Ehrenfest Statistical Dynamics in Chemistry: Study of Decoherence

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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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Page 2 of 19

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


1,2

we in-

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• 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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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)



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


in the initial distribution (47).

If

Page 13 of 19

0.8 0.6 0.4 0.2 0

1

Eigenvalues

1

Eigenvalues

1

Eigenvalues

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


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



•

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


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


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-



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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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


Ehrenfest Statistical Dynamics in Chemistry: Study of Decoherence

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