OK, thank you very much.
Yeah, I would like to, first of all,
thank the organizers for giving me the opportunity
to talk about the work that we are doing at the theory
division here at CFALE-DASI and for putting together
this nice workshop with so kind of different backgrounds
from the people.
And the first thing that I would like to do
is to acknowledge the co-workers
of this, of what I will present, Cheng Li.
He's a PhD student.
Lamin Majet is a postdoc in our group.
And of course, Troy and Sandra, you know him.
OK, so let me just, first of all, in a couple of slides,
give a little bit of background of basically
the molecular Hamiltonian and what
we understand by non-ideopathic coupling molecules,
because the audience is very heterogeneous.
And of course, to some of you, this is already very trivial.
So this would be the full molecular Hamiltonian.
You see here the nuclear kinetic energy, electronic kinetic
energy, electron-electron interaction,
electron-nuclear interaction, and nuclear-nuclear interaction.
And this can be written in this way,
in which all these terms are collected
into what is called this electronic Hamiltonian,
and then the kinetic energy here.
And if one solves the time-independent Schrodinger
equation with this Hamiltonian, then
one has what is called the clump-nuclear Schrodinger
equation, and one gets, then, potentials
that depend on the position of the nuclei.
And these potentials, the so-called potential energy
surfaces, this is very much at the heart of our understanding
of chemistry, of the motion of molecules, of their dynamics.
And we can think of molecules as moving on,
so basically of the nuclei, moving
on top of these potentials and surfaces.
This is possible because in many situations,
the kinetic energy of the nuclei is much smaller
than the energies in this Hamiltonian here.
So this is pretty much like a small perturbation.
And then this separation makes sense.
These functions that we get, so these electronic wave
functions in which the positions of the nuclei
enter as parameters, they can be then used
to define an ansatz, in which, again, well,
these are the functions where the nuclei enter parametrically,
and these are the nuclear wave functions.
And if we plug this into the time-dependent Schrodinger
equation, into the full time-dependent Schrodinger
equation, then we will get one term in which, basically, here
we have the kinetic energy of the nuclei,
these potentials in which the nuclei moves.
So this is the potential energy operator.
And then we have these coupling terms
here, which are related to the kinetic energy of the nuclei,
basically.
If we forget about this, or we just neglect it,
then we will have this picture in which the nuclei move
on independent potential energy surfaces.
And this is true in many situations,
but there will be situations in which these couplings are
essential.
Basically, these, and we'll see this in a second.
So these couplings, they take this form.
These are differential operators that
diverge at the points of electronic degeneracy.
We can see that this, basically, well,
this is the scalar coupling.
This is very often neglected.
And basically, this term is the one
that, if we see its dependence, it's
basically the gradient of this operator,
and then sandwiched between two electronic wave functions,
and depends on the difference.
So if we have a point in which two potential energy surfaces
are degenerate, basically, these couplings diverge.
And this gives rise to the concept
of conical intersections.
So this would be one of such points, or seams.
I mean, these conical intersections
are in a space which is n minus 2-dimensional,
where n is the internal degrees of freedom of the molecule.
So if we have two degrees of freedom,
then they are really a point.
And when the nuclei move close to these areas,
close to this point of intersection,
then these terms are important, and they
induce transitions between the different electronic states.
And this is then, basically, there
is a lot of photochemistry, which
is related to phenomena, basically,
of electronic decay through conical intersections.
So if we want to treat these situations theoretically,
if we do quantum dynamics, and I will
show quantum dynamical examples, and also
examples based on classical mechanics.
In quantum mechanics, what one usually does,
I don't want to go into many details here,
but one takes this representation
of the Schrodinger equation, and one transforms it
to a, for it is called diabatic, in which these differential
operators are transformed into potential operators.
And there are different ways to do this.
And I will just briefly show it later for an example.
And the idea is that one has a position-dependent
transformation that transforms us
from these diabatic potentials to the diabatic representation.
This is a much nicer representation here.
There are no diversion couplings.
And this is much more adequate than for quantum dynamics,
because basically, the wave packets,
they are spread over all space.
And they would see the divergence points,
and this would make lots of problems.
So this representation is much better for quantum dynamics.
I will also show results based on what
is called surface-hopping simulations.
I also don't want to go into many details.
But another way to deal with these problems
is just to use classical trajectories for the nuclei
and account for the transitions between the different electronic
states.
This means that one is solving nuclear dynamics,
so classical dynamics for the electrons, where
one gets the force as the gradient of these potential
energy surfaces.
And then these potential surfaces
can be obtained by any quantum dynamics method, sorry,
electronic structure method, multi-configurational methods,
or, for example, for the ground state Hartree-Fock or DFT
would be adequate.
And then all these goes into different methods
to treat how the trajectories jump from one trajectory
to the other.
One scheme that we have been using
is the Landau-Zena scheme, which is quite simple,
because it depends only on the potentials
and on the velocities of the nuclei on the potentials.
Now we have also other methods implemented.
But as I said, I mean, I don't want to go into much detail.
Just mention that when we do these kind of simulations,
then we use the adiabatic representation,
because then these potentials are directly
obtained from quantum chemistry codes.
And this is the natural representation
when one deals with classical trajectories.
OK.
So now that I have given away the theoretical background
of what we are doing, let me just show again,
this picture was this morning in the talk of Joachim Ulrich
here.
So the calculations that we do are motivated by experiments
done at FLASH.
And FLASH is this linear FEL, which has these operation
parameters.
And Joachim already motivated why it's
interesting to look at molecules at FLASH.
I would just add up that basically with FLASH, what
we accomplish is we do ionizations
that are mostly valence ionizations.
And with these, we have molecules
where I would still say they still
behave in a way that one can think about chemistry.
So one removes an outer electron and this molecule will move.
So we can use pump to induce chemistry,
and then maybe also probe it with FLASH.
When we remove very deep electrons,
actually, this will cause coulomb explosions.
This is not so much in the chemical regime.
But with FLASH, we can really study chemistry,
autovalence chemistry.
So Joachim briefly mentioned this morning acetylene.
This is one of the works that we've been working on.
I will only present it here in this transparency
and then move to another topic.
What they did is, as he introduced this morning,
they used FLASH to remove an autovalence electron.
Then they looked, they waited for a certain time
until the proton moves.
And then they broke the system again with another pulse
and looked at the occurrence of fragments
that contain the two protons on the same side of the molecule.
And then they concluded that when one excites
into this cationic state or when ionizes
into this cationic state, the system undergoes ultrafast
isomerization.
This is what we also see from our simulations.
But also, a conical intersection is involved
that will bring the system down again to the ground state,
to the cationic ground state.
And these we have simulated.
And basically, we came up with a picture
in which these two elements go hand in hand.
And we are still improving on these.
We are still working on the problem.
And there will be more papers to come on this
in the near future.
But I would like to discuss today
about another system that has been studied at FLASH.
And as you will see later, I think
this comes very nicely into the topic of this workshop.
Because we will see correlations between electron holes
and protons moving.
This is an experiment that we were also
OK, the alkyl molecules involved.
This is by the group of Andreas Wolf in Heidelberg.
And what they did at FLASH is they took this molecule,
the protonated water dimer.
This is a stable system in the gas phase.
It's a closed-shell system.
It's just two water molecules with an extra proton.
And they ionize this system with FLASH.
It's written here.
And then they look at basically the fragmentation channels
that one gets.
This is the fragmentation channel
that is seen with the largest probability.
There are also other fragmentation channels
that are seen.
And in these experiments, there was no time resolution.
This is only one photon experiment.
The pulse was attenuated to make sure
that one is not doubly ionizing.
And that's it.
They just ionize and look at the fragments.
Now, let's look at the energy levels.
Basically, we have here these outer valence orbitals,
or, well, these outer valence states.
We are concentrating, basically, on these states here.
I will show the orbitals that are involved later.
There is a gap of about five electron
balls with these two states here.
So they cannot really talk to each other in terms of couplings.
So you can see here the possible decay channels
once you have ionized.
You can go into hydronium plus protonated water.
Or you can go into this free-body channel up here.
It's easy to estimate the energies of these channels.
This is just still in the ground state of the system.
If you separate the, basically, if you just
pull away the two waters, one will take the proton.
The other one will just remain a neutral water.
And this is associated with the energy of about 1.5 electron
volts.
If you include here the ionization potential of water,
you end up with the energy of this channel,
about 14 electron volts.
And if you look at what is the proton affinity of water,
then you can estimate or you can get, basically,
this energy is about 21.2 electron volts.
And this would still be below this second state here.
So we are going to center on the dynamics induced
by ionizing into these states.
And in principle, this channel seems to be open.
And we'll see later what happens with this channel.
OK.
So if you see, these are basically the orbitals
that contain the hole, the electron hole,
in the leading configuration for the electronic states.
And these orbitals have been computed
for a geometry of the system that
corresponds to the ground state.
So this would be the proton in the middle of the two water
molecules, and the oxygen-oxygen distance about 2.5 angstrom.
And you can see the topology of the potential G
surface as a function of the position of the proton
as a function of the inter-oxygen distance.
And you can see here, these regions of intersections
that, so this is basically conical intersections, one
here and one here.
And this is related to the fact that when
you remove an electron from one of the water molecules,
if you move the proton to one side,
basically the hole tends to localize at the other side.
So this is the reason why these two curves cross in the middle.
If you go to this upper potential here,
this is a potential G surface that corresponds
to a state in which, well, this is, again, the electron hole.
And here, you are removing the electron
from the orbitals that form the hydrogen bond.
So basically, now you have an electron hole, which
is around the central proton.
So you have to think that there is a positive charge
associated to this proton.
And then you have an electron hole surrounding it.
And when you do that, the proton moves in a more or less
shallow potential.
And at some point, if the proton moves enough to one
of the sides, it will, again, so this state
will become unstable.
And you will have a conical intersection.
And basically, the charge will then localize.
So this is a conical intersection related
to the fact that when the proton moves to one side,
in the end, the hole tends to localize
in the other water molecule.
So we have here an interplay between the position
of the proton and where the holes are located.
For this, we have done, so basically,
we have done quantum dynamics on top of these potentials
in surfaces.
So basically, this would be the set
of 15 internal division of freedom
of the protonated water dimer.
And we have selected the subset of seven division of freedom.
These are the ones that, in the ground state,
are the lowest frequencies.
So basically, we are simulating with the coordinates
proton between the two water molecules,
proton out of the axis, oxygen-oxygen distance,
internal rotation of the water molecule,
and this vacuum, so basically, this pedagogicalization
angles of the water molecules.
And these coordinates are just frozen at the ground state
equilibrium distances.
Then the potential energy surface that we are using,
we have computed at the Casse CF level.
So this is a good correlated energy,
so good correlated method for excited states.
And we have done this in a subspace.
And for one, it's just giving which
we have a low quality method for the whole space,
and then a high quality method for a subspace.
And in this way, we can get the whole potential
in a seven dimensional grid.
All right, then we need to take these potentials
and bring them to the diabatic representation.
I'm just going to skip through this quickly,
but the main idea is that we use a diabatic representation that
is forced to reproduce the diabatic potentials
in all regions of the potential energy surface.
And this is because we have very large amplitude motions.
And this means that far from the conical intersections,
we also need to reproduce the potential energy surfaces well.
So this is basically how these potentials are constructed.
But the main ingredient is we have
a model for the conical intersections.
We diagonalize this model.
Then we get some potentials that we don't use for anything.
We just use the transformation matrix.
And then you take the true potentials
and transfer them back to the representation
that we use for the dynamics.
This has been already used in the past and works well.
So let's have a look at the results.
So now we start looking at a diabatic dynamic.
This means we have the full Hamiltonian
that I showed at the beginning.
But now we switch off the couplings
and start the quantum dynamics with the coupling switched off.
And then we see that if we start from the lowest
a diabatic state, we have a very fast Coulomb explosion.
This is the Coulomb explosion channel.
This is when the wave packet is placed
under the conical intersection and just leaves very quickly.
If we put the wave packet in the second state,
so this means above the conical intersection,
it will mostly stay.
So the proton will stay localized in the middle.
And there will be a small leak out, but very small.
And basically, the proton stays in the central regions.
If you remember, if you had the double cone,
this means that the wave packet stays in the upper part
of the double cone.
And then if we put the wave packet
in the third state of the cation,
then it really doesn't move at all.
It just stays there confined and there is no Coulomb explosion.
So from this, you can already see
that in order for the system to dissociate
and to find the Coulomb explosion panel,
you need the non-diabatic effects.
So you need the coupling between the potential energy
bases.
OK, now we use the full Hamiltonian,
so the couplings are in now.
And if we start from the lowest of the three states,
then we have also a very fast Coulomb explosion.
If we start from the second one, we also
have a very fast Coulomb explosion.
You see that basically the system is,
the protons are going to one water molecule
or to the other water molecule.
If we start from the third state,
then we have a certain probability
that the proton remains at the center.
And if you look at the, I mean, this
is just a cut in a 2D space.
These are 2D densities, but the other two of them
are also there.
If one looks at the full wave function,
one sees that when the proton is here in the middle,
actually it has left through the perpendicular direction.
So this means that if one is in this state,
the proton has a chance to leave perpendicular to the system.
And this would be a free body channel.
So the one that I introduced at the very beginning.
So it seems that if we start from the upper state,
we have enough energy to live through this free body channel.
Let me just show very quickly the photoionization spectrum
that we could compute also from our calculations.
So again, this here are the potentials.
And this is the full photoionization spectrum
computed from the autocorrelation function
of the quantum dynamical calculations.
And we have these peaks.
In order to assign them, what we did
is we started the calculation again without couplings
and looking at then at the spectra,
starting from every of these potentials.
And you can see this huge, this very wide peak here
corresponds to when the wave packet starts here
on the red curve, so on the red potential.
And this is the very fast.
So this is a very fast Coulomb explosion.
So there is no time for resolving
any vibrational structure, and this is just a wide peak.
This peak here corresponds to when one, and this and this,
B and C, corresponds to when one puts the wave packet
on the green curve.
And in this case, there is some chance of the wave packet
to vibrate within this potential.
And this will generate some vibrational structure.
And this bump here corresponds to the little bit
of leaking that was in this green curve that
could live through here.
And this peak here corresponds to when
we put the wave packet here on the upper curve.
And then it will vibrate here when
we will see some vibrational structure.
But that's it.
So this is the photo-instantion spectrum.
And the good thing is that when discussing
with experimentalists with Andreas Volff,
now they want to do a new campaign of measurements
in which way they will try to put the photon energy of flash
as close as possible to here so that they hopefully
can record the photo-instantion spectrum
by looking at the energy of the photoelectrons.
And then we will see if our calculations agree
with what they can measure.
So this is very useful information
for the experimentalists working on the system.
Now, is this the whole story?
I mean, I just showed you that the three-body channel is open
if we start from the upper potential.
It seems that there is enough energy.
So what we are doing is we are also
working on classical approaches, because we
want to go to larger systems.
And since we have these programs,
really we decided to run also classical trajectories
on this system in which you can also do the quantum dynamics.
And here is the kinetic energy release
calculated from these classical trajectories,
starting also from different vibrational states.
So let me just drive you shortly through this data.
Here we have this curve is the kinetic energy
released for a quantum dynamical calculation.
We see that it is in good agreement with these upper
curves, and this is the total energy of the two monomers.
So basically, these curves correspond
to the sum of the vibrational energy of monomer one,
so all the energy of the vibrating molecule.
This is the kinetic energy of the center of mass,
so basically from the velocity of this water
molecule moving in space.
And this is then the same thing for monomer two.
And this curve here correspond to the kinetic energy only
coming from the motion of the centers of mass.
And we see that no matter from which electronic state
we start, we end up with the same final velocity
of the water molecules.
But the internal energy content is very different.
So when we start from this higher electronic state,
then we get a much higher energy content.
And this basically means that when
we start from higher electronic states,
there is a very fast energy transfer
into vibrations, which is fast enough
that the water molecules are still close to each other
when this energy transfer has almost competed,
and then they start to move away from each other.
And the final velocity, the terminal velocity,
let's say, in these electronic states,
then turns up to be the same.
So this is a first indication that there
is a very fast energy transfer from electronic degrees
of freedom into vibrational degrees of freedom.
And this is something that should be measurable
experimentally.
I mean, if the experimentals record kinetic energy
releases, one should be able to see something like this.
All right.
OK, so now let me show you.
You may say, OK, you are comparing
seven-dimensional quantum dynamics
and full-dimensional classical dynamics, right?
So the point is that in the classical dynamics calculations,
we never saw the free body channel.
And the explanation is because the energy is taken so quickly
by the water molecules, and there
is nothing of energy for the product to live perpendicular.
To investigate this, what we did is we did classical dynamics,
but now with constraint, basically
with constraints in the equations of motion.
So basically using exactly the same model
for the quantum dynamics and for the classical dynamics
calculations.
And what we will see here is a constrained classical dynamics
starting from the second electronic state.
And you see the proton moves, and the water molecules
are totally rigid.
And then the system now has enough energy
to really live through the perpendicular.
We had never seen this kind of channel
when the water molecules can adapt to the basically
when the calculation is full-dimensional
and they can take vibration energy from the system.
Now here, we will see exactly the same trajectory,
but without constraints.
And so the same initial conditions,
but now the water molecules can start to vibrate.
And you see that they are doing scissoring motions
and vibrational motions, which are quite dramatic.
So they have taken quite a lot of vibrational energy
on the order of one to two EVs.
And then there is no way for the three-body channel.
So basically, we come to the conclusion
that we have this three-body fragmentation channel, which
is energetically open.
If you remember from the very beginning of the talk,
this channel was open from the energy of the cationic state
of, let's say, of the upper cationic state
that we are considering.
But there is no way to reach it kinetically.
The energy flows too quickly into vibrations,
and then this channel cannot be reached.
So OK.
Now, the next molecule that we are considering is the trimer.
And the reason for this is because we
see in a read in the dimer this interplay between the proton
moving to one direction, the electron hole moving
into the other direction, and all these non-adiabatic effects
that are involved in the curves crossing.
With this system, then we want to continue.
Basically, we want to generalize these and try
to see generalities.
And we want to even to do a bit larger
clusters in the near future.
But let me show you what we have on this molecule
that we have started to work shortly.
So now, again, we have a set of orbitals
that correspond to removing electrons
from different parts of the system.
So basically, at 16 electron volts,
we would be removing electrons from these orbitals
in the water molecules, in the external water molecules.
Now, a little bit higher, these are
orbitals that are involved in the hydrogen bonding
of the system.
So they would come a bit deeper.
Then we have this orbital that is sitting
on the central water molecule.
And this corresponds to removing an electron
from the place where the positive charge is really
sitting in the ground state of the system.
So if we look at the ground state,
there's a positive charge density in the central water
molecule.
And this explains this higher ionization potential.
And then this would be the final state.
And we can now look at the potential surfaces
as a function of the proton position
along this axis for this system.
And now we start to find here very interesting
antibiotic effects.
This is basically the two-dimensional potential
that one obtains by moving these two protons
and keeping the rest fixed.
And we see here a mass of curve crossings.
And this is now just a cut in which we just
move one of the two.
And basically, what we see here, this
would be, for example, basically a cationic state
of the system in which the charge, in which the electron has
been removed from one of the external water molecules.
And now we start to move the proton, for example,
this proton in direction to this water molecule.
And we will see that very quickly.
Now we will go through a series of curve crossings.
And this is basically now, OK, the proton comes closer.
Now this state becomes unstable because this positive charge
is approaching a place where there is really
a positive charge.
So now we have here a curve crossing, and then again,
and again, and again.
And at the end, we have here, up here,
we would have the positive charge here,
and this proton very close to this charge.
So we see basically that the position of the proton
is strongly coupled to what, to where, basically, the hole is
more stabilized in the system.
The same thing can be done by coming from up here.
This would correspond to making the hole in the central water
molecule when the protons are still here.
This has a very high energy.
And as we move the proton away from the central water
molecule, this diabatic state would be stabilized.
You can see how nicely the diabatic states reach
these points of intersection.
One can pretty much per hand diabatic in the system.
We could go to the data and just reassign the points such
that then we get diabatic curves.
It's really a very dramatic example that I like a lot.
Now what we can do is we can look at the dynamics.
We have already a few hundreds of trajectories
starting from different electronic states,
which I just want to show one example in which we start
the dynamics from the hole in the central water molecule,
the one that is holding already the positive charge
at the beginning.
So the positive charge is really with a proton.
And then we just let it go.
Here I show the electronic population.
This is an average of about 100 trajectories.
So at the beginning, the whole electronic population
corresponds to the electronic state that has this hole.
And then it will decay very quickly.
So in about less than 20 femtoseconds,
this state is totally repopulated.
And then other states will get populated and depopulated.
We see here, and in the end, we end up pretty much
basically in the ground electronic state
of this cationic manifold.
And this is something that happens in a time scale
of about 50 femtoseconds, so very quick.
And we just look now here at the dynamics.
So as I said, remember the electronic hole starts here.
And this is also where, let's say,
the positive charge corresponding to the proton also
is.
And now this starts.
So at the beginning, let's say the protons are trying
to find their way through the crossings.
And now we see that basically the system has decided
that the excess proton localizes on this side
and the electron hole localizes on this water molecule.
We have, as I said, hundreds of trajectories.
And one thing that is ahead of us in the next phase
is basically to analyze all these trajectories
and look for other possible dissociation channels
and also look at what happens when
we start from other electronic states.
But this one is particularly interesting because
of the fact that proton and electron hole
are localized at the beginning in the same position.
So with this, I reached the conclusions of the work.
So we say that for these protonated water clusters,
so they exhibit upon ionization strongly correlated electron
hole and proton motions, which are coupled
via non-allebatic effects.
We have also seen that non-allebatic pathways
are involved in the fragmentation of the clusters.
And they channel the electronic excitation energy
in an ultra-fast and efficient way into a rational motion.
I mean, we have seen that even they
can quench channels that would be energetically open.
We also see, I mean, what we are going to do in the future
is look at larger clusters.
So the question is, will they fragment
or will the two charges separate fast and far enough
to avoid fragmentation?
I mean, now we have just two or three water molecules.
So it's quite clear that we have a very high positive charge
density.
And the system will mostly basically cool and explode.
But we have enough water molecules.
It can be that the proton and hole can move enough,
so fast enough, into the, let's say,
into the outer parts of the cluster.
And then the system would remain stable.
These are things that we want to find out.
And basically, what would be nice to have, as I said
at the beginning, the experiments
of having conducted are not time-dependent.
There is no time information on this kind of systems.
We are providing these via our simulations.
But it would be nice to see experiments
that can follow the coupled evolution of the electron
hole and the excess proton in a time-resolved way.
And with this, I have reached the end.
And thank you very much for your attention.
Thank you.