1. Introduction
The field of the chemical physics is very broad. Among the main goals of the field is to study the properties and dynamics of molecular systems, including intra-and inter-molecular energy transfer processes leading to dissociation of excited molecules. This goal has been achieved via a large number of theoretical and experimental studies. Time resolved spectroscopic and pump-probe methods, in both the frequency and the time domains, have been especially useful for providing data to test theoretical methods [
1,
2]. These methods have been applied to the vibrational predissociation of an extensive variety of van der Waals (vdW) complexes composed of three or more atoms with a range of bond energies, atomic masses, and vibrational frequencies. Due to the weakness of the vdW interactions in these systems, the constituents retain their chemical integrity upon complex formation so the energy transfer mechanism can be easily identified and studied at the state-to-state level [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16]. The NeBr
van der Waals molecule has been particularly useful for these studies since the dynamics is on the boarder line for which classical and quantum methods are comparably useful. This allows us to investigate in particular the applicability of the trajectory surface hopping (TSH) method [
8,
14,
15,
16]. The TSH method has been widely used [
17,
18,
19,
20,
21,
22,
23], demonstrating its validity and efficiency for a variety of molecular dynamics problems. For this study of the vibrational predissociation of NeBr
, the diabatic potential energy surfaces are formed by the interaction of the Ne atoms with the
Br
vibrational levels. The couplings between surfaces are provided by the van der Waals potential. The TSH results obtained for NeBr
vibrational predissociation are compared to previous work using other methods [
8,
9,
10]. We also implement the “kinetic mechanism” [
15] to interpret the results of the TSH simulation. This method considers two mechanism for transferring energy from the vibration of the diatom to the van der Waals modes. The first mechanism corresponds to a direct vibrational predissociation (VP) transfer of the dissociative coordinate. The second mechanism involves preliminary energy transfer to the non-dissociative van der Waals modes, followed by intramolecular vibrational redistribution (IVR). After IVR has taken place, the cluster is cooled by expelling the rare gas atoms, a process called IVR-evaporative cooling (EC). The values of the kinetic rate constants which characterize these elementary steps are determined by fitting the results of the TSH simulation to the analytical expressions for the time evolution of the NeBr
concentration. The simplicity of our treatment and its relatively low computational cost will allow it to be extended to systems with more degrees of freedom (e.g., more rare gas atom), thus offering an attractive and different alternative to purely classical treatments.
The paper is organized as follows: in
Section 2, we discuss the procedure of the TSH method as well as the computational details. We also describe our implementation of the kinetic mechanism. In
Section 3, results are presented and discussed through figures and tables.
Section 4 summarizes the conclusions that can be drawn from this work. In the
Appendix A, we describe our procedure to compute the transition probabilities in considerable detail.
2. Theory and Methods
We use Jacobi coordinates
to describe the NeBr
triatomic complex, with
r being the bond length of Br
,
R the intermolecular distance from the Ne atom to the center of mass of the dihalogen and the angle between
and
vectors. These definitions are shown in
Figure 1. Calculations are performed considering total angular momentum null (
), with
being the angular momentum of Br
and
the orbital angular momentum, a well justified constraint while studying photodissociation events.
The classical Hamilton function corresponding to this three degree of freedom model of the complex is written as
where
is the van der Waals interaction between the Ne atom and the Br
molecule,
and
is the Morse’s potential between Br-Br atoms. The quantities
and
stand for inverse of reduced masses for the Br
and NeBr
, respectively. The values
,
,
,
,
and
are from [
24].
For the quantum treatment, we have used as quantum coordinate the Br vibration and the other variables are treated as classical coordinates; for our case these are R and . A potential energy surface (PES) is defined by a single state of the quantum degree of freedom (r). It is on this surface that the trajectories for the classical degrees of freedom evolve. Quantum transitions are modeled by hops of the trajectories from one surface to another. These are governed by the evolution of the multicomponent time-dependent vibrational wave function .
As we have commented before, our surfaces are diabatic (see
Figure 2) and the couplings are provided by the van der Waals potential. This is different from the usual trajectory surface hopping TSH treatment where the transitions occur between electronic adiabatic surfaces, in well defined regions of avoided crossings.
2.1. TSH Method in the Diabatic Representation
For our triatomic system, we can write the Hamiltonian as follow:
where
is the quantum part depending on the vibrational coordinate
r of Br
,
is the Hamiltonian describing the classical degrees of freedom and
is the interaction operator which couples the quantum and classical degrees of freedom.
In our case,
describes the vibrational motion of Br
in the
B state,
and
Finally,
has the form
If we do a variable change
to simplify notation, the averaged Hamiltonian can be written as:
where
1 if
and 0 if
is the Kronecker’s delta and
is the vibrational energy for Br
obtained by solving
where
operator corresponds to the function
defined in Equation (
6).
We have used Bode’s method [
25] for the integration of these equations. This subroutine is very important because we have the Morse potential depending of
and
, and we want to know the averaged effect of
r for a vibrational level. It is the source of the PES and the coupling among them is due to non diagonal elements
.
The equations motionscover on the
surface are defined as follows (taking into account that
=
,
):
The state vector
describing the vibration of Br
is obtained by solving the time-dependent Schrödinger equation
is written as
where the sum is over all
v states of
with energy
,
. The complex variable indicates the amplitude of each vibrational level over the total wave function. This is known as the semiclassical expansion of the electronic wave function. Replacing (Equation (
15)) in (Equation (
14)), we obtain:
Replacing
, multiplying
on the left, and taking into account
then yields, respectively:
and
The
coefficients satisfy
and for each vibrational level we determine the population as follows:
where
is the density matrix.
The transition probabilities from the current state
v to all other states
during the time interval
are computed using the surface hopping probability (see
Appendix A for more detail):
The initial conditions for the classical trajectories are selected randomly for a total energy corresponding to the zero-point of the complex NeBr
, for a particular vibrational state of the Br
molecule. The component of the momentum that is parallel to the quantum state coupling vector is only taken into account to adjust, in order to conserve, the total energy [
26,
27]. In our case, quantum transitions occur between diabatic surfaces, defined for each vibrational level Br
. These surfaces are coupled by the NeBr
potential. Momenta are adjusted during a surface hop using
where
and
are the classical momenta which correspond to the classical coordinate
R after and before the transition, respectively. The value of
is obtained by imposing the total energy conservation after the transition. In this way, the angular momentum is also conserved. Energy conservation imposed using:
Taking into account
,
, replacing
and taking
and
, we obtain:
with the energy conservation satisfying:
If , then there is not a real solution for this equation and the hop cannot occur. In this case, it is called a frustrated hop.
If , the hop can occur, and the rescaling factor () is computed as:
when a hop occurs, we reset the wave function employing the “instantaneous decoherence” (ID) approach [
26]
∀
2.2. Treatment of Frustrated Hop
When a frustrated hop occurs, we activate “
” prescription [
28]. Specifically, when a frustrated hop is encountered, the following quantities are computed:
where is the nuclear momentum of the trajectory and “” is the gradient of the target vibrational state v, and are the projection of the nuclear momentum and the force of the target vibrational state along the hopping vector h, respectively. If and have the same sign, the target vibrational state accelerates the trajectory along h. Otherwise, if the two quantities have opposite signs, the target vibrational state causes a delay in the trajectory of the Ne. For that, we use the follow criterion for frustrated hop.
2.3. Kinetic Mechanism
The kinetic mechanism allows us to understand the path followed by our system as it relaxes and dissociates (see
Figure 3). We consider Br
(intermediate state detected in the TSH simulation) as a sum of two contributions: a short-lived contribution coming from the VP process, which we denote by Br
Ne
, and a longer-lived contribution coming from the IVR process denoted as Br
Ne
. Therefore, only the sum is taken into account to fit the kinetic rate constants. We include the Br
Ne
intermediates in the direct VP process, dividing each direct VP step into two processes characterized by the rate constants
and
for the loss of the first vibrational quantum, and
and
for the loss of the second one.
We fit the data obtained in the simulation by using the next procedure:
where:
2.4. Computational Details
In the methodology involved, two important stages contribute:
This step consists of propagating the dynamics of the system by evolving classically the nuclear motion on the potential energy surface. For this, employ the Adams Bashfort method [
25], initiating with the method of Runge Kutta 4th order. We obtain, from the system of Equations (
10)–(
13), the momentum and coordinate of the Ne, the
angle and the angular momentum of the system in Jacobi coordinates (see
Figure 1). These coordinates are defined as follows:
r is the distance between the diatomic constituents,
R is the distance of the noble gas to the mass of the chemical bound and
is the angle between
r and
R. In the method, some physical considerations of importance are imposed for obtaining a consistent result, such as the conservation of total energy. We consider that our system is dissociated beyond a certain maximum distance. For this, we take R
Å and a maximum time which the system will remain bounded, t
ps. Taking into account the above, we use an integration step equal to
ps, which leads to
integration cycles and ensuring a total energy conservation error of less than
cm
.
Simultaneously, in the other simulation thread, the Equation (
20) is integrated, obtaining the
coefficients. The
coefficients are the weight of each vibrational level (if
) during the dynamics of the system and
(if
) indicate the coherence between states.
In order to apply the population conservation, the sum of the populations has to be equal the unity. We also incorporate a method for the hop decision (it is known as the Fewest Switches algorithm, see
Appendix A). Another important aspect is the rescaling of the momentum to preserve the total energy of the system when the hop occurs.
To obtain the average interaction potentials of van der Waals, the Bode method is implemented. It is very important to define the average potentials and crossings between curves (see
Figure 2).
This stage consists of performing a fitting to initial, intermediate and final populations during the simulation. The complexity of doing this is that the parameters are shared and therefore obtaining these depends on the statistical behavior for each initial vibrational level. We followed the steps from the reference [
29], which is a generalization of the Marquardt method for multiple equations and shared parameters.
3. Results and Discussion
The first two columns of
Table 1 report the rate constants obtained by fitting the kinetic model to the TSH results for lifetimes and intermediate state dynamics (see Equation (
31a)) using
Table 1 shows the rate constants obtained from the fitting of the TSH results to the kinetic mechanism. From this table we get much information about the path followed and the time spent in each initial, intermediate and final state. If (
) or (
), the system remains more time in the intermediate state before the dissociation occurs. However, if (
) or (
) occurs, the complex breaks its bond faster than otherwise. Moreover, for the lower vibrational levels, we obtained larger values for
. This means that just after hopping, the molecule is dissociated. On the other hand, for the higher vibrational level there is a great probability of following an IVR process (
), and the others follow a VP process
. In addition, there is a competition between
and
. When
, the system is dissociated, losing a vibrational quantum number. In another case, the system is submitted to an IVR process again. In
Figure 4, we show the lifetimes obtained.
In
Figure 5, we show the process which the system followed until its dissociation.
In
Figure 5, we can see that from vibrational level
the loss of two quanta through the VP
process is significant. This causes the system to take longer to dissociate. Particularly for vibrational levels
, the predominant process is IVR-EC
.
Through the simulation, we can know at what vibrational level the system is, and we do not need algorithms to obtain it. This is very good because the TSH method itself gives us that information. In the following figure, we present the exit channel (statistically) for each initial vibrational level.
As we can see in
Figure 6 that the TSH results are in agreement with previous theoretical and experimental results. Other very important observables are
and
. The first one is the averaged rotational energy. The second one is the Br-Br angular momentum. Both quantities are calculated when the molecule is dissociated.
As we can appreciate in
Figure 7, we report our results in comparison with experimental ones and previous results.
In
Figure 8, we represent the rotational distributions of Br
after dissociation. We can see that there is a peak in the range of j = 20–24 (
Figure 8c). This is an effect that we can see from
21 to
26, with the loss of two vibrational quantum numbers. This means that, for these vibrational levels having greater rotational excitation, the system takes a longer time to dissociate. Therefore, it does not have enough energy to break the vdW bond. They are non-reactive van der Waals modes and, as a direct consequence, the lifetimes are longer.
To analyze the conservation relation, we compute the correlation matrix for all vibrational levels under study. As we discussed before for the lowest vibrational levels (
16–20), the fragmentation process occurs when the system loses one quantum energy. For those vibrational levels, more correlation is found for the
and
parameters. As we have shown in
Figure 9, the parameter
shows a strong correlation with those parameters where the process involves two losses of quantum energy. Then,
plays an important role because it links the number of quantum energy losses in the fragmentation of the system (
).
4. Conclusions
We have investigated the vibrational predissociation process for the NeBr
system by using TSH method. In our simulation, we studied a range of vibrational levels (from
to
). We found that the larger values for
correspond to the lower vibrational levels. This is in correspondence with the dissociated of the molecule occurs just after hopping surface. As we comment in
Section 3, there is a competition between VP and IVR processes. The TSH method is a robust methodology for the study of molecular fragmentation for this kind of system. This affirmation could be justified through
Figure 6,
Figure 7 and
Figure 8. In these figures, we showed our results in comparison with the previous theoretical results and the experimental ones. As we can appreciate, the agreements are very good.
Perspectives
This method is a powerful tool for dealing with this type of system and can be extended to more complex structures (e.g., more vdW interactions). We continue with this work and we are checking that the strongest coupling belongs to . In that case, we must consider fewer surfaces for dynamics than before.