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Dynamics
Ehrenfest statistical dynamics in chemistry: study of decoherence effects José Luis Alonso, Pierpaolo Bruscolini, Alberto Castro, Jesús ClementeGallardo, Juan Carlos Cuchí, and Jorge Alberto Jover-Galtier J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00511 • Publication Date (Web): 26 Jun 2018 Downloaded from http://pubs.acs.org on June 27, 2018
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Journal of Chemical Theory and Computation
Ehrenfest statistical dynamics in chemistry: study of decoherence eects J. L. Alonso,
†,‡
P. Bruscolini,
†,‡
¶
A. Castro,
J. Clemente-Gallardo,
†,‡
J. C. Cuchí,
§
∗,†,‡,k
and J. A. Jover-Galtier
†Departamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, ES 50009
Zaragoza, Spain ‡Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, Mariano Esquillor s/n, Edicio I+D, ES 50018 Zaragoza, Spain ¶BIFI-Fundación ARAID, Universidad de Zaragoza, Edicio I+D-Campus Río Ebro, Mariano Esquillor s/n, ES 50018 Zaragoza (SPAIN) §Departament d'Enginyeria Agroforestal, ETSEA-Universitat de Lleida, Av. Alcalde Rovira Roure 191, ES 25198 Lleida, Spain kCentro Universitario de la Defensa de Zaragoza, Academia General Militar, crta. de Huesca s/n, ES 50090 Zaragoza, Spain E-mail: jorge.jover@bi.es Abstract
states of the electronic Hamiltonian.
In previous works, we introduced a geometric
1
route to dene our Ehrenfest Statistical Dy-
Introduction
namics (ESD) and we proved that, for a simple toy-model, the resulting ESD does not pre-
The Schrödinger equation for a combined sys-
serve purity.
tem of electrons and nuclei is generally too com-
We now take a step further: we
basis
in the
plex and involves too many degrees of freedom
Ehrenfest Statistical Dynamics (ESD) model
to be solvable, neither analytically nor by nu-
by considering some uncertainty in the degrees
merical methods.
of freedom of a simple but realistic molecu-
made, one of the most important and success-
lar model, consisting of two classical cores and
ful being the classical approximation for a sub-
one quantum electron. The Ehrenfest model is
set of the particles.
sometimes discarded as a valid approximation
dynamical (HQCD) models are therefore neces-
to non-adiabatic coupled quantum-classical dy-
sary and widely used.
namics because it does not describe the deco-
we discussed how these HQCD models are built.
herence in the quantum subsystem.
However,
Most approaches can be described in two steps:
any rigorous statistical analysis of the Ehren-
rst, a partial deconstruction of the quantum
fest dynamics, such as the described ESD for-
mechanics (QM) of the total system (electrons
malism, proves that decoherence exists. In this
and nuclei) which simplies the model, and
article, decoherence in ESD is studied by mea-
then a reconstruction that aims to recover the
suring the change in the quantum subsystem
essential properties of the total Schrödinger
purity and by analysing the appearance of the
equation lost in the deconstruction process.
investigate decoherence and pointer
basis
Approximations need to be
Hybrid quantum-classical
116
In a previous work
2
to which the system decoheres,
HQCD models in the literature present at
which for our example is composed by the eigen-
least two levels of deconstruction. The rst one,
pointer
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called Born-Oppenheimer molecular dynamics
construction proceeds by allowing the system
(BOMD), is far away from the total Schrödinger
to perform certain specially designed stochas-
equation for electrons and nuclei, as electrons
tic jumps between adiabatic states.
are assumed to remain in the ground state for
jumps cannot however be well justied from
all times.
rst principles.
The second one, closer to the total
These
Another relevant algorithm,
Schrödinger equation, is called Ehrenfest Dy-
widely used in Molecular Dynamics (MD), is
namics (ED). In ED the nuclei are still classical
the decay of mixing formalism by Truhlar and
(as in BOMD) but the electrons are allowed to
coworkers.
populate excited states. A recent review on the
tion stops at the ED and the reconstruction
topic
is developed by adding terms in the dynam-
15
discusses these two approaches. The de-
11,16
In this method, the deconstruc-
construction simplies the model by forcing the
ics which introduce decoherence.
separability of the quantum states of the nuclei
malism, one considers an ensemble of hybrid
(which are later considered as classical) and the
quantum-classical systems and computes the
electrons,
even if this separability cannot be
dynamics of the quantum subsystem using two
preserved exactly by the evolution of an inter-
components: one arising as the fully coherent
acting quantum system. Therefore the decon-
solution to the Liouville-von Neumann equation
struction ignores entangled states of the quan-
and one,
tum nuclear and electronic degrees of freedom.
decoherence (see expression (34) below).
17
One could naively conclude that this implies a
ad-hoc,
In this for-
that incorporates electronic
There are other studies and proposals to
unitary evolution for the electronic subsystem,
tackle decoherence eects.
thus being the cause for the preserved purity in
would like to mention Bittner and Rossky,
the ED evolution. This reasoning is erroneous,
Neria and Nitzan,
as generic ED evolution is not unitary for the
and Subotnik.
electronic subsystem.
2
7,8
22
Among them, we
Schwartz and coworkers
21 23
Particularly in the last two
Nevertheless, as proved
approaches, one can nd the idea of consid-
in Theorem 1 of the cited work, ED does pre-
ering statistical mixtures of hybrid quantum-
serve purity, and no decoherent eects can be
classical systems in order to study the problem
found.
of decoherence. For example, Subotnik used in
The second step in the denition of HQCD
his works the formalism of the partial Wigner
models, the reconstruction of the dynamics, is
transform, introduced in the context of MD by
much more complicated.
Nielsen and coworkers,
Many dierent pro-
2426
to represent the hy-
posals tackle the diculty of re-incorporating
brid quantum-classical system; by adding some
into HQCD models the essential properties of
extra variables into the picture, it is possible
the total Schrödinger equation that have been
to describe in an ecient way the decoherence
lost.
eects of some systems.
One of these properties is the decoher-
ence phenomenon in the electronic subsystem.
A completely dierent route is the one fol-
2730
In the context of HQCD, decoherence is under-
lowed by Abedi and coworkers,
stood as the fact that the neglected wave func-
scribes an exact factorisation of the total wave
tions in the classical limit (i.e.
those of the
function. Then, the classical limit for the nu-
nuclei) rapidly lose overlap in time, leading to
clear part can be taken, and recently the ap-
the destruction of superpositions in the quan-
pearance of decoherence in the resulting dy-
tum subsystem, i.e. forcing the electronic wave
namics has been analyzed.
functions into a mixture of pure states. pure states form the so-called the quantum sybsystem.
1820
3,5
which pre-
31
Such
In this work, we take a step back from these
for
approaches, and examine how uncertainty in
pointer basis
the initial conditions may aect Ehrenfest dy-
Dierent approaches aim to address the re-
namics. In order to adress this issue, a formula-
construction of quantum dynamics with quite
tion of dynamics in terms of statistical distribu-
dierent tools. In J. C. Tully's Trajectory Sur-
tions is necessary. In previous works
face Hopping (TSH) algorithms,
for example,
troduced a geometric route to dene our Ehren-
the deconstruction goes to BOMD and the re-
fest Statistical Dynamics (ESD). We now take
12
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1,2
we in-
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Journal of Chemical Theory and Computation
• NQ
a step further by investigating decoherence in ESD. It is important to notice that, due to the
external (valence) electrons,
interaction between classical cores and quan-
•
tum electrons in a molecular model, cores act the molecule. In this setting, it is thus possible
decoherence hypothesis , 1820
~r1 , . . . , ~rNQ
tors
and
respectively.
system is given by (atomic units will be used hereafter)
fest dynamics causes the existence of a pointer basis for the quantum subsystem.
H=−
we proved that, for a
simple toy-model,the resulting dynamics does not preserve purity.
~ 1, . . . , R ~N , R C
The Hamiltonian operator for this molecular
In
other words, the interaction described in Ehren-
2
cores that
and cores will be denoted by 3-dimensional vec-
tum system selects a set of orthogonal projec-
In our previous work
NC
In the following, the coordinates of electrons
by
which the environment of a non-isolated quantors onto the Hilbert space of the system.
which are coupled to the
will be considered classical.
as an environment to the quantum subsystem of to consider the
quantum particles, typically the most
Now, and after carefully
NC X 1 ∇2J + He (R), 2M J J=1
(1)
tem. We will see that decoherence does occur in
~ 1, . . . , R ~ N ) ∈ R3NC , and where R = (R C MJ is the mass of the J -th core and ∇2J is the ~ J . The opLaplacian operator with respect to R erator He (R), called the electronic Hamiltonian
ESD and we will compute the changes in purity
of the molecule, depends parametrically on the
and the appearance of pointer basis associated
core positions:
with
dening the meaning of decoherence in the context of HQCD models, we apply ESD to a realistic model: a diatomic, isolated molecular sys-
to the decoherence phenomenon.
NQ 1 X 2 X ZJ ZK He (R) = − ∇ + ~J − R ~ K| 2 j=1 j J 0,
(36)
hφj (R)|φk (R0 )i ' δjk ,
∀R, R0 ∈ D.
(38)
Z = 1 for the simulation of the ionised We have chosen the value α = 3 a.u.,
sider a basis for the electronic states formed by
which reproduces approximately the experi-
eigenstates of the electronic Hamiltonian at a
mental properties of the Na2 neutral molecule.
reference position
We take dimers.
It is useful for the problem at hand to con-
R0
of the cores:
With this new potential, it is possible to con-
B = {φj := φj (R0 ) | j = 0, 1 . . .}.
sider the eigenvalue equation for the electronic
(39)
Hamiltonian: For any
He (R)|φj (R)i = Ej (R)|φj (R)i, j = 0, 1, 2 . . .
He (R)
(37) with
E0 (R) ≤ E1 (R) ≤ . . .
R ∈ D,
the electronic Hamiltonian
is approximately diagonal in this basis,
which simplies the computations.
The eigenstates of
The
next
step
in
the
description
of
the
the electronic Hamiltonian are assumed to be
problem is the analysis of the dynamics de-
normalised.
termined by the Ehrenfest model (5).
Figure 1a shows the rst eigen-
values of this operator.
For each set
R,
the
given initial conditions
(R0 , P0 , ψ0 ),
For
integra-
eigenvectors of the electronic Hamiltonian form
tion of the Ehrenfest model gives the evolution
a suitable basis for the Hilbert space of elec-
(R(t), P (t), ψ(t))
tronic states.
degrees of freedom. Initial parameters
It is possible to simplify the problem by re-
P0
stricting the positions of the cores. In the folafter denoted by
D,
are taken along the
for all
lowing, the set of positions of the cores, here-
of the quantum and classical
x axis,
R0 and so that R(t) ∈ D
t ≥ 0.
With these tools, it is interesting to analysis
is restricted to positions
the adiabaticity of the Ehrenfest model when
axis, and such that the intercore
applied to the described example. For this pur-
distance varies only between 11 and 17 a.u. Fig-
pose, let us choose as initial quantum state any
R
along the
x
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Journal of Chemical Theory and Computation
of the eigenstates of the electronic Hamiltonian (j) at R0 , i.e. ψ (0) = φj (R0 ). We have computed (j) numerically the projections hφk (R(t))|ψ (t)i
t
for dierent values of these projections for
with
right hand side turns out to be negligible.
Figure 1c represents
j = k = 0, 1, 2.
3.2 Statistical uncertainty for a single dimer: Decay of purity and pointer basis
We
j = k . As for each R ∈ D basis {φj (R)} are orthogonal
imately value 1 when
to each other, the rest of projections are approximately zero.
How-
ever, in our examples, the second term in the
have found that these projections take approxelements in the
d~Jjk (R) the non-adiabatic couplings.
We are interested in the appearance of decoher-
This leads to the following
ence eects: purity changes and pointer basis.
approximation
For this reason, let us consider that the uncertainty in the initial conditions of our dimer is
ψ (j) (0) = φj (R0 ) ⇒ hφk (R(t))|ψ (j) (t)i ' δjk , ∀t.
described by the following probability density
(40)
function:
We can conclude that, for the considered ex-
FQC (ξ, ψ; 0) =
ample, the system behaves in an approximately adiabatic way.
N X
1 N
! δ(P − Pj,0 )
(44)
j=1
δ(R − R0 )δ(ψ − ψ0 ).
Summing up, the two relevant approximations found for this model are (38) and (40).
R = (R1 , R2 )
with
lution of generic initial conditions by ED (5) (as
tion and momenta, respectively, of the two cores
long as positions
R(t)
in our model, and
of the cores stay in the
ψ
the posi-
the quantum state of the
valence electron. Observe that the only uncer-
D):
allowed set
and
P = (P1 , P2 )
Both can be combined to approximate the evo-
tainty is assumed in the classical degrees of free-
ψ(0) =
X
c j φj ,
dom, while the quantum subsystem is taken in
c0 , c1 , . . . ∈ C
the initial state
j
⇒ ψ(t) '
X
Z
cj exp −i
t
Ej (R(t0 ))dt0 φj .
x
(41)
taken along the
(neither exact nor approximate) is not a general
that a value of
property of ED, as the Ehrenfest equations inis written at each
ψ(t) =
t as X
1
axis for the given quantum state
plained above, initial momenta
It is important to notice that adiabaticity
troduces non-adiabatic couplings.
For simplicity, the
R0
the equilibrium position of the cores along the
0
j
ψ0 ∈ MQ .
initial ionic positions are also xed, with
MQ .
Pj,0
As ex-
should be
x axis. Numerical results show N = 41 is high enough to pro-
vide good statistical results.
If the state
The evolution of the probability distribution can be written as
cj (t)φj (R(t)),
(42)
N 1 X δ(R − Rj (t)) FQC (ξ, ψ; t) = N j=1
j then from the Ehrenfest equation (5) it is im-
δ(P − Pj (t)) δ(ψ − ψj (t)) ,
mediate to compute that
(45)
d i cj (t) = Ej (R(t))cj (t) dt " # X X P~J jk −i ck (t) · d~J (R(t)) , M J J k ∂ jk ~ dJ (R) = φj (R) φ (R) . ~J k ∂R
From this expression we can write the corre-
(43)
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sponding density matrix as:
following
Z t 1 0 0 E0 (Rj (t ))dt φ0 ψj (t) ' √ exp −i 2 0 Z t 1 0 0 + √ exp −i E1 (Rj (t ))dt φ1 , 2 0
|ψihψ| ρ(t) = dµQC (ξ, ψ)FQC (ξ, ψ; t) hψ|ψi MC ×MQ Z
=
N 1 X |ψj (t)ihψj (t)| . N j=1 hψj (t)|ψj (t)i
(49)
(46)
for
Let us now analyse some simple examples,
Substituting in (46) it is possible
subsystem:
changes in purity and the existence of a pointer
1 ρ(t) ' (|φ0 ihφ0 | + |φ1 ihφ1 | 2 ei∆1 (t) + ei∆2 (t) |φ0 ihφ1 | + 2 e−i∆1 (t) + e−i∆2 (t) + |φ1 ihφ0 | , 2
basis for the molecule. As a rst approach to the problem, consider in the initial probability distribution (44) that the initial quantum state is an eigenstate of the
ψ0 = φk ∈ B .
j = 1, 2.
to estimate the density matrix of the quantum
which will become useful in order to predict the
electronic Hamiltonian, i.e.
Page 12 of 19
An
approximation to the density matrix of the sys-
(50)
tem can be obtained by substituting (41) in (46) for every set of initial conditions
(R0 , Pj,0 , ψ0 ).
where we have dened the quantities
Observe that, due to the small variation of the eigenstates of
He (R)
Z
and to the approximate
∆j (t) =
adiabaticity of the dynamics, eigenstates of the
t
gj (t0 )dt0 ,
j = 1, 2.
0
(51)
gj (t) = E1 (Rj (t)) − E0 (Rj (t)),
electronic Hamiltonian are approximately constant (except for a global phase). As a conse-
gj (t)
quence, the purity of the quantum subsystem is
Observe that
close to 1 for all the evolution. Observe that,
ground state and the rst excited state energies
although no decoherence seems to occur in this
for each initial conditions at each time t. Notice
example, this is the rst hint of the existence of
that the density matrix
a pointer basis for the molecule.
sum of two projectors, each one projecting onto
extract some conclusion on the behaviour of the
N =2
Even if the dierence
tribution:
1 δ(P − P1,0 ) + δ(P − P2,0 ) 2 δ(R − R0 ) δ (ψ − ψ0 ) ,
exponent
(52)
|g1 (t) − g2 (t)| between the
ψ1 (t), ψ2 (t), equations
the with
(R0 , P1,0 , ψ0 ), (R0 , P2,0 , ψ0 )
initial
|ψ1 (t)i
|ψ2 (t)i
and
scription of the quantum state. We can verify this assumption by computing the spectral de-
ρ(t),
dened
in (15). The dependence of its eigenvalues with
(48)
time is presented in Figure 2a. We notice that
quantum
determined
Thus, the dif-
composition of the density matrix
1 ψ0 = √ (φ0 + φ1 ) . 2 (41),
non-negligible.
will become periodically relevant for the de-
where the initial quantum state is chosen as
to
∆j (t)
ference between the vectors
(47)
Ehrenfest
j = 1, 2.
gaps is very small, integration in time makes the
FQC (R, P, ψ; 0) =
jectories
corresponds to the
χj (t) = e−i∆j (t) φ0 + φ1 ,
and we take the fol-
lowing expression for the initial probability dis-
According
ρ(t)
the subspace generated by the vector
Consider now a second example. In order to system, we choose
are the gaps between the
by
the number of eigenvalues dierent from zero
tra-
is indeed a function of time, depending on the
the
orthogonality of vectors
conditions
χ1 (t)
and
χ2 (t).
A more general analysis can be done by ex-
respectively, are the
tending the above computation to generic values of
N
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in the initial distribution (47).
If
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0.8 0.6 0.4 0.2 0
1
Eigenvalues
1
Eigenvalues
1
Eigenvalues
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Journal of Chemical Theory and Computation
0.8 0.6 0.4 0.2
0.8 0.6 0.4 0.2
0 0
10
20
30
40
50
0 0
Time (1000 a.u.)
10
20
30
40
50
0
10
Time (1000 a.u.)
(a)
20
30
40
50
Time (1000 a.u.)
(b)
(c)
(a) Evolution of eigenvalues of ρ(t) for the case of a quantum initial condition correspondN = 2. (b) Evolution of eigenvalues of ρ(t) for the case of a quantum condition corresponding to 2 eigenstates and N = 3. (c) Evolution of the eigenvalues of ρ(t)
Figure 2:
ing to 2 eigenstates and initial
for the case of a quantum initial condition corresponding to 2 eigenstates and
N = 41.
Notice that
small non-zero eigenvalues appear also in this case.
matrix tends to
the quantum state is approximated by (41), the density matrix of the quantum subsystem has
ρ(t) →
the following form:
1 (|φ0 ihφ0 | + |φ1 ihφ1 |) 2 N 1 X −i∆j (t) + e |φ0 ihφ1 | + ei∆j (t) |φ1 ihφ0 | , 2N j=1
generic case can be computed based on the identication of the pointer basis with eigenstates of the electronic Hamiltonian, as detailed above. If the initial state is a linear superposition of
(53)
rity tends to a value of
generated by
1/n.
For generic linear
combinations, the asymptotic value of purity
−i∆j
φ0 + φ1 ,
j = 1, . . . , N.
can be computed as:
(54)
ψ0 =
Notice that the corresponding sum of projectors tend to zero when
N
n
such states, with equal coecients, then the pu-
which is a sum of projectors onto the subspaces
adding
(55)
The asymptotic value of the purity for a
ρ(t) '
χj (t) = e
1 (|φ0 ihφ0 | + |φ1 ihφ1 |) , 2
N
X
grows, since we are
j
time-dependent vectors of norm one,
⇒ρ(t) →
moving with dierent velocities, in the linear space generated by
φ0
and
φ1 .
cj φj , c0 , c1 , . . . ∈ C X
|cj |2 |φj ihφj |,
(56)
j
The case for
P(ρ(t)) →
N = 3 is represented in Figure 2b. The case N = 41 is represented in Figure 2c. Notice how
X
|cj |4 .
j
we obtain two eigenvalues approximately equal
Observe that the complex phase of the coe-
and a series of remaining eigenvalues represent-
cients is irrelevant for the computations, as the
ing the eect of the change of the position of the and the non-vanishing part from the projector
eigenstates of the electronic Hamiltonian can 0 always be redened as cj φj = |cj |φj . Additionally, the above computations show
sum, with a negligible weight. Indeed, for large
the existence of pointer basis in ESD. Accord-
cores in the spectrum of the Hamiltonian
values of
N,
He (R)
coecients in the sum in (53) are
ing to the meaning of decoherence in the con-
3,5
an approximately random set of complex num-
text of HQCD,
bers with modulus one, their sum being zero.
basis appears as the set of stable states under
Thus, for large enough values of
t,
the density
described above, the pointer
the non-unitary evolution of the system. From (41), it can be concluded that eigenstates of the
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•
electronic Hamiltonian, i.e. elements in the basis
B,
are for ESD the elements in the pointer
Page 14 of 19
Case A: The initial quantum state is an eigenstate of the electronic Hamiltonian:
basis of the quantum system.
ψ0 = φ1 .
Summarising, we have observed that the clas-
(57)
sical subsystem (the cores), acting as an envi-
•
ronment onto the valence electron, produce a
Case B: The initial state is a linear su-
dynamics that, for times longer than the deco-
perposition of two eigenstates of the elec-
herence time, selects a small set of the possi-
tronic Hamiltonian:
ble quantum states.
In this particular exam-
1 ψ0 = √ (φ0 + φ1 ) , 2
ple, these are the eigenstates of the electronic Hamiltonian. These conclusions can be a starting point for a full analysis of pointer basis in
•
molecular systems and in ESD, which could be
Case C: The initial state is a linear superposition of three eigenstates of the elec-
performed by considering more general initial
tronic Hamiltonian:
parameters and larger molecular systems. Ob-
1 ψ0 = √ (φ0 + φ1 + φ2 ) , 3
serve that this is in agreement with the decoherence hypothesis,
19
(58)
as the pointer basis does not
(59)
depend on the initial state of the quantum subsystem. It is also important to notice that the
The results plotted in Figure 3a are in agree-
results here obtained are characteristic of ESD.
ment with the behaviour given in (56).
Other decoherent mechanisms will present dif-
leads to changes in the purity of the quantum
ferent features and the pointer basis might ap-
states,
pear in a completely dierent fashion.
preserving.
1,2
ample.
3.3 Numerical simulations of an ionised dimer
ESD
unlike standard ED, which was purityLet us analyse in detail each ex-
Case A reproduces the case in which
the initial quantum state is an eigenstate of the electronic Hamiltonian at of Section 3.2).
R0 (see the beginning
The purity is approximately
In order to illustrate the results in Section 3.2,
constant and equal to 1 for all the evolution. On
we have computed explicitly the changes in the
the contrary, cases B and C show a sharp drop
purity of a simple model of an ionised dimer,
on the purity (on the left side of Figure 3a),
as described above.
and reach approximately stationary values of
As initial conditions, we
take a certain initial state
ψ0
1/2 and 1/3, respectively.
of the quantum
The decoherence time can be read from the
subsystem. For this state, the equilibrium po-
R0
of the cores are taken as their initial
plots as the time it takes for the system to reach
positions. The nuclei are allowed to move only
a stationary value. This denition is of course
along the axis of the molecule and keeping their
not precise, but in any case this time is approx-
center of mass stationary. Some uncertainty is
imately 1200 and 2500 a.u. for cases B and C,
allowed in the values of the initial momenta, −5 ranging from 0 to 40 × 10 a.u., in both di-
respectively.
rections. With these values, the kinetic energy
molecule, which has been computed to be ap-
of the cores is never large enough to dissociate
proximately 30000 a.u.
sition
This time is very short in com-
parison with the typical oscillation time of our
Observe that the numbers
the dimer. Thus, dierent regimes of oscillation
N
of dierent tra-
are considered. In conclusion, the uncertainty
jectories considered in the mixtures are essen-
in the initial conditions is given in the form of
tial in order to understand the behaviour of the
(44).
system. Figures 3b and 3c represents the purity
Three dierent choices for the quantum sub-
of cases B and C, respectively, for dierent num-
systems are considered. In each case, the initial
bers of trajectories, chosen in increasing order
position of the cores is always the equilibrium
of initial speeds of the cores. It can be inferred
position for the given quantum state:
that a low number of trajectories causes large
ACS Paragon Plus Environment 14
Page 15 of 19
Purity
1
0.8
1 0.8 0.6 0.4 10 20 0 10 20 30 30 40 50 40 N Time (1000 a.u.)
0.6 Purity
(b)
0.4
0.2
Purity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Journal of Chemical Theory and Computation
Case A Case B Case C
0 0
10
20
30
40
1 0.8 0.6 0.4 0.2 10 20 0 10 20 30 30 40 50 40 N Time (1000 a.u.)
50
Time (1000 a.u.)
(a)
(c)
Figure 3: Three dierent cases, labelled A, B and C, are considered. For each case, the initial state of the valence electron is xed by (57), (58) and (59), respectively. The cores are initially at the equilibrium position for the corresponding electronic state, and with dierent initial speeds along the molecule axis, as described in the text. The system evolves according to ESD. the purity of the density matrix stable asymptotic value of
1/n,
ρ(t)
with
n
Evolution of
the number of eigenstates of the electronic Hamiltonian
whose linear superposition determine the initial state of the quantum subsystem. of the purity of the density matrix
(a)
for the proposed cases. The purity reaches an approximately
ρ(t)
with respect to time and the value of
initial quantum state corresponds to (58) in
(b)
and to (59) in
(c).
(b-c)
N
Evolution
in (44).
The
It can be observed that the
decoherence time and the asymptotic value of the purity stabilise for larger values of
N.
oscillations in the purity, while adding trajecto-
metric route to dene the Ehrenfest Statisti-
ries causes the stabilisation of the value of the
cal Dynamics (ESD). We proved that, for a
purity after the decoherence time. If much large
simple toy-model, the resulting ESD is purity
cases were considered, these oscillations are ex-
non-preserving. Now, after having carefully de-
pected to disappear.
Decoherence time would
ned the meaning of decoherence in the context
then determine when the system has become a
of hybrid quantum-classical dynamics (HQCD),
mixture of states from the pointer basis, and the
we have tested out that, in ESD, decoherence
decoherence hypothesis introduced in Section 1
does appear.
is satised.
19
In fact, in this article we have proved that, when applied to a realistic molecular model,
4
ESD denes a noncoherent evolution for the
Conclusions
electronic degrees of freedom of the molecule. In our computations, the state space encod-
Non-adiabatic transitions and decoherence are
ing the quantum degrees of freedom is consid-
two of the most important phenomena in chem-
ered as the nite dimensional vector space ob-
ical dynamic reactions. To which extent does a
tained by considering the values of the elec-
given theoretical model include each eect is
tronic wave-function on a grid, instead of the
a complicated question. Several methods have been developed to deal with both notions. In previous works,
1,2
more usual approach for Ehrenfest dynamics
15
that considers the truncated expansion in the
we introduced a geo-
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Page 16 of 19
electronic Hamiltonian eigenbasis. It is impor-
is purity non-preserving: A necessary in-
tant to notice that this choice and the intrinsic
gredient for decoherence.
(i.e., tensorial) nature of our formalism ensures
2012,
that our results are completely independent of any choice of basis for the system.
(3) Bedard-Hearn,
Thus, in
Chem. Phys. 2005, 123, 234106.
phe-
nomenon have been investigated. In particular,
J.
(4) Landry, B. R.; Subotnik, J. E. Commu-
the purity of the electronic state has been ob-
nication:
served to decrease in a short time, reaching an
Standard surface hopping pre-
dicts incorrect scaling for Marcus' golden-
asymptotic value. Also, we have observed the
rule rate: The decoherence problem can-
existence of a pointer basis, which in our exam-
not be ignored.
ple turns out to be composed of the eigenstates
J. Chem. Phys. 2011, 135,
191101.
of the electronic Hamiltonian of the molecular system.
E.;
tions with nuclear-induced decoherence.
realistic situations. decoherence
R.
tum/classical molecular-dynamics simula-
non-coherent phenomena which are observed in the
Larsen,
algorithm for nonadiabatic mixed quan-
in the simple language of Ehrenfest dynamics,
of
J.;
stochastic decoherence (MF-SD): A new
tistical nature of the model is able to encode,
features
M.
Schwartz, B. J. Mean-eld dynamics with
this intrinsic way, we have proved how the sta-
Some
137, 054106.
J. Chem. Phys.
Notice that this fact cannot be
(5) Larsen,
R.
E.;
Bedard-Hearn,
M.
J.;
Exploring
role
of
predicted for other systems and situations (i.e.
Schwartz,
situations in which the evolution is not approx-
decoherence in condensed-phase nonadia-
imately adiabatic).
batic dynamics:
In conclusion, the present
B.
J.
the
A comparison of dier-
paper shows that decoherence and pointer ba-
ent mixed quantum/classical simulation
sis can be observed in simple models evolving
algorithms for the excited hydrated elec-
under ESD. Further studies are required in or-
tron.
der to extend this analysis to more complex
20066.
J. Phys. Chem. B 2006, 110, 20055
systems.
Acknowledgement
(6) Prezhdo, O. V. Mean eld approximation
We are deeply grateful
for the stochastic Schrödinger equation.
Chem. Phys. 1999, 111, 83668377.
to the referee for all his comments that, in our opinion, have helped us to greatly improve our work.
J.
The authors have received support by
(7) Subotnik, J. E. Augmented Ehrenfest dy-
the research grants E24/1 and E24/3 (DGA,
namics yields a rate for surface hopping.
J. Chem. Phys. 2010, 132, 134112.
Spain), MINECO MTM2015-64166-C2-1-P and FIS2017-82426-P, MICINN FIS2013-46159-C3-
(8) Subotnik, J. E.; Shenvi, N. A new ap-
2-P and FIS2014-55867-P. Support from scholarships
B100/13
(DGA)
and
proach to decoherence and momentum
FPU13/01587
rescaling in the surface hopping algorithm.
(MECD) for J. A. J-G is also acknowledged.
J. Chem. Phys. 2011, 134, 024105.
Authors acknowledge the use of Servicio General de Apoyo a la Investigación-SAI, Univer-
(9) Subotnik, J. E.; Shenvi, N. Decoherence
sidad de Zaragoza.
and surface hopping: when can averaging over initial conditions help capture the effects of wave packet separation?
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Journal of Chemical Theory and Computation
Graphical TOC Entry
For Abstract use only Title: Ehrenfest statistical dynamics in chemistry: study of decoherence eects Authors: J. L. Alonso, P. Bruscolini, A. Castro, J. Clemente-Gallardo, J. C. Cuchí, and J. A. Jover-Galtier
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