A pronounced vibrational state dependence of photoelectron angular distributions observed in chiral photoionization experiments is explored using a simple, yet realistic, theoretical model based upon the transiently chiral molecule H_{2}O_{2}. The adiabatic approximation is used to separate vibrational and electronic wavefunctions. The full ionization matrix elements are obtained as an average of the electronic dipole matrix elements over the vibrational coordinate, weighted by the product of neutral and ion state vibrational wavefunctions. It is found that the parity of the vibrational Hermite polynomials influences not just the amplitude, but also the phase of the transition matrix elements, and the latter is sufficient, even in the absence of resonant enhancements, to account for enhanced vibrational dependencies in the chiral photoionization dynamics.

## I. INTRODUCTION

A first point of departure for any discussion of vibrational features in molecular photoionization spectroscopy is the Franck-Condon (FC) Principle under which vibrational and electronic degrees of freedom are effectively deemed to be uncoupled. Breakdown of these FC expectations are, however, well established for both vibrational cross-section behavior^{1} and vibrationally dependent photoelectron angular distributions (PADs).^{2,3} Most commonly, such FC-breakdowns have been associated with the presence of continuum resonances, although a non-FC vibrational dependence has also been reported for molecule frame PADs at energies below that of an established shape resonance.^{4} A common theoretical explanation for these phenomena has emerged that explicitly allows for the variation of the electronic dipole matrix elements along the vibrational coordinate.^{1,3,5}

Contini *et al.*^{6} provided the first experimental data suggesting a non-FC vibrational level dependence could be observed in a photoelectron circular dichroism (PECD) experiment. PECD depends on the detection of a forward-backward scattering asymmetry in the lab-frame PAD when a chiral molecule is photoionized using circularly polarized radiation.^{7} A subsequent high resolution PECD experiment on the chiral molecule Methyloxirane revealed there to be, in fact, a very strong PAD dependence on the ion vibrational state reached.^{8} Most dramatically, at low photon energy the forward-backward PAD asymmetry of ∼6% was observed to completely reverse direction when a weak vibrational mode was excited, and this occurs in a region where no shape resonance has been either predicted^{9,10} or observed.^{8}

As a prelude to a full theoretical simulation of the methyloxirane multi-mode photoionization dynamics, a simplified, yet realistic model based upon Hydrogen Peroxide (Fig. 1) is developed here. H_{2}O_{2} is the simplest chiral molecule (although rapid tunneling through the symmetric double well torsional barrier of the ground state allows, in practice, for the enantiomeric (*g*^{±}) forms to rapidly interconvert or racemize). The H_{2}O_{2} vibrations display a strong local mode character, with the O and H atom motions effectively decoupled, suggesting a pseudo-diatomic model can reasonably be adopted for treating vibrations. The results here take advantage of this to simplify the vibrational treatment to that of a 1-dimensional harmonic displacement along, e.g., the

_{2}O

_{2}has served as a model prototype for coherent chiral control scenarios.

^{11}

## II. THEORY AND MODEL

The general form of electron angular distribution expected from ionization of a randomly oriented sample is

where σ is the integrated cross-section, *p* is the helicity of the light (*p* = ±1 for, respectively, left- and right-circularly polarized light (CPL), and *p* = 0 linear polarization), the P_{j} are Legendre polynomials, and θ is measured with respect to the light propagation direction (CPL) or electric vector (linear polarization). The first Legendre polynomial evaluates to cos θ and can thus introduce a forward-backward asymmetry, but its coefficient,

^{7,12,13}

Vibrational effects are incorporated into these matrix elements following an approach that has been successfully applied to linear molecule photoionization by a number of authors.^{1,3,5,14,15} Using the adiabatic approximation, electronic and nuclear functions are separated (viz.,

_{i}and

*Q*.

*M*

_{i, f}, is further assumed to vary negligibly with

*Q*so that Eq. (2a) can be written

A MP2/6-31G^{*} optimized geometry (H-O-O-H dihedral angle 121.2°,

*g*

^{+}conformer of H

_{2}O

_{2}were used as a reference for the photoionization dynamics calculations. CMS-Xα calculations at fixed displaced geometries were used to obtain incoming wave

**S**-matrix normalized continuum photoelectron partial wave functions from which radial dipole matrix elements

^{13,15}These were then averaged over the chosen vibrational coordinate, weighted by the product

## III. RESULTS

The following results consider ionization of the HOMO. Examples of the harmonic O-O stretching vibration functions and their projection are presented in Fig. 2. The neutral and cation vibrational frequencies were taken to be 929 cm^{−1} and 1073 cm^{−1} as given by MP2/6-31G^{*} calculations, but the displacement of the cation equilibrium geometry along *Q* was treated as an arbitrarily adjustable parameter.

Figure 3 first shows typical results of the fixed geometry CMS-Xα calculations for selected

^{*}shape resonance below 2 eV—and then

*only*for the shortest

*r*-averaged matrix elements closely resemble the post-calculation averaged means ⟨σ(

*r*)⟩

_{r}and ⟨

*b*

_{j}(

*r*)⟩

_{r}. Both these averages are virtually indistinguishable from the fixed geometry results at

*r*

_{e}except for a small perturbation within 1 eV of threshold (attributable to the short

^{*}shape resonance).

Vibrational state specific results (*v* = 0 → *v*^{+}) for four arbitrary assumed displacements of the ionic potential along the O–O vibrational coordinate appear in Figure 4. Also shown, for reference, are the zero point motion averaged curves included in Fig. 3. The vibrational branching ratios are seen to be flat, and spread to encompass higher vibrational levels at greater displacements of the ion potential, both in accordance with qualitative FC arguments and expectations. On the other hand a clear non-FC vibrational state dependence is demonstrated by the chiral *b*_{1} parameters, even in the absence of any continuum resonance. One might assume this is expected since a positive relative displacement of the ion *r*_{e} would skew the sampling towards vertical ionization occurring at longer

*b*

_{1}curves in Fig. 3 suggest this should provide an upwards shift for

*b*

_{1}curves in Fig. 4 relative to the solid reference curve. The converse applies for negative ion potential displacements. But while the

*v*

^{+}= 0, 1 results for assumed ±0.12 Å offsets (panels (c) and (d)) behave this way, the

*v*

^{+}= 2, 3 curves shift in an opposing sense. The behavior of the

*b*

_{1}(

*v*

^{+}= 1) results for smaller equilibrium displacements ±0.032 Å (seen in panels (a) and (b)) is even more dramatic, and contrary to the simple expectation outlined above.

Although not shown here, very similar vibrational PECD behavior is calculated for, e.g., the O-H stretching mode and for realistic anharmonic functions

## IV. DISCUSSION

In the foregoing it is the chiral *b*_{1} parameters that display the greater sensitivity to assumed geometry, as previously established.^{16–18} We have previously argued that this is because *b*_{1} depends on the sine of the relative phase between interfering partial waves, while *b*_{2} partially depends on its cosine (and the integrated cross-section, contains *no* phase information).^{7,17} Because a sine varies much more rapidly than the relatively flat cosine function for small angles, *b*_{1} is expected to be much more sensitive to small structure-induced changes in scattering phase of adjacent partial waves. One thus infers here that the phases, at least, are varying with *Q* to account for the vibrational dependence. Even so, the variation seen in *b*_{1}(*v*^{+} = 1) for small ion equilibrium displacements is surprising, seemingly exceeding the variability in the fixed geometry *b*_{1} results (Fig. 3).

The stretching vibrational modes and harmonic functions are fully symmetric with respect to the symmetry operations of the molecular point group. Nevertheless, the odd vibrational wavefunctions (actually the Hermite polynomials) display odd parity with respect to reflection through *Q* = *r*_{e} and this antisymmetry is approximately maintained, at least for small displacements, in the term

*v*

^{+}= 1 into parts over the positive and negative loops of

As a simple test of whether this accounts for the unexpectedly large variations in the chiral angular distributions, calculations for small relative geometry displacements have been repeated using

*b*

_{1}does then revert towards the reference norm. For larger geometry displacements the parity effect is already reduced since the pseudo-antisymmetry of

*b*

_{1}variations less extreme.

## V. CONCLUSIONS

A simple pseudo-diatomic model calculation for ion vibrational state specific PECD shows that for small displacements of the ion potential curve the dichroism parameter,

*b*

_{1}parameters for

*v*

^{+}= 0, 1 that have comparable magnitudes (few %), but opposite sign can result—exactly as observed experimentally for methyloxirane.

^{8}This suggests, but does not prove, a common vibrational phase effect to explain the experimental observations. If correct, vibrational parity in excited neutral levels should also play a similar role, and more extensive resolved

*v*→

*v*

^{+}experiments would be revealing.

Finally, it is worth noting that even if these more dramatic predictions are generally applicable they apply essentially to small equilibrium displacements, i.e., to vibrational modes traditionally associated with weak FC factors. In unresolved experiments, these extreme behaviors would probably be masked by transitions with stronger FC factors, so that fixed equilibrium geometry calculations should remain reliable for such cases.