Adiabatic theorem and Born Oppenheimer Approximation

I try to understand the BO approximation here.

If we walk in mountain, some times, quit a bunch of mosquitoes like bees would fly around your head. You keep on hiking but the bees follow you by swirling around your head (and making noise too...really annoying).

Now, in molecule or materials, the nucleus can be given analogy of your head. The electrons can be given an analogy of bees around the head of a hiker. Here, the movement of the hiker's head (nucleus) does not affect the movement of bees (electron movement). This analogy may be useful for a kindergartner. But, for me, it is not enough. How to understand this really? Let us break this big problem in to pieces.

Another usual textbook discussion is the following. The mass of proton is 1837 times heavier than electron. Thus, nucleus is much much heavier than electron. Because of this, we can consider the nucleus is fixed in space and electrons are the only moving particles. This also doesn't makes sense to me. Just the higher mass consideration is not a satisfying argument to me.

So, I need a more intuitive as well as rigorous argument.

Let us start with Adiabatic theorem.

If a quantum system is changed gradually, the final state has the same state as the original state. Griffiths Quantum Mechanics beautifully explains this quantum adiabatic theorem. What if we change the system drastically?

Now, come to Born-Oppenheimer approximation. In BOA, we make use of this adiabatic theorem. Right? Where it is used? How it is used to prove it in original paper? Is there a simpler version (text book version) to understand the proof now?

Inserting symbols in MSWorld/PowerPoint

Many times, we may try to add an Angstrom (Å and also other symbols) symbol or degree celciums etc. But, we may need to search a lot in the insert Symbol option provided by MS Office (World and PowerPoint)

Here is the way how you can simply located those symbols.

Font: Select Normal Text
In the subset select Latin: Subset
You can find Angstrom symbol.

A straight forward way is copy the symbols here and paste in your document.

Character code is "00C5"

Hohenberg-Kohn theorems

I assume you know nothing about the Hohenberg-Kohn theorems. Here is the link for the paper that form the basis for the Kohn-Sham Density functional theory. Now, you download the paper and find skimming through the paper. You may not understand anything. Cool! It is not just you.

Here is a journey to understand Hohenberg-Kohn theorems.

The title is "Inhomogeneous electron gas". What we understand from this? We know electrons. What is electron gas? Collection of large number of electrons which behave like free electrons. Right?

Gas is defined as follow:
"a substance or matter in a state in which it will expand freely to fill the whole of a container, having no fixed shape (unlike a solid) and no fixed volume (unlike a liquid)."

Now, we may consider such gas which consist of electrons. However, materials have have nucleus which are positively charged and attract electrons and forms some weired kind of patterns/orbits/etc in the real materials. So, the electrons are not homogeneous inside a material and they must be in-homogeneous. Thus the title give some intuition on the subject of the paper.

Let us jump on the the formalism section.

First the authors start with the definition of the Hamiltonian and how it is actually written in different contributions added together.

H = T + U + V

Here, T, U and V stands for kinetic energy, electron-electron repulsion energy, and V is the electron-nucleus attractive potential energy.

List of Computational Physics and Computational Chemistry or Density Functional theory (DFT) Courses

Here is the list of Computational Physics and Computational Chemistry Courses that would be useful for learning and doing research in condensed matter physics or chemistry or biology.

LABORATORY OF COMPUTATIONAL PHYSICS (link here)
academic year 2015/16
Teacher: Maria Peressi

PY 502, Computational Physics, Fall 2018 (link here)
Department of Physics, Boston University
Instructor: Anders Sandvik

554027 Modern Fortran Programming for Chemists and Physicists (link here)
Trainer: Pekka Manninen

PhD Course and Lecture Series in Density Functional Theory  (link here)
Lecturer: Rickard Armiento, Linkoping University,

Materials simulations: Introduction to Density Functional Theory (2019) (link here)

Quantum Chemistry by Prof. Stephan P. A. Sauer (YouTube Videos Playlist)
Department of Chemistry, University of Copenhagen.

Summer Lecture Series in Theoretical Chemistry 2012: Density Functional Theory (YouTube link)
Prof. Sherrill, Georgia Institute of Technology.

Scientific Programming and Numerical Computation
from Wu-ting Tsai, National Taiwan University.

Working with OpenMP

Here I will discuss the OpenMP and its usage in high performance computing using different languages.

Here are some links to learn and apply OpenMP.

OpenMP on Ubuntu (from medium.com)

OpenMP is the language or API that is used to communicate with different processors (or cores) in the same node (note that a node is a collection of number of processors). Any high performance computing use OpenMP because to use more than a processor for a single job, you need to use OpenMP.

Save this program in a file helloSerial.c and run.


{example program with serial running which include the time taken to complete}


Now, check this program helloParellel.c

{simple parellel program which include the time take to complete}

The parallel program is faster. Why?


Some frequently asked questions.

What is a processor?
What is a node?
What is a thread?
What is a core?
What is dual core in laptops?

Python blogs follow

https://scientific-python-101.readthedocs.io/numpy/math.html

How to remove a coloumn from multiple coloumn in Linux

I have a list of coordinates (x,y,z). In addition to that, there are additional columns which are necessary.  How to remove this?

AWK can be used in a simple manner.

USE:

awk '{print $2, $3, $4}' inFile.txt > coordinates.xyz

This will create the file coordinates.xyz with the necessary coordinates.

White background in Jmol, Iqmol and Avogadro

For publication, you may need white background structures. Here is how you can do it.

Jmol

Right click on the screen
Select Color
Select Background
Select the color (white)
Now you will get the white background

How to read all the questions and answers without registering (without username and password)

When you search for a question in google, you come up with a first link from Quora. You click it and you read the answer. Suddenly, you find another question from quora (on the side bar) and you want to read it too. But, the problem is that the site would ask you to register to read more.

To avoid registration, you can do following.

1. Search in normal mode
2. Open the first link normally
3. Now, if you want to read more of the Quora questions, right-click on the link and open in incognito mode. In this way, you can read all the questions.

It won't work if you do everything in incognito-mode.

How to highlight the novelty of your research in a research paper to publish in a journal?

In any peer-reviewed journal, the first question asked would be "What is the novelty of your work". It is important to ask this question to yourself and answer in your research introduction.

A quote from an answer given by Baba in Quora: "Virtually 99.999+% of the research work is incremental. The new elements they bring is either in the method, connecting the dots in a new way that was not previously thought of before, or observation of an effect that was not seen before. This does not dilute the achievement of the authors of the work. However, taking already existing building blocks and making a new figurine out of the said blocks can be argued that in itself is not novel. It is a highly skilled art form, and not easy."

Here are the ways to find out the novelty of your work and also some tips to improve your paper writing process.

1. Do a through literature survey and find out what are the key differences between your work and and research already done.
2. Systematically analyze the already published papers, reference books, patents, conference papers, etc and find out what should be done.
3. Establish that the results you got has never reported before or the method you have used is new in case what your work is improving a methodology.
4. Write the paper when you start the work. Use the writing process as a tool to generate ideas and discover ideas and fuel to your thought process.
5. Use tools like Mendeley or any other paid to write the paper for reference management.
6. Use LaTeX to format your paper. In mathematics, if you submit paper written using word processors, it may be considered unprofessional. I remember some incident related to this and I will write about this later (probably this is related to solving a famous mathematical problem). When the number of equations, figures, references get more and more, you would end up with difficulty if you use word processors. So, you need LaTex.
7. In the results and discussion, discuss the results obtained by other authors and discuss how your result is new or adds more to the field. You also generate new insights based your results and already reported results.
8. In the conclusion highlight the results obtained and insights advances in your field or study.