Nonlinear resonance and torsional dynamics

H. W. Schranz and M. A. Collins

1

Nonlinear resonance and torsional dynamics: model simulations of HOOH and CH3OOCH3


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.


3

Molecules of the type ABBA are chosen because the symmetry restricts the number

4


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


The torsional potential is taken to be quadratic in the torsion angle


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


III.

KINEMATIC COUPLING


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


and in R1 and R3. This means that a Taylor expansion of the G-matrix elements

10


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



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


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


.

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


16


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


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 )!


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



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


21

discussion of Eq. (3.11), since the kinetic energy operator contains no coupling


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


23


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


25

anharmonic coupling terms required in the Hamiltonian for n : 1 resonance (which


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


and potential energy


28

REFERENCES

M VM(Q) = ∑ k i δQ i 2 i=1

.

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E ≥ V M (Q )

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(A.5)

and C1 is a constant independent of E and Q.

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by W. L. Hase (JAI Press, New York, 1992), p. 1.

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=

⌠ (M-2)/2 ⌡ dQ [E-V M (Q)]

.

(A.6)

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√ π

(L-1)! L 2!

aL

, a>0 ,

(A.7)

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29


30

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31

Phys. 96, 2034 (1991).

32


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


Table III. Potential energy surface parameters for CH3OOCH3 models Anharmonic model (AD1) rOC/Å

Harmonic model (HD1) rOC/Å

1.42

34


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


36


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


328.6

sym. stretche

100

ω/cm-1

torsion


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









37


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


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


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

41


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


43

FIGURE CAPTIONS


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.


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.