A theoretical and experimental study of the double photoionisation of molecular bromine and a new double ionisation mechanism

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Chemical Physics 343 (2008) 270–280 www.elsevier.com/locate/chemphys

A theoretical and experimental study of the double photoionisation of molecular bromine and a new double ionisation mechanism Timo Fleig a, David Edvardsson a

b,*

, Simon T. Banks c, John H.D. Eland

c

Institute of Theoretical and Computational Chemistry, Heinrich Heine University, Universita¨tsstrasse 1, D-402 25 Du¨sseldorf, Germany b ¨ rebro University, Fakultetsgatan 1, S-701 82 O ¨ rebro, Sweden Department of Natural Sciences, O c Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom Received 27 April 2007; accepted 10 August 2007 Available online 19 August 2007

Abstract Complete double photoionisation spectra of Br2 have been measured at 30.4, 37.9 and 40.7 nm. The spectra are compared with high level relativistic calculations of the state energies and potential energy surfaces of Br2þ 2 carried out within a four-component relativistic framework. Excellent agreement is obtained between experimental and theoretical relative state energies which allows for an assignment of the main part of the spectrum. Spectroscopic constants and tunnelling lifetimes have been determined for quasi-bound states. From the experimentally determined electron distributions we deduce that a new form of dissociative autoionisation plays a major role in double photoionisation of bromine and other molecules where certain atomic ions are major fragmentation products.  2007 Elsevier B.V. All rights reserved. Keywords: Double photoionisation; Bromine; Relativistic; Ab initio; Configuration interaction

1. Introduction The bromine molecule is frequently chosen as a model target for spectroscopic examination and it has special importance as a probable intermediate in the effective destruction of tropospheric ozone by bromine compounds [1]. Although the doubly charged bromine molecular ion Br2þ 2 has long been known to be stable on a microsecond timescale, its spectrum is relatively unexplored, both experimentally and theoretically. The double ionisation energy (DIE) of bromine was first measured by electron impact as 30.0 eV [2]. Most recently a spectrum of the doubly charged cation was reported in a TPEsCO study [3] which located the first three electronic states, giving the adiabatic DIE as 28.25 eV and the vertical value as 28.55 eV. On the theoretical side most of the studies have been limited to the neutral Br2 molecule and its ground state [4–6]. The first study of excited states in Br2 and the elec*

Corresponding author. Tel.: +46 (0)8 16 23 73. E-mail address: [email protected] (D. Edvardsson).

0301-0104/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.08.007

tronic states in Brþ 2 was carried out by Balasubramanian [7]. More recently, Visscher et al. investigated ground state properties of the dihalogens including Br2 [8]. To the best of our knowledge no theoretical relativistic calculations have been presented for the doubly charged bromine molecule and its excited states. In this paper, we report complete photoionisation spectra taken by the TOF-PEPECO method [9], where the energies of all electrons emitted either singly or in pairs are recorded without discrimination by energy or angle. The spectrum is interpreted by high level relativistic molecular structure calculations of both ground- and excited-state properties. Excited states are notoriously difficult to describe accurately – especially for doubly charged molecules where an accurate treatment of electron correlation is crucial. Different methods have been employed in this paper to investigate in detail the expected accuracy of calculated spectroscopic properties. Specifically, the dependence of vertical excitation energies for excited states on the one-particle basis set and level of correlation have been studied.

T. Fleig et al. / Chemical Physics 343 (2008) 270–280

In addition to the spectrum of the doubly charged ion and its interpretation, we report a new form of dissociative double ionisation. The experimental phenomena engendered by this mechanism have been observed in complete double photoionisation spectra of a number of other molecules, and are explained for the first time here. The remainder of this paper is organized as follows: Section 2 gives a description of the computational methods used. Next, we briefly summarize the TOF-PEPECO technique. Sections 4 and 5 include the theoretical and experimental results, respectively. Section 6 includes a discussion and finally conclusions are given. 2. Computational details All electronic structure calculations have been carried out with a local version of the four-component relativistic program package DIRAC [10]. For the Configuration Interaction (CI) calculations, we have used the relativistic double-group CI program LUCIAREL [11,12] and for the Multi-Reference Coupled Cluster (MRCC) calculations we employed the state-selective approach by Olsen [13,14] which has recently been extended to the general relativistic framework [15]. Using these methods we can employ a spin– orbit free (SOF) Hamiltonian and a fully relativistic Hamiltonian including spin–orbit terms. All approaches exploit time-reversal symmetry at the one-particle level so that spinors form degenerate (Kramers) pairs. The spinors are optimized in average-of-configuration open-shell calculations to ensure a balanced description of the different electronic states that arise from the valence orbital manifold. We have used the uncontracted basis sets sp-pVDZ/sppVTZ/sp-apVTZ [10] together with the ANO-RCC set [16]. The correlation treatment ranges from CompleteActive-Space (CAS) expansions with eight electrons in six active Kramers pairs, CAS(8in6), to MR singles and doubles (SD) CI including core-valence (CV) correlation from both the 3d- and 4s-shells. The virtual spinor space has been truncated at an energy of 10 a.u. in all calculations. This is routinely performed when using uncontracted basis sets and does not introduce any appreciable errors in our calculations, which we have tested for cut-off values up to 50 a.u. The effect of spin-other-orbit terms has also been studied. Two different Hamiltonians have been employed: The Dirac–Coulomb (DC) and the Infinite-Order TwoComponent (IOTC) [17]. The IOTC Hamiltonian is based on an atomic mean-field approximation for two-particle integrals and includes the spin-other-orbit operator which originates from the Gaunt operator in four-component relativistic theory. All calculations were performed in the double group of the D2h symmetry group. In the spinor optimisation, we averaged over all configurations arising from distributing the 8 (10 for the neutral molecule) valence electrons in six valence Kramers pairs. The nuclear Gaussian exponent for an atom of charge 35 was, in these calculations, equal

271

to 2.4305454351 · 108 and the speed of light was 137.0359998 a.u. Potential energy curves were calculated at a series of dif˚. ferent internuclear separations ranging from 1.9 to 8.0 A Smooth functions were obtained using a cubic spline interpolation over a mesh of 62,000 points. Vibrational wavefunctions, eigenvalues, Franck–Condon factors and tunneling lifetimes were obtained by solving the radialnuclear Schro¨dinger equation using the program LEVEL 8.0 by LeRoy [18]. The reduced mass was calculated using isotopic abundance averaged atomic masses. In order to allow for ease of comparison with the neutral ground state, we did not correct the reduced mass to account for the loss of two electrons. Good convergence was found with an ˚ . In calculating the Frank– integration step of 0.0001 A Condon factors we employed a simple Morse potential for the neutral ground state. We used the spectroscopic ˚ , xe = 325.321 cm1 and xexe = parameters Re = 2.285 A 1 1.0774 cm to determine the parameters for the Morse ˚ 1 and De = 24557.67 cm1. function, giving b = 1.598 A The program LEVEL employs a semiclassical method for determining resonance widths and, therefore, lifetimes. In addition to this approach, we have calculated a small number of widths using the time-delay matrix method first introduced by Smith [19]. This quantum method relates the time-delay due to a resonance to the energy derivative of the scattering matrix. For a single potential the S matrix reduces to just one element which is evaluated over a range of energies, the widths and lifetimes being evaluated as discussed in [20]. For these calculations the S matrix was determined by applying plane wave boundary conditions to an R matrix which was propagated from small R according to the algorithm of Stechel et al. [21]. Spectroscopic constants were determined from the interpolated potential energy curves. 3. Experimental method In the TOF-PEPECO technique [9], photoionisation is by pulsed light of line-selected wavelength from atomic discharges in He or Ne. Recent improvements to the pulsed lamp made light pulse widths of about 3 ns available for this work at a repetition rate of 13 kHz, improving the energy resolution for high-energy electrons. The light intersects an effusive jet of target molecules on the axis of a magnetic bottle in a region of strong (0.8 T), divergent magnetic field. All emitted electrons are forced by the field to follow the field lines of a long (5.5 m) solenoid to a distant detector, where their time of flight (TOF) is measured. The design is based on the principles of magnetic bottle spectrometer design originally stated by Kruit and Read [22]. Bromine has a powerful effect on surface potentials, and its introduction to the apparatus generally caused calibration drifts. A long period of acclimatisation was necessary before the series of runs at different wavelengths. To minimise the effects of shifts during each run, sections of data recorded in time sequence were recalibrated individually

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T. Fleig et al. / Chemical Physics 343 (2008) 270–280

by autocorrelation to the spectra acquired immediately after or before calibration runs on oxygen at 58.4 nm, used as a standard. 4. Theoretical results 4.1. Calibration and benchmarks In Table 1, we present results of the calculations on the 1 þ 0þ g ð Rg Þ ground state of the neutral bromine molecule with a variety of different basis sets and correlation methods. As can be inferred from the table, large basis sets and a treatment of electron correlation at the MRCC level of theory is needed in order to get excellent agreement between theory and experiment. The deviation from experimental values is ˚ for the equilibrium internuclear distance and only 0.005 A 1 1.5 cm for the harmonic frequency using the very large ANO-type basis set and CV correlation from both the 3d- and 4s-shells. However, the computational demands for excited states in general and states of the dication in particular are much too large to allow for a treatment using this very accurate method. Thus, in the calculations of potential energy curves of electronic states in the dication we therefore had to choose a basis set with DZ quality and a treatment of electron correlation which only includes CV correlation from the 4s-shell. We can therefore expect ˚ in these errors in equilibrium bond distances of 0.1 A calculations. Single-point calculations of vertical excitation energies can be performed at a much higher level of theory. Calcu˚ were performed for the lowest lations at R = 2.28105 A four gerade and six ungerade states with the basis sets sppVDZ/sp-pVTZ/sp-apVTZ and ANO-RCC. In these calculations the eight valence electrons were correlated in a CAS(8in6) reference space with single–double (SD) excitations into the virtual space. From these calculations we deduced a maximum basis set error of 0.015 eV for vertical excitation energies at this level of electron correlation. The uncorrelated excitation energies show errors of up to 0.08 eV – especially for higher-lying excited states of ungerade symmetry. For the lowest set of gerade states we have been able to correlate up to 32 electrons (S24 Table 1 Bond lengths and harmonic frequencies for neutral Br2 using various basis sets, including (SO) and neglecting (SOF) spin–orbit interaction, and correlating the respective shells in core-valence (CV) fashion in addition to CAS expansions for valence electrons ˚) Method Correlation Re (A xe (cm1) DZ SO MRCI TZ SOF MRCI TZ SO MRCI TZ SOF MRCI TZ SOF MRCC TZ SOF MRCC ANO-RCC SOF Experiment [34]

(CV (CV (CV (CV (CV (CC (CV

4s CAS) 4s CAS) 4s CAS) 3d, 4s CAS) 3d, 4s CAS) 3d, 4s CAS) 3d, 4s CAS)

2.353 2.315 2.318 2.307 2.305 2.306 2.286

282.7 308.0 306.3 311.4 315.8 312.7 326.8

2.281

325.3

CAS8in6 SD) and the differences in vertical excitation energies compared to calculations with only 12 correlated electrons do not surpass 0.015 eV. We ascribe the fact that the deviations with respect to the level of electron correlation are small to the average-of-configuration approach in the spinor optimisation which leads to a balanced description of ground and excited states, all of which are essentially described by the CAS configurations. For nuclear charges less than 40 the dominance of oneelectron spin–orbit over two-electron spin–orbit terms does not necessarily prevail. In the determination of excitation energies, however, differential effects due to the neglect of two-electron spin–orbit contributions may be small. Calculations at the correlated level with both 8 and 12 correlated electrons using the DC and the IOTC Hamiltonians show that the spin-other-orbit interaction typically reduces the vertical excitation energies slightly. We observe a maximum deviation of 0.01 eV for the lowest five excited states which shows the accuracy of the DC Hamiltonian in the present study. 4.2. Electronic state energies The calculated state energies relative to the 0þ g ground state in Br2þ are shown in Table 2 together with the leading 2 Slater determinant expansion coefficients. The determinants are combinations of Kramers ‘‘up’’ and ‘‘down’’ creator strings. The Kramers pairs given are the ones significant for the occupation of the determinant irrespective of Kramers projection. The remaining electrons doubly occupy the lowest other Kramers pairs not explicitly given. It should be noted that the two Kramers pairs pg (four spinors) are not degenerate due to spin–orbit coupling at the one-particle level. The same applies for the pu spinors. The Kramers pair k-projection r and p is therefore only an approximate notation. We denote p Kramers pairs according to kx as p1/2 and p3/2, respectively. From the analysis of the composition of the electronic wavefunctions we have identified five different double ionisation types from the neutral ground state depending on the removed electrons and the cation configurations. These are shown in Table 3. An accurate theoretical determination of the double ionisation energy (DIE) is difficult. In these calculations, the treatment of correlation is more important than spin– orbit coupling which introduces only a small shift in energy for the ground states. Even at a very high level of theory using a spin–orbit free SD24 CAS(10in6) SD 32 with the very large ANO-RCC basis set, the calculated value is only 26.94 eV. 4.3. Potential energy curves Fig. 1 displays the calculated potential energy curves for the nine lowest states of Br2þ 2 . The four lowest electronic states supporting quasi-bound states are, in increasing energy order, determined to be 0g(1), 1g, 2g and 0g(2). Vibrational energy levels, tunnelling lifetimes and

T. Fleig et al. / Chemical Physics 343 (2008) 270–280

273

Table 2 ˚ ) and leading CI configurations at the CAS(8in6) SD (10 a.u. cutoff) level Calculated vertical transition energies (R = 2.285 A Tv (eV)

Omegaa

Configurationsb

6.303 6.222 6.150 6.067

4g 0u 3g 0g

6.060

1g

5.964

2g

5.933 5.827 5.709

1g 1u 0u

5.690

0u

5.595

1u

5.514

2u

5.456 5.359 5.249 5.195

3u 2g 2u 1u

5.178

0u

5.153 5.147 5.072 5.014 4.986 4.938

3g 1g 0g 2u 4g 0g

4.934

1g

4.894

0g

4.860

0g

4.831 4.761 4.631 4.584

0g 2g 1g 1g

4.528 4.304

2g 1g

4.316

0g

4.287

0g

4.184

2g

3.619 3.427 3.421 3.419 3.293

1u 1g 0u 1u 0g

0:96r1g p1u3=2 p1g3=2 r1u ð0:45  0:45Þp1u1=2 p1g1=2 þ ð0:36  0:36Þp1u3=2 p1g3=2 0:73p1u1=2 p1g1=2 p1g3=2 r1u þ 0:52p1u3=2 p1g1=2 p1g3=2 r1u ð0:40 þ 0:40Þp2g1=2 p1u1=2 r1u ð0:36 þ 0:36Þp2g3=2 p1u1=2 r1u 0:34p2g1=2 p1u3=2 r1u  0:32p1u1=2 p1g1=2 p1g3=2 r1u þ0:36p2g1=2 p1u1=2 r1u þ ð0:40  0:43Þp2g3=2 p1u1=2 r1u 0:63p2g3=2 p1u3=2 r1u þ 0:42p2g1=2 p1u3=2 r1u þ0:36p1u1=2 p1g1=2 p1g3=2 r1u 0:72r1g p1g1=2  0:58r1g p1g3=2 ð0:87  0:32Þp1u1=2 p1u3=2 p1g1=2 r1u ð0:38 þ 0:38Þp1u1=2 p1u3=2 p1g3=2 r1u ð0:34 þ 0:34Þp2u1=2 p1g1=2 p2g1=2 r1u ð0:38  0:38Þp1u1=2 p1u3=2 p1g3=2 r1u þð0:35  0:35Þp2u1=2 p1g1=2 p2g1=2 r1u 0:35p2u1=2 p1g1=2 p2g3=2 r1u þ 0:39p1u1=2 p1u3=2 p1g3=2 r1u þ0:44p2u1=2 p2g1=2 p1g3=2 r1u 0:33p2u3=2 p2g1=2 p1g3=2 r1u ð0:47  0:32Þp2u1=2 p2g1=2 p1g3=2 r1u 0:35p2u3=2 p2g1=2 p1g3=2 r1u þ0:30p1u1=2 p1u3=2 p1g1=2 r1u þ0:37p1u1=2 p1u3=2 p1g3=2 r1u ð0:74  0:57Þp1u1=2 p1u3=2 p1g3=2 r1u ð0:72  0:62Þp1u1=2 p1g1=2 p1g3=2 r1u 0:85r1g p1g1=2 p1g3=2 r1u 0:53r1g p2g1=2 r1u þ 0:34r1g p2g3=2 r1u þð0:42 þ 0:42Þr1g p1g1=2 p1g3=2 r1u ð0:44 þ 0:44Þr1g p2g1=2 r1u þð0:35 þ 0:35Þr1g p1g1=2 p1g3=2 r1u 0:54p1u1=2 p1g1=2 p1g3=2 r1u þ 0:77p1u3=2 p1g1=2 p1g3=2 r1u 0:70p1u1=2 p1u3=2 0:59p2u1=2 p2g1=2 p2g3=2 þ 0:42p2u3=2 p2g1=2 p2g3=2 0:56p1u1=2 p1g3=2  0:57p1u3=2 p1g1=2 0:94p1u3=2 p1g1=2 p1g3=2 r1u ð0:36  0:36Þr1g p1u1=2 þð0:49  0:49Þp1u1=2 p1g1=2 p1g3=2 r1u 0:35p1u3=2 p1g1=2 p1g3=2 r1u 0:37p2g3=2 p1u1=2 r1u þ 0:45p1u1=2 p1u3=2 0:31p2u1=2 p2g1=2 p2g3=2 þð0:52 þ 0:52Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:53  0:53Þr1g p1u1=2 þð0:33  0:33Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:58 þ 0:58Þr1g p1u1=2 0:46p1u1=2 p1g1=2 p1g3=2 r1u  0:61p1u3=2 p1g1=2 p1g3=2 r1u 0:50r1g p1g1=2  0:71r1g p1g3=2 0:37p1u3=2 p1g1=2 p1g3=2 r1u þ 0:39p1u1=2 p1g1=2 p1g3=2 r1u 0:35p2g1=2 p1u1=2 r1u  0:60p2g1=2 p1u3=2 r1u ð0:53  0:63Þr1g p1g3=2 þ 0:32p1u1=2 p1g1=2 p1g3=2 r1u ð0:39 þ 0:39Þp1u3=2 p1g1=2 p1g3=2 r1u þð0:30 þ 0:30Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:53  0:53Þp1u3=2 p1g1=2 p1g3=2 r1u ð0:29  0:29Þp2g3=2 p1u1=2 r1u ð0:53 þ 0:53Þp1u3=2 p1g1=2 p1g3=2 r1u ð0:30 þ 0:30Þp2g3=2 p1u1=2 r1u ð0:35  0:38Þp2g1=2 p1u3=2 r1u þð0:34  0:31Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:32  0:34Þp1u3=2 p1g1=2 p1g3=2 r1u ð0:59  0:32Þp1g3=2 r1u þ 0:34p1g1=2 r1u ð0:67  0:68Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:48  0:48Þp1u3=2 p1g3=2 þ ð0:37  0:37Þp1u1=2 p1g1=2 ð0:39  0:39Þp1u3=2 p1g1=2 þ ð0:39  0:39Þp1u1=2 p1g3=2 ð0:37  0:37Þp2u1=2 p1g1=2 r1u þ ð0:39 þ 0:39Þp1u3=2 p1g1=2 p1g3=2 r1u þð0:32  0:32Þp1u1=2 p1g1=2 p1g3=2 r1u

Ionisation typec B

C

D D B (E) (D) (D) E E C

C

B B

(continued on next page)

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T. Fleig et al. / Chemical Physics 343 (2008) 270–280

Table 2 (continued) Tv (eV)

Omegaa

Configurationsb

Ionisation typec

3.289

0g

3.186

1g

3.118

2g

3.095 2.994 2.781 2.658 2.657 2.133 2.112 1.854 1.774 1.418 1.365 1.059 0.622 0.149 0.000

3g 2u 1u 0u 0u 1u 0u 1u 2u 3u 0u 0g 2g 1g 0þ g

ð0:37 þ 0:37Þp2u1=2 p1g1=2 r1u  ð0:39 þ 0:39Þp1u3=2 p1g1=2 p1g3=2 r1u þð0:31 þ 0:31Þp1u1=2 p1g1=2 p1g3=2 r1u ð0:35  0:36Þp2g1=2 p1u3=2 r1u ð0:30  0:30Þp1u3=2 p1g1=2 p1g3=2 r1u ð0:43 þ 0:42Þp2g1=2 p1u3=2 r1u ð0:30 þ 0:29Þp2g3=2 p1u3=2 r1u þð0:33 þ 0:34Þp1u3=2 p1g1=2 p1g3=2 r1u 0:96p1u3=2 p1g1=2 p1g3=2 r1u 0:87p1g3=2 r1u  0:26p1u1=2 p1u3=2 p1g1=2 r1u 0:71p1g1=2 r1u þ 0:46p1g3=2 r1u ð0:62 þ 0:62Þp1g1=2 r1u ð0:62 þ 0:62Þp1g1=2 r1u ð0:90  0:32Þp1u1=2 p1g1=2 ð0:66 þ 0:66Þp1u1=2 p1g1=2 0:65p1u3=2 p1g1=2 þ 0:66p1u1=2 p1g3=2 0:68p1u1=2 p1g3=2  0:67p1u3=2 p1g1=2 0:96p1u3=2 p1g3=2 ð0:67 þ 0:67Þp1u3=2 p1g3=2 0:81p2g3=2 þ 0:33p2g1=2 0:89p1g1=2 p1g3=2  0:32p1u1=2 p1u3=2 p2g1=2 p2g3=2 ð0:87 þ 0:31Þp1g1=2 p1g3=2 0:84p2g1=2  0:37p2g3=2  0:23p2u1=2 p2g1=2 p2g3=2

B B B B B B A A A A

Only coefficients larger than 0.3 are included. a The parity +/ is only given where an unambiguous identification could be made. b A parenthesis (a + b) denotes that b is the coefficient of the determinant where all Kramers pairs have been Kramers ‘‘barred’’ respective to the determinants corresponding to coefficient a. c A parenthesis (X) for the ionisation type indicates that there is significant contribution of such a configuration. Otherwise the state is dominated by the respective configuration.

Table 3 Double ionisation types from the neutral ground state

5

Removed electrons

Dication configuration

Label

p2g

r2g p4u p2g

A

p1g  p1u

r2g p3u p3g

B

r1g  p1g

r1g p4u p3g

C

p2u

r2g p2u p4g

D



p1u

r1g p3u p4g

4 2u(1) 3.5

3u(1) 0u(1)

3

E

Franck–Condon factors from vertical transitions from the v = 0 level of the neutral ground state in Br2 are listed in Table 4. Spectroscopic constants for these states are listed in Table 5. The 0g(1) and 1g states display similar features with ˚ and each supporting potential wells centred around 2.22 A 20 or more vibrational levels. This ties in with the calculated Franck–Condon factors, which are largest for the low vibrational levels on the excited potential with a very rapid fall-off for higher levels, indicating a small difference in equilibrium internuclear distances between the neutral and doubly charged states. The values of Re for the two ˚ lowest states in Br2þ 2 are ca. 0.06 A shorter than the corresponding value of the ground state in the neutral system. This can be understood since the 0g(1) and 1g states mainly arise from a removal of the two antibonding pg electrons (ionisation type A). The 2g and 0g(2) electronic states are

0u(2) 1u(1)

E / eV

r1g

4.5

2.5

0g(2)

2

2g(1)

1.5 1g(1) 1 0.5

0g(1)

0 –0.5 1.5

2

2.5

3

3.5

4

R/Angstroms Fig. 1. The nine lowest potential energy curves for the Br2þ system 2 calculated at the SD CAS(8in6) level of theory.

T. Fleig et al. / Chemical Physics 343 (2008) 270–280

275

Table 4 Calculated relative vibrational energies, lifetimes, widths and Franck–Condon factors for electronic states in Br2þ 2 Electronic state

Vibrational level

Relative energy (eV)

s (s)

C (cm1)

FC-factor

0g(1)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.000 0.038 0.075 0.112 0.149 0.184 0.220 0.254 0.288 0.321 0.354 0.385 0.416 0.447 0.476 0.504 0.531 0.557 0.582 0.605 0.625

– – – – – – – – – – – – – – – 1.39 · 103 5.59 · 101 3.60 · 104 3.81 · 107 7.17 · 1010 3.24 · 1012

– – – – – – – – – – – – – – – 3.83 · 1015 9.50 · 1012 1.48 · 108 1.39 · 105 7.41 · 103 (7.65 · 103)a 1.64 (1.67)a

4.90 · 101 4.00 · 101 1.02 · 101 7.41 · 103 9.48 · 106 2.77 · 105 6.03 · 107 1.17 · 107 9.53 · 1010 2.85 · 1010 1.43 · 1010 1.09 · 1010 1.77 · 1012 4.84 · 1011 1.46 · 1011 5.56 · 1013 6.02 · 1012 3.01 · 1012 1.40 · 1013 2.89 · 1013 6.00 · 1013

1g(1)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.131 0.171 0.209 0.248 0.285 0.323 0.359 0.396 0.431 0.466 0.501 0.535 0.568 0.600 0.632 0.663 0.693 0.722 0.749 0.776 0.800 0.821

– – – – – – – – – – – – – – – 3.40 · 102 3.17 · 101 4.55 · 104 1.02 · 106 3.63 · 109 2.28 · 1011 4.33 · 1013

– – – – – – – – – – – – – – – 1.56 · 1014 1.67 · 1011 1.17 · 108 5.22 · 106 1.46 · 103 2.32 · 101 (2.49 · 101)a 12.3 (12.7)a

4.01 · 101 4.19 · 101 1.55 · 101 2.31 · 102 1.01 · 103 1.40 · 107 3.68 · 106 4.51 · 108 5.61 · 109 1.01 · 1010 1.45 · 1010 2.53 · 1012 7.26 · 1011 1.64 · 1011 4.41 · 1012 9.18 · 1012 1.01 · 1012 5.00 · 1013 1.21 · 1012 3.89 · 1013 1.29 · 1016 1.24 · 1013

2g(1)

0 1 2 3 4 5 6 7 8 9 10 11

0.626 0.658 0.689 0.719 0.749 0.778 0.805 0.832 0.858 0.882 0.905 0.926

– – – – – – 2.11 3.19 · 103 7.98 · 106 3.48 · 108 2.84 · 1010 4.92 · 1012

– – – – – – 2.52 · 1012 1.67 · 109 6.65 · 107 1.53 · 104 1.87 · 102 (1.94 · 102)a 1.08 (1.16)a

9.67 · 101 2.56 · 102 6.79 · 103 6.56 · 105 1.18 · 104 2.67 · 106 9.86 · 107 3.90 · 107 1.15 · 108 5.03 · 109 5.17 · 109 1.07 · 109

0g(2)

0 1 2 3 4 5

1.092 1.118 1.142 1.165 1.185 1.208

2.60 · 101 1.06 · 101 3.59 · 101 2.06 · 101 5.41 · 102 1.01 · 102 (continued on next page)

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Table 4 (continued) Electronic state

a

Vibrational level

Relative energy (eV)

6 7 8 9 10 11 12 13 14 15

1.198 1.216 1.223 1.228 1.233 1.238 1.241 1.244 1.246 1.248

s (s)

C (cm1)

FC-factor 3.18 · 103 8.14 · 104 1.53 · 104 1.38 · 105 1.10 · 106 1.32 · 105 2.38 · 105 2.77 · 105 2.56 · 105 1.68 · 105

Calculated using the time-delay method.

Table 5 Calculated spectroscopic constants for electronic states in Br2þ 2 ˚) State Re (A xe (cm1) xexe (cm1) 0g(1) 1g 2g 0g(2)

2.222 2.215 2.270 2.338

313 323 265 215

3.363 3.074 3.928 5.292

shifted to slightly larger internuclear distances compared to ˚ , respectively, indithe 0g(1) and 1g states, 2.27 and 2.34 A cating a less effective screening of the nuclear charges. These states also display more shallow potential wells. For the 0g(2) state we were not able to converge the calculations outside the top of the potential barrier. The calculated vibrational spacings of the quasi-bound levels is 34 meV and most of these levels can for all practical purposes be considered as stable. Only the highest levels in these states have tunnelling lifetimes on a microsecond time-scale or shorter. For levels close to the top of the barrier, where the semi-classical approximation should be less accurate, we have also used the time-delay method to calculate resonance widths. These results are included in Table 4. The potential energy curves of the five lowest states with ungerade symmetry were all found to be purely repulsive. This indicates that the character of the pu electrons is strongly bonding since these states are dominated by configurations of the type r2g p3u p3g corresponding to type B ionisation. All the ungerade curves show a plateau starting at ˚. around 2.5 A 5. Experimental results A complete map of the electron pair energies in double photoionisation of bromine at 30.4 nm is shown as Fig. 2. In this form of representation the vertical axis is the quantity hm  (E1 + E2), where the summed electron energy is subtracted from the photon energy to give the final dication energy relative to the neutral ground state. The horizontal scale shows the lower electron energy of the pair; no information is lost because the data are symmetrical about the diagonal where E1 = E2. Maps recorded

Fig. 2. Greyscale image of the electron pair map for double photoionisation of Br2 at 30.4 nm (40.814 eV).

at the other wavelengths show the same characteristics, as far as the energy ranges allow. The main features are horizontal ridges, representing states of the nascent doubly charged ions and concentrations of intensity along and between the ridges at low energies of the slower electron. Spectra of the doubly charged ions at different wavelengths and extracted in different ways from the complete maps are given as Figs. 3 and 4, and the principal electronic states are listed in Table 6, as identified by comparison with the calculations and spectra of cognate molecules. 5.1. Spectra of Br2þ 2 Double photoionisation spectra of bromine at 40.7, 37.9 and 30.4 nm (30.5, 32.7 and 40.8 eV) are shown in Fig. 3. Even at the lowest photon energy, where energy resolution is best, vibrational structure is not seen, presumably because the energy resolution is not high enough. In fact the calculated vibrational spacing of about 34 meV in the

T. Fleig et al. / Chemical Physics 343 (2008) 270–280

277

Table 6 Experimentally determined peak positions from the Br2þ spectrum 2 together with theoretical values, ionisation type (see text) and symmetry of electronic states Experiment

Description

Theorya,b

Type

Symmetry

28.39 28.53 28.91 29.38

Shoulder Peak Strong peak Peak

30.3

Broad band

31.6

Broad peak

32.6 32.89

Shoulder Strong peak

33.57

Peak

34.22

Peak

28.390 28.539 29.012 29.449 29.775 29.808 30.164 30.244 30.502 30.523 31.809 31.811 32.918 33.021 33.221 33.250 33.284 33.324 33.328 33.404 33.462 33.537 34.323 34.612

A A A A B B B B B B B B C C E E D D E B D D C B

0g 1g 2g 0g 0u 3u 2u 1u 0u 1u 1u 0u 2g 1g 0g 0g 0g 1g 0g 2u 0g 1g 1g 0u

Fig. 3. Complete double photoionisation spectra, showing the Br2þ 2 states populated at 30.4, 37.9 and 40.7 nm.

a The calculated energies are given at the centre of the Franck–Condon ˚. region corresponding to an internuclear distance of 2.28105 A b The absolute energies have been adjusted to the experimental peak at 28.39 eV.

Fig. 4. Energy distributions of the lower energy electrons of pairs from Br2 at 30.4 nm corresponding to ionisation energy ranges of (a) 32– 32.5 eV, (b) 30.7–31.2 eV and (c) 28.2–29.6 eV. Range (c) covers the first three bands in the dication spectrum while (a) and (b) are located in gaps between the strong double ionisation peaks.

bound states is indeed substantially less than the best experimental resolution of about 50 meV. There are eight distinct bands at 30.4 nm and the first three peaks are clearly seen at both the longer wavelengths. As in Cl2 and I2 double ionisation, the lowest states of the 3  1 1 þ dication are expected to be 3 R g0 , Rg1 , Dg and Rg , all 2 arising from pg ionisation, as expressed using conventional K–S coupling notation. The two components of the 3 R g state are expected to be close in energy, and to contribute a single band to the spectra. This is confirmed by the calculations which predict a separation of only 149 meV. In the spectrum taken at 30.4 nm the first band has a distinct shoulder before the peak, and we attribute the shoulder to the unresolved lowest state. At the longest

wavelength, where resolution should be best, the second and third bands have sharp edges at low energy and gentle tails on the high energy sides; this is a normal sign that the adiabatic ionisation ((0, 0) transition) is accessible. These identities of the four lowest states found by the full relativistic calculations, all found to be quasi-bound, indeed agree with the qualitative expectations. In Table 6 and Fig. 5, we show a comparison between calculated state energies and experimental features in the spectra. 5.2. A new double photoionisation mechanism A particular feature of the map and of the electron distributions is the existence of steps in intensity as a function of the electron energy, for broad ranges of the double ionisation energy (see Fig. 4). Such steps have been seen in electron pair maps for double photoionisation of other compounds, but have not been interpreted before. They are particularly clear in this case of bromine, and the step energies near 0.4 and 1.4 eV are recognised as excitation energies of atomic bromine ions, Br+; 3P1,0 (0.39, 0.48 eV) and 1D (1.41 eV). The double ionisation spectra of several other compounds show similar steps, all of which prove to be at the lowest excitation energies of their atomic ion fragments, S+, I+, Cl+, N+ and O+. Below the steps there are more or less smooth and flat distributions to lower electron energies.

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T. Fleig et al. / Chemical Physics 343 (2008) 270–280 300

2g || 1g 0g 0g |

250

2g 0u 1u

|

200

Intensity

|

150

1g 0g ||

100

0g 1u 2u 1u | || 0u | 3u 0u ||

50

0 26

28

30

32

34

36

38

40

42

eV Fig. 5. Double photoionisation spectrum of Br2. The assignment of the bands are given in terms of the final electronic states in the dication. The theoretical values have been adjusted so that the 2g state corresponds to the strong peak at 28.91 eV.

Because the excitation energies are those of atomic fragments, it is clear that the mechanism of double ionisation giving rise to these electrons must involve real or latent dissociation. The energies at the steps are presumably those of electrons formed in autoionisation of excited neutrals producing ground state atomic ions. The fact that the electrons have continuously distributed energies, often from near zero up to the atomic ion energies, might possibly be explained by a continuous range of autoionising states of the neutral atoms. This would make the mechanism very similar to the dissociative autoionisation mechanism already clearly recognised in double photoionisation of many O, N, S and I compounds [23–28], where sharp peaks are seen in the electron energy distributions. However, for free atoms the continuous range of excited states required for such a mechanism simply does not exist. An alternative, and probably better explanation is based on the idea that high Rydberg states of the superexcited neutral atoms may be formed with energies very near the excited ion limits. Such states could then autoionise to the dissociative part of the ground state surface over a range of internuclear distances beyond the potential energy barrier shown in Fig. 1. It is not possible to confirm this mechanism on the evidence available at present, but we think the attribution to high Rydberg states is at least highly probable. It may be relevant that excited neutral atoms in high Rydberg states just below the ionisation thresholds have relatively long lifetimes allowing time for fragment separation. It is notable that in contrast to the well established dissociative autoionisation mechanism, which often leads to double ionisation below the threshold for stable dication

formation [23–26,29], this new mechanism seems to operate mainly at higher energies. Only two cases of the characteristic edges and continuous electron energy distributions have come to light below molecular double ionisation thresholds, in N2O (N+ edge) and NO2 (O+ edge). The difference may be partly ascribed to the proposed requirement for high Rydberg states, necessarily at higher energy, rather than the very low ones (usually n* = 3) which give rise to the structured electron spectra in the established mechanism. The steps and continuous electron energy distributions characteristic of this new mechanism have been observed in spectra of a select group of compounds with an outer atom of Br, S, I, Cl, N or O, including OCS, CS2, H2S, ICl, ICN, CH3I, CH3Cl, N2O, NO2 [30] and I2 [31]. A common feature of these cases is that the atomic ion is a major product of the double ionisation concerned. But in other cases where the same atomic ions are formed in abundance by double ionisation, no such phenomena are seen, that is for O2, CO, NO, CO2, N2 (HCN), nor for larger molecules such as CH2I2 where they might be expected. Not all the accessible atomic ion excited states give rise to the edges even in the cases where the phenomenon is seen; bromine is unusual in the clear visibility of two edges possibly representing three limiting ionic states. Detailed interpretation of this new mechanism must probably await both more advanced theory and more advanced experiments. On the experimental side it would be useful to know the kinetic energies carried by the atomic ions in dissociative double ionisation by this mechanism. For diatomic molecules like Br2 there is a limited set of possibilities, because the energy balance is completely

T. Fleig et al. / Chemical Physics 343 (2008) 270–280

determined for atomic products by the state of the other ion. The lowest double ionisation energies (hm  E1  E2) at which autoionisation electrons associated with Br+ (3P0) and Br+(1D) are clearly detected (30 and 31 eV) would allow the other Br+ ion to be in any of its states up to and including 1S. No higher states of the Br+ ions are accessible. In the electron pair map for Br2 at 37.9 nm, the lower energy step is visible, as at 30.4 nm. Similarly the maps for OCS and other molecules [30] show the same step structures at 30.4 and 25.6 nm. This means that the origin of the phenomenon cannot be a superexcited neutral molecule. It must involve a superexcited singly ionised intermediate state or states, because in all the electron pair maps which show the phenomenon, it seems to occur over a fixed range of ionisation energy. The spread of energies of the slower electron must be compensated by different kinetic energy releases, as mentioned above, which may (but does not necessarily) imply a range of interatomic distances at the moment of autoionisation. 6. Discussion The agreement shown between the relative energies of the theoretical dication states and the positions of peaks and shoulders in the experimental spectrum is very good indeed, and gives confidence in the accuracy of all aspects of the calculations. The Franck–Condon factors and resulting band shapes also agree very closely with observation, particularly on the fact that the bands show sharp leading edges in the best resolved spectra, corresponding to intense (0, 0) and (1, 0) vibrational transitions. This gives confidence in the theoretical equilibrium bond distances in the dication states, which are slightly shorter for the first three states than the bond in ground-state neutral Br2. This bond contraction on double ionisation can be understood as the result of competition between the effects of the removal of two antibonding electrons on the one hand, and the Coulomb repulsion on the other. In the comparable iodine double ionisation the bond becomes slightly longer [31], while in the case of chlorine the ground state dication bond lengths calculated by Peyerimhoff et al. [32] are distinctly shorter than in the neutral molecule. From this comparison it seems that the antibonding power of the pg electrons, which diminishes in the larger molecules, is a more potent influence than the Coulomb repulsion. For vibrational levels close to the top of the potential barriers we note good agreement between resonance widths calculated using the quantum time-delay matrix method and those obtained using a semiclassical approximation. This gives confidence in the values for the more tightly bound states for which the semiclassical method is likely to be an even better approximation. Given the accuracy of the semiclassical approach, we highlight the much greater ease of its use over the fully quantum approach. Accurately determining the time-delay matrix requires con-

279

vergence with respect to both the size of the underlying basis and the mesh size of the energy grid. The latter is extremely important given that for long-lived states the peaks in the time-delay matrix are so narrow that a remarkably fine mesh is required to accurately describe them. In the present calculations we diagonalized the Hamiltonian in a DVR basis obtained with 2000 particle-in-abox functions (see Muckerman [33]). Overall, the process is relatively time consuming and labour intensive given the high degree of accuracy offered by the extremely fast semiclassical method. The potential energy curves in Fig. 1 all appear to converge towards the lowest cation pair dissociation limit, Br+(3Pe) + Br+(3Pe) at 25.6 eV, neglecting spin–orbit effects. This allows us to predict the kinetic energy releases in dissociations from the states of Br2þ populated in 2 photoionisation. The calculations show that the levels of the first four electronic states do not dissociate by tunnelling on measurable timescales so the first dissociations should be observed at energies near the top of the calculated barriers, at around 29.7 eV ionisation energy. The kinetic energy release is thus calculated to be 4.1 eV. We expect higher excited states to dissociate to excited atomic products, as in I2, and it will be such states that can be involved in the new dissociative double ionisation mechanism discussed above. The dissociation of the lowest states to ground state products explains why no signs of the new mechanism are seen at energies below about 30 eV. 7. Conclusions The TOF-PEPECO technique has been used to investigate double valence photoionisation of the Br2 molecule. The part of the double photoelectron spectrum ranging from 28 to 34 eV has been interpreted using relativistic four-component calculations. Most observable structure in the spectrum can be attributed to transitions to electronic states in the dication differing at most by two spin–orbitals relative the ground state of neutral Br2. This type of study combines the difficulties of electron correlation effects in excited states, high density of states, and the need for a treatment of special relativity on equal footing. This becomes particularly obvious in the vertical excitation spectra where spin–orbit splitting is seen to have the same order of magnitude as state separations within an electronic configuration which are predominantly due to electron correlation effects. First-order spin–orbit splittings are in the order of 0.5 eV, and even second-order splittings may be as large as 0.2 eV. We have shown that relativistic double group CI methods are competitive approaches for the description and interpretation of photoionisation spectra, comprising an alternative to the fock-space coupled cluster method or the algebraic diagrammatic construction (ADC). On the experimental side, the evidence for a new dissociative double photoionisation mechanism is presented. The theoretical understanding of the dynamics of this pro-

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cess, involving highly excited states of the monocation and dication at long internuclear distances is still beyond the reach of the present theory and provides a challenge for future calculations. References [1] D. Xu, J.S. Francisco, J. Huang, W.M. Jackson, J. Chem. Phys. 117 (2002) 2578. [2] J.T. Herron, V.H. Dibeler, J. Chem. Phys. 32 (1960) 1884. [3] A.J. Yencha, S.P. Lee, G.C. King, in: Proceedings of the XXIV ICPEAC Conference, 2005. [4] P. Schwerdtfeger, L.V. Szentpaly, K. Vogel, H. Silberbach, H. Stoll, H. Preuss, J. Chem. Phys. 84 (1986) 1606. [5] J. Andzelm, M. Klobukowski, E. Radzio-Andzelm, J. Comput. Chem. 5 (1984) 146. [6] L.R. Kahn, P. Baybutt, D. Truhlar, J. Chem. Phys. 65 (1976) 3826. [7] K. Balasubramanian, Chem. Phys. 119 (1988) 41. [8] L. Visscher, K.G. Dyall, J. Chem. Phys. 104 (1996) 9040. [9] J.H.D. Eland, O. Vieuxmaire, T. Kinugawa, P. Lablanquie, R.I. Hall, F. Penent, Phys. Rev. Lett. 90 (2003) 053003. [10] DIRAC04, A Relativistic Ab initio Electronic Structure Program, release dirac04.0 (2004), written by H.J.Aa. Jensen, T. Saue, L. Visscher, with contributions from V. Bakken, E. Eliav, T. Enevoldsen, T. Fleig, O. Fossgaard, T. Helgaker, J. Laerdahl, C.V. Larsen, P. Norman, J. Olsen, M. Pernpointner, J.K. Pedersen, K. Ruud, P. Salek, J.N.P. van Stralen, J. Thyssen, O. Visser, T. Winther. [11] T. Fleig, J. Olsen, L. Visscher, J. Chem. Phys. 119 (2003) 2963. [12] T. Fleig, H.J.Aa. Jensen, J. Olsen, L. Visscher, J. Chem. Phys. 124 (2006) 104106. [13] J. Olsen, J. Chem. Phys. 113 (2000) 7140. [14] J.W. Krogh, J. Olsen, Chem. Phys. Lett. 344 (2001) 578. [15] T. Fleig, L.K. Sørensen, J. Olsen, Theo. Chem. Acc. 118 (2) (2007) 347.

[16] MOLCAS, A Program Package for Computational Chemistry. G. ˚ . Malmqvist, B.O. Roos, U. Ryde, V. Karlstro¨m, R. Lindh, P.-A Veryazov, P.-O. Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Comput. Mater. Sci. 28 (2003) 222. [17] H.J.Aa. Jensen, M. Ilias, Two-component Relativistic Methods based on the Quaternion Modified Dirac Equation: From the Douglas– Kroll to the Barysz-Sadlej-Snijders Infinite Order, unpublished. [18] R.J. Le Roy, LEVEL 8.0: A Computer Program for Solving the Radial Schro¨dinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663, 2007, see http://leroy.uwaterloo.ca/programs. [19] F.T. Smith, Phys. Rev. 118 (1960) 349. [20] L. Celenza, W. Tobocman, Phys. Rev. 174 (1968) 1115. [21] E.B. Stechel, R.B. Walker, J.C. Light, J. Chem. Phys. 69 (1978) 3518. [22] P. Kruit, F.H. Read, J. Phys. E16 (1983) 313. [23] J.H.D. Eland, S.S.W. Ho, H.L. Worthington, Chem. Phys. 290 (2003) 27. [24] J.H.D. Eland, Chem. Phys. 294 (2003) 171. [25] Y. Hikosaka, J.H.D. Eland, Chem. Phys. 299 (2004) 147. [26] M. Hochlaf, J.H.D. Eland, J. Chem. Phys. 120 (2004) 6449. [27] A.E. Slatter, T.A. Field, M. Ahmad, R.I. Hall, J. Lambourne, F. Penent, P. Lablanquie, J.H.D. Eland, J. Chem. Phys. 122 (2005) 084317. [28] J.H.D. Eland, J. Electron Spectrosc. Related Phenom. 144–147 (2005) 1145. [29] M. Ahmad, P. Lablanquie, F. Penent, J.G. Lambourne, R.I. Hall, J.H.D. Eland, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 3599. [30] Unpublished work, Oxford laboratory. [31] D. Edvardsson, A. Danielsson, L. Karlsson, J.H.D. Eland, Chem. Phys. 324 (2006) 674. [32] P.G. Fournier, J. Fournier, F. Salama, D. Sta¨rk, S.D. Peyerimhoff, J.H.D. Eland, Phys. Rev. A34 (1986) 1657. [33] J.T. Muckerman, Chem. Phys. Lett. 173 (1990) 200. [34] National Institute of Standards and Technology, web pages, Constants of Diatomic Molecules.