Exercise 1

Part of the course Computational Chemistry.

When starting with computational chemistry, it is important to gain some intuition for the sheer size of the problem and the scales involved. Luckily for us, in many cases the formal complexity can be greatly reduced by clever approximations - in fact these approximations are what most of computational chemistry is about. We will discuss the different assumptions and qualitative properties of the resulting models over the course of the semester.

Task 1.1: Dimensions

Consider the molecular Hamiltonian, which via the Schrödinger equation can be used to calculate the properties of a system.

• Which properties does the Hamiltonian depend on (ignore spin for a moment)?
• How many dimensions does the Hamiltonian have for a molecule of 10 atoms?
• How many degrees of freedom, i.e. independent dimensions, does the Hamiltonian have for any molecule of 10 atoms?

How do the answers change if you consider the wave function instead?

Task 1.2: Compute resources

Assume two computational methods (A and B) being able to describe molecules. A scales with $N^3$ and B scales with $\exp (N)$ where N is the number of electrons in the system. If a notebook can evaluate benzene for method A and the nitrogen dimer for method B, what is the limit for a compute cluster which has about 10.000 times the compute resources? Calculate the number of electrons and give examples of systems illustrating the limits. Either write a python function generalizing this problem or solve it once for this question.

Task 1.3: Assessing the Models

The energy of an atom can be estimated to be $-\frac{1}{2}{Z}^{7/3}$ (in atomic units). If a H-C bond is about 400 kJ/mol in energy and a C-C bond is about 350 kJ/mol, then which approximate percentage of the total energy of ethane is in the bonds, i.e. the part where all the chemistry happens? What does that say about the accuracy requirements?