Richard Zare

L Duggan

M Raunhardt

M Schäfer

U Hollenstein

T P Softley

F Merkt

PFI-ZEKE photoelectron and high resolution photoionization spectra of ND 3 with MQDT simulations

Keywords: ZEKE spectroscopy, Rydberg spectroscopy, Multichannel quantum defect theory

The rotationally resolved PFI-ZEKE (pulsed-field ionization zero-kinetic energy) photoelectron spectrum of ND 3 has been recorded at the v 2 + = 0, 1 and 2 thresholds, by single-photon VUV excitation from the ground-state. An assignment of the 2 2 ~A X     1 1 ~A X  spectrum is presented, allowing a determination of the adiabatic ionization potential of ND 3 (82261.7(15) cm -1 ) and of the rotational constants and positions of the v 2 + = 0, 1 and 2 vibrational levels of the ND 3 + ion. These parameters are used in a multichannel quantum defect theory (MQDT) simulation of the highresolution photoionization spectrum in the range 82100 -83300 cm -1 [(R. Seiler et al., J. Chem. Phys. 118 10024 (2003)]. A good simulation and an assignment of around 80% of the lines of the ND 3 photoionization spectrum are achieved, verifying the validity of the molecular constants derived from the PFI-ZEKE photoelectron spectrum and also the quantum defect parameters derived from ab initio calculations.

(1) Introduction

The ammonia molecule has been, for many years, a model polyatomic system for studies of photoionization dynamics and photoelectron spectroscopy. The photoionization from the ground state is characterized by a large geometry change from pyramidal to planar leading to the characteristic long progression in the out-ofplane (umbrella) bending mode in the photoelectron spectrum [1]. The photoelectron spectrum of NH 3 has been studied with rotational resolution using pulsed-field ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy, both with VUV excitation from the ground state [2 -4], and using (2 + 1) resonance-enhanced excitation via the B ~state [5]. A mass-analyzed threshold ionization (MATI) spectrum was obtained via the B ~and C ~states [6,7]. There have also been numerous resonance-enhanced multiphoton ionization (REMPI) studies of the Rydberg states [8 -14] including some double-resonant studies of higher Rydberg states, and very recently a rotationally resolved study of photoelectron angular distributions has been reported [15].

Very high resolution photoionization spectra of four isotopomers of ammonia NH 3 , NH 2 D, NHD 2 and ND 3 , were reported by Seiler et al [2]. In comparison with previous studies of the photoionization and photoelectron spectra of NH 3 , these spectra, recorded using a VUV laser source of bandwidth 0.008 cm -1 , contain very detailed information on the interactions between the various Rydberg series, and between Rydberg series and continua (including dissociation continua). To assist in the interpretation of these spectra, an ab-initio R-matrix calculation of quantum defect parameters for ammonia has been carried out, including the calculation of the geometry dependence of the quantum defects in the  2 -umbrella-bending coordinate, and these parameters have been used to perform multichannel quantum defect theory (MQDT) simulations of the high-resolution photoionization spectrum of NH 3 [16]. To 3 further test the validity of these quantum defect parameters and the MQDT formalism used for calculating the spectra [6,7,16], it is desirable to extend the MQDT calculations to the other isotopomers, ND 3 , NH 2 D, and ND 2 H, for which highresolution photoionization and photoelectron spectra were also reported in [2,17].

Isotopic substitution does not change the electronic structure and therefore the same quantum defect matrix should be suitable for the simulation of all isotopomers.

However, isotopic substitution does change the ionization potential, the nuclear spin statistics, and the rotational constants and vibrational frequencies of the neutral species and the molecular ion. These differences will not only change the rovibrational limits of each Rydberg series, but also the interactions between the Rydberg series, having strong effects on the line positions, spectral intensities and autoionization linewidths.

In recent work we reported an assignment of the PFI-ZEKE photoelectron spectra of the v 2 + = 3 bands of NH 2 D and NDH 2 and were able to use the information on rovibrational thresholds derived from these spectra to identify several Rydberg series in the photoionization spectra [17]. The ND 3 photoionization spectrum reported in [2] has not been assigned to date, the main difficulty being the uncertainty in the ionization potential of ND 3 ; the best value reported previously was 82280  40 cm -1 [8]. The rotational constants for the ND 3 + ion are also known to much lower precision than those of NH 3 + [18], and therefore the identification of the rovibrational Rydberg series limits in the photoionization spectrum of ND 3 is challenging.

In the present paper we report new experimental PFI-ZEKE photoelectron spectra of ND 3 which allow an accurate determination of the ionization potential, as well as the spectroscopic parameters of the ion. The accurate energies of the various ionization thresholds for specific vibration-rotation levels of the ion could then be (2) PFI-ZEKE photoelectron spectra of ND 3

(2.1) Experimental

The PFI-ZEKE photoelectron spectra of ND 3 + in the region of the v 2 + = 0, v 2 + = 1 and v 2 + = 2 ionization thresholds were recorded using an experimental apparatus similar to that described in Ref. [19]. In brief, the spectra were recorded following single-photon VUV excitation from the lowest rotation-inversion levels of the ground neutral state in a skimmed pulsed molecular beam of ND 3 gas. Tunable VUV radiation in the wavenumber range 82000 -85000 cm -1 was produced by resonance-enhanced difference-frequency mixing  VUV = 2 1 - 2 in a krypton gas jet using the (4p) 5 1 lists the magnitudes of the pulses used for the three vibrational bands recorded; the differences of the fields employed reflect the very different intensities of the three observed bands. The first pulse served the purpose of eliminating prompt electrons and field ionizing the highest Rydberg states. It was immediately followed by a second pulse with opposite polarity which produced the electrons that were detected in the PFI-ZEKE photoelectron spectra. A resolution of 2 -3 cm -1 was achieved in the spectra which was sufficient to resolve the rotational structure of the photoelectron bands of ND 3 . The field-induced shifts of the ionization potential arising from these pulses were calculated using the procedure described in

Ref. [20] and are listed in Table 1. The VUV wavenumber was determined by calibrating the wavenumbers of the two dye lasers. This calibration was achieved by recording the laser-induced fluorescence spectrum of I 2 and optogalvanic spectra of Ar and Ne. The estimated accuracy of the ionization thresholds determined in this work is 1 cm -1 .

(2.2) Results

The experimental PFI-ZEKE photoelectron spectra obtained following single-photon excitation from the neutral vibronic ground state are shown in panel (a) of Figs. 1, 2 and 3. The Franck-Condon factor is highly unfavourable for the v 2 + = 0 band relative to bands of higher v 2 + , hence a spectrum of lower signal-to-noise ratio is obtained at that threshold. As is usual in PFI-ZEKE photoelectron spectroscopy, the wavenumbers of the observed peaks (when corrected for the field ionization shiftsee above) correspond to the differences between the energies of the rovibrational ionization thresholds in question and the initial levels of the excitation, i.e. where I 0 is the adiabatic ionization energy, G(v 2 + ) is the vibrational energy in the  2 inversion (umbrella bending) mode of the ion (the only mode excited in these spectra)

) , ( ) , ( ) ( 2 0 K J F K N F G I h            v  , (1)
and the F terms are given by the rigid-rotor expression

2 ) ( ) 1 ( ) , ( K B C N BN K N F     . ( 2
)
for the rotational energies of the ion [(N, K) = (N + , K + )] and neutral ground state

[(N, K) = (N, K)] . The values of F(N, K) for the 1 1 ~A X  ground state of ND 3 used
in the simulations are taken from reference [21].

The general rovibronic selection rules for the photoionization (or for transitions to Rydberg states) of polyatomic molecules derived in Ref. [22] can be expressed as

* e rve rve           , (3)
where   rve and rve    represent the rovibronic symmetries of the cationic and neutral states in the molecular symmetry group D 3h (M), e  the symmetry of the photoelectron wave function, (i.e. A 1 ' and A 1 '' for even- and odd- partial waves, respectively) and *  the dipole moment representation (i.e. A 1 '' in D 3h (M)). For ND 3 these rovibronic selection rules can be written as

E E A A A A             2 2 1 1 (4)
for even- partial waves and

E E E E A A A A A A A A                         2 2 2 2 1 1 1 1 (5)
for odd- partial waves.

A schematic diagram of the rotational energy levels of the out-of-plane bending mode of ND 3 is of a 2 '' symmetry, out-of-plane bending levels with an even (odd) number of vibrational quanta transform as A 1 ' (A 2 '') and have the same rovibronic symmetry labels as the 2 0 (2 1 ) levels. Distinct rovibronic selection rules and an overall different rotational structure are thus expected for transitions from the ground neutral state of ND 3 to out-of-plane bending levels of ND 3 + with even and odd numbers of vibrational quanta.

The general rovibronic photoionization selection rules in Eqs. (3) to ( 5) are rather unrestrictive and the full set of transitions predicted using them exceeds the number of transitions actually observed. To assist with the assignment of the spectra, simulations were carried out based on the orbital ionization model described in Ref. [23], which itself is an extension of the Buckingham, Orr and Sichel (BOS) diatomic photoionization model [24], to predict the strongest lines. The intensity distributions of the spectra displayed below the experimental spectra in panels (b) of Figs. 1, 2 and 3 were calculated using Equation ( 6) [23],

. ) 1 ( ) ( 1 2 1 ) , , , , ( 2 1 , 2 1 , 2                                                          E E F F C Q C K N K N v I    (6)
In Equation ( 6), '' and C represent the Boltzmann factor (including nuclear-spin statistical weight) and a proportionality constant, respectively, and '' represents the orbital angular momentum quantum number of the different terms of a single-centre expansion (with expansion coefficients

      C ) of the molecular orbital '' out of which the photoelectron is ejected.   , E F   is a radial integral and                         K K N N N Q    ) 1 2 ( ) ( , (7)
where the term in parentheses represents a 3j-symbol (see Ref. [23] for more details). ) the sum in Eq. ( 6) reduces to a single term

2 0 1 ) 1 2 ( exp ) , , , , (                                           K K N N g N T k E C K N K N v I B  (8)
where C   is a constant, '' stands for the spin statistical factor and g'' is the degeneracy factor of the ground state rotational levels, i.e. 2N''+1 for K = 0 and 2(2N''+1) for K 0 levels. The simulations presented in the panels (b) of Figs. 1, 2 and 3 were made using Eq. ( 8) assuming a rotational temperature of 15 K.

The pattern of peaks calculated using the orbital ionization model for v 2 + = 0 (Fig. 1b) is roughly the same as for v 2 + = 2 (Fig. 3b), because the vibronic symmetry is the same in both cases. The pattern is different for v 2 + = 1 (Fig. 2b) however, because the a 1. The best-fit values of the adiabatic ionization energy, the vibrational intervals and the rotational constants of ND 3 + are given in Table 5 and are compared to the best values from the literature, or to values extrapolated from the literature (e.g., from the work of Kraemer and Špirko [25] in which ab-initio calculations are combined with a fit to experimental data to calculate a potential energy surface for NH 3 and ND 3 from which spectroscopic constants can be derived). The experimental rotational constants and vibrational intervals fall within 0.1 cm -1 and 3 cm -1 , respectively, of the literature values. The rotational constants for the v 2 + = 0, 1 levels can also be compared to values obtained from high resolution REMPI spectra of the ' C 3p (n = 3) Rydberg state [26], also shown in Table 5. The good agreement between these values reflects the expected similar geometry between the Rydberg state and the ion. For the purpose of assigning the photoionization spectrum of ND 3 , the most significant improvement in parameters derived from these measurements is in the experimental value of the adiabatic ionization potential (82261.7 (7) cm -1 ) , which is now known with almost two orders of magnitude higher precision than the previous best value.

The K = 0 lines in Tables 2 to 4 originating from levels with K = 0 are of symmetry type A  A and are uniquely associated with even-ℓ photoelectron partial 4). The other lines are of type E  E and could in principle be described as associated with either even-ℓ or odd-ℓ partial waves because transitions involving near-degenerate E and E levels are not resolved. [Note that although we refer to 'photoelectron partial waves' here, the actual excitation in a PFI-ZEKE photoelectron experiment is to a high Rydberg state with a given ℓ value rather than a true continuum]. As discussed previously [2,17], even-ℓ partial waves are likely to dominate the spectra and, in fact, all of the observed transitions can be explained by using even-ℓ partial waves alone. Four K = 0 transitions are unambiguously identified that can only be explained assuming even-ℓ photoelectron partial waves, and these include strong transitions in all three vibrational bands, most notably the 1 0

A 1 ''  0 0 A 1 ' transition at 82270.4 cm -1 in the v 2 + = 0 band, the 0 0 A 1 '  1 0 A 1 '' transition at 82936.4 cm -1 in the v 2 + = 1 band and the 1 0 A 1 ''  0 0 A 1 ' transition at 83665.7 cm -1 in the v 2 + = 2 band
. This predominance of even-ℓ partial waves in the single-photon ionization of ND 3 substantiates our previous findings derived from the spectra of NH 3 [2], ND 2 H and NH 2 D [17]. In general the strongest features are

identified as 0 ' '      K K K .
The two (v 2 + = 1) or three (v 2 + = 0 , 2) strongest features predicted by the orbital ionization model for each vibrational band are also observed as strong features in the experimental spectra (Figs. 1, 2 and 3) but with some differences in the relative intensities. As discussed above, several additional transitions are observed for which no intensity is predicted by the orbital ionization model, particularly in the v 2 + = 1 and 2 bands. The spectra calculated using the relaxed selection rules shown in panel (c) of Figs. 1 to 3 contain nearly all of the observed features, but not always with the correct relative intensities. The inclusion of higher-ℓ components in the simulations (Eq. ( 6))

is therefore unlikely to represent the direct ionization cross-sections. One factor contributing to some of the intensity discrepancies may be the uncertainty about the relative populations of the neutral molecule rotational levels in the supersonic jet. For NH 2 D and ND 2 H it was previously found [17] that there was significant nuclear-spin-state inter-conversion in the supersonic expansion, attributable to an initial clustering near the jet orifice. The simulations shown in Figs.

1 -3 assume a rotational temperature of 15 K; changing the temperature or the degree of nuclear-spin-state inter-conversion does not improve the fit especially for the weak features observed on the high wavenumber side of the v 2 + = 1 and 2 bands.

The most likely explanation for the differences to the predictions of the orbital ionization model lies in the channel interactions between different Rydberg series. As has been well documented elsewhere [6,7,27,28] the pseudo continuum of high-n

Rydberg states detected in PFI-ZEKE photoelectron experiments below a particular ionization threshold may interact with either bound Rydberg series converging to higher energy thresholds, or the continua associated with lower energy thresholds, and thereby borrow intensity from the stronger transitions to those interacting channels.

Such channel interactions are not incorporated into the BOS orbital ionization model, but have been explained in our previous study of the (2 + 1′) MATI spectra of NH 3 via the B ~and C ~states using multichannel quantum defect theory [6,7]. which is also believed to involve transitions to a group of v 2 + = 3 autoionizing states, and the features at the high wavenumber end of the v 2 + = 2 band of the photoelectron spectra may also be "borrowing" intensity from the overlapping autoionizing features.

Such intensity borrowing, however, can only occur if the autoionizing features share a common ground state with the perturbed line of the PFI-ZEKE spectrum.

(3) MQDT simulation of the photoionization spectrum of ND 3

The spectroscopic constants derived from the rotationally resolved PFI-ZEKE photoelectron spectra of ND 3 were used to carry out a simulation of the photoionization spectrum using Multichannel Quantum Defect Theory (MQDT). The methodology employed is described in more detail elsewhere [16] and follows closely the theory described in references [6,7,29,30]. In this scattering-theory approach the long-range 'channels' define the set of possible quantum states of the system when the electron is at long range from the ion core.

          N N i i K J E D J E 2 ) ( 2 ) ( ) 1 2 ( 3 4 ) (    , (9)
where ) (

) ( E D N i
is the reduced transition dipole moment between the initial state and the final channel i ,  is the fine structure constant, and  is the photon energy (atomic units). The energy-dependent reduced transition dipole moments are related to the transition dipole moments in the rovibronic eigenchannel representation,

) ( N D  , using the expression,            A D T D N i N i i ) ( i i ) ( e e . ( 10
)
In this equation,  labels one of the N O independent solutions of the Schrödinger equation at this energy E, and α labels the eigenchannels defined in the expansion

. ) ( ) (         E E A ( 11
)
The eigenchannel wavefunction   is given by

  i g f U i i i i     ) tan(     , ( 12
)
where f i and g i are regular and irregular radial Coulomb functions, and i is the ion core and electronic angular momentum parts of the wavefunctions which characterize each channel i.  α is the eigenchannel quantum defect and U iα is the transformation matrix between the long-range basis and the eigenchannel basis.

               i K N K N lN N T i N Q K J d NK Q Q K K N U D         ) ' ' ' ' ( ) ( ) ( ) ( ) ( ' ' ' ' ' ' ' ' d ) ( v v . ( 13
)    K N Q v and K N Q       v
are vibrational wavefunctions and the factors in square brackets in Eq. ( 13), including the vibrational overlap integral, are treated as variable input parameters in the present calculations.

To determine the transformation matrix U iα in Eqs. ( 12) and ( 13) the rovibronic quantum defect matrix in the long-range basis i i 

 is constructed by relating it to a set of input quantum defects defined in the close-coupled Hund's case (b) basis,

  , ' ' ) ( d ' ' ) ' ( ' ' , ) ( ) ( ' ;                             K K N N N K N K N i i K N K Q Q Q Q K K N       v v v v (14 )
where the

       ,
matrix elements are treated as input parameters, determined in this work principally by ab-initio calculation [16]. The quantum defect matrix i i   is then diagonalized, as in ref [29], to determine the U iα elements as the eigenvectors and the eigenchannel quantum defects  α as the eigenvalues.

The T matrix in Eq. ( 10) is given by

F o r P e e r R e v i e w O n l y          A U E T N N i i )] ( [ cos ) ( ) ( ) (     (15)
and the coefficients A  and scattering phases   are determined from the generalized eigenvalue equation [29]:

ΛA ΓA ) tan(  closed for )] ( sin[ ) ( ) ( i U N i N i i          open for ) sin( ) ( ) ( i U N N i i       closed for 0 i i    open for ) cos( ) ( ) ( i U N N i i       . ( 16
)
 i is the effective principal quantum number in channel i at the total energy E .

The expression in the square brackets in Eq. ( 14) determines the coupling between channels associated with different vibrational thresholds of the ND 3 + ion, and is thus responsible for vibrational autoionization. The coupling between channels with different v + quantum numbers is assumed, for the purposes of these calculations, to be negligible for 1    v . For the 1    v matrix elements, rather than integrating over the vibrational wavefunction, we use the following harmonic approximation to the matrix elements [31] (which effectively ignores the very small dependence of the vibrational wavefunctions on the rotational quantum numbers N + and K + ); where v + is the vibrational quantum number of the closed channel undergoing autoionization, and  is the vibrational frequency. There is also a symmetry selection rule applying to these off-diagonal matrix elements which restricts the number of  - pairs for which the derivative in Eq. ( 17) needs to be evaluated [16]. Note that this rule is applied in D 3h symmetry in which, for example, p and d correlate to the same representation. Hence the non-zero elements include both pairs with opposite parity (e.g., p -d) as well as one pair with the same parity (d -d). Equation ( 17) ignores all vibrational modes other than the out-of-plane bend and is a purely harmonic approximation. Evaluation of ( 17) requires knowledge of how the short-range electronic quantum-defect matrix elements

2 1 2 1 ) ( 8 d d ' ) ( d 2 ' ' ) ( ' ' ) ( ' ' '                         v v v            c h Q Q Q Q Q e K N K N v v       (17)
      
vary with the  2 + normal vibrational coordinate at the equilibrium geometry. The variation is determined in this work by ab-initio calculation as described elsewhere [16]. For the matrix elements diagonal in the v + quantum number, we set the integral over the vibrational coordinate equal to the value at the planar equilibrium geometry, assuming only a small variation of the quantum defects over the range of geometries over which the vibrational wavefunctions of the low vibrational levels extend.

The diagonal quantum defects employed in the MQDT calculation      have been slightly adjusted from the ab-initio values to give a better fit to the photoionization spectrum of NH 3 [16] and are listed together with the off-diagonal elements and the off-diagonal vibrational integrals (from Eq (17)) in Table 6. The best match to the experimental intensities is obtained by setting all the transition dipole moment parameters in Eq. ( 13) to zero, except for those involving  = 0. Note however that because there is substantial  = 0 - = 2 mixing in this system the Rydberg states populated have a mixture of ns and nd character. The rotational temperature in the supersonic expansion for the experiments of Ref [2] was estimated to be 5 -10 K, and therefore the spectrum is calculated as a sum of partial photoionization cross sections out of the 6 lowest energy levels of ND 3 which are the only ones significantly populated. The channels included in the calculation were those with v + = 0 to 3 and  = 0 to 2. The partial photoionization cross section calculation (Eq. ( 9)) is carried out for a specific total angular momentum N, and this, together with symmetry considerations [7], restricts the number of rotational quantum numbers N + K + that need to be included. Fig. 5 compares the MQDT simulation obtained with an overview of the experimental photoionization spectrum from reference [2]. The simulation has been colour coded according to the initial rotational level from which the transitions originate. The strong features observed at either end of the experimental spectrum are well reproduced in the simulated spectrum and much of the structure observed in the central region of the spectrum is also reproduced. Figs. 6 and7 give expanded views of two regions of the spectrum. The degree of qualitative match between simulation and experiment is extremely sensitive to the diagonal elements of the quantum defect matrix used as input and to the positions of the ro-vibrational ionic thresholds. Most of the features in the calculated spectrum are identified as autoionizing levels belonging to series converging on the v 2 + = 1 state of the ion, but a number of transitions belonging to higher vibrational levels (v 2 + = 2 and 3) have also been identified as shown in Fig. 5, including the n = 10, v 2 + = 2, 3 0 A 1 ''  0 0 A 1 ' transition, three transitions (n = 9, 11, 13) of the v 2 + = 2, 1 0 A 1 ''  0 0 A 1 ' series and the intense n = 9, v 2 + = 3, 1 1 E''  1 1 E' transition at the high-wavenumber end of the spectrum.

These vibrational assignments were made by allowing the transition moments to one Overall the assignment and detailed identification of Rydberg series in the v 2 + = 1 region of the ND 3 high-resolution photoionization spectrum is less straightforward than for the NH 3 spectrum [2]. The region near the v 2 + = 0 threshold, where series were readily identified in the photoionization spectrum of NH 3 , is quite weak for ND 3 , and at higher wavenumber there are no obvious series appearing in between the v 2 + = 0 and v 2 + = 1 thresholds (whereas for NH 3 the 1 0 A 2 '  0 0 A 2 '' series dominates this region of the spectrum). Nevertheless, it is possible to make use of the likely correlation between the transitions that are strong in the PFI-ZEKE photoelectron spectra of ND 3 and those series that have high intensity in the photoionization spectra.

As illustration, Figure 8 displays one long progression that can be assigned as a series converging to the v 2 + = 1, 1 1 E'  2 1 E rovibronic threshold which corresponds to a strong transition in the v 2 + = 1 PFI-ZEKE photoelectron spectrum at 82967 cm -1 . The 1 1 E''  1 1 E' transition also appears with high intensity in the v 2 + = 1 band of the PFI-ZEKE photoelectron spectrum at 82946 cm -1 and a series converging to this threshold can be identified in the photoionization spectrum of ND 3 from n = 17 -23, and is well reproduced in the MQDT simulation shown in Fig. 7.

The series and fractions of series identified, assigned and simulated in the high-resolution photoionization spectrum of ND 3 are summarised in Table 7. The central region of the spectrum between 82450 cm -1 and 82850 cm -1 is chosen for illustration and shown with series assignments in Fig. 8. A large percentage of the strong spectral features can be assigned by the MQDT simulation, which is a powerful tool in the interpretation of such complex photoionization spectra. The work described in this paper, when coupled with our previous work in references [2,6,7,16,17], provides information on the several ro-vibrational ionization thresholds of four isotopomers of ammonia. The band origins for the observed transitions are collected in Table 8. Rotational constants and vibrational intervals for the ND 3 + ion have also been derived from experiment for the first time. The underlying motivation for this work was to extend the assignment of the highresolution photoionization spectrum, and MQDT calculations carried out on NH 3 to the photoionization spectrum of ND 3 . The good agreement using the same set of electronic parameters for both isotopomers, despite the very different rotational and vibrational constants, nuclear spin statistical weights and vibronic dynamics indicates the reliability of these parameters. Whereas the agreement between the measured and calculated line positions is good, the intensities are not described so well by the calculations. We attribute the discrepancies to the fact that predissociation has not been included in the calculations, even though it is known that many Rydberg states of ammonia are predissociative.

Conclusions

The ammonia molecule is a model system with which to study photoionization in polyatomic molecules and reveals many fascinating features; the studies presented in this work and references [2,16,17] have substantially extended the understanding of the highly excited states of ammonia, while still leaving room for detailed considerations of the predissociation of these states in the future, and of the competition with autoionization.

(v 2 + = 2)  1 1 ~A X 
transition of ND 3 . The lines are assigned using the notation

K K N N        (b)
Simulation of the PFI-ZEKE photoelectron spectrum using the orbital ionization Ref. [2]) and the MQDT simulation (top trace) calculated using quantum defects from reference [16]: the separate contributions from each initial rotational state are colour coded. photoionization spectrum (bottom trace, ref [2]). The simulated spectrum was computed using quantum defects from reference [16]: the v 2 + = 1 region is dominated by a series of transitions originating from the 1 1 E' ground state level converging to photoionization spectrum (bottom trace, ref [2]). The simulated spectrum was computed using quantum defects from reference [16]: Table 1: Two-pulse sequences used to record the PFI-ZEKE photoelectron spectra of ND 3 . The shifts of the ionization thresholds were estimated as described in reference [20].

the v + = 1, 1 1 E'  2 1 E'' rovibronic threshold.
v 2 + =
Table 2: Assignment of rotational lines (in the notation

K K N N        ) of the v 2 + = 0
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.7 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and (2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5.

Table 3: Assignment of rotational lines (in the notation

K K N N        ) of the v 2 + = 1
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.5 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and (2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5.

F

K K N N        ) of the v 2 + = 2
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.45 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and ( 2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5. [26]. Other values are determined as follows:

(a) Theoretically determined from an adjusted ab-initio potential surface [25]. (b) Infrared spectrum of ND 3 + recorded in solid neon [31]. (c) Calculated using the equation . d) Calculated using the C 2 -C 1 and C 1 -C 0 values in reference [25]. (e) Experimental measurement from REMPI studies in ref [8].

) ( 0 2 1 ) ( 0 3 3 NH ND C C  (
Table 6: Diagonal and off-diagonal elements of the quantum defect matrix used in the MQDT simulations (from ref [16]). The final row lists the vibrationally integrated off-diagonal elements as given in Eq (17). Ref. [2]) and the MQDT simulation (top trace) calculated using quantum defects from reference [16]: the separate contributions from each initial rotational state are colour coded. photoionization spectrum (bottom trace, ref [2]). The simulated spectrum was computed using quantum defects from reference [16]

1 1 E  1 1 E'' v 2 + = 3, n = 9 n = 9 11 13 10 12 0 0 A 1   3 0 A 1  v 2 + = 2, n = 10 0 0 A 1   1 0 A 1  v 2 + =
0 0 A 1 '  3 0 A 1  v 2 + = 2, n = 10 1 1 E'  1 1 E'' v 2 + = 1 25 20 n =15 30 n = 15 20 25 n = 15 20 0 0 A 2   1 0 A 1  v 2 + = 1 0 0 A 1 '  1 0 A 1  v 2 + = 2, n = 11 Wavenumber /cm -1 1 1 E  2 1 E v 2 + = 1
K K N N        ) of the v 2 + = 0
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.7 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and (2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5.

F

K K N N        ) of the v 2 + = 1
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.5 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and (2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5.

F

K K N N        ) of the v 2 + = 2
band of the PFI-ZEKE photoelectron spectrum of ND 3 . The reported line positions in the first column have not been corrected for the shift due to the pulsed field ionization. The field-free thresholds can be obtained by adding 1.45 cm -1 to each of the numbers in column 1 (using the data presented in Table 1.) The calculated wavenumbers calc  are obtained from Eqs. ( 1) and (2) using the best fit rotational constants, vibrational intervals and the ionization energy given in Table 5. 1395.2 (20 ) 1393 (a) Rotational constant B + (cm [26]. Other values are determined as follows:

(a) Theoretically determined from an adjusted ab-initio potential surface [25]. (b) Infrared spectrum of ND 3 + recorded in solid neon [31]. (c) Calculated using the equation . [25]. (e) Experimental measurement from REMPI studies in ref [8]. [16]). The final row lists the vibrationally integrated off-diagonal elements as given in Eq (17).

  photoionization spectrum of ND 3 using MQDT.

2 P 1 / 2 5p' [ 1 / 2 ]

 21212 (J = 0)  (4p)6 1 S 0 two-photon resonance at 2 1  = 98855.071cm -1 . The tunability of the VUV radiation was achieved by scanning the frequency  2 . The fundamental frequencies  1 and  2 were generated from the output of two pulsed dye lasers (Lambda Physik, Scanmate 2e), the first one being frequency tripled. The resultant VUV radiation had a bandwidth of about 0.4 cm -1 . The ND 3 gas sample was diluted in a 1:4 mixture with argon and introduced into the spectrometer via a pulsed supersonic expansion. This expansion was crossed by the VUV radiation beam at right angles in the middle of a set of six cylindrical electrodes used to apply the pulsed electric fields. PFI-ZEKE photoelectron spectra were recorded by monitoring the wavenumberdependent electron signal produced by delayed pulsed-field ionization of very high Rydberg states located below the successive ionization thresholds. A two-pulse electric field sequence was employed for field ionization in order to improve the

1 1 ~A

 11 X  vibronic ground state of ND 3 and of the first two out-of-plane bending levels (2 0 , 21 ; the notation implies state of ND 3 + is presented in Fig.4. Because the

  2 '' symmetry (D 3h ) of the  2 vibrational mode results in a different set of allowed rotational transitions. To assign the experimental spectra, the strong rotational features in the simulated vibrational bands are identified with the strong features in the experimental spectra. Lining up the simulated and experimental spectra allows an initial value for the sum ) rotational constants B + and C + of the ion can then be determined by a leastsquares fitting method to improve the fit. At the low temperature of the supersonic expansion only low rotational states are populated and centrifugal distortion effects are negligible.Following a preliminary analysis as described above it was recognised that numerous transitions were observed which could not be identified using the orbital ionization model. The simulations shown in panel (c) of Figs. 1, 2 and 3 assume a relaxation of the orbital restrictions allowing transitions from orbital components of f to the p character. As discussed below, such relaxation does not necessarily reflect the underlying direct ionization cross sections, but was useful in assigning additional lines. Tables 2, 3 and 4 give the assignments and observed line positions of the rotational features in each vibrational band. The wavenumbers of the observed bands lie slightly below the actual ionization thresholds by an amount given by the field-induced shifts listed in Table

1 Page

 1 photoelectron spectra by detecting ions produced by autoionization as a function of

  photoionization cross sections into the set of N O open channels, summed over all

Figure 1

 1 Figure 1 (a) PFI-ZEKE photoelectron spectrum of the

Figure 2

 2 Figure 2 (a) PFI-ZEKE photoelectron spectrum of the

Figure 3 :

 3 Figure 3: (a) PFI-ZEKE photoelectron spectrum of the

  . 8). (c) Simulation based on a less restrictive photoionization model. (d) Experimental photoionization spectrum of ND 3 .

Figure 4 : 2 +(

 42 Figure 4: Energy level diagram showing schematically the rotational levels of the

Figure 5 :

 5 Figure 5: Comparison of the photoionization spectrum of ND 3 (bottom trace, from

Figure 6 :

 6 Figure 6: Expanded view of the MQDT simulation (top trace) of the ND 3

Figure 7 :

 7 Figure 7: Expanded view of the MQDT simulation (top trace) of the ND 3

Figure 8 :

 8 Figure 8: Example of series assignments based on the MQDT calculations in a

Figure 1 :

 1 Figure 1: (a) PFI-ZEKE photoelectron spectrum of the

Figure 2 ( 29 Figure 3 :Figure 4 :Figure 5 :

 229345 Figure 2 (a) PFI-ZEKE photoelectron spectrum of the

Figure 7 :

 7 Figure 7: Expanded view of the MQDT simulation (top trace) of the ND 3

  Calculated using the C 2 -C 1 and C 1 -C 0 values in reference

Table 4 :

 4 Assignment of rotational lines (in the notation

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	25

Table 5 :

 5 Spectroscopic parameters of ND 3 /ND 3 + . Parameters labelled 'Expt' are from the present work, while those labelled 'REMPI' are from the 2 + 1 REMPI studies of the C  ~state of ND 3

Table 7

 7 Summary of series or parts of series identified in the high-resolution photoionization spectrum of ND 3 .

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 8

 8 Ionization energies (band origins for transitions to specific vibrational thresholds) derived from the present work and references[2,16,17] for NH 3 , NH 2 D, NHD 2 and ND 3 .

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	Page 31 of 43	Molecular Physics Molecular Physics
	Red: 0 0 A 2 ''
	Orange: 1 0 A 2 '
	Blue: 1 1 E''
	Green: 1 1 E'
	Pink: 0 0 A 1 '
	Purple: 1 0 A 1 ''	2
	Red: 0 0 A 2 ''
	Orange: 1 0 A 2 '
	Blue: 1 1 E''	MQDT simulation
	MQDT simulation o F r Green: 1 1 E' Pink: 0 0 A 1 ' Purple: 1 0 A 1 '' F o r
	P P
	e e	Experimental
	wavenumber /cm -1 w v R e r e i e O n l y Figure 6: Expanded view of the MQDT simulation (top trace) of the ND 3 photoionization spectrum (bottom trace, ref [2]). The simulated spectrum was computed using quantum defects from reference [16]: the v 2 + = 1 region is dominated by a series of transitions originating from the 1 1 E' ground state level converging to 31 n = 26 30 29 28 27 wavenumber /cm -1 e r R e v i e w O n l the v Experimental y
	URL: http://mc.manuscriptcentral.com/tandf/tmph

+ = 1, 1 1 E'  2 1 E'' rovibronic threshold.

Table 1 :

 1 Two-pulse sequences used to record the PFI-ZEKE photoelectron spectra of ND 3 . The shifts of the ionization thresholds were estimated as described in reference[20].

	Page 34 of 43

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Table 2 :

 2 Assignment of rotational lines (in the notation

Table 3 :

 3 Assignment of rotational lines (in the notation

Table 4 :

 4 Assignment of rotational lines (in the notation

Table 5 :

 5 Spectroscopic parameters of ND 3 /ND 3 + . Parameters labelled 'Expt' are from the present work, while those labelled 'REMPI' are from the 2 + 1 REMPI studies of the C  ~state of ND 3

		-1 ) Expt	5.350 (20)	5.230 (20)	5.070(20)
		Literature	5.347 (a)	5.214 (a)	5.078 (a)
		REMPI	5.361 (1)	5.232 (1)
	Rotational constant C + (cm -1 ) Expt	2.675 (50 )	2.700 (50 )	2.730 (50)
		Literature	2.6238 (c)	2.6538 (d)	2.6828 (d)
		REMPI	2.626 (1)	2.646 (1)
	Adiabatic ionization	Expt	82261.7 (15 )	82948.2(15) 83656.9(15)
	energy (hc.cm -1 )	Literature	82280 (40) (e)

Table 6 :

 6 Diagonal and off-diagonal quantum elements of the electronic quantum defect matrix used in the MQDT simulations (from ref

	Page 39 of 43	Molecular Physics

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Table 8

 8 Ionization energies (band origins for transitions to specific vibrational thresholds) derived from the present work and references[2,16,17] for NH 3 , NH 2 D, NHD 2 and ND 3

	Page 40 of 43

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Acknowledgements

We are grateful for financial support of this work to the Swiss National Science Foundation (project No 200020-125030), to the Petroleum Research Fund of the American Chemical Society, and to the Engineering and Physical Sciences Research Council.

Molecule Ion

J'' K''

Table 7 Summary of series or parts of series identified in the high-resolution photoionization spectrum of ND 3 .


Richard Zare

Foudhil Bouakline
email: foudhil.bouakline@googlemail.com

Stuart C Althorpe

Pascal Larregaray

Laurent Bonnet

Strong geometric-phase effects in the hydrogen-exchange reaction at high collision energies: II. Quasiclassical trajectory analysis

Keywords: Foudhil Bouakline, Suart Althorpe, Pascal Larregaray, hydrogen-exchange reaction, geometric phase, quasiclassical trajectories, topology and homotopy, nearside-farside analysis hydrogen-exchange reaction, geometric phase, quasiclassical trajectories, topology and homotopy, nearside-farside analysis

Strong geometric-phase effects in the hydrogen-exchange reaction at high collision energies: II. Quasiclassical trajectory analysis.

INTRODUCTION

The dynamics of the H + H 2 system and its isotopic variants has been extensively studied both theoretically and experimentally for many decades [1]- [34]. After so many years of study, leading to very good agreement between experimental and quantum mechanical results [12,25,26], and an unprecedented understanding of the reaction at the state-to-state level [28], it is tempting to say that this prototype reaction is fully understood and there is little to learn about it. However, there are still some issues which continue to be discussed such as quantum bottleneck states [20], and also the different reaction mechanisms governing this system [17][18][19], such as the recently discovered 'tug-of-war' mechanism [31,32] in H + D 2 inelastic scattering.

Another particularly important feature of this system is its well characterized (Jahn-Teller) conical intersection (CI) seam [35]- [37], connecting the electronic ground state potential energy surface to the first excited state surface by a hyperline passing through the nuclear equilateral triangle geometries. As such, it has been used as a benchmark for understanding non-adiabatic reaction dynamics [13,30] around a CI.

The main role of a conical intersection is to act as a funnel allowing population transfer between the upper and lower adiabatic electronic states via non-radiative transitions [START_REF]Conical Intersections: Electronic Structure, Dynamics and Spectroscopy[END_REF][START_REF] Worth | [END_REF]. Another accompanying quantum effect is the so-called geometric (or Berry) phase (GP) [40,41], which occurs even if the nuclear motion is confined to the ground adiabatic electronic state and avoids the neighbourhood of the CI. The GP is the sign-change acquired by the adiabatic electronic wave function when the nuclei complete an odd number of loops around the CI. This affects the nuclear dynamics by introducing a corresponding sign-change in the continuity boundary condition of the nuclear wave function [42]- [45], which cancels out the sign acquired by the adiabatic electronic wave function, required by the singlevaluedness of the total wave function. The effect of the GP on the nuclear wave function of a bound-state system is well understood [45]: It brings changes in the quantum numbers and energy levels with respect to those computed with normal boundary conditions, and this has been observed in many Jahn-Teller molecules [46]- [49]. However, our understanding of such effects in non-bound systems has only become clear recently, owing to detailed reactive scattering calculations and experiments on the hydrogen-exchange reaction [4]- [34], and also on the predissociation of the upper cone-states of the H 3 molecular system [50]- [START_REF] Bouakline | The Jahn-Teller Effect: Fundamentals and Applications for Physics and Chemistry[END_REF].

Mead [43] was the first to predict GP effects in the H + H 2 reaction using symmetry arguments. He showed that the GP changes the relative sign of the inelastic and reactive scattering contributions to the fully symmetrized cross sections, whenever the nuclear wave function does not encircle the CI when unsymmetrized. Later work by Kuppermann and co-workers [5], [55]- [58] using multivalued basis functions approach to include the GP, considered more general effects where the unsymmetrized wave function may encircle the CI. Their calculations predicted strong GP effects in the differential cross sections (DCS), but these predictions were neither reproduced by later calculations that omitted the GP [9,11,24], nor by experiments [8,10]. Detailed experimental results agreed quantitatively well with theoretical predictions ignoring the GP boundary condition. This controversy stimulated further work by Kendrick [11,59,60], who performed time-independent quantum calculations and used a different method to include the GP, namely the Mead-Truhlar vector potential approach [42]. He found that GP effects appear in the scattering observables at specific values of the total angular momentum J, but cancel on summing over all the partial waves to give the state-to-state integral cross sections (ICS), and also on summing over a limited number of J (0 ≤ J ≤ 10) to give the low impact parameter DCS.

Subsequent calculations by Juanes-Marcos et al. [24,61,62], using a timedependent wave packet method [14] to solve the nuclear Schrödinger equation and the vector potential approach to include the GP, confirmed these results and extended them to high impact parameters to give the fully converged DCS. They found that the DCS do show small GP effects, which cancel on integrating over all the scattering angles to give the ICS. In addition to these calculations, Althorpe and co-workers [24,61,63] were able to explain these findings using topological arguments originated in the late 1960s in Feynman path integral theory in order to explain the analogous Aharonov-Bohm effect [64]- [66]. They demonstrated that the central idea is very simple and can be derived even without path integrals, using symmetry arguments [24,67]. They showed that the nuclear wave function can be unwound and separated, without any approximation, into two components, each of which contains all the Feynman paths that loop in the relevant sense around the CI. These two components correspond to two different reaction mechanisms in the H + H 2 reaction as shown in figure 1; one passes over one transition state (1-TS) and the other over two transition states (2-TS). The interference between these two components governs the extent to which scattering observables are affected by the GP. In addition, the topological approach of Althorpe [63] predicts that the two reaction paths (1-TS and 2-TS) scatter their products into different regions of space, which causes the cancellation of the GP in the ICS. In a recent work [30], some of us extended the previous calculations to higher total energies (up to 4.5 eV), well above the energy minimum of the CI seam occurring at 2.74 eV, and took full account of the nonadiabatic coupling to the first excited electronic state. We found little contribution of the excited state to the state-to-state dynamics even at very high energies, and obtained accurate predictions of state-to-state scattering observables using just the ground state surface with inclusion of the geometric phase. The effect of the GP in the state-to-state probabilities, at energies above 3.5 eV, were found to be stronger than at low energies, and to survive the coherent sum over the partial waves to produce features in the state-to-state DCS, which could be detected in an experiment with an angular resolution of 20 • . However, these effects completely cancel out in the ICS, owing to differences in the slopes of the phases of the 1-TS and 2-TS scattering amplitudes.

The aim of the present article is to investigate the origin of this cancellation of GP effects in the ICS. We do this by carrying out quasiclassical trajectory (QCT) calculations, which allow a more direct and straightforward probing of the reaction mechanisms. We compare the quantum and QCT results in some detail, by decomposing the cross sections into 1-TS and 2-TS contributions, and also, further, into nearside and farside contributions. The methodology used is described in Sec. 2. The results are presented and discussed in Sec. 3. Section 4 concludes the article.

METHODOLOGY

Extraction of the one-and two-transition state mechanisms

Here we summarize the topological approach [24,63,67] which allows us to unwind the nuclear wave function and extract from it the 1-TS and 2-TS Feynman paths. Once the unwound parts are extracted, one can use the 1-TS and 2-TS components to interpret the dynamics and explain the effect of the geometric phase on the different scattering observables. The basic idea of the topological approach is that the nuclear wave function can be split into two parts, Ψ e and Ψ o . One component (Ψ e ) is the portion of the wave function which contains all the Feynman paths that complete an even number of clockwise (and an odd number of counterclockwise) loops around the CI; the other part (Ψ o ) is the sum over all the odd clockwise (and even counterclockwise) looping Feynman paths. The only effect of the GP is to change the relative sign of these two components. In general, it is extremely difficult to extract reaction paths from a nuclear wave function; however, this approach is easy to implement and it only requires [62] the computation of the nuclear wave functions with the GP boundary condition (Ψ GP ) and with normal boundary conditions ignoring the GP (Ψ NGP ). Using either symmetry arguments or Feynman paths [63,67], one can show that: where α is any angle such that α = 0 → 2π describes a closed loop around the CI. It is clear from this equation that GP effects will be noticeable only if the even and odd looping components (Ψ e and Ψ o ) interfere. From the previous equation, it is trivial to extract the even and odd looping wavefunctions without any approximation:

Ψ NGP (α) = 1 √ 2 [Ψ e (α) + Ψ o (α)] Ψ GP (α) = 1 √ 2 [Ψ e (α) -Ψ o (α)] (1)
Ψ e (α) = 1 √ 2 [Ψ NGP (α) + Ψ GP (α)] Ψ o (α) = 1 √ 2 [Ψ NGP (α) -Ψ GP (α)] (2)
In the H + H 2 reaction, these two components (Ψ e and Ψ o ) correspond to paths that pass over one transition state (1-TS) and two transition states (2-TS), respectively. So, the terms "Ψ e paths" and "1-TS paths" (and "Ψ o paths" and "2-TS paths") are equivalent, and will be used interchangeably. Figure 1 illustrates this terminology for the hydrogen-exchange reaction, where the three nuclei are treated as indistinguishable particles. If the reaction enters the encirclement region from the A+BC asymptotic arrangement and exits into the AC+B arrangement, it is obvious that the 1-TS paths make less than one loop around the CI in the clockwise sense, so they are assigned a winding number n = 0; the 2-TS paths make less than one revolution, but in the anticlockwise sense, and are assigned a winding number n = -1, following the convention of refs. [63,67]. Therefore, the 1-TS Feynman paths are contained in Ψ e and the 2-TS paths in Ψ o . The even and odd components of the wave function also contain higher winding numbers, but these are ignored as it is highly unlikely that the nuclei pass over three or more transition states in the H + H 2 reaction.

Since the GP affects the relative phases of Ψ e and Ψ o , but not these functions individually [62], this topological approach allows us to treat the 1-TS and 2-TS mechanisms as two separate reactions giving rise to their own scattering observables, neither of which is affected by the geometric phase. Consequently, one can make direct comparisons between scattering observables obtained independently from these two components with the corresponding QCT predictions. Separating out the QCT 1-TS and 2-TS mechanisms is easy, since one just counts how many times a given trajectory crosses a transition state (see Fig. 1). For instance, considering trajectories which enter the encirclement region from the A+BC asymptotic arrangement (region I) and exit into the AC+B arrangement (region II), 1-TS trajectories are the ones ultimately crossing the I/II separation line, whereas 2-TS trajectories ultimately cross the III/II separation line. (Because of the indistinguishability of the three nuclei, an analog analysis is performed for trajectories ending up in the C+AB asymptotic arrangement). Almost all trajectories cross a given separation line only once, though complex trajectories involving recrossing exist, but are statistically irrelevant.

We should emphasize that the unwinding technique given by Eq.2 applies to the exact wave function, and the paths we extract from it are different classes of Feynman paths, not classical trajectories. This approach is rigorous, and applies even when the system can access the conical intersection seam (as is done here) and has enough energy to hop between the two surfaces [68].

The GP and NGP wave functions of Eq.2 were calculated using just the ground adiabatic electronic state surface for all total energies. Previous work [13,30] has shown that the excited state does not contribute appreciably to the dynamics, even at energies above the CI energetic minimum. Previous work [24,30,62] at lower energies (E = 2.3 eV) found that Ψ e and Ψ o scattered into different hemispheres (which was the reason that the GP dephased so efficiently in the ICS): Ψ e scattered into the "nearside", Θ = +θ hemisphere (see Fig. 2), and Ψ o into the 'farside', Θ = -θ hemisphere. This identification was made by noting that the slope of the phase of the Ψ e scattering amplitude decreased as a function of θ, whereas the slope of the phase of Ψ o increased with θ. It relied on the fact that each of Ψ e and Ψ o was either nearside or farside scattered, and gave a slope that varied more or less linearly with θ (see Fig. 3). At the higher energies (E = 4.3 eV) that we investigate here, this simple analysis is no longer possible, since the phases of the Ψ e and Ψ o scattering amplitudes behave in a more complicated way (see Fig. 3), suggesting that Ψ e and Ψ o probably contain a mixture of nearside and farside components.

In the present work, we carry out a more systematic nearside-farside analysis of Ψ e and Ψ o (at E = 4.3 eV), by making use of semi-classical techniques [START_REF] Child | Molecular Collision Theory[END_REF][START_REF] Dobbyn | [END_REF] for decomposing each of Ψ e and Ψ o scattering amplitudes into nearside-and farsidescattering subamplitudes. It is well known that, in the semiclassical limit (J sin θ ≫ 1), the state-to-state scattering amplitude becomes

f n ′ ←n (E, θ) ≃ f [+] n ′ ←n (E, θ) + f [-] n ′ ←n (E, θ) (3)
where

f [+]
n ′ ←n and f

[-]

n ′ ←n represent the nearside and farside components, respectively. n and n ′ are the composite quantum numbers of the reactant and product diatoms, E is the energy of the system and θ is the scattering angle. The two components of the scattering amplitude are given by

f [±] n ′ ←n (E, θ) = 1 2ik vj J (2J + 1)e J[±] k ′ k (π -θ)S n ′ ←n (J, E) (4)
and

e J[±] k ′ k (π -θ) = [ 1 2π(J + 1 2 )sinθ ] 1/2 exp{±i[(J + 1 2 )]θ + 1 2 π(k ′ -k) - 1 4 π]} (5)
where v and j are the vibration and rotation quantum numbers of the reactant diatomic molecule, k and k ′ are the projections of j on the initial and final relative velocity vectors, respectively. k vj denotes the magnitude of the atom-diatom approach momentum, J the total angular momentum quantum number, and S n ′ ←n the scattering matrix elements. The decomposition in Eq.( 3) was applied separately to Ψ e and Ψ o , with the result that the total scattering amplitude f n ′ ←n was separated into four separate pieces:

f [e][+] n ′ ←n (1-TS, nearside-scattered), f [e][-] n ′ ←n (1-TS, farside-scattered), f [o][+] n ′ ←n (2- TS, nearside-scattered) and f [o][-]
n ′ ←n (2-TS, farside-scattered). The same decomposition was also applied to the QCT trajectories. As already mentioned, it is easy to determine whether a trajectory has traversed one or two transition states, and also to determine whether it is nearside or farside-scattered (by testing the sign of the component of the final atom-diatom relative velocity vector parallel to the impact parameter, which is nearside if positive and farside if negative). This analysis was made for all final quantum states (v ′ , j ′ ), making possible detailed comparisons between quantum and QCT results.

Quantum mechanical calculations

The calculations reported here made use of GP and NGP scattering matrix elements obtained from calculations reported previously [30]. Both sets were calculated using the BKMP2 [71] ground electronic state potential energy surface (PES), with inclusion of the diagonal non-Born-Oppenheimer correction. The calculations were performed using the reactant product decoupling (RPD) method [14], and the geometric phase was incorporated by the Mead-Truhlar vector potential approach [42]. To converge the cross sections for the H + H 2 (v = 0, j = 0) → H 2 (v ′ , j ′ ) + H reaction over total energies ranging from 1.25 eV to 4.5 eV, we found it necessary to include all partial waves up to total angular momentum quantum number J = 55.

The hydrogen nuclei are treated as distinguishable particles, ignoring the requirement that the nuclear wave function be antisymmetric under exchange of two 1 H nuclei. Thus, our theoretical product distributions and cross sections correspond to the experimentally measurable values only for the odd-j product states.

Quasiclassical trajectory calculations

The quasiclassical trajectory (QCT) calculations were carried out using the same BKMP2 potential energy surface as the quantum calculations. The QCT methodology was adapted from refs. [START_REF] Truhlar | Atom-Molecule Collision Theory[END_REF][START_REF] Halvick | [END_REF]; it uses a standard Monte Carlo sampling of the initial conditions, and the step-adaptive Adams method to integrate the Hamilton equations, where conservation of total energy and total angular momentum were carefully checked. The calculations were performed for two different total energies (2.3 eV and 4.3 eV), where a total of 5 × 10 6 trajectories were propagated at each energy. The rovibrational quantum numbers (v ′ , j ′ ) of the product outgoing diatomic molecule were assigned, for a given rotational angular momentum, by calculating the exact vibrational action. The Gaussian-weighted (GW) binning method [74]- [76], was employed to extract product distributions, using a full width-at-half-maximum of 0.05 for the Gaussian functions. Unlike the standard histogram binning (SB) method, where the final vibrational action of each trajectory is rounded to its nearest integer, which may lead to inappropriate results, the GW procedure circumvents this problem by weighting each trajectory by a Gaussian-like coefficient, such that the closer the final vibrational action is to an integer, the larger the coefficient. It has been proven [74]- [76] that the GW procedure gives more realistic results than the SB technique, especially when the total energy disposal in the products is too low for the classical and quantum densities of vibrational states to be equal. However, test calculations on our system showed negligible differences between the GW and SB techniques for only high final vibrational states and for a total energy of 2.3 eV, and almost the same results for a total energy of 4.3 eV. The separation into 1-TS and 2-TS trajectories was done by converting to the hyperspherical coordinate system of Kuppermann [77], and counting how many times each trajectory crossed a transition state. The resulting 1-TS and 2-TS scattering cross sections were then further decomposed into nearside and farside-scattered components, to permit direct comparison with the quantum results (obtained as in Sec. 2.2). 3. RESULTS AND DISCUSSION

Overview and summary of previous results

Quantum results were obtained from the S-matrix elements of ref.

[30] at energies 2.3 and 4.3 eV as described in Sec. 2. The overall effect of the GP in the 2-4.5 eV total energy range is summarised in Fig. 4. At 2 eV, the contribution from 2-TS scattering is very small indeed, although it does make small differences to some of the DCS by interfering with the 1-TS scattering, as described in refs. [30,62]. By 3.5 eV, however, the 2-TS scattering is now of roughly equal importance to the 1-TS scattering (the overall ratio of 1-TS:2-TS at 4.3 eV is 3:1). This results in GP effects in the DCS as described in ref. [30] and illustrated in Fig. 5 (for one particular final state). At 2.3 eV, the 2-TS component is very small indeed and produces no noticeable contribution to the DCS (although for some other final states it does make small differences [30,62]). At 4.3 eV, the interference between the 1-TS and 2-TS contributions produces strong GP effects in the DCS.

Comparison with QCT results and nearside-farside decomposition

Figures 6 and7 show comparisons between the quantum and QCT integral cross sections (rotational product distributions for each v ′ ) at total energies of 2.3 and 4.3 eV. At 2.3 eV, the QCT and quantum results are in good overall agreement, and reproduce results reported earlier in ref. [62]. At 4.3 eV, the QCT and quantum results are also in good agreement, although the QCT overestimate the bimodal shape of the product distributions at v ′ = 0, 1. In the quantum calculations, this shape is barely evident, as two shallow maxima (see Fig. 7), but in the classical calculations the maxima are pronounced. An explanation for the above discrepancy is given when the separate 1-TS and 2-TS distributions are compared (at 4.3 eV) in Fig. 8. As mentioned above, the separation of these two classes of mechanism is easy to make, and involves no approximation. At 2.3 eV, the results (not shown) are in reasonable agreement, reproducing those of ref. [62]. At this energy, only about 2 × 10 -4 % of the trajectories pass over two transition states, so the 2-TS contribution to the ICS is very small. At 4.3 eV, however, the 1-TS and 2-TS contributions are of roughly the same magnitude. Figure 8 shows that the 1-TS and 2-TS QCT results reproduce well the corresponding quantum results. The bimodal shape is seen to be a direct consequence of the fact that the 1-TS rotational distributions are hotter than the 2-TS distributions at low v ′ , resulting in two peaks, which are more pronounced in the QCT case. However, the agreement between the quantum and classical 1-TS and 2-TS distributions is on the whole very good.

This agreement is seen to continue when each of the 1-TS and 2-TS cross sections is decomposed further, into nearside and farside contributions. In the 1-TS cross sections, the farside contribution to both the quantum and classical results is negligible (and not shown here). In the 2-TS cross sections, the nearside and farside contributions are of equal magnitude. Figure 9 shows that the quantum and QCT predictions of these quantities agree on the whole very well. As v ′ increases, the importance of the farside contribution increases; at v ′ = 0 the nearside contribution is dominant, at v ′ = 3 the farside contribution is roughly twice that of the nearside.

Origin of the GP cancellation in the ICS

By this stage in the analysis, we have split the state-to-state ICS into 4 pieces indicating the sense in which the trajectory winds about the CI and the hemisphere into which it scatters, and have found excellent agreement between the quantum and QCT results. This gives us confidence that the QCT results provide a faithful description of the dynamics, subject to the caveat that the QCT rotational distributions will tend to be hotter than the quantum distributions. We can therefore use the QCT results to explain why the difference between the 1-TS and 2-TS scattering is such that, at 4.3 eV, the GP effects that are present in the state-to-state (quantum) DCS cancel out in the corresponding ICS.

Let us first remind ourselves why a similar cancellation occurs at 2.3 eV. Figure 10 shows the angular distribution of trajectories as a function of total angular momentum (i.e. initial impact parameter). It is clear that the majority of the 1-TS trajectories scatter into the nearside (Θ = +θ) hemisphere, whereas the majority of the 2-TS scatter into the farside (Θ = -θ) hemisphere. As a result, the 1-TS-2-TS overlap integral is highly oscillatory as a function of θ, which causes the GP effects (in the quantum results) to average out to a very small value on integrating over θ to give the ICS.

Figure 11 shows the angular distribution at 4.3 eV. This is consistent with Fig. 9 in showing that the 2-TS trajectories now scatter into both the nearside and farside hemispheres. However, it is clear from this plot that the nearside 2-TS trajectories scatter into a different region of angular phase-space to the 1-TS trajectories. The 2-TS nearside trajectories scatter from relatively low impact parameters (J = 0 → 23) into the forward direction θ = 0 → 60 • , whereas the 1-TS trajectories scatter from high impact parameters (J = 30 → 47) into this range of angles. As a result, the 1-TS-2-TS overlap integral is still highly oscillatory, and integrating over θ to give the (quantum) ICS causes an efficient averaging out to zero of the GP effects in the DCS.

A full characterisation of the mechanisms responsible for the 1-TS and 2-TS scattering at 4.3 eV is difficult (not least because it does not make sense to talk about neatly defined 'reaction mechanisms' at such high energies), and will not be attempted here. Nevertheless, we can pick out some general features from Fig. 11 that give clues into how the dynamics at 4.3 eV differ from the much better understood dynamics at 2.3 eV. First, the 1-TS trajectories no longer scatter into the recoil direction θ ∼ 180 • , even from low impact parameters. This is not surprising, since the simple recoil mechanism that produces scattering into these angles at lower energies is unlikely to operate at energies as high as 4.3 eV. Second, the low impact (J < 25) 2-TS trajectories behave very differently from the high impact (J > 25) trajectories. The former scatter in roughly equal proportions into the nearside and farside hemispheres, with low v ′ (< 2) favouring the nearside hemisphere, and high v ′ the farside (see also Fig. 9). The high impact 2-TS trajectories, by contrast, scatter entirely into the farside direction. This suggests that the latter are closer to the direct 'insertion' mechanism followed by the lower-energy 2-TS trajectories, in which the attacking H atom first weakens, and then passes through the reactant H-H bond.

CONCLUSIONS

We have carried out detailed comparisons of quantum and QCT reactive scattering cross sections for the hydrogen-exchange reaction at high energies (4.3 eV), and found excellent agreement between the two sets of results. This agreement continues to hold when the cross sections are decomposed into contributions from 1-TS and 2-TS reaction paths (which procedure is rigorous and based on topology), and when the latter are further decomposed into nearside and farside-scattering contributions (which decomposition is approximate in the quantum calculations, and makes use of a semiclassical approximation to the scattering wave function).

As a result, we can trust the QCT results to give a faithful description of the quantum scattering at these high energies. In particular, inspection of the QCT angular distributions answers the question raised by previous work [30], as to why GP effects in the high energy (> 3.5 eV) DCS cancel out in the ICS. Unlike at lower energies (< 3.5 eV), the 1-TS and 2-TS paths both scatter products into the nearside hermisphere; however, the nearside 2-TS paths scatter into a different region of angular phase-space than the 1-TS paths, and thus result in a 1-TS-2-TS interference term which is oscillatory as a function of θ. As a result the GP effects cancel out in the ICS.

Further work will be needed to characterise the nature of the reaction mechanisms at these higher energies. We have shown here that the QCT calculations are sufficient to describe these mechanisms, including those properties that are responsible for the cancellation of GP effects in the ICS.

The results of this paper confirm those of ref. [30] in predicting that the best way to detect GP effects experimentally in the hydrogen-exchange reaction is to measure the the state-to-state DCS at high energies (> 3.5 eV). Measurements of the ICS at these energies will not contain noticeable GP effects, although measurement of the v ′ = 0 rotational distribution would allow an experiment to distinguish between 1-TS and 2-TS products.

F

A + H B H C → H A H C + H B .
The H 3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann (Ref. [77]), in which the equilateral-triangle geometry of the CI is in the center (×) and the linear transition states ( ‡) are on the perimeter of the circle. The angle α is the internal angular coordinate which describes motion around the CI. The dashed (red) lines separate the three different atom-diatom channels.

Figure 2: Diagram illustrating "nearside" scattering (positive deflection angles Θ) and "farside" scattering (negative deflection angles). The arrow (chains) represents the initial approach direction of the reagents in the center-of-mass frame; the gray rectangle represents the spread of impact parameters in the initial plane wave.

Figure 3: (Color online). Phases of the (v = 0, j = 0) → (v ′ = 2, j ′ = 1, Ω ′ = 0) 1-TS (solid line) and 2-TS (dashed line) scattering amplitudes.

Figure 4: (Color online). (v = 0, j = 0) → (v ′ , j ′ ) quantum state-to-state ICS (solid line), 1-TS (dashed line) and 2-TS (dashed-dotted line) contributions.

Figure 5: (Color online). (a) State-to-state DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) computed using the lower adiabatic PES including the diagonal nonadiabatic correction without (solid line) and with (dashed line) the GP. (b) State-tostate DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) obtained by extracting the 1-TS (solid line) and 2-TS (dashed line) contributions to the GP and non-GP scattering amplitudes. For clarity, the 2-TS DCS are shown multiplied by a factor of 200 for a total energy of 2.3 eV.

Figure 6: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 2.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).

Figure 7: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).

Figure 8: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) 1-TS and 2-TS state-to-state ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).

Figure 9: (Color online). Quantum mechanical and QCT "nearside" and "farside" components of the 2-TS state-to-state ICS at 4.3 eV total energy.

  Geometric Phase Effects in the Hydrogen-Exchange Reaction

5 2. 2 .

 52 Geometric Phase Effects in the Hydrogen-Exchange Reaction Nearside-Farside analysis

  Geometric Phase Effects in the Hydrogen-Exchange Reaction 7

Figure 1 :

 1 Figure1: Schematic representation of the 1-TS (solid) and 2-TS (dashed) reaction paths in the reaction H A + H B H C → H A H C + H B . The H 3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann (Ref.[77]), in which the equilateral-triangle geometry of the CI is in the center (×) and the linear transition states ( ‡) are on the perimeter of the circle. The angle α is the internal angular coordinate which describes motion around the CI. The dashed (red) lines separate the three different atom-diatom channels.Figure2: Diagram illustrating "nearside" scattering (positive deflection angles Θ) and "farside" scattering (negative deflection angles). The arrow (chains) represents the initial approach direction of the reagents in the center-of-mass frame; the gray rectangle represents the spread of impact parameters in the initial plane wave.Figure3:(Color online). Phases of the (v = 0, j = 0) → (v ′ = 2, j ′ = 1, Ω ′ = 0) 1-TS (solid line) and 2-TS (dashed line) scattering amplitudes.Figure4: (Color online). (v = 0, j = 0) → (v ′ , j ′ ) quantum state-to-state ICS (solid line), 1-TS (dashed line) and 2-TS (dashed-dotted line) contributions.Figure5: (Color online). (a) State-to-state DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) computed using the lower adiabatic PES including the diagonal nonadiabatic correction without (solid line) and with (dashed line) the GP. (b) State-tostate DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) obtained by extracting the 1-TS (solid line) and 2-TS (dashed line) contributions to the GP and non-GP scattering amplitudes. For clarity, the 2-TS DCS are shown multiplied by a factor of 200 for a total energy of 2.3 eV.Figure6: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 2.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure7: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure8: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) 1-TS and 2-TS state-to-state ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure9: (Color online). Quantum mechanical and QCT "nearside" and "farside" components of the 2-TS state-to-state ICS at 4.3 eV total energy. Figure 10: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 2.3 eV total energy. Note that all the 2-TS and only 10000 1-TS trajectories are plotted.Figure 11: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 4.3 eV total energy. Note that, for clarity, only 10000 trajectories are plotted for each type.

Figure 10 :

 10 Figure1: Schematic representation of the 1-TS (solid) and 2-TS (dashed) reaction paths in the reaction H A + H B H C → H A H C + H B . The H 3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann (Ref.[77]), in which the equilateral-triangle geometry of the CI is in the center (×) and the linear transition states ( ‡) are on the perimeter of the circle. The angle α is the internal angular coordinate which describes motion around the CI. The dashed (red) lines separate the three different atom-diatom channels.Figure2: Diagram illustrating "nearside" scattering (positive deflection angles Θ) and "farside" scattering (negative deflection angles). The arrow (chains) represents the initial approach direction of the reagents in the center-of-mass frame; the gray rectangle represents the spread of impact parameters in the initial plane wave.Figure3:(Color online). Phases of the (v = 0, j = 0) → (v ′ = 2, j ′ = 1, Ω ′ = 0) 1-TS (solid line) and 2-TS (dashed line) scattering amplitudes.Figure4: (Color online). (v = 0, j = 0) → (v ′ , j ′ ) quantum state-to-state ICS (solid line), 1-TS (dashed line) and 2-TS (dashed-dotted line) contributions.Figure5: (Color online). (a) State-to-state DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) computed using the lower adiabatic PES including the diagonal nonadiabatic correction without (solid line) and with (dashed line) the GP. (b) State-tostate DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) obtained by extracting the 1-TS (solid line) and 2-TS (dashed line) contributions to the GP and non-GP scattering amplitudes. For clarity, the 2-TS DCS are shown multiplied by a factor of 200 for a total energy of 2.3 eV.Figure6: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 2.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure7: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure8: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) 1-TS and 2-TS state-to-state ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure9: (Color online). Quantum mechanical and QCT "nearside" and "farside" components of the 2-TS state-to-state ICS at 4.3 eV total energy. Figure 10: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 2.3 eV total energy. Note that all the 2-TS and only 10000 1-TS trajectories are plotted.Figure 11: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 4.3 eV total energy. Note that, for clarity, only 10000 trajectories are plotted for each type.

Figure 11 :

 11 Figure1: Schematic representation of the 1-TS (solid) and 2-TS (dashed) reaction paths in the reaction H A + H B H C → H A H C + H B . The H 3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann (Ref.[77]), in which the equilateral-triangle geometry of the CI is in the center (×) and the linear transition states ( ‡) are on the perimeter of the circle. The angle α is the internal angular coordinate which describes motion around the CI. The dashed (red) lines separate the three different atom-diatom channels.Figure2: Diagram illustrating "nearside" scattering (positive deflection angles Θ) and "farside" scattering (negative deflection angles). The arrow (chains) represents the initial approach direction of the reagents in the center-of-mass frame; the gray rectangle represents the spread of impact parameters in the initial plane wave.Figure3:(Color online). Phases of the (v = 0, j = 0) → (v ′ = 2, j ′ = 1, Ω ′ = 0) 1-TS (solid line) and 2-TS (dashed line) scattering amplitudes.Figure4: (Color online). (v = 0, j = 0) → (v ′ , j ′ ) quantum state-to-state ICS (solid line), 1-TS (dashed line) and 2-TS (dashed-dotted line) contributions.Figure5: (Color online). (a) State-to-state DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) computed using the lower adiabatic PES including the diagonal nonadiabatic correction without (solid line) and with (dashed line) the GP. (b) State-tostate DCS for (v = 0, j = 0) → (v ′ = 2, j ′ = 1) obtained by extracting the 1-TS (solid line) and 2-TS (dashed line) contributions to the GP and non-GP scattering amplitudes. For clarity, the 2-TS DCS are shown multiplied by a factor of 200 for a total energy of 2.3 eV.Figure6: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 2.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure7: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure8: (Color online). Quantum (closed square solid line) and QCT (closed circle solid line) 1-TS and 2-TS state-to-state ICS at a total energy of 4.3 eV, for (v = 0, j = 0) → (v ′ , j ′ ).Figure9: (Color online). Quantum mechanical and QCT "nearside" and "farside" components of the 2-TS state-to-state ICS at 4.3 eV total energy. Figure 10: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 2.3 eV total energy. Note that all the 2-TS and only 10000 1-TS trajectories are plotted.Figure 11: (Color online). Density plot of the correlation between the deflection angle and the total angular momentum J for 1-TS (open squares) and 2-TS (open circles) QCT trajectories at 4.3 eV total energy. Note that, for clarity, only 10000 trajectories are plotted for each type.

Figure 1 .

 1 Figure 1.

  Figure 5.

Figure 7 .

 7 Figure 7.

Figure 11 .

 11 Figure 11.

Acknowledgements F.B. and S.C.A. acknowledge the support of this work by the UK Engineering and Physical Sciences Research Council (EPSRC). P.L. acknowledges fruitful discussions with Eckart Wrede.