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I have been recently reading a lot on the quantum mechanical theory regarding Density Functional Theory, DFT and Time-Dependent Density Functional Theory, TDDFT (Oscillatory and Rotatory Strengths in particular) in order to understand how first principle calculations fundamental theory work.

I get that DFT is used to calculate the ground-state configuration of a system and TDDFT gets you the excited states useful for spectra determination and that there have been developed several algorithms in order to make calculations more efficient (timewise).

But I still can't answer myself in a short way how does each one works (math aside).

If you were to explain them to a person that is not necessarily familiar with quantum mechanics, how would you do it?

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In DFT the electron density is the superstar. It is a function that depends only on three spatial coordinates and the total electron density could be spin polarized (spin is difficult to explain correctly to someone not familiar with QM). In QM, the wavefunction contains all the information about the system, however it depends, at least, on $3N$ coordinates. So, in principle, it would be easier to use the electron density to explain the properties of a $N$ electron system.

Now, Hohenberg and Kohn proved that the electron density of the most stable configuration of the system (the ground state) and its wavefunction can be used alternatively as full descriptions of the system. From the electron density we can calculate the number of electrons, of the $N$ electron system, and looking for cusps in the density and how fast it drops to deduce where are the nuclei and their charges. Having the composition of our system, we can solve the Schrödinger Equation. To find the minimum energy of the system we should look at the minimum of the energy integral among the $N$-electron functions that are normalized and any of them giving a specific density $\tilde{\rho}$ with $N$ electrons, then we go over all density distribution that sums up (integrate) to $N$, choosing the density that minimizes the energy as the exact ground state density, $\rho$. If we consider the potential external to electrons apart from the kinetic and electronic repulsion energies, then we can define a universal potential for all electronic systems.

The difficult part is knowing how this universal potential is. It seems like an impossible task, because it is similar to solve exactly the $N$ electron Schrödinger Equation. Actually, Kohn and Sham proposed a fictitious system of non-interacting electrons with a density equal to the ground state density of the real system. Since the Kohn-Sham electrons don't interact with themselves, it is necessary to solve only the one-electron equations (simpler to solve). The "error", or correlated part, of the energy (or whatever property) is contained in a remainder called the exchange-correlation energy. This energy is accounted for in different forms with appropriate exchange-correlation functionals.

The current description of DFT only works for ground states, or static systems. The inclusion of time-dependent processes and excited-state properties of electronic systems has motivated the development of TDDFT, being the time-dependent density the superstar now. TDDFT is very different from DFT in two main aspects: there is no minimum principle, so we have to worry about memory, causality and initial states, and TDDFT is about the dynamics of the system.

The fundamental theorem of TDDFT is due to Runge and Gross, and it can be said that is partially similar to the Hohenberg-Kohn theorem (the difference is the time-dependency). For solving the equations, there exists a scheme similar to Kohn-Sham DFT, with non-interacting electrons, but now we have to solve the time-dependent KS equation with an initial condition (the static KS orbitals at $t_0$). This means that we propagate the initially occupied single-particle orbitals via TDKS equations.

I think that DFT and TDDFT in a nutshell, with math aside, could be explained in that form.

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  • $\begingroup$ Just regarding the last sentence, what do you mean to solve the time-dependent KS equation with an initial condition? What parameters do you need to take into account to impose an initial condition? Is the initial condition orbital dependent only? $\endgroup$ – C. Alexander Jan 31 '20 at 15:57
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    $\begingroup$ @C.Alexander If we have a time-dependent external potential that is switched at time $t>t_0$ on a system in its ground state at $t_0$, then the initial KS wavefunction, density and external potential can be obtained from the static KS equations (as normal DFT) at $t_0$. After this time, the TD potential starts to influence the system and we have to solve the TDKS equations with the initial condition that the orbitals at $t_0$ are the solutions of the static KS equations, i.e., a propagation of the occupied KS orbitals from $t_0$ to $t_1$. Also, this requires an approx of the TD potential. $\endgroup$ – Verktaj Jan 31 '20 at 17:39
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Try to read original papers by Kohn. There are two relatively solid theorems about DFT:

  • Density functional exists
  • Density functional has a minimal value on ground state electron configuration

This two facts are essential but they provide only existence without any insight how to solve many electron problem. In my opinion, W. Kohn describes DFT in a very simple way in his nobel lecture. Also, in my view, if you want to understand "mathematical moments", you should (at least) understand how to rewrite many electron problem in terms of DFT.

Then, I emphasize that DFT was historically considered only for non-dynamical systems. It seems that any other methods (as TDDFT) provides only an approximate description (for instance, TDDFT use linear-response approximation) of a dynamical system. To understand how it works, it seems useful to start from this original paper. Also, you can try this paper. Finally, in my humble opinion, you should understand QM on a basic level to work with DFT, it is important.

If you ask about algorithms in a specific package for TDDFT (like VASP), it seems that nobody knows but developers.

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  • $\begingroup$ I guess that a math aside explanation is hard to work, I understand now from what I've read that QM is necesarry to provide a non empty/vague idea of the methods. On the other hand, I understand that one can build analogue time dependent KS and HK theorems, how approximate are they? $\endgroup$ – C. Alexander Jan 21 '20 at 16:02

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