# 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

## 1 Answer

Time-evolution of conceptual DFT quantities has been considered starting, I think, with Chattaraj ~2000. (I imagine there is some earlier work by Ghosh and/or Harbola, but I do not know a reference.) Example references: IJQC v91 633 (2003); J. Phys. Chem. A (Feature article) v21, 4513 (2019); Chapter 13 in "Theoretical Aspects of Chemical Reactivity" (ed. by Toro-Labbe, Elsevier, 2007). The issue is discussed (with more references) in the recent round-table perspective article Theoretical Chemistry Accounts v139, 36 (2020).

I do not really see much issue at a superficial level, though for the purpose of describing chemical reactions the more typical approach is that, instead of considering time-dependence, instead one looks at how reactivity indicators change along a reaction path (which, in the limit of zero temperature, represents a sort of "leading line" about which reactive trajectories cluster). This is primarily represented in the work of Chattaraj and especially Toro-Labbe (1990s-present). Looking at chemical reactivity indicators along a minimum-energy-path (functionally but not quite mathematically equivalently, the IRC) tends to give a clearer picture of the driving forces of a reaction because one can never be certain that a given reactive trajectory (time-dependent path) is representative, and it is likely that there is some "sloshing" around the reactant/product wells that may obfuscate the interpretation. The Toro-Labbe group, especially, has made excellent use of this strategy. Of course, there are always the necessary caveat that correlation and causality are inequivalent.

For photochemistry (e.g., the work of Christophe Morell) or truly dynamic (e.g. scattering) processes, it seems more interesting to me to look at the time-dependent quantities. At a superficial level, one can derive an analogue of the reactivity indicators using the same strategy, namely differentiation of the energy (of the ground or any other stationary state) with respect to the number of electrons and/or the (possibly time-dependent) external potential. This was done by Chattaraj and Poddar (1990s), Ayers & Parr (~2000), and Davidson, Johnson, and Yang (in a different context, ~2010). For a nondegenerate stationary state, the chemical potential and all other reativity indicators are cleanly defined as responses (and the same treatment that Cardenas, Cedillo, Bultinck, and Ayers developed for quasi-degenerate states could be extended), only requiring a suitable (Gorling, Nagy, Levy, Ayers) definition of the time-independent excited-state DFT functional. (The Theophilou, Gross, Oliviera, Kohn, Fromager, etc. ensemble excited-state-functional could be used too.) At a formal level, though, the same types of "working formulas" one would ordinarily use would still hold.

To make treatments of time-dependent reactivity indicators more rigorous one would need to leave the usual energy-based formulations behind, and move to the action (cf. the work of van Leeuwen); matters may be much more complicated (but if one wishes to be rigorous about differentiation in DFT, matters are always complicated anyway). My intuition is that, but for some mathematical niceties/footnotes, the "working formulas" of conceptual DFT would remain essentially unchanged.

• +1. Thank you Paul for this contribution to the community! There's a lot of helpful information there! – Nike Dattani Sep 26 '20 at 4:56
• +1 Very interesting insight. I don't know much about this topic, but could you in principle move to a quasi-energy formulation (as described in a different context) instead of an action based formulation? – Tyberius Sep 26 '20 at 20:54