What is the significance of Fractional Occupation Numbers in DFT?

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Discussion Overview

The discussion centers on the significance and physical interpretation of Fractional Occupation Numbers in Density Functional Theory (DFT). Participants explore various contexts in which fractional occupation numbers are applied, including their implications for charge delocalization and numerical integration techniques in computational methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire about the physical significance of fractional occupation numbers in DFT, seeking references for deeper understanding.
  • One participant draws an analogy comparing fractional occupation to an average family size, questioning the meaning of a fractional entity in a physical context.
  • Another participant explains that fractional occupation numbers can be used as a numerical integration technique for metals, enhancing convergence when integrating over a Fermi surface.
  • It is proposed that fractional occupation numbers may indicate charge delocalization, suggesting that a state expected to be fully occupied could instead be fractionally occupied, potentially reflecting either a physical reality or an error in DFT.
  • Participants mention that fractional occupations might be intentionally simulated to analyze energy dependence on occupation numbers, emphasizing the need for a correct description that includes a vanishing second derivative of the energy function.

Areas of Agreement / Disagreement

Participants express varying viewpoints on the interpretation and implications of fractional occupation numbers, indicating that multiple competing views remain and the discussion is not resolved.

Contextual Notes

Some limitations are noted, such as the potential for fractional occupation numbers to reflect either physical realities or errors in DFT, and the dependence on specific scenarios for their utility.

supw
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What is the significance/physical interpretation of Fractional Occupation Numbers in DFT?
 
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Please provide a reference - though it's probably an average occupation.
Like the average family having 2.6 kids - what is the significance of a fractional kid?
 
@Simon :: Fractional Occupation Numbers with reference to Density Functional Theory.
 
Yes. I know. Answer stands.
By "reference" I mean like a book of an article you can show me that talks about the fractional occupation number, so I know the context and I'm not just guessing.
 
Fractional occupation is often used as a numerical integration technique for metals. If you have a Fermi surface and you're integrating over a discrete grid of points, the abruptness of the change in the occupation results in very slow convergence wrt the grid density. Using fractional occupations that smoothly go from 1 to 0 from the inside to the outside of the Fermi surface can be used to speed convergence.
 
There are different scenarios where fractional occupation numbers are useful. Here I give two:

1- They may indicate charge delocalization. So, a state which is supposed to be fully occupied is now fractionally occupied and the rest is spread over several ions. This situation could be the reality or an inherent error in DFT.

2- Fractionally occupied states my be intentionally simulated to check the energy dependence on the occupation number regarded as a continuous function. A correct description requires a vanishing second derivative for this function. Check figure 1 of this paper:
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.085202
 

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