Connecting level statistics and reaction dynamics: Mode selective decay of resonant states for HXXH-type molecules

Chemical Physics 154 ( 199 1) 63-84 North-Holland

Connecting level statistics and reaction dynamics: mode selective decay of resonant states for HXXH-type molecules B. Hartke ’ Institut ftirphysikalische Chemie der Universitiit Wiirzburg, Marcusstr. 9-11. W-8700 Wiirzburg, Germany

W. Karrlein Siemens-Nixdorf-Informationssysteme AG, Otto-Hahn-Ring 6. W-8000 Miinchen 83, Germany Received 2 October 1990; in final form 5 February 199 1

Molecular complexity, defined as the number of degrees of freedom and couplings between them, is a crucial issue when investigating the dynamical properties of molecules. It is often found that a system tends to behave in a statistical way when the couplings are effective. By freezing some degrees of freedom within a molecule, it is possible to study the effects of the remaining degrees of freedom, especially with respect to their interactions. The strength of these interactions can be correlated to structural features such as bond angles or atomic masses separating vibrationally excited bonds. Important systems for investigations along these lines are HXXH-type molecules (here HCCH and HNNH). As predicted in recent statistical analyses of energy level distributions, there should be significant differences in their dynamical behaviours. It is found that HNNH displays high selectivity of quasidegenerate resonances, whereas linear HCCH molecules show a rapid internalenergy redistribution. By comparison of HNNH with a less complex model system H{NN}H (where the N==Nunit is reduced to a pseudoatom with twice the mass ofone N atom) a blocking effect of the NN fragment is discovered, that is not found in the systems HCCH and H{CC}H. This blocking effect of an additional bond is in contrast to a dynamical acceleration effect found recently for the CO bond in formaldehyde.

1. Introduction Mode selectivity plays an important role in the understanding of elementary chemical reactions [ 1,2 1. It is characterized by a non-monotonic correlation of reactivity and energy. At first sight, this is equivalent to a strong contradiction to the older statistical RRKM viewpoint, where an increased energy leads to an acceleration of chemical reactions. RRKM theory assumed that IVR causes localized energy (e.g., in one single vibrational mode) to spread over all other degrees of freedom before the reaction can take place. Mode selectivity usually requires mechanisms that hinder rapid IVR and/or benefit fast reactive processes. In this sense, rapid IVR and mode selectivity appear to exclude each other at first sight. This, however, is not true in every case. Recent ex’ Present address: University of California, Los Angeles, Dept. of Chemistry and Biochemistry, 405 Hilgard Av., Los Angeles, CA 90024- 1569, USA.

periments by Polik et al. [ 3 ] and statistical analysis by Miller et al. [ 4,5 ] on formaldehyde have shown that rapid IVR and mode selectivity might “exist” at the same time. (In effect IVR, rapid or slow, exists unless the different modes are completely uncoupled.) Whether one or the other effect is observed depends only on the “magnification” one uses to look at the system. While investigating the dynamical behaviour of individual energy levels (a high magnification), Polik et al. [ 3 ] have found selective non-statistical behaviour of the decay rates of resonant states. On the other hand (at lower magnification), when looking at a sample of energy levels [ 4,5 1, a statistical or monotonic dependence of the decay rates of energy could be discovered. This is in accordance with studies of Levine [6] on unimolecular decay times. In the present study, we want to focus on mode selective phenomena in the unimolecular radical dissociation of HX (X=C, N) bonds. These phenomena can be grouped into three types [ 7 ] : ( 1) characteristic patterns of highly excited vi-

0301-0104/91/% 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)

64

B. Hartke,

K Karrlein

/Connecting

brational wavefunctions, leading to (2) decay rates that differ by orders of magnitude, even for near-degenerate resonant states, and (3) selective breakage of one of several isotopical bonds. All of these effects are described in great detail in ref. [ 7 1, where many references can be found. Here, the first two phenomena are of interest to us. There are two characteristic structures of highly excited wavefunctions: local and hyperspherical. The first have been known and understood for a long time [ 8-13 1. Local wavefunctions are characterized by their extension into one of the product exit valleys of the potential surface [ 9,10 1. This corresponds to investing almost all of the available energy into one single mode (the promoting mode) to create a highly excited (hot) bond. Hyperspherical wavefunctions have been predicted by Manz et al. [ 14-221. They have been experimentally detected only very recently [ 23-28 1. The structure of hyperspherical modes is characterized by an arc in the coordinates of the bond distances, perpendicular to the product valleys. The total vibrational energy is distributed equally among several modes. Therefore hyperspherical wavefunctions are more compact. It has been found that there are great differences of decay times of different H-X (X=C, N, 0) bonds [ 16-1 f&20,21,29]. Recently, it has been possible to simulate the experimental findings of Polik et al. [ 3 ] for formaldehyde using a quite simple model Hamiltonian [ 301. By use of the same Hamiltonian, it was possible to show [ 3 1 ] that the vibrational energy levels are as close to the Wigner distribution as Miller [ 4,s ] suggested based upon experimental results. In ref. [ 3 11, similar statistical investigations of the energy levels of HNNH, H{NN}H, HCCH, and H{CC}H are made. (Note that {XX} symbolizes a pseudoatom with twice the mass of one X atom.) These calculations are based upon a model (see section 2 ), which considers all HX and XX stretches but freezes all other degrees of freedom. Additionally, only adjacent bonds are allowed to couple, and bond angles are frozen to their equilibrium values. (Therefore HXXH can be called a three-dimensional system, i.e. one with three degrees of freedom, and H{XX}H a two-dimensional system.) With this model Hamiltonian there are significant differences between HNNH/H{NN}H and HCCH/H{CC}H: whereas the energy levels of the first systems follows

level statistics

and reaction

dynamics

the Poisson distribution (which means they are only weakly correlated), the acetylene-based systems follow the Wigner distribution (which means high correlations between them). These findings can be discussed in terms of two features of the model systems: the bond angle HXX or H{XX}H and the mass of the central atoms X or {XX} [ 3 11. Therefore, we postulate that there will be differences in the dynamical behaviour of the highly excited vibrational resonances of the HX bonds: the diimin-based systems should show regular structured wave functions, as the different modes are more or less uncoupled; consequently, the IVR is slowed down. Selective decay dynamics should be favoured because of this fact. In contrast, the highly correlated energy levels of the acetylene systems should cause irregular structures and irregular dynamics, as IVR is rapid in this molecular system. Our chosen model systems have been studied to different degrees. Diimin has been subject to only a few investigations. Classical trajectory calculations [ 32,331 predict a fast flow of energy (400 fs) in the range of 2 eV (excitation energy), resulting in a redistribution of about 40% of the initially localized energy. The trajectories have been called chaotic [ 321. The initiation of this chaotic movement and the rate of the flow of energy depend on the bending motions [ 331. The torsion is coupled to the HN stretch and to the HNN bending vibration and therefore acts as a crucial factor to enhance the flux [ 341. These calculations suggest a Wigner-like behaviour and irregular dynamics. As the results presented in this paper are based upon quantum mechanical calculations, they are complementary to those previous results in refs. [ 32-341. Experiments with HNNH are difficult because of the instability of the HNNH molecule. Isomerization has been studied with classical trajectories [ 3 5 1, where Coriolis couplings between several stretching, bending, and torsional motions are identified to support isomerization. Acetylene, however, has been investigated to a great extent, but the results are still controversial; e.g. considering the (experimental) value of the dissociation energy of the HC bond [ 36-45 ] to radical products. Several coordinate systems are discussed with respect to their applicability to describe the dynamics of HCCH. Fleming and Hutchinson et al. [ 461 are in favour of “mixed” coordinates (bond and rotating

65

B. Hartke, W. Karrlein/Connecting levelstatisticsand reactiondynamics

local coordinates) that are energy dependent. According to these authors, a pure local coordinate system is doomed to fail because of the high zero-order couplings. However, in refs. [ 47-561 the use of local coordinates leads to a correct reproduction of the frequencies of the HCCH spectra. Again in contrast, Scherer et al. conclude, that neither normal nor local coordinates are suitable to describe the overtone spectra [ 57 1 correctly. Many investigations concentrate on the simulations of stimulated emission spectra (SEP) and their dynamical implications [ 32,33,38,58-741. An interesting attempt to explain the chaotic clumps found in the SEP spectra [ 66-7 1 ] in terms of a temporal hierarchy of different processes [ 34,42,58,59,73] was found. It was thus possible to analyze surprising differences of HCCH and DCCD spectra [ 72,741. All in all, this short overview shows that acetylene is a molecule of high interest. As HNNH is a topologically equivalent molecule to HCCH, it is surprising that there are hints, based upon a statistical analysis [ 3 11, to a significantly different dynamics, which is the subject of this report. In the next section, the model Hamiltonians and techniques are presented. Section 3 is devoted to the results, which are discussed and summarized in section 4.

ti=&+&,+i~++a,+&~

(2.la)

for three-dimensional systems HXXH (note that there are no direct coupling terms for the bonds a and b as they are not adjacent ) and h=i;,+r;,

(2.lb)

+&,

for two-dimensional systems H{XX}H, where the bond c is neglected and replaced by a pseudoatom {XX> with twice the mass of atom X. Therefore the bonds a and b are coupled, since they become adjacent. The bond Hamiltonians for dissociative HX bonds are described as Morse oscillators [21,75,77,78]. &= $

+Dyf-D, I

i=a,b,

(2.2)

where pi is the reduced mass of bond i and yi=l-exp(-BAqi).

(2.3)

Here Aqi is the bond dilatation Aqi =qi - qio with equilibrium bond length qio and /3 is the Morse parameter. The vibrationally cold XX bonds - in threedimensional systems - are modelled as harmonic oscillators:

2. Methods and techniques

j&

2.1. Hamiltonians

Finally, the coupling terms are approximated as

The naming of atoms, bonds and angles are made according to the following scheme: H-X-X-H++1”314b2. a As internal coordinates, we restrict ourselves to stretching coordinates of bond a to c: qa, q,, and qC. The bond angles are frozen at their equilibrium values. Coupling is allowed only for adjacent bonds [ 2 1,3 1,75,76 1. For convenience, capital letters indicate operators, wavefunctions etc. of the three-dimensional systems, small letters those for two-dimensional systems and italic small letters those for individual bonds and couplings. The model Hamiltonians are

2

+0.5k,Aq,z.

i,_= COS(%j) - 1 PiPj Y lJ m

(2.4)

(2.5)

which means that there is only a kinetic coupling of adjacent bonds. 2.2. Solving the Schriidinger equation for twodimensional systems The two-dimensional Schriidinger equation is solved by transforming it into an algebraic eigenvalue problem [ 2 1,75 1. For this purpose, the eigenfunctions ~/m~&of the H{XX}H systems are determined as a linear combination of symmetrized products of Morse wavefunctions (LCSPM) [ 14181.

B. Hartke, W. Karrlein /Connecting level statistics and reaction dynamics

66

(2.6)

I~~+)=n,~(Ik).I~),fl~),Ik),)

3

(2.7)

where I k)i is the ith Morse wavefunction of the individual HX bond: Ailk);=eilk)i.

(2.8)

The normalization constants nw= 1 for k#l and Gk-- 1/fi for k=l, which occurs only for + symmetry. With nmaxas the largest vibrational quantum number of the HX oscillator, there are d=0.5(n,,,+l)(n,,,+lfl)

(2.9)

basis functions I kf f ) in ( 2.6 ) . For the present cases nmax=17 (HN) and 27 (HC); the dimensions are ( symmetry) d(NH)=136

and

(2.10)

d(CH)=351.

By insertion of (2.6) into the Schriidinger equation (2.11)

4 Wmn* > =emnt I Wmn+ > the algebraic eigenvalue problem hc,,k

(2.12)

=emnf omnf

is solved using conventional diagonalization algorithms of the EISPACK library [ 791. The coefficient c with largest weight c2 is used to label the states with vibrational quantum numbers m and n [ 2 1,75 ] c,,k,,,+

= mkp &,,,~

(2.13)

.

2.3. Solution of the Schriidinger equation for threedimensional systems The strategy of evaluating eigenfunctions for the three-dimensional systems is similar to the one used with two-dimensional systems [ 75 1. Here the ansatz (2.6) is expanded in terms of harmonic wavefunctions of the XX bonds,

I ~mnfv)=

C C G+u,mn~vIk~+u) kl

u

,

(2.14)

where )u ) c is the z&hharmonic wavefunction of bond c and an eigenfunction of the Hamiltonian (2.4). The corresponding algebraic eigenvalue equation is

, HC mn?v---%n~vCmn~v

(2.16)

with the dimensions determined by (cf. eq. (2.9) ) D=d(v,,,+

1) ,

(2.17)

where v,,, is the highest harmonic quantum state included in the sum (2.14). The labeling of states is in analogy to the rule for two-dimensional systems. The time needed for diagonalization of a matrix scales with the cube of its dimension. Therefore, a method is needed to minimize the dimension of the Hamilton matrix in ( 2.16 ) , by reducing the basis sets of the Morse and the harmonic wavefunctions. In addition to others, cf. [ 53,56,80-891, a method for this purpose has been put forward in ref. [ 75 ] for CH20. As there is no coupling between both promoting modes HN in HNNH (cf. eq. ( 2.1a) ) , this model has to be modified when applied to molecules of the HXXH-type. Neglecting the XX bond in a first step (see the neglect of CO for CHIO [ 75 ] ) would induce a coupling of the HX bonds that will not persist within the full three-dimensional systems. The modified reduction procedure is a three-step process: Step 1. The Schriidinger equation (2.16) is solved using all 136 LCSPM basis functions (kllf: ) plus the two lowest harmonic basis functions, I0), and I 1) C. The resulting Hamilton matrix is of dimension 272. (As the HN oscillators are decoupled, restriction of the harmonic basis set to only IO), would lead to a diagonal matrix without any couplings, in which we are interested. Thus, a reduction would lead to an incorrect basis with respect to the coupling. ) Step 2. State-selective reduction of the LCSPM basis functions I kl? ) using the strategy of ref. [ 7 5 1. It is important to be aware of the following point: every basis function (klf > appears twice in the summation (2.14), once combined with IO) Cand once with I 1),. It is filtered out only if both expansion coefftcients CkikOand &, are considered unimportant for fulfilling the convergence

with Ikl+u)=lkl?)lu),,

(2.15)

ckl,

(O,l),mn~

(0.1)

=~klf(O,l),mn+(O,I)

(2.18a)

B. Hartke, W Karrlein/Connecting levelstatisticsand reactiondynamics

for e-+0, where t is a convergence parameter, see ref. [751, and Lnfc~,lj ad Ckj, (0.1 ),mn+ (0.1) are exact values using the full basis of 272 wavefunctions. Step 3. Higher harmonic basis functions 1u), with u& 2 are included gradually until convergence is satisfied: E mnk Vmm. --Emn~v c Wf

(2.18b)

>

for v,,,+v= cx),where the reduced basis of (2.18a) is used. Because of the large number of Morse oscillators in the CH bond [ 27 1, the described strategy cannot be applied to HCCH. The dimensions of the Hamilton matrix to be diagonalized exceed all the available computer facilities. Therefore, we used the method of triatomic fragments in molecules (TAFIM ) . This is convenient because of acetylene, where there is only one triatomic fragment, namely HCC. For further details of TAFIM see ref. [ 761. Irrespective of what reduction procedure ( [ 75 ] or TAFIM [ 761) is used, a reduced basis set of harmonic and Morse wavefunctions is obtained. By use of this set eq. (2.14) can now be approximated by c* kl

I*

Z(kl+u,mnfvi~~+~)

9

(2.19)

u

where 1k) , 11)) and I u) are elements of the reduced basis. This ansatz (2.19) leads to a much smaller eigenvalue problem (2.20)

IX P?l~flJ=G?I~*“L”+”

compared to (2.16). The tilde indicates reduced properties. 2.4. SOD-FFTpropagation

of resonant states

Resonant states dissociate with characteristic lifetimes r=, defined by [ 7,901 (!&(t=O)

Because of the small coupling coefficient cos cw/m (cf. eq. (2.5)), it is true that the eigenfunctions of (2.6) and (2.19) are good approximations to the exact resonant states provided their energy is only slightly above the dissociation threshold. The lifetimes can therefore be evaluated using the eigenfunctions I%,,* > and I Y,,,n+v):

3

u,mn f “imax =Ckl+u,mn+o

1 ~mn+v)=

67

I P=(t))

=ex (2.21)

The basis used to calculate the eigenfunctions of our Hamiltonian is incomplete: as continuum states cannot be described exactly. For bound states the basis would be exact, if the coupling was zero [ 78,901.

lnl( ~mndt=o) I ymdt))

12=--tlk+,

. (2.22)

In order to obtain I Ym,,f .( t ) ) the timedependent Schriidinger equation ifi-$ I Ymnfv> =m ymnrlv>

(2.23)

is solved numerically by the SOD-FFT propagation method [90-921. It is therefore necessary to represent the wavefunction IY,,,n+v)on a discrete rectangular spatial grid, with qi=qi,o+kAqi,

k=O, l,***,k,.

(2.24)

The propagation of (2.23) is carried out using equidistant time steps t,,=nAt.

(2.25)

The spatial and temporal step size Aq and At are determined by refs. [ 90-93 ]

Aqd Wprrmx and At d ZIIE,,,,, ,

(2.26)

where pmpxis the maximum momentum conjugate to the coordinate q and E,._ is the largest eigenvalue of the discrete Hamiltonian.

3. Resolts Two issues of mode selectivity mentioned in section 1 will be subject to our studies: (A) Do characteristic patterns of highly excited vibrational wavefimctions, such as local and hyperspherical structures, persist when increasing the complexity of the system? (B) Can the selective dynamics of local and hyperspherical resonances (found for systems such as CH2, CH20, and H20 [ 21,30,75] ) be established for our systems? With our two- and three-dimensional systems, we are able to compare the dynamical behaviour for dif-

68

B. Ha&e, W. Karrlein /Connecting level statistics and reaction dynamics

Table 1 Parameters of the model systems Atomic masses a) (atomic units) ‘)

m(H) m(N) m(C)

1.0079 14.0067 12.011

Bond angle (Y‘) (deg)

106 180 180

Equilibrium bonds a> length (4) c,

4.Y4b 4e

1.9559 2.3622

(HNNH)

2.003 15 2.26771

(HCCH)

Parameters a) of Morse oscillators

D (eV) Bkl (ao’)

3.212 1.214

(HNNH)

5.5981 0.96837

(HCCH)

Coupling constant b-C)

~JJ%l~l

6.96x lo-’

(HNNH)

9.61 x lo-’

(HCCH)

(HNNH) (HCCH) (two-dimensional systems)

a) Parameters taken from ref. [ 381; the same parameters are used for the two-dimensional systems, if not stated otherwise. b, Coupling constant, see eq. (3.1). ‘) Conversion factors: 1 amu= 1822.88734au, 1 u,,=O.52917706~ lo-“‘m, 1 E,,=27.211608eV, 1 atuz2.41888428x lo-“s. Table 2 Results for two-dimensional systems System

State n)

Type b,

Dimension c)

e (eV) *)

Statistical weight c,

s f’

YS’

Figure

H{CC}H

023023+ 226226+ 10121012+

I I I I

351 378 351 378 351 378

0.0629 0.0629 0.9492 0.9492 1.7512

0.2770 0.2769 0.4191 0.4191 0.9570

1.7979

0.9475 0.9476 0.9014 0.9014 0.6630 0.4752

1.0376

0.9543 0.9544 0.8810 0.8810 0.1949 0.0919

la lb 2a 2b 3a 3b

136 136

0.6668 1.0327

0.9569 0.8891

0.2588 0.4879

0.7633 0.1633

4a 4b

HWN]H

21267-

h h

I h

a) States are labeled with the quantum numbers m, n of (2.13) and with the symmetry ( + ) b, I, local; h, hyperspherical. ‘) Number of basis functions (dimension of the eigenvalue problem (2.12 ) ). ‘) Energy above the dissociation threshold. ‘) Statistical weight, see (2.13). f, See (3.5). I) See (3.7).

ferent complexities (i.e. number of degrees of freedom). Therefore we group our results according to issues (A) and (B), dealing first with two-dimensional systems followed by the three-dimensional ones. The model parameters of two- and three-dimensional systems are summarized in table 1. As mentioned there are two important restrictions within our model: (a) only adjacent bonds are allowed to couple via the kinetic coupling constant cos CQ k,= -

m

’

(3.1)

(cf. eq. (2.5) and table 1) and (b) only stretching motions are considered. The usage of harmonic oscillator for double (NN) and triple (CC) bonds in the threedimensional systems is justified by assum-

ing these bonds to be vibrationally cold. Dissociation of them is not considered here. 3.1. Characteristic patterns of vibrational wavefunctions (A) Table 2 contains important information about all the selected vibrational energy levels of the two-dimensional systems. Note that all the energies are scaled such that the minimum energy of the educts H{XX)H equals -D, in other words, the classical dissociation threshold is 0 eV. The minimum energy for resonant states is given by E,. = C 8,

(3.2)

where i runs over all bonds in the product species and ep is their individual zero-point energy. For the two-

8. Hartke, W. hivreiin /Connecting be1 stat&tiesand reactiondynamics

69

E=0,9492eV mn+= 2 26+ bxai

1

-9&b--

8

i

Fig. 1. Highly excitedstates mn+ ofHfCC)H withenergies.&,,f: the wavefunctions are represented by ~ui~otour plots of ~,,,~t(q~,&)=+_ixmax (i=i, 2,3,4,5) (agl) with vibrational quantum numbers and symmetry mn +-. The potential energy is also shown as equicontour plots V(q,, qa) = _+ixO.25D (i= 1, 2, ..,, 7). The axes q. and & are the internal coordinates of the highly excited H-X bonds. (a) tocal mode mn+ =O 23-, max=1.06;(b)localmodemn~=O23+,max~i.06.

dimensional systems, ( 3.2 ) reduces to the zero-point energy of the H(XX) ascillator e*(H{NN))=0.1909 e”( H(K)

eV,

) =0.2034 eV .

(3.3)

Figs. 1-4 show equicontour plots of highly excited vibrational wavefunctions. The following conclusions can be drawn from these results: ( 1) In all systems there are regular structured vibrational wavefunctions according to the Hose-Taylor criterion [ 941. (2) The classification into local or hyperspherical wavefunctions is made either by visual inspection of the plots or by regarding different qualitative measures: the dominant expansion coefficient, the “entropy” of a state f 75 f , and the energy distribution

Fig. 2. See caption to fig. 1. (a) Local mode mnrt=226--, max-0.60; (b) local mode mnrt =226+, max=O.60.

coefficient y. The local wavefunctions are characterized by one highly excited bond, with the others being cold, cf. figs. 1,2 and 4a. By symmetrization (at the Cr-axis), the frontier lobes point towards the radical pmducts. A hyperspherical excitation is charaeterized by a more or less equal dist~bution of excitation energy. Thus circular patterns are obtained, see figs. 3 and 4b. The local distribution of energy leads to the dominance of one single-basis wavefunction I nzn k ) in the expansion of eq. (2.6 ) . In other words, the corresponding linear coeffkient is close to unity, When the vibrational energy is hyperspherically distributed, the expansion coefficient is smaller (compared to the local ones}, implying significant dis~butions of more basis wavefunctions 1klk ) f 1mn+ >. In general, it is true that l%~&f~> khyp?Rph.

1’

*

(3.4)

Another useful measure for the local/hyperspherical nature of a wavefunction is the entropy, defined as missing information [ 75 ]

B. Hartke, W.Karriein/Connecting levelstatisticsand reactiondynamics

70

HtNNIH

HfCC}H

; ~!

t

%‘Qt

I

1 4

i

1

-

Fig. 3, See caption to fg. mn+=1012-, m&=1.30, mn_+=lOI2+,max=1.30.

s= -

C Ickkmnf

qjao-

1. (a) Hypersphericai (b) h~~phe~~

I2 v

I2 lnIckkmnf

t mode mode

(3.5)

kl

provided a pure state for which 1cHf ,mn+ }’ = S,,, s is zero. On the other hand, an equally distributed state with 1cw+ ,,,,,* 12= 1/n (where n is the numbr of basis wavefunctions in (3.5) ) leads to s=ln( n). Consequently the following inequality is valid:

The third measure (energy distribution) byeq. (3.7) Y= 5

ickljr(fr,mnt

,2(*-E

.

is defined

(3.7)

>

The values of y are between zero and unity. The smaller y, the more equal is the distribution of vibrational energy among the bonds. Here the following inequality holds 0 < yhypersph.

-= %&I

<

1 *

hypersph.

1

5

(3.8)

-v+i-

4

Fi 4. See caption to fig. 1, but this is for the system HfNN)H. (a) Local mode mnrt =2 12-, max=O.96; (b) h~~phe~~l modemnf=67-,max=1.33.

As can be seen from the corresponding values listed in table 2 the classification of the selected states to be local or hyperspherical is confirmed by the criteria mentioned above. These different measures provide different and independent information about the characteristic pattern of a vibrational wavefunction. However, they are not to be taken as quantitative measures. The label “local excitation” is primarily nothing else but a statement about the ~st~bution of vibrational energy between two (or more) bonds. As long as we have the coordinate system of bond coordinates, it is an empirical fact that the linear expansion (2.6) (and of course (2.14), too) is governed by just one dominant coeffkient. Therefore eq. ( 3.3) depends on the coordinate system chosen, Eqs. (3.6) and (3.8 ), however, do not depend on a specific system, as they are sums over all coeffkients. (3) The wavefunctions of H{CC}H are of gerade (figs. lb, 2b, 3b) and ungerade (figs. 1a, 2a, 3a) symmetries. An interesting observation can be made

B. Hark

W. Karrlein /Connecting level statistics and reaction dynamics

by inspection of these figures: local wavefunctions of different symmetry have the same structures (figs. 1, 2)) in contrast to hyperspherical wavefunctions of different symmetry, fig. 3, which show significant differences. This can’be explained as follows: the arc of a hyperspherical wavefunction is situated on the potential energy surface in such a way that a cut along this arc yields a double-minimum potential. At the barrier height of this double-well potential, ungerade hyperspherical wavefunctions have a node, whereas the corresponding wavefunctions of gerade symmetry have a lube. This region is of high energy causing deviations from the characteristic hyperspherical pattern. The higher in energy the hyperspherical wavefunction is, the less this effect should be noticeable. This effect can be seen, regarding the numerical criteria and the energy of the state 1ylo IZf ) in table 2. Local wavefunctions, whose lobes are situated in the double wells (equivalent to the product exit valleys) are not affected by the barrier, and thus they do not show differences. One local and one hyperspherical resonant state of HNNH and HCCH are selected to study the regularity of highly excited vibrational wavefunctions in threedimensional systems. When selecting these wavefunctions, we are guided by the corresponding ones in two-dimensional systems (see above). Tables 3 and 4 document the convergence tests (2.18 ) in order to build up a reduced basis set to be used in the expansion (2.19). Table 3 shows the convergence (2.18a) according to the reduction param-

11

eter E [ 751. As was mentioned in section 2, only two harmonic basis functions have been used, 10) and 11) . Both states are governed by the contribution of two basis wavefunctions. It is interesting to notice that the sum of the vibrational quanta in the secondary contributions 157- 1) for l!P~,_,) and 1210- 1) for I Y2u,,,_o) are more or less equal to the primary contributions. Furthermore, the distribution of the quanta in these secondary basis wavefunctions are equal to that in the primary ones. The main contributors to the local wavefunctions are of a local nature; the same holds for hyperspherical wavefunctions. The maximum number of basis wavefunctions 306 is reduced to 66 and 72 for e = 10m6, respectively; see table 3. Table 4 shows the convergence (2.18b), i.e. the expansion of the harmonic basis above u> 1 for fixed E. It can be seen that convergence is reached for a harmonic basis ranging from u=O to 6 and 8, respectively. For comparison, the converged calculations (with respect to the harmonic basis) are repeated using a stronger tolerance value e = 1O- ‘, which means a larger Morse basis, cf. the last column in table 4. Table 4 shows that there are four secondary basis wavefunctions for the local target state (with C-kl u,mn_“> 0.1) , but only one for the hyperspherical target state, for converged results. The dominant coefftcient (and consequently its statistical weight) of the local state is lower than the one for the hyperspherical state. This is an exception of the qualitative rule (3.4). Again, as above, the nature of the second-

Table 3 Convergence tests for HNNH states with a fixed harmonic basis v=O, 1 State a)

212-O

Energy (eV) w Dimension =) t *’

0.702569 44 5

ij-

Expansion coefficients f,

u

67-O 0.702570 66 6 =)

0.702570 98 7

67-O 57- 1 212-o 210-l

0.9840 0.1290

0.9840 0.1290

1.065548 46 5

1.065545 72 6 =)

1.065545 94 7

0.9771 -0.2011

0.9777 -0.2011

0.9777 -0.2011

0.9840 0.1290

‘) States are labeled with the quantum numbers m, n of (2.13) and with the symmetry ( f ). ‘) Energy above the dissociation threshold. c, Number of reduced basis functions (dimension of the eigenvalue problem (2.20) ) . ‘) Tolerance value shown as a negative decade logarith, see (2. Isa). ClConvetSed results. I) Coefftcient with 1c I> 0. 1 are shown.

72

3. Ha&e,

W. Karriein / &o~~~ting level statistics and reaction dynamics

Table 4 Convergence tests for HNNH states with fixed tolerance value e State a) Energy (eV) ‘) Dimension ‘) t d, UC’

0.702669 99 6 2

ij-

Expansion coefficients 8)

u

212-O 2 io- I 211- 1 2 9-2 2 9-4

0.9729 0.1590 0.1092 -0.1007

0.702548 132 6 3

0.9823 0.1260

State a) Energy (eV) b, Dimension ‘) td’ UC)

1.065704 72 6 2

ij-

Expansion coefficients *)

u

67-O 57- 1

0.702619 165 6 4

0.9774 0.1454 0.1046

212-O 0.702706 198 6 5

0.702745 231 6 6

0.702743 297 6 8”

0.702720 441 I 8

0.9583 0.1670 0.111 -0.1313 0.1126

0.9450 0.1753 0.1134 -0.1640 0.1367

0.9457 0.1749 0.1133 -0.1626 0.1357

0.9530 0.1697 0.1118 -0.1421 0.1190

67-O

0.9740 -0.2129

1.065711 96 6 3

0.9737 -0.2135

1.065712 120 6 4

1.0657 12 144 6 5

1.065712 168 6 6 r’

1.065712 329 7 6

0.9737 -0.2135

0.9737 -0.2135

0.9737 -0.2135

0.9737 -0.2135

a>States are labeled with the quantum numbers m, n of (2.13) and with the symmet~ ( rt ). bt Energy above the dissociation threshold. ClNumber of reduced basis functions (dimension of the eigenvalue problem (2.20) ). d, Tolerance value shown as negative decade logarithm, see (2.18a). ‘) Harmonic basis functions with quantum numbers up to v. ‘) Converged results. s) Coefficient with 1cl 80.1 are shown.

ary basis functions contributing is either local or hyperspherical, according to the nature of the target state. Convergence tests for three-dimensional resonances of HCCH are shown in tables 5 and 6. The reduction method used for this model system is TAFIM [ 76 1. Here, we present the convergence for the target state 1Y226_-o) as an example. For further details see ref. [ 761. To model the local target state, the two fragmental wavefunctions 1y2 a) and 1vz6 o) are needed; note that the first quantum numbers corresponds to the HC bond in the HCC fragment. Table 5 shows the convergence using all 27 Morse basis wavefunctions of the HC oscillator with an increasing harmonic basis; this step corresponds to (2.18b). Again all secondary contributions are of an equal nature to their target state. Because of the low excitation of 1yz o} only one other basis function ap pears with a si~i~~t cont~bution. Convergence is thus easily reached for v=O, 1, 2, 3. This is different for the highly excited fragment target state 1ty26o).

Because of the effective couplings a large basis of harmonic functions is obtained. Convergence is slower and less accurate. Table 6 shows the convergence for fixed harmonic basis sets; this corresponds to (2.18a). This step means a reduction of the Morse basis set. The last columns show the result without reduction for comparison. At last we arrive at reduced basis sets for each bond. (a) The local target state 1Y226_-o): Morse basis wavefunctions for bond a,

IO>, Il>,..., l6),

(3.9a)

b,

14), IS>, .... 126) >

(3.9b)

and harmonic basis wavefunctions for bond c,

IO>, ll>,...,

Ill> f

(3.9c)

(b) Hyperspherical target state I Yf10i2_o): Morse basis wavefunctions for bond

B. Hartke, W. Karrlein /Connecting level statistics and reaction dynamics

73

Table 5 Convergence tests for the HCC fragment with respect to the harmonic basis State a) Energy (eV) b, Dimension c, vd’

- 4.60877 1 54

iu

Expansion coefficients r)

20 11

0.9449 -0.3169

- 4.608429 81 2

1

0.9419 -0.3203

State ‘) Energy (eV) ‘) Dimension ‘) v *’

-0.007772 270 9

iu

Expansion coeffkients r)

250 26 0

0.2576 0.9127

20 - 4.608435 108 3”

-4.608435 135 4

-4.608435 432 15

0.9420 -0.3202

0.9420 -0.3202

0.9420 -0.3202

26 0 -0.007753 297 10

0.2567 0.9127

-0.007863 324 115

-0.007897 351 12

-0.007896 378 13

0.0078 10 432 15

0.2655 0.9107

0.2486 0.8623

0.2679 0.9057

0.2609 0.9121

‘) States are labeled with the quantum numbers m, n of (2.13) and with the symmetry ( f ). b, Energy above the dissociation threshold. =) Number of reduced basis functions (dimension of the eigenvalue problem (2.20) ). d, Harmonic basis functions with quantum numbers up to v. ‘) Converged results. f, CoefIicient with )cl 20.2 are shown.

Table 6 Convergence tests for the HCC fragment with a fixed harmonic basis State ‘) Energy (eV) b, Dimension c, td’ UC’

-4.608387 11 4 3

-4.608430 19 5 3

iu

Expansion coefficients s)

20 11

0.9419 -0.3207

0.9419 -0.3203

State ‘) Energy (eV) b, Dimension ‘) Cd) v r’

-0.007863 211 7 =’ 11

iu

Expansion coefficients g,

25 0 26 0

0.2655 0.9106

20 - 4.608434 24 6” 3

- 4.608435 29 7 3

-4.608335 108

0.9420 -0.3202

0.9420 -0.3202

0.9420 -0.3202

3

26 0

a) States are labeled with the quantum numbers m, n of b, Energy above the dissociation threshold. ‘) Number d, Tolerance value shown as negative decade logarithm, ‘) Converged results. 8) Coefficients with 1c I> 0.2 are

-0.007863 235 8 11

0.2655 0.9107

-0.007863 256 9 11

-0.007863 324

0.2655 0.9107

0.2655 0.9107

11

(2.13) and with the symmetry ( k ). of reduced basis functions (dimension of the eigenvalue problem (2.20) ). see (2.18a). ‘) Harmonic basis functions with quantum numbers up to v. shown.

74

B. Hake,

W. Karrlein /Connecting level statistics and reaction dynamics

Table 7 Results for the three-dimensional systems System

State 8)

HNNH

212-o 67-O

1 h

226-O 1012-O

1

HCCH

Type b,

h

Dimension ‘)

e (eV) *)

Statistical weight c,

s f’

YB’

.Figure

297 252

0.7027

1.0657

0.8943 0.9479

0.5393 0.2388

0.7613 0.1317

5a 5b

1680 1848

0.9757 1.8199

0.5466 0.9042

1.1884 0.4526

0.7933 0.1620

6a 6b

a) States are labeled with quantum numbers m, n of (2.13) and with the symmetry ( k ). b, 1, local, h, hyperspherical. ‘) Number of basis functions (dimension of the eigenvalue problem (2.12 ) ). d, Energy above the dissociation threshold. e, Statisticalweight,see (2.13). f, See (3.5). *‘See (3.7).

a,

lo>, ll>,-,

115))

(3.9d)

b,

II>>

118) >

(3.9e)

12>,...,

and harmonic c,

basis wavefunctions

IO>, ll>,...,

IlO>.

for bond (3.9f)

Table 7 summarizes the results for the three-dimensional resonant states. By comparison with the zero-point energy of the radical products HCC and HNN EO(HNN)

co.2857

eV

E’(HCC)

=0.3342

eV

all states in table 7 are resonant. Figures 5 and 6 show two-dimensional cuts through the three-dimensional vibrational wavefunctions at the equilibrium position of the multiple bond (CC or NN). The axes shown are HX bond distances. Fig. 5 (HNNH) shows perfect local and hyperspherical patterns. The corresponding patterns for HCCH in fig. 6 are somewhat diffuse. This optical impression is confirmed by the numerical values listed in table 7. The statistical weight of the local HCCH resonance is lower than the one for the hyperspherical state. This is not observed for the pair of HNNH resonances. The sizes of the basis sets to describe these HCCH resonances are at their lower boundaries, in spite of the immense number of basis functions of more than 2000. This is a consequence of the strong couplings between the adjacent oscillators. Another consequence is the average bond distance of the multiple bonds [ 75 ]

(Aqc)=(‘v,,-,IAq,l’y,,-,). The values are for HCCH

(3.10)

4

1

bl

I -q,

lq--

4

Fig. 5. Highly excited states mn k v of HNNH with energies E,,,,+; The wavefunctions are represented by equicontour plots of Fmy,,,(q., qb, Aq=)= +iXmax (i= 1,2, 3,4, 5) (ac’.5) with vibrational quantum numbers and symmetry mn k u. The plots are shown as two-dimensional cuts through the three-dimensional wavefunctions at Aqc= 0, the equilibrium position of the multiple bond. The axes q. and qb are the internal coordinates of the highly excited H-X bonds. (a) Local mode mn+ v=2 12 - 0, max=2.17;(b)hypersphericalmodemn+u=67-0,max=2.98.

B. Hartke, W Karrlein/Connecting levelstatisticsand reactiondynamics

and for HNNH both

KC H Q E=O. 9757ev mn-v = 2 26-

( Aqc) = O.OOOOao .

0

The couplings cause desymmetrization of the cold XX multiple bonds, and therefore a deviation from ( Aqc) is observed. The highly excited local CH stretch causes a more compressed CC bond than the corresponding hyperspherical stretch. This effect might be the result of the loss of electron density at the carbon atoms, that is higher for local excitations. Again, the weak couplings in the HNNH model system do not have any influence on the expectation value of the NN bond length.

IOiXll

51== E=1.8199eV mn-v = 10 12-

3.2. Selective dynamics of local and hyperspherical

J

1

-

qaQ--

resonant states (B)

Tables 8 and 9 show the grid parameters and results of the solution of the time-dependent Schrijdinger equation, using eqs. (2.23 )-( 2.26). We first turn to the two-dimensional systems, cf. table 8. Figs. 7a-d and 8a-d show the time evolution of local (7a,b and 8a,b) and hyperspherical (7c,d and 8c,d) resonant states. In both systems local resonances show a significant flow of probability density into the potential product exits, which means elongation and consequently dissociation of the H{XX} bonds. It is seen that one quantum of the remaining product H{ XX} bond disappears. It is shifted into the dissociating bond. By this energy flow, the high excited bond has now enough energy to dissociate. Res-

5

Fig. 6. See caption to fig. 5, but this is for the system HCCH. (a) The local mode mn + v= 2 26 - 0, max= 1.18; (b) hypersphericalmodemnfv=1012-O.max=2.67.

local

( Aqc ) = -0.0043a.

hypersph.

(Aqc ) =

75

,

O.O003a, ,

Table 8 Parameters and results of propagation for two-dimensional systems System

State ‘)

HWV

2 12-

Aqab’

N==)

Aa

81

0.1

N,,

Al d,

t (ps) e,

Norm”

e (eV) g,

r (ps) h,

Figure

0.4

0

0.666773 0.666820 1.032661 1.032630

0.3590

la lb 7c ld

0.949070 0.949070 1.759296 1.759296

30.87

6 7-

0.1

64

0.1

64

0.4

0 1.045

0.999300 0.061296 0.999487 0.987051

226-

0.1

128

0.1

128

0.25

10 12-

0.1

81

0.1

81

0.25

0 0.653 0 0.653

0.999088 0.999088 0.999378 0.999378

0.1

81

I .045

H{CC}H

67.304

1167.2

8a 8b 8c 8d

a) Quantum numbers m, n of the dominating basis function, (2.13). b, Grid constants in atomic units, see (2.24). =) Number of grid points. d, Time step of the propagation, see (2.25). ‘) Time of propagation in ps. ‘) Conservation of norm, see (3.10). ‘) Conservation of energy, ‘see (3.10). ‘) Decay times in ps; mathematically exact values are shown here.

B. Hartke, W Karrlein /Connecting

16

levelstatistics and reaction dynamics

Table 9 Parameters and results of the propagation for three-dimensional systems System HNNH

HCCH

State a’

Aqa,bb’

Na,s ‘)

Aqc

N,

Atd’

t(Ps) c,

norm f,

e(eV) g,

r(ps) h’

Figure

0 0.968 0 0.968

0.999370 0.996455 0.999545 0.999483

0.797527 0.191441 1.160476

353.18

9a 9b 9c 9d

0 0.097 0 0.097

0.997835 0.978354 0.998611 0.907074

1.140421 1.128811 2.022895 2.005400

2 12- 0

0.09

81

0.04

32

0.4

6 I-O

0.09

64

0.04

32

0.4

226-

0

0.08

162

0.04

32

0.4

10 12-

0

0.08

72

0.04

32

0.4

35158.1

1.160468 -

10a IOb

-

IOC

IOd

a) Quantum numbers m, n of the dominating basis function, (2.13). b, Grid constants in atomic units, see (2.24). ‘) Number of grid points. d, Time step of the propagation, see (2.25). ‘) Time of propagation in ps. ‘) Conservation of norm, see (3.10). g, Conservation of energy, see (3.10). h, Decay times in ps; mathematically exact values are shown here.

of this kind are called Feshbach resonances [ 7,161. Hyperspherical resonances, although higher in energy than the local ones, do not show any change of their shapes within the time of propagation and the graphical resolution. It would be necessary to transfer more than one vibrational quantum from the remaining product bond into the dissociative bond. This, however, is a process that would take a long time. The decay times are determined using (2.22 ), see table 8 and figs. 7e and 8e. Local resonances decay faster than the near-degenerate hyperspherical resonant states. The decay times differ by about two orders of magnitude. By conservation of norm and energy (see eqs. (3.11) ), the quality of the propagations can be estimated: onances

( y(t)

I Wf+ Al) >

(3.1 la)

for the norm, and (Y’(t)IfiIy(~+At)>

(3.11b)

for the energy. The energies of all the resonances are stable, see table 8. As the propagation of 1v, Iz_ ) is carried out up to 1 ps, there is a significant flow of probability density, as the decay time is only one third of the propagation time. Therefore, the norm conservation is no longer possible. The exponential decay shown in fig. 7e suggests that the approximation to exact resonances is excellent for H{NN}H. The situation is different for H{CC}H. Fig. 8e shows that there are

oscillations for the hyperspherical state. The decay time is greater than 1 ns. An exact evaluation is not possible, as the FIT-SOD method is only applicable to processes that take place on the femtosecond and on the picosecond time scales. Therefore, the value for the decay time is only a lower boundary [ 29,901. The decay behaviour of the local state of H{CC)H is very irregular. This is the result of the strong, effective couplings, in contrast to the corresponding local decay of H{NN}H. This behaviour is predicted in ref. [ 3 1 ] by a statistical analysis of the vibrational energy levels. The propagation of the three-dimensional resonances is summarized in table 9. Note that HNNH resonances are propagated for ten times longer than resonances of HCCH. The reason will be clear from figs. lob and 10d. Even within this small time period of almost 100 fs, there is significant loss of initial structure. Table 9 documents that there is poor energy and norm conservation for both wavefunctions. In fig. 1Oe the evaluation of (2.22) is shown. Large oscillations are displayed and therefore the calculation of decay times makes no sense. The main reason behind this chaotic behaviour is surely an incomplete basis set. The condition for appliance of (2.22) small couplings between zero-order states - is not fulfilled for HCCH. This non-fulfillment of eq. (2.22) has been predicted by statistical analysis of the energy levels for HCCH [ 3 11. In ref. [ 3 1 ] it has been found that the HCCH energy levels follow the Wigner distribution, which means that they are highly correlated. With this result, the prediction in ref. [ 3 1 ]

B. Hartke, @I k’lrrlein /Connecting level statistics and reaction dynamics

,

9

t

mn- = 2 12t =o.oops

qb’%

I

,.

8

mn-

EleV

rips

-- I 8 53 % $ f-2. T 0.0

-tips-

1.0

Fig. 7. Decay of vibrational resonances F~,,+ (q*, qb, t), specified by vibrational quantum numbers and symme~ mnrt , energy E, and decay time r. The dynamics is shown by snag&tots at t=O and at the end of the propagation period. Contours represent equidensities Iv,,,,+_12=0.005, 0.05,0.1,0.2,0.4,0.6,0.8, 1.0 (a;*). For the potential ~ui~atiur lines see fis. 1. (a) Local resonance of H{NN)H mnf =2 12- at t=O ps; (b) local resonance of H{NN}H mnk =2 12- at f= 1.045 ps; (c) hyperspherical resonance of H(NN}H at t= I.045 ps. (e) Mode selective decay of vibrationally mnIt=67-atf=O~,(d)hyperrphericiiireson~~ofH~NN)Hmn+=67excited resonances of the two-dimensional system H{NN}H demonstrated by semilogarithmic plot of the squares of the overlaps, cf. eq. (2.21), versus time t (dashed lines) fitted to exponential decay curves, cf. eq. (2.22) (continuous line). Quantum numbers mn, energies E, and decay times rare also specified. The dashed lines coincides with the line fitted to eq. (2.22).

confirmed: HCCH shows internal reorganization processes caused by the high correlation of its vibrationai levels, which is the result of the linear structure of the molecule. In contrast to HCCH, HNNH is predicted to show is

regular dynamics [ 3 11. Table 9 documents good conservation of norm and energy. Figs. 9a, b, c, d show, that the characteristic structures are preserved during propagation. By evaluation of (2.22) it is found that the local resonance decays faster (by a

8. Hartke, W. Karrlein/Connecting levelstatisticsand reactiondynamics

78

HtCC}H 9

1

i mi7-

Elf?V

-t/ps-----

-

qofy---

9

rips

.6

Fig. 8. See caption to fig. 7, but this is for the system H{CC)H. (a) Local resonance of H{CC)H mn+ =226- at t=O ps; (b) local resonance of H{CC)H mn+ =226- at k0.653 ps; (c) hyperspherical resonance of H{CC)H mnrt = IO12- at t-0 ps; (d) hyperspherical resonance ofH(CC}H mn+ = 10 12- at ho.653 ps. (e) See caption to fig. 7e, but here for the system H{CC)H.

factor of 100) than the hyperspherical one. It is, however, surprising that we obtain decay times that indicate long-lived resonant states; cf. table 9 and fig. 9e. As already mentioned above, the decay times of such high values can only be understood as lower boundaries. In figs. 9b and d no flow of probability density is found, because of the stability of the resonances. The oscillations seen in fig. 9e are of lower size, than those in fig_ 1Oe for HCCH; note the range

at the axis. Ad~tion~ly, these oscillations are regular.

4. Discussion For our model systems, we only consider highly excited HX bonds. All other degrees of freedom are either cold (all multiple bonds) or fixed to their equilibrium positions (all angles). Because of the

B. Hartke, W. darrlein /Connecting level statistics and reaction dynamics HNNH

79

HNNH a

t

C

E=0.7975eV mn-v = 2 12- 0 t=o.oops

Ej-

E= 1.1605eV mn-v=6 7- 0 t = 0.00 ps

t

qb'aO

qda(

b

d

t=o.97ps T= 353.3ps

t = 0.97ps T = 35158ps

1 8 OT t N-

mn-v

FleV

u1~.6’7:01116’i”“.~~“

0

-t/ps-

q,lao-

6

Ips 35158

.9

Fig. 9. Decay of vibrational resonances Y,,,,*;.( q., q,,, Aq,, t) shown as a two-dimensional cut at Aq,=O (equilibrium bond distance), specified by the vibrational quantum numbers and symmetry mn + u, the energy E, and the decay time t The dynamical process is shown as snapshots at t=O and at the end of the propagation period 1.For the contour values see figs. 7a, b. (a) Local resonance of HNNH mnfv=2 12- 0 at t=O ps; (b) local resonance of HNNH mn+v=2 12- 0 at k0.968 ps. (c) Hyperspherical resonance of HNNH mnfv=6 7- 0 at t=O ps; (d) hyperspherical resonance of HNNH mnfv=6 7- 0 at t=O.968 ps. (e) See caption to fig. 7e, but here for the three-dimensional system HNNH.

coupling terms included in our model Hamiltonian, we observe two characteristic structures depending on the distribution of the vibrational energy: local modes (the energy is localized in one bond) and hyperspherical modes (the energy is distributed among several bonds). We want to interpret both issues mentioned at the beginning of section 3, from a statistical point of view.

4.1. Characteristic patterns of vibrational wavefunctions (A) In the two-dimensional systems H{NN}H and H{CC}H local and hyperspherical modes have been found. Their classification according to graphical and qualitative numerical criteria is established. The highly excited states are regular. Wavefunctions of

E=l.lMN@V mn-v -_ 2 26- 0

i = o,oop5

gerade and ungerade s~mmei~es~ e.g. I wmn+> and 1tpmn_) are usually separated by an energy gap of 1O-* eV I75 ] +For H jCC)H, the h~p~~pbe~~~ pair with mn= 10 12 shows a gap of AE=O.Q467 eV. In sectian 3, the interaction of the circu;lar shape of the h~~~he~~ wayefu~ctio~ and the to~~~~ of the ~o~ent~a~surface is discussed as the main source of this large separation. It would be interesting to know

whether this splitting of ~~~~be~~ ~bmt~~~s ca51n be detected in a spectrum. For hypersphericai wavefunctions of low excitation, however, the gap should become smailer. The circular structure of such a wa~efun~t~on is compact and approaches the norm& mode wa~efu~c~o~. Simu~tan~usl~ the ~tentia~ energy surface becomes harmonic in this energy range. Therefore, the effect

B. Hartke, W. Karrlein /Connecting level statistics and reaction dynamics

of symmetry splitting should only be possible for excitations in the region of the dissociation threshold. The introduction of a third degree of freedom has significant effects on the structures of the local and hyperspherical wavefunctions. The regularity of HCCH wavefunctions is lowered by the strong mixing of zero-order states [ 3 1] ; in contrast, HNNH wavefunctions show very regular structures. Both are confirmed by numerical measures. The question of whether regular patterns of highly excited wavefunctions are preserved by the addition of one further degree of freedom cannot be answered with a simple yes or no. On the one hand, the pair H{NN}H and HNNH is another example (cf. [ 75 ] CH2 and CHzO ) where an additional vibration does not wash out the regularity of local and hyperspherical wavefimctions. On the other hand, H{CC}H and HCCH are examples that do noi keep the regularity. At this point it will be helpful to have a look at the statistical results. Using the same Hamiltonians, the study [ 3 1 ] has shown that there are significant differences, especially for HCCH and HNNH. The energy levels of HNNH show only low correlations. In other words, the coupling strength between both excited HN bonds is low. This is different for HCCH, where highly correlated energy levels have been found. This dramatic effect is surprising, considering the small mass differences between N and C, together with the bond angle HNN. It has to be emphasized at this point that there is no direct coupling of both HX bonds at each end of the molecule incorporated in our model. With these facts in mind, the question about the effect of the additional degrees of freedom and their consequences for the regular structured vibrational wavefunctions, can now be answered. Both molecules, HNNH and HCCH, are examples of the opposite effects of a multiple bond. Although the Hamiltonians in our model are the same, the small differences in masses and structure lead to different energy level statistics and different patterns of wavefunctions. In turning the arguments, it should be possible to predict the regularity of vibrational wavefunctions from the results of statistical analysis. A high correlation of energy levels correlates with less regular patterns of the corresponding wavefunctions. The following subsection will show that there is also a correspondence between statistics and molecular dynamics.

81

4.2. Selective decay of local and hypersphericar resonances (B) The resonant decays of local and hyperspherical wavefunctions of H{NN}H show an exponential dependence. They are different from the decay behaviour of states of the H{CC}H system, that do not obey eq. (2.22). This is the result of the strong couplings of zero-order states in H{ CC}H, cf. ref. [ 3 11, that are higher than in H{NN}H. For H{NN}H resonances, selectivity of local versus hyperspherical decay is found. Near degenerate, but energetically lower, local resonances decay faster (by a factor of 100 ) than hyperspherical ones. This is in contrast to the RRKM prediction. As is discussed in section 1, this conclusion does not imply a non-RRKh4 behaviour when a sample of states is considered, cf. refs. [ 3-5 1. In the three-dimensional system HCCH, whose energy levels are highly correlated [ 3 11, the approximation necessary for (2.22) does not hold anymore. The Wigner distributed vibrational energy levels are connected by strong zero-order couplings. Initial regular structured contours loose their shape and begin to spread over the phase space. In ref. [ 3 11, HNNH is shown to be a system, whose energy levels are correlated only to a small extent. The regularity of the initial wavefunctions is a first proof. It can be predicted that the local and hyperspherical resonances should decay regularly. In section 3 it is shown that the regularity of the initial wavefunctions is not destroyed, when being propagated. Again there is the same line of arguments as stated above, cf. section 4.1. Starting from the same model Hamiltonian for HNNH.and HCCH, one arrives either at rapid IVR (cf. HCCH ) or highly selective decay dynamics (cf. HNNH). Therefore we can connect statistical and dynamical results to state: a high correlation of energy levels corresponds with fast IVR, whereas a low correlation of energy levels corresponds to bond-selective decay dynamics. Both statements are only caused by some minor differences of HCCH and HNNH, such as mass and shape. Surprisingly, the decay times for both local and hyperspherical resonances of HNNH are very long; up to nanoseconds. It is not possible to evaluate an exact decay time in the nanosecond regime using the SODFFT method. This would require long time propagations. SOD-FFT can thus provide lower boundaries of the decay times.

B. Hartke, W Karrlein /Connecting level statistics and reaction dynamics

82

The interpretation of these stable resonances in HNNH uses the mechanism of Feshbach resonances. In order to dissociate a highly excited bond, a transfer of at least one vibrational quantum from another less excited bond is necessary. This quantum is transformed into the translational energy of the products. Therefore, it is necessary for the donor of a vibrational quantum and the acceptor (both are vibrational oscillators) to be coupled directly. For example, this is fulfilled in the two-dimensional systems and also in the three-dimensional system CHzO [ 301. In our model of HNNH both the donor and acceptor (bonds a and b) are decoupled by the NN fragment. This would require a two-bond transfer process (from bond a to c and then to bond b) that takes a very long time. This mechanism prolongs both the decay times of the local and hyperspherical states. An excited NN fragment should lead to an increase of the decay times. It would also be of great interest to see the effects of couplings that range over more than one bond. Within such a model Hamiltonian, both promoting modes in HNNH would be coupled. Since these long-range couplings are small, this should not lead to a great increase in the dissociation. In contrast to CH20, where the additional third bond CO has an accelerating effect on the dynamics of CH bond dissociation, the additional bond NN acts in an opposite way: it slows down the dissociation dramatically. It could be demonstrated that known correlations between zero-order states lead to very different but predictable effects. Whereas the addition of the CC triple bond causes a fast internal redistribution of vibrational energy on the femtosecond time scale, an addition of NN to form the topologically equivalent molecule HNNH leads to hindered but selective dynamics. It can be concluded that a necessary condition to observe selective dynamics of highly excited vibrational resonances is a small coupling of zero-order states ( HCCH is an opposite example ) . Furthermore, the bond to be dissociated (the acceptor bond in the Feshbach mechanism) should be adjacent to a (highly) excited donor bond. Acknowledgements

B.H. and W.K. are thankful to Prof. Dr. J. Manz

(Universitiit Wiitzburg) for catalyzing this work. Financial support from the Deutsche Forschungsgemeinschaft and the Studienstiftung des deutschen Volkes are gratefully acknowledged. All calculations were carried out at the Leibniz Rechenzentrum, Munich on the CRAY YMP 4-432 and the CYBER 180-995/E. W.K. thanks the Siemens-NixdofiInformationssysteme AG for generous technical support.

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