^{1}, A. B. Alekseyev

^{2}, G. G. Balint-Kurti

^{3}, M. Brouard

^{1,a)}, Alex Brown

^{4}, R. J. Buenker

^{2}, E. K. Campbell

^{1}and D. B. Kokh

^{2}

### Abstract

A fully quantum mechanical dynamical calculation on the photodissociation of molecular chlorine is presented. The magnitudes and phases of all the relevant photofragment *T*-matrices have been calculated, making this study the computational equivalent of a “complete experiment,” where all the possible parameters defining an experiment have been determined. The results are used to simulate cross-sections and angular momentumpolarization information which may be compared with experimental data. The calculations rigorously confirm the currently accepted mechanism for the UV photodissociation of Cl_{2}, in which the majority of the products exit on the C ^{1}Π_{1u } state, with non-adiabatic couplings to the A ^{3}Π_{1u } and several other Ω = 1 states, and a small contribution from the B state present at longer wavelengths.

This work was supported by the U.K. EPSRC (to M.B. *via* Programme Grant No. EP/G00224X1), the E.U. (to M.B. *via* FP7 EU People ITN project 238671) and the Deutsche Forschungsgemeinschaft (to R.J.B. *via* Grant No. BU 450/21-2). Valuable discussions with Dr G.A.D. Ritchie are gratefully acknowledged. A.B. thanks the Natural Sciences Engineering Research Council of Canada (Discovery Grant) for financial support.

I. INTRODUCTION

A. Notation

B. Previous experimental work

1. Absorptionspectrum and dissociation dynamics

2. Branching ratios and translational anisotropy

3. Atomic alignment measurements

4. Atomic orientation measurements

C. Previous theoretical studies

II. METHOD

A. Potentials

B. Dynamics

III. RESULTS

A. Absorption cross-sections

1. Total absorption cross-section

2. Partial cross-sections and phases

B. Evolution of the Ω = 1 wavepackets

C. Branching ratio and spatial anisotropy parameters

D. Angular momentumpolarization parameters

1. Ground state (Cl + Cl) polarization parameters

2. Low order orientation of excited fragments of the (Cl* + Cl) channel

3. The effect of isotopic substitution

4. Alignment of ground state Cl in the excited (Cl + Cl*) channel

5. High order orientation in the (Cl + Cl) channel

6. Adiabatic results

7. Effect of molecular rotation

IV. SUMMARY AND CONCLUSION

### Key Topics

- Photodissociation
- 42.0
- Polarization
- 40.0
- Dissociation
- 21.0
- Absorption spectra
- 18.0
- Non adiabatic reactions
- 16.0

## Figures

Adiabatic potential energy curves of Cl_{2} (right)^{15} important for dissociation in the first absorption band (left).^{53} At short wavelengths the C ^{1}Π_{1u } state (green) dominates with the state (blue) becoming more important as the wavelength increases.

Adiabatic potential energy curves of Cl_{2} (right)^{15} important for dissociation in the first absorption band (left).^{53} At short wavelengths the C ^{1}Π_{1u } state (green) dominates with the state (blue) becoming more important as the wavelength increases.

Adiabatic correlation diagram linking the *atomic* Cl(^{2}P_{ J }) states (right) with the *molecular* states in Hunds case (c) and (a) (middle), and molecular orbitals (left). The numbers give the occupancy of the 5σ_{ g }2π_{ u }25 molecular orbitals. The five states believed to be important in the photodissociation are also labelled (A, C, B, 3, and 4). Adapted from Refs. 6,8,18,19.

Adiabatic correlation diagram linking the *atomic* Cl(^{2}P_{ J }) states (right) with the *molecular* states in Hunds case (c) and (a) (middle), and molecular orbitals (left). The numbers give the occupancy of the 5σ_{ g }2π_{ u }25 molecular orbitals. The five states believed to be important in the photodissociation are also labelled (A, C, B, 3, and 4). Adapted from Refs. 6,8,18,19.

Diagram of the excited potential energy curves believed to be important in the photodissociation of Cl_{2}.

Diagram of the excited potential energy curves believed to be important in the photodissociation of Cl_{2}.

Top panel: |*c* _{ jk, u }(*R*)|^{2} parameters, where *j* corresponds to the C ^{1}Π_{1u } state. These quantities correspond to the relative contributions of the different diabatic states to C ^{1}Π_{1u }. When the internuclear separation *R* is low, the C ^{1}Π_{1u } state has an almost one-to-one correspondence with the C ^{1}Π_{1u } state, but at larger *R* considerable spin-orbit mixing of the diabatic states occurs. Bottom panel: *R* dependence of the adiabatic transition dipole moments for the bright states of Cl_{2} in the Franck–Condon region.

Top panel: |*c* _{ jk, u }(*R*)|^{2} parameters, where *j* corresponds to the C ^{1}Π_{1u } state. These quantities correspond to the relative contributions of the different diabatic states to C ^{1}Π_{1u }. When the internuclear separation *R* is low, the C ^{1}Π_{1u } state has an almost one-to-one correspondence with the C ^{1}Π_{1u } state, but at larger *R* considerable spin-orbit mixing of the diabatic states occurs. Bottom panel: *R* dependence of the adiabatic transition dipole moments for the bright states of Cl_{2} in the Franck–Condon region.

Top panel: Diabatic potentials for the electronically excited states used in the photodissociation dynamics calculation. The diabatic states are defined by neglecting the spin-orbit couplings. Bottom panel: *R*-dependence for the spin-orbit couplings to the C^{1}Π_{ u } state.

Top panel: Diabatic potentials for the electronically excited states used in the photodissociation dynamics calculation. The diabatic states are defined by neglecting the spin-orbit couplings. Bottom panel: *R*-dependence for the spin-orbit couplings to the C^{1}Π_{ u } state.

Simulated values for the total absorption cross-section. Experimental data points from the NIST database^{53} provided for comparison.

Simulated values for the total absorption cross-section. Experimental data points from the NIST database^{53} provided for comparison.

Upper panel: Partial cross-sections for the full photodissociation calculation. Lower panel: Close-up of states correlating to the (Cl + Cl*) exit channel.

Upper panel: Partial cross-sections for the full photodissociation calculation. Lower panel: Close-up of states correlating to the (Cl + Cl*) exit channel.

Graphs showing Φ_{ mn } = cos [ϕ_{ m } − ϕ_{ n }], where ϕ_{ m } and ϕ_{ n } are the fragment channels *m* and *n*, as a function of photon energy. The top panel corresponds to phase differences between Ω = 1 states, and the bottom panel corresponds to interferences between Ω = 1 and Ω = 0 states. See text for details.

Graphs showing Φ_{ mn } = cos [ϕ_{ m } − ϕ_{ n }], where ϕ_{ m } and ϕ_{ n } are the fragment channels *m* and *n*, as a function of photon energy. The top panel corresponds to phase differences between Ω = 1 states, and the bottom panel corresponds to interferences between Ω = 1 and Ω = 0 states. See text for details.

Wavepackets for the Ω = 1 adiabatic potentials, sampled at different points in time. The *y*-axis scales are normalized to the initial wavepacket in the C ^{1}Π_{1u } state.

Wavepackets for the Ω = 1 adiabatic potentials, sampled at different points in time. The *y*-axis scales are normalized to the initial wavepacket in the C ^{1}Π_{1u } state.

Top panel: Branching ratio between ground and excited state Cl as a function of wavelength. Bottom panel: Calculated energy dependence of the β(*E*) parameter for fragments in the excited channel. The dashed lines in each panel were obtained using the *R*-dependent transition dipole moments, the continuous lines were with a constant transition dipole moments.^{32}

Top panel: Branching ratio between ground and excited state Cl as a function of wavelength. Bottom panel: Calculated energy dependence of the β(*E*) parameter for fragments in the excited channel. The dashed lines in each panel were obtained using the *R*-dependent transition dipole moments, the continuous lines were with a constant transition dipole moments.^{32}

Energy dependence of the polarization parameter of the Cl* fragment, for photodissociation to the (Cl + Cl*) channel. Experimental results, including error bars, from Ref. 8 are provided for comparison.

Energy dependence of the polarization parameter of the Cl* fragment, for photodissociation to the (Cl + Cl*) channel. Experimental results, including error bars, from Ref. 8 are provided for comparison.

Top panel: Energy dependence of the polarization parameter for the ^{35}Cl( ^{2} *P* _{1/2}) photofragments as obtained from the full QM dynamical calculations (red continuous line). Experimental results from Kim *et al.* ^{9} (open circles) are shown for comparison, along with the semi-classical results of Asano and Yabushita (green dotted line).^{14} Bottom panel: as for the top panel but showing the QM dynamical results for the ^{37}Cl( ^{2} *P* _{1/2}) photofragments (continuous blue line). The fully quantum dynamics results from this work are compared with the semi-classical results of Asano and Yabushita (black dotted line).^{14,16} The experimental results (open green triangles) are taken from Alexander *et al.* ^{8} Note that in both panels the sign of the theoretical data have been inverted.

Top panel: Energy dependence of the polarization parameter for the ^{35}Cl( ^{2} *P* _{1/2}) photofragments as obtained from the full QM dynamical calculations (red continuous line). Experimental results from Kim *et al.* ^{9} (open circles) are shown for comparison, along with the semi-classical results of Asano and Yabushita (green dotted line).^{14} Bottom panel: as for the top panel but showing the QM dynamical results for the ^{37}Cl( ^{2} *P* _{1/2}) photofragments (continuous blue line). The fully quantum dynamics results from this work are compared with the semi-classical results of Asano and Yabushita (black dotted line).^{14,16} The experimental results (open green triangles) are taken from Alexander *et al.* ^{8} Note that in both panels the sign of the theoretical data have been inverted.

Top panel: Coherent polarization parameter for the Cl fragment in the (Cl + Cl*) channel. Middle and bottom panels: Orientation polarization parameters for the Cl atoms produced in the ground (Cl + Cl) channel.

Top panel: Coherent polarization parameter for the Cl fragment in the (Cl + Cl*) channel. Middle and bottom panels: Orientation polarization parameters for the Cl atoms produced in the ground (Cl + Cl) channel.

## Tables

Correspondence between the mixed Hund's case (a)/(c) labels employed here, and their Hund's case (c) equivalents.

Correspondence between the mixed Hund's case (a)/(c) labels employed here, and their Hund's case (c) equivalents.

Laboratory frame alignment parameters reported from previous studies by Brouard and co-workers,^{12} Rakitzis *et al.*,^{26} Bracker *et al.*,^{25} and Rakitzis and Kitsopoulos^{28} at 308, 320, and 355 nm, respectively, for the Cl photofragments in the ground state product channel, and from Samartzis *et al.* ^{27} and Rakitzis *et al.* ^{26} for the excited state product channel. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Laboratory frame alignment parameters reported from previous studies by Brouard and co-workers,^{12} Rakitzis *et al.*,^{26} Bracker *et al.*,^{25} and Rakitzis and Kitsopoulos^{28} at 308, 320, and 355 nm, respectively, for the Cl photofragments in the ground state product channel, and from Samartzis *et al.* ^{27} and Rakitzis *et al.* ^{26} for the excited state product channel. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Molecular frame alignment parameters reported from previous studies by Brouard and co-workers,^{12} Rakitzis *et al.*,^{26} Bracker *et al.*,^{25} Rakitzis and Kitsopoulos,^{28} and Samartzis *et al.* ^{27} and data are shown for the Cl fragments in the ground state product channel (top) while data are shown for the excited state product channel (bottom). Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Molecular frame alignment parameters reported from previous studies by Brouard and co-workers,^{12} Rakitzis *et al.*,^{26} Bracker *et al.*,^{25} Rakitzis and Kitsopoulos,^{28} and Samartzis *et al.* ^{27} and data are shown for the Cl fragments in the ground state product channel (top) while data are shown for the excited state product channel (bottom). Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Laboratory frame orientation parameters reported from previous studies by Kim *et al.* ^{9} and Alexander *et al.* ^{8} Note that the authors used the values of the spatial anisotropy, β, from the work of Samartzis *et al.* ^{27} to calculate their orientation moments. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Laboratory frame orientation parameters reported from previous studies by Kim *et al.* ^{9} and Alexander *et al.* ^{8} Note that the authors used the values of the spatial anisotropy, β, from the work of Samartzis *et al.* ^{27} to calculate their orientation moments. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Molecular frame orientation parameters reported from previous studies by Kim *et al.* ^{9} and Alexander *et al.* ^{8} Note that the authors used the values of the spatial anisotropy, β, from the work of Samaratzis *et al.* ^{27} to calculate their orientation moments. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Molecular frame orientation parameters reported from previous studies by Kim *et al.* ^{9} and Alexander *et al.* ^{8} Note that the authors used the values of the spatial anisotropy, β, from the work of Samaratzis *et al.* ^{27} to calculate their orientation moments. Errors (1σ) in the final digit(s) are given in parenthesis where appropriate.

Parameters used in the propagation algorithms.

Parameters used in the propagation algorithms.

Definitions of the molecular frame polarization parameters^{26,48} in terms of expressions for the dynamical functions, *f* _{ K }(*q*, *q* ^{′}),^{10} and expressions for the laboratory frame polarization parameters of Picheyev *et al.* ^{46,51} The *V* _{ K }(*J*) are *J*-dependent normalization factors, defined elsewhere.^{35,58} Adapted from a more extensive table presented in Ref. 46. Note that *V* _{2}(*J*) = 2.795 for *J* = 1.5.

Definitions of the molecular frame polarization parameters^{26,48} in terms of expressions for the dynamical functions, *f* _{ K }(*q*, *q* ^{′}),^{10} and expressions for the laboratory frame polarization parameters of Picheyev *et al.* ^{46,51} The *V* _{ K }(*J*) are *J*-dependent normalization factors, defined elsewhere.^{35,58} Adapted from a more extensive table presented in Ref. 46. Note that *V* _{2}(*J*) = 2.795 for *J* = 1.5.

Table of polarization parameters for photodissociation to the (Cl + Cl) channel. Columns from left to right: polarization parameter being determined, wavelength of the experiment performed, results from experiment,^{8,12,26,28} results from the adiabatic calculation, results from the full calculation, limiting values of the polarization moment.

Table of polarization parameters for photodissociation to the (Cl + Cl) channel. Columns from left to right: polarization parameter being determined, wavelength of the experiment performed, results from experiment,^{8,12,26,28} results from the adiabatic calculation, results from the full calculation, limiting values of the polarization moment.

Table of polarization parameters for photodissociation to the (Cl + Cl*) channel, for the Cl* atoms. Experimental results of Alexander *et al.*,^{8} results of the adiabatic calculation, results of the full calculation, and the theoretical limiting values of the polarization moments are provided for comparison. Note that the for the sign of the results from the full calculation have been reversed.

Table of polarization parameters for photodissociation to the (Cl + Cl*) channel, for the Cl* atoms. Experimental results of Alexander *et al.*,^{8} results of the adiabatic calculation, results of the full calculation, and the theoretical limiting values of the polarization moments are provided for comparison. Note that the for the sign of the results from the full calculation have been reversed.

Table of appropriate correction factors to convert the experimental polarization moments to their axial recoil values (their limiting values in the absence of parent molecule rotation). Note that the γ values are estimate on the basis of the observed deviations in β from the limiting value of −1. The *s* _{2} factor has not been included because it is unaffected by parent molecule rotation, and so its correction factor in all instances would be 1.

Table of appropriate correction factors to convert the experimental polarization moments to their axial recoil values (their limiting values in the absence of parent molecule rotation). Note that the γ values are estimate on the basis of the observed deviations in β from the limiting value of −1. The *s* _{2} factor has not been included because it is unaffected by parent molecule rotation, and so its correction factor in all instances would be 1.

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