Understanding Self-Consistent Solutions in Kohn-Sham Equations

  • Thread starter saray1360
  • Start date
  • Tags
    Self
In summary, self-consistent solution in the context of Kohn-Sham equations refers to the condition where the density computed on a particular iteration of the KS loop is equal to the density computed on the previous iteration. This ensures consistency in the equations and implies that there is at least one solution. It is an important aspect in the study of electronic structure and has been extensively researched by various scientists.
  • #1
saray1360
57
1
what does self consistent solution mean in for example Kohn-sham equations?
 
Physics news on Phys.org
  • #2
The equations of a linear system are consistent if they possesses a common solution, and consistency implies at least one solution.

See - http://en.wikipedia.org/wiki/System_of_linear_equations#Consistency

Self-consistency is the condition that [tex]F[n(\vec{x})]\,-\,n(\vec{x}) = 0[/tex].

See - section 4.4 of http://people.ccmr.cornell.edu/~muchomas/P480/Notes/dft/dft.pdf
from http://people.ccmr.cornell.edu/~muchomas/P480/

See also the original Kohn-Sham paper - "Self-Consistent Equations Including Exchange and Correlation Effects," by W. Kohn and L. J. Sham, Physical Review 140, A1133-A1138 (1965).
http://www.cem.msu.edu/~cem883/topics_pdf/Kohn_Sham.pdf

and

"Inhomogeneous Electron Gas," by P. Hohenberg and W. Kohn, Physical Review 136, B864-B871 (1964).
http://prola.aps.org/pdf/PR/v136/i3B/pB864_1

See also Electronic structure By Richard M. Martin

This might also be of interest - KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations
http://portal.acm.org/citation.cfm?id=1499096.1499099
 
Last edited by a moderator:
  • #3
It means that on the n-th iteration of the KS loop, the density that you compute from the KS states is equal to the density computed on the (n-1)-th iteration.
 

Related to Understanding Self-Consistent Solutions in Kohn-Sham Equations

1. What is a self consistent solution?

A self consistent solution is a solution to a mathematical or physical problem in which all the variables are consistent with each other. This means that each variable is dependent on the others and there are no contradictions or discrepancies in the solution.

2. Why is a self consistent solution important in science?

A self consistent solution is important in science because it ensures that the mathematical or physical model accurately reflects the real world. It allows scientists to make reliable predictions and draw meaningful conclusions from their data.

3. How is a self consistent solution achieved?

A self consistent solution is achieved by using iterative methods, where the solution is refined through a series of calculations until a consistent set of values is reached. It may also involve adjusting the initial assumptions or parameters of the model until the solution is consistent.

4. What are some examples of self consistent solutions?

Examples of self consistent solutions include the calculation of electron orbitals in an atom, the determination of planetary orbits in the solar system, and the modeling of fluid flow in a pipe system. These solutions are all dependent on multiple variables and must be consistent to accurately represent the physical systems they describe.

5. What challenges arise when seeking a self consistent solution?

One of the main challenges in seeking a self consistent solution is balancing accuracy and complexity. As the number of variables and interactions increase, the calculations required to achieve a self consistent solution become more complex and time-consuming. Additionally, some systems may have multiple self consistent solutions, making it difficult to determine which is the most accurate representation of the real world.

Similar threads

  • Atomic and Condensed Matter
Replies
1
Views
673
  • Atomic and Condensed Matter
Replies
0
Views
860
Replies
2
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Atomic and Condensed Matter
Replies
2
Views
2K
  • Atomic and Condensed Matter
Replies
6
Views
4K
  • Atomic and Condensed Matter
Replies
4
Views
2K
Replies
9
Views
9K
  • Atomic and Condensed Matter
Replies
0
Views
864
Back
Top