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Quantum equilibration of the double-proton
transfer in a model system porphine†
Cite this: Phys. Chem. Chem. Phys.,
2020, 22, 22332
Guillermo Albareda,
Josep Maria Bofill,
Ivano Tavernelli f
*ab Arnau Riera,c Miguel González, bd
Iberio de P. R. Moreira, bd Rosendo Valero
be
bd
and
There is a renewed interest in the derivation of statistical mechanics from the dynamics of closed
quantum systems. A central part of this program is to understand how closed quantum systems, i.e.,
in the absence of a thermal bath, initialized far-from-equilibrium can share a dynamics that is typical to
the relaxation towards thermal equilibrium. Equilibration dynamics has been traditionally studied with a
focus on the so-called quenches of large-scale many-body systems. We consider here the equilibration
of a two-dimensional molecular model system describing the double proton transfer reaction
in porphine. Using numerical simulations, we show that equilibration indeed takes place very rapidly
Received 3rd June 2020,
Accepted 7th September 2020
(B200 fs) for initial states induced by pump–dump laser pulse control with energies well above the
synchronous barrier. The resulting equilibration state is characterized by a strong delocalization of the
DOI: 10.1039/d0cp02991b
probability density of the protons that can be explained, mechanistically, as the result of (i) an initial
state consisting of a large superposition of vibrational states, and (ii) the presence of a very effective
rsc.li/pccp
dephasing mechanism.
1 Introduction
There is currently a renewed interest in the derivation of
statistical mechanics from the dynamics of a closed quantum
system.1 In this approach, instead of assuming a priori that the
system is in some mixed state, such as e.g. a micro-canonical
ensemble, one describes it at all times using a pure state. One
then seeks to show that, under reasonable conditions, the
system behaves as if it was described by a statistical ensemble.
a
Max Planck Institute for the Structure and Dynamics of Matter and Center for
Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany.
E-mail: guillermo.albareda@mpsd.mpg.de, guillealpi@gmail.com
b
Institute of Theoretical and Computational Chemistry, Universitat de Barcelona,
Martı́ i Franquès 1, 08028 Barcelona, Spain
c
ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and
Technology, Castelldefels (Barcelona), 08860, Spain
d
Departament de Ciència de Materials i Quı́mica Fı́sica, Universitat de Barcelona,
Martı́ i Franquès 1, 08028 Barcelona, Spain
e
Departament de Quı́mica Inorgànica i Orgànica, Universitat de Barcelona,
Martı́ i Franquès 1, 08028 Barcelona, Spain
f
IBM Quantum, IBM Research – Zurich, 8803 Rüschlikon, Switzerland
† Electronic supplementary information (ESI) available: (S1) Dynamics of the twoproton density for the initial state C0(R1,R2;2.2a0). (S2) Dynamics of the twoproton density for the initial state C0(R1,R2;1.9a0). (S3) Dynamics of the
two-proton density for the initial state C0(R1,R2;1.75a0). (S4) Dynamics of the
two-proton density for the initial state CG(R1 + 2.2a0,R2 2.2a0). (S5) Dynamics of
the two-proton density for the initial state CG(R1 + 2.2a0,R2). (S6) Dynamics of the
two-proton density for the initial state CG(R1 + 2.2a0,R2 1.0a0). See DOI: 10.1039/
d0cp02991b
22332 | Phys. Chem. Chem. Phys., 2020, 22, 22332--22341
In this way the use of statistical mechanics can be justified
without introducing additional external degrees of freedom,
such as e.g. thermal ‘‘baths’’.
A central part of this programme has been to understand the
process of equilibration, i.e., how a constantly-evolving closed
quantum system can share aspects typical to the relaxation
towards a static equilibrium. The main insight relies on the
fact2–4 that, if measurements are limited to small subsystems or
restricted sets of observables, then ‘‘typical’’ pure states of large
quantum systems are essentially indistinguishable from thermal states. It can then be shown5,6 that under very general
conditions on the Hamiltonian and for nearly all initial states,
the system will eventually equilibrate, in the sense that an
(again, restricted) set of relevant physical quantities will remain
most of the time very close to fixed, ‘‘equilibrium’’ values.
Quantum equilibration has been recently put within reach
of experimental verification. Fueled by enormous improvements in experimental techniques it is now feasible to control
quantum systems with many degrees of freedom. This is
particularly true for the development of techniques to cool
and trap ultracold atoms with optical lattices (giving raise to
effective lattice systems)7–11 or suitable confinement potentials
(that lead to effective low-dimensional continuous systems).12–15
Similarly, systems of trapped ions16,17 allow us to precisely
study the physics of interacting systems in the laboratory.18–23
In such highly controlled settings, equilibration and thermalisation dynamics has been studied both experimentally24–28 and
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numerically, often with a focus on the so-called quenches,
i.e. rapid changes of the Hamiltonian.29–37
While the most common notion of equilibration applies to
thermodynamically large systems, we are here interested on the
more general notion of equilibration on average1 (viz., a timedependent property equilibrates on average if its value is for
most times during the evolution close to some equilibrium
value). This notion of equilibration applies also to small
systems and we use it here to characterize the dynamics of
an isolated molecular system. We consider a model system
describing the double proton transfer reaction in the electronic
ground state of porphine, a paradigmatic system in which the
making and breaking of H-bonds occurs in an anharmonic
Born–Oppenheimer potential-energy surface. Despite the
approximations involved, the two dimensional analytical
potential energy surface describing the coupled linear motions
of the two protons serves well for the proof of principle
discussed here. As it will be shown, for initial states with
energies above the synchronous double-proton transfer energy
barrier involving a large coherent sum of vibrational states, the
isomerization dynamics can reach a long-lived quasi-stationary
regime with associated physical quantities, such as the mean
position or momentum, that remain close to fixed, ‘‘equilibrium’’,
values.
2 Definitions
^
^
hAðtÞi
¼ hcðtÞjAjcðtÞi
¼
ðT
0
^
hcðtÞjAjcðtÞidt:
(1)
If the infinite-time average fluctuation of hÂ(t)i around Ā is
small, then we say that the observable A equilibrates.
This notion of equilibration is compatible with the recurrent
and time reversal invariant nature of unitary quantum
dynamics in finite dimensional systems. To see that, let us
assume a closed system whose state is described by a vector in a
Hilbert space of dimension dT and whose Hamiltonian has a
dE
P
^¼
Ek jEk ihEk j, where Ek are its
spectral representation H
k¼1
X
k;l
^ l :
ck cl eiðEk El Þt Ek jAjE
(2)
The long-time average in (1) is then:
A ¼
X
k
^ k ;
jck j2 Ek jAjE
(3)
1 Ð T iðEk El Þt
e
dt. Therefore,
T 0
if the system relaxes at all, it must be to the following density
matrix:
where we have used that dk;l ¼ limT !1
o¼
X
k
jck j2 jEk ihEk j:
(4)
As noted in ref. 29, the result in (4) can be thought as stating the
prediction of a ‘‘diagonal ensemble’’, |ck|2 being the corresponding weights. For a non-integrable system, this ensemble is
generally expected to reduce to a thermal ensemble.
Sufficient conditions for equilibration in the sense defined
above can be then defined in terms of the so-called effective
dimension. Roughly speaking, the effective dimension is
a measure of the number of significantly occupied energy
eigenstates, and is defined as:
deff 1 : ¼
Equilibration is achieved when a measurable quantity, after
having been initialised at a non-equilibrium value, evolves
towards some value and then stays close to it for an extended
amount of time. More precisely, given some observable A and a
system of finite but arbitrarily large size, if its expectation value
hÂ(t)i equilibrates, then it must do so around the infinite time
average,1
1
A ¼ lim
T!1 T
of  can be written as:
X
k
jck j4 ¼ Tr o2
(5)
In ref. 5, 6 and 38 it is shown that for any Hamiltonian with
non-degenerate gaps,§ a large effective dimension is sufficient
to guarantee that equilibration will be attained. Note that if the
initial state is taken to be an energy eigenstate, the resulting
effective dimension is one, while the one resulting from a
uniform coherent superposition of dT energy eigenstates to
different energies is dT.
Provided that a given observable equilibrates, it was recently
shown40 that the equilibration time can be related with a
dephasing time td as:
Teq B td = p/sG.
(6)
where sG is the standard deviation of the energy gaps, Ga =
Ej Ei, when weighted by their respective relevances qa, i.e.:
sG2 : ¼
X
qa Ga2 ;
(7)
a
where a A G = {(i, j):i, j A {1,. . .dE}, i a j}, and qa : ¼ jva j2
P 2
vb
b
energies and |Eki are eigenstates of Ĥ.‡ Choosing units such
that h = 1, the state at time t is then given by jcðtÞi ¼
P iE t
ck e k jEk i with ck hEk|c(0)i, and the expectation value
with va = cj*Ajici/sA, and Aij := hEi|A|Eji are the matrix elements of
A in the energy eigenbasis. The denominator sA = amax amin is
the range of possible outcomes, being amax(min) the largest
(smallest) occupied eigenvalue of Â.
‡ Note that the sum runs over dE r dT terms, since some eigenspaces can be
degenerate. If the Hamiltonian has degenerate energies, we choose an eigenbasis
of Ĥ such that the initial state, |c(0)i, has non-zero overlap with only one
eigenstate |Eki for each distinct energy.
§ Concerning the assumption of the Hamiltonian having non-degenerate gaps, it
is shown in ref. 39 that as long as there are not exponentially many degeneracies
the argument stays the same.
k
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3 Porphine model
The above introduced concepts are now used to characterize
the non-integrable model Porphine designed by Smedarchina
et al.41 to describe the switch from synchronous (or concerted)
to sequential (or stepwise) double-proton transfer.42,43 This
model accounts for the motions of two protons (labeled 1 and 2)
along coordinates R1 and R2, respectively, from the domains of the
reactant (R) to the product (P) (see Fig. 1). The PES model is41
i
2
2
U0 h
V ðR1 ; R2 Þ ¼ 4 R21 D20 þ R22 D20 4GD20 R1 R2
D0
(8)
þ 2Gð2 þ GÞU0 ;
where G is the coupling parameter which formally represents the
interaction between the two hydrogen bonds, 2D0 is the transfer
distance of each proton, i.e., the distance between its two equilibrium positions, and U0 is the barrier height for single-proton
transfer. The parameter U0 = 0.473 eV has been fitted in ref. 41 in
order to account for the experimental results of nuclear magnetic
resonance and laser-induced fluorescence measurements of
ref. 44–46. The other two parameters, D0 = 1.251a0 and G = 0.063
are based on density functional theory calculations of Smedarchina et al.47 at the B3LYP/6-31G* level. The resulting 2D model
PES is illustrated in the background of Fig. 1. The barriers are
labeled TS (‘‘transition states’’) for two alternative reaction paths.
The reaction can lead from the reactant R via alternative transitions states TS to the intermediates (I), and subsequently via the
other two TS to the product P. In addition, Fig. 1 shows a central
Fig. 1 Double proton tranfer of the model porphine. The protons move
along coordinates R1 and R2. The four snapshots represent the transfer of
the two protons from reactant (R) to product (P), sequentially along
intermediate states (I) involving four transition states (TS), or simultaneously through a second order saddle point (SP2). In the background:
Potential energy surface for the model porphine, eqn (8), adopted from
ref. 41–43. The equidistant values of the contours range from 0 eV, for
the potential minima of the reactant (R) and product (P) configurations,
to 5 eV. The corresponding energies of the local minima for the intermediates (I), of the four barriers labeled TS, and of the second order saddle
point (SP2) are 0.238, 0.600, and 1.069 eV, respectively.
22334 | Phys. Chem. Chem. Phys., 2020, 22, 22332--22341
maximum (that corresponds to a saddle point of second order)
labeled SP2. The competing synchronous reaction mechanism
leads from the reactant R via SP2 to the product P.
The model potential in eqn (8) is symmetric with respect to
the diagonals R1 = R2. It accommodates nearly degenerate
doublets of eigenstates Cv+(R1,R2) and Cv(R1,R2), with energies
below the barriers TS, plus higher excited states. We then chose
our initial state to be of the general form:
C0(R1,R2;DR) = C0,R(R1 + DR,R2 + DR),
(9)
where DR represents a shift along the diagonal R1 = R2 and
1
C0;R ðR1 ; R2 Þ ¼ pffiffiffiðC0þ þ C0 Þ is a superposition state that
2
represents the localized ground state wavefunction of the
reactant, with C0+(R1,R2) and C0(R1,R2) representing the lowest doublet (n = 0), i.e., HCn(R1,R2) = EnCn(R1,R2), where H is
the full Hamiltonian of the system, i.e.:
H(R1,R2) = TR1 + TR2 + V(R1,R2),
(10)
2 @ 2
h
the kinetic energies of the two protons,
2M @Rk2
M = mp the proton mass, and V(R1,R2) the scalar potential given
with TRk ¼
in eqn (8).
Due to the symmetry of the model potential in eqn (8), the
localized ground state wavefunction of the reactant is equivalent to the localized ground state wavefunction of the product.
Therefore, the initial states in eqn (9) are, in fact, representative
of two equivalent configurations where the two protons are
placed along the reactant–product diagonal (see Fig. 2).
The localization, eqn (9), of the reactant wavefunction may
be generated by symmetry breaking, for example, by selective
deuteration of the C4N rings for the reactant; this would not
change the PES, but it would induce localization of the reactant
wavefunction due to the slightly heavier mass that is associated
with the reactant compared to the product.42 Also, shifts of
initial wavefunctions from equilibrium to non-equilibrium
positions may be induced, for example, by means of pump–
dump laser pulse control,48 as designed by Tannor and
Rice.49–51 Essentially, the ultrashort pump pulse transfers the
molecule from the electronic ground state to an excited state.
Here, the system evolves from the original configuration until
Fig. 2 Due to the symmetry of the model potential used in this work,
i.e., eqn (8), the states in eqn (9) are representative of the two equivalent
hydrogen configurations in (a) and (b).
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it is shifted to the target position. Finally, the dump pulse
brings the wavepacket back to the electronic ground state, thus
preparing the initial state for the subsequent reaction. Analogous shifts of the original wavefunctions to nonequilibrium
positions have been demonstrated by means of laser pulse
control, by Kapteyn, Murnane, and co-workers.52
4 Equilibration time
To gain some insight into the behaviour of the effective
dimension and the equilibration time in porphine, we consider
the case of initial states defined in eqn (9) and the type of
projective position measurements:
R̂ = |R1ihR1| # I2 + I1 # |R2ihR2|.
(11)
Ð1
In eqn (11), I1ð2Þ ¼ 1 dR1ð2Þ R1ð2Þ R1ð2Þ are identity operators
in the position representation acting, respectively, in the subspaces defined by R1 and R2. Whereas the position operator in
eqn (11) involves an infinite Hilbert space, the relevant energy
subspace is finite in practice (for numerical purposes) and
hence it is possible to apply the machinery introduced above.
We evaluate the effective dimension deff in eqn (5) and
the equilibration time Teq in (6) for different values of DR A
[2.7,2.7] (see Fig. 3). For that, we first computed va = cj*Rjici/sR,
where Rij := hEi|R̂|Eji and sR = Rmax Rmin has been chosen to
pffiffiffiffiffi
be sR = 3sgs, and sgs ¼ 1 25 is the dispersion of the localized
ground state wavefunction of the reactant.
In addition to deff, which represents an equilibration criterion,
and Teq, which provides an estimate of the velocity of the
equilibration process, we found convenient to show also these
two concepts in a compact way by introducing an effective
equilibration time (or effective equilibration speed):
T~eq ¼
Teq
:
deff 1
(12)
Note that, by definition, even if Teq is small, T̃eq can still be close
to infinite for initial states being eigenstates of the observable of
interest (i.e., deff B 1).
The resulting values of deff, Teq and T̃eq are plotted in Fig. 3.
We identified three different regions: (i) in green, regions where
equilibration is expected to occur fast, (ii) in blue, regions
where equilibration is expected to occur slowly, and (iii) in
red, regions where equilibration is not expected to occur.
Interestingly, region (iii) corresponds to the regime where our
initial state can be written as the superposition of a very small
number of energy eigenstates. Note that for DR E 0 then C0 is
just the sum of two energy eigenstates C0+ and C0. The
number of energy eigenstates required to define the initial
wavefunction increases as one departs from the global minima.
Regions of slow equilibration (in blue) are the result of a nontrivial interplay between Teq and deff. Note, for example, the
relative minimum shown by Teq at DR = 0 that corresponds to
the situation where the initial wavefunction would fall equally
to the left and right wells. In regions where equilibration is
expected to occur fast (in green), the equilibration time Teq
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Fig. 3 Effective dimension deff (top panel), equilibration time Teq (mid
B
panel), and effective equilibration time T eq (bottom panel), all in dashed
red line. The results are for the initial states defined in eqn (9) and different
values of DR for the type of measurement defined in eqn (11) using a square
grid of size 151a0 with a grid spacing of 0.08a0. The Born–Oppenheimer
potential-energy surface is also plot in solid black line for reference. Three
different regions are identified in the bottom panel: (i) in green, regions
where equilibration is expected to occur rapidly, (ii) in blue, regions where
equilibration is expected to be attained slowly, and (iii) in red, regions
where equilibration is not expected to occur.
decreases almost monotonically. This is due to the fact that the
barrier leads to a very effective dephasing mechanism at
energies close or above the global maximum.
It is important to emphasize that the quantities deff, Teq and
T̃eq are to be understood only as qualitative descriptors of
equilibration, as they are defined using rather heuristic arguments. Therefore, as it will be shown in the next section, these
quantities do not provide an accurate quantitative estimation of
the equilibration time, but do provide us with a good guess.
5 Equilibration dynamics
We now aim to study the dynamics that leads to equilibration
for initial states induced by pump–dump laser pulse control of
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the form in eqn (9). Specifically, we choose DR = 2.2a0 such
that the initial state lies in the rapid equilibration regime
(see Fig. 3). The mean energy of this initial state is rather high
(4.885 eV) and well above the TS and SP2 barriers (respectively
0.600 eV and 1.069 eV).
Starting with C0(R1,R2;2.2a0) (see Fig. 4a), the initial synchronous mechanism of the first forward reaction is characterized by a rapid dispersion of the wavepacket and relief
reflections of the broadened wavepacket from wide regions of
the steep repulsive wall of the PES close to the minimum
of the product that leads to the switch from the sequential to
the synchronous mechanism (see Fig. 4b). This situation
is reminiscent of the near field interference effect arising
when periodic diffracting structures are illuminated by highly
coherent light or particle beams.53
The time dilation supported by continuous wavepacket
dispersion leads to a strong proton delocalization. An apparently
‘‘chaotic’’ flux is however coherently directing the recovery of the
concerted double proton transfer at t \ 100 fs. Due to the
dephasing between partial waves, the grid structure of the probability density associated to the sequential double proton transfer
progressively dilutes into what is reminiscent of a stationary state,
showing a series of minima disposed perpendicular to the diagonal
R1 = R2 (see Fig. 4c). The long-lived quasi-stationary state, formed
already at B150 fs, lasts beyond 500 fs (see Fig. 4d) and it gives rise
to the equilibration of the position operator in eqn (11) as well as of
the momentum operator P̂ = |P1ihP1| # I2 + I1 # |P2ihP2|. The full
evolution dynamics can be found in S1 (ESI†).
The above dynamics result into an effective equilibration of
the phase-space variables. In Fig. 5 (top panel) we show the
Fig. 4 Dynamics of the two-proton density |C(R1,R2,t)|2 at four different
time snapshots for the initial state C0(R1,R2;2.2a0). The initially welllocalized nuclear density gives way later to a strong proton delocalization
along the synchronous pathway that is maintained for at least 1 ps.
22336 | Phys. Chem. Chem. Phys., 2020, 22, 22332--22341
Paper
Fig. 5 Equilibration of the position (solid black line) and momentum
(dashed blue line). In the inset: Ensemble average of the phase space as
a function of time. Upper panel: The initial state, C0(R1,R2;2.2a0), has an
energy of 4.885 eV, i.e., 3.7 eV above the SP2. Middle panel: The initial
state, C0(R1,R2;1.9a0), has an energy of 1.905 eV, i.e., 0.8 eV above the
SP2. Bottom panel: The initial state, C0(R1,R2;1.75a0), has an energy
of 1.185 eV, i.e., 0.1 eV above the SP2. The values of deff, Teff, and
B
T eff for the three different initial conditions DR(a0) = {2.2,1.9,1.75}, are
respectively deff = {29.89,33.39,48.2}, Teq = {35.12 fs,59.27 fs,72.75 fs}, and
B
T eq = {1.17 fs,1.77 fs,1.51 fs}.
differences R% hR̂(t)i and P% hP̂(t)i as a function of time for the
initial state C0(R1,R2;2.2a0). Equilibration is achieved once these
differences become small ({1) and stay small for a lapse of time
that is comparable to the relevant time-scale of the dynamics.54
The equilibration of the position and momentum yields a
phase-space portrait (see the inset in Fig. 5 top panel) that is
reminiscent of the phase-space portrait of a damped harmonic
oscillator. Interestingly, this phase-space dynamics is attained
here without any loss (nor gain) of energy.
In Fig. 5 (middle and bottom panels) we show the dynamics
of R% hR̂(t)i and P% hP̂(t)i for another two different initial
states, viz., C0(R1,R2;1.9a0) and C0(R1,R2;1.75a0), with energies 1.905 eV and 1.185 eV (i.e., B0.8 eV and B0.1 eV above the
SP2 barrier) respectively. As we decrease DR and thus approach
the equilibrium position DR = 0, the energy of the initial state
gets closer to the top of the synchronous barrier. This results
into a dynamics that departs more and more from an equilibration process. Note, for instance, that while the phase-space
portrait in Fig. 5 (mid panel) is still reminiscent of some
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damped dynamics, the phase-space diagram in Fig. 5 (bottom
panel) is clearly not, as it does not converge to equilibrium. For
even smaller DR values, the initial states fall down into
the minimum of the well, where equilibration is no longer
expected to occur. The full dynamics for the initial states
C0(R1,R2;2.2a0), C0(R1,R2;1.9a0), and C0(R1,R2;1.75a0) can
be found in S1, S2, and S3 (ESI†) respectively.
6 Discussion
One might think that provided there are at least two degrees of
freedom with non-negligible coupling between each other, one can
expect one to act as a bath for the other and vice versa, both leading
to the damping of the mutual dynamics. As a result, one may
encounter that observables reach some equilibrium value. While
this hypothesis might sound reasonable, it is incorrect. First of all,
two particles with identical masses (e.g., two protons) can hardly act
one as a bath for the other. Second, even a system with a single
degree of freedom can equilibrate. See for example the works in
ref. 54 and 55. In order a quantum system to equilibrate there is no
need for a bath. That is the main point that differentiates equilibration from thermalization. Also, working with initial conditions
far-from-equilibrium within anharmonic regions of the potentialenergy surface does not guarantee equilibration. Otherwise
equilibration should be observed for almost every molecule
out-of-equilibrium. This is definitely not the case.
To gain more insight into the physical mechanism that
yields equilibration, we here take advantage of the interpretation value of Bohmian mechanics.56–61 It is possible to reinterpret the quantum formalism as describing particles following
definite trajectories, each with a precisely defined position at
each instant in time. In this interpretation, called Bohmian
mechanics, the trajectories of the particles are guided by the
wavefunction. This allows for phenomena such as double-slit
interference, as has been investigated experimentally for single
photons.62–64 Note that this is different from semi-classical
trajectories,65–67 other generalized hydrodynamic approaches,68
and also from the Feynman path formalism of quantum
mechanics.69 For example, in contrast to the Feynman formalism,
Bohmian mechanics says that each quantum particle in a given
experiment follows a trajectory in a deterministic manner. Thus,
much of the intuition of classical mechanics is regained.70 In any
case, let us insist that we use an ensemble of exact Bohmian
trajectories (evaluated from the wavefunction solution of the timedependent Schrödinger equation) with the only intention to
provide a different viewpoint on the process of equilibration that
is more ‘‘digestible’’ for an adherent of classical mechanics and
that it is at the same time compatible with the expectation values
associated to the exact quantum dynamics.
We consider an ensemble of Bohmian trajectories {Ra(t)} =
a
{R1(t),Ra2(t)} initially sampled from the probability distribution
P(R1,R2) = |C0(R1,R2,2.2a0)|2 and whose time evolution is
defined as
ðt
(13)
Ra ðtÞ ¼ Ra ðt0 Þ þ vðRa ðt 0 Þ; t 0 Þdt 0 ;
t0
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Fig. 6 Time evolution of a randomly chosen set of five hundred Bohmian
trajectories Ra1(t) sampled from |C0(R1,R2,2.2a0)|2.
where
vðRa ðtÞ; tÞ ¼
h
1 @C 1 @C
;
:
Im
M
C @R1 C @R2 Ra ðtÞ
(14)
In Fig. 6 we show the time evolution of an ensemble of
five hundred Bohmian trajectories Ra1(t) sampled from
|C0(R1,R2,2.2a0)|2. An initially orchestrated oscillatory dynamics
with an amplitude that extends over the two wells is rapidly
(at around B50 fs) substituted by apparently incoherent oscillations with a very small amplitude. Not much later, at around
150 fs, the trajectories fall into a dynamics that is highly
localized and homogeneously distributed. Of particular interest
is the fine structure of the Bohmian trajectories, characterized
by high frequency components and rapid shifts in their direction.
This quantum mechanical motion arises from strong timedependent quantum force fields (originating from quantum
interferences) which essentially dominate the dynamics and are
responsible for shifting the direction of trajectories away from
that dictated by the classical force field.
As already anticipated in Section (4), the above dynamics can
be seen as the result of a very efficient dephasing mechanism
that leads to a diagonal ensemble. The barrier in between the
two wells induces a continuous time dilatation supported by
wavepacket dispersion that yields strong interference effects
between different parts of the wavepacket. Eventually, these
interferences are responsible for freezing the transfer of the two
protons and lead to the localization of the corresponding
trajectories within the concerted pathway (see Fig. 6). This
equilibration dynamics corresponds to the picture where the
two protons become localized at certain configurations along
the concerted transfer reaction path, as if they were associated
to a stationary state.
In view of the above results, we may conclude that there are
two effects that must coexist to find equilibration:
(i) Initial states consisting of a large superposition of energy
eigenstates.
(ii) The presence of a very effective dephasing mechanism.
Note that these two conditions are not easily met in general,
but can be found for the porphine molecule and probably also
for other related molecules, under certain initial conditions.
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More specifically, for the porphine model considered in this
work, condition (i) is met for initial states with energies
well above the SP2 barrier, and the dephasing mechanism in
condition (ii) is originated by the SP2 barrier.
Concerning condition (i), there are other types of initial
states, different from those defined in eqn (9), that can be
considered as potentially leading to the equilibration of the
canonical variables R̂ and P̂. Very general initial conditions can
be defined, for example, by introducing the wavefunction
CG(R1 + DR1,R2 + DR2), where CG(R1,R2) represents a Gaussian
approximation to the localized ground state wavefunction
pffiffiffiffiffi
of the reactant, with a dispersion sgs ¼ 1 25 and centered
at the origin of coordinates. In particular, for DR1 = DR2, the
wavepacket CG(R1 + DR1,R2 + DR2) is centered along
the diagonal R1 = R2 that leads from one Intermediate to the
other (i.e., perpendicular to the diagonal R1 = R2 that leads from
the reactant to the product). Due to the symmetry of the model
potential in eqn (8), the state CG(R1 + DR1,R2 DR1) around the
Intermediate minima, may correspond to four equivalent hydrogen atoms distributions (see Fig. 7). Starting from the initial
state CG(R1 + 2.2a0,R2 2.2a0), the resulting dynamics is
analogous to that shown in Fig. 4 but the probability density
that corresponds to Fig. 4c and d extends now along the
diagonal R1 = R2 instead of R1 = R2 and there is a notable
increase of the probability density along the sequential paths.
The associated behaviour of hR̂i and hP̂i is not significantly
different from the results in Fig. 5. Specifically, the results in
Fig. 8 illustrate that the present definition of ‘‘quantum equilibration’’ in eqn (1) allows relaxation to states that are far from
the traditional equilibrium scenarios, which are centered close
Fig. 7 Due to the symmetry of the model potential defined in eqn (8), the
initial states CG(R1 + DR1,R2 DR1) localized around the intermediate
minima may correspond to the four equivalent proton distributions in
(a–d).
22338 | Phys. Chem. Chem. Phys., 2020, 22, 22332--22341
Paper
Fig. 8 Equilibration of the position (solid black line) and momentum
(dashed blue line) for the initial state CG(R1 + 2.2a0,R2 2.2a0). In the
inset: Ensemble average of the phase space as a function of time. The
initial state, CG(R1 + 2.2a0,R2 2.2a0), has an energy of 5.10 eV, i.e., 4.03 eV
above the SP2.
to the potential minima. The full dynamics for the initial state
CG(R1 + 2.2a0,R2 2.2a0) can be found in S4 (ESI†).
Other initial states, with resulting dynamics occurring outside the two main diagonals (concerted mechanism of proton
transfer), can be also considered. Should the wavepacket be
displaced along only one degree of freedom, one would likely
observe the sequential mechanism of the proton transfer. In
Fig. 9 and 10, we show the dynamics of hR̂i and hP̂i (and the
corresponding expectation value dynamics of hR̂1,2i and hP̂1,2i)
that result from the initial states CG(R1 + 2.2a0,R2) and CG(R1 +
2.2a0,R2 1.0a0) respectively. The full dynamics for the initial
states CG(R1 + 2.2a0,R2) and CG(R1 + 2.2a0,R2 1.0a0) can be
found respectively in S5 and S6 (ESI†).
Fig. 9 Top panel: Equilibration of the position (solid black line) and
momentum (dashed blue line) for the initial state CG(R1 + 2.2a0,R2). In
the inset: Ensemble average of the phase space as a function of time. The
initial state, CG(R1 + 2.2a0,R2), has an energy of 3.08 eV, i.e., 2.01 eV above
the SP2. Bottom panel: Dynamics of hR̂1,2i and hP̂1,2i.
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Fig. 10 Top panel: Equilibration of the position (solid black line) and
momentum (dashed blue line) for the initial state CG(R1 + 2.2a0,R2
1.0a0). In the inset: Ensemble average of the phase space as a function of
time. The initial state, CG(R1 + 2.2a0,R2 1.0a0), has an energy of 2.85 eV,
i.e., 1.78 eV above the SP2. Bottom panel: Dynamics of hR̂1,2i and hP̂1,2i.
As it is apparent from all the above dynamics, the symmetry
of the initial state does not seem to play a crucial role. Instead,
it seems to be the energy of the initial state in comparison with
the height of the SP2 barrier what determines whether or not
equilibration will take place. As the energy of the initial
wavepacket is decreased and gets closer to the height of the
SP2 barrier, it takes longer to equilibrate. Overall, this is in
consonance with the theoretical predictions in Fig. 3.
In general, our results are in accordance with the idea that
the diagonal and microcanonical ensembles give the same
predictions for the relaxed value,29 i.e.:
X
X
^ k ¼ 1
^ a ;
A ¼
jck j2 Ek jAjE
Ea jAjE
(15)
N
E0 ;DE a2DE
k
where E0 is the mean energy of the initial state, DE is the half-width
of an appropriately chosen energy window centred at E0, and the
normalization is the number of energy eigenstates with energies in
the window [E0 DE,E0 + DE]. Thermodynamical universality is
evident in this equality: although the left-hand side depends on the
details of the initial conditions through the set of coefficients ck, the
right-hand side depends only on the total energy, which is the same
for many different initial conditions. Explanations for the above
universality have been proposed based on at least three possible
scenarios.29 Here we have proposed a possible explanation based
on (i) and (ii) above. Specifically, in our example, E0 and DE do
depend on the height of the SP2 barrier.
7 Conclusions
To summarize, we have shown that quantum equilibration,
mostly studied for many-body systems made of a large number
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of particles, can also play a role in isolated molecular systems
that involve a few number of degrees of freedom. Specifically,
we have seen that a two-dimensional model system describing
the double proton transfer in porphine can present equilibration dynamics for a wide range of far-from-equilibrium
initial states that are compatible with a pump–dump preparation scheme. We have seen that the resulting equilibration state
is reached in B200 fs and that it is associated with zero mean
position and momentum of the two protons. This equilibration
state is characterized by a strong delocalization of the probability density of the protons, which can appear either along the
concerted or sequential transfer paths depending on the symmetry of the initial conditions.
By picking up a number of very general initial conditions, we
have shown that the symmetry of the initial state does not play
a crucial role. Instead, it is the energy of the initial state in
comparison with the height of the SP2 barrier what determines
whether or not equilibration will take place and how long it will
take to develop. We have thus concluded that there are two
main facts that must coexist to find equilibration:40 (i) initial
states consisting of a large superposition of energy eigenstates,
and (ii) the presence of a very effective dephasing mechanism.
For the porphine model considered in this work, condition
(i) is met for initial states with energies well above the SP2
barrier, and it is the SP2 barrier itself that facilitates the
fulfillment of (ii). This conclusion is compatible with the idea
that the diagonal and microcanonical ensembles give the same
predictions for the relaxed value for a wide range of initial
conditions.29
Relying on the de Broglie–Bohm interpretation of quantum
mechanics, we have shown that the equilibration process in
porphine can be associated with the picture where, due to strong
interferences, the two protons become localized at certain configurations along the concerted and sequential transfer paths as if
they where associated to a stationary state. This result may help,
for example, in the interpretation of experimental data. That is,
the close connection between Bohmian mechanics and weak
values64,71,72 provides an operational interpretation of the above
described equilibration mechanism based on trajectories that
could be put within reach of experimental verification.62,63
Let us insist that Bohmian trajectories have been introduced in
this work with the intention to gain insight into the physical
mechanism that yields equilibration. However, the conclusion
that equilibration is actually taking place is based solely on the
analysis of the time-evolution of the position and momentum
expectation values (which are not dependent on the particular
interpretation choice).
Witnessing equilibration dynamics in the laboratory for
an arbitrary type of molecules could be feasible thanks to the
development of femtosecond spectroscopy73,74 and to the most
recent technological advances that allow characterizing wavepacket dynamics at an experimental level in a wide variety of
molecular systems.75–77 However, whether quantum equilibration dynamics in an isolated porphine molecule might be
actually observed will strongly depend on the validity of the
approximations considered in the employed model. The model
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porphine considered here does not include the effect of the
heavy atom vibrations of the molecular scaffold,78 and therefore,
while it served well for a proof of principle, the excitation of
specific heavy atom vibrations might have a sizable influence on
the proton transfer dynamics. Yet, avoiding possible effects coming from the scaffold dynamics could be possible. Because of the
bistability in the position of the two protons, which can occupy
different, energetically equivalent positions (tautomerization),
porphines have been devised as molecular switches.79–82 In this
respect, there exist an extensive literature (see e.g. ref. 80) on the
control of the scaffold structural freedom by means of surface
anchoring81,83 or molecular assembling.82,84
Finally, let us mention a recent work where the third law of
thermodynamics is investigated for a 1D model boron rotor
B13+.55 In ref. 55 the entropy production reaches a stationary
(or quasi-stationary) value (3.017). This value, however, does
not correspond to the equilibrium value (3.401), which is far
beyond the variance of the fluctuations (0.101). In the authors
words: ‘‘this points to incomplete equilibration’’. Furthermore,
this quasi-equilibration process occurs in approximately 10 picoseconds. At the time scale of a few picoseconds it is difficult to
imagine that the coupling of the rotation/pseudo-rotation of the
boron rotor to the rest of degrees of freedom will not modify the
dynamics.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
The authors acknowledge financial support from the Spanish
Ministerio de Economı́a y Competitividad (Projects No. CTQ201676423-P and No. PID2019-109518GB-I00), the Generalitat de
Catalunya (Project No. 2017 SGR 348), and the Spanish Structures
of Excellence Marı́a de Maeztu program (through grant
MDM-2017-0767). G. A. acknowledges also financial support
from the European Union Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie Grant Agreement No. 752822. Open Access funding provided by the Max
Planck Society.
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