07. Computational Chemistry

Hee, hee...hei, hei...

Heisenberg is out for a drive when he's stopped by a traffic cop. The cop says:

"Do you know how fast you were going?

Heisenberg replies: "No, but I know where I am".

Goals

• • Use computational chemistry to confirm VSEPR predictions of molecular geometry

  • Compute the molecular orbitals of small molecules to understand the difference between pi/sigma, and bonding/antibonding molecular orbitals.

Background

Computational Chemistry

Computational chemistry is a subfield of chemistry that uses computers to predict properties of molecules without doing experiments in a lab. The major type of computational chemistry that we will be exploring here involves using ab initio calculations to predict molecular geometries, energies, and molecular orbitals of small molecules. The term ab initio means that absolutely no data from experiment enters into these calculations; instead, a computer program makes a lot of approximations to try and solve the Schrödinger Equation, which is the equation that governs the energies and probability densities of electron clouds in atoms and molecules.

Ab initio calculations are really important, and ubiquitous in all subfields of chemistry:

• Organic chemists can calculate the molecular orbitals of a molecule they are studying to figure out which part of a large molecule will react

• When designing a new drug, organic chemists and biochemists can see if the drug will bind to its biological target using computational chemistry

• Chemists who work on solar cells and OLED technology can get an idea for how a new material might absorb or emit light using computation

All of these use cases have the benefit of getting an idea of whether or not a new molecule or a new material even has a possible chance at working towards its intended application, before wasting lots of time and money going into a lab to make it!

Computational Methods and Basis Sets

It has been proven that it is impossible to come up with an exact solution to the Schrödinger Equation for anything other than an atom with one electron--i.e., a hydrogen atom. Computers can get close, but it takes an extremely long amount of time. To make this concrete: getting near-perfect molecular orbitals and energies for the aspirin molecule, which contains 21 atoms and as far as drugs go is rather small, will take the world's fastest supercomputer about 500 years. The time it takes to run a calculation is called "computational cost". Molecules with more atoms take exponentially longer than that to run--or, in other words, have exponentially higher cost.

What this means in practice is we have to come up with approximations in order to practically run calculations on the larger molecules that are of interest to us in terms of their applications.

There are two main ways we make out calculations more approximate to make them run-able in a shorter amount of time:

(1) Choice of Approximate Method

Many approximate methods to solve the Schrödinger Equation have been developed by computational chemists over the years. Their details are way beyond the scope of this course, but it's important to categorize them into a few buckets. As you go down the list, the methods get most costly (i.e., slower), but more accurate:

• Semiempirical methods. These methods use data from experiments to simplify a lot of the calculations and make them go really, really fast. However, because experimental data is only input for a handful of molecules, these methods are usually very inaccurate for newer molecules. These days, semiempirical methods are rarely used, because better methods exist.

• Hartree-Fock (HF) methods. Hartree-Fock is the earliest and simplest approximation that most of the other approximations on this list built on top of. These days, pure Hartree-Fock is rarely used, because better methods exist.

• Density Functional Theory (DFT) methods. These methods take the whole molecule's electron density and use it to figure out what orbitals need to come together to make up that electron density. This is the most important and widely-used class of methods in chemistry, because it has a perfect tradeoff of speed versus accuracy.

  • • To use a DFT method, one must specify a density functional, which is a set of instructions for the computer regarding how to treat the electrons in a molecule. Commonly-used density functionals include: B3LYP, PBE0, and wB97X. (n.b. density functionals are usually named after the people who invented them; for example, B3LYP was invented scientists named Becke, Lee, Yang, and Parr!)

• Post Hartree-Fock methods. These methods tend to be more accurate than DFT methods, but substantially slower. Examples include: MP2 and quantum Monte Carlo.

For this lab, we will be using the B3LYP functional of DFT. This method is quite widely used in chemistry, because it offers an excellent trade-off between accuracy and cost!

(2) Choice of Basis Set

The term basis set means "the set of atomic orbitals that go into calculating the molecular orbitals". For example, the valence MO diagrams that you have been constructing in class use a simple basis set consisting of just the atomic 1s, 2s, and 2p orbitals of each atom.

In principle, though, atoms have an infinite amount of orbitals! Remember the hydrogen atom: the principle quantum number n can be 1, 2, 3, ...or any integer up to infinity! This is not practical for a computer. Therefore, we have to cut off some of the really high-n orbitals. This should be OK, because as you go higher up in n, the orbitals stop having electrons in them, so they become less important to the overall calculation.

Generally speaking, bigger basis sets mean more accurate, but more costly (i.e., slower), calculations. From smallest to biggest, some common basis sets include:

• STO-3G (this is a minimal basis set that only includes filled orbitals. You have been using something similar to this in lecture to construct your MO diagrams!)

• 3-21G

• 6-31* (this is considered the minimum acceptable basis set for research-quality results)

• 6-311++G**

• aug-ccpvtz

In this lab, we will be using the 6-31G* basis set, because again it provides a good compromise between accuracy and speed!

IMPORTANT: When running calculations on multiple molecules, you MUST choose the SAME approximate method and basis set for ALL molecules that you are studying!! If you do not do this, you will end up with results that are impossible to interpret.

A summary of approximate methods and basis sets is given in the diagram below:

Geometry Optimization

Whenever you run an ab initio calculation, you need to provide three things as an input to the calculation:

• An approximate 3D structure for the molecule

• The molecule's overall charge and spin multiplicity (n.b. "spin multiplicity" = (# of unpaired electrons) + 1)

• A set of instructions for the computer, including the chosen approximate method and basis set

With just these things, the computer will calculate the energy and orbitals corresponding to whatever molecule you give it!

However, this is usually NOT what we want to do. As hinted at above, whenever you use a Lewis structure to guess at the 3D structure of a molecule based on VSEPR theory, this is only an approximation to the true structure. The first step in ANY computational chemistry calculation is to run a "geometry optimization", to find the ACTUAL 3D structure of a molecule. To do this, we need to tell the computer one more instruction:

• Do a geometry optimization

With this option selected, the computer will optimize all the bond lengths and bond angles to get the lowest-energy 3D structure! The bond lengths and bond angles after a geometry optimization are usually extremely accurate, and often match up quite well versus experiment!

Frontier Molecular Orbitals

When computing the molecular orbitals of a molecule, there are two orbitals that are quite important. They are:

• Highest Occupied Molecular Orbital (HOMO) = the highest-energy orbital that has electrons in it

• Lowest Unoccupied Molecular Orbital (LUMO) = the lowest-energy orbital that does not have electrons in it

The HOMO and LUMO are always right next to each other in energy. These orbitals are called 'frontier molecular orbitals' and determine quite a bit about the reactivity of a molecule! In general, chemical reactions involve electrons moving from the HOMO of one reactant to the LUMO of the other reactant!

Experiment:
The experimental protocol is outlined on the 'Results' sheet for this lab. You do not need to complete a lab notebook for this lab activity. By its very nature, computational chemistry is 100% replicable: if you put the same inputs into the computer, you will get the same outputs, 100% of the time!