# What is the relationship between quantum mechanics and computational chemistry?

Sep 28, 2016

Quantum mechanics is extremely central to computational chemistry. Many methods in computational chemistry require that we understand the math behind such quantum mechanical systems as atoms and their electrons, and so on.

The big hurdle to understanding quantum mechanics IS the math.

Here are two examples.

PHYSICAL REALISM

One major dilemma is that the wave function which describes the system must be physically realistic. That means we have to define our variables such that the math represents something real. That means we need boundary conditions.

Some examples of those:

• It's comforting to be able to observe something without changing its configuration merely by the action of observing it. This thing that we can observe multiple times and see the same thing over and over again is an eigenvalue. The Schrodinger equation is an "eigenvalue problem".

To have an eigenvalue, you must have a corresponding eigenfunction (you operate on it as much as you want, and it'll spit back that function), and ${e}^{u}$ satisfies that requirement very well.

• Anything we can represent using periodic functions, like $\sin \theta$ and $\cos \theta$, must satisfy the relationship of cyclic equality.

So let's say, $\sin \theta = \sin \left(\theta + 2 \pi\right)$, or $\Phi \left(\phi\right) = \Phi \left(\phi + 2 \pi\right)$, etc. Some might call that situation a "particle on a ring".

• The simplest system, the particle in a box, has bounds at $0$ and $L$, i.e. the length of the box. We say that an electron vanishes at the bound of our system. Our "box" could be, say, a conjugated $\pi$ system like benzene or 1,3,5-hexatriene.

And the reason why we need to bother with all this is that we know it, but does the computer know it? We have to define it so the computer can do what we want it to do.

ELECTRON-ELECTRON REPULSION

Another of the most troublesome issues is electron-electron repulsion.

In three-dimensional space, for a mere two-electron helium atom, the Hamiltonian operator which corresponds to the energy at each energy level, is:

$\hat{H} = {\sum}_{i = 1}^{2} \left({\hat{K}}_{e , i}\right) + {\hat{K}}_{n} + {\sum}_{i = 1}^{2} \left({\hat{V}}_{n e , i}\right) + {\hat{V}}_{e e}$

where $e$ means electron, $n$ means nucleus, $n e , i$ means interaction between the nucleus and electron $i$, and $e e$ means electron-electron interaction. $K$ and $V$ are the kinetic and coulombic repulsion energies, respectively.

Since the nucleus is massive compared to the electron, its kinetic energy, by the Born-Oppenheimer Approximation, can be neglected with little error to the calculation of the ground-state energy of an atom.

So, ${\hat{K}}_{n}$ goes away.

$\hat{H} = {\sum}_{i = 1}^{2} \left({\hat{K}}_{e , i}\right) + {\sum}_{i = 1}^{2} \left({\hat{V}}_{n e , i}\right) + {\hat{V}}_{e e}$

And with the fuller version of the Hamiltonian it looks more like this:

hatH = -ħ^2/(2m_e) grad_1^2 - ħ^2/(2m_e) grad_2^2 - e/(4piepsilon_0vecr_1) - e/(4piepsilon_0vecr_2) + e^2/(4piepsilon_0|vecr_1 - vecr_2|)

where ${\nabla}^{2}$ is defined as the Laplacian of the system, which is the second-order derivative for each coordinate in the system. For example, in Cartesian coordinates, it is defined as:

${\nabla}^{2} = \frac{{\partial}^{2}}{\partial {x}^{2}} + \frac{{\partial}^{2}}{\partial {y}^{2}} + \frac{{\partial}^{2}}{\partial {z}^{2}}$

The electron-electron repulsion is usually what is most difficult to model in a situation like this, because the electrons are moving in three-dimensional space, so they have the freedom to instantaneously repel each other at any given moment, pseudo-randomly.

So modeling this effect in a computer, which functions in a fairly linear fashion, is extremely difficult. Basically, computers are fairly predictable. Electrons, not so much.

The point is, it gets super complicated, and you have to know your quantum mechanics very well to model the quantum mechanical world in computational chemistry.