# Tag Info

24

These are a few extra points to complement Andrew Rosen's comprehensive response: To be absolutely clear, typical DFT calculations are not performed at 0K, a better description of what happens is that they are performed "for a static crystal". Static crystal means that the atoms are fixed at their crystallographic positions (which is what a typical DFT ...

19

You are correct that KS-DFT, strictly speaking, involves calculations of a potential energy surface at 0 K. However, if you accept that the density functional approximation you are using is sufficiently accurate, it is not too difficult of a stretch to go from 0 K to finite temperature conditions for an application of interest. The key assumption is that the ...

18

As you mention, there are many empirical dispersion corrections for density functional theory. Generally, the term "DFT-D" refers to a generic dispersion-corrected density functional calculation, regardless of the specific method used for the dispersion correction used. The D3 dispersion model is a specific dispersion correction method and is now something ...

14

Kieron Burke and co-workers have shown that in many cases one can get better results by using the HF density as input to a DFT-XC evaluation of the energy, as opposed to using the DFT-XC to generate a self-consistent density. This has been termed density-driven error, as opposed to functional-error. SE is especially problematic in cases where an electron is ...

14

This is a very interesting question and to the best of my knowledge there is not yet a conclusive answer. As the chemistry stackexchange threads linked by @Tyberius already state Kohn-Sham orbital energies and Kohn-Sham orbitals are often used in the chemistry community to interpret computational findings, because empirically this works ok. But there are ...

13

At least I was able to find three other forms of DFT listed as: Lattice density-functional theory: It's suitable for modeling the lattice gas, binary liquid solutions, order-disorder phase transformations. Orbital-free density functional theory: It's less accurate than KS but it's much faster. Strictly-Correlated-Electrons density functional theory: It's an ...

13

It comes down to the fact that HF/KS both are variational method. This short article by Julien Toulouse gives a great description of ways to compute static/dynamic response properties. Here, I'll just summarize the relevant portion. We can compute derivatives of the energy with respect to any variable $x$ as: $$\frac{dE}{dx}=\frac{\partial E}{\partial x}+\... 13 Noncollinear magnetism means that the orientation of the magnetization varies in space. Examples for such structures are magnetic domain walls, spin spirals, or magnetic skyrmions. To describe these systems one has to consider the Kohn-Sham wave functions as spinors$$\Psi_\nu(\mathbf{r}) = \begin{pmatrix} \psi_\nu^{\uparrow}(\mathbf{r}) \\ \psi_\nu^{\...

12

"However, it is notorious due to the exponential wall" That is completely true, though there's indeed some methods such as FCIQMC, SHCI, and DMRG that try to mitigate this: How to overcome the exponential wall encountered in full configurational interaction methods?. The cost of FCIQMC still scales exponentially with respect to the number of ...

12

The main issue here is that the calculated energy is a potential energy, that's why the absolute value is useless. As potential energy always depends on a reference system, you can only use the absolute value in comparison (as you do the calculations using exactly the same conditions) or in quantities that are obtained by substractions of these energies like ...

12

For the "why" you should use the tetrahedron method when computing the density of states, check out this paper on exactly that topic! In short, other approaches (e.g. Gaussian smearing) will occasionally obscure (and artificially introduce) some features of the density of states. As for how the method works, quoting from the paper directly about ...

11

Kohn-Sham DFT may only be rigorous at zero temperature, but at nonzero temperature, Kohn-Sham-Mermin DFT is an equally rigorous replacement. There are two major differences Rather than deriving the orbital equations from a minimization of the energy, $E$, one minimizes the free energy $F = E - TS$, where $S$ is the entropy. A practical consequence is ...

11

The band gap problem in DFT is not just due to approximate exchange-correlation functionals--it is a reflection of the fact that the Kohn-Sham (K-S) orbitals are a mathematical construction of a non-physical, non-interacting system of electrons that yields the true ground state charge density of the real many-body system. In exact DFT, the derivative of the ...

11

Orbital-Free Density Functional Theory (OFDFT) is, as the name suggests, an attempt to work with DFT without using the Kohn-Sham (KSDFT) approach of expressing the density as a sum of non-interacting orbital densities. In the KS formulation, the kinetic energy functional explicitly depends on the choice of orbitals, whereas in principle the true energy ...

11

If the ion-ion interaction contributes a constant term to the Hamiltonian $H$, then our new Hamiltonian is $H+C$. The eigenvalue of a constant is just itself, so we have: $$\tag{1} (H + C )\psi = (\epsilon + C)\psi$$ So if your DFT code calculates only $\epsilon$ (the energy if you neglect the ion-ion interaction), it is easy to obtain the energy with the ...

11

Using your notation, the definition for the universal functional is $$F_{HK}[\rho] = \left< \psi_0[\rho] \right| \hat{T} + \hat{W} \left| \psi_0[\rho] \right>,$$ where $\hat{T}$ and $\hat{W}$ are kinetic and electron-electron interaction operators, respectively. This definition is possible because of the one-to-one mapping between densities and their ...

11

First of all, let me emphasize that it is more appropriate to speak of KS equations (plural), which you correctly denoted by an index $i$ in your post. This index goes over all KS orbitals (i.e. single-particle wavefunctions) of the system. Additionally, as you mentioned, these equations have the same form as the single-particle Schrödinger equation. And ...

10

Basically, this is the split between restricted, unrestricted, and generalized Hartree-Fock (or Kohn-Sham) theory. In the restricted theory, both the spin-up and spin-down electrons occupy the same spatial orbital: $\psi_{2n}({\bf r}) = \phi_n ({\bf r}) |\uparrow \rangle$, $\psi_{2n+1}({\bf r}) = \phi_n ({\bf r}) |\downarrow \rangle$. This is the Ansatz you ...

10

Add more information to @Nike Dattani's answer: The matter can be viewed as a set of ions and electrons. The Kohn-Sham equation listed in your post aims to solve the electronic part. As for the ionic part, which is usually treated classically in the framework of Newton's mechanics. The ion-ion potential or force can be calculated with the empirical method (...

10

It's determined by fitting. You optimize the functional (i.e. the coefficients therein) to yield the lowest errors possible. See e.g. the wB97M(2) functional for an example of a recent double hybrid.

10

A few more papers to add to Michael's answer: What Do the Kohn-Sham Orbitals and Eigenvalues Mean Relationship of Kohn–Sham eigenvalues to excitation energies

10

Your question is on absolute energies, but all of your physical examples are relative energies. One could also add properties to the list; however, these are again not determined by the total energy, but rather as its derivatives with respect to some perturbation... which are again energy differences, just now between the perturbed and unperturbed system. ...

9

As you note, the interacting electrons and the Kohn-Sham non-interacting electrons have the same density. How is this possible when the Hamiltonians for the two systems are so different? The answer is that the Kohn-Sham potential—the potential felt by the non-interacting electrons—is constructed very carefully. Namely, if we want to remove electron-electron ...

8

I guess this will go in the answer slot, it is a bit long for a comment. DFT typically has quite a bit less spin contamination than HF (attributed to the inclusion of correlation). One issue however is that DFT tends to favor electron delocalization, and conveniently HF tends to the opposite. This is part of why hybrid DFT methods get improvements, the ...

8

I would like to emphasise a few aspects that seem to be a little bit between the lines in the other answers. Density functional theory is based on the fact that observables of an interacting-electron system can in principle be obtained from its ground-state electron density. The Kohn-Sham system is a means of obtaining this density (and a few other objects ...

7

A couple of comments perhaps to clarify some points. The first Hohenberg-Kohn theorem states that the one-electron density determines uniquely the energy, for a non-degenerate ground state. The second H-K theorem states that this density can be obtained by employing the variational principle, i.e. the density is obtained by minimizing the energy as a ...

7

As Nike has already mentioned, Slater determinants occur in KS-DFT in much the same way that they do in Hartree-Fock as a way to combine a set of one-electron orbitals into an antisymmetric many-electron wavefunction. However in DFT, as the name suggests, we are typically more interested in the density. The Hohenberg-Kohn theorems show that DFT can, in ...

7

From the Wikipedia article on Kohn-Sham equations: "As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals." So the answer to your question: "Are slater determinants found in practice in Kohn-Sham Density Functional theory?" is ...

7

As far I understand, to search for spin-polarized solutions, the DFT code will have to solve two coupled Kohn-Sham equations, one for each of the two spin species. There is both direct and indirect coupling between the two Kohn-Sham equations . The direct coupling comes about by the dependence of the effective potential in the Kohn-Sham equation ...

7

To add to the comprehensive answer by Kevin J. M., an example of a class of systems in which the use of a hybrid functional can lead to radically different band structure characteristics compared to a semilocal DFT, is in the area of topological materials. In this paper the authors show that semilocal DFT incorrectly predicts whether a material is ...

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