Introduction

As I already revealed in the last post, I intend to have several projects with Density Functional Theory on this blog. I already have a simple project on GitHub, about a ‘quantum dot’1 with volumetric visualization of orbitals with VTK.

I thought that exposing some theory in a separate post would be nice for further references, so without further ado, here it is.

Links On this blog

T…

Density Matrix Renormalization Group

Introduction

As promised in the Numerical Renormalization Group post, I implemented a Density Matrix Renormalization Group program. As I start writing this post, the program is already on GitHub1. It’s quite basic, currently it is implemented only for Heisenberg modelchains, for both spin 1/2 and 1. It runs only for even number of sites and symmetries are not used to speed up computation. Also it…

Introduction

After some posts about the theory it is time to present the Hartree-Fock program1. You might find the previous posts useful, along with the links in there: How to solve a quantum many-body problem, The Hartree-Fock method.

Here is the program in action (the final version might have minor differences):

Although it…

Introduction

Last time we ended up with a simplified Hamiltonian:

and having the variational principle for help. It turns out that this simplified Hamiltonian is not simple enough. In this post I’ll expose one method that allows one to do calculations for quite complex quantum systems, the Hartree-Fock method. Because the subject is quite large, obviously I cannot cover it into detail in a blog…

Variational principle with two by two symmetric matrix

[Click here for a PDF of this post with nicer formatting]

I pulled [1], one of too many lonely Dover books, off my shelf and started reading the review chapter. It posed the following question, which I thought had an interesting subquestion.

Variational principle with two by two matrix.

Consider a \( 2 \times 2 \) real symmetric matrix operator \(\BO \), with an arbitrary normalized trial vector