “build” an MO in general: 1. Assign a set of AOs to each atom in a molecule: Computational MO Theory e.g. C can get 1s 2 2s 2 2p 2  a 1s a 2s and a set of 2p’s  {1s, 2s, 2p}  Minimal basis set for C Nothing is stopping us from giving C more orbitals though, say  {1s, 2s, 2p, 3d} The bigger the basis set for each atom, the more accurate predictions we get Calculation take a lot longer too!

Now You Try Qualitatively guestimate the MOs and Energetic ordering for CO (carbon monoxide). Assume a minimal basis set. C has 6e- and O has 8e-.

To “build” an MO in general: 2. Write each MO as: Computational MO Theory Linear combination of atomic orbitals (LCAO) approximation AO“weight” of AO This says: For each MO, add up the contributions from each AO There are as many MOs as AOs used to build them

Put a basis set on each atom: Computational MO Theory N H H H H H H H H H H H H H H H H H H H H H H H H H {1s} on H’s {1s,2s,2p} on C’s and N’s Augment with a set of p’s on H’s Augment with a set of d’s on C’s and N’s ~340 basis functions

So for this molecule with the chosen basis set: 340 AOs 340 MOs Computational MO Theory N 170 e- 85 of the MOs are occupied (up to 2 e-/MO) 255 of the MOs are “virtual” (empty) The highest occupied MO (HOMO) and the lowest unoccupied MO (LUMO) is where most of the chemistry happens Valence MOs

There are 340  i for this molecule in the chosen basis set Computational MO Theory 340 term in the sum too The C m,i are called MO coefficients C m,i can be arranged in a matrix C But how do we find C??

3. Inset the LCAO expansion into the Schrodinger equation: Computational MO Theory Left multiply by   * Rearrange Integrate Abbreviate

3. Inset the LCAO expansion into the Schrodinger equation: Computational MO Theory Abbreviate Schrodinger equation in matrix form HC = SCE Hamiltonian matrix MO Coefs. “Overlap” matrix MO energies Different approximation for elements of H and S result in different quantum chemistry methodologies

4. Solve for C and E by “diagonalization” Computational MO Theory HC = SCE HC = CE C T HC = E Orthonormalize the AO basis: Left multiply by C -1 = C T E is a diagonal matrix of MO energies Eigenvalues of H C is said to diagonalize H Eigenvectors of H

5. Hardest thing for us to do is put together H Computational MO Theory 11 22 nn 11 22 nn … … … H =

6. Huckel approximation Goals are to: 1.Cut down on the number if H  and S  to evaluate 2.Easily evaluate the H  and S  that remain Computational MO Theory Assume: 1.Only p z AOs are important. Neglect everything else 2.There is only 1 valence electron/atom 6 valence e-

6. Huckel approximation The only integrals we have to evaluate are for nearest neighbours: Computational MO Theory Nearest neighbour integrals

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = 11 22 33 44 55 66 11 22 33 44 55 66 H =                 

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = 11 22 33 44 55 66 11 22 33 44 55 66 S =             0  0  

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = HC = SCE HC = CE C T HC = E Build H in the computer and send it to a diagonalization routine

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #1

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #2

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #3

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #4

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #5

6. Huckel approximation Computational MO Theory This model works great for conjugated carbon molecules We’ll set  = 0 and  = H = [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C = E = MO #6

Now You Try Write out the Hamiltonian for: From C sketch the MOs. Without looking at E, write an MO energy diagram. Fill it with (Huckel) valence electrons. What would happen to bond lengths for this molecule in the first excited state? Why? [,1] [,2] [,3] [,4] [,5] [,6] [1,] [2,] [3,] [4,] [5,] [6,] C =