Matrices

Another little bit of math related stuff for all you people who like to think about it. It's from here and it took me forever to find out what they were talking about.

To start with (and to use this nice LATEX tool) what do we have:

A time dependent vector:



And we define the expected value operator <> which means averaging over time.

The question is what is represented by the following two expressions.





No, in this case the T's cannot just be ignored as I would usually do.

---

The first one is a matrix like

is a matrix.

The covariance matrix, in other words:


The second expression is a number, the sum of all the variances or the trace of the covariance matrix (which stays invariant with a similarity transformation).


As far as implementation goes, I can only stress how nice numpy is. All I had to do was the parsing. Numpy quietly converts my 66x19553 matrix into its covariance matrix and diagonalises it.

Multi reference analysis

I have to summarize one more of those lectures from the conference. That in spite of the fact that readers aren't giving me very much incentive to do so ... Anyway, I'll talk about multireference analysis (MRA), again nothing very chemical but pretty nice and I want to review it.

The talk was from Janos Pipek. I don't find a reference but I'll write a short story about it.

Every starting chemistry student is shocked after hearing that in spite of having this nice variational principle everything is only based on AOs and those AOs are all based on Gaussians. The way out is wavelet MRA, for example with a Daubechie function:

It has many nice properties:

It becomes zero after a given distance.

Those functions next to each other are orthogonal.

It is easy to refine the basis if better results are needed for some areas. The large functions can be accessed as linear combinations of refined functions.

It seems that integrals can be easily evaluated.

That makes it a nice candidate for a basis function. They use it for jpeg compression and I am sure that they will soon use it in theoretical chemistry.