Diagnóstico de la situación de los migrantes en Cuenca

Our analysis of the electron dynamics now narrows to the molecular level, investigating the excitation process that occurs for several individual molecules along the proton trajectory. TDKS and the adiabatic KS orbitals for condensed phases, especially those with a significant amount of disorder such as most liquids, provide the challenge of delocalized single-particle electronic state, however. Simply observing the change in occupation of a particular adiabatic state over time (i.e. 𝐶!" 𝑡 = !!!!!" 𝜓! r,𝑡 𝜑! r ) does not provide much physical insight as each KS orbital is likely to contain several water molecules and represents a convolution of individual-molecule responses. To isolate the response of each individual water molecule, we developed an analysis based on maximally localized Wannier functions (MLWF) [165,166]. Developed by Marzari et al., MLWFs are generated by identifying the unitary transformation that when applied to a set of wavefunction (e.g KS orbitals) generates a new set of single-particle wavefunctions, 𝑛 , whose spread defined as Ω= 𝑛 r! _{𝑛 − 𝑛 r 𝑛} !

! is minimized. This approach is common in the computational

Figure 4.13 | Maximally localized Wannier functions calculated for liquid water. (A) The Wannier centers (i.e. the center of mass of electron density) are represented in blue. Each water molecule has four Wannier centers corresponding to its eight valence electrons. (B) The four Wannier functions nearest to a water molecule’s oxygen atom are visualized. The Wannier functions for water can be interpreted as lone pairs (red and green) and OH bonds (yellow and blue).

In Figure 4.13 we demonstrate the localization of single-particle wavefunctions for liquid

water. The blue dots in the figure correspond to the geometric centers of the wavefunctions. It is immediately apparent that Wannier functions fall in clusters of four (8𝑒!_{) around each}

water molecule. Two of these MLWF centers fall along the OH bonds while the other two are in a position similar to the lone pairs of the water molecule. In good agreement with this physical insight, the MLWFs when visualized (Figure 4.13b) also appear to describe electrons and bonds and lone pairs of water separately. The MLWFs for each molecule in liquid water has an average spread of ~0.49Å2_{. This makes MLWFs an invaluable tool to identify both the}

total changes to electron density per molecule, but further broken down into contributions from lone pairs and OH bonds.

To identify the changes to occupations associated with a specific molecule, we first generate the MLWFs for the adiabatic basis (𝑡 =0). Again, because the bulk water nuclei essentially do not move throughout the simulation, the MLWFs for the ground state provide a

good reference system for bulk water throughout the entire RT-TDDFT simulation. Using these MLWFs of the adiabatic basis, we then project the TDKS orbitals directly onto these localized electronic states. Recall that the ground-state electron density can be computed as

𝜌 𝑟 = 𝜓_! r,𝑡= 0 !

! = 𝜑! r

!

! = ! 𝑤! r ! where 𝜓! are the occupied TDKS

orbitals, 𝜑_! are the occupied adiabatic KS wavefunctions, and 𝑤_! are the MLWFs generated by applying a unitary transformation to the valence band of 𝜑_!. We use the coefficients associated with the projection to determine the hole population for each MLWF according to

H!! 𝑡 = 𝑑𝑘 Ω !" 𝑓!,! 1− 𝑤! r 𝜓!,!(r,𝑡) ! ! Eq. 4.7

where 𝑓!,! is the fixed occupation of the TDKS wavefunction and k-point integration is performed if necessary. In this work, we only calculate the MLWFs for the occupied KS orbitals. This is because there is well-known problem of non-uniqueness when calculating MLWFs to conduction band states [165]. In this work, we are only able to use this analysis to comment on the hole generation in the valence band of liquid water. The ionization process that was revealed in the previous section, however, suggests that a MLWF analysis for the excited-state electrons would most likely be ill suited anyway as they are delocalized throughout the full simulation cell. More details of the approach we developed can be found in Appendix A.

Our choice of individual molecules to investigate is motivated by the fact that only molecules falling along the path (< 2Å) of the projectile are likely to be ionized. We apply our MLWF analysis to three molecules at the beginning, middle, and end of the simulations that showed greatest hole generation (Figure 4.12a,b,c). We plot the hole populations for each

of the three molecules in liquid water in Figure 4.14. The total hole population for each

molecule is determined by summing the contribution for each of its four MLWFs according to Eq. 4.7. Consistent with observations of the SEP densities (Figure 4.12), the response of an

individual molecule does not occur until the projectile has approached its position in the simulation cell. The sudden hole generation for each of the molecules is consistent with the position of the molecule along the proton trajectory. Additionally, we plot separately the contributions of lone pairs (yellow) and OH bonds (green). It is immediately obvious from the simulations that there is a significantly greater contribution to excitation from the lone pairs than from the OH bonds. From our analysis, we determined that the average contribution of lone pairs is 70.1 ± 23.2%. We attribute the difference in the contribution to the orientation and proximity of the water molecule as the proton travels by it. Those molecules whose lone pair electrons are oriented towards the proton, for example, are more likely to be ionized.

4.4.3 Gas phase vs. condensed phase response: hydrogen bonding, extended electronic