talk.txt

% INTRODUCTION TO IDEA CODE

# SUMMARY: Many Body Problem
* "Ideal goal": the interacting many body Hamilitonian,  
$$ \mathcal{H} = - \frac{1}{2}\sum_{i}^{N_{e}}\nabla_{i}^{2} - \sum_{i}^{N_{e}} \sum_{I}^{N_{n}} \frac{Z_{I}} {\abs{\vec{r}_{i} - \vec{R}_{I} }}   +\frac{1}{2}  \sum_{i}^{N_{e}} \sum_{j \neq i}^{N_{e}} \frac{1} {\abs{\vec{r}_{i} - \vec{r}_{j} }} $$ 
$$  + \frac{1}{2}  \sum_{i}^{N_{n}} \sum_{J \neq   I}^{N_{n}} \frac{Z_{I}Z_{J}} {\abs{\vec{R}_{I} - \vec{R}_{J} }} -  \sum_{I} \frac{1}{2 M_{I}}\nabla^{2}_{I}.$$
$$ \mathcal{H}\Psi = \mathcal{E}\Psi$$
* Unreachable for all but the **SIMPLEST** of systems unfortunately, 
* We are forced to introduce various approximations, 
* BO, 
* HF ( & Slater Determinants $\Psi=\Phi$)  (works for single electron system..), however: (no correlation, exchange only for spin, ...), 

[comment2]: <>('DFT')
# Side-Stepping the problem: Reduced Quantities
* Using Density $n(r)=\sum_i \delta(r-r_i)$
* Scales well, generalizable to different levels of theory:
  - DFT
  - Density Matrix ($\gamma(r,r')$)
  - Greens functions  ($G(r,r',t,t')$)
* DFT:
  - One to one correspondence between the external potential and the ground state wavefunction 
   $$V_{ext} \Leftrightarrow  |\Psi_0> $$
  - Any observable is a FUNCTIONAL of the density
* DFT does not however tell us how to get the ground state density, 
  - We have to introduce KS-DFT, 

[comment3]: <>('KS-DFT')
# KS-DFT
* Kohn-Sham DFT uses an auxilliary system
* We set out to find the density of a real *interacting* system using the ground state of a *non-interacting* system,
 $$ n_0(r) = n_{os}(r) $$
* Some more ingredients were considered
   - Correlation
   - Exchange,
* this (KS-DFT) is a key approach (though an approximation) for understanding physics of condensed matter. 

[comment4]: <> ( 'Test your knowledge' )
# Testing Our Understanding
* The core idea is that true understanding of a concept, especially in physics, is demonstrated by the ability to apply it mathematically and make predictions.
* It's not useful to state a law like newtons law without using it to make a prediction we can test, 
* So comes the next activity, testing our understanding, of
  - Exact many body case
  - Approximate cases 
* We stated this at the begining: 
  - **Unreachable** for all but the **SIMPLEST** of systems **unfortunately**
  - this means we can then test our undestanding on these *simple* systems, this is good for us, 

[comment5]: <> ('Testing knowledge II')
# Simple  
* How simple is simple? 
  - 1/2/3 electron systems, these can be solved exactly, and approximately, and we can then compare!
  $$ \left[ - \frac{\nabla_r^2}{2} + V_{ext}(r_i) \right] \Psi(r) = E \Psi(r)  $$
  $$ \sum_{i=1}^{2} \left[ - \frac{\nabla_i^2}{2} + V_{ext}(r_i) \right] \Psi(r) = E \Psi(r)  $$
* The physics we can learn remains the same, 
  - the density/occupation
  - the potential that corresponds to the density
  - ...
* So we can now use a *numerical* tool to do this, which implements both types of scenarios we have in mind,
  - Exact case
  - Approximate case,
* For the approximate case, we need to have access to multiple approximations, 
  - For discussion and exercises, we can limit to those we have met in the past few days
  - these are HF, KS-DFT.

[comment6]: <> ('IDEA CODE')
# What is iDEA code?
* iDEA = interacting Dynamic Electrons Approach,

    ![iDEA](./img/idea-logo.png){width=30%} 

* Python code that offers both exact and approximate approaches to quantum mechanics. 
* Among the goals of iDEA is to help users understand when popular approximations used in practical quantum theory calculations may be unreliable and why, (*Exactly what we need*)

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[comment7]: <> ('IDEA CODE II')
# Installation and setup

    ![iDEA](./img/idea-logo.png){width=30%} 
* Where can you get iDEA code?

```{ .bash }
pip install iDEA-latest
git clone  https://github.com/iDEA-org/iDEA.git
cd iDEA && pip install -e .
```

[comment8]: <> ('iDEA code III')
# What can we Compute with iDEA?
* Exact solution of the many-electron problem by solving the static and time-dependent Schrödinger equation, including exact exchange and correlation.
* Exact solutions which approach the degree of exchange and correlation in realistic systems.
* Free choice of external potential that may be time-dependent, on an arbitrarily dense spatial grid, for any number of electrons with any spin configuration.
* ...
* various approximate methods
 - Non-interacting electrons
 - Hartree theory
 - Hartree-Fock
 - LDA
 - Hybrids
... 

[comment9]: <> ('iDEA code IV')
# What can we Compute with iDEA?
* Observables:
  - density
  - density matrix
  - exchange energy
  - exchange potential $V_{xc}$
  - external energy
  - external potential $V_{ext}$
  - hartree energy
  - hartree potential $V_{H}$
  - kinetic energy (T)
  - single particle energy


[comment10]: <> ('iDEA code V')
# DEMO
* iDEA has a built in system we can start with, this is the two electron atom
* *Important*: we are in 1D, along the x-axis, (for 1-3 electrons) this simplicity gives us the possibility to understand the physics exactly if needed
* **Hello World** with iDEA, 
```{ .python .numberLines startFrom="1"}
import iDEA
system = iDEA.system.systems.atom
ground_state = iDEA.methods.interacting.solve(system, k=0)
n = iDEA.observables.density(system, state=ground_state)
E = ground_state.energy

import matplotlib.pyplot as plt
print(E)
plt.plot(system.x, n, 'k-')
plt.show()
```

[comment11]: <> ('iDEA code VI')
# Exercise
* Reproducing a paper: 
  - Advantageous nearsightedness of many-body perturbation theory contrasted with Kohn-Sham density functional theory*
    https://arxiv.org/pdf/1812.02661
* Study Fig 1. in  the paper, and for a asymetric quantum double well potential, the exercise is to work out the solution and answer the physics of  the results viz;
  - What is the difference between the exact and non interacting density n?
  - What is the difference between the external potential vs the $V_{s}$?
  - Work out the exact $V_{H}$ and the exact $V_{XC}$ and intepret the gap shift. 
* It will be illustrative for you to consider the exercise of 7th july while exploring this problem.

[comment12]: <> ('iDEA code V')
# Exercise
$$\left[-\frac{\nabla^{2}}{2}+V_{H}(\mathbf{r})+V_{\text {ext }}(\mathbf{r})\right] \varphi_{i}(\mathbf{r})+V_{x c}(\mathbf{r}) \varphi_{i}(\mathbf{r})=\epsilon_{i} \varphi_{i}(\mathbf{r})$$
vs 
$$ \left[-\frac{\nabla^{2}}{2}+V_{H}(\mathbf{r})+V_{\text {ext }}(\mathbf{r})\right] \varphi_{i}(\mathbf{r})+\int d \mathbf{r}^{\prime} V_{x}(\mathbf{r}, \mathbf{r}^{\prime}) \varphi_{i}\left(\mathbf{r}^{\prime}\right)=\epsilon_{i} \varphi_{i}(\mathbf{r})$$


# BROAD GOALS
* Procedure: 

    ![](./img/tik.png){width=90%} 

* Approximations:
  - non-interacting
  - KS-DFT , TDDFT, (LDA,..)
  - Landau-Buttiker
  - Hybrids
  - Many body peturbation theory (HF, GW, COHSEX, GW+ssc)