# How can reactivity indices be calculated in a time-dependent scheme?

Many reactivity descriptors can be obtained from ground state (or static) DFT as energy derivatives respect to the number of electrons, $$N$$, and the external potential, $$v(\mathbf{r})$$, like chemical potential, $$\mu=\left(\frac{\partial E}{\partial N}\right)_{v(\mathbf{r})}$$, global hardness, $$\eta=\left(\frac{\partial^2 E}{\partial N^2}\right)_{v(\mathbf{r})}$$, Fukui function, $$f(\mathbf{r})=\left(\frac{\partial^2 E}{\partial N\delta v(\mathbf{r})}\right)=\left(\frac{\partial\rho(\mathbf{r})}{\partial N}\right)_{v(\mathbf{r})}$$ and the linear response function, $$\chi(\mathbf{r},\mathbf{r}')=\left(\frac{\delta^2 E}{\delta v(\mathbf{r})\delta v(\mathbf{r}')}\right)_N=\left(\frac{\delta\rho(\mathbf{r})}{\delta v(\mathbf{r}')}\right)_N$$.

My question is about the time-dependent regime. For instance, in the definition of $$\mu$$ and $$\eta$$ the derivatives are with $$v(\mathbf{r})$$ constant, however, in TDDFT the external potential changes with time. Also, for $$f(\mathbf{r})$$ and $$\chi(\mathbf{r},\mathbf{r}')$$, the derivatives should be, presumably, respect to $$v(\mathbf{r},t)$$, i.e., they evolve in time. The latter interpretation seems rather evident because the electrons are subject to a time-dependent potential and, thus, the electron density changes with time and also its derivatives.

Considering that local properties could change through a, for instance, chemical reaction, I suppose that knowing the time evolution of Fukui functions or the linear response function should give some insight about what is happening, from another perspective. But this topic doesn't seems to be very attractive due to the short literature that exist and I don't understand the reason.

How can these descriptors be calculated in a time-dependent scheme? Do global descriptors ($$\mu$$ and $$\eta$$) need a redefinition? Is there a problem with TD reactivity indices in general?

• Is it possible to elaborate a bit more on your question? You are asking if these definitions still make sense physically for TDDFT or not? Or you are asking if the formula remains the same or it will be changed? – Alone Programmer May 4 '20 at 2:50
• @AloneProgrammer aren't that two questions connected? If those definitions still make sense implies that the formula remains the same. On the other hand, if the definitions aren't useful in TDDFT, then the formulas should be changed. – Verktaj May 4 '20 at 3:25
• Yes, I agree. I think it depends... For example $\mu$ the chemical potential, in my opinion, and based on thermodynamics, it doesn't make sense to have a time-dependent chemical potential, but it could be argued that we monitor this variable and we hopefully looking for its equilibrium to find the chemical potential that makes sense thermodynamically... I'm not sure about the other variables though. – Alone Programmer May 4 '20 at 3:31
• @AloneProgrammer Can we talk about equilibrium quantities in an excited state? The equilibrium would be the grand state occupation of electrons. I would argue most of these derivatives are not well defined in excited state: intuitively in equilibrium an extra charge would go somewhere around the Fermi level, and chemical potential definition reflects this. In excited state, do the electron go to excited levels or to the hole created by excitation? Some descriptors showing where new electron density created and where deficient may be relevant to reactivity, but different formula needed. – Greg Jul 4 '20 at 15:12
• @Greg What do you mean equilibrium quantities in an excited state? You need to wait to reach equilibrium and then measure the thermodynamical quantities... otherwise, it doesn't make sense to have them in any state other than equilibrium... My opinion here is a bit conservative but certainly you could find lots of other information for studying non-equilibrium thermodynamics that could be applied on excited states as well. – Alone Programmer Jul 4 '20 at 15:20