Predissociation of state selected Br2+ cations

High resolution ion imaging methods have been used to carry out a systematic investigation of the wavelength dependence of the recoil anisotropy of Br+(3P2) fragments resulting from one photon dissociation of state selected Br2+ cations in both spin–orbit components of their 2Πg ground state and with both v″ = 0 and 1.

The resonance structure so discerned is found to be concentrated in the energy gap between the ground [Br(2P3/2) + Br+(3P2)] and first excited [Br(2P3/2) + Br+(3P1)] dissociation limits, and to converge with increasing energy in a manner consistent with it being associated with a series of predissociating vibrational levels in a bound potential that correlates with the first excited dissociation asymptote.

This resonance structure has been interpreted by performing spin–orbit averaged ab initio electronic structure calculations for all ungerade excited states of Br2+ associated with the …σgπuπgu* valence space, incorporating spin–orbit effects semi-empirically, and then propagating wavepackets on the coupled diabatic potential energy curves so derived.

These model calculations succeed in reproducing all of the trends observed experimentally, and provide much new insight into the non-adiabatic couplings amongst the various excited states of this textbook open-shell system.


The past year has seen first reports of the use of velocity map ion imaging methods1–4 in studies of the photofragmentation of state selected molecular cations like Br2+5 and CF3I+.6

In both cases, the parent ion of interest was prepared by 2 + 1 resonance enhanced multiphoton ionisation (REMPI), using photons of frequency ν1, and then photolysed with a second, tuneable pulsed laser (of frequency ν2).

As the Br2+ study demonstrated, judicious choice of the REMPI frequency can allow preparation of the parent ion with electronic, fine structure and vibrational quantum state specificity.5

Imaging the Br+ fragment ions that result following absorption of a ν2 photon then affords a particularly direct visualisation of the velocity (i.e. speed and angular) distribution not just of the Br+ fragment ion but also, through energy and momentum conservation considerations, of the undetected partner species (Br in this case).

Analysis of images recorded at numerous photolysis wavelengths in the range 372 < λ2 < 432 nm led to a precise determination of the bond strength of the Br2+ cation, in both spin–orbit components of its ground electronic state: D0[79Br2+(X 2Π1/2,g)] = 23 528.1 ± 0.6 cm–1 and D0[79Br2+(X 2Π3/2,g)] = 26 345 ± 2 cm–1, and hence a value for the spin–orbit splitting in the ground state Br2+ cation: A = –2817 ± 3 cm–1 – in excellent accord with that derived in a recent two-colour 2 + 1′ pulsed field ionisation (PFI) study of Br2.7

The Br2+ cation has an inverted 2Πg ground state deriving from the electronic configuration …(σg)2u)4g*)3u*)0 (henceforth written simply in terms of the respective orbital occupancies, i.e. 2430).

The PFI study has provided the most precise values for the ionisation energies for both spin–orbit components of this state: IE = 84 828 ± 2 cm–1 [2Πg,3/2] and 87 648 ± 2 cm–1 [2Πg,1/2].7

The current state of knowledge regarding the excited electronic states of Br2+ is limited.

Information regarding the A state of Br2+ has been derived from photoelectron (PE) spectroscopy,8,9via analysis of the A → X dispersed emission spectrum following fixed frequency excitation10 and, at higher resolution, by fluorescence excitation11 and velocity modulation absorption spectroscopy.12

All of these studies identified the A state as an inverted 2Πu state, but also found surprising differences in the vibrational frequencies and anharmonicity constants associated with the two spin–orbit components.

This quite striking spin–orbit dependence of the vibrational frequencies was rationalised, qualitatively at least, by two different sets of ab initio calculations.13,14

These showed the adiabatic potentials for the Ω = 3/2,u and Ω = 1/2,u states, after due allowance for correlation effects and spin–orbit interactions, to be very different.

Specifically, in the case of the Ω = 1/2,u state both sets of calculations indicated a dominant role for the configuration 2340(2Π1/2,u) in the vertical Franck–Condon region but also revealed an avoided crossing with the 4Σ–u state13 and/or the 2Σ+u14 states arising from the (2421) configuration at slightly larger internuclear separations.

The resulting potential for the adiabatic Ω = 1/2,u state is thus flatter (and the levels it supports more closely spaced) than that for the 2Π3/2,u state which interacts, at more extended geometries, with the 2421(2Δ3/2,u) state.

Our previous imaging studies of the photodissociation of state-selected Br2+ cations following one photon absorption focussed on the energetics of the fragmentation process,5 but also revealed striking wavelength dependent variations in the recoil anisotropy of the ground state Br + Br+ products.

Following convention,15 this recoil anisotropy can be quantified in terms of an anisotropy parameter, β, defined via the following expression for I(v,θ), the variation in product flux (with recoil velocity v) as a function an θ, the angle between the detection axis and the electric vector of the photolysis laser radiation, εphot, I(v,θ) ∝ f(v){1 + βP2(cos θ)}.f(v) is the speed distribution of the detected photofragment and P2(x) = (3x2 – 1)/2 is a second order Legendre polynomial.

Here we report the results of further systematic studies of the way in which the fragment recoil anisotropy varies with photolysis wavelength, for Br2+ molecules in both spin–orbit components of the 2Πg ground state and with both v″ = 0 and 1.

The plots of βversus photolysis wavelength so obtained show clear resonance structure.

This is found to be concentrated in the energy gap between the ground [Br(2P3/2) + Br+(3P2)] and first excited [Br(2P3/2) + Br+(3P1)] dissociation limits, and to converge with increasing energy in a manner consistent with it being associated with a series of predissociating vibrational levels in a bound potential that correlates with the first excited dissociation asymptote.

To understand this resonance structure better we have conducted further ab initio electronic structure calculations for all ungerade excited states of Br2+ associated with the …σgπuπgu* valence space, incorporated spin–orbit effects retrospectively, using a semi-empirical approach, and then propagated wavepackets on the resulting coupled potential energy curves.

These model calculations successfully reproduce all of the experimental observations, and provide much new insight into the non-adiabatic couplings amongst the various excited states of this textbook open-shell system.


The velocity map imaging spectrometer used in this work has been described previously.16

A pulsed, skimmed, supersonic beam of Br2 molecules (50 Torr seeded in 1 atm of Ar carrier gas) directed along the Z-axis, was crossed at right-angles by the output of two pulsed tuneable lasers that counter-propagate along the X-axis.

Laser 1, an Nd:YAG pumped frequency doubled dye laser (Spectra-Physics GCR-250 plus Sirah Cobra Stretch, output bandwidth <0.1 cm–1 in the visible), was used to prepare Br2+ ions by one colour 2 + 1 REMPI.

Wavelengths (in air) and the corresponding vacuum wavenumbers of all photons used in this study were measured using a wavemeter (Coherent, WaveMaster).

Ion preparation wavelengths used here were λprep ∼ 272.967 nm, 263.012 nm, 274.318, and 264.333 nm.

The first two of these are resonant (at the two photon energy) with the v′ = 1 levels of the Rydberg states of 79Br2 labelled [3/2]4d;0g and [1/2]4d;0g state, respectively,17 while ionisation at the latter two wavelengths benefits from (non-isotopomer selective) resonance enhancement by the v′ = 0 levels of the same two states.

These shorthand descriptors imply that the resonance enhancing Rydberg states comprises a d electron orbiting the 2Πg ion core with Ω = [3/2] or [1/2], that the quantum number for the total electronic angular momentum along the bond axis, Ω = 0, and that the overall state symmetry is gerade.

The resulting ions were photolysed with the output of laser 2, a second tuneable laser system (Spectra-Physics GCR-170 plus PDL-2 dye laser), which provided linearly polarised radiation in the wavelength range 333 < λphot < 379 nm (for photolysis of 2Πg,3/2 parent ions) and 336 < λphot < 425 nm (for 2Πg,1/2 ions), with εphot perpendicular to the molecular beam axis and parallel to the plane of the detector (εphot‖Y).

Typical pulse energies employed in the present studies were 10–25 µJ and 100–150 µJ for lasers 1 and 2, respectively.

Both pulses were focussed into the interaction region with 20 cm f.l. lenses, with the laser pulse used to induce ion fragmentation delayed ∼5 ns relative to the Br2+ ion preparation pulse.

The resulting ions were extracted, along Z, under velocity map imaging conditions using electric fields custom designed so as to optimise the spectrometer resolution.

860 mm downstream from the interaction volume the ion cloud impinges on the front face of a position sensitive detection system consisting of a pair of microchannel plates and a phosphor screen, which is read out by a CCD camera equipped with a fast intensifier (Photonic Science) that is gated to the time-of-flight (TOF) of 79Br+ ions.

As in our previous Br2+ photolysis study5 it was necessary to discriminate against unwanted background 79Br+ signal arising from laser 1 alone.

This was achieved by inserting a Pockels cell (Cleveland Crystals, model QX 1020 comprising a KD*P crystal, no index matching fluid and enclosed by uncoated spectrosil B windows) and a linear polarizer in the beam path immediately before the lens with which λphot is focused into the interaction region.

The linear polariser was aligned so as to transmit λphot in its ‘natural’ polarisation state, i.e. with εphot‖Y.

Application of the appropriate half-wave retardation voltage, Vλ/2, for the KD*P crystal at the photolysis wavelength of interest causes εphot to be rotated through 90°; in this state the fixed polarizer blocks λphot.

In this way, λphot was switched on and off on successive laser shots, synchronous with the 10 Hz repetition rate of lasers 1 and 2.

Each ion image resulting from a single laser shot was processed with an event counting, centroiding algorithm provided with the commercial camera software DaVis (LaVision) running on a PC, and the resulting counts accumulated in two alternate buffers according to whether λphot was present or not.

Our initial investigations of the photolysis wavelength dependence of β simply involved accumulating signal from typically 104 laser shots at any one λphot value in the two buffers, computing the difference image, then reconstructing the corresponding 3-D velocity distribution as described previously5,16 using an algorithm based on the filtered back-projection method of Sato et al.,18 and then repeating the whole process at quite widely spaced wavenumber intervals.

This provided a gross overview of the way in which the recoil anisotropy of the various product channels varied with λphot, but also hinted at the presence of much finer structure in the βversusλphot spectrum.

A faster data collection procedure was required in order to investigate this fine structure.

In the subsequent generation of experiments, images for each status at any given λphot were thus accumulated for just 300 laser shots (the minimum number that was found, experimentally, to be needed in order to generate an image of sufficient signal to noise to allow definition of the anisotropy parameter, β, to a precision of ∼0.1).

The images so obtained were directed to a second computer, and the 3-D velocity distribution reconstructed from the difference image, ‘on-the-fly’, while the PC controlling the experiment stepped the wavenumber of the photolysis laser by a user specified amount, and began accumulating the next data set.

The results of many such accumulations are spectra of β as a function of photolysis wavelength, as illustrated in the following section.

Results and discussion

The remainder of this paper is divided into sub-sections devoted to, in turn, images of the Br+ fragments and the analysis of such images and their wavelength dependence, high quality ab initio calculations for the lower lying ungerade states of Br2+ that could contribute to absorption from the X 2Πg ground state, the inclusion of spin–orbit effects (in both the parent ion and the asymptotic products) and the diabatisation of the ab initio potentials using a semi-empirical approach and, finally, interpretation of the experimental observations in the light of wavepacket propagations using the spin–orbit resolved potentials so derived.

Product recoil anisotropy spectra

Fig. 1 shows difference images of the 79Br+ fragments resulting from 79Br2+ parent ion formation by 2 + 1 REMPI at 263.012 nm (38 009.7 cm–1) and subsequent photolysis at phot = 23 900, 26 000, 28 100 and 29 800 cm–1, respectively.

As shown previously,5 REMPI at this excitation wavelength yields X2Πg,1/2 ions in their v+ = 1 vibrational level; dissociation of these parent ions to form ground state products (process (2)) accounts for the ring of greatest radius in each image.

79Br2+(2Π1/2,g)v+=1 +  → 79Br+(3P2) + 79Br(2P3/2)

This ring expands with increasing photon energy and additional rings appear in the images.

In order of increasing appearance threshold, most of these extra features can be associated with, respectively, the equivalent fragmentation originating from the ground spin–orbit state of the parent ion 79Br2+(2Π3/2,g)v+=1 +  → 79Br+(3P2) + 79Br(2P3/2), and the analogues of process (1) and (2) but yielding 79Br+ ions in their spin–orbit excited 3P1 and 3P0 states, i.e.79Br2+(2Π1/2,g)v+=1 +  → 79Br+(3P1) + 79Br(2P3/2)and79Br2+(2Π3/2,g)v+=1 +  → 79Br+(3P1) + 79Br(2P3/2),and79Br2+(2Π1/2,g)v+=1 +  → 79Br+(3P0) + 79Br(2P3/2)and79Br2+(2Π3/2,g)v+=1 +  → 79Br+(3P0) + 79Br(2P3/2).

Also shown in Fig. 1 are 2-D slices through the corresponding reconstructed 3-D velocity distributions and the speed distributions derived therefrom.

Peaks in the respective speed distributions attributable to the various dissociation pathways are labelled accordingly.

Most of the additional weak features can be attributed to excitations from the v+ = 0 and 2 levels of 79Br2+(2Π1/2,g) which are formed with low probability in the 2 + 1 REMPI process.

The weak faster ring in the 23 900 cm–1 image, for example, is due to photolysis of 79Br2+(2Π1/2,g)v+=2 parent ions to yield ground state products.

A couple of points regarding the high velocity resolution of these fragment ion images merit reiteration.

Firstly, preparation of the 79Br2+(2Π1/2,g)v+=1 parent ions involves 2 + 1 REMPI at a wavelength within the predissociation broadened Q branch of the [1/2]4d;0g ← X;0g (v′ = 1 ← v″ = 0) two photon transition.

The relatively narrow homogeneous width of each of the Q branch lines ensures that excitation at the peak of the band envelope will select a relatively narrow sub-set of rotational levels (centred at J″ ∼ 5) from within the full Boltzmann distribution of populated rotational states at the prevailing beam temperature (estimated at 12 K).

The observation that all peaks in the speed distributions are well described by single narrow Gaussian functions can then be understood most readily by assuming that the final one photon ionisation step from the [1/2]4d;0gv′ = 1 level exhibits a high propensity for transitions that conserve the rotational angular momentum of the core, thereby resulting in formation of ions in a narrow distribution of rotational states with quantum number N+ ∼ 5.

Second, in contrast to traditional ion imaging experiments, wherein the neutral fragment of interest is ionised by REMPI in order that it can be imaged, the fragment of interest in the present study is created as an ion.

Thus, providing that the parent ion densities are kept sufficiently low to preclude measurable space charge effects, the observed ion recoil distribution should arise solely from the parent ion fragmentation of interest, without any of the blurring due to the mutual recoil of the ion and the photoelectron that can become significant when the 2 + 1 REMPI results in significant photoelectron kinetic energy release.19

The displayed images exhibit variations in the angular anisotropy of any given ring with phot.

This is illustrated at one level of detail in Fig. 2, which shows the angular dependence of the signal associated with the fastest velocity sub-group in each of the four images shown in Fig. 1, and is summarised in the global plot of βversusphot shown in Fig. 3.

All of the measured angular distributions are described well by eqn. (1).

Such is consistent with, though not definitive justification for, an assumption that prevails throughout the present analysis – namely, that the depolarising effects of the nuclear spin and the time delay between formation and photolysis of the parent ion will suffice to negate any possible alignment of the Br2+ ion formed in the REMPI process.

The data points in Fig. 3 were obtained by analysing many such images recorded at quite widely spaced (typically 300 cm–1) intervals in phot.

We first concentrate on the outermost ring (channel (2)).

Just above threshold (phot ∼ 23 160 cm–1), the associated β parameter is close to zero.

This is simply a reflection of the fact that the image size is comparable to the (finite) image resolution, possibly compounded by any breakdown of the axial recoil approximation.20

As soon as the image radius becomes large relative to the experimental resolution, β exhibits a value close to +2, characteristic of excitation via a parallel transition (i.e. an excitation in which the transition moment lies parallel to the internuclear axis).

As phot increases, however, β is seen to decline, to ∼0.5 at phot ∼ 28 100 cm–1, before starting to increase again.

Channel (4), the energetic onset of which is at phot ∼ 26 300 cm–1, shows generally similar behaviour.

In this case, however, β declines to a value approaching –1 (characteristic of excitation via a pure perpendicular transition) at phot ∼ 28 100 cm–1, as is clearly evident in the image displayed in Fig. 1(c), before again beginning to increase.

The corresponding βversusphot plots obtained when photolysing (non isotopomer selected) Br2+(2Π1/2,g)v+=0 parent ions show no comparable node: in this case the measured β for the channel leading to products (2) is ∼1.2 across the investigated range 25 400 ≤ phot ≤ 29 000 cm–1, while β for the channel yielding products (4) is seen to increase gently from a value of ∼0.3 at phot = 26 900 cm–1 to ∼0.7 at phot = 29 000 cm–1.

Analysis of such images also allows determination of the photolysis wavelength dependence of the branching fraction Γ, here defined as where σBr+Br+ and σBr+Br+* are the relative yields of ground (channel (2)) and spin–orbit excited Br+(3P1) (channel (4)) products arising in the photodissociation of vibrational state selected Br2+ ions.

This branching fraction is found to remain relatively constant, at Γ ∼ 0.42, for Br2+(2Π1/2,g)v+= 0 parent ions photolysed in the range 26 900 ≤ phot ≤ 29 000 cm–1.

For Br2+(2Π1/2,g)v+=1 ions, Γ is ∼0.25 at phot = 26 900 cm–1, increases gradually to ∼0.5 at phot = 28 700 cm–1, and thereafter declines to ∼0.4 at phot = 29 600 cm–1 (the highest wavenumber investigated).

More careful investigations reveal the presence of high frequency structure within the βversusphot plots, specifically in the wavenumber region between the ground and first excited dissociation limits.

This is illustrated in Figs. 4 and 5, which show βversusphot scans for the ground state product channel (2) following excitation of Br2+(2Π3/2,g) parent ions in, respectively, their v+ = 0 and 1 levels.

Both spectra were obtained by recording and analysing difference images recorded at 3.6 cm–1 wavenumber intervals.

Both spectra display a similarly complex pattern of resonances that, superficially at least, appear as dips against a positive background close above the first dissociation threshold but are more reminiscent of an interference or ‘beat’ pattern at higher phot.

Focussing (arbitrarily) on the minima in these spectra, we find that the inter-resonance intervals decrease essentially linearly with increasing phot (i.e. as for a classic Birge–Sponer plot) – suggesting that the resonances should be viewed as a series of vibrational levels in a bound, but predissociating, excited state potential.

Linear extrapolation of the v+ = 0 data set suggests a convergence limit ∼3200 cm–1 above the ground state dissociation energy (cf. the 3136.4 cm–1 separation between the ground (3P2) and first excited (3P1) spin–orbit states of Br+).

Comparison between the spectra shown in Figs. 4 and 5, after offsetting the latter by the v+ = 1 – v+ = 0 wavenumber separation, confirms that they reveal a common set of resonances.

Analogous resonances can be observed when photolysing spin–orbit excited Br2+(2Π1/2,g)v+=0 ions.

This is illustrated in Fig. 6, for the case of ground state 81Br+ fragment ions.

The displayed spectra reveal an apparent anti-correlation between maxima in the βversusphot plot obtained by photolysis of Br2+(2Π3/2,g)v+=1 parent ions and minima in the corresponding plot obtained by exciting Br2+(2Π1/2,g)v+=1 ions.

A more detailed discussion of the nature of the predissociating resonances revealed in these spectra is reserved until later, pending derivation of potential energy curves for the various ungerade excited states of Br2+.

Before concluding this Section, however, it is worth considering the β dependent sensitivity of βversusphot plots and their relationship to absorption versusphot spectra with which we are all more familiar.

The following definition of β where σ and σ are the probabilities of, respectively, parallel and perpendicular excitations leading to the fragment states of interest, provides a convenient starting point.

Given eqn. (9), it follows that and that Eqns. (10) and (11) highlight the non-linear nature of βversusphot plots, which show maximum sensitivity to fractional changes in σ or σ when β = 0.5 but, as expected, are insensitive to small changes at the limiting values of β.

Ab initio calculations

A series of high-level ab initio calculations have been performed using the MOLPRO 200221 electronic structure package running on a Beowulf cluster of networked PCs.

All of the calculations were performed in the Abelian D2h sub-group of the full D∞v point group in which the symmetry species Σ+g, Πu, Σ–g, Σ+u, Πg, Σ–u are represented by Ag, B3u + B2u, B1g, B1u, B2g + B3g and Au, respectively.

A spin restricted Hartree–Fock calculation was first performed on the ground (1Σg+) state of the neutral Br2 molecule, and the Hartree–Fock orbitals so derived were used as the starting point for a complete active space self consistent field (CASSCF)22,23 calculation on the ground (2Πg) state of the Br2+ molecular ion.

Following this, for each internuclear separation a further CASSCF calculation, state averaged over states with the same spin and symmetry was performed.

The active space in these calculations was taken to be the full valence space of [core](1σg)2(1σu*)2(2σg)2(1πu)4(1πg*)3(2σu*)0, occupied by 13 electrons.

The core orbitals, although constrained to be doubly occupied in all configurations, were fully optimised.

An internally contracted multi-reference configuration interaction (MRCI)24,25 calculation was then performed to include the effects of dynamical electron correlation, which allowed the determination of accurate adiabatic potential energy curves.

The Davidson correction26 was applied to the MRCI energies in order to account for the contribution that quadruple excitations make to the correlation energy.

Scalar relativistic corrections to the two-electron integrals were included using the Douglas–Kroll approximation.27

All of the above calculations were performed using the contracted correlation-consistent polarised valence quadruple zeta (cc-pVQZ) basis set developed by Dunning and co-workers.28–30

The diagonal spin–orbit matrix elements were calculated for both the ground (X 2Πg) and the excited A 2Πu electronic state, around their respective equilibrium geometries, using MRCI wavefunctions.

The full Breit-Pauli spin–orbit Hamiltonian was used to calculate matrix elements between configuration state functions in which no external (virtual) orbitals were occupied.

For external configurations, a mean field analogue was employed.

The generally contracted basis set employed in the earlier spin–orbit averaged MOLPRO calculations could not be used to evaluate the spin–orbit integrals; the segmented cc-pVTZ basis set obtained from the EMSL basis set library31 was used instead.

Fig. 7 shows the spin–orbit averaged MRCI adiabatic potential energy curves for the X 2Πg ground state and the first eight excited ungerade states of Br2+ which can arise from a one-electron excitation within the valence space (the specific electronic configurations are shown in Table 1).

Also shown are the calculated spin–orbit resolved sub-states of the X 2Πg state.

The general forms of these various spin–orbit averaged potentials, their relative energetics and their respective equilibrium bond lengths are in good accord with the earlier calculations of Boerrigter et al.14

To compare the calculated well depth with the experimentally determined ground state bond dissociation energy, De, it is necessary to remove the effects of the spin–orbit averaging.

The minimum of the X 2Πg,3/2 spin–orbit state will lie below that of the spin–orbit averaged potential by an energy equal to half of the calculated spin–orbit splitting, A. The asymptotic energy also needs to be corrected, since the calculated asymptote lies at the average energy of the Br+(3PJ) + Br(2PJ) atomic limit.

This correction factor (C) is given by where the term energies E are taken from the NIST Atomic Spectra Database32 and the pre-multipliers are the appropriate (2J + 1) electronic degeneracy factors.

The suitably corrected value for the calculated dissociation energy De[Br2+(2Πg,3/2)] will thus be given by: This spin–orbit resolved well-depth (eqn. (13)) agrees very well with the experimentally derived value De[Br2+(2Πg,3/2)] = 26 528 cm–1 (ref. 5) as does the calculated spin–orbit splitting, A = –2819 cm–1 (cf. the experimental value of –2817 ± 4 cm–1).

The calculated value of the spin–orbit splitting for the A 2Πu state (A = –2274 cm–1) is in good accord with early estimates obtained from analysis of the He I photoelectron spectrum of Br233 and of the Br2+(A → X) dispersed emission spectrum,10 but contradicts more recent estimates obtained via threshold PE spectroscopy of Br2(∼–3460 cm–1 (ref. 9)) and from analysis of high resolution Br2+(A ← X) absorption spectra (–710 cm–1 (ref. 12)).

Incorrect assignment of the upper state vibrational quantum numbers appears to be the most likely explanation for both sets of discrepancies.

We have also calculated transition dipole moments from the ground state to various of the excited states, as a function of internuclear separation r.

Absorption to the A2Πu state is dominant, with a transition dipole moment of ∼2 D at the ground state equilibrium geometry (re ∼ 2.19 Å).

By way of comparison, the corresponding transition dipole moments to the 2Δu, 2Σ+u and 2Σ–u excited states at the same value of r are calculated to be, respectively, ∼0.01, ∼0.05 and ∼0.03 D.

Inclusion of spin–orbit effects using a semi-empirical approach

The assignment of the resonance structures revealed in section 3.1 requires knowledge of the asymptotic behaviour of the molecular states at large r.

The Br(core,4p5) + Br+(core,4p4) combination leads to a total of 90 spin–orbit sub-states for each of g and u inversion symmetry.

Inclusion of all of these for the full range of internuclear separation was not possible with the available ab-initio software.

However, the spin–orbit matrix elements will be dominated by one-centre integrals that can be taken semi-empirically from the known atomic term values.

A model hamiltonian was therefore constructed, in which the ab-initio spin–orbit averaged potential energy curves of section 3.2 were first simulated using empirical matrix-elements, as functions of r, in a basis of coupled products of atomic wave-functions used with spin–orbit averaged term values (eq. (12)).

The calculations were then repeated with the inclusion of the atomic splittings.

Note that in this and the following section the energies are expressed relative to that for the lowest dissociation limit (2P3/2 + 3P2) which, henceforth, we shall define as E = 0 cm–1.

The sub-states for Br(core,4p5)2P can be represented by Russell-Saunders coupling with a spin–orbit splitting of –3685.24 cm–1 between 2P3/2 and 2P1/2 (ζ = –2457 cm–1).

However, for Br+(core,4p4) the term values of E(3P2) = 0 cm–1, E(3P1) = 3136.4 cm–1, E(3P0) = 3837.5 cm–1 depart significantly from the Landé interval rule, requiring an intermediate coupling basis34 for 3P, 1D and 1S with ζ = –2806 cm–1 and F2 = 1581 cm–1.

For both g and u inversion symmetry the molecular basis can be partitioned into 21 functions for Ω = ±1/2, 15 functions for Ω = ±3/2, 7 functions for Ω = ±5/2 and 2 functions for Ω = ±7/2, each consisting of from 2 to 20 orbital products.

For example, the combination of 2P3/2(m = 1/2) and 3P1(m = 0) gives the following Ω = 1/2,u molecular function: in which A and B denote the two Br atoms, and each primitive basis function and the m quantum numbers are expressed for compactness in terms of the holes in the complete atomic shells rather than the occupied orbitals.

The molecular hamiltonian matrix made use of a zero-differential overlap (ZDO) approximation in a hybrid LCAO model.

The sums of the Br and Br+ atomic term values contributed to the diagonal elements in the coupled basis, whereas all molecular terms were expressed in terms of 8 functions of r in the primitive basis of coupled orbital products.

Coulomb integrals α(r) for each pσ or pπ electron were represented by Rydberg functions35 with three parameters.

Valence bond exchange integrals A(r) for each sigma or pi electron pair, and molecular two-centre pσ,pπ and pπ,pπ′ exchange integrals, were represented by simple exponential functions (two parameters each).

Finally, resonance integrals β(r) for single uncoupled pσ or pπ electrons were also represented by simple exponential functions (two parameters each).

The 18 parameters in this model were optimised by least squares fitting to the ab initio spin–orbit averaged potential energy curves for the states denoted as 2430(2Πg), 2340(2Πu), 2421(4Σ–u, 2Δu, 2Σ+u and 2Σ–u) and 1431(4Πu), which represents the minimum set of states necessary to determine all these parameters: (most higher states have very mixed configurations).

Any portion of a potential curve lying higher than 0.07 hartrees above the lowest dissociation asymptote was excluded from the fit, which gave a standard deviation of 0.004 hartrees (880 cm–1).

These semi-empirical potential curves repeat the pattern of Fig. 7 for the relative energies and crossing points of states of differing Λ, S symmetry.

With inclusion of atomic spin–orbit coupling the Br2+2Πg splitting becomes –2744 cm–1, i.e. deviating from the experimental value by <3%, thus lending validity to the model hamiltonian.

The calculations also reproduce the complicated pattern of avoided crossings around the minimum energy for the lower u spin–orbit sub-states, as calculated by Boerrigter et al. using a limited CI expansion and spin–orbit matrix elements taken over from a Hartree–Fock–Slater calculation.14

We now examine these potential curves over the full range of internuclear separation.

Interpretation of these curves was aided by evaluation of the radial variations of the electronic coupling matrix elements 〈ψi|∂/∂r|ψj〉, which exhibit localised peaks highlighting avoided crossings in a diabatic Λ, Σ, Ω basis.

Fig. 8 shows the three lowest adiabatic curves for Ω = 3/2,u (solid lines) and the lowest curve for Ω = 5/2,u (dashes).

Vertical lines join pairs of Ω = 3/2,u curves at the points of maximum change of diabatic character, indicating avoided crossings of 4Σ–u with 2Πu at r = 2.20 Å, and of 2Πu with 2Δu at r = 2.76 Å.

The calculated interstate separations at these avoided crossings are 1291 cm–1 and 1817 cm–1 respectively, both of which are sufficiently large that the dynamics for these states will be predominantly adiabatic.

Note that the curve for 2Δ5/2,u is almost degenerate with the third 3/2,u curve at short r, but with the second 3/2,u curve at long r, in keeping with the strong avoided crossing for 3/2,u.

All of these curves converge to the lowest dissociation limit (2P3/2 + 3P2), although the highest has a very small barrier at large r that is much too low to be responsible for the observed resonance structure.

All other 3/2,u and 5/2,u states lie too high in energy to be of relevance in the present context.

The pattern of potential curves for Ω = 1/2,u (Fig. 9) is more complicated than for Ω = 3/2,u and 5/2,u.

The diabatic 2Πu curve is subject to three avoided crossings with 2Σ+u at r = 2.59 Å (ΔV = 1232 cm–1), with 2Σ–u at r = 2.99 Å (ΔV = 695 cm–1), and a complex weaker interaction between r = 3.28 and 3.76 Å with the lowest of a group of repulsive curves of mixed configurations (ΔV = 326 cm–1).

Furthermore the 2Σ+u and 2Σ–u states (which cross near r = 2.4 Å in the spin–orbit averaged ab initio calculations, Fig. 7) become strongly mixed in this region due to their mutual interactions with the 2Πu diabatic state.

The lowest four adiabatic curves converge on the lowest dissociation limit (2P3/2 + 3P2), whereas the fifth (principally 2Πu at long range) converges to the second dissociation limit (2P3/2 + 3P1).

Thus this latter curve has the properties necessary to support transient bound vibrational states converging on this higher limit, which could be predissociated by non-adiabatic interaction with the lower states.

In the diabatic limit, parallel (ΔΩ = ΔΛ = 0) transitions from the X 2Πg ground state of Br2+ to the lower excited states will be allowed for 2Π3/2,g → 2Π3/2,u and 2Π1/2,g → 2Π1/2,u.

Fig. 8 indicates that the vertical parallel transition for the Ω = 3/2 adiabatic states lies at E ∼ –4000 cm–1, notwithstanding the fact that the 2Π3/2,u and 4Σ–3/2,u states are strongly mixed in the Franck–Condon region (r″e = 2.19 Å).

For the Ω = 1/2 adiabatic states (Fig. 9) the vertical parallel transition lies at E ∼ –1800 cm–1.

In both cases, therefore, the parallel component of the transition intensity for excitation from low v″ levels of the X state will be tailing off around this dissociation limit.

Perpendicular (ΔΩ = ΔΛ = ±1) transitions in the diabatic limit are allowed for 2Π3/2,g → 2Δ5/2,u and 2Π1/2,g → 2Δ3/2,u, and from both 2Π3/2,g and 2Π1/2,g to 2Σ–1/2 and 2Σ+1/2,u.

(Note that in terms of signed values of Ω the latter are 2Π±1/2,g to 2Σ–∓1/2,u and 2Σ+∓1/2,u).

The corresponding vertical transitions from low v″ levels to the adiabatic states all peak at E > 0.

Thus, despite the low integrated intensity of the perpendicular transitions, it is to be expected that the anisotropy parameter β will show a generally downward trend with increasing energy above the dissociation threshold, as observed.

Wavepacket propagations on the spin–orbit resolved potentials

It is apparent from the above discussion that the Ω = 3/2,u and 5/2,u curves cannot support the levels responsible for the observed resonance structure.

Attention is therefore restricted to calculating the spectrum of Ω = 1/2,u states, using time-dependent wavepacket propagations.

It is most convenient for this purpose to use a hamiltonian based on diabatic states, as this involves potential coupling between states (multiplicative) rather than dynamic coupling (differential) as required for an adiabatic basis.

The 4Σ–1/2,u state has no crossings with other states, and was not considered further.

Diabatic curves for the 2Π1/2,u, 2Σ–1/2 and 2Σ+1/2,u states were represented by Rydberg functions35 of the form: V(r) = –De{1 + c1(r – re)}exp{c1(r – re)}, with localised off-diagonal coupling terms deduced from the calculated values of 〈ψi|∂/∂r|ψj〉 and the adiabatic separations (ΔV) at the diabatic crossing points.

These were optimised by least-squares refinement (Fig. 10).

The weak coupling between 2Π1/2,u and the repulsive state was then dropped from consideration (but see below).

Vibrational wavefunctions for v″ = 0 or 1 were used as initial wavepackets, and projected in turn onto each of the 2Π1/2,u, 2Σ–1/2 or 2Σ+1/2,u diabatic curves.

Propagation was continued for 2000 time steps to τ = 4.24 ps.

Excitation spectra were calculated from the auto-correlation functions, and action spectra for dissociation from correlation functions monitored at r = 6.4 Å for exit on each of the three diabatic potentials.36

In general after this time interval there remained an oscillating wave associated with the fully bound vibrational states below the dissociation limit, whereas the flux in the exit channels had almost vanished.

Excitation from v″ = 0

Following excitation to 2Π1/2,u the outgoing wavepacket follows mainly adiabatic dynamics on the lowest state (potential I), as expected given its large energetic separation from adiabatic potential II.

The absorption spectrum calculated from its auto-correlation function consists of transitions to about 50 vibrational levels converging on the lowest dissociation limit, with peak intensity at E ∼ –2000 cm–1, followed by an exponentially decaying continuum for a further 4000 cm–1 (Fig. 11(a)).

These levels and the continuum are supported by the adiabatic potential I. The action spectrum for dissociation consists of this same continuum, with a superimposed series of very weak narrow peaks indicating some (weak) sampling of the higher potentials (Fig. 12(a)).

The spectra following excitation to 2Σ–1/2 are dramatically different.

The absorption spectrum peaks in a continuum above the second dissociation limit, at E ∼ 3800 cm–1 (Fig. 11(b)).

The spectrum at E < 0 cm–1 exhibits a long series of weak peaks that are energetically coincident with those observed for excitation to potential I (Fig. 11(a)).

At energies above –1500 cm–1 these are interspersed by a number of additional peaks associated with a higher potential curve.

Between the first and second limits (calculated energy separation ΔE = 3139 cm–1) there is a series of strong peaks converging on the higher limit, but with unusual line profiles.

Close to E = 0 cm–1 the peaks are differential in shape, first rising then falling with increasing frequency.

At E ∼ 750 cm–1 the peaks are symmetrical, whereas at still higher energies the peaks are again differential but with opposite phase.

This latter pattern is repeated in the action spectrum for dissociation (Fig. 12(b)).

Examination of wavefunctions driven into resonance at some of these resonant frequencies demonstrates that they are supported by adiabatic potential III.

The interaction between states supported by potentials II and III is stronger than that between potentials I and II, for two reasons.

Firstly, the minimum energy separation at the avoided crossing is less than half the corresponding energy separation between potentials I and II.

Secondly, the difference in the slopes of potentials II and III is much smaller than that between potentials I and II in the relevant interaction regions.

As we show later, these calculations provide a detailed understanding of the spectra observed experimentally.

But first, why do the profiles of the peaks vary so much?

For this we must look to the Beutler–Fano theory of interfering resonances,37 in which each lineshape is described by a parameter qv.

qv is a function of three matrix elements; the transition moments to the bound state 〈ψbχv|µ|ψ0χ0〉 and to the resonant component of the continuum 〈ψcχc|µ|ψ0χ0〉, and an interaction matrix element 〈ψbχv|H|ψcχc〉: The transition moments are determined in the Franck–Condon region, and are smooth functions of v′.

The interaction term, in contrast, is determined in the region of the avoided crossing where the nuclear wavefunctions χv(r) and χc(r) are highly oscillatory, and beat against each other because of the difference in kinetic energy.

In the adiabatic representation this term can be partitioned as38 where the inner integral was determined above and is a localised function (approximately Lorentzian) around the avoided crossing.

This matrix element was evaluated as a function of the vibrational energy using a WKB approximation for the radial wavefunctions, and is displayed in Fig. 13.

(For fully bound vibrational levels, Ev < 0, this interaction results in perturbations, rather than predissociation).

Note that this matrix element changes sign around E = 750 cm–1, being negative below and positive above value.

Thus qv will be negative at E < 750 cm–1, which results in a differential line-shape +ve to –ve, infinity at E = 750 cm–1, which gives a normal Lorentzian line-shape, and positive at E > 750 cm–1, resulting in a differential line-shape –ve to +ve, as in Fig. 12(b).

The precise details of the line-shapes and their variations with E are very sensitive to the shapes of the potential curves II and III over the whole region from the inner radial turning points to beyond the avoided crossing, and to the relative moments of the two possible perpendicular transitions.

The spectrum following excitation to 2Σ+1/2,u contains the same general features as for 2Σ–1/2,u, but the peak intensity is shifted to E ∼ 7300 cm–1 (Fig. 11(c)).

It shows a progression of weak vibrational peaks between the first two dissociation limits, which all have normal line-shapes since they are excited directly (qν → ∞), but their intensities are too low to make any significant contribution to the overall spectrum.

Excitation from v″ = 1

The lower state vibrational wavefunction χ1(r) has a node and spans a wider range of r than for v″ = 0.

These features are reflected in the Franck–Condon intensity distributions.

Other than this, the calculated spectra possess all the same features as described above for v″ = 0 (see Fig. 14, and also Figs. 12(c) and 12(d)).

However, the degree of overlap between the spectra arising from transitions from χ1(r) to 2Σ+1/2,u and to 2Σ–1/2 is now much greater in the region of interest.

This raises the possibility of a further interference, between the 2Πg → 2Σ+1/2,u and 2Πg → 2Σ–1/2,u transition amplitudes.

Further initial wavepackets were therefore constructed in which excitation was to both curves II and III, first in-phase and then out-of-phase.

The calculated action spectrum following out-of-phase excitation was found to have the same general appearance as in the case of excitation to potential II only (i.e. similar to Fig. 12(d)), albeit with some changes in relative peak intensities.

In the case of in-phase excitation, however, the differential line-shapes, and thus qv, were found to be reversed in the energy range (1500 < E < 3000 cm–1) where the two excitations overlap significantly.

We return to this point in the next section.

Finally we return to consideration of the outermost avoided crossing, which has been neglected in the above calculations.

This is the weakest interaction of all, for which the diabatic approximation will be the best zero-order model.

Consequently its effect is to cause local perturbations both in the frequencies and the intensities of peaks at E ∼ 1500 cm–1.

Such interactions are likely to account for the observed irregularity of spacings in the experimental spectra ∼1000 cm–1 below the second dissociation limit.

The anisotropy parameter β

2Π3/2,g → 2Π3/2,u and 2Π1/2,g → 2Π1/2,u excitations both give rise to long progressions of parallel transitions with intensity distributions that peak within the bound regions of the respective upper state potentials, with the peak for 2Π3/2,u lying ∼2000 cm–1 further below the dissociation threshold than that for 2Π1/2,u.

Above this limit both progressions merge into continua that fall in intensity with increasing frequency.

As Figs. 12(a) and 12(c) show, the 2Π1/2,g → 2Π1/2,u continuum carries a progression of weak resonances as a result of coupling with bound Ω = 1/2,u levels supported by potential III.

The corresponding 2Π3/2,g → 2Π3/2,u continuum displays no such resonances, of course, since the calculation includes no bound Ω = 3/2,u levels at these energies with which it might interfere.

In contrast, 2Π3/2,g → 2Σ–1/2 and 2Π1/2,g → 2Σ–1/2 excitations both give rise to structured perpendicular transitions, rising in intensity with increasing frequency.

The transitions 2Π3/2,g → 2Δ5/2,u and 2Π1/2,g → 2Δ3/2,u will make further unstructured perpendicular contributions to the continuum in the same spectral regions as transitions to 2Σ–1/2.

The ab initio transition moments indicate that the integrated intensities of the parallel transitions are much higher than for the perpendicular transitions (section 3.2).

Thus theory would predict that the recoil anisotropy parameter, β, for the ground state products should fall with increasing frequency, and that it should display sharp resonance structure with opposite phases to those observed in absorption.

Qualitatively, this is precisely what is observed, as is the variation with energy of the resonant line-shapes.

All of the experimental spectra (Figs. 3, 4 and 5) show an overall downward trend in β value with increasing energy, though the reversal of the trend in Fig. 3 at  > 28 100 cm–1 suggests the involvement of higher states not considered here.

Quantitatively the ratio of these transition intensities, and thus β, must depend on the respective transition dipole moments and Franck–Condon factors and thus on the values of both Ω″ and v″.

The tail of the Franck–Condon intensity distribution for the 2Π1/2,g → 2Π1/2,u parallel transition will extend further above E = 0 cm–1 than will the tail associated with the 2Π3/2,g → 2Π3/2,u transition.

Furthermore, for both values of Ω″, this parallel contribution to the total transition intensity will extend further for v″ = 1 than for v″ = 0 (compare Figs. 11(a) and 14(a) for Ω = 1/2).

Thus β would be predicted to be highest near this threshold for the case of excitation from Ω″ = 1/2 and v″ = 1, and higher for the case of excitation from Ω″ = 3/2, v″ = 1 than when exciting from Ω″ = 3/2, v″ = 0.

Reference to Figs. 3–5 demonstrates that each of these predictions is borne out experimentally.

We now turn to consider the resonances evident in the βversus plots displayed in Figs. 4–6.

The present modelling shows the resonance states to be supported by adiabatic potential III, which is predominantly 2Σ+1/2,u in the vertical Franck–Condon region.

These resonances are predicted to show energy dependent Beutler–Fano absorption line-shapes, on account of interference with continuum states associated with adiabatic potential II.

Thus in the case of excitation from X2Π3/2,g, for example, the total absorption probability, σ, would be predicted to include a parallel contribution to 2Π3/2,u (that varies smoothly with ), a smooth perpendicular excitation to 2Δ5/2,u, a perpendicular excitation to 2Σ–1/2,u (which would introduce both positive and negative, dependent, contributions to σ), and a further dependent contribution associated with excitation to 2Σ+1/2,u which, as we saw previously, should be most important in the case of excitations from v″ = 1.

This dependence in the mix of parallel and perpendicular contributions to σ will manifest itself in βversus plots such as those shown in Figs. 4 and 5.

The precise details of the line-shapes and their variation with E will be sensitively dependent upon the precise shapes of potential curves II and III over a wide range of r, and on the relative transition moments of the two perpendicular excitations to 2Σ states, but the model calculations are able to capture the trends seen in the present experimental measurements.

Finally, we address the observation that the interference structures seen when exciting from the v″ = 1 levels of X 2Π3/2,g and X 2Π1/2,g have opposite phase (Fig. 6).

Examination of single configuration electronic wavefunctions for the participating states reveals that the transition dipoles for 2Π3/2,g → 2Σ–1/2,u and 2Π3/2,g → 2Σ+1/2,u have the same sign, whereas those for 2Π1/2,g → 2Σ–1/2,u and 2Π1/2,g → 2Σ+1/2,u have opposite signs.

Hence we conclude that this, at first sight, surprising anti-correlation is attributable to interference between the transition dipoles to the 2Σ–1/2,u and 2Σ+1/2,u states, as discussed in section 3.3, modulated by further interference at the avoided crossing.


We have described the results of high resolution imaging studies of the Br+(3P2) fragment ions resulting from photodissociation of Br2+ molecular ions prepared, state selectively, by 2 + 1 REMPI.

Image analysis reveals that the fragment recoil anisotropy is a very sensitive function of photolysis wavelength.

The observed resonances all lie in the energy gap between the ground [Br(2P3/2) + Br+(3P2)] and first excited [Br(2P3/2) + Br+(3P1)] dissociation limits, and converge with increasing energy in a manner consistent with their being associated with a series of predissociating vibrational levels in a bound potential that correlates with the first excited dissociation asymptote.

The structure has been rationalised by performing spin–orbit averaged ab initio electronic structure calculations for all ungerade excited states of Br2+ associated with the …σgπuπgu* valence space, incorporating spin–orbit effects semi-empirically, and then propagating wavepackets on the coupled diabatic potential energy curves so derived.

The model calculations are able to reproduce all of the trends observed experimentally and provide much new insight into the non-adiabatic couplings amongst the various excited states of this text-book open-shell system.

Specifically, our interpretation requires parallel and perpendicular contributions to the total absorption, the relative strengths of which depend on Ω″ and v″ as governed by the Franck–Condon principle.

The resonance line-shapes arise as a result of Fano interference effects, the details of which are sensitively dependent upon the precise shapes of potentials supporting the interacting bound and continuum states, and on the relative transition moments to these states.

βversus plots are a particularly sensitive means of revealing such interferences since, unlike linear absorption spectroscopy, they enable discrimination between states (and transitions) of different polarisation.

Somewhat similar frequency dependent fluctuations in fragment recoil anisotropy have been reported for the case of (rotationally state selected) NO(X) fragments resulting from near threshold dissociation of NO2, but the multi-dimensional nature of the potential energy surface(s) involved precluded any comparably detailed interpretation of those observations39.