Time-Dependent Density Functional Theory looked like a mess when it was first explained to me. I probably made the face of a person who smells sushi for their first time, wondering if this is some sort of bad joke.
After all, quantum mechanics is supposed to be a theory of non-commuting observables that evolve in a linear fashion. TD-DFT is nothing like that, yet it claims to reproduce all the same effects. Fishy indeed.
TD-DFT first focuses on the density of the wave function, in particular, the position basis of it. This is relevant for chemical calculations where it is very important to know where are the electrons. Of course, the density is one of many observables that are relevant, but TD-DFT makes it stand out by letting this observable evolve by means of a functional of itself. In other words, you don’t fully evolve the wave function by means of an operator, but instead you have a very complicated, non-linear functional that takes as its input the density and lets it evolve. In practice, since the functional is non-linear, in practice, the evolution is done iteratively.
Runge and Gross proved that if you only cared about the evolution of one observable, the density, this procedure is equivalent to the full quantum mechanical evolution. In other words, you can map the evolution of a particular observable the wavefunction under Schrodinger’s equation into a functional of the same observable.
What you gain from this approach is a computational speedup. The prize paid is that writing the exact functional is actually a very hard problem, at least as hard as doing the full quantum mechanical evolution. However, in practice, approximated functional can be written down and used for real calculations that can predict properties for real materials. This technique is widely used, mostly as a black box toolkit used by many physical chemists around the world.
In our latest papers, we were able to show that this mapping can be also performed for open quantum systems instead of just Schrodinger’s equation. First, we developed the general theory of how the Runge-Gross theorem can be generalized, placing it in context of previous incomplete attempts. This paper was published in PCCP as a Hot Article. In it, we discuss how the theorem works even in the highly non-Markovian regime of an open quantum system.
In our second paper, we take this even further. The evolution of an observable of an open quantum system can be mapped to a functional for a close system. At first, this seemed counter-intuitive. After all, you cannot map the evolution of an open system into a closed system.
O te peinas o te haces rolos.
However, if you only care about one observable, and you are willing to use non-linear functionals, this can be done consistently, for just that observable. Since most of the code written for TD-DFT was for closed systems, our results shows that those techniques could be used to model open quantum systems. We feel that new chemical calculations with thermodynamic effects can now be explored with this theory.
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Dr. Strangelove: It is not only possible, it is essential.
![\left[ \frac{d}{dt}\mathbf{P}^\mathcal{S}_t\right]_{t=\tau} = 0\; \Leftrightarrow \; \left[\rho^\mathcal{S}_\tau\otimes I^\mathcal{E},\rho^\mathcal{SE}_\tau\right] =0](http://www.minustwofish.com/wp-content/cache/tex_605f730deb8085cb598e41ac42a06b43.png)
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Agosta and Di Ventra for Markovian OQS [Phys. Rev. Lett. 2007, 98, 226403] and discuss its domain of validity. We suggest ways to expand their scheme, but also propose a novel KS scheme where the auxiliary system is both closed and non-interacting. This scheme is tested numerically with a model system, and several considerations for the future development of functionals are indicated. Our results formalize the possibility of practising TD-CDFT in OQS, hence expanding the applicability of the theory to non-Hamiltonian evolutions.
