Download Classical phase-space analysis of vibronically coupled systems

Classical phase-space analysis of vibronically coupled systems
Stefan Dilthey† , Bernhard Mehlig∗ and Gerhard Stock†
† Institute
of Physical and Theoretical Chemistry, J. W. Goethe University, Marie-Curie-Str. 11,
D-60439 Frankfurt, Germany
∗ Complex Systems, Physics & Eng. Physics, Chalmers/GU, 41296 Gothenburg, Sweden
(November 29, 2001)
Abstract
We use the mapping-formalism to obtain a classical analogon of a quantummechanical vibronic-coupling Hamiltonian. For this Hamiltonian, a detailed
phase-space analysis is performed, enabling us to gain an intuitive physical
picture of the quantum dynamics. We propose a simple classical model for
the time-dependence of the populations on diabatic/adiabatic potentials. The
results are compared to numerical quantum-mechanical calculations. It turns
out that the classical model describes the quantum recurrences well, up to
times of the order of the Heisenberg time.
1
I. INTRODUCTION
Driven by significant progress in nonlinear dynamics, the investigation of the correspondence of quantum and classical dynamics represents an active field of research [1,2]. For
example, there has been considerable interest in generalising intrinsically classical concepts
to the realm of quantum mechanics. Prominent examples include the description of dynamics in phase space, the condition of ergodicity, and the notion of deterministic chaos.
On the other hand, one may be interested in the classical description of quantum systems
which do not have an obvious classical analogon. For example, consider the well-known
spin-boson problem, that is, an electronic two-state system (the spin) coupled to one or
many vibrational degrees of freedom (the bosons) [3]. Exhibiting nonadiabatic transitions
between discrete quantum states, the model apparently defies a straightforward classical
treatment.
In order to incorporate quantum degrees of freedom (DoF) into a classical formulation, a number of mixed quantum-classical models have been proposed. Following the work
of Landau, Zener, and Stückelberg [4–6], for example, one may employ a “surface-hopping”
ansatz to describe nonadiabatic transitions between coupled potential-energy surfaces [7–13].
Alternatively, a quantum-classical description may be derived by starting with a quantummechanically exact formulation for the complete system and performing a partial classical
limit for the heavy-particle DoF. This procedure is not unique, however, since it depends
on the particular dynamical formulation chosen as well as on the specific way to achieve
the classical limit. Well-known examples are the Ehrenfest mean-field limit in the wavefunction formulation [14–19], the quantum-classical Liouville equation for the density operator [20–25], and the stationary-phase approximation in the path-integral formulation [26,27].
Since electronic and nuclear dynamics are treated on a different dynamical footing, however,
quantum-classical models may not necessarily provide a satisfying classical picture of nonadiabatic dynamics. For example, it is not clear how to define vibronic surfaces-of-section
(SOS) or vibronic periodic orbits.
To overcome this problem, one may invoke a classical model for the electronic DoF.
This can be achieved, for example, by modeling the quantum-mechanical spin in terms of a
classical angular momentum [28–30], or by exploiting the formal equivalence of Schrödinger’s
equation for an N -level system and Hamilton’s equation for N classical oscillators [31–33].
Employing furthermore Ehrenfest’s classical limit to the nuclear DoF, one formally obtains
a classical treatment of both electronic and nuclear DoF. Most notably, this ansatz was
pursued in various “classical models of electronic DoF” due to McCurdy, Meyer and Miller
[28,33–36]. These formulations are in many aspects similar to a mixed quantum-classical
mean-field description. However, they allow us to define and to analyse the phase space of
the complete vibronic problem [37–40].
Although the idea of a classical analogon of quantum DoF is conceptionally appealing,
the approach is not completely satisfying from a theoretical point of view. Starting out with
an approximate classical (rather than an exact quantum-mechanical) formulation, there
are two interrelated problems: The nature of the approximations involved is difficult to
specify and the formulations are not unique, i.e., various analogies result in different classical
models. To avoid these problems, the equivalence of discrete and continuous DoF should
be established on the quantum-mechanical (rather than on a classical) level. This can be
2
achieved, for example, by employing quantum-mechanical bosonisation techniques such as
the Holstein-Primakoff transformation [41] and Schwinger’s theory of angular momentum
[42]. Representing spin operators by boson operators, the discrete quantum DoF are hereby
mapped onto continuous variables. Since the latter posses a well-defined classical limit, the
problem of a classical treatment of discrete quantum DoF is bypassed. Exploiting this idea,
recently a “mapping approach” to the semiclassical description of nonadiabatic dynamics
has been proposed [43,44]. The approach consists of two steps: A quantum-mechanical exact
transformation of discrete onto continuous DoF (the “mapping”) and a standard classical
or semiclassical treatment of the resulting dynamical problem. On a semiclassical level, this
means to evaluate the Van-Vleck Gutzwiller propagator [1], on a purely classical level, this
means that the observables of interest are evaluated with standard quasiclassical sampling
techniques.
In recent works, the semiclassical [43–46] and quasiclassical [47,48] evaluation of the
mapping formulation has been investigated in detail. In extension to the usual quantumclassical ansatz, it has be shown that the classical limit of the mapping formulation contains a
zero-point energy term, that accounts for quantum fluctuations in the electronic DoF [47,48].
As a consequence, the classical evaluation of the mapping formulation generally leads to a
considerably higher level density and to a more efficient nonadiabatic relaxation behavior.
In particular, detailed analytical [49] and numerical [47,48] studies have shown that the
classical mapping method in many cases represents a significantly better approximation to
the exact quantum result than the conventional ansatz.
In this work we are concerned with a phase-space analysis of the classical dynamics
exhibited by a mapped vibronic-coupling problem. The outline of the paper is as follows.
Section II briefly reviews the mapping formalism and applies the formulation to a spin-boson
model with a single vibrational mode. The parameters of the model are chosen to reflect
the situation of a photo induced electron transfer promoted by a high-frequency vibrational
mode. Studying the corresponding SOS under various conditions, Sec. III is concerned with
a detailed analysis of the classical phase space. In particular, the periodic orbits of the
vibronic problem are investigated in some detail [50]. Section IV demonstrates that these
vibronic periodic orbits may show up as recurrences in various time-dependent observables
that reflect the nonadiabatic quantum dynamics of the system. Section V concludes.
3
II. MAPPING FORMALISM
As discussed in Ref. [44], there are various ways to achieve a mapping of discrete onto
continuous DoF, including several variants of the Holstein-Primakoff transformation [41],
Schwinger’s theory of angular momentum [42], and the representation through spin coherent
states [27,51]. Here, we briefly review the Schwinger-type bosonisation of an N -level system
[43], and then focus on the classical limit of this formulation in the case of a vibronically
coupled two-state system.
A. Quantum-mechanical theory
Consider a N-level system with basis states |ψn i (n, m = 1, . . . , N ) and the Hamiltonian
H=
X
n,m
hnm |ψn ihψm |.
(2.1)
In order to represent this system by N oscillators, we introduce the following mapping
relations for the operator and the basis states [43]
|ψn ihψm | → a†n am ,
|ψn i → |01 , . . . , 1n , . . . , 0N i.
(2.2a)
(2.2b)
Here an , a†m are harmonic-oscillator creation and annihilation operators with commutation
relations [an , a†m ] = δn,m and |01 , . . . , 1n , . . . , 0N i denotes a harmonic-oscillator eigenstate
with a single quantum excitation in the mode n. According to Eq. (2.2a), the bosonic
representation of the Hamiltonian (2.1) is given by
H=
X
hnm a†n am ,
(2.3)
n,m
It is easy to show that the mapping of the operators (2.2a) preserves the commutation
relations and leads to the exact identity of the electronic matrix elements of the propagator
(h̄ ≡ 1)
hψn |e−iHt |ψm i = h01 . . . 1n . . . 0N |e−iHt |01 . . . 1m . . . 0N i.
(2.4)
The image of the N -level Hilbert space is the subspace of the N -oscillator Hilbert space with
a single quantum excitation. This “physical” subspace is invariant under the action of any
operator which results by the mapping (2.2a) from an arbitrary N -level system operator.
B. Classical limit
The mapping Hamiltonian (2.3) contains products of non-commuting operators, which
should be symmetrised in order to perform the classical limit. Equivalently, one may introduce Cartesian electronic variables
√
X̂n = (a†n + an )/ 2,
(2.5a)
√
†
Pˆn = i(an − an )/ 2,
(2.5b)
4
thus yielding
c=
H
1
2
X
n
hnn (X̂n2 + P̂n2 − 1) +
X
hnm (X̂n X̂m + P̂n P̂m ),
(2.6)
n6=m
where we have used hats on top of the operators in order to emphasize that Eq. (2.6) is
still an exact quantum-mechanical expression. The classical limit of Eq. (2.6) can then be
obtained by simply changing from the Heisenberg operators X̂n , P̂n to the corresponding
classical functions Xn , Pn
H=
1
2
X
n
hnn (Xn2 + Pn2 − 1) +
X
hnm (Xn Xm + Pn Pm ).
(2.7)
n6=m
As is well known, the semiclassical Van-Vleck-Gutzwiller approximation is exact if the
Hamiltonian is quadratic. A semiclassical calculation employing the classical Hamiltonian
(2.7) with the initial (or boundary) condition (2.2b) therefore yields the exact quantummechanical result.
It is interesting to compare the classical limit of the mapping formalism [Eq. (2.7)] to
previous formulations, which employ a classical model in order to describe the discrete electronic DoF [28–40]. Since the correspondence is achieved at the classical level, the latter
formulations cannot establish a quantum-mechanical equivalence of discrete and continuous
representations. As a consequence, the classical model Hamiltonians neglect the commutator
P
[an , a†n ] = 1, which results in the − 21 n hnn term in the quantum Hamiltonian (2.6) describing the zero-point energy excitation of the electronic oscillators [47]. When trying to achieve
meaningful semiclassical quantization conditions for their classical “electron-analog” model,
Meyer and Miller where the first to realize this deficiency of the classical formulation
√
√ [33]. As
a remedy, they subsequently invoked “Langer-like modifications” (i.e. 2nk → 2nk + 1)
to the off-diagonal elements of the Hamiltonian function, which retain the zero-point energy
term mentioned above. As is shown in sec. IV A, the proper inclusion of electronic zero-point
energy excitation is crucial for overall performance of the classical approach [47,48].
C. Constants of motion
In the following we wish to focus on the case of an electronic two-level system, which
requires two oscillator DoF within the mapping formalism. In various applications, however,
it would be advantageous to reduce the number of DoF to a single electronic oscillator. With
this end in mind, we notice that the transformation relations (2.2) map the identity operator
in the discrete Hilbert space onto the constant of motion
X
n
a†n an = 1.
(2.8)
Eq. (2.8) simply states that the total electronic population is conserved. The existence of
this constant of motion is utilized by the Holstein-Primakoff transformation to eliminate one
boson DoF [41]. In the case of a two-level system with h12 = h21 , this leads to
HHP = h11 (1 − a† a) + h22 a† a + h12 a†
5
q
1 − a† a +
q
1 − a† a a .
(2.9)
While the classical Hamiltonian pertaining to the Schwinger representation is well defined
[see Eq. (2.7)], the classical limit of Eq. (2.9) is quite ambiguous. This is because the squareroot operator in Eq. (2.9) is only well defined in the physical subspace, but otherwise leads
to an imaginary contribution to the Hamiltonian [44].
Alternatively, one may try to eliminate one electronic DoF on the classical level. To this
end, it is advantageous to change to classical action-angle variables {nk , qk } defined by
√
2nk + 1 eiqk = Xk + iPk .
(2.10)
The classical version of Eq. (2.8) then seems to imply that
X
nk = 1.
(2.11)
k
P
The latter relation, however, only holds for the classical ensemble average, i.e., k hnk i =
1. Requiring (2.11) for each individual trajectory of the ensemble, in fact, represents an
approximation. Employing this approximation, one may eliminate one DoF by introducing
the variables n = n2 = 1 − n1 , q = q2 − q1 [33]. This yields the Hamiltonian
q
H = (1 − n)h11 + nh22 + h12 (2n + 1)(3 − 2n) cos q,
(2.12)
which represents the two-level system in terms of two classical variables {n, q}. Allowing
P
for k nk = N , where N is a constant that may be different for each trajectory and has the
mean value hN i = 1, one may omit the approximation (2.11), thus obtaining a formulation
with three variables.
We note in passing that unitary transformations such as (2.10) and the performance
of the classical limit do not commute in general. For example, there exist no quantum
Hamiltonian corresponding to Eq. (2.12). This is because of well-known difficulties in the
definition of a proper quantum-mechanical phase operator q̂ [52]. Rewriting the classical
phase q in (2.12) in terms of Cartesian variables, on the other hand, one obtains
√
1
1
H = (3 − X 2 − P 2 )h11 + (X 2 + P 2 − 1)h22 + h12 X 4 − X 2 − P 2 ,
2
2
(2.13)
which does possess a corresponding quantum Hamiltonian.
D. Model system
To apply the above formalism to vibronically coupled molecular systems, we identify the
|ψn i with diabatic electronic states and the hnm with operators of the nuclear dynamics. As a
simple but nontrivial model system we adopt an electronic two-state system (n = 1, 2) with
a single vibrational mode and constant interstate coupling h12 = h21 = g. The diagonal
electronic matrix elements hnn can be written as a sum of kinetic energy and diabatic
potentials
hnn = 12 ωp2 + Vn ,
Vn = 21 ωx2 + κn x.
6
(2.14)
Here x denotes the dimensionless position of the vibrational mode, ω is its vibrational
frequency, and κn denotes the linear coordinate shift in the electronic state |ψn i. Choosing
κ1 = −κ2 ≡ κ, the model (2.14) is equivalent to the well-know spin-boson problem with a
single vibrational mode [3].
For interpretational purposes it is instructive to change from the diabatic electronic
representation with basis states |ψn i to the adiabatic representation with basis states
|ψnad i =
X
m
Snm |ψm i.
(2.15)
The unitary transformation S = (Snm ) diagonalises the diabatic potential matrix to give
the corresponding adiabatic potential-energy curves
W1/2 = 21 (V1 + V2 ) ∓
1
2
q
(V2 − V1 )2 + 4g 2 .
(2.16)
Fig. 1 shows the diabatic (full lines) and adiabatic (dashed lines) potential-energy curves
of a model with the parameters g = 0.1 eV, ω = g, κ = 0.5g. At x = 0, the diabatic
potentials are seen to intersect, while the adiabatic potentials exhibit an avoided crossing.
Note that the diabatic potentials interchange (V1 ↔ V2 ) upon reflection at the axis defined by
x = 0. The adiabatic potentials, on the other hand, are symmetric with respect to this axis.
For the periodic-orbit analysis given below it is helpful to note that the adiabatic potentials
are well described by harmonic oscillators with the vibrational frequencies ω1ad = 0.082 eV
and ω2ad = 0.118 eV. This corresponds to the vibrational periods of 35 fs on the upper and
50 fs on the lower adiabatic potential. The vibrational period corresponding to the diabatic
frequency ω = 0.1 eV is T = 41 fs.
Within the mapping formulation, the classical limit of the spin-boson problem may be obtained by (i) representing the two diabatic electronic states |ψn i by boson operators X̂n , P̂n
and (ii) changing from both electronic and vibrational Heisenberg operators to the corresponding classical functions. Changing to electronic action-angle variables and employing
the approximation (2.11), we finally obtain
H=
q
ω 2
(p + x2 ) + (2n − 1)κx + g (2n + 1)(3 − 2n) cos q.
2
(2.17)
The Hamiltonian (2.17) is used in the phase-space analysis of the classical dynamics
presented in the next section. Hereby, we are particularly interested in the situation that
the system is initially prepared in into a nonequilibrium vibronic state. Such a preparation
of the system can be achieved, for example, by assuming that the molecule is impulsively
excited from its electronic and vibrational ground state to the optically bright electronic state
|ψ2 i. As is schematically illustrated in Fig. 1, the photo excitation creates a vibrational
wave packet, which evolves on the coupled potential-energy surfaces, thereby undergoing
nonadiabatic transitions [53]. In the present calculations we have assumed a Gaussian initial
distribution centered at x(0) = 3, which corresponds to the ground state of the harmonicoscillator potential.
7
III. ANALYSIS OF THE CLASSICAL DYNAMICS
In the following we analyse the classical dynamics of our model (2.17) by examining
Poincaré SOS [54]. The SOS are obtained by fixing q and E and plotting the points of
intersection of the classical trajectories with this surface (q = 0, q̇ > 0). In subsection
III A we discuss SOS for different values of q and E, and examine their symmetries and the
development of regular and chaotic regions when the parameters are varied. In subsection
III B we describe the properties of some of the shortest periodic orbits found on the SOS.
A. Surfaces-of-section
Our aim is to understand the dynamics of the classical Hamilton function (2.17) obtained
by the mapping-formalism for our model potential. Fig. 3 (a) shows a SOS for q = 0 (and
q̇ > 0) at E = 0.65eV= V2 (x = 3) + 21 ω. This corresponds to an excitation onto the upper
diabatic potential curve at x = 3. 1 Plotted are the nuclear degrees of freedom (x,p). The
boundaries mark the energetically accessible phase space and are discussed in more detail
in appendix V A.
The SOS in Fig. 3 (a) shows that our model exhibits mixed classical dynamics: Most of
the area of the energetically available phase space belongs to chaotic motion, but there are
also some islands of integrability. The dynamics in our model is of the type first described by
Percival. [55] The islands of regular motion contribute in two ways to phase-space averages
of dynamical correlation functions: the islands contribute directly, as discussed by Percival,
as parts of phase-space volume to be integrated over. Furthermore, however, dynamical
correlations corresponding to the irregular (ergodic) part are significantly influenced by
trajectories sticking to hierarchical phase-space structures at the boundaries of the islands
[56]. Thus, in order to discuss possible recurrences in the quantum dynamics, it is necessary
to understand the dynamics on the regular islands.
In order to obtain a better overview of the phase space and to ensure that we do not
miss important features by restricting the discussion to the value q = 0, further plots of
surfaces-of-section are presented for different values of the angle-coordinate q. We observe a
deformation of the integrable islands with changing q (Fig. 3), leading to an again symmetric
situation for q = π.
Since the potentials (Fig. 1) are symmetric, the SOS exhibits this symmetry, too. To
be more specific, we have to distinguish between two different symmetries: First, there is a
time-reversal symmetry, that is p → −p and q̇ → −q̇ at the same time. But as we restrict
ourselves to SOS in the direction q̇ > 0 this cannot be seen in our plots. Second, the
transformation x → −x results in an exchange of the potentials, i.e. n → 1 − n. q̇ depends
on 1 − 2n and changes the sign under this symmetry operation. As a result, the surfacesof-section at q = 0 for two different directions of q̇ are connected by mirror symmetry with
respect to x = 0.
1 The
variable x is dimensionless because all units are absorbed in the parameters of the model.
8
In a next step, the energy of the system has been varied and surfaces-of-section are
shown in Fig. 4 for q = 0, q̇ > 0 and E = 0.1, 0.2, 0.3, 0.65, 1.0 and 1.5eV. As energy is
increased we observe two opposite effects: First, the resulting phase space projects onto the
SOS in the approximate form of an ellipse because at high energies the motion is governed
by the oscillator potential. Second, the ellipse consists of a narrow band of irregular motion.
Decreasing the energy on the other hand leads to a higher ratio of integrable islands compared
to the chaotic part of the SOS. At very low energies, E ≤ 0.2eV, the qualitative picture
changes: the motion is mostly integrable and the forbidden area in the middle of the ellipse
has vanished (see V A). Coming back to the energy E = 0.65eV which corresponds to our
quantum initial condition, we show the positions of some tori and isolated periodic orbits
in the SOS (q = 0), cf. Fig. 5. We have found two isolated orbits of period 1, labeled as
Ia and Ib, with their fixed points lying on the p = 0-axis and x0 = 3.33 and x0 = −2.75
respectively. The period-2-orbits (IIa and IIb) are found at p0 = ±2.35 and the fixed points
of the period-3-orbits build a triangle in the SOS with Ia in the center of it. An enlargement
of this large island is plotted in Fig. 5 (b) and the fixed points of the orbits are marked. In
the following section we discuss these orbits in more detail. Their properties are listed in
table I.
B. Periodic orbits
Some of the periodic orbits found for the energy E = 0.65eV are shown in Fig. 6 (compare
table I). Plotted on the left side is the coordinate n against the nuclear degree of freedom
x. As pointed out above, n represents the diabatic population of the upper electronic state
ψ2 . The same is done in the adiabatic representation and the results are shown on the right
hand side of Fig. 6. In the following, the interpretation of the graphs will be discussed in an
exemplary fashion in some detail for the Ib orbit: It is self-retracing and oscillates between
two turning points at x ≈ ±2.75. The diabatic potential changes whenever the point x = 0
is crossed. For positive x, the potential V2 is populated, while negative x correspond to
population of V1 . This means that the dynamics always takes place on the upper potential
(see Fig. 1), which results in the adiabatic picture in an oscillation on the upper adiabatic
potential V2ad . Correspondingly the period time of 36.4fs for the Ib-orbit is comparable to
the period T2ad = 35fs obtained by a quadratic fit of the upper adiabatic potential.
The two periodic orbits IIa and IIb show just the opposite behavior: Starting for positive
x on the lower diabatic potential V1 they change to V2 while moving to negative values of
x. So they both are mainly localized on the lower adiabatic potential and have a period of
46.3fs in good agreement with the calculated value of T1ad = 50fs (see sec.II).
All orbits reflect the symmetry of the potentials. In the adiabatic representation they are
symmetric with respect to the axis x = 0, while the diabatic orbits exhibit point symmetry
around (x = 0, n = 0.5) since reflection at x = 0 corresponds to a change of the diabatic
potential surface. Therefore every diabatic periodic orbit possesses this symmetry-point or,
alternatively, there is a second orbit which is the result of a π-rotation of the old one. The
same is true for the adiabatic representation and the axis-symmetry. An example for this
are the orbits IIa and IIb. The initial values of this partner-orbit are (x1 , p1 ) = (−x0 , −p0 ).
Note that such a partner-orbit also exists for Va and VIIIa but is not shown here.
Comparing the higher period orbits III, V and VIII with Ia, it is obvious that their
9
dynamics is of the same type. Therefore they can all be thought as being composed of Ia
with slightly changed coordinates of the turning points. As a consequence the time of such
a period-k orbit which belongs to Ia is approximatively k times higher than the basic period
of Ia (39.2fs).
The question arises to which extent the periodic orbits in Fig. 6 found for the energy
E = 0.65eV depend on the choice of the energy E. To examine this question, the orbit Ia
has been observed for different energies in an iterative procedure: Its fixed point (x0 , p0 )
on the SOS belonging to the old energy is the starting point of the search-algorithm at the
new energy. The step size has been chosen to be ∆E = 0.01eV. The result is shown in
Fig. 7 where the parametric dependence of the period T is plotted over the energy-range
0.3eV< E < 1.0eV. The curve in Fig. 7 is smooth and the period T exhibits only a weak
dependence on the energy E.
The other period-1-orbit Ib cannot be traced to energies higher than Ec = 0.65eV since at
this energy, the starting point [x0 (Ec ), p0 (Ec )] collides with the boundary of accessible phase
space.
10
IV. CLASSICAL PHASE-SPACE ANALYSIS OF NONADIABATIC QUANTUM
DYNAMICS
In this section we compare quantum results with the classical approximation. As a first
example we discuss in subsection IV A the integral level density N (E) and show that it is
necessary to account for the whole electronic zero point energy in a classical calculation.
Subsection IV B is dedicated to time-dependent observables like diabatic and adiabatic population on the upper potential.
A. Level density
As a first test of the classical formulation, we consider the integral level density N (E)
of the model defined above, that is, the number of eigenstates at energy E. Quantummechanically, this quantity is defined as
N (E) = Tr Θ(H − E),
(4.1)
where Θ represents the Heaviside step function. Classically, N (E) can be approximated
through the calculation of the classical phase-space volume enclosed by the energy shell, i.e.,
NC (E) =
1
(2πh̄)f
Z
Z
dq dp Θ [E − H(q, p)] ,
(4.2)
with the coordinates q = {q1 , q2 , ..., qf } and their conjugated momenta p = {p1 , p2 , ..., pf },
where f denotes the number of DoF of the system under consideration. The integral level
density determines the relaxation dynamics of a system. Thus, it is clear that a useful
classical formulation should provide a good approximation of the quantum-mechanical level
density. As is shown in the Appendix, the classical level density NC (E) of the Hamiltonian
(2.17) can be evaluated in analytical form for energies which are high enough. In our model,
the condition
E ≥ (1 + Γ)g,
(4.3)
must be fulfilled. Here Γ stands for the electronic zero point energy (ZPE) and takes values
between 0 (if no zero point energy is considered) and 1 (if one accounts for the whole ZPE).
With the parameters of our model this threshold in (4.3) is Ethr = 0.1eV or Ethr = 0.2eV,
respectively. In the energy range above Ethr we get
!
(1 + Γ)2 κ2
1+Γ
E+
.
NC (E) =
ω
6
ω
(4.4)
For energies below this value, we cannot perform the phase-space integral analytically. The
classical level density NC (E) is therefore calculated numerically for E < Ethr . Fig. 2 shows
the quantum staircase function N (E) as obtained by diagonalisation of the Hamiltonian,
compared with two classical approximations. If no zero point energy is considered (Γ = 0),
we obtain the dotted curve which underestimates the quantum data. The dashed curve on
the other hand accounts for the whole ZPE (Γ = 1) and fits N (E) very well. This indicates
11
that the full inclusion of electronic zero point energy is necessary for an accurate classical
description. It should be noted that the linear energy dependence of N (E) predicted by
Eq.(4.4) only holds from the 4th energy level on.
Finally, Eq.(4.4) can be generalized to f nuclear degrees of freedom:


f
2 X
1+Γ  f
κ2i
f −1 (1 + Γ)
NC (E) = Qf
E + E
+ O(E f −2 ) .
6
ω
f ! i=1 ωi
i
i=1
(4.5)
In comparison to the well-known result for f uncoupled harmonic oscillators
Nosc (E) =
Ef
f!
Qf
i=1
(4.6)
ωi
the leading term in (4.5) has an additional factor of (1 + Γ).
B. Time-dependent nonadiabatic dynamics
In the following we consider the time-dependent dynamics of the model with an initial
density (operator) ρ̂0 = ρ̂(t = 0) of the form
ρ̂0 = |ψ2 i|φ0 ihφ0 |hψ2 |,
(4.7)
where hφ0 | is the initial nuclear wave function. This preparation of the system can be
achieved, for example, by assuming that the molecule is impulsively excited from its electronic and vibrational ground state to the optically bright electronic state |ψ2 i. As is
schematically illustrated in Fig. 1, the photo excitation creates a vibrational wave packet,
which evolves on the coupled potential-energy surfaces, thereby undergoing nonadiabatic
transitions [53]. In the present calculations we have assumed a Gaussian initial distribution
centered at x(0) = 3.
A useful quantity to describe this photoinduced non-Born-Oppenheimer dynamics is the
time-dependent population probability of the initially excited diabatic electronic state [58,53]
P di (t) = Tr (ρ̂(t)|ψ2 ihψ2 |) .
(4.8)
As discussed elsewhere [53], transitions between diabatic electronic states are important for
the interpretation of spectroscopic data. This is because in the vicinity of a surface crossing the electronic dipole transition operator is only smooth in the diabatic representation.
The adiabatic representation, on the other hand, is unique and is often advantageous for
the interpretation of nonadiabatic relaxation processes. The corresponding time-dependent
population probability of the upper adiabatic electronic state can be defined as
P ad (t) = Tr ρ̂(t)|ψ2ad ihψ2ad | .
(4.9)
In what follows, we show that the most significant features of the dynamics, as far as the
observables (4.8) and (4.9) are concerned, are already found by a simple classical approximation. The quantum recurrences can be understood in an intuitive and classical physical
picture. The expectation value of an observable Â(t) in the Heisenberg picture reads
12
h
hA(t)iQ = Tr ρ̂Â(t)
and in a classical approximation
1
hA(t)iC =
2π
Z
dx0
Z
i
dp0 ρ0 (x0 , p0 ) A[x(t), p(t)]
(4.10)
(4.11)
setting h̄ = 1. x(t), p(t) denote the coordinates of a trajectory at time t with initial conditions x0 , p0 . These initial conditions are sampled with a classical density function ρ0 (x0 , p0 )
realising the quantum-mechanical initial conditions ρ̂0 of (4.7).
The classical approximation (4.11) is adequate for short times. It must certainly fail for
beyond which quasi-periodic motion (due
times larger than the Heisenberg time tH = 2πh̄
∆
to the discreteness of the energy levels with mean spacing ∆) takes over. Moreover, quasiclassical calculations cannot account for the quantum-mechanical phases and therefore must
miss interference effects.
Our aim is now to further approximate (4.11) by extracting the relevant quasi-periodic
motion. In order to describe the quantum recurrences with our classical model we argue
that the integrable islands (cf. Fig. 3) represent the most significant contributions to the
dynamics of the observables of interest. The contribution of an integrable island can roughly
be approximated by the periodic orbit in the centre of it [56,57]. This is possible, if the
properties of the orbit remain nearly unchanged while varying the starting point from the
centre to the edge of the integrable island. In this case all trajectories started in this region
show a similar quasi-periodic behaviour which can be represented by the shortest periodic
orbit. A comparison of the periodic orbit Ia and those which are localized in its surrounding
(Fig. 6) underlines the great influence of Ia and indicates that the main features of the
dynamics can be found in this single orbit. The same is true for the other integrable islands
and their periodic orbits.
Accounting therefore in the calculation of (4.11) only for a finite number K of periodic
orbits [x(k) (τ ), p(k) (τ )] with x(k) (τ ) = x(k) (τ + T (k) ) and p(k) (τ ) = p(k) (τ + T (k) ), the expectation value of hA(t)iC can be approximated as a weighted sum of the contributions of single
orbits:
K
P
hA(t)iC ≈
hAk (t)i × wk with
k=1
hAk (t)i =
1
T (k)
TZ(k)
0
dτ
Z
dx0
Z
dp0 δ[x0 − x(k) (τ )] δ[p0 − p(k) (τ )]
ρ0 (x0 , p0 ) A[x(t), p(t)]
=
1
T (k)
T (k)
Z
dτ ρ0 [x(k) (τ ), p(k) (τ )] A[x(k) (t + τ ), p(k) (t + τ )].
(4.12)
0
The weight-factors wk account for the contributions of each orbit, i.e. they represent the
phase-space weight of the integrable islands the respective orbit belongs to. However, since
the phase-space volume of the integrable islands is not known to us and because of the
influence of different return times, the areas in the SOS only approximate these weights.
Therefore we restrict ourselves to an estimation of these prefactors, keeping in mind that
13
the exact dynamics can only be qualitatively reproduced by this ansatz. Nevertheless, it
should be mentioned that the choice of the wk of course does not influence the time of the
recurrences in our approximation.
In our model we considered the quantum-mechanical initial condition of excitation into
the upper diabatic potential energy surface (i.e. n0 = 1) at x0 = 3 and with p0 = 0. The
2
2
nuclear part of the density function is ρN
0 (x0 , p0 ) = exp[−(x − x0 ) − (p − p0 ) ] and the
electronic part has the same structure with the variables Xi , Pi introduced in the mapping
formalism. The transformation to the action-angle variables n, q used in our calculations
N
E
leads to ρE
0 (n) = exp[−2(n − n0 )]. We have ρ0 (x0 , p0 , n0 ) = ρ0 × ρ0 .
The results are shown in Figs. 8 and 9, where only the orbits Ia and Ib are taken into
account in the sum. Note that the period-2 orbits do not contribute since their starting
point is on the lower potential surface. As explained above the higher period orbits show
mainly the same behaviour as Ia. In Figs. 8 (a) and (b) the value of hn(t)i according to
(4.12) is plotted for the two period-1 orbits. Due to the consideration of the electronic zero
point energy the classical coordinate n takes values between − 12 Γ and 1 + 21 Γ. As the whole
ZPE is considered (Γ = 1 , cf. IV B) we have n ∈ [− 12 ; 32 ] which has been mapped into the
range of the quantum-mechanical observable [0, 1] for the purpose of a better comparability 2
. We add the contributions of the two single orbits with an estimated weight-factor of 20 for
Ia and compare the result with the quantum-mechanical one in Fig. 8 (c). These quantum
results are obtained by wave-packet propagation-propagation [59] and exhibit oscillations
in the diabatic population probability of the upper state. After an initial decay there are
several recurrences with a period time of ≈ 35fs modulated by three times faster oscillations
of ≈ 12fs. This indicates that the wave-packet often changes the diabatic potential energy
curve.
We notice a very good agreement of the periodic orbit-signal up to times of t = 100fs
. The quantum data shows a superposition of an initial decay within the first 25fs and a
quasi-periodic dynamics. The classical approximation models this oscillating part very well
and the positions of the peaks are reproduced correctly. On the other hand, the initial
decay of hn(t)i is not displayed correctly because the periodic orbit is not forced to start on
the upper diabatic potential curve as the quantum-mechanical wave-packet is. This initial
condition only enters in the classical density function ρ0 (x0 , p0 , n0 ) but since the orbit itself
never reaches the value n = 1 this can not be expected for the final result.
We conclude that the quantum recurrences can be modeled by the two shortest periodic
orbits only, and the time scales can be explained in the classical picture. The 35fs oscillation
corresponds to the period of the upper adiabatic potential and the shorter 12fs fluctuations
are Rabi oscillations belonging to the energy gap of the potentials. These Rabi oscillations
arise in the classical case as a fast movement in the n, q-subspace. Depending on the relative
phase to the slower oscillations in the x, p-subspace the periodic orbit Ia or Ib is obtained.
The same calculations are performed for another observable, the population on the
adiabatic upper potential (Fig. 9). In this case the oscillating structure of the quantummechanical curve is found to be at least qualitatively correctly reproduced, but the modu-
2 The
contributions of each orbit has been normalized by
as in Figs. 9 (a) and (b).
14
R Tk
0
dτ ρ0 in Figs. 8 (a) and (b) as well
lation of the amplitudes cannot be seen in the periodic orbit result. This is not surprising
because the use of only two orbits leads to a quasi-periodic structure in the observable.
To account for the quantum-mechanically obtained amplitudes one would have to consider
probably more periodic orbits with a longer period time.
All these results pertain to the fixed energy E0 = 0.65eV, but the quantum-mechanical
situation of a wave packet starting on one of the diabatic potentials implies an uncertainty
of the energy. This is well-known from standard quasi-classical calculations where the initial
value n0 = 1 is fixed and the nuclear coordinates x0 , p0 are sampled according to ρN
0 (x0 , p0 )
leading to a different energy E for every single trajectory with the probability p(E) =
exp[−(E − E0 )2 ]. Analysing periodic orbits leads to an opposite situation: now the energy
of every orbit is fixed, and the value of n depends on E, x, p and q. Therefore it is clear that
the condition n(0) = n0 cannot be fulfilled.
If the uncertainty of energy was taken into account, an averaging over all energies with
the weighting factor p(E) would have to be done. This would lead to the problem of finding the ’same’ periodic orbits at different energies, as it was already mentioned in the last
section.
It is obvious that the superposition of contributions at different energies with slightly different periods (cf. Fig. 7) leads to interference effects and a damping since the orbits are not
exactly in phase. It should be noted that this averaging over the energy leads to slightly
better results than the E = 0.65eV -contribution of Ia alone.
15
V. CONCLUSIONS
In this paper we have constructed a classical analogon of a quantum-mechanical vibroniccoupling problem, by means of the so-called mapping-formalism [43,44]. We have performed
a detailed phase-space analysis of the corresponding classical Hamiltonian, describing local and global features of the classical dynamics, for realistic parameter values. Our approach provides a means of estimating the quantum-mechanical level density, and suggests
an intuitive physical picture of the quantum dynamics of populations on diabatic/adiabatic
potential surfaces. We have compared our classical results to fully quantum-mechanical
calculations of the integrated level density and of the time evolution of diabatic/adiabatic
populations.
The results are as follows: First, the classical Weyl estimate of the level density, NC (E),
fits the quantum result very well provided the zero-point energy is included in full. Second,
for the parameter values considered above, the time dependence of diabatic/adiabatic populations can be modeled using a simple classical ansatz for the regular dynamics pertaining to
integrable islands in phase space (for times smaller than the Heisenberg time): the dynamics
is dominated by the shortest vibronic periodic orbits in the centre of the most important
integrable islands.
16
APPENDIX
A. On the energetically accessible phase space
In this section the boundaries of the accessible area in the surfaces-of-section (e.g. Fig. 3)
are discussed. The energetically accessible phase-space area for every fixed value of n and q
is calculated as follows. We get the function p = p(x) with the parameters E, n, q:
2
1
2 ωp
q
= E − 12 ωx2 − (2n − 1)κx − g cos q (2n + 1)(3 − 2n).
(5.1)
If the SOS at fixed q is plotted in the x, p-plane the dependency on n is hidden. We consider
p = p(x, n) and must look for the extrema
of p (or p2 ) with respect to n. Introducing
q
∆(x) = 12 (V2 (x) − V1 (x)) and W (x) = ∆(x)2 + g 2 , we have
q
(2n − 1)g cos q = ∆(x) (2n + 1)(3 − 2n)
(5.2)
and finally obtain the value of n for extremal p:
nextr =
∆(x)
1
±q
,
2
∆(x)2 + (g cos q)2
(5.3)
∆(x)
which reduces in the case q = 0 or q = π to nextr = 21 ± W
.
(x)
Inserting this back into (5.2) shows which sign in (5.3) leads to a contradiction:
±q
2g cos q
∆(x)2 + (g cos q)2
=
q
(2nextr + 1)(3 − 2nextr )
(5.4)
Since both square roots are positive by defintion the sign of cos q decides which solution in
(5.3) is the right one. If cos q is positive, the lower sign leads to a contradiction and only
the upper one can be used. As the second derivative of p2 at nextr is positive, this value of
n belongs to a minimum of p2 which corresponds to the large forbidden area in the middle
of each plot (Fig. 4) for energies E > 0.2eV. The analytic expression reads
2
1
2 ωpmin
= E − κ1 x − 21 ωx2 − ∆(x) − 2W (x).
(5.5)
The outer frontier of the accessible phase space is found as the maximum of p2 at the
boundary n = − 12 Γ = − 12 or n = 1 + 21 Γ = 32 , respectively:
2
1
2 ωpmax1
2
1
2 ωpmax2
= E − κ1 x − 21 ωx2 + ∆(x)
= E − κ2 x − 21 ωx2 − ∆(x).
(5.6)
(5.7)
On the other hand if cos q is negative, everything is the other way round. The lower sign
in (5.3) is the right one, belonging to a maximum of p2 :
2
1
2 ωpmin
= E − κ1 x − 21 ωx2 − ∆(x) + 2W (x)
(5.8)
The other boundaries (5.6) and (5.7) are now minima marking the forbidden area in the
middle of the SOS.
17
B. On the choice of the correct root in calculating n 0
Consider a SOS at q = 0 taken for fixed energy E. The choice of the coordinates x and
p imply together with the energy conservation the allowed value for n which is given by
q
E − 12 ω(x2 + p2 ) − κ1 x − 2∆(x)n = g (2n + 1)(3 − 2n).
(5.9)
We obtain two possible solutions for n:
n1/2 =
q
1
2
A(x)∆(x)
+
g
±
4W (x)2 − [A(x) − ∆(x)]2
2W (x)2
(5.10)
with the abbreviaton A(x) = E − 21 ω(x2 + p2 ) − κ1 x. To decide which of these solutions
contains the right root, they are again inserted into equation (5.2) leading to
"
q
#
A(x)
W (x)2 q
− 1 . (5.11)
(2n1/2 + 1)(3 − 2n1/2 ) + g
± 4W (x)2 − [A(x) − ∆(x)]2 = −
∆(x)
∆(x)
Evaluation of both sides with the chosen coordinates x, p shows which of the solutions is
consistent. It is also possible that both solutions do not lead to contradictions or, on the
other hand, that no solution at all fulfills this condition. This is the case if the starting
point lies outside the energetically accessible phase space.
C. On the calculation of NC (E)
In our model the calculation of NC (E) according to (4.2) requires 4 integrations, corresponding to one nuclear and electronic DoF, respectively:
1
NC (E) =
(2π)2
+∞
Z
dp
−∞
+∞
Z
Z2π
1+ Γ
2
dx dq
−∞
0
Z
dn Θ[E − H(p, x, q, n)],
(5.12)
−Γ
2
where Γ accounts for the electronic zero point energy and the setting h̄ = 1. First, we
introduce polar coordinates for the nuclear DoF in the following way:
"
(2n − 1)κ
R =p + x−
ω
#
"
(2n − 1)κ
.
tan(α) = p/ x −
ω
2
2
#2
(5.13)
(5.14)
Now the Hamiltonian (2.17) reads
H=
q
ω 2
κ2
R − (2n − 1)2 + g (2n + Γ)(2 + Γ − 2n) cos q,
2
ω
(5.15)
and is independent on α. Therefore this integration simply gives 2π and the energetical
limit for the R-integration is
18
Rmax =
s
q
2
κ2
E − g cos q (2n + Γ)(2 + Γ − 2n) + (2n − 1)2
ω
2ω
!1
2
(5.16)
leading to
1
NC (E) =
(2π)
Z2π
1+ Γ2
dq
0
Z
dn
1 2
R (q, n).
2 max
(5.17)
−Γ
2
Rmax must be positive. If the energy E is very small, it can happen that some values of cos
q lead to a negative expression under the square root in (5.16). Therefore the limits of the
q-integration would be restricted to some n-dependent value, making an analytic evaluation
impossible. Thus we suppose in what follows that E is big enough to bypass this difficulty.
In this case the q-integration from 0 to 2π is easy and leads to a polynomial in n:
1
NC (E) =
ω
1+ Γ
2
Z
dn
"
#
κ2
E + (2n − 1)
,
2ω
2
(5.18)
−Γ
2
which finally results in Eq. (4.4).
The multidimensional case is treated analogously by defining polar coordinates for every
nuclear DoF like in (5.13).
19
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21
TABLES
orbit
Ia
Ib
IIa
IIb
IIIa
IIIb
Va
VIIIa
x0
3.330
-2.725
-3.599
3.599
3.163
3.261
2.9216
2.995
period T [fs]
39.2
36.4
46.3
46.3
117.8
117.8
196.6
314.4
p0
0
0
2.352
-2.352
-0.275
0
0
0.2
Ljapunov-exponent
0
0
0
0
0
0.186
0
0
TABLE I. starting points (at q = 0) and period of some periodic orbits
22
FIGURES
1.2
t=0
energy [eV]
0.9
0.6
0.3
0
−3
0
x
3
FIG. 1. Diabatic potentials (solid lines) and adiabatic potential (dashed lines). The wavepacket
symbolizes the excitation at x = 3.
23
30
20
N
10
0
0
0.5
1
1.5
energy [eV]
FIG. 2. Energy dependence of the staircase function N (E): quantum result (solid curve),
compared with the classical one which is analytically derived, considering the whole zero point
energy (dotted curve) or neglecting the zero point energy (dashed curve), respectively.
24
(a)
3
p
0
−3
(b)
3
p
0
−3
(c)
3
p
0
−3
−3
0
3
x
FIG. 3. Surfaces-of-section for the energy E = 0.65eV at different values of q: (a) q = 0 ; (b)
q = π2 ; (c) q = π
25
3
p
(a)
(b)
(c)
(d)
(e)
(f)
0
−3
3
p
0
−3
3
p
0
−3
−3
0
3
6
x
−3
0
3
6
x
FIG. 4. Surfaces-of-section at q = 0 for different energies E: (a) 0.1eV ; (b) 0.2eV ; (c) 0.3eV ;
(d) 0.65eV ; (e) 1.0eV ; (f) 1.5eV
26
4
(a)
II
2
p
0
Ib
−2
II
−4
−3
0
3
x
(b)
0.5
p
0
Ia
Va
IIIb
VIIIa
IIIa
VIIIb
−0.5
Vb
3
3.5
4
x
FIG. 5. Surface-of-section at q = 0 and E = 0.65eV: (a) whole area and (b) zoom of important
region
27
diabatic
1
adiabatic
Ia
Ia
Ib
Ib
IIa
IIa
IIb
IIb
n
0
1
n
0
1
n
0
1
n
0
1
IIIa
IIIa
IIIb
IIIb
Va
Va
VIIIa
VIIIa
n
0
1
n
0
1
n
0
1
n
0
−3
0
x
328
−3
0
x
FIG. 6. Periodic orbits at the energy E = 0.65eV
3
period time [fs]
44
42
40
38
36
0.4
0.5
0.6
0.7
energy [eV]
0.8
0.9
1
FIG. 7. Period T of the orbit Ia as a function of the energy E
29
1
(a)
n 0.5
0
1
(b)
n 0.5
0
1
(c)
n 0.5
0
0
50
100
150
time [fs]
FIG. 8. Contribution of the period-1-orbits Ia (a) and Ib (b) to the population probability
on the upper diabatic level when started on the upper potential curve; (c): Comparison of their
weighted sum (solid line) with the quantum result (dashed line). Note that the Heisenberg-time
tH is 125fs.
30
1
n 0.5
(a)
0
1
n 0.5
(b)
0
1
n 0.5
(c)
0
0
50
100
150
time [fs]
FIG. 9. Contribution of the period-1-orbits Ia (a) and Ib (b) to the population probability
on the upper adiabatic level when started on the upper potential curve; (c): Comparison of their
weighted sum (solid line) with the quantum result (dashed line). Note that the Heisenberg-time
tH is 125fs.
31