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JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 1
1 JULY 1999
Flow of zero-point energy and exploration of phase space
in classical simulations of quantum relaxation dynamics.
II. Application to nonadiabatic processes
Uwe Müller and Gerhard Stock
Faculty of Physics, University Freiburg, D-79104 Freiburg, Germany
~Received 10 March 1999; accepted 7 April 1999!
The unphysical flow of zero-point energy ~ZPE! in classical trajectory calculations is a consequence
of the fact that the classical phase-space distribution may enter regions of phase space that
correspond to a violation of the uncertainty principle. To restrict the classically accessible phase
space, we employ a reduced ZPE ge ZP , whereby the quantum correction g accounts for the fraction
of ZPE included. This ansatz is based on the theoretical framework given in Paper I @G. Stock and
U. Müller, J. Chem. Phys. 111, 65 ~1999!, preceding paper#, which provides a general connection
between the level density of a system and its relaxation behavior. In particular, the theory establishes
various criteria which allows us to explicitly calculate the quantum correction g. By construction,
this strategy assures that the classical calculation attains the correct long-time values and, as a
special case thereof, that the ZPE is treated properly. As a stringent test of this concept, a recently
introduced classical description of nonadiabatic quantum dynamics is adopted @G. Stock and M.
Thoss, Phys. Rev. Lett. 78, 578 ~1997!#, which facilitates a classical treatment of discrete quantum
degrees of freedom through a mapping of discrete onto continuous variables. Resulting in negative
population probabilities, the quasiclassical implementation of this theory significantly suffers from
spurious flow of ZPE. Employing various molecular model systems including multimode models
with conically intersecting potential-energy surfaces as well as several spin-boson-type models with
an Ohmic bath, detailed numerical studies are presented. In particular, it is shown, that the ZPE
problem indeed vanishes, if the quantum correction g is chosen according to the criteria established
in Paper I. Moreover, the complete time evolution of the classical simulations is found to be in good
agreement with exact quantum-mechanical calculations. Based on these studies, the general
applicability of the method, the performance of the classical description of nonadiabatic quantum
dynamics, as well as various issues concerning classical and quantum ergodicity are discussed.
© 1999 American Institute of Physics. @S0021-9606~99!01925-X#
I. INTRODUCTION
which, for example, discard trajectories not satisfying predefined criteria. However, most of these techniques share the
problem that they manipulate individual trajectories, whereas
the conservation of ZPE should correspond to a virtue of the
ensemble average of trajectories.
In a preceding paper,7 henceforth referred to as Paper I,
we have proposed an alternative strategy to tackle the ZPE
problem. The basic idea is that an appropriate classical description should explore at least roughly the same phasespace volume as the quantum description. Since in general
this condition does not hold ~for then there would be no ZPE
problem to begin with!, it has been suggested to use a reduced ZPE ge ZP (0< g <1) to accomplish this goal. Performing a theoretical analysis that connects level densities
and the long-time relaxation behavior of a quantum system,
we have established several criteria which may be employed
to determine the optimal quantum correction g. Although the
connection between ZPE excitation and relaxation dynamics
has been discussed previously,4 this analysis has provided
clear criteria how much the ZPE should be reduced in order
to get the correct quantum-mechanical relaxation behavior.
In Paper I we have introduced the general theory and studied
simple model problems which allow for an analytical treat-
Quasiclassical trajectory calculations are a wellestablished tool to simulate the dynamics of molecular
systems.1 In this approach, the evaluation of the dynamics is
performed on a purely classical level ~i.e., \50), while the
quantum nature of the system is mimicked through a sampling of the probability distribution of the quantummechanical initial state. As a consequence, quantum fluctuations such as the zero-point energy ~ZPE! are introduced
which would vanish on a purely classical level. Although its
necessity for a successful classical description of quantum
dynamics is generally accepted, this somewhat inconsistent
strategy may give rise to spurious flow of ZPE.2–6 This is
because in classical mechanics energy can flow between
modes without restriction, while in quantum mechanics each
oscillator mode must hold an amount of energy that is larger
or equal to its ZPE. This flaw of the quasiclassical description may result in quite unphysical behavior such as ZPEinduced dissociation of the system. Numerous approaches
have been proposed to fix the ZPE problem.2–6 These include
a variety of ‘‘active’’ methods @i.e., the flow of ZPE is controlled and ~if necessary! manipulated during the course of
individual trajectories3# and several ‘‘passive’’ methods4,5
0021-9606/99/111(1)/77/12/$15.00
77
© 1999 American Institute of Physics
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78
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
U. Müller and G. Stock
ment. However, these models do not undergo relaxation and
are therefore not suited to directly reveal the connection between level densities and the relaxation behavior observed in
a dynamical simulation. To illustrate the concept and to demonstrate the capability of the approach, this work presents
detailed numerical studies of multimode molecular systems
that undergo irreversible relaxation.
As a serious challenge of the method, we adopt a recently introduced classical description of nonadiabatic quantum dynamics.8,9 In this formulation, an N-level system is
mapped onto a system of N oscillators which are constrained
to their ground and first excited state, thus rendering the
correct treatment of ZPE flow crucial. After briefly reviewing the mapping formalism and the main results of Paper I,
computational results are presented for two types of molecular systems: First, we are concerned with the relaxation behavior of strongly coupled few-mode systems. Hereby, a
two-state three-mode model of pyrazine undergoing ultrafast
internal conversion is adopted, which has been employed by
several authors as a standard example of ultrafast electronic
relaxation.10–16 The model is amenable to a complete
quantum-mechanical evaluation of all quantities of interest,
thus allowing for a comprehensive discussion of all criteria
suggested in Paper I. Furthermore, we consider the relaxation behavior of weakly coupled many-mode systems,
whereby various spin-boson-type models modeling photoinduced electron transfer are employed.17–19 Based on these
studies, we discuss the practical applicability of our concept
to constrain the flow of classical ZPE, the general performance of the mapping approach to classically describe nonadiabatic relaxation processes, as well as various issues concerning classical and quantum ergodicity.
II. RELAXATION IN CLASSICAL AND QUANTUM
MECHANICS
Let us consider an isolated nondegenerate quantummechanical system H, which at time t50 is prepared in the
nonequilibrium state r 0 . The system may be characterized
by its level density
D ~ e ! 5Tr d ~ e 2H ! ,
~2.1!
and the spectral density S( e ) associated with the initial
preparation r 0
S ~ e ! 5Tr r 0 d ~ e 2H ! .
t→`
E
^ P k& 5
E
de S ~ e ! ^ A e & ,
~2.3!
^ A e & 5TrS AD ~ q, e ! /D ~ e ! ,
~2.4!
D ~ q, e ! 5TrB d ~ e 2H ! .
~2.5!
de S ~ e ! D k ~ e ! /D ~ e ! ,
D k ~ e ! 5Tru c k &^ c k u d ~ e 2H ! ,
~2.6!
~2.7!
where D k ( e ) is the state-specific level density pertaining to
state u c k & .
The expressions above point out the importance of level
densities in the description of relaxation processes: Since any
observable can be represented by projectors P k , it is sufficient to know the associated state-specific level densities
D k ( e ) to calculate the microcanonical average. This is to
say, that any approximation that manages to reproduce the
quantum-mechanical spectral density S( e ) as well as the
level densities D k ( e ) will yield correct long-time mean values. Denoting the classical approximations to the above expressions by the index C, one may thus define the following
conditions for the equivalence of classical and quantum
long-time limits:7
S C ~ e ! 5S ~ e ! ,
~2.8!
D kC ~ e ! /D C ~ e ! 5D k ~ e ! /D ~ e ! ,
~2.9!
D C ~ e ! 5D ~ e ! .
~2.10!
Let us first consider condition Eq. ~2.10!. Adopting a
one-dimensional notation ~e.g., x5 $ x j % , j51 . . . f ), the classical level density reads
D C~ e ! 5
~2.2!
Furthermore, let us consider an observable A(q) which,
after undergoing relaxation, is assumed to become stationary
at long times. We shall assume that the observable only depends on a ~in general multidimensional! ‘‘system coordinate’’ q, thus decomposing the trace operation in a trace
over system and bath variables, i.e., Tr5TrS TrB . As shown
in Paper I, we then obtain for the long-time mean value of A
^ A & [ lim Tr r ~ t ! A5
The mean value ^ A & at long times is given as integral over
the spectral density S( e ) and the microcanonical average
^ A e & . The latter can be expressed in terms of a ‘‘coordinatespecific’’ level density D(q, e ) normalized to the overall
level density D( e )5TrS D(q, e ). As discussed in Paper I,
the derivation of Eq. ~2.3! exploits ~i! von Neumann’s criterion for quantum ergodicity,20 stating that the time average
and the phase average of an arbitrary observable are equivalent for a nondegenerate system, and ~ii! that the time average and the long-time limit of an observable coincide if the
observable becomes stationary at long times.
As a simple example, let us assume that the observable
A describes the projection on the state u c k & , i.e., A5 P k
[ u c k &^ c k u . The probability to find the system at long times
in state u c k & is then given by
E
dx dp
~2p\! f
d @ e 2H C ~ x,p !# ,
~2.11!
that is, D C ( e ) measures the area of the energy surface of the
classically accessible phase-space volume. The integral level
density, that is, the level number function
N~ e !5
E
e
0
dé D ~ é ! 5Tr U ~ e 2H ! ,
~2.12!
is therefore directly proportional to the phase-space volume
that is energetically accessible to the system. Deviations of
the classical and quantum-mechanical level density therefore
reflect the fact that the phase space explored by the system
may be quite different in quantum and classical
mechanics.21,22 For example, the quantum-mechanical phasespace distribution may enter regions corresponding to quan-
© Copyright American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Insitute of Physics
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
Flow of zero-point energy. II
tum tunneling which are not accessible to the classical system. On the other hand, the classical phase-space distribution
may enter regions of phase space that correspond to a violation of the uncertainty principle. A famous example of the
latter phenomena is the ZPE problem mentioned above.
Stating that an appropriate classical description should
explore the same phase-space volume as the quantum description, condition @Eq. ~2.10!# therefore may be considered
as a minimal requirement for a successful classical description of quantum relaxation. The equivalence @Eq. ~2.9!# of
the normalized state-specific level densities, on the other
hand, is a much stronger condition. It requires that quantum
and classical level densities should coincide not only in the
average but in a specific subspace of phase space as well. To
discuss the last condition @Eq. ~2.8!#, recall that the spectral
density S( e ) corresponds to the level density weighted by
the initial distribution @Eq. ~2.2!#. It is therefore equivalent to
the energy distribution of the system which, at least for large
systems, is typically rather diffuse and structureless. In this
case, a classical approximation of the spectral density is usually appropriate if Eq. ~2.10! holds and if a suitable classical
initial distribution r 0 has been employed.
To restrict the classically accessible phase space according to the rules of quantum mechanics, we have proposed to
invoke quantum corrections to the classical calculation. The
considerations are based on the formulations of Thiele23 and
others,24 who exploited the inverse Laplace transform of the
partition function in order to calculate the level number
N( e ). At the simplest level of the theory, these corrections
have been shown to correspond to including only a fraction g
(0< g <1) of the full ZPE into the classical calculation.
Adopting appropriate quasiclassical initial conditions, this
modification can also be employed in a dynamical trajectory
calculation.
It is important to note that this procedure also suggest a
solution in cases where no reference calculations exist. That
is, if we a priori know the long-time limit of an observable,
we can use this information to determine the quantum correction. For example, let us assume that the system may be
described by N energetically well-separated states that
couple to a many-dimensional bath. Since for large times and
low temperatures the system is expected to completely decay
in its adiabatic ground state, the ZPE correction g then may
be determined by requiring that
^ P ad
k & C 5 d k,0 ,
~2.13!
where P ad
k denotes the projector on the kth adiabatic state.
Since different observables are associated with different
parts and projections of phase space, however, it should be
noted that the latter criterion does not strictly guarantee that
other long-time averages are reproduced correctly.
III. CLASSICAL DESCRIPTION OF NONADIABATIC
QUANTUM DYNAMICS
There are a variety of mixed quantum-classical formulations that combine a quantum-mechanical description of the
light particles of the system ~e.g., the electrons! with a classical description of the heavy particles ~e.g., the nuclei!25–31
The dynamical coupling of the quantum-mechanical and
79
classical subsystems is thereby achieved through a meanfield ansatz or the assumption of instantaneous ‘‘hops’’ between electronic potential-energy surfaces. To avoid dynamical inconsistencies associated with this ad hoc combination
of quantum and classical mechanics, recently a ‘‘mapping
approach’’ to the classical description of nonadiabatic dynamics has been proposed.8 In this formulation the problem
of the classical treatment of discrete quantum degrees of
freedom is bypassed by transforming the discrete quantum
variables to continuous variables. The approach thus consists
of two steps: an exact quantum-mechanical transformation of
discrete onto continuous degrees of freedom ~the ‘‘mapping’’! and a standard semi- or quasiclassical treatment of
the resulting dynamical problem. This section briefly reviews
the mapping formalism of Ref. 8 and discusses its quasiclassical evaluation.
A. Mapping approach
Let us consider an N-level system described by the
Hamiltonian
H5
h nm u c n &^ c m u .
(
n,m
~3.1!
The discrete N-state system ~3.1! can be represented by N
continuous degrees of freedom through the mapping
relations8
u c n &^ c m u °a †n a m ,
~3.2a!
u c n& ° u 0 1 . . . 1 n . . . 0 N& .
~3.2b!
a n ,a †m
Here
are the usual oscillator creation and annihilation
operators with commutation relations @ a n ,a †m # 5 d n,m and
u 0 1 . . . ,1n . . . ,0N & denotes a harmonic-oscillator eigenstate
with a single quantum excitation in the mode n. Introducing
Cartesian variables X n 5(a †n 1a n )/ A2 and P n 5i(a †n
2a n )/ A2, the bosonic representation of the Hamiltonian Eq.
~3.1! reads
H5 21
h nm ~ X n X m 1 P n P m 2 d n,m ! .
(
n,m
~3.3!
The mapping @Eq. ~3.2!# preserves the commutation relations
of the operator basis and leads to the exact identity of the
electronic matrix elements of the propagator
^ c n u e 2i/\Ht u c m &
5 ^ 0 1 . . . 1 n . . . 0 N u e 2i/\Ht u 0 1 . . . 1 m . . . 0 N & ,
~3.4!
i.e., the Hamiltonians in Eqs. ~3.1! and ~3.3! are fully equivalent when used as generators of quantum-mechanical time
evolution. In the case of a two-level system, the formalism is
equivalent to Schwinger’s theory of angular momentum.32
To apply the formulation to vibronically coupled molecular systems, we identify u c n & with electronic states and
h nm 5h nm (x,p) with operators of the nuclear dynamics,
where x5 $ x j % , p5 $ p j % , ( j51 . . . M ). Depending solely on
N1M continuous degrees of freedom, the Hamiltonian @Eq.
~3.3!# has a well-defined classical analog. Since the mapping
formalism is quantum-mechanically exact, the approach allows us—without any further approximations—to extend
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80
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
U. Müller and G. Stock
well-established techniques of classical trajectory calculations to nonadiabatic problems on coupled potential-energy
surfaces.
The semiclassical evaluation of the mapping approach
via the Van Vleck–Gutzwiller approximation22 has been discussed in detail in Ref. 8. In this work, we are concerned
with the quasiclassical evaluation of the approach which has
been studied in Ref. 9. In short, the transition to classical
mechanics is performed by changing from Heisenberg operators y k (t) (y k 5X n , P n ,x j ,p j ) obeying Heisenberg’s
equations of motion (iẏ k 5 @ y k ,H # ) to the corresponding
classical functions obeying Hamilton’s equations ~e.g., ẋ k
5 ] H/ ] p k ). Since the classical approximation is employed
to the complete system @Eq. ~3.3!#, the mapping approach
naturally provides a consistent and completely equivalent dynamical treatment of quantum $ X n , P n % and classical $ x j ,p j %
degrees of freedom.
It is noted that the idea of a consistent classical treatment
of electronic and nuclear variables was anticipated by Meyer
and Miller in their ‘‘classical electron-analog models.’’28
However, the quantum-classical analogies employed in Ref.
28 are not unique and involve additional approximations.9 In
contrast, the mapping formalism establishes an equivalence
of discrete and continuous degrees of freedom on a quantummechanical level and thus uniquely determines the Hamiltonian as well as the initial conditions. In recent works on the
semiclassical description of nonadiabatic dynamics,33,34
Miller and co-workers use a modified formulation of their
original model,28 which is identical to the classical limit of
the mapping approach proposed in Ref. 8.
For the computational studies on nonadiabatic molecular
dynamics discussed below, we consider electronic two- and
three-state systems with diagonal matrix elements
h nn ~ x,p ! 5E n 1
1
2
2
(j @ v j p 2j 1 v j ~ x j 1 k (n)
j /v j! #.
~3.5!
Here E n denotes the vertical transition energy of the diabatic
state u c n & , and v j and k (n)
j / v j represent the vibrational frequency and the state-dependent linear coordinate shift of the
jth vibrational mode, respectively. Depending on the model
considered, the off-diagonal diabatic coupling h nm 5h mn is
either assumed to be constant or to be linear in a single
vibrational mode ~see below!. Furthermore, it is assumed
that the system is initially in its upper electronic state ~say,
u c 2 & ), while the nuclear degrees of freedom are in thermal
equilibrium with respect to the unshifted Hamiltonian h 0
5 12 ( j v j (p 2j 1x 2j )
r ~ 0 ! 5 u c 2 &^ c 2 u e 2 b h 0 /Tr e 2 b h 0 .
~3.6!
As discussed in Ref. 12, this initial condition corresponds to
the photoexcitation of the system from an initial electronic
state ~e.g., u c 0 & ) to the upper state u c 2 & .
The mapping Hamiltonian @Eqs. ~3.3! with ~3.5!# describes the ubiquitous situation of M low-frequency oscillators ~the nuclear degrees of freedom $ x j ,p j % ) coupled to N
high-frequency oscillators ~the electronic degrees of freedom
$ X n , P n % ). The quantum nature of the electronic oscillators
manifests itself in the fact that only single excitations
u 0 1 . . . 1 n . . . 0 N & may occur @cf. Eq. ~3.2b!#. Quantum mechanically, the latter condition is clearly fulfilled due to the
structure of the Hamiltonian @Eq. ~3.3!#. However, this virtue
does not apply to the classical counterpart of Eq. ~3.3!, since
the ZPE excitation of the high-frequent electronic oscillator
becomes an essential effect. For this reason, the classical
evaluation of the mapping formalism represents a stringent
test for the simple concept of a constant quantum correction
proposed in Paper I.
B. Quasiclassical evaluation
A key quantity in the discussion of nonadiabatic relaxation dynamics12 is the time-dependent population probability of the initially excited diabatic electronic state u c n &
P di
n ~ t ! 5Tr r ~ t ! u c n &^ c n u .
~3.7!
According to the mapping relations @Eq. ~3.2a!#, the quasiclassical expression for the diabatic population probability
reads
P di
n ~ t !5
E
1
dG r n r e @ X 2n ~ t ! 1 P 2n ~ t ! 21 # ,
2
~3.8!
where dG5dx dp dX dP and the initial state @Eq. ~3.6!# of
the quantum system is represented through the phase-space
distributions r e (X, P) and r n (x,p) of the electronic $ X, P %
and nuclear $ x,p % degrees of freedom, respectively. For interpretative purposes it is often instructive to also consider
adiabatic electronic population probabilities P ad
n (t)
ad
5Tr r (t) u c ad
n &^ c n u , which are related to the diabatic quantities through a unitary transformation.12 The quasiclassical
evaluation of adiabatic populations has been discussed in
Ref. 15.
As is stated by Eq. ~3.2b!, the electronic initial state u c n &
is mapped onto the harmonic-oscillator eigenstate
u 0 1 . . . 1 n . . . 0 N & containing a single quantum excitation in
the mode n. Also, the nuclear initial distribution @Eq. ~3.6!#
is given in terms of harmonic-oscillator eigenstates. Employing Wigner’s formalism,35 the phase-space distribution of a
harmonic oscillator at temperature T is given by
r 0 ~ x,p ! 5
a 2 a /2(x 2 1p 2 )
e
,
2p
~3.9!
where a 5tanh(\ v /2k B T). Since this distribution only depends on the action variable n5 21 (x 2 1p 2 ) of the oscillator,
it is convenient to change from Cartesian variables (x,p) to
action-angle variables (n,q) via p1ix5 A2ne iq and sample
n from Eq. ~3.9! and q from the uniform distribution,
respectively.1 Several limiting cases of Eq. ~3.9! are of interest. For \→0 we obtain Boltzmann’s distribution ( a
5\ v /2k B T), and for T→0 we obtain the ground-state distribution of the harmonic oscillator ( a 51). Furthermore,
one may Taylor-expand Eq. ~3.9! around its mean value a /2,
thus obtaining in lowest order
r 0 ~ n ! 5 d ~ n2 a /2! ,
~3.10!
which is sometimes referred to as action-angle initial distribution.
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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
Flow of zero-point energy. II
81
It is straightforward to introduce the quantum corrections discussed above into the initial distributions in Eqs.
~3.9! and ~3.10!. The reduction of the ZPE content of an
oscillator from 21 \ v to 21 g \ v simply amounts to the replacement of a → a / g . Similar modifications can be introduced to
the Wigner transform of more general distribution. Since the
mapping Hamiltonian also contains electronic ZPE, furthermore, the ZPE term e ZP5 21 ( n h nn in Eq. ~3.3! needs to be
replaced by the reduced ZPE g e ZP for consistency reasons.
IV. RELAXATION OF STRONGLY COUPLED
FEW-MODE SYSTEMS
A. Model and computational methods
As the first example we consider a two-state three-mode
model of the S 1 (n p * ) and S 2 ( pp * ) states of pyrazine,
which has been adopted by several authors as a standard
example of ultrafast electronic relaxation.10–16 Taking into
account two totally symmetric modes and a single nontotally
symmetric mode, Domcke and co-workers have identified a
low-lying conical intersection of the two lowest excited singlet states of pyrazine, which has been shown to trigger internal conversion and a dephasing of the vibrational motion
on a femtosecond time scale.10,12 Exhibiting complex electronic and vibrational relaxation dynamics, the model represents a challenge for an approximate description. It has therefore been used as a test example for time-dependent
self-consistent-field13 and coupled-cluster calculations14 as
well
as
for
various
mixed
quantum-classical
approaches.15,16,30 The parameters of the model Hamiltonian
@Eq. ~3.5!# are E 1 54.02 eV, E 2 55.03 eV for the vertical
(2)
excitation energies, v 1 50.1262 eV, k (1)
1 50.037 eV, k 1
(1)
(2)
520.2534 eV, v 2 50.074 eV, k 2 520.1054 eV, k 2
50.1488 eV for the vibrational frequencies and coordinate
shifts of the two totally symmetric modes, and v C 50.1178
eV, l50.2618 eV for the vibrational frequency and the coupling constant of the single nontotally symmetric mode, that
couples the two electronic states via the diabatic matrix elements h 125h 215lx C . For a general discussion of vibroniccoupling systems and the associated nonadiabatic relaxation
dynamics, see, for example, Refs. 12 and 36.
To obtain quantum-mechanical reference calculations,
the Hamiltonian has been represented by a direct product
basis constructed from diabatic electronic states and
harmonic-oscillator states ~see Ref. 12 for details!. Since the
computation of state-specific level densities N k ( e ) requires
eigenvalues and eigenstates of the system, the resulting
Hamiltonian matrix of dimension of 7040 has been diagonalized using standard routines.37 Classical trajectory calculations for the mapping Hamiltonian @Eqs. ~3.3! with ~3.5!#
have been performed using a standard Runge–Kutta routine
with adaptive step size. To obtain converged results for the
Monte Carlo evaluation @Eq. ~3.8!# of the electronic population probabilities, ensemble averages over ~i! 2000 trajectories for action-angle initial conditions and ~ii! 5000 trajectories for Wigner initial conditions proved sufficient for all
models considered. Since there is no ZPE problem associated
with the low-frequent nuclear variables of this model, no
FIG. 1. ~a! Total level number N( e ), ~b! normalized state-specific level
number N 1 ( e )/N( e ), and ~c! spectral density S( e ) as obtained for the twostate three-mode model of pyrazine. Thick lines show exact quantummechanical results, thin lines display classical results for the limiting cases
g 50, 1 ~full lines! and the intermediate cases g 50.44 ~dashed lines!; and g
5 0.68 ~dotted lines!.
ZPE correction needs to be applied to the vibrational modes
of the system.
B. Level density
Let us start with the discussion of the level density and
related energy-dependent quantities of the system. Figure 1
shows ~a! the total level number N( e ), ~b! the normalized
state-specific level number N 1 ( e )/N( e ) pertaining to the diabatic electronic state u c 1 & , and ~c! the spectral density S( e )
of the two-state three-mode model of pyrazine. The thick
lines are exact quantum-mechanical results, the thin lines
display various classical results. Figure 1~a! demonstrates the
effect of electronic ZPE excitation on the classical evaluation
of the total level number. While the limiting cases of g 50
~lower line! and g 51 ~upper line! qualitatively fail to match
the quantum reference, the classical calculation with the ZPE
correction g 50.68 ~dotted line! is seen to represent a quite
accurate approximation to the quantum results. The latter
value of the ZPE correction has been determined by requiring optimal agreement of N C ( e ) and N( e ) in the energy
range 4.5 eV, e ,5.5 eV.
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82
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
The normalized state-specific level number N 1 ( e )/N( e )
is shown in Fig. 1~b!. ~Note that N 2 /N512N 1 /N.) The
state-specific data show the same qualitative behavior as the
total level numbers, i.e., the limiting cases g 50 and g 51
under- and overestimate the quantum data, while an intermediate value of the quantum correction ( g 50.44, dashed line!
matches the quantum results quite well. It is noted, however,
that requiring optimal agreement of state-specific level numbers results in a different ZPE correction as requiring optimal
agreement of total level numbers. As a consequence, the
classical results optimized to reproduce N k /N ( g 50.44,
dashed line! underestimate the total level number ('41% at
e 55 eV!, while the results optimized to reproduce N ( g
50.68, dotted line! overestimate the state-specific level number ('7% at e 55 eV!.
The spectral densities shown in Fig. 1~c! have been obtained for the initial preparation @Eq. ~3.6!# at zero temperature. Up to a constant energy shift e 0 5 21 ( j v j , this spectral
density is thus equivalent to the S 0 →S 2 absorption spectrum
of the three-mode model of pyrazine. The finite resolution of
the quantum spectrum ~thick line! has been obtained by convolution of the stick spectrum with a Lorentzian line width
of 265 cm21 . All spectral densities have been normalized to
1. Classical results are shown for g 50.68 ~dotted line! and
for the limiting cases g 50 and g 51 ~thin lines!. The data
have been obtained by histogramming the energy distribution
r ( e ) of the classical system sampled from action-angle initial conditions. For g 50, insertion of the electronic initial
conditions into the mapping Hamiltonian Eq. ~3.3! yields H
5h 22 , i.e., one obtains the spectral density of the threedimensional harmonic oscillator h 22 . For g .0, the classical
system is prepared in a mixture of both electronic states,
which results in a spectral density that qualitatively mimics
the quantum data. Increasing the value of the quantum correction is seen to result in a larger width of the spectral
density, although this quantity appears to be not as sensitive
to the choice of g as the level density. The classical result for
g 50.68 is seen to closely match the high frequency part of
the quantum spectrum, whereas it fails to reproduce the partitioning of the quantum spectrum into the main absorption
band centered at e 'E 2 and a vibronic intensity-borrowing
band at e 'E 1 .
C. Relaxation dynamics
Let us now turn to the central issue of this article, that is,
the connection between the level densities and the relaxation
behavior of the system. To this end, Fig. 2 shows in thick
lines quantum-mechanical results for the ~a! diabatic and ~b!
adiabatic electronic population probabilities of the initially
excited electronic S 2 state, as well as ~c! the mean position
^ x 2 & of one of the totally symmetric vibrational modes. The
diabatic electronic population exhibits an ultrafast initial decay on a time scale of '50 fs, which is followed by pronounced recurrences of the electronic population. The adiabatic electronic population, on the other hand, is seen to
decay within only '20 fs and exhibits only minor recurrences. As discussed in Refs. 10–12, the conical intersection
of the model affects a complex interplay of electronic and
vibrational dynamics: The coherent vibrational motion trig-
U. Müller and G. Stock
FIG. 2. Time-dependent ~a! diabatic and ~b! adiabatic electronic excitedstate populations and ~c! vibrational mean positions as obtained for the
two-state three-mode model of pyrazine. Thick lines show exact quantummechanical results, thin lines display classical results for the limiting cases
g 50 ~dotted lines! and g 51 ~full lines!.
gers the internal-conversion process which is reflected in the
recurrences of the diabatic electronic population. In turn, the
internal conversion affects an ultrafast dephasing of the vibrational dynamics which is reflected in the damping of the
mean positions as shown in Fig. 2~c!.
To get a first impression of the capability of the classical
description, Fig. 2 shows classical results for the limiting
cases g 50,1 employing action-angle initial conditions. As
anticipated in the discussion above, the limiting cases g 50
~dotted lines! and g 51 ~thin lines! significantly under- and
overestimate the true electronic relaxation dynamics, respectively. In particular, the adiabatic population probability for
g 51 is seen to assume negative values for times larger than
20 fs. This somewhat surprising artifact is explained by the
fact that within the mapping formalism the electronic population probability is directly proportional to the mean energy
content ^ 21 V nn (X 2n 1 P 2n ) & of the corresponding electronic oscillator. The finding of negative adiabatic population probabilities therefore means that the energy content of the corresponding electronic oscillator drops below the ZPE.
In contrast to these limiting cases, Fig. 3 shows ZPEcorrected classical results, which have been obtained for
g opt50.44. As previously discussed, this ZPE correction has
been obtained by requiring optimal agreement of classical
and quantum diabatic state-specific level densities. Therefore, the quantum-mechanical long-time limit of the diabatic
population should be reproduced by the classical calculation.
For times larger than 400 fs, the quantum-mechanical diaba-
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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
Flow of zero-point energy. II
83
FIG. 4. Percentage of trajectories that individually violate the ZPE conditions for the lower ~full line! and upper ~dotted line! electronic oscillator.
FIG. 3. Same as Fig. 2, except classical results are shown for g 50.44,
whereby action-angle ~full lines! and Wigner ~dashed lines! initial conditions have been employed.
tic population is found to fluctuate around P di
2 (`)50.33,
while the classical calculation for g opt50.44 give P di
2 (`)
50.30. Considering that condition @Eq. ~2.9!# can be
matched only qualitatively @cf. Fig. 1~b!#, the agreement is
quite satisfying.38 Furthermore, it should be pointed out that
the ZPE-corrected results do not only yield the long-time
limit of the diabatic population but also reproduce the entire
time evolution of the electronic and vibrational observable
under consideration. In particular, the data nicely match the
coherent transients of the diabatic population and the vibrational motion.
To study the effects of the initial phase-space distribution on the relaxation dynamics, we have also performed
classical calculations employing Wigner initial conditions.
While for g 50 action-angle and Wigner initial conditions
are identical by definition, the opposite limiting case of g
51 is found to yield similar results ~not shown in Figs. 2 and
3!. In particular, we also observe spurious flow of ZPE indicated by negative values of the adiabatic population probability. The dashed lines in Fig. 3 display the ZPE-corrected
Wigner simulations for g opt50.44. The data are again similar to the action-angle results, however reproduce the coherent beating of the signals only qualitatively. Within the mapping approach, action-angle initial conditions generally have
been found superior to Wigner initial conditions.
Let us now discuss the alternative criteria derived above,
which require ~i! the equivalence of classical and quantum
total level density @Eq. ~2.10!# and ~ii! that the system localizes in its adiabatic ground state @Eq. ~2.13!#. It is interesting
to note that both criteria result in the same ZPE correction of
g50.68. As already suggested by the state-specific level density, the overall relaxation is somewhat exaggerated in this
case, although both criteria guarantee positive population
probabilities. In other words, the ‘‘minimal’’ criteria Eqs.
~2.10! and ~2.13! might not give the best possible classical
description, but at least cure the ZPE problem.
We have studied various other two-state models exhibiting electronic relaxation including a model of the internalconversion process in benzene cation40 and few-mode spin
boson-type models of intramolecular electron transfer.41 In
all cases, the ZPE correction determined via the total level
densities were found to be somewhat larger than the one
determined via the state-specific level densities. Furthermore, in all cases the latter criterion yielded the correct longtime limits of the diabatic population. The best ZPE correction to reproduce the overall electronic and vibrational
relaxation dynamics was found to lie in between the two
cases, usually closer to the upper limit determined via the
total level density.
In conclusion, Figs. 1–3 provide a clear numerical demonstration of the connection between the level densities and
relaxation behavior discussed in Sec. II. The spectral density
defines the energy range of interest for a given preparation
r 0 of the system. For these energies the ZPE correction g is
chosen by requiring that either the total or the state-specific
level densities of the quantum system is reproduced by the
classical approximation. In any case, this ~independently determined! quantum correction cures the ZPE problem of
negative population probabilities. The best agreement of
classical and quantum dynamics is typically achieved for an
intermediate value of the quantum correction.39 This finding
is also remarkable in the light of the fact that the conditions
derived in Sec. II; ~i! merely state the equivalence of equilibrium values ~not of the dynamics! and ~ii! are only fulfilled approximately.
Finally, it may again be emphasized that a dynamically
consistent treatment of classical ZPE flow should refer to the
behavior of the ensemble rather than to behavior of individual trajectories.2 This fact is illustrated in Fig. 4, which
shows the percentage of trajectories that individually violate
the adiabatic ZPE conditions 21 (X̃ 21 1 P̃ 21 2 g ).0 ~full line!
and 21 (X̃ 22 1 P̃ 22 2 g ).0 ~dotted line!. In the average, about
half of the trajectories are found to disregard these conditions
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84
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
U. Müller and G. Stock
for g 50.44, although the ensemble of trajectories has been
shown to reproduce the correct quantum dynamics. Furthermore, only 4% of all trajectories never violate this condition.
V. RELAXATION OF SPIN-BOSON SYSTEMS
In the example above, the short-time dynamics of a molecular system is governed by a few high-frequency intramolecular vibrational modes that strongly couple to the electronic transition; a situation that presumably is generic for
the optical response of a polyatomic molecule on a femtosecond time scale.12 In the following we wish to consider the
opposite case where the response of a molecular system is
mainly determined by the interaction of a few-level system
with many weakly coupled vibrational modes of the environment; a typical example for this case are intermolecular
electron-transfer reactions in the condensed phase. The standard model to describe this type of dynamics is a two-level
system that is bilinearly coupled to an harmonic bath.17 Employing this so-called spin-boson model, we are able to compare our classical studies to numerically exact path-integral
calculations.31
As a first example, we adopt a spin-boson model recently discussed by Makri and Makarov.18 It consists of a
two-level system with interstate coupling h 125h 215g[1
and bias E 2 2E 1 52g in the low-temperature limit k BT
50.2g, and assumes an Ohmic spectral density of the bath
J( v )[ ( j k 2j d ( v 2 v j )5 p /2a v e 2 v / v c with a Kondo parameter a 50.1 and a cutoff frequency v c 57.5g. In the classical calculations the bath has been simulated by including
250 harmonic modes whose frequencies are equally distributed between 0.01< v <10. As the cutoff frequency of the
bath is higher than the electronic bias, the quantum correction to the ZPE should be employed to both electronic and
nuclear degrees of freedom. To determine the optimal ZPE
correction we employ criterion Eq. ~2.13!, that is, we require
that the system relaxes completely in its adiabatic electronic
ground state.
For this model, Fig. 5 shows quantum ~big dots! and
classical ~thin lines! results of the ~a! adiabatic and ~b! diabatic excited-state populations. Employing action-angle initial conditions, the classical results have been obtained for
the limiting cases g 50 ~dashed lines! and g 51 ~dotted
lines! as well as for the optimal ZPE correction g opt50.6
~full lines!. While the latter results for P ad
2 (t) decay to zero
by construction, the uncorrected ( g 51) results for the adiabatic population probability are seen to assume negative values, thus clearly exhibiting spurious flow of ZPE. Interestingly, the correct adiabatic behavior imposed on the classical
system is directly reflected in the true diabatic relaxation
dynamics. As shown in Fig. 5~b!, the ZPE-adjusted result
with g opt50.6 are in excellent agreement with quantum reference data, while the limiting cases g 50 and g 51 considerably under- and overestimate the diabatic population probability, respectively. Note that the case g 50 corresponds to
Boltzmann initial condition for the nuclear degrees of freedom. The complete lack of ZPE excitation in the bath modes
is seen to result in a significant underestimation of the damping of the coherent oscillations.
FIG. 5. Time-dependent ~a! adiabatic and ~b! diabatic electronic excitedstate populations as obtained for a dissipative two-state spin-boson model.
Exact quantum results ~big dots! are compared to classical results ~thin
lines! for the limiting cases g 50 ~dashed lines! and g 51 ~dotted lines! and
for g opt50.3 ~full lines!. ~c! Diabatic electronic excited-state populations
obtained for Wigner initial conditions with g 51 ~dotted line! and g opt
50.6 ~full line! and Boltzmann initial conditions with g opt50.14 ~dashed
line!.
As a further example of the effects of initial conditions,
Fig. 5~c! displays diabatic population probabilities as obtained for ~i! Wigner initial conditions with g 51 ~dotted
line! and with the optimal ZPE correction g opt50.6 ~full
line! and ~ii! Boltzmann initial conditions for the nuclear
degrees of freedom and action-angle initial conditions for the
electronic degrees of freedom with g opt50.14 ~dashed line!.
The optimal ZPE corrections again have been determined
with the aid of criterion Eq. ~2.13!. Note that Wigner and
action-angle initial conditions approximately result in the
same optimal ZPE correction, while Boltzmann initial conditions result in a much lower value of g. This is because in
the latter case the spectral density is shifted because of the
missing nuclear ZPE, thus probing the level densities of the
system in a much lower energy regime @cf. Eq. ~2.3!#. The
ZPE-corrected Boltzmann results are seen to attain the correct long-time limit but largely overestimate the coherent
beating of the diabatic population. The Wigner calculations,
on the other hand, resemble the action-angle results. While
the ZPE-adjusted results reproduce the quantum data quite
well, the results for g 51 exhibit negative adiabatic population probabilities. It is noted that Miller and co-workers have
recently presented Wigner calculations for g 51 which were
in good agreement with quantum results.34 The examples
chosen, however, e.g., various spin-boson models with zero
electronic bias, are hardly sensitive to ZPE excitation in the
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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
FIG. 6. Comparison of quantum ~thick lines! and classical ~thin lines! diabatic electronic populations as obtained for a dissipative three-state spinboson model describing ~a! sequential and ~b! superexchange electron transfer.
electronic degrees of freedom and are therefore reproduced
as well by classical simulations with g 50.44
Let us finally consider an example which demonstrates
the limits of the proposed procedure. To model chargetransfer dynamics in photosynthetic reaction centers, Sim
and Makri have recently reported exact long-time pathintegral simulations.19 Employing various spin-boson-type
systems comprising the initial state u c 1 & , the intermediate
state u c 2 & , and the final state u c 3 & , they investigated the
effects of sequential ~i.e., E 1 .E 2 .E 3 ) and superexchange
~i.e., E 2 .E 1 .E 3 ) electron-transfer mechanisms. As a representative example, Fig. 6 shows results obtained for ~a! the
proposed sequential model of wild-type reaction centers and
~b! the superexchange model for a mutant reaction center.
Both models are simulated at room temperature T5300 K
and employ an Ohmic spectral density with a Kondo parameter a 51.67 and a cutoff frequency v c 5600 cm21 . The sequential model is characterized by the vertical excitation energies E 1 50, E 2 52400 cm21 , E 3 522000 cm21 , and the
diabatic couplings h 12522 cm21 , h 235135 cm21 . The parameters of the superexchange model are E 1 50, E 2
52000 cm21 , E 3 52630 cm21 , and h 125h 235240 cm21 .
The quantum results ~thick lines! of the sequential model
~a! are seen to completely decay in the diabatic electronic
ground state of the system. For short times, furthermore, the
model predicts transient electronic population in the intermediate state. In the superexchange model ~b!, on the other
hand, there is a nonvanishing long-time population of the
initial diabatic state, whereas the intermediate state is hardly
ever populated. Using action-angle initial conditions, we
have performed classical mapping simulation with ZPE correction ~a! g 50.6 and ~b! g 50.34. The ZPE corrections
have been calculated via Eq. ~2.13! and are only employed to
Flow of zero-point energy. II
85
the electronic variables. In general, the classical simulations
are seen to predict a somewhat too slow kinetics. In the case
of the sequential model the long-time limits of the diabatic
populations are correctly reproduced, while this is only
roughly the case for the superexchange model. Increasing the
initial ZPE excitation results in faster kinetics but also in
negative diabatic and adiabatic population probabilities. For
example, in the case of the sequential model the limiting
di
cases g 50 and g 51 result in ^ P di
2 & '0.2 and ^ P 2 &
'20.2, respectively. In that sense, the ZPE-adjusted results
can be regarded as the optimal classical results.
There are several reasons for the relatively poor performance of the classical description in Fig. 6. First, the ZPE
problem becomes more serious when the number of quantum
states increases, because each mapped quantum state will
contribute its ZPE to the system.7 The main problem, however, seems to be that the classical description may be nonergodic ~see below!, while the quantum description is ergodic according to the definition given above. As a
consequence, the assumptions underlying our strategy are
not met, and the simple quantum correction proposed does
not necessarily guarantee a similar exploration of phase
space in classical and quantum mechanics. Apart from issues
associated with the ZPE problem, one might also suspect that
the presence of tunneling represents a difficulty for the classical approach. It should be stressed, however, that the effects of electronic tunneling are, at least in principle, included in the formulation. Due to the mapping of the
electronic states onto harmonic oscillators, tunneling between electronic states corresponds to energy transfer between coupled oscillators, which is readily described by the
classical description.
VI. CLASSICAL AND QUANTUM ERGODICITY
As previously pointed out, the derivation of Eq. ~2.3! is
based on von Neumann’s criterion for quantum ergodicity
and on the fact that the time average and the long-time limit
of an observable coincide if the observable becomes stationary at long times. Performing the classical limit of Eq. ~2.4!,
we obtain for the classical microcanonical average
^ A C~ e ! & 5
* dx dp A C ~ x,p ! d @ e 2H C ~ x,p !#
,
* dx dp d @ e 2H C ~ x,p !#
~6.1!
which is readily recognized as the standard classical expression resulting from the assumption of ergodic mixing.20 To
study to what extent the assumption of classical mixing is
fulfilled by the model systems under consideration, we compare classical results for the equilibrium diabatic population
^ P di
2 & obtained by explicit trajectory calculations @Eq. ~3.8!#
to the results for ^ P di
2 & obtained from Eqs. ~2.3! and ~6.1!.
Details on the calculation of multidimensional phase-space
integrals for the mapping Hamiltonian Eq. ~3.3! can be found
in the Appendix. For the two-state three-mode model of
pyrazine, the latter calculations yield ^ P di
2 & 50.38, 0.30, 0.22
for the ZPE corrections g 50, 0.44, 1, respectively. The excellent agreement with the simulated long-time limits reveals
that the classical model of pyrazine is indeed a mixing sys-
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86
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
tem. Furthermore, this finding suggests to determine the ZPE
correction g opt directly by evaluating criterion @Eq. ~2.13!#
via Eq. ~6.1!.
That ergodicity is not an exceptional but rather a generic
property of strongly coupled few-modes systems is supported by the works of Köppel, Cederbaum, and
co-workers.42 Employing similar multimode vibroniccoupling models, they found that the vibronic relaxation dynamics results in complex classical motion with mixed
regular/irregular phase-space structures, which are reflected
in largely irregular quantum statistical properties of the corresponding molecular spectra. It is noted that the long-time
limit of the diabatic and adiabatic electronic populations can
also be estimated by calculating the phase-space averages
pertaining to uncoupled adiabatic potential-energy surfaces.
As shown by Manthe and Köppel, this simple ansatz yields
quite accurate results for numerous vibronically coupled
systems.11
The situation is somewhat different for spin-boson models comprising many weakly interacting modes. Here the initial nonequilibrium preparation of the system does not lead
to an excitation of the bath, because the available energy per
bath mode goes to zero for an infinite bath. Bath modes with
a frequency higher than the temperature therefore remain in
their vibronic ground state. The question whether the classical system is ergodic or not, therefore strongly depends on
the nuclear initial conditions imposed. Employing, for example, Boltzmann initial conditions which only allow for
thermal excitations, the classical spin-boson system under
consideration are ergodic, that is, dynamical calculations and
phase-space averages do agree. Using action-angle or
Wigner initial conditions for the bath modes, on the other
hand, the high-frequent modes tend to keep the ZPE initially
included, thus undergoing quasiperiodic motion. As a consequence, the system is not ergodic and the long-time limits
obtained by trajectory calculations may be different from
those calculated via phase-space averages.
Finally, it is instructive to make contact to the work of
Heller, who addressed the question of quantum localization
versus quantum ergodicity.21 To this end, he defined the ratio
F5N* /N` , whereby N` denotes the energetically accessible phase-space volume and N* represents the phase-space
volume actually visited by the system under a given initial
preparation. That is, a small value of F indicates localization,
while F approaching one indicate ergodicity. As an estimate
of N` , one may employ a maximum-entropy analysis43 or
calculate the phase-space volume that is accessible after the
initial decay of the autocorrelation function of the system.21
Alternatively, one may use the total level number of the system at the maximal energy e max assessed by the spectral
density, i.e., N` 5N( e max ). As suggested by Heller, the actually visited phase-space volume is given by N*
5( ( n p 2n ) 21 , whereby the sum goes over all Franck–Condon
factors p n describing the initial preparation of the system.
For the two-state three-mode model of pyrazine we thus obtain N ` '103 and N* '200, that is, approximately one fifth
of the energetically accessible phase space is actually visited
by the system. Comparing this value to F5 31 obtained in the
case of Gaussian random eigenfunctions,21 the two-state
U. Müller and G. Stock
three-mode model of pyrazine is found to approach the limit
of a quantum-mechanically ergodic system.
VII. CONCLUSIONS
Classical trajectory calculations comprising highfrequent degrees of freedom can be very sensitive to the
amount of ZPE excitation included in the simulation. That is,
including no ZPE considerably underestimates the relaxation, while including the full ZPE may lead to serious artifacts due to unrestricted flow of ZPE. In this work, we have
pursued the natural ansatz of employing a reduced ZPE
g e ZP . Establishing a connection between the level density
of a system and its relaxation behavior, we have obtained
several criteria @Eqs. ~2.8!–~2.10!# which allow us to explicitly calculate the quantum correction g. By construction, this
strategy assures that the classical calculation attains the correct long-time values and, as a special case thereof, that the
ZPE is treated properly.
As a stringent test of this concept, we have adopted the
mapping formulation of nonadiabatic quantum dynamics.8
Resulting in negative population probabilities, the quasiclassical implementation of this theory significantly suffers from
spurious flow of ZPE. Employing a molecular two-state
three-mode model undergoing ultrafast electronic and vibrational relaxation, the computational results shown in Figs.
1–3 provide a comprehensive numerical demonstration of
the connection between level densities and relaxation behavior. That is, the ZPE problem of negative population probabilities indeed vanishes, if the quantum correction g is chosen such that the quantum-mechanical state-specific level
densities are reproduced within the energy range defined by
the spectral density. Moreover, the complete time evolution
of the classical simulations is found to be in very good agreement with exact time-dependent wave-packet propagations.
The latter finding is particularly remarkable in the light of
the fact that the criteria imposed merely guarantee the
equivalence of equilibrium values, but do not establish the
equivalence of the dynamics.
Employing the concept introduced above to the simulation of complex systems that defy a quantum-mechanical
treatment, one has to establish a criterion to determine g that
does not rely on a quantum reference calculation. A possibility is to utilize long-time mean values of the system that
are known a priori. The validity of this scheme has been
demonstrated by considering the quantum and classical relaxation behavior of several spin-boson-type models describing photoinduced electron transfer. To calculate the quantum
correction, we have assumed that the system decays completely in its adiabatic ground state. Assuming furthermore
classical ergodicity, the ZPE correction is readily obtained
via Eqs. ~2.13! and ~6.1!, i.e., without any dynamical simulations. Employing this procedure, the ZPE-adjusted simulations are indeed found to be the best possible classical description in the sense that the ZPE is treated properly and the
time-dependent quantum dynamics is reproduced.
It should be stressed that, at least for the systems considered, it is essential to employ criteria as well as quantum
corrections that exclusively refer to the ensemble average
rather than to the behavior of individual trajectories. As has
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J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
Flow of zero-point energy. II
been demonstrated in Fig. 4, most of the trajectories in ZPEadjusted simulations individually violate the ZPE condition.
However, these simulations correctly account for the mean
ZPE content and also reproduce to a large extent the true
quantum dynamics. The ZPE content of individual trajectories therefore has no physical significance in nonequilibrium
simulations.
Finally, it is instructive to compare the ZPE-adjusted
classical mapping calculations to other trajectory-based
methods that describe nonadiabatic quantum dynamics,25–31
most notably the classical-path ~or mean-field! model26 and
the surface-hopping model.27 As shown in Ref. 9, the
classical-path method emerges from the classical limit of the
mapping formalism if the ZPE excitations of the quantum
oscillators are neglected, i.e., for g 50. This limiting case has
been found to generally underestimate the true quantum relaxation, particularly if more than two quantum states are
involved.9,25 Although the surface-hopping technique in
some case represents an improvement over the classical-path
model,25,27,30 it has been found to yield rather similar results
when applied to the description of nonadiabatic bound-state
processes.16 For example, both methods significantly fail to
reproduce the correct branching ratio at the ~avoided! curve
crossing of the spin-boson model shown in Fig. 5, while the
ZPE-adjusted mapping results are in excellent agreement
with quantum reference data. Since all these methods in
some sense overstress the classical limit ~e.g., ZPE excitation
in the mapping approach, classically forbidden transitions in
surface-hopping calculations!, however, it is not easy to predict which method will perform better for a given problem.
where N n 5 21 (X 2n 1 P 2n 2 g ) and e ZP5 21 (12G) ( j v j . As explained above, we have included the reduced ZPE 21 G v j and
1
2 g V n into the nuclear $ x j ,p j % and electronic $ X n , P n % degrees of freedom, respectively. Here (0<G, g <1) and we
restrict ourselves to the limiting cases G51 and G5 g .
To facilitate the nuclear integrations, we perform a
change of nuclear variables:
x̃ j 5 Av j /2~ x j 1 d x j ! ,
~A5!
p̃ j 5 Av j /2p j ,
~A6!
thus yielding for Eq. ~A1!:
A ~ e ! 5C M
E
F
dG 8 A ~ G 8 ! d e 2
G
(j ~ p̃ j 2 1x̃ j 2 ! 2E X P ,
x̃ j 5R f j ~ w i , u k !
~ i51•••M ! ,
~A8!
p̃ j 5Rg j ~ w i , u k !
~ k52•••M ! ,
~A9!
R 25
(j ~ p j 2 1x̃ j 2 ! ,
~A10!
where the hyper-radius R corresponds to the nuclear energy
of the system and w i P @ 0,2p ) and u k P @ 0,p /2) represent
generalized polar and azimuth angles, respectively. The
quantities f j ( w i , u k ) and g j ( w i , u k ) are given by trigonometric functions. Performing the R integration, we finally obtain
E
21
dX dP dV R 2M
max J ~ V ! A ~ X, P,V,R max ! ,
~A11!
ACKNOWLEDGMENTS
We thank Adrian Alscher and Michael Thoss for numerous helpful discussions. This work has been supported by the
Deutsche Forschungsgemeinschaft.
APPENDIX
where we have defined V5 w 1 ... w M u 2 ... u M , R max
5Ae 2E X P U( e 2E X P ), and J(V) denotes the Jacobi determinant of the transformation. Assuming, furthermore, that
the observable A does not depend on the nuclear coordinates,
the V integration can be performed to give
The purpose of this appendix is to outline the calculation
of phase-space integrals of the form
E
dG A ~ G ! d @ e 2H ~ G !# ,
~A7!
where C M 5 @ 2 p M 11 (P j v j ) # 21 and dG 8 5dX dP dx̃ dp̃. We
introduce generalized spherical coordinates45
A ~ e ! 5C M
A~ e !5
87
~A1!
where G comprises the phase-space variables of the problem.
For simplicity, we restrict ourselves to the case of an electronic two-state system with constant interstate coupling.
The generalization to an N-state system with coordinatedependent interstate coupling is straightforward. The classical mapping Hamiltonian Eq. ~3.3! can be written as (\
[1)
A ~ e ! 5Ć M
E
dX dP A ~ X, P !~ e 2E X P ! M 21 U ~ e 2E X P ! ,
~A12!
21
with Ć M 5 @ (M 21)!(P j v j ) p /M # . Since the total electronic population is preserved ( ( n N n 51), Eq. ~A12! reduces to a two-dimensional integration over electronic
action-angle variables.9 The structure of the phase-space integral for an electronic N-level system is completely equivalent to Eq. ~A12! and requires a 2(N21)-dimensional integration. The calculation of phase-space averages of
observables explicitly depending on nuclear coordinates, in
general, requires a Monte Carlo evaluation of Eq. ~A11!.
M
H5
1
2
(
j51
dx j5
v j @ p 2j 1 ~ x j 1 d x j ! 2 # 1E X P ,
(
N n k (n)
j /v j ,
(
N n E n 2 12
n51,2
~A2!
~A3!
M
E X P5
n51,2
(
j51
v j d x 2j 1g ~ X 1 X 2 1 P 1 P 2 ! ,
~A4!
1
See, for example, L. M. Raff and D. L. Thompson, in Theory of Chemical
Reaction Dynamics, edited by M. Baer ~Chemical Rubber, Boca Raton,
FL, 1985!, Vol. 3; Advances in Classical Trajectory Methods, edited by
W. L. Hase ~Jai, London, 1992!, Vol. 1.
2
For a general discussion and an overview of existing methods, see Y. Guo,
D. L. Thompson, and T. D. Sewell, J. Chem. Phys. 104, 576 ~1996!; C.
Schlier, ibid. 103, 1989 ~1995!; T. D. Sewell, D. L. Thompson, J. D.
Gezelter, and W. H. Miller, Chem. Phys. Lett. 193, 512 ~1992!.
3
J. M. Browman, B. Gazdy, and Q. Sun, J. Chem. Phys. 91, 2859 ~1989!;
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88
J. Chem. Phys., Vol. 111, No. 1, 1 July 1999
W. H. Miller, W. L. Hase, and C. L. Darling, ibid. 91, 2863 ~1989!; G. H.
Peslherbe and W. L. Hase, ibid. 100, 1179 ~1994!; D. A. McCormack and
K. F. Lim, ibid. 106, 572 ~1997!; M. Ben-Nun and R. D. Levine, ibid. 105,
8136 ~1996!.
4
T. Uzer, B. D. MacDonald, Y. Guan, and D. L. Thompson, Chem. Phys.
Lett. 152, 405 ~1988!; D. Lu and W. L. Hase, J. Chem. Phys. 91, 7490
~1989!.
5
G. Nyman and J. Davidson, J. Chem. Phys. 92, 2415 ~1990!; A. J. C.
Varandas, ibid. 99, 1076 ~1993!; A. J. Marks, ibid. 108, 1438 ~1998!.
6
A semiclassical treatment of the ZPE problem has been suggested in R.
Alimi, A. Garcı́-Vela, and R. B. Gerber, J. Chem. Phys. 96, 2034 ~1992!.
7
G. Stock and U. Müller, J. Chem. Phys. 111, 65 ~1999!, preceding paper.
8
G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 ~1997!; M. Thoss and G.
Stock, Phys. Rev. A 59, 64 ~1999!.
9
U. Müller and G. Stock, J. Chem. Phys. 108, 7516 ~1998!.
10
R. Schneider, W. Domcke, and H. Köppel, J. Chem. Phys. 92, 1045
~1990!.
11
U. Manthe and H. Köppel, J. Chem. Phys. 93, 1658 ~1990!.
12
W. Domcke and G. Stock, Adv. Chem. Phys. 100, 1 ~1997!.
13
M. D. Prasad, Chem. Phys. Lett. 194, 27 ~1992!.
14
G. S. Latha and M. D. Prasad, J. Chem. Phys. 105, 2972 ~1996!.
15
G. Stock, J. Chem. Phys. 103, 2888 ~1995!.
16
U. Müller and G. Stock, J. Chem. Phys. 107, 6230 ~1997!.
17
U. Weiss, Quantum Dissipative Systems ~World Scientific, Singapore,
1993!.
18
N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4600 ~1995!.
19
E. Sim and N. Makri, J. Phys. Chem. 101, 5446 ~1997!.
20
I. E. Farquhar, Ergodic Theory in Statistical Mechanics ~Wiley, London,
1964!.
21
E. J. Heller, Phys. Rev. A 35, 1360 ~1987!.
22
M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics ~Springer,
New York, 1990!.
23
E. Thiele, J. Chem. Phys. 39, 3258 ~1963!.
24
P. C. Haarhoff, Mol. Phys. 6, 337 ~1963!; 7, 101 ~1963!; W. Forst and Z.
Prás̆il, J. Chem. Phys. 51, 3006 ~1969!; M. R. Hoare and T. W. Ruijgrok,
ibid. 52, 113 ~1970!.
25
For a recent overview, see articles in Quantum and Classical Dynamics in
Condensed Phase Simulations, edited by B. J. Berne, G. Ciccotti, and D.
F. Coker ~World Scientific, Singapore, 1998!.
26
N. F. Mott, Proc. Cambridge Philos. Soc. 27, 553 ~1931!; J. B. Delos and
W. R. Thorson, Phys. Rev. A 6, 720 ~1972!; G. D. Billing, Chem. Phys.
Lett. 30, 391 ~1975!; R. B. Gerber, V. Buch, and M. A. Ratner, J. Chem.
Phys. 77, 3022 ~1982!; D. A. Micha, ibid. 78, 7138 ~1983!.
27
J. C. Tully and R. K. Preston, J. Chem. Phys. 55, 562 ~1971!; M. F.
Herman, ibid. 81, 754 ~1984!; J. C. Tully, ibid. 93, 1061 ~1990!; D. F.
Coker and L. Xiao, ibid. 102, 496 ~1995!; J.-Y. Fang and S. HammesSchiffer, ibid. 107, 8933 ~1997!.
U. Müller and G. Stock
28
H.-D. Meyer and W. H. Miller, J. Chem. Phys. 70, 3214 ~1979!; 71, 2156
~1979!; 72, 2272 ~1980!.
29
P. Jungwirth and R. B. Gerber, J. Chem. Phys. 102, 6046 ~1995!; T. J.
Martinez, M. Ben-Nun, and R. D. Levine, J. Phys. Chem. 100, 7884
~1996!; A. Ferretti, A. Lami, and G. Villani, J. Chem. Phys. 106, 934
~1997!; C. C. Martens and J.-Y. Fang, ibid. 106, 4918 ~1997!.
30
Mixed quantum-classical descriptions of nonadiabatic dynamics associated with conical intersections have been discussed in Refs. 15 and 16 and
in H.-D. Meyer, Chem. Phys. 82, 199 ~1983!; A. Ferretti, G. Grannucci,
A. Lami, M. Persico, and G. Villani, J. Chem. Phys. 104, 5517 ~1996!; M.
S. Topaler, T. C. Allison, D. W. Schwenke, and D. G. Truhlar, J. Chem.
Phys. 109, 3321 ~1998!; 110, 687 ~1999!.
31
Mixed quantum-classical descriptions of spin-boson relaxation dynamics
have been discussed in Refs. 16,34, and 44.
32
J. Schwinger, in Quantum Theory of Angular Momentum, edited by L. C.
Biedenharn and H. V. Dam ~Academic, New York, 1965!.
33
X. Sun and W. H. Miller, J. Chem. Phys. 106, 6346 ~1997!.
34
X. Sun, H. Wang, and W. H. Miller, J. Chem. Phys. 109, 7064 ~1998!.
35
E. Wigner, Phys. Rev. 40, 749 ~1932!; M. Hillery, R. F. O’Connell, M. O.
Scully, and E. P. Wigner, Phys. Rep. 106, 122 ~1984!.
36
H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59
~1984!.
37
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.
Numerical Recipies, 2nd ed. ~Cambridge University Press, Cambridge,
1992!.
di
38
We have determined P di
2 (`) by averaging P 2 (t) for 500 fs <t<1500 fs.
di
The result for P 2 (`) can be further improved by fitting the state-specific
level densities in the full energy range 4.0 eV, e ,5.5 eV. This results in
g 50.35 and P di
2 (`)50.32.
39
A similar conclusion was reached empirically in Ref. 15. Employing the
classical electron analog model ~Ref. 28!, it was found that the relaxation
behavior sensitively depends on the ‘‘Langer-like’’ modification of the
classical model. Within the mapping formulation, these modifications
have been identified to correspond to the ZPE of the quantum variables
~Ref. 9!.
40
H. Köppel, Chem. Phys. Lett. 205, 361 ~1993!.
41
B. Wolfseder and W. Domcke, Chem. Phys. Lett. 259, 113 ~1996!.
42
D. M. Leitner, H. Köppel, and L. S. Cederbaum, J. Chem. Phys. 104, 434
~1996!; T. Zimmermann, H. Köppel, and L. S. Cederbaum, ibid. 91, 3934
~1989!.
43
F. Remacle and R. D. Levine, J. Chem. Phys. 98, 2144 ~1993!; V. B.
Pavlov-Verevkin, and J. C. Lorquet, ibid. 104, 1362 ~1996!.
44
G. Stock, J. Chem. Phys. 103, 1561 ~1995!.
45
A. O. Barut and R. Raczka, Theory of Group Representation and Applications ~World Scientific, Singapore, 1986!.