How does DFT handle degenerate eigenvectors?

In summary, Density Functional Theory (DFT) addresses degenerate eigenvectors by incorporating symmetry considerations and using methods like the generalized Kohn-Sham equations. When multiple eigenstates correspond to the same energy level, DFT can utilize symmetries to construct a linear combination of these states to form a unique, non-degenerate representation. Additionally, techniques such as the use of projection operators or enforcing constraints during the self-consistent field calculations can help manage degeneracy, ensuring accurate electronic structure predictions.
  • #1
Mart1234
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DFT and degeneracy
I have a question about how DFT (density functional theory) handles degenerate states. The Hamiltonian in DFT is a functional of the electron density defined via ##n(\mathbf{r})=\sum^N_{k=1}|\psi_k(\mathbf{r})|^2##. However, say I have a pair of degenerate states. Then any linear combination of these two states is also a solution to the Kohn-Sham equations and the electron density based on the above definition seems to be not well defined. How does DFT address this? Is there a constraint for degenerate states which picks a proper orientation?
 
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  • #2
Hm, each of the degenerate states has it's own density. E.g. if degeneracy is due to rotational degeneracy, as an example, psi might be a p orbital. As they are degenerate, the orbital can point in any direction and the corresponding electronic density will be concentrated along the same direction.
 

FAQ: How does DFT handle degenerate eigenvectors?

1. What is density functional theory (DFT)?

Density functional theory (DFT) is a computational quantum mechanical modeling method used to investigate the electronic structure of many-body systems, particularly atoms, molecules, and the condensed phases. It simplifies the complex many-body problem by using electron density as the primary variable instead of wave functions, allowing for efficient calculations of properties and behaviors of systems.

2. What are degenerate eigenvectors in the context of DFT?

Degenerate eigenvectors occur when two or more eigenstates of a quantum system have the same energy level. In DFT, this can arise in systems with symmetrical properties or in certain geometries where multiple electronic configurations yield the same energy, leading to multiple valid solutions represented by these degenerate states.

3. How does DFT handle degenerate eigenvectors when calculating electronic properties?

DFT typically employs a method known as "symmetry adaptation" to handle degenerate eigenvectors. By recognizing the symmetry of the system, DFT can combine the degenerate states into a single representation, allowing for a more accurate description of the electronic structure. This can involve averaging properties or constructing linear combinations of the degenerate states to represent the system effectively.

4. What are the implications of degenerate eigenvectors for DFT calculations?

The presence of degenerate eigenvectors can complicate DFT calculations, particularly in the determination of ground state properties and response functions. It may lead to challenges in convergence and could affect the accuracy of predicted electronic properties, necessitating careful consideration of the symmetry and degeneracy in the modeling process.

5. Can degenerate states affect the accuracy of DFT results?

Yes, degenerate states can impact the accuracy of DFT results. If the degeneracy is not properly accounted for, it can lead to incorrect predictions of energy levels, electronic distributions, and other properties. Therefore, it is crucial for researchers to identify and appropriately handle degenerate eigenvectors to ensure reliable outcomes in DFT studies.

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