systems studied have involved heavy atom (X) - hydrogen stretching motion, with ... frequency from other modes, namely torsion and out-of-plane bending. Normal mode calculations show that the lowest ... Here we do not include any cubic or ..... barriers. This feature extends over a wide energy range and thus suggests that.
H. W. Schranz and M. A. Collins
1
Nonlinear resonance and torsional dynamics: model simulations of HOOH and CH3OOCH3
H. W. Schranz and M. A. Collins
2
because they are well separated in frequency from these lower energy motions. Mixing of these other motions with X - H stretching in the overtone states implies IVR in the sense that a wave packet of such states might correspond to X - H stretching
Harold W. Schranz and Michael A. Collins Research School of Chemistry, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
at some time and angle bending at another. There are other types of molecular motion which are often well separated in frequency from other modes, namely torsion and out-of-plane bending. Normal mode calculations show that the lowest frequency mode of a tetra-atomic molecule
ABSTRACT
is often characterised as almost pure torsion or out-of-plane bending. For this reason, many studies of the large amplitude dynamics of macromolecules allow
Simple models of the vibrational dynamics of HOOH and CH3 OOCH 3 are
only torsional motion with rigid bond lengths and angles [22-24]. However, the
investigated by classical trajectory methods. Nonlinear resonances due to
overtone states of a "torsion mode" may be close in energy to the fundamentals,
kinematic coupling between the torsional motion and symmetric bond bending
overtones or combination levels of the remaining molecular vibrational modes.
are found to have significant dynamical effects in some cases. The timescales
Whether these states will "mix" to a significant extent depends on the magnitude of
and magnitudes of these energy transfer processes are examined.
appropriate terms in the Hamiltonian. Such mixing has certainly been invoked in attempts to explain features in various infra-red spectra [2]. Moreover, recent experimental [3] and theoretical studies [3,4] of the spectroscopy and dynamics of p-
I. INTRODUCTION
fluorotoluene have invoked models of the vibrational level mixing arising from methyl torsion-vibration coupling. Despite the low methyl torsion barriers, the IVR
The process of intramolecular vibrational energy redistribution (IVR) is of great significance in a wide variety of physical, chemical and biochemical systems
rates [3] among ring modes are two orders of magnitude greater than those in a similar molecule (p-difluorobenzene) without the methyl group.
[1-21]. In recent years, many experimental and theoretical papers have considered
The purpose of this paper is to report classical simulations (numerical
IVR with particular reference to its impact on spectroscopy [1-5], laser induced
experiments) of a number of models of HOOH and CH3OOCH3 (with united methyl
chemical reaction [7-17] and unimolecular decomposition [16-21]. Many of the
groups) which probe the importance of nonlinear resonance interactions between
systems studied have involved heavy atom (X) - hydrogen stretching motion, with
the torsion and higher frequency modes. Recently, Spears and Hutchinson [10]
reference to the local versus harmonic character of the overtone states and with
studied the torsional dynamics of trans-diimide focussing on low order and
reference to the effect of Fermi resonance mixing this motion with bond bending
combination resonances between the torsion and the stretches and bends. Here,
modes [7-15].
the nonlinear resonances studied include the familiar case of Fermi resonance
Normal mode calculations usually show that X - H stretching modes are not
(2 : 1 frequency matching) [25] as well as 3 : 1, 4 : 1, and combination resonances.
mixed with other types of valence coordinate motion like angle bending or torsion
Tetra-atomics of this type are chosen as the simplest molecules which allow torsion.
H. W. Schranz and M. A. Collins
3
Molecules of the type ABBA are chosen because the symmetry restricts the number
4
H. W. Schranz and M. A. Collins
II. MODEL HAMILTONIANS
of effective interactions. The hydrogen and methyl substituents illustrate some important mass effects. Although the coupling of torsion with angular momentum is a priori important, this initial investigation is restricted to the case of zero angular momentum for the sake of clarity.
A. Potential energy surfaces A molecular potential energy surface which describes vibrational motion about a stable equilibrium is usually written in terms of valence coordinates. This
Coupling of torsion with other motions may well be caused by anharmonic
description is superior to say an expansion in Cartesian coordinates because it
terms in the molecular potential energy. Such terms may vary considerably in
takes advantage of the character of chemical bonding (and is invariant to rotation or
magnitude from one molecule to another. Here we do not include any cubic or
inversion of the molecule). For example, the diagonal force constant for a bond
higher order potential energy couplings between torsion and other modes. The
stretch is usually much larger than associated off-diagonal terms.
dynamics reported here only involves kinematic coupling.
Clearly then, the
generally accepted that an appropriate set of valence coordinates result in a Taylor
timescales for energy transfer reported here may not apply to a particular molecule.
expansion for the potential energy about equilibrium which is more accurate at a
However, the models studied herein provide a measure of the coupling that might
given order than the corresponding expansion in Cartesian coordinates. Thus, an
be expected in the "generic" case where no particularly important anharmonic
expansion of the potential in curvilinear normal coordinates is superior to an
couplings arise in the potential energy. Since such potential energy couplings
expansion in Cartesian normal coordinates [29].
could either reinforce or oppose the coupling due to the unavoidable kinematic effects, we expect the models to provide a measure of the "average" interaction
Also, it is
For simplicity, the potential energy surface of an ABBA type tetra-atomic is taken to be a sum of simple valence terms
between torsion and other modes. Nonlinear resonance involving torsion is potentially important in
V(R1,R2,R3,θ1,θ2,τ) = VR1(R1) + VR2(R2) + VR3(R2) + Vθ1(θ1) + Vθ2(θ2) + Vτ(τ) ,
understanding the simplest type of chemical reaction, conformational isomerisation
(2.1)
[10,22-24,26-28]. Moreover, the very large amplitude of torsional motion, coupled
where the internal valence coordinates {R1 ,R 2 ,R 3 ,θ 1 ,θ 2 ,τ} are defined in Fig. 1.
with the periodic character of relevant terms in the Hamiltonian, suggests there may
The bond stretching and angle bending potentials are quadratic:
be some dynamical effects which are novel in comparison with previous studies. The details of the models are described in sections II and III. Methods are
kR VRi(Ri) = 2 i (Ri-Re)2
,
(2.2)
kθ Vθi(θi) = 2 i (θi-θe)2
.
(2.3)
described in section IV. Calculations are described in section V and discussed in section VI.
In some simulations a related form of Vθ(θi) is employed: γθ Vθi(θi) = 2 i (cosθi-cosθe)2 .
(2.4)
5
H. W. Schranz and M. A. Collins
The torsional potential is taken to be quadratic in the torsion angle
H. W. Schranz and M. A. Collins
6
coordinates to the normal modes are given in Tables V-VII for models typical of the range studied here.
k Vτ(τ) = 2τ (τ-τe)2
,
(2.5)
The above surfaces provide only a first approximation to the potential energy surfaces for either HOOH [30] or CH3 OOCH 3 [36]. They provide a reasonable description of the frequencies and the character of the normal modes in tetra-
or as a cosine series truncated at the Mth term
atomics of this type so that the qualitative behaviour of energy redistribution within M
Vτ(τ) =
∑ a m cos m τ
.
(2.6)
the molecules can be ascertained.
m=0
B. The kinetic energy The potential parameters for the HOOH and CH3OOCH3 models are given in Tables I–IV. Parameters for HOOH are based on those of Getino et al. [30]. The
In Cartesian coordinates {x,p} the total kinetic energy of a molecule of N atoms is of a simple quadratic form
"anharmonic models" for HOOH and CH3 OOCH 3 are composed of a sum of harmonic stretch terms, cosine angle bends [see Eq. (2.4)] and a truncated cosine series for the torsion [see Eq. (2.6)].
2 2 2 1 N p xk + pyk + pzk T= 2 ∑ mk k=1
(2.7)
There is some controversy concerning the equilibrium value of the torsion angle in CH3OOCH3 [31-34]. Recent theoretical [33] and experimental studies [34]
The classical trajectories described below have been evaluated in Cartesian
have concluded that the equilibrium torsion angle is close to 119° (though it lies in a
coordinates using this form of the kinetic energy and the potential of Eq. (2.1) with
very shallow well). Here, the equilibrium geometry for CH3OOCH3 is based on the
the valence coordinates as explicit functions of the Cartesians. The nonlinear
results of Haas and Oberhammer [34]. The anharmonic torsional potential was
transformation from Cartesian to valence coordinates ensures that even though
obtained from the work of Koput [35] and the force constants for the stretch and
both potential energy and kinetic energy are separable as written in Eqs. (2.1) and
angle bend terms were adjusted to yield normal mode frequencies close to those
(2.7), the motion is nonseparable.
reported by Budenholzer et al. [36]. The anharmonic torsional potentials for HOOH
To understand the coupling of these valence motions, it is useful to write the
and CH3 OOCH 3 are shown in Figs. 2a and 3a. Because of the low barrier for
Hamiltonian in terms of the valence coordinates and their conjugate momenta so
motion through the trans (τ = 180°) configuration, shown in Fig. 3a, it was
that the kinematic coupling of these valence coordinates and momenta is explicit.
considered appropriate that the harmonic model for CH3OOCH3 has an equilibrium torsion angle of 180°. In order to examine the dynamical consequences of various nonlinear resonances, these force constants have been varied to the model values shown in Tables II and IV. The normal mode frequencies and the contribution of symmetry
7
H. W. Schranz and M. A. Collins
III.
KINEMATIC COUPLING
H. W. Schranz and M. A. Collins
and for the torsion
Gττ =
A. Coupling terms Handy and coworkers [37] have derived the quantum kinetic energy operator in terms of the valence coordinates of Fig. 1, so the corresponding, rather lengthy, classical equivalent is not reproduced herein. In internal (valence) coordinates and momenta {Q,P} the kinetic energy T is given by
1 T=2
8
3N-6 3N-6 ∑ ∑ G ij P iP j i=1 j=1
.
(3.1)
1 1 1 1 1 1 [ + ]+ [ + )] sin2 θ 1 µ1R12 µ2R22 sin2 θ 2 µ2R22 µ3R32 2cosθ1 2cosθ2 1 1 2 µ2R22 sin2 θ 1 m2R1R2 sin2 θ 2 m3R2R3 2cosθ1 cosθ 2 cosτ sin θ 1 sin θ 2 µ2R22 2cosθ 1 cosτ 2cosθ2 cosτ m R R m R1R2 , 3 2 3 2 sinθ 1 sinθ 2 sinθ 1 sinθ 2 +
(3.5)
where µ1 = [1/m1+1/m2]-1, µ2 = [1/m2+1/m3]-1 and µ3 = [1/m3+1/m4]-1. The well-known "resonant interaction" between two modes of motion arises
The G-matrix [38] elements Gij are functions of the internal coordinates. For the
from terms in the Hamiltonian which allow either mode to act as an "external driving
present models of HOOH and CH3OOCH3 the off-diagonal G-matrix elements are
force" on the other, where the driving force oscillates at the natural frequency of the
usually very small (see the ensemble averages shown in Tables VIII and IX), so we
driven mode [10,39]. For example, a term in the potential energy proportional to
focus on the diagonal G-matrix elements. For the bonds
δQ1δQ2 (where the deviation from equilibrium is defined as δQi = Qi - Qie) would be a 1 : 1 resonant interaction if the modes with coordinates Q1 and Q2 had the same
1 GRiRi = , i=1, . . . ,3 µi
;
(3.2)
natural frequencies. The 2 : 1 stretch - bend resonant interaction which has been much studied in recent times [7-15] can be seen in Eqs. (3.3) and (3.4). If one expands the bond lengths in Eqs. (3.3) or (3.4) about their equilibrium values, one
for the bond angles
sees that Eq. (3.1) contains terms like Pθ2δR. If angle bending forms part of modes 2cosθ 1 1 1 G θ1θ1 = + µ1R12 µ2R22 m2R1R2
;
(3.3)
of motion which oscillate at half the frequency of the stretching motion, then terms like this constitute resonant driving of angle bending modes by bond stretching modes and vice versa.
2cosθ2 1 1 + -m R R G θ2θ2 = 2 2 3 2 3 µ3R 3 µ2R 2
;
(3.4)
The rich structure of Eq. (3.5) implies that many modes may be coupled to the torsion motion by nonlinear resonance. However, we note that since torsion is normally the lowest frequency mode, an n : 1 resonant interaction requires terms in the Hamiltonian which are nth order in the deviation of τ from its equilibrium value and first order in another coordinate.
Symmetry places restrictions on these
couplings. For ABBA type molecules, the Hamiltonian is symmetric in θ 1 and θ 2,
9
H. W. Schranz and M. A. Collins
and in R1 and R3. This means that a Taylor expansion of the G-matrix elements
10
H. W. Schranz and M. A. Collins
significant for HOOH due to kinematic effects.
(and the potential energy surface) about equilibrium can contain terms like
For CH3OOCH3, or any peroxide with heavy substituents, the τ dependence
δθ1 + δθ2, but not δθ1 - δθ2 [rather (δθ1 - δθ2)2]. This means that the terms allowing
of Gττ is quite significant. The cosine term in Eq. (3.6) may be expanded about
n : 1 resonance between the symmetric bend and torsion can be of the form δτn (δθ1
equilibrium as
+ δθ 2 ), which is overall (n+1)th order of "smallness" in terms of deviation from 1
cosτ = cosτ e - δτ sinτ e - 2 δτ2 cosτe + ...
equilibrium. In contrast, n : 1 resonant interaction of asymmetric bending and torsion is allowed only through terms of the form δτ2n (δθ1 - δθ2)2, which is of overall order 2n+2.
Hence, the coupling of torsion with asymmetric bending and
and for various values of τe we note that
asymmetric stretching is expected to be very weak in ABBA type molecules.
cosτ = - δτ + 1 δτ3 - ... ,
τe = 90°,
cosτ = - 1 - √23 δτ + 41 δτ2 + ... ,
τe = 120°,
cosτ = -1 + 1 δτ2 + ... ,
τe = 180°.
6
Note also that Eq. (3.5) may be written in the form
2
2
Gττ = f1(R1,R2,R3,θ1,θ2) +f2(R1,R2,R3,θ1,θ2) cosτ .
(3.6)
(3.10)
(3.11)
Thus, given the required frequency ratios, 3 : 1 resonances could be important for molecules with τ e = 90°, 3 : 1 and 4 : 1 resonances could be important for molecules with τ e = 120° and 4 : 1 resonances could be important for molecules
At equilibrium for HOOH (see Table I)
with τ e = 180°. Gττ = 2.389 + 0.036 cosτ amu-1 Å-2
.
(3.7)
Of course, since torsion is often a rather "floppy" mode, the torsion angle may take values far from τ e and the use of the first few terms in Eq. (3.11) could be
For the "harmonic" and "anharmonic" CH3OOCH3 models of Table III,
misleading.
Such large amplitude torsion may then give rise to unexpected
resonant effects. Moreover, the frequency of the torsional motion is expected to be Gττ = 0.180 + 0.043 cosτ amu-1 Å-2
,
(3.8)
Gττ = 0.290 + 0.119 cosτ amu-1 Å-2
,
(3.9)
a sensitive function of the torsion energy.
and B. Torsion frequency Since the diagonal G-matrix elements are dominant for the molecules under respectively. Note that bond lengths are taken in Å, mass in amu and energy in cm-
study, we can approximate an isolated valence mode by a Hamiltonian including
1 and the resulting time unit is 0.914 ps. The part of Gττ that is independent of τ is
only the diagonal kinetic term and potential terms, e.g. for the torsional mode,
larger than the term associated with cosτ, particularly for HOOH. This indicates that the light hydrogen atoms yield an effective mass for torsion which is only weakly
H(τ,Pτ) = 21 GττPτ2 + V(τ) .
(3.12)
dependent on the torsion angle. Consequently, since the only large terms involving torsion are proportional to Pτ2, only 2 : 1 resonant interactions are expected to be
Given energy Eτ in the torsional mode, the effective classical frequency is
11
H. W. Schranz and M. A. Collins
H. W. Schranz and M. A. Collins
12
modes over a wide energy range. At higher energy, where torsion is effectively τ+
νeff(Eτ) = [
∫dτ
τ-
√
2 G ττ
1 Eτ-V(τ) √
]-1
.
(3.13)
hindered rotation, there is a strong variation of effective classical frequency with energy. It is likely that classical resonance will be difficult to achieve unless the resonance is very strong and broad (it is known that some 2 : 1 resonances are sufficiently strong to allow substantial energy flow even when tens of cm-1 off
In the harmonic approximation,
resonance [7,8,10,11]). Moreover, we note that the classical frequency is only close ωeff(Eτ) = 2π νeff(Eτ) =
kτG ττ , √
(3.14)
to the average quantum energy gap in the region above the cis barrier where odd and even torsion wavefunctions are nearly degenerate.
where Gττ is evaluated at equilibrium. Note that when this frequency is very close
In preliminary simulations we perform calculations on molecular models with
to the lowest frequency given by a normal mode analysis of the whole molecule, the
harmonic torsional modes in order to distinguish which types of resonance are
assumption of an isolated torsion fundamental is justified. At higher energy, the
possible from kinematic coupling. Subsequent simulations incorporate the more
torsional potentials of Figs. 2a and 3a yield energy dependent frequencies as
realistic torsion potentials so that the effect of frequency variation is included.
shown in Figs. 2b and 3b. This variation of frequency with energy is of central importance to the question of resonance interaction with other modes. To examine the relevance of
IV. METHODS
the classical frequency to quantum dynamics, the energy levels of HOOH and CH 3 OOCH 3 with rigid bond lengths and bond angles have been evaluated [40], and the gaps between adjacent energy levels are shown in Figs 2b and 3b.
In this section the methods employed for the initial state selection and integration of the classical trajectories are described. Ensembles of classical
Clearly, the classical torsion frequency is closely correlated with the quantum
trajectories are used to generate averages of dynamical variables which can be
energy gaps between adjacent levels. We note that the classical frequency is zero
analysed for information on the timescale and extent of the intramolecular
at energies corresponding to the potential energy maxima at the trans and cis
vibrational energy redistribution.
configurations (see Figs. 2a and 3a). A notable feature of Figs 2b and 3b is the relatively flat variation of frequency with energy, apparent in both the classical
A. Initial state selection
frequency and quantum energy gaps, between the dips caused by the trans and cis
In order to determine the importance of nonlinear resonances in facilitating
barriers. This feature extends over a wide energy range and thus suggests that
IVR it is useful to observe the dependence of the IVR upon the manner in which the
resonant interaction at this "plateau" frequency could give rise to a large amount of
energy is initially partitioned amongst the various molecular vibrational modes.
energy transfer. Correspondingly, the "ladder" of torsion state energies is regular in
Firstly, a normal mode analysis was carried out at the equilibrium geometry
this region and may be commensurate with the "ladder" of levels due to other
[38]. The Cartesian coordinates and momenta of each atom were displaced from equilibrium as follows. Each of the five highest frequency modes (bond stretching
13
H. W. Schranz and M. A. Collins
and bending modes with frequency νi) were assigned an energy by specifying the classical equivalent of a quantum number, ni, such that the energy of that mode is given by Ei = hνi (ni+1/2). The six lowest frequency normal modes, corresponding to
14
H. W. Schranz and M. A. Collins
.
and subtracted from the total velocities to yield corrected velocities qnri:
.
. .
qnri = qi – qri ,
(4.5)
overall rotation and translation, were assigned zero energy. The sixth highest frequency normal mode which corresponds to torsion provides a very poor
A scale factor λ is then calculated
description of this large amplitude motion. Initially, this mode is assigned a given quantum number as above, but the phase of this mode is fixed randomly at two values which correspond to momentum only.
λ =
An ensemble of Cartesian
E V – V(q) K(p)
,
(4.6)
coordinates and momenta were then evaluated from the normal mode eigenvectors
from the current values of the kinetic energy K(p) and potential energy V(q) and if
using random phases for each normal mode (except torsion) in standard fashion.
λ>0 the current velocities q nri can be scaled to satisfy the desired value of the
The assigned total vibrational energy should then be Ev:
vibrational energy EV
.
6
EV =
∑
.
Ei
.
(4.1)
.
q′nri = λ1/2 qnri.
(4.7)
i=0 At non-zero energies, the normal mode approximation is in error and it is necessary to scale the momenta to achieve a desired total vibrational energy EV and to eliminate spurious components of angular momentum. The total molecular
Otherwise, the current configuration must be rejected and a new selection of random phases is made. In practice, such rejections were rare. This initial ensemble is not randomly distributed in the direction of the eigenvector corresponding to torsion. To correct this bias, each initial state was
angular momentum around the centre of mass is
integrated for a random time up to half the period of the torsional mode in order to N
J=
∑
ri × p i
,
(4.2)
randomise the phase of this mode.
The resultant Cartesian coordinates and
momenta make up the ensemble of initial conditions for the simulations reported
i=1
below. During this initialisation procedure, at most minor leakage (5-10%) of and the angular velocity vector of the molecule, ω, may be obtained as
torsional mode energy into the other modes was observed (at the highest excitations studied). This procedure was satisfactory as long as the coupling
ω = I-1 J
,
(4.3)
between the torsional mode and the other modes was sufficiently small, as is the case with the HOOH models and most of the harmonic CH3OOCH 3 models. Note
where I is the instantaneous, configuration dependent, moment of inertia tensor
that, in the case of some models of CH3 O O C H 3 , the normal mode primarily
[41]. The rotational velocity vectors for each atom are calculated from
associated with the torsional motion also had small but significant components of
.
qri = ω × ri
other modes such as OO stretch and symmetric bend and symmetric stretch. ,
(4.4)
The
15
H. W. Schranz and M. A. Collins
16
H. W. Schranz and M. A. Collins
method above is a relatively standard quasiclassical procedure in that the modes
Here the G-matrix elements are evaluated from Eqs (3.2)-(3.5) at the instantaneous
which are initially in their ground states are allocated zero-point energy
values of the coordinates during a trajectory, while the internal momenta P are
[16,19,42], although this approach is still the subject of debate [43].
calculated from the Cartesian momenta p by inverting
p = B TP
B. Classical trajectory method
,
(4.11)
In the present study, Newton's equations of motion were integrated using the Verlet leapfrog algorithm [44]. Ensembles of 100-1000 classical trajectories were
where B is the Wilson B matrix [38]. For computational convenience, the angle
run using a vectorised code on a Fujitsu VP2200. Trajectories were normally
variables θ 1 and θ 2 were replaced by cosθ 1 and cosθ 2 with consequent minor
integrated for times ranging from about 1 ps to 10 ps. A satisfactory timestep was ∆t
adjustment to the B matrix and the G matrix elements of Eqs. (3.3) and (3.4). As
= 0.0914 fs for which energy conservation was satisfied to better than 1 part in 105.
shown in Tables VIII and IX, the off-diagonal G matrix elements are usually small, so that the energies evaluated in Eqs (4.8) to (4.10) should provide a reasonable
C. Analysis of the dynamics
measure of the energy associated with each type of motion.
The energy associated with each type of molecular vibration is measured
The valence mode energies are averaged over the ensemble of trajectories
during the trajectories as follows. At sufficiently low energy (and low excitation of
to yield a set of time dependent energies denoted by . The behaviour of these
the torsional mode), the normal mode approximation is moderately accurate. The
functions gives an indication of the rate and extent of the intramolecular energy flow
energy of each normal mode is then given from the projection of the cartesian
and the strength of any resonant interaction between modes. Since the number of
˚ on the normal mode eigenvectors [19,38]. At higher coordinates q and velocities q
vibrational modes is small, and does not constitute a set of "bath" states, it is not
energies, we estimate an energy associated with each "valence mode". The energy
possible to quantify the rate of IVR by a single parameter such as a rate constant.
for each valence mode is defined as the diagonal component of the kinetic energy
We summarise the character of the intramolecular energy transfer
(when expressed in terms of internal momenta) plus the corresponding potential
calculating the average energy transferred into a mode at time t = T relative to time t
term,
=0 ERi = 21 GR R PR 2 + VRi(Ri) , i i i
i = 1,2,3
,
(4.8)
Eθi = 21 GθiθiPθi2 + Vθi(θi)
i = 1,2
,
(4.9)
,
= -
.
by
(4.12)
Some insight into the extent of IVR for different models can be gauged from examining the microcanonical probability density P(E,τ) for the torsion angle where
1 2
Eτ = GττPτ2 + Vτ(τ)
.
(4.10)
energy E is available to be shared amongst the torsional mode and a subset of the other vibrational modes. For the particular case of a harmonic torsional mode in microcanonical equilibrium with M-1 other harmonic vibrational modes it is possible
17
H. W. Schranz and M. A. Collins
to derive the general expression for the probability density as (see Appendix)
PM(E,τ) =
1 (M-1)!
√ π
2 (2M-3)/2 (2E kτ - δτ )
)M-1 (2E kτ
2M-3 ( 2 )!
H. W. Schranz and M. A. Collins
18
V. CALCULATIONS
Simulations of the models described in Tables I-IV are reported. For .
(4.13) convenience, the various models are referred to by the abbreviations used in these tables.
The theoretical probability density for the specific case of a torsional mode A. HOOH with harmonic torsion
modelled by an isolated simple harmonic oscillator (M=1) is
The potential parameters for HOOH are based on those of Table I, but the 1 P 1(E,τ) = π
1
√
2E - δτ2 kτ
,
(4.14)
value of the harmonic torsion force constant kτ has been varied over a range such that the torsion frequency lies in a 1 : 2, 1 : 3 or 1 : 4 ratio with the symmetric or
and for a harmonic torsional mode in microcanonical equilibrium with another
asymmetric bends as shown in Table II. Initial states with single and multiple quanta excitations in the torsion and/or the bends were investigated. Consistent with the
harmonic mode (M=2) is
qualitative arguments of Section III, the magnitude of IVR due to 3 : 1 and 4 : 1 2 P2(E,τ) = π
√
2E 2 kτ - δ τ 2E kτ
resonance is small for HOOH. .
(4.15)
ω τ : ω sb = 1 : 2 Extensive energy transfer takes place for the case of ωτ : ωsb = 1 : 2 . Even for single quantum excitation of the torsion, nearly 200 cm-1 is transferred from the
Note that the isolated mode density in (4.14) is bimodal whereas for the two mode
symmetric bend to torsion within 1 ps. The dynamics for torsional excitations
case in (4.15) it is unimodal. As the number of degrees of freedom M to which the
ranging from 0 to 15 quanta is shown in Fig. 4 (model HH2 of Table X). For
torsion is strongly coupled increases, PM(E,τ) becomes increasingly peaked around
example, with the insertion of 15 torsional quanta, about 5000 cm-1 (45%) of the
δτ = τ-τe = 0. Thus, the shape of PM(E,τ) reveals the number of modes to which the
torsional energy is transfered on a subpicosecond timescale.
torsion is strongly coupled on the timescale of observation of the dynamics.
ω τ : ω asb = 1 : 2
Probability densities, calculated as time and ensemble averages, are reported below and compared with these limiting cases.
For the model with ω τ : ω asb = 1 : 2 (model HH3 of Table X), the energy transfer is rather small and is due to an off-resonant 2 : 1 symmetric bend : torsion interaction. ω τ : ω sb = 1 : 3 One or two quanta inserted in the torsion mode gives virtually no energy transfer (see Fig. 5a and model HH4 of Table X). When 5 quanta of torsional energy are inserted, there is a loss of torsional energy of 229 cm-1 in 0.5 ps.
19
H. W. Schranz and M. A. Collins
H. W. Schranz and M. A. Collins
20
Varying the O __ O stretch force constant and ω τ (models HH5 and HH6), and
angle bending force constant). As Fig. 2b shows, the torsion frequency is roughly
monitoring the energy in the symmetric bend mode shows that this energy transfer
constant only in the energy range 600 cm-1 to 2400 cm-1. There ωτ : ωsb ≈ 6 : 1. No
is due to ωτ : ωsb = 1 : 3 resonance rather than a combination resonance, ωτ + ωoo =
substantial energy loss is observed from the torsion mode, even for initial excitation
ωsb. However, inserting one or two quanta in the symmetric bend mode apparently
of up to 40 quanta.
leads to a concerted, though weak, excitation of both torsion and OO stretch, presumably via a combination resonance (Fig. 5b).
For both torsional and
symmetric bend excitations, significant correlation coefficients were calculated for
C. CH3 OOCH 3 with harmonic torsion Here we consider the behaviour of harmonic models of CH3OOCH3 with n : 1
energy transfer between the bending and torsional valence modes. In addition, for
ratios of the symmetric or asymmetric bend frequencies to the torsion frequency:
the case of symmetric bend excitation, there was an additional strong correlation
ω τ : ω sb = 1 : 2 When ω τ : ω sb = 1 : 2 (model HD2 of Table XI), even single quantum
with the OO stretch valence mode. The weakness of this 3 : 1 resonant interaction makes it difficult to eliminate
excitation of the symmetric bend (Fig. 7) leads to significant loss of energy from the
other possible causes, though a 2 : 1 resonance interaction is off resonant by 493
bending modes and rapid uptake by the torsion mode on a timescale of 0.65 ps.
cm-1 and does not appear to be responsible.
The normal mode energies shown in Fig. 7b clearly indicate that the resonant
ω τ : ω asb = 1 : 3
interaction is between the symmetric bend and torsion mode. This was further
Calculations were performed with initial excitation of the torsion mode. No
demonstrated by inserting many quanta of excitation into the torsion or symmetric
subsequent excitation of the asymmetric bend was observed, consistent with the
bend modes, as shown in Fig. 8. For initialisation with torsional excitation, Fig. 8a,
symmetry arguments above. A small amount of energy transfer from the torsion to
we see very rapid energy transfer out of the torsional mode on a timescale as short
the symmetric bend was observed; an off-resonant version of the above symmetric
as 0.17 ps. For initialisation with symmetric bend excitation, Fig. 8b, the energy
bend : torsion interaction (model HH7 of Table X).
transfer out of the symmetric bend mode occurs on a somewhat longer timescale of
ω τ : ω sb = 1 : 4 and ω τ : ω asb = 1 : 4
0.27 ps.
The absence of any significant 1 : 4 interaction between either of the bends
ω τ : ω asb = 1 : 2
and the torsional mode is clearly illustrated in Figs 6a and 6b for models HH8 and
Again, no energy transfer between torsion and asymmetric bending is
HH9 of Table II (see Table X), respectively, where, upon 1,2 or 5 quanta of torsional
observed, even when ωτ : ωasb = 1 : 2 (model HD3 of Table XI) and up to 40 quanta
excitation, no significant IVR involving the torsional mode is observed.
are inserted in the torsion mode. In this case, energy transfer between torsion and symmetric bending is observed even though the 1 : 2
B. HOOH with anharmonic torsion For the model of HOOH with anharmonic torsion (model AH1 of Table I),
frequency matching is
31 cm-1 off resonance. ω τ : ω sb = 1 : 3
there is little IVR observed to or from the torsion mode on a 2ps timescale, when
Even for torsional excitation as high as 40 quanta, there is no net transfer of
the ratio of the harmonic frequencies ranges between 1 : 2 and 1 : 4 (by altering the
energy out of this mode (model HD4 of Table XI). This is consistent with the
H. W. Schranz and M. A. Collins
21
discussion of Eq. (3.11), since the kinetic energy operator contains no coupling
H. W. Schranz and M. A. Collins
22
picoseconds.
terms appropriate to 3 : 1 resonance when the equilibrium torsion angle is 180o .
The simulated probability densities correspond to time and ensemble
Figure 9 demonstrates the energy transfer out of the torsion mode when the
averages of trajectories with 0, 5 and 40 quanta of torsional excitation. The
equilibrium torsion angle is 120o (model HD5 of Table XI). Fig. 9a shows that there
theoretical probability densities are those for an isolated single harmonic oscillator
is significant energy transfer for initial excitation of five or more quanta on a 2 ps
and for two harmonic oscillators which are in microcanonical equilibrium. The
timescale. Fig. 9b shows that when the 3 :1 resonance is detuned by ± 30 cm-1,
former densities are bimodal whereas the latter are unimodal. Thus, on the basis of
energy transfer slows dramatically.
the shape of P(E,τ) a clear distinction can be made between models with strong,
ω τ : ω sb = 1 : 4
little or no IVR involving the torsional mode.
In this case (model HD7 of Table XI), with very high initial excitation of the
For low levels (zero-point energy) of torsional excitation, the simulated
torsion, there is considerable energy transfer on a timescale longer than a
probability densities in Figs. 12b and 12c are clearly bimodal and provide a good
picosecond. For example, with 40 quanta of torsional excitation, there is a loss of
match to the theoretical densities for an isolated torsional mode Eq. (4.14); there is
2300 cm-1 of energy from the torsional mode in about 3 ps (Fig. 10a). However,
very little IVR involving the torsional mode for models HD7 and HD8. The
even 10 quantum excitation of the symmetric bend does not produce any significant
corresponding result in Fig. 12a for model HD2 is also bimodal but somewhat less
excitation of the torsion within 90 ps (for the parameters of model HD7). It appears
of a match to the M=1 theoretical result; the shape is in fact intermediate between
that torsional excitation is necessary to activate the 4 : 1 resonant interaction. Fig.
the M=1 and M=2 theoretical predictions reflecting the slight amount of energy
10b shows that when the 4 : 1 resonance is detuned by ± 5 cm-1, energy transfer
transfer involving the torsional mode even at this low level of torsional excitation.
slows dramatically.
For higher levels of torsional excitation, the simulated probability densities in
If the equilibrium bond angles are altered from their values of θe=120° to 90°
Fig. 12a become unimodal quite rapidly and provide an excellent match to the M=2
(model HD8 of Table XI and Fig. 11) the magnitude of the IVR following torsional
theoretical predictions, Eq. (4.15); the torsion shares energy strongly with only one
excitation is significantly decreased. This is consistent with the type of kinematic
other mode, the symmetric bend. For model HD7 the simulated probability densities
coupling implicit in Eq. (3.5); for 90o bond angles, there is no relevant linear term in
in Fig. 12b become approximately unimodal only at much higher excitations than
δ(θ1+θ2) in the kinetic energy.
required for model HD2; the 4:1 resonance requires higher energies to couple the
Comparison of probability densities
torsion and symmetric bend. On the other hand, for model HD8, the simulated
In Fig. 12 we plot the simulated and theoretical probability densities P(E,τ) for the torsion angle (see Section IV C) for three models of CH3OOCH 3 which show
probability densities in Fig. 12c remain bimodal as the required coupling terms allowing facile IVR between the torsion and symmetric bend are absent.
different degrees of energy transfer as observed above: model HD2 has been shown to exhibit subpicosecond IVR due to a strong 2:1 resonance; model HD7
D. CH3 OOCH 3 with anharmonic torsion
exhibits significant energy transfer on a picosecond timescale due to a 4:1
Calculations have been performed for a model of CH3 OOCH 3 with an
resonance; model HD8 exhibits no energy transfer on a timescale of several
anharmonic torsion potential. This model has a markedly different geometry from
H. W. Schranz and M. A. Collins
23
H. W. Schranz and M. A. Collins
24
the harmonic models treated above. Notably, the equilibrium torsion angle occurs at
hindered rotor. There the torsion frequency is a rapidly varying function of the
119° rather than 180°, as shown in Fig. 3a. From the considerations discussed in
energy, so that one cannot usefully consider frequency matching.
section III, it is possible that 2 :1, 3 : 1 and/or 4 : 1 resonances between the torsion
Fig. 14 summarises the energy transfer observed for models (AD3, AD4, AD5
and symmetric bend could play a role. One problem encountered in characterizing
and AD6 of Table IV) which have ωτ : ωsb in the range 1 : 2.1 to 1 : 4.8 (for harmonic
the resonant behaviour is the significant energy dependence of the effective
frequencies). This corresponds to approximate ratios in the range 1 : 1.6 to 1 : 3.9
frequency of the torsional mode as can be seen from Fig. 3b. The normal mode with
for torsion energies in the range 1500 to 4500 cm-1. Figs 14a and 14b show the
the greatest torsional character has a frequency of 73.5 cm-1 but the effective
average energy in the torsion mode following excitation with 60 quanta, at short and
frequency of the isolated torsional mode is 81.5 cm-1 at zero torsional energy. The
long times, respectively. We note that energy transfer is rapid and extensive near a
effective frequency varies rapidly below 1500 cm-1 and above 4500 cm-1. There is a
ratio of 1 : 2, as in Fig. 13. However, as the ratio rises to 1 : 4.1 (1 : 3.4 for torsion
region from 1500 to 4500 cm-1 where the effective frequency is 89 cm-1 to within 10
energies in the range 1500 to 4500 cm-1) energy transfer is almost eliminated on a
cm -1 and it is plausible that in this region 2 : 1 and sufficiently strong higher order
10 ps timescale. As the frequencies rises even further to 1 : 4.8 (1 : 3.9 for torsion
resonances may be effective. Results are reported for models in which the angle
energies in the range 1500 to 4500 cm-1) energy transfer is again observed. Fig.
bending force constant γθ in Eq. (2.4) has been fixed at various values which result
14c shows that the extent of energy transfer is reflected in the probability densities
in a spread of frequency ratios, ωτ : ωsb.
P(E,τ), which show an increase and then a decrease in bimodal character as the
Results are shown in Fig. 13 for a model with an approximate 1 : 2.4 ratio
frequency ratio rises.
between the harmonic torsion and symmetric bend frequencies (model AD2). This corresponds to an approximate ratio of 1 : 1.9 for torsion energies in the range 1500 to 4500cm-1 . Initial excitation of up to 100 quanta was inserted in the torsional
VI. DISCUSSION AND CONCLUSIONS
mode. Clearly, at low excitations (less than 20 quanta) the torsion and symmetric bend do not exchange significant amounts of energy. At higher excitations, above
When the torsion potential is quadratic in δτ = τ - τe, energy transfer has been
20 quanta, the energy transfer appears much stronger. So strong, that some
observed for a number of nonlinear resonances involving torsion and symmetric
energy transfer takes place within the initial randomisation time for the torsion (see
bending. In this case, the torsional motion is characterised by a single frequency.
Section IV A). For example, with 60 quanta, 979 cm-1 energy is transferred in about
For the models of HOOH, energy transfer due to 2 : 1 and 3 : 1 resonances has
0.11 ps. We also observe energy transfer into the symmetric stretching of the OCH3
been observed. As indicated by the qualitative arguments concerning the G-matrix
bonds on a timescale of several picoseconds. At much higher excitations, above 60
elements, nonlinear resonances are much stronger when the peroxide substituents
quanta, the energy transfer appears to become less coherent. This slowdown in
are more massive. Thus, more extensive energy transfer due to 2 : 1, 3 : 1 and 4 : 1
IVR seems to coincide with the "cusp region" of the effective torsional frequency
resonances has been observed for CH3OOCH3.
curve at Eτ=4800 cm-1(see Fig. 3b). At and above these energies, the torsional
The most effective nonlinear resonance clearly occurs when ωτ : ωsb = 1 : 2,
mode has sufficient energy to surmount the cis (τ = 0) barrier and becomes a
as is predicted by the arguments in Section III concerning the order of the
H. W. Schranz and M. A. Collins
25
anharmonic coupling terms required in the Hamiltonian for n : 1 resonance (which
H. W. Schranz and M. A. Collins
26
pair of modes, the torsion and the symmetric bend, is strongly involved.
are of order n+1 in deviations from equilibrium). The relative effectiveness of other
It is interesting to note how accurately the classical torsion frequency
nonlinear resonances is also well correlated with the symmetry arguments of
estimates the quantum energy gaps below the cis barrier in Figs 2b and 3b (the
Section III, as demonstrated by the degree of energy transfer observed for different
classical frequency is approximately the average energy gap above the cis barrier).
equilibrium bond and torsion angles.
We have seen that the kinematic coupling terms in the Hamiltonian are sufficiently
Some weak energy transfer has been observed between the symmetric bend
large to produce energy transfer classically in the energy range where the torsion
and OO stretch, and between the symmetric bend and symmetric OCH3 stretch over
frequency is approximately constant. It is possible then that the energy levels of the
several picoseconds. However, the most extensive energy transfer occurs between
symmetric bending mode may be commensurate with every second energy level of
the torsion and symmetric bending modes on the timescale of 0.1 to 10 ps, even
the torsion mode, for energies between the trans and cis barriers. These two factors
when the modes are significantly off exact 1 : 2 resonance.
suggest that there may be significant mixing of the overtone states of the torsion and
When the torsion potential is anharmonic, the torsional motion is not characterised by a single frequency. As shown in Figs 2b and 3b, this frequency varies significantly with energy.
symmetric bending over a wide energy range. The quantum dynamics of coupled torsion and symmetric bending is the subject of current research.
However, we see that for both HOOH and
CH3OOCH3, the torsion frequency only varies slowly in the energy range between the trans and cis barriers. Hence, nonlinear resonance may play a role in energy transfer in this energy range.
ACKNOWLEDGEMENTS
Indeed, the strongest nonlinear resonance
mechanism, 2 : 1 for CH3 OOCH 3 , is effective in producing energy transfer from torsion to symmetric bending in this energy range.
We gratefully acknowledge an allocation of computer time on the Fujitsu FACOM VP-2200 of the Australian National University Supercomputer Facility.
There is some indication (see Fig. 14) that 4 : 1 resonance also leads to energy transfer from torsion to symmetric bending for the anharmonic model of CH3OOCH3.
APPENDIX
Complementary information on the extent of IVR to that provided from Consider a set of M harmonic vibrational modes described by the analysis of ensemble averages of mode energies is furnished by examining the shape of the torsional probability density P(E,τ). It is clear that for conditions of rapid
Hamiltonian
IVR between the torsion and the symmetric bend the simulated densities become
HM(P,Q) = TM(P) + VM(Q) ,
(A.1)
M TM(P) = ∑ G ii P i 2 , i=1
(A.2)
unimodal and that for little energy transfer the distributions are bimodal. It is also with kinetic energy noteworthy that the simulated densities are not as strongly peaked as the theoretical densities PM (E,τ) characterized by M>2 [see Eq. (4.13)]. This is strong evidence to support our contention that when IVR occurs in these models, only a
27
H. W. Schranz and M. A. Collins
and potential energy
H. W. Schranz and M. A. Collins
28
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M VM(Q) = ∑ k i δQ i 2 i=1
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(A.5)
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Thus, the microcanonical probability density PM (E,Q) of a configuration Q given a
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,
by W. L. Hase (JAI Press, New York, 1992), p. 1.
[E-VM(Q)](M-2)/2
=
⌠ (M-2)/2 ⌡ dQ [E-V M (Q)]
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(A.6)
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√ π
(L-1)! L 2!
aL
, a>0 ,
(A.7)
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be obtained by integration of Eq. (A.6) over Q1 , ... , QM-1 subject to the constraint
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H. W. Schranz and M. A. Collins
31
Phys. 96, 2034 (1991).
32
H. W. Schranz and M. A. Collins
TABLES
[44] L. Verlet, Phys. Rev. 159, 98 (1967); M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, (Clarendon Press, Oxford, 1987), p. 78.
Table I. Potential energy surface parameters for HOOH models
[45] R. K. Pathria, Statistical Mechanics, (Pergamon Press, Oxford, 1972), Chapter 2.
Anharmonic model (AH1)
Harmonic model (HH1)
[46] H. W. Schranz, S. Nordholm and G. Nyman, J. Chem. Phys. 94 (1991) 1487;
rOH/Å
0.965
rOH/Å
0.965
H. W. Schranz, J. Phys. Chem. 95 (1991) 4581.
rOO/Å
1.452
rOO/Å
1.452
[47] I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th
θHOO/degrees
100
θHOO/degrees
100
Edition, (Academic Press, New York, 1980), p. 81.
τ/degrees
119
τ/degrees
119
423570
kOH/(cm-1Å-2)
423570
230498
kOO
/(cm-1Å-2)
230498
51387
γOOH/cm-1
51387
a0/cm-1
302.713
kτ/cm-1
2097
a1/cm-1
1208.474
a2/cm-1
1122.084
a3/cm-1
-170.820
kOH/(cm-1Å-2) kOO
/(cm-1Å-2)
γOOH/cm-1
Table II. Range of harmonic HOOH models based on that in Table I. Model
Frequencies
Altered Parameters
HH2
ωsb : ωτ = 2 : 1
kτ = 6849
HH3
ωab : ωτ = 2 : 1
kτ = 6207
HH4
ωsb : ωτ = 3 : 1
kτ = 3044
HH5
ωτ+ωOO = ωsb ; ωsb : ωτ ≠ 3 : 1
HH6
ωsb : ωτ = 3 : 1 ; ωτ+ωOO ≠ ωsb
kτ = 3024, kOO = 206587
HH7
ωab : ωτ = 3 : 1
kτ = 2759
HH8
ωsb : ωτ = 4 : 1
kτ = 1712
HH9
ωab : ωτ = 4 : 1
kτ = 1552
kτ = 3658
33
H. W. Schranz and M. A. Collins
Table III. Potential energy surface parameters for CH3OOCH3 models Anharmonic model (AD1) rOC/Å
Harmonic model (HD1) rOC/Å
1.42
34
H. W. Schranz and M. A. Collins
Table V. Normal mode frequencies and the percentage of symmetry coordinate contributionsa to each normal mode for typical HOOH models (AH1 and HH1).
1.42 ω/cm-1
torsion 99.98
Percentage of symmetry coordinate asym. sym. OO stretch asym. stretchd bendc bend b
sym. stretche
rOO/Å
1.46
rOO/Å
1.46
θCOO/degrees
105.2
θCOO/degrees
120
409
τ/degrees
119
τ/degrees
180
938
kOC/(cm-1Å-2)
2×10 5
kOC/(cm-1Å-2)
2×10 5
1407
kOO/(cm-1Å-2)
1.3×105
kOO/(cm-1Å-2)
1.3×105
1479
γOOC/cm-1
5×10 4
kOOC/cm-1
5×10 4
3895
a0/cm-1
-1074.60
kτ/cm-1
2800
3896
a1/cm-1
1433.17
a2/cm-1
2149.20
that symmetry coordinate when the molecule is distorted from equilibrium along the
a3/cm-1
922.84
eigenvector for each normal mode.
0.02 93.1
6.8 99.99
6.8
0.01 93.1
0.01 0.01
0.04 99.99
0.05
99.93
a Defined as the percentage of the total potential energy due only to terms involving
Table IV. Range of CH3OOCH3 models based on those in Table III. Model
Frequencies
Altered Parameters
HD2
ωsb : ωτ = 2 : 1
kτ = 6611
HD3
ωab : ωτ = 2 : 1
kτ = 4686
HD4
ωsb : ωτ = 3 : 1
kτ = 2938
HD5
ωsb : ωτ = 3 : 1
kτ = 2906, τe = 120°
HD6
ωab : ωτ = 3 : 1
kτ = 2082
HD7
ωsb : ωτ = 4 : 1
kτ = 1653
HD8
ωsb : ωτ = 4 : 1
kτ = 1823, θe = 90°
HD9
ωab : ωτ = 4 : 1
kτ = 1172
AD2
ωsb : ωτ = 2.4 : 1
γθ = 9200
AD3
ωsb : ωτ = 2.1 : 1
γθ = 6336
AD4
ωsb : ωτ = 2.9 : 1
γθ = 14256
AD5
ωsb : ωτ = 4.1 : 1
γθ = 32076
AD6
ωsb : ωτ = 4.8 : 1
γθ = 47000
b δθ1- δθ2; c δθ1+ δθ2; d δR1 - δR3; e δR + δR3. 1
35
H. W. Schranz and M. A. Collins
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H. W. Schranz and M. A. Collins
Table VI. Normal mode frequencies and the percentage of symmetry coordinate
Table VII. Normal mode frequencies and the percentage of symmetry coordinate
contributionsa to each normal mode for a typical harmonic model of CH3OOCH 3
contributionsa to each normal mode for a typical anharmonic model of CH3OOCH3
(HD1).
(AD1).
ω/cm-1
torsion
127
100
Percentage of symmetry coordinate asym. sym. OO stretch asym. stretchd bendc bend b
328.6
sym. stretche
100
ω/cm-1
torsion
Percentage of symmetry coordinate asym. sym. OO stretch asym. stretchd bendc bend b
73.5
99.2
0.1
0.7
360.1
0.8
17.0
73.0
390.3
28.3
53.1
18.6
392.8
795.0
59.1
39.6
1.3
785.7
933.3 1020.7
100 12.6
7.3
99.1 78.6
936.9 80.1
973.5
9.1 0.9
20.7 0.9
4.3
sym. stretche
0.7 99.1
5.5
90.2
a Defined as the percentage of the total potential energy due only to terms involving
a Defined as the percentage of the total potential energy due only to terms involving
that symmetry coordinate when the molecule is distorted from equilibrium along the
that symmetry coordinate when the molecule is distorted from equilibrium along the
eigenvector for each normal mode.
eigenvector for each normal mode.
b δθ1- δθ2; c δθ1+ δθ2; d δR1 - δR3; e δR + δR3. 1
b δθ1- δθ2; c δθ1+ δθ2; d δR1 - δR3; e δR + δR3. 1
37
H. W. Schranz and M. A. Collins
Table VIII. Average G-matrix elements for HOOH.a
R1
R2
R1
1.063
-0.011
R2
0.125
R3
R3 0.000
θ1
θ2
τ
0.041
0.091
0.007
-0.011
0.061
0.060
0.000
1.063
0.019
0.041
0.006
1.126
0.032
0.048
1.126
0.047
θ1 θ2 τ
Table X. Average energy transfer for the torsional valence mode in the harmonic HOOH models for given excitations in the torsion and bending modes {nτ,nab,nsb}. Model HH1 (Table I) HH2 ωsb : ωτ = 2 : 1
2.393
a For bond lengths in Å, mass in amu and energy in cm-1 .
These values are
averages over time of an ensemble of classical trajectories for model
HH3 ωasb : ωτ = 2 : 1
HH1 initialised with zero-point vibrational energy.
Table IX. Average G-matrix elements for CH3OOCH3.a θ2
τ
0.032
0.032
-0.001
-0.011
-0.039
0.033
0.000
0.129
0.032
0.032
-0.001
0.114
0.065
-0.001
0.114
-0.001
R1
R2
R1
0.129
-0.031
0.000
0.125
R2 R3 θ1
R3
θ1
θ2 τ
a For bond lengths in Å, mass in amu and energy in cm-1 .
0.174
HH4 ωsb : ωτ = 3 : 1
HH5 ωτ+ωOO = νsb ωsb : ωτ ≠ 3 : 1
HH6 ωsb : ωτ = 3 : 1 ωτ+ωOO ≠ ωsb
These values are
averages over time of an ensemble of classical trajectories for model HD1 initialised with zero-point vibrational energy.
38
H. W. Schranz and M. A. Collins
HH7 ωasb : ωτ = 3 : 1 HH8 ωsb : ωτ = 4 : 1 HH9 ωasb : ωτ = 4 : 1
nτ,nab,nsb 0,0,0 1,0,0 2,0,0 5,0,0 0,0,0 1,0,0 2,0,0 5,0,0 10,0,0 15,0,0 0,1,0 0,0,1 0,0,0 1,0,0 2,0,0 5,0,0 10,0,0 15,0,0 0,1,0 0,0,1 0,0,0 1,0,0 2,0,0 5,0,0 0,1,0 0,0,1 0,0,2 0,0,0 1,0,0 2,0,0 5,0,0 0,1,0 0,0,1 0,0,2 0,0,0 1,0,0 2,0,0 5,0,0 0,1,0 0,0,1 0,0,2 0,0,0 1,0,0 2,0,0 5,0,0 0,0,0 1,0,0 2,0,0 5,0,0 0,0,0 1,0,0 2,0,0
/cm-1 t = 0.5 ps t = 5.0 ps 2.6 -0.1 -1.4 -8.0 61.1 77.1 -93.9 -1082.1 -2308 -2259.8 10.7 91.7 26.2 -63.1 -58.7 -107.7 -807.4 -1936.9 95.4 237.8 3.6 48.5 93.3 -229.1 12.2 58 39.5 153.5 250.3 20.1 21.8 16.5 11.8 7.5 47.6 46.1 37.2 66.5 -1.0 28.5 62.9 -293.1 8.2 6.5 36.8 5.9 1.1 -21.9 -95.2 1.1 -4.2 7.1 38.5 -0.8 5.6 -1.1
18.5 88.3
H. W. Schranz and M. A. Collins 5,0,0
-37.8
39
H. W. Schranz and M. A. Collins
40
Table XI. Average energy transfer for the torsional valence mode in the harmonic CH3 OOCH 3 models for given excitations in the torsion and bending modes {nτ,nab,nsb}.
H. W. Schranz and M. A. Collins Table XI.
Model HD2 ωsb : ωτ = 2 : 1
HD3 ωasb : ωτ = 2 : 1
HD4 ωsb : ωτ = 3 : 1
HD5 ωsb : ωτ = 3 : 1 τe = 120°
HD7 ωsb : ωτ = 4 : 1
HD8 ωsb : ωτ = 4 : 1 θe = 90°
nτ,nab,nsb 0,0,0 1,0,0 2,0,0 5,0,0 10,0,0 20,0,0 30,0,0 40,0,0 0,1,0 0,0,1 0,0,2 0,0,5 0,0,10 0,0,20 0,0,0 1,0,0 2,0,0 5,0,0 10,0,0 20,0,0 30,0,0 40,0,0 0,1,0 0,0,1 0,0,0 1,0,0 2,0,0 5,0,0 10,0,0 20,0,0 30,0,0 40,0,0 0,1,0 0,0,1 0,0,0 1,0,0 5,0,0
t = 0.5ps 72.3 -14.1 -178.7 -581.7 -774.3 -539.4 -1393 -2697 62.3 333.1 142.7 717.1 1486.6 1684.3 0.5 -3.5 0.1 -32.6 -366.6 -904.6 -402.3 -415.8 8.1 12.4 0.2 2.4 4.5 5.9 -15.2 -128.9 -164.6 -167.6 2.8 1.9 -2.3 2.7 -12.1
10,0,0 20,0,0 30,0,0 40,0,0 0,0,0 1,0,0 5,0,0 10,0,0 20,0,0 30,0,0 40,0,0 0,1,0 0,0,1 0,0,2 0,0,0 1,0,0 5,0,0
-79.1 -356.0 -713.9 -1040.7 -0.8 2.7 9.4 12.4 -87.6 -274.8 -350.4 1.2 0.7 1.9 0.8 -1.3 -0.7
10,0,0 20,0,0 30,0,0 40,0,0
-22.1 -145.2 -240.9 -325.9
/cm-1 t = 5.0ps
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H. W. Schranz and M. A. Collins
Table XII. Average energy transfer for the torsional valence mode in an anharmonic CH 3 OOCH 3 model (AD2) for given excitations in the torsion and bending normal modes {nτ,nab,nsb}. nτ,nab,nsb 0,0,0 1,0,0 5,0,0 10,0,0 20,0,0 30,0,0 40,0,0 50,0,0 60,0,0 70,0,0 80,0, 0 100,0,0
6.0 49 -7.0 -715.3 -1511 -2257.8 1.4 3.6 1.7 -37.2 -45.8 -108.0 -224.6 -73.8
42
/cm-1 t = 0.5ps 13 -1.0 -178 -266 -374 -580 -698 -736 -840 -1000 -1092 -1248
H. W. Schranz and M. A. Collins
43
FIGURE CAPTIONS
H. W. Schranz and M. A. Collins
44
Figure 6. Average torsion energy versus time, for initial torsional excitation of 0 (m), 1 (r), 2 (∆) or 5 (l) quanta, for harmonic models of HOOH: a) HH8, ωsb : ωτ = 4 : 1;
Figure 1. The labelling scheme used for the internal valence coordinates. The
b) HH9, ωasb : ωτ = 4 : 1.
torsion angle τ is the angle between the planes A1 B 2 B 3 and B2 B 3 A 4 where the atoms are numbered in sequence.
Figure 7. Average energy versus time for a harmonic model of CH3OOCH 3 (HD2, ω sb : ω τ = 2 : 1) following single quantum excitation of the symmetric bend: a)
Figure 2. a) The torsion potential of Eq. (2.6) for the HOOH torsion (model AH1 of
valence mode energies: OOC bond angles (l,n ) and torsion angle (s ); b) normal
Table I); and b) the corresponding effective classical frequency (cm-1). The filled
mode energies: symmetric bend (l), asymmetric bend (n), torsion (s).
circles (l) are energy gaps between adjacent levels of the isolated torsion mode [40] versus the average of the energy levels. The filled triangles (s ) are the average
Figure 8. Average torsion energy versus time for a harmonic model of CH3OOCH3
of successive energy level gaps for energies above the cis (τ = 0) barrier.
(HD2, ωsb : ωτ = 2 : 1) following: a) 0 (m), 1 (r), 2 (◊), 5 (∆), 10 (l), 20 (n), 30 (u) or 40 (s ) quanta of torsional excitation, or b) 0 (m ), 1 (r), 2 (∆ ), 5 (l), 10 (n ) or 20 (s )
Figure 3. a) The torsion potential of Eq. (2.6) for the CH3 OOCH 3 torsion (model
quanta of symmetric bend excitation.
AD1 of Table II); and b) the corresponding effective classical frequency (cm-1). The filled circles (l) are energy gaps between adjacent levels of the isolated torsion
Figure 9. Average torsion energy versus time for a harmonic model of CH3OOCH3
mode [40] versus the average of the energy levels. The filled triangles (s ) are the
(HD5, ωsb : ωτ = 3 : 1) with τe = 120o, given: a) 0 (m ), 1 (r), 5 (◊), 10 (∆), 20 (l), 30
average of successive energy level gaps for energies above the cis (τ = 0) barrier.
( n ) or 40 (u ) quanta of torsional excitation; b) for torsional frequencies at 3 : 1 resonance [ω τ = 130 cm-1 (n ) as in (a)]; below resonance [ω τ = 119 cm-1 (l)]; and
Figure 4. Average torsional energy versus time for a harmonic model of HOOH
above resonance [ωτ = 139 cm-1 (s)].
(HH2, ωsb : ωτ = 2 : 1). Initial torsional excitation: 0 (m), 1 (r), 2 (∆), 5 (l), 10 (n) and 15 (s) quanta.
Figure 10. Average torsion energy versus time for a harmonic model of CH3OOCH3 (HD7, ω sb : ω τ = 4 : 1), given: a) 1 (m ), 5 (r), 10 (∆ ), 20 (l), 30 (n) or 40 (s) quanta
Figure 5. Average energy versus time for a harmonic model of HOOH (HH4,
of torsional excitation; b) for torsional frequencies at 4 : 1 resonance [ωτ = 97.6 cm-1
ω sb : ω τ = 3 : 1). a) Torsional energy for initial torsional excitation of 1 (m ), 2 (r),
(l) as in (a)]; below resonance [ω τ = 96.4 cm-1 (n )]; and above resonance [ω τ =
and 5 (∆ ) quanta. b) O_ O bond energy (m ), OOH bond angle energies (r,∆ ) and
98.8 cm-1 (s)].
torsion angle energy (l), for an initial excitation of two quanta in the symmetric bending normal mode.
Figure 11. Average torsion energy versus time for a harmonic model of CH3OOCH3 (HD8, ωsb : ωτ = 4 : 1), with θe = 90° given: a) 1 (m ), 5 (r), 10 (∆), 20 (l), 30 (n) or 40 (s) quanta of torsional excitation.
H. W. Schranz and M. A. Collins
45
Figure. 12. Simulated and theoretical torsional probability densities P(E,τ) versus torsion angle for three models of CH3 OOCH 3 : a) HD2; b) HD7; and c) HD8. Initial torsional excitations are 0 (l), 5 (n) and 40 (s) quanta. Theoretical predictions are given for an isolated harmonic torsion, Eq. (4.14), (solid lines) and for a harmonic torsional mode in microcanonical equilibrium with another harmonic mode, Eq. (4.15), (dotted lines).
Figure 13. Average torsion energy versus time for an anharmonic model of CH3OOCH3 (model AD2, ωsb : ωτ ≈ 2.4 : 1), given 0 (m ), 1 (r), 5 (◊), 10 (∆), 20 (l), 30 (n), 40 (u), 50 (s), 60 (m ⋅ ), 70 (\r), 80 +(r ) or 100 (⋅r) quanta of torsional excitation.
Figure 14. Average torsion energy versus time for a) short times, b) long times and c) torsional probability densities P(E,τ) for a range of anharmonic models of CH3OOCH3 with 60 quanta of torsional excitation and ωsb : ωτ ≈ 2.1 : 1 (model AD3) (l); 2.9 : 1 (model AD4) (n ); 4.1 : 1 (model AD5) (u ); and 4.8 : 1 (model AD6) (s ). Because of extremely rapid energy transfer out of the torsion on a subpicosecond timescale, these results were generated without complete randomisation of the torsional mode (except for the sign of the torsional momentum) but include randomisation of the other vibrational modes.