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Complex autocorrelation function and energy spectrumby classical trajectory calculations

Petra R. Zdanskaa)

Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic,Flemingovo 2, 166 10 Prague 6, Czech Republic

Nimrod Moiseyev Department of Chemistry and Minerva Center of Nonlinear Physics in Complex Systems, TechnionIsrael

Institute of Technology, Haifa 32000, Israel

Received 14 April 2004; accepted 8 July 2004

A quasiclassical method which enables evaluation of complex autocorrelation function from

classical trajectory calculations is proposed. The method is applied for two highly excited

nonlinearly coupled harmonic oscillators in regimes prevailed either by regular or chaotic classical

motions. A good agreement of classical and quantum autocorrelation functions is found within short

Ehrnfest time limit. Fourier transforms of the autocorrelation functions provide moderate resolvedenergy spectra, where classical and quantum results nearly coincide. The actual energy levels are

obtained from approximate short-time autocorrelation functions with the help of filter

diagonalization. This study is a follow up to our previous work P. Zdanska and N. Moiseyev, J.Chem. Phys. 115, 10608 2001, where the complex autocorrelation has been obtained up to overallphase factors of recurrences. 2004 American Institute of Physics. DOI: 10.1063/1.1787489

I. INTRODUCTION

Our present study is motivated by a quest for a method

suitable for calculation of rovibronic spectra of nearly clas-

sical polyatomic systems, such as chromophores bound in

van der Waals clusters,1 5 van der Waals matrices,6,7 etc.

Calculation of rovibronic spectra of polyatomic systems rep-

resents a hardly tractable problem. Fully quantum treatment

is possible for systems up to four to nine degrees of

freedom,8,9 although for special cases systems up to 24 de-

grees of freedom have been calculated.10,11 Another possibil-

ity is mixed quantum-classical approach where a small im-

portant part of a system is simulated in quantum way, while

the influence of the rest of the system is approximated by

classical dynamics.12,13 The third type of approach uses a full

semiclassical treatment.1417 Our present concern is a quasi-

classical approach known as Wigner phase-space WPSmethod.18,19 All degrees of freedom are treated on the same

footing, i.e., by classical mechanics, and quantum mechanics

comes into the calculation only through the initial choice of

positions and momenta of classical trajectories. All quantities

are calculated without a specific evaluation of the semiclas-

sical phase, which makes the WPS method promising for

larger systems than those allowed by quantum and other

semiclassical approaches.

The other motivation of our study is a theoretical puzzle

posed by the WPS approach. The WPS method is capable of

very accurate approximations to cross sections19 or mean-

field potentials,20 namely, quantities which are dependent on

quantum time-dependent wave functionnot merely on its

absolute value, but the phase, too. Therefore one asks,

whether complex quantum amplitudes, such as the autocor-

relation function, can be obtained from WPS propagations,

and if yes, then how? One possibility is to map time-

dependent phase-space density onto a wave function and

thus to calculate the autocorrelation function in Hilbert

space.21 Yet the backward mapping of classical evolved

phase-space density includes a conceptual problem, as long

as classical and quantum evolutions result in general case in

very different phase-space densities.19 Thus the introduced

error may be hard to estimate in unknown systems. Counter

to ones intuitive notions, the expectation values calculated

from the classical evolved density are rigorous approxima-tions to quantum expectation values,22 due to, loosely speak-

ing, vanishing of the oscillatory part of the quantum density

upon the integration. Therefore we suggest to utilize expec-

tation values of the product Hamiltonianevolution operator

rather then the backward mapping of the classical evolved

phase-space density, in order to calculate the phase of the

autocorrelation function.

Our paper is organized as follows: In Sec. II we summa-

rize the theory for the WPS approach needed for derivations

of new formulas. In Sec. III we summarize the results of our

previous study:23 We show that the phase of autocorrelation

function can be defined using expectation values of the prod-uct operator of Hamiltonian and evolution operator. This

definition results in the phase integration which has to go

through regions in time domain of zero amplitude, therefore

the phase of individual recurrences is obtained only up to the

relative phases between recurrences.23

In Sec. IV we propose a solution to overcome this diffi-

culty. We define an auxiliary cross correlation function with

a frozen wave packet which propagates on a classical peri-

odic orbit. The cross-correlation function does not vanish

which allows for integration of the phase, and coincides withaElectronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 13 1 OCTOBER 2004

61750021-9606/2004/121(13)/6175/11/$22.00 2004 American Institute of Physics

Downloaded 14 Jan 2005 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
http://dx.doi.org/10.1063/1.1787489

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the required autocorrelation function at periodic time inter-

vals. This designed coincidence enables us to obtain the rela-

tive phase between recurrences, and thus the complete auto-

correlation function can be calculated: phase and amplitude.

We demonstrate our techniques for the model Pullen-

Edmonds Hamiltonian of two nonlinearly coupled harmonic

oscillators24,25 in Secs. V and VI. Our calculations give an

evidence that the presented method works in regular and cha-

otic regimes alike, yet the calculations differ in the effortrequired. While in the regular case Sec. V a single periodicorbit is sufficient to follow the periodic motion, in the cha-

otic regime one has to define multiple periodic orbits to fol-

low individual recurrences Sec. VI.The main importance of obtaining the autocorrelation

function is that it can be used to obtain the energy spectrum,

therefore in Sec. VII we demonstrate the application of Fou-

rier transform and filter-diagonalization2629 methods to cal-

culate the energy spectra from classical approximations to

autocorrelation functions. In Sec. VIII we present our con-

clusions.

II. WIGNER PHASE-SPACE APPROACH

The WPS method is a quasiclassical approach to quan-

tum dynamical problems, consisting of three steps: i Theinitial phase-space density P is obtained as a Wigner trans-

form of the initial wave function ,

Pq,p1

2 F dqe i/pq

qq/2*qq/2, 1

where F is the system dimension. P is represented as the

ensemble of phase-space points,q

i,p

i. Phase-space points

are preferentially distributed in the region ofP such that the

weight, wjP(qj ,pj), is ascribed to every randomly gener-

ated point qj ,pj; the weight is compared with the ran-domly generated number, NmaxP, and the point is se-lected ifw iN. ii Phase-space points, qi ,pi, are evolvedin time by classical propagation. The obtained trajectories,

qi( t),pi(t), represent the time-dependent classical phase-space density P( t). iii The quantum expectation values,

A( t), defined as

A te i/H tAei/H

t 2

are approximated by the phase space integrals,

A t dq dpAq,pPq,p; t, 3where A corresponds with A through the Wigner-Weyl trans-

form of the operator A given by

Aq,p dqe i/pqqq/2Aqq/2 4 dpe i/qppp/2App/2. 5

In practice, A( t) is evaluated by Monte Carlo integrationover classical trajectories,

A tiAqi t,pi t . 6

Following this, the survival probability C(t)2 definedas the absolute value squared of the autocorrelation function

C(t),

C teiH t/, 7

is obtained for A being the initial density operator, ;the survival probability is approximated within the WPS

method by

C t2iPqi t,pi t. 8

III. PHASE OF AUTOCORRELATION FUNCTION

Calculation of the complex autocorrelation function is

considered fully within the realm of the WPS method,

namely, the phase is calculated using expectation values,

which are approximated by Eqs. 3 and 6. Let the phase,

C(t), of the autocorrelation function be defined as

C tC te iC t. 9

By differentiating the identity C(t)/C*(t)exp2iC(t) we

obtain the expression for the time derivative of the phase

given by

C

t1RH , 10

where

He iH

t/ PH eiH t/

C2

, 11

and P is the density operator defined as

P . 12

Equation 11 allows us to calculate the phase using the time-dependent expectation values, which in turn are accessible

by the WPS approach using Eqs. 3 and 6. The survivalprobability, C2, is obtained using the known WPS approxi-mation given by Eq. 8. The nominator is defined as the

expectation value PH(t) of the product operator AH PH .

Following the WPS approach, we calculate the Wigner-

Weyl transform AH of the operator AH using Eqs. 4 and5. It is convenient to calculate separately the potential and

kinetic contributions, AHAVAT , where AV is the

Wigner-Weyl transform of AV PV and AT is the Wigner-

Weyl transform of AT PT. AV is calculated using Eq. 4 as

long as AV includes the potential operator commuting with

coordinate representation,

AV dqe i/pqqq/2*qq/2Vqq/2. 13

The potential operator is expanded in Taylor series such that,

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Vqq/2n0

qn

2 nn!

dnV

dqn. 14

This result is substituted to Eq. 13. We use the followingproperty of Fourier transform,

dqe i/pqqq/2*qq/2qn

i n

nP

pn, 15

and we obtain the series for AV given by,

AVn0

in

2 nn!

dnV

dqn

nP

pn. 16

The Wigner-Weyl transform of AT is obtained in the same

manner using the momentum representation, Eq. 5,

ATn0

i n

2n

n!

dnT

dpn

nP

qn

. 17

The Wigner-Weyl transform AH ofAH is a complex function

given by the series for its real and imaginary parts,

RAHn0

2

4

n 1

2n !

d2nVdq2n

2nP

p2n

d2nT

dp2n

2nP

q2n , 18

IAH

2 n0

2

4

n 1

2n1 !

d2n1Vdq2n1

2n1

P

p2n1

d2n1T

dp2n1

2n1

P

q2n1 . 19

We obtain the WPS approximation for H by substitutionsinto Eq. 11using Eqs. 6, 18, and 19 we obtain theWPS approximation of the nominator, while Eq. 8 is usedfor the denominator,

H iAHqi t,pi t

iPqi t,pi t. 20

The phase of the autocorrelation function is calculated

using Eq. 10 such that,C

t1

iRAHqi t,pi t

iPqi t,pi t. 21

The phase of the autocorrelation function is calculated

by numerical integration using the phase derivative over

time. In cases where the autocorrelation function consists of

well separated recurrences see, e.g., Fig. 2, the phase de-rivative between the recurrences cannot be calculated for the

reason of numerical inaccuracy. Therefore in such cases, the

phase is known for individual recurrences only up to a con-

stant phase factor.23 The solution of this difficulty is ad-

dressed in Sec. IV, where we define and evaluate a cross-

correlation function, which coincides with the

autocorrelation function at a single point within each recur-

rence.

A. Inconsistency within WPS approach: Theharmonic approximation

In this section we discuss inconsistent expressions for

autocorrelation function, which are derived using different

algebraical approaches. Let us demonstrate this point for thecase of autocorrelation amplitude, C2. The derivation of thephase derivative, Eq. 10, is applicable also in the case ofautocorrelation amplitude; we obtain

lnC

t1

iIAHqi t,pi t

iPqi t,pi t. 22

Another way to obtain the time derivative of the autocorre-

lation amplitude, is by differentiating Eq. 8; doing this andusing Hamiltons equations of motion we obtain the expres-

sion,

lnC

t

1

2

iAqi t,pi t

iPqi t,pi t , 23

where

AdV

dq

P

p

dT

dp

P

q. 24

By comparison of Eqs. 22 and 23 one can see that Eq.

23 is obtained from Eq. 22, when only the first term ofthe series for IAH in Eq. 19 is included.

Similar discussion is possible for the time derivative of

the autocorrelation phase. An alternative to the WPS ap-

proximation, Eq. 21, is the expression given by,

Ct iHqi t,pi t

Pqi t,pi t

iPqi t,pi t. 25

The proof of Eq. 25 can be found in Ref. 23. Again, by

comparison of Eqs. 21 and 25 one can see that Eq. 25 isobtained from Eq. 21, when only the first term of the seriefor RAH , Eq. 19, is included.

The existence of different formulas for same quantities,

all consistent with the WPS assumption Eq. 6, representsthe inherent error of the WPS approximation. One can see

that Eq. 23 is obtained from Eq. 22 using the quadraticapproximation, therefore the WPS approximation to autocor-

relation amplitude is exact in the case of harmonic oscillator.

On the other hand, Eq. 25 is given by the linear approxi-mation to Eq. 21. As a consequence, using of Eq. 25results in values ofC/t which are incorrect for harmonic

oscillator.

In order to clarify the role of the quadratic term in cal-

culation of the phase derivative, let us consider the case of a

coherent Gaussian wave packet given by,

1

F/4exp qq0

2

2

i

q"p0 , 26

where the harmonic Hamiltonian is given by, Hp2/2

q2 /2. Using Eq. 1 we obtain the corresponding Wignerdistribution given by,

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P1

Fexp qq0

2

pp0 2

. 27

The Hamiltonian H and the Wigner distribution P are sub-

stituted into Eq. 18. RAH can be written as the sum ofzero-order and second-order terms, RAHPHPH

(2 ),

where H(2 ) is given by

H 2 F

2

1

2 pp0 2qq02. 28

The contribution of the zero-order term to C/t, as de-

fined by Eq. 25, is given by

C

t

H

F

4

q0p0

2. 29

The contribution of the second term is calculated using the

equation,

C

t

H 2

iH 2 qi t,pi tPqi t,pi t

iPqi t,pi t, 30

and is given by

C

t

H 2

F

4. 31

The sum of the zero- and second-order contributions leads to

the correct result,

C

t

F

2

q0p0

2, 32

which is in aqreement with the quantum-mechanical defini-

tion given by Eqs. 10 and 11. It is evident from this

discussion that the second-order term is responsible for one

half of the zero-point energy shift in the case of coherentharmonic motion. It is thus expected that an error of about

one half of the zero-point energy is caused by omitting the

quadratic term in series Eq. 18.

IV. CROSS-CORRELATION FUNCTION

In this section we show how over-all phase factors of

autocorrelation recurrences can be obtained. Let us consider

a quasiperiodic motion with the approximate period T. Then

we may assume that a recurrence in autocorrelation function

occurs for time tT. The idea is to define the new function

(t) such that (tNT) and therefore, after T, 2T, . . .

etc. periods the values of C(tNT) are equal to X(tNT),

CNTXNT, N0,1,..., 33

where X( t) is defined as

X t teiH t/. 34

The idea is to define (t) in such a way that there are norevivals in X( t) and therefore unlike C(t), X( t) does not

drop to zero between the recurrences. In this way we avoid

numerical difficulties in extracting time-dependent phase de-

rivatives between recurrences where C(t)0. To assure thiscondition, it is reasonable to define ( t) as the frozen wave

packet moving according to the classical periodic contour.

X(t) represents a cross-correlation function, however, notice

that in this case the left vector (t) depends on t parametri-

cally, in contrast to the most common definition of cross-

correlation function.

A. Classical periodic contour

We define an approximate classical periodic contour

orbit, qc(t),pc(t) , such that qc(0),pc(0 )qc(T),pc(T) . Except for approximately following the

motion of the classical phase-space density, the periodic con-

tour can be arbitrary; here we suggest the following proce-

dure:

i positions and momenta of evolving trajectories areweighted by Pqi(T),pi(T) , to give a large preference to

those contributing to a given autocorrelation recurrence. Fur-

ther we confine to trajectories falling within a small energy

interval E0EHq i(t),p i( t)E0E; this is espe-cially reasonable in cases when the wave function bifurcates

due to time evolution. The first estimate of the classical con-

tour according to this procedure is given by

qc0 t iqi tPqi T,pi T i

iPqi T,pi T i,

35

pc0 t ipi tPqi T,pi T i

iPqi T,pi T i,

where i1 for E0EH q i(t),p i( t)E0E and i0 otherwise. qc0(t),pc0(t) satisfy the condition of ap-proximately following the classical phase-space packet.

ii The second condition is that the contour must form a

closed loop, which is accomplished by using the back-and-

forth Fourier transform:

qc tn

expi 2ntT

0

T

dt exp i 2ntT

qc0 t , 36where n acquires small integer values pc( t) is obtained simi-

larly.

B. Frozen wave packet

We define a wave function (t) given by

te i/pctpc 0 q

ei/qctqc 0 p

, 37

that represents a frozen wave packet which keeps the shape

of the initial wave function while its position is changedaccording to the classical periodic contour, qc(t),pc(t).

The wave packet (t) acquires the important property,

(NT) . ( t) is given by

t1

F/4exp qq0qc 0 qc t

2

2

i

qp0pc 0 pc t

i

p0qc 0 qc t

38



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for the case of initial Gaussian wave packet defined by

Eq. 26.

C. Absolute value and phaseof cross-correlation function

Calculation of complex cross-correlation function X( t)

by the WPS approach is similar as in the case of the auto-

correlation function. The absolute value square X2 is ob-tained as the expectation value of the density operator,

P t t t , 39

such that

X t2iPqi t,pi t;t , 40

where

Pq,p; t dqe i/pqqq/2;t*qq/2;t. 41

The phase, X, of the cross-correlation function is de-

fined as

X tX te iX t. 42

By differentiating the identity X(t)/X*(t)exp2iX(t), we

derive the expression for the time derivative ofX given by

X

t

1

RHpcRq

qRppc 0 pc t , 43

where the notation a

, aH ,q ,p, stands for

ae iH

t/ PaeiH t/

X2. 44

The WPS approximation to a is obtained as follows: the

nominator, Pa, is evaluated using Eq. 6, where A

Aa is given by the Wigner-Weyl transform of Aa

Pa; and the denominator, X2, is evaluated using

Eq. 40,

a iAaqi t,pi t

iPqi t,pi t. 45

The Wigner-Weyl transforms, AH , Aq , and Ap , arecomplex functions whose only real parts are used to calculate

the cross-correlation phase Eq. 43. These are listedbelow:

RAHPHn1

2

4 1

2n !

d2nVdq2n

2nP

p2n

d2nT

dp2n

2nP

q2n ,

RAqPq, 46

RApPp.

The WPS approximations for the integrals RH , Rq ,and Rp , are given by

RH iRAHqi t,pi t

iPqi t,pi t,

Rq iPqi t,pi tqi t

iPqi t,pi t, 47

Rp iPqi t,pi tpi t

iPqi t,pi t.

To summarize, the complex cross-correlation function

X( t) is calculated using the WPS approach. The absolute

value X is obtained using Eq. 40, while the phase X isobtained by numerical integration of its time derivative,

X/t, which is given by Eqs. 43 and 47. The cross-correlation function X(t) can be evaluated numerically in the

time interval from t0 to tNT, as long as it does not

include revivals nor drops to zero. As discussed before, this

is not true for the autocorrelation function C(t) which can be

evaluated from classical trajectory calculations for individual

recurrences only up to the overall phase factors.23

Using theequivalence Eq. 33, we obtain the correct phase of the au-tocorrelation function at T, 2T,... from the calculated clas-

sical cross-correlation function. Thus the missing overall

phases of autocorrelation recurrences are determined and the

classical complex autocorrelation function is obtained.

V. ILLUSTRATION FOR REGULAR SYSTEM

The Pullen-Edmonds Hamiltonian is given by

Vq12/2q 2

2/2q 12q2

2,

48Tp 1

2/2p 22/2.

It represents two nonlinearly coupled harmonic oscillators.

For energies beyond a given energy limit , the system

becomes mixed regular and chaotic. For the chosen,0.05, the energy limit 10 a.u. The initial wave packet

is given by the Gaussian function, Eq. 26. The initial

position of in the phase space is q0 4 a.u., 2.957 a.u.,p0 0,0. The corresponding energy expectation value is 21

a.u., which is set far in classical mixed regular-chaotic re-

gime. The prevailed motion for this wave packet is regular,

as shown in Ref. 23. The system is illustrated in Fig. 1. The

contour plot of potential illustrates its nonlinearity which dis-

plays itself by deviation from circular curvature. The wave

packet, which is evolved, approximately, for one classicalperiod, deforms and starts to develop nodes, but stays local-

ized.

We confine our study to a short-time propagation, for

three classical periods. This is approximately the Ehrnfest

time for the studied system, where all trajectories acquire the

same Maslov indices at a time, as shown in Ref. 23. Once

this condition is satisfied, the WPS approximation given by

Eq. 6 is correct. The quantum-mechanical propagation,which serves as the benchmark for our study, allows us to

calculate the exact autocorrelation function Eq. 7 for thistime period. The autocorrelation function C(t) includes three

well distinguished recurrences, as shown in Fig. 2.



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The classical trajectory simulation is performed for an

ensemble of trajectories, qi( t),pi(t) , which are chosen by

means of preferential sampling of the initial phase-space

density, P. P is defined by Eq. 27.

A. Cross-correlation function

The definition of the cross-correlation function is based

on a notion of classical periodic contour orbit,

qc(t),pc(t), with the period T, which approximately rep-

resents the classical periodic motion of the phase-space den-

sity. The classical periodic contour is calculated using Eqs.

35 and 36. qc( t) is illustrated in Fig. 1.The cross-correlation function X(t) is defined by Eqs.

3438. For an illustration, we evaluate the cross-correlation function using quantum mechanics, Fig. 3. Un-

like the autocorrelation function, X( t) does not climb downto zero, nor creates recurrences. This property is essential for

its accessibility by the classical trajectory method, which is

addressed below. In the classical trajectory method, the

cross-correlation function is a means to obtain the phase of

the autocorrelation function at time points tNT, where the

two functions coincide. These points are emphasized in both

Figs. 23.

B. Classical trajectory calculationof cross-correlation function

The absolute value of cross-correlation function is evalu-

ated using Eq. 40. The Wigner-Weyl transform of , P ,

is obtained by substitution of Eq. 38 to Eq. 41,

P1

Fexp qq0qc 0 qc t

2

pp0pc 0 pc t2

. 49

The absolute value as obtained from classical trajectory cal-

culation is shown as part of Fig. 5. The comparison of quan-

tum and classical results for X2, Figs. 3 and 5, shows anexcellent agreement.

The phase of the cross-correlation function X is calcu-

lated using numerical integration of the time derivative,

X/t. The time derivative of X is calculated from the

integrals RH , Rq , and Rp , Eq. 43. Theirquantum definition is given by Eq. 44, while their WPSapproximations are given by Eqs. 47. The quantum andclassical results for Rq and Rp are displayed in Figs.

4a and 4b. Clearly, the WPS approximations forRq and Rp given by lines are in an excellent agreement

with the quantum results given by points.Let us discuss in detail the classical approximation of the

quantity RH in the light of the results obtained in Sec.III A. The substitution of Pullen-Edmonds Hamiltonian Eq.48 to Eqs. 46 and 47, leads to the expression

RH iAHqi t,pi t]Pqi t,pi t

iPqi t,pi t, 50

where AHHH(2 ), is the sum of the classical Hamiltonian

H and the quadratic term H(2 ). The latter is given by,

FIG. 1. One-period wave packet evolution. The Pullen-Edmonds potential is

drawn by dashed contours. The wave packet is shown at times t10, t22.3 a.u., t33.5 a.u., and t44.65 a.u. The evolved wave packet inter-

sects the closed contour Eqs. 3536, which serves in the definition of

the cross-correlation function.

FIG. 2. Autocorrelation function calculated by exact quantum propagation

for the time interval 0 t15.5 a.u. The absolute value and the real part are

denoted by black and gray lines, respectively. The values for t

T,2T,3T are emphasized by dots.

FIG. 3. Cross-correlation function by quantum propagation for the time

interval 0t11 a.u. The absolute value and the real part are denoted by

black and gray lines, respectively. The values at tT,2T,3T, where thecross-correlation function coincides with the autocorrelation function, are

emphasized.



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H 2 F

2

1

2 pp0 2qq02

f

F

qf2

f

F/2pfpf0

2

qf2

. 51

As discussed in Sec. III A, the WPS approximation for His not unique, and AH can be substituted either by Hamil-

tonian only, AHH, or by a series, represented here by the

sum, AHHH(2 ). In Fig. 4c we show these two approxi-

mations lines compared with the quantum result points.We can conclude that both approximations provide similar

results, close to the quantum benchmark. The contribution of

H(2 ) is relatively small due to high expectation value of en-

ergy compared to the zero-point energy.

The classical complex cross-correlation function is dis-

played in Fig. 5. The quantitative comparison of the quantum

and classical cross-correlation functions is given in Table I,

where we list the phase of the cross-correlation functions at

points tT,2T,3T. quant is the phase as obtained by quan-tum calculation; class is the phase as obtained by classical

calculation in the quadratic approximation, AHH

H

(2 )

;and classH is the phase as obtained by classical calculation in

the linear approximation, AHH. It is clear that class is a

good approximation to quant , while the error of classH is

very large. classH includes the error 2 at t3T. Con-

sequently, energy spectrum corresponding to the linear ap-

proximation includes the error estimated by, /( 3T)

0.45 a.u. This value is close to the half of zero-point en-ergy; this result is in agreement with the role of the quadratic

term, H(2 ), in the case of harmonic oscillator, as discussed in

Sec. III A.

C. Classical trajectory calculationof autocorrelation function

The absolute value of the autocorrelation function

C2(t) is obtained using Eq. 8. As shown in Fig. 6, theautocorrelation function includes three recurrences, which

appear in each classical period T. We are able to evaluate the

time derivative of the phase C, given by Eq. 21, using

classical trajectory calculations. The phase, C , is evaluated

from C/t by numerical integration and is known for in-

dividual recurrences up to overall phase factors. To adjust the

overall phase factors, it is necessary to determine the phase

of the autocorrelation function for at least one time-point per

FIG. 4. Rq , Rp , and RH , defined by Eq. 44 calculated by

quantum and classical trajectory methods. a Rq1 circlesquantum,solid lineclassical and Rq2 trianglesquantum, dashed lineclassical, b Rp1 circlesquantum, solid lineclassical and Rp2trianglesquantum, dashed lineclassical, c RH circlesquantum, dashed lineone-term classical approximation, solid linetwo-

term classical approximation.

FIG. 5. Cross-correlation function obtained by classical trajectory calcula-

tions for the time interval 0 t11 a.u. The absolute value and the real part

are denoted by black and gray lines, respectively. The points tNT, which

are used to obtain the phase of the autocorrelation function, are emphasized.

FIG. 6. Autocorrelation function obtained by classical trajectory calcula-

tions for the time interval 0 t15.5 a.u. The absolute value and the real

part are denoted by black and gray lines, respectively. The points tNT,

which are obtained from the phase of the cross-correlation function, are

emphasized.

TABLE I. Phase of the cross-correlation function at points tT,2 T,3 T,

where the cross-correlation function coincides with the autocorrelation

function in regular regime. quant exact quantum values; class,

classM classical quadratic and linear approximations, respectively; err-error

of the classical values, errclassquant/2.

t quant class err classH

T4.625 0.226 0.295 3.45% 0.543

2T9.255 0.428 0.440 0.60% 0.967

3T13.885 0.428 0.602 8.70% 0.304



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recurrence by another calculation. These points are deter-

mined by the cross-correlation function for the time points

tNT, where C(NT)X(NT), Eq. 33. The particularphase values at points tT,2T,3T are listed in Table I.Thus, the complex autocorrelation function obtained entirely

from classical trajectory calculations is shown in Fig. 6.

VI. ILLUSTRATION FOR CHAOTIC REGIME

Application of the new method in chaotic regime is pos-

sible with a small modification, as shown in this section. For

illustration, we use the same type of Hamiltonian Eq. 48,

where the initial Gaussian wave packet is centered at non-

zero momentum position, here we choose q0 4 a.u., 1.645 a.u. and p0 0, 4 a.u.. When the corre-

sponding phase-space packet Eq. 27 is propagated, its

large portion occupies chaotic sea, as can be shown with the

help of Poincare surface of section, Ref. 23.

In regular regime, we could identify a quasiperiodic mo-

tion with a period T, causing a sequence of equally spaced

recurrences. The chaotic motion, on the other hand, is char-

acterized by bifurcations of the phase-space density, there-fore recurrences are multiplied and appear at irregular time

spacing. This becomes evident after a close look at quantum

autocorrelation function, Fig. 7. The first recurrence appears

for t5 a.u. In place of a second recurrence at t10 a.u.,three small recurrences are found, indicating a bifurcation to

three different quasiperiodic motions. At t14 a.u. there is a

large recurrence, the period of which corresponds to three

approximate classical periods, but the recurrence time is by

1 a.u. shorter than expected.

The bifurcations of the time-evolved phase-space density

must be reflected by different periodic contours for each re-

currence. The total phase of recurrences is obtained for each

recurrence separately, from cross-correlation functions oftheir own periodic contours. Within the time interval 0 t15.5 a.u. we define five periodic contours according to Eqs.

35 and 36, with periods T1 , . . . , T5 . The correspondingfive classical cross-correlation functions are displayed in

Figs. 8a 8e. In Table II we list the classical and quantumphase at time points tT1 , . . . ,T5. It is clear that the clas-sical calculations provide a good approximation to the cha-

otic autocorrelation function with the exception of the sec-

ond recurrence, where the error of nearly 70% is obtained. It

is assumed that this error is caused by the interference of

trajectories with different Maslov indices the Ehrnfest crite-

rium is not perfectly satisfied. However, the phase error

FIG. 7. Quantum autocorrelation function for chaotic regime. Classical pe-

riods T1 , . . . ,T5 for individual quasiperiodic motions are emphasized.

FIG. 8. Classical cross-correlation functions for chaotic regime. Each recur-

rence in the autocorrelation function is created by a different quasiperiodic

motion, therefore the phase of each recurrence is obtained using a different

cross-correlation function. Five cross-correlation functions are displayed.

TABLE II. Phase of the cross-correlation function at points t

T1, . . . ,T3, where the cross-correlation functions coincide with the auto-

correlation function in chaotic regime. quant exact quantum values; classclassical quadratic approximation; errerror of the classical values, err

classquant/2.

t quant class err

T15.32 0.179 0.141 1.87%

T29.27 0.77 0.592 68.11%

T310.80 0.277 0.427 7.52%

T411.57 0.0508 0.164 5.65%

T514.31 0.294 0.377 4.16%



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does not take a great effect in the corresponding energy spec-

trum due to the relatively small amplitude of the second

recurrence, as shown below.

Calculation of the classical autocorrelation amplitude

and time derivative of the autocorrelation phase is done ac-

cording to Sec. III with no alteration for the chaotic regime.

The absolute phases of individual autocorrelation recur-

rences are obtained from the corresponding cross-correlation

functions, at points tT1 , . . . ,T5. The classical autocorre-lation function for chaotic regime is shown in Fig. 9. The

agreement between the classical and quantum results Fig. 7is remarkable.

VII. ENERGY SPECTRUM FROM CLASSICALAUTOCORRELATION FUNCTION

The predominant role of the autocorrelation function in

most calculations is that its Fourier transform gives energy

spectra together with the corresponding Franck-Condon am-

plitudes I(E) as given by the equation,

IER0

dtexpi/EtC t. 52

Limitation of our method to short-time propagation allowsevaluation of moderate resolved energy spectra using Eq.

52, which are shown in Figs. 10a and 10b. The classicalapproximation is in qualitative agreement with quantum

spectra in both regular and chaotic cases. The spectral inter-

val 18.5 a.u.E23.5 a.u. is shown in Figs. 11a and 11b;we can see that the spectral peaks appear near the occupied

energy levels, which are shown by vertical lines.

The actual energy levels can be obtained even from

short-time propagations with the help of filter-

diagonalization method.8,2628 Here we apply the filter-

diagonalization method to classical autocorrelation functions

in regular and chaotic regimes in order to demonstrate the

ability of the WPS approach to approximate the exact quan-

tum energy levels.

The present implementation of the filter-diagonalizationmethod is described below:

i The Fourier transforms of the time-dependent wave

packet,

k0

tmax

dt eiHEkt/ 53

are used as basis functions to find the eigenvalues of the

evolution operator, U()eiH t/. tmax15.5 a.u. is the time

of the propagation, for which the autocorrelation function

has been calculated; 0.05 a.u., is a short-time interval,

which gives an optimal convergence of the filter-

diagonalization method. The evolution and overlap matricesare given by,

Ukk0

tmax

dt dte iEkEkt/C tt,

54

Skk0

tmax

dt dte iEkEkt/C tt.

ii The basis functions k are defined by energy grid

points Ek , which are selected to follow a bell distribution

around the energy window center E0 . The energy window

width is given by the standard deviation 1.67 a.u. of en-

ergy grid points Ek , k1160. The precision of the calcu-

FIG. 9. Classical autocorrelation functionchaotic regime.

FIG. 10. Moderate resolved spectra as obtained by Fourier transform of the

classical gray and quantum black autocorrelation functions for the wholerange of occupied energies. a regular regime, b chaotic regime.

FIG. 11. Moderate resolved spectra as obtained by Fourier transform of the

classical gray and quantum black autocorrelation functions for a smallenergy interval. The actual quantum energy levels and Franck-Condon over-

laps are shown by vertical lines. a regular regime, b chaotic regime.



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lated energy levels is estimated as the standard deviation of

few three to eight filter diagonalizations performed for dif-ferent values of E0 , 19a.u.E023.2 a.u..

The energy levels found in regular regime, falling into

interval 18.5 a.u.E23.5 a.u., are listed in Table III anddisplayed in Fig. 12a. The seven significantly populated

states are extracted both from quantum and classical autocor-relation functions. The standard deviations are of orders

103 104 a.u., which is an error significantly smaller than

energy spacing between populated levels. The error given by

short-time propagation is also by one to two orders of mag-

nitude smaller than the difference between the quantum and

classical calculations given by the third column in TableIII. We may thus conclude that the WPS approximation hasbeen capable of prediction of energy levels in regular regime

with the absolute error being 0.010.03 a.u.

We attempt the calculation of exact energy levels using

the filter-diagonalization method in chaotic regime for the

energy interval 18.5 a.u.E23.5a.u., however due to

small energy spacing of occupied states the energy levelscannot be obtained. The obtained energies are listed in Table

IV and displayed in Fig. 12b. One can see that the standarderror of the filter-diagonalization method 0.010.03 a.u.is often larger than the energy spacing of the occupied levels

in chaotic regime, which signifies the incomplete representa-tion of Hamiltonian due to short-time propagation. It is thus

characteristic that the standard error of the obtained energy

levels is almost one order of magnitude larger in chaotic case

compared to regular regime.

VIII. CONCLUSIONS

We propose a quasiclassical method for calculation of

complex autocorrelation functions within Ehrnfest time limit.

The method is tested for a model of two nonlinearly coupled

harmonic oscillators for regular and chaotic regimes, and

proves to work in good quantitative agreement with quantum

benchmarks.We formulate an approximation of complex autocorrela-

tion function which is fully consistent with the Wigner

phase-space approach, namely, uses the rigorous quasiclassi-

cal expression for expectation values. We suggest a way to

overcome a problem of lost overall phases of separate recur-

rences, which arise using the presented method Ref. 23.The lost phase factors are obtained by calculation of a cross-

correlation function which serves as a link between t0 and

tT, where T is the approximate classical period.

Using the approximate complex autocorrelation function

we attempt to calculate the energy spectrum of the system

with the help of filter-diagonalization method. We demon-

strate that the Wigner phase-space approach is capable ofprediction of energy levels of the given nonlinear system in

regular regime with a very good precision.

One of the main obstacles for practical application of the

Wigner phase-space method, and one of the main challenges

of this approach can be expressed in words of Garashchuk

and Tannor,21 Although we have been able to calculate the

phase to the same accuracy as the amplitude, in our experi-

ence the method is accurate only through intermediate times;

without further modification the classical Wigner method is

not competitive with methods based on the VVG van Vleck-Gutzwiller propagator expression. Italics added. The so-lution may hint toward separation of autocorrelation recur-

FIG. 12. Energy levels, as obtained by filter-diagonalization method bot-

tom compared with Franck-Condon amplitudes top. The quantum and

classical results are marked by black and gray color, respectively. The finite

width of peeks is given by the standard error calculated for multiple filter-

diagonalization calculations. a regular regime, b chaotic regime.

TABLE III. Energy levels obtained by the filter-diagonalization method

from the quantum and classical short-time autocorrelation functions in regu-

lar regime. Eclassclassical results; Equant quantum results; absolute dif-

ference between quantum and classical values, EclassEquant.

Eclass Equant

19.08740.0012 19.06500.0006 0.0224

19.78580.0103 19.81810.0001 0.0323

20.45120.0016 20.42880.0015 0.0224

21.1399

0.0180 21.1651

0.0008 0.025221.82660.0007 21.81210.0020 0.0145

22.49920.0021 22.52910.0007 0.0299

23.21820.0025 23.21180.0014 0.0064

23.87710.0005 23.90740.0007 0.0303

TABLE IV. Energy levels obtained by the filter-diagonalization method

from the quantum and classical short-time autocorrelation functions in cha-

otic regime. Eclassclassical results; Equantquantum results; absolute dif-

ference between quantum and classical values, EclassEquant .

Eclass Equant

18.5750.015 18.5570.022 0.018

18.9240.016 18.8940.027 0.031

19.3610.025 19.3280.036 0.032

19.807

0.020 19.778

0.041 0.02920.2480.003 20.2120.024 0.037

20.7080.015 20.7000.014 0.008

21.1230.015 21.1030.005 0.020

21.5300.017 21.5190.025 0.011

22.0100.027 21.9990.026 0.010

22.4060.019 22.3860.027 0.020

22.8300.023 22.8140.028 0.016

23.3070.021 23.3090.018 0.002



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rences using the calculation of their Maslov index; the

preliminary discussion of this problem is found in Ref. 23.

ACKNOWLEDGMENTS

This work has been supported by the Center for Com-

plex Molecular Systems and Biomolecules, which is funded

by the Czech Ministry of Education Grant No. LNO0A032,and completed within the framework of the research project

Grant No. Z4 055 905.

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