Please what is self consistent theory and which resource can i use to learn everything about bself consistent theory and DFT theory
Dear Chinedu Onyeke,
Regarding DFT there are many wonderful review. One of those that I personally found inspiring is freely available at
https://arxiv.org/abs/cond-mat/0211443
by Prof. K. Capelle. You might find many also by K. Burke and eventually books by E.K.U. Gross and G. Vignale for example. Each of them can give you a in-deep survey of Density Functional Theory.
On the other hand, Self Consistent method is a mathematical/numerical technique to find solution of non-linear equation. The idea, I believe somewhat dates back to Newton and SCF is essentially an iterative method. Given an equation like
F(x)=x
you insert an educated guess in the left hand side, evaluate a new x, and plug it back in the F(x) to get a new value. And so on, until the difference between some function of x in a certain number of subsequent steps is not small enough (for esample, the total density or the total energy are routinely used). The solution you obtained in this way is a self-consistent solution. For any given non-linear problem, the SCF method is probably not the most efficient but it is a workhorse that can work in many different situations and initial conditions. The SCF is routinely used in DFT codes to solve the non-linear Kohn-Sham equation for the electronic wavefunctions.
I hope this helps,
Roberto
PS: Clearly the implementation of the SCF method in current codes is more refined than the extreme simplification I used here. But my aim was to give you an idea of the method and make clear that there is nothing particular about it.
The Self-Consistent Field (SCF) method is a computational approach employed in quantum chemistry for the determination of the electronic structure of molecules. It is a fundamental approach for solving the electronic Schrodinger equation, which defines the behavior of electrons in a molecular system.
The fundamental concept underlying the Self-Consistent Field (SCF) method is to iteratively solve the Schrodinger equation to determine a set of molecular orbitals and their corresponding occupation numbers that minimize the electronic energy of the system. The procedure entails the subsequent stages:
1. Initial Guess: Start with an initial guess for the molecular orbitals.
2. Calculate Electron Density: Use the initial orbitals to calculate the electron density.
3. Solve Schrodinger Equation: Use the electron density to solve the Schrodinger equation and obtain a new set of molecular orbitals.
4. Update Electron Density: Calculate a new electron density based on the updated orbitals.
5. Convergence Check: Check whether the orbitals and electron density have converged to a stable solution. If not, return to step 3 and repeat the process.
The iterations persist until a state of self-consistency is achieved, whereby the electronic energy and molecular orbitals exhibit negligible alterations between iterations.
To better understand the methodology with the equations, visit this link for greater clarity: https://www.youtube.com/watch?v=hr9zndnd1Jk
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