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 H2O2. 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.

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 behavior1 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 predicted9,10 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. H2O2 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 H2O2 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

$r_{\sf OO}$
rOO coordinate. Although O–H stretching, wagging, and torsional modes (in particular, the anticipated change to a planar structure in the cation) are thereby ignored this provides a plausible model for an initial investigation of generic effects in vibrationally resolved PECD. Following the same reasoning and philosophy, H2O2 has served as a model prototype for coherent chiral control scenarios.11 

FIG. 1.

The g+ transient chiral structure of H2O2. The only element of symmetry is the C2 rotational axis, z.

FIG. 1.

The g+ transient chiral structure of H2O2. The only element of symmetry is the C2 rotational axis, z.

Close modal

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

\begin{equation}I(\theta )= \frac{\sigma }{4\pi } \big(1+b^{\lbrace p\rbrace }_{1}{\rm P}_1(\cos \theta )+ b^{\lbrace p\rbrace }_{2}{\rm P}_2(\cos \theta )\big),\end{equation}
I(θ)=σ4π1+b1{p}P1(cosθ)+b2{p}P2(cosθ),
(1)

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 Pj 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,

$b^{\lbrace p\rbrace }_{1}$
b1{p}⁠, is necessarily zero for all but chiral molecules and CPL, and so conventionally this term has often been omitted. The theory allowing the coefficients in Eq. (1) to be calculated from partial wave radial dipole matrix elements has been presented and discussed elsewhere.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.,

$\Psi (r;Q)\break=\psi (r;Q)\mathcal {X}(Q)$
Ψ(r;Q)=ψ(r;Q)X(Q)⁠) allowing the photoionization matrix elements,
$T_{i,f,v,v^+}$
Ti,f,v,v+
, to be written as
\begin{eqnarray}T_{i,f,v,v^+} &=& \int \mathcal {X}_{f,v^+}(Q) M_{i,f}(Q) \mathcal {X}_{i,v}(Q)dQ,\end{eqnarray}
Ti,f,v,v+=Xf,v+(Q)Mi,f(Q)Xi,v(Q)dQ,
(2a)
\begin{eqnarray}\hspace*{-1.5pc}M_{i,f}(Q)&=& \big\langle \psi _{f,{\vec{k}}}^{(-)}({\bf r};Q) \mid {\hat{\mu }} \mid \psi _{i}({\bf r};Q) \big \rangle _{\bf r} ,\end{eqnarray}
Mi,f(Q)=ψf,k()(r;Q)μ̂ψi(r;Q)r,
(2b)
where
${\hat{\mu }}$
μ̂
is the electric dipole operator, and ψi and
$\psi _{f,{\vec{k}}}^{(-)}$
ψf,k()
are the neutral and continuum (ionized) state electronic wavefunctions which have only a parametric dependence on the nuclear coordinate, Q.
$\mathcal {X}_{i,v}$
Xi,v
and
$\mathcal {X}_{f,v^+}$
Xf,v+
are the corresponding vibrational wavefunctions. In the FC approximation the electronic matrix element, Mi, f, is further assumed to vary negligibly with Q so that Eq. (2a) can be written
$T_{i,f,v,v^+} = M_{i,f} \int \mathcal {X}_{f,v^+}(Q) \mathcal {X}_{i,v}(Q) dQ$
Ti,f,v,v+=Mi,fXf,v+(Q)Xi,v(Q)dQ
, and electronic and nuclear motions are effectively decoupled.

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

$r_{\sf OO}= 1.468$
rOO=1.468 Å) and vibrational analysis for the neutral g+ conformer of H2O2 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
$f_{\nu }^{l,m}(Q)$
fνl,m(Q)
were obtained, as described previously.13,15 These were then averaged over the chosen vibrational coordinate, weighted by the product
$\mathcal {X}_{i,v}(Q)\mathcal {X}_{f,v^+}(Q)$
Xi,v(Q)Xf,v+(Q)
, as implied by Eq. (2a), using a 21-point numerical quadrature. Vibrational state specific
$b^{\lbrace p\rbrace }_{j}$
bj{p}
parameters could then be calculated.

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.

FIG. 2.

Harmonic vibrational wavefunctions of H2O2 neutral (a), cation (b), and the product

$\mathcal {X}_{v=0}\mathcal {X}_{v^+}^+$
Xv=0Xv++ (c) for an assumed +0.032 Å equilibrium displacement in the ion. Matrix elements are evaluated at the fixed geometries marked in (a).

FIG. 2.

Harmonic vibrational wavefunctions of H2O2 neutral (a), cation (b), and the product

$\mathcal {X}_{v=0}\mathcal {X}_{v^+}^+$
Xv=0Xv++ (c) for an assumed +0.032 Å equilibrium displacement in the ion. Matrix elements are evaluated at the fixed geometries marked in (a).

Close modal

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

$r_{\sf OO}$
rOO around the neutral equilibrium separation. Detailed examination confirms the absence of resonances except for a σ* shape resonance below 2 eV—and then only for the shortest
$r_{\sf OO}$
rOO
. (For most geometries the correspondingly longer wavelength of the electron wave resonantly trapped between the O atoms shifts the resonance energy to below the ionization threshold.) Also included are the results obtained after averaging the matrix elements with the weighting
$|\mathcal {X}_{v=0}(r_{\sf OO})|^2$
|Xv=0(rOO)|2
, i.e., sampling the zero point motion for unspecified ion vibrational state. These curves obtained from r-averaged matrix elements closely resemble the post-calculation averaged means ⟨σ(r)⟩r and ⟨bj(r)⟩r. Both these averages are virtually indistinguishable from the fixed geometry results at re except for a small perturbation within 1 eV of threshold (attributable to the short
$r_{\sf OO}$
rOO
σ* shape resonance).

FIG. 3.

Fixed geometry CMS-Xα calculations for the HOMO photoionization from H2O2(v = 0) at several different O-O separations. An averaged result (see text) is also included.

FIG. 3.

Fixed geometry CMS-Xα calculations for the HOMO photoionization from H2O2(v = 0) at several different O-O separations. An averaged result (see text) is also included.

Close modal

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 b1 parameters, even in the absence of any continuum resonance. One might assume this is expected since a positive relative displacement of the ion re would skew the sampling towards vertical ionization occurring at longer

$r_{\sf OO}$
rOO⁠. The fixed geometry b1 curves in Fig. 3 suggest this should provide an upwards shift for b1 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 b1(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.

FIG. 4.

Calculated vibrational branching ratios (BR), chiral (

$b^{\lbrace +1\rbrace }_{1}(v^+)$
b1{+1}(v+)⁠) and anisotropy (
$b^{\lbrace 0\rbrace }_{2} \equiv \beta$
b2{0}β
) parameters for the HOMO photoionization from H2O2(v = 0) for four arbitrary (short and long, positive and negative) assumed cation equilibrium displacements Δre along the
$r_{\sf OO}$
rOO
coordinate.

FIG. 4.

Calculated vibrational branching ratios (BR), chiral (

$b^{\lbrace +1\rbrace }_{1}(v^+)$
b1{+1}(v+)⁠) and anisotropy (
$b^{\lbrace 0\rbrace }_{2} \equiv \beta$
b2{0}β
) parameters for the HOMO photoionization from H2O2(v = 0) for four arbitrary (short and long, positive and negative) assumed cation equilibrium displacements Δre along the
$r_{\sf OO}$
rOO
coordinate.

Close modal

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

$\mathcal {X}$
X⁠. As a further example, Fig. 5 shows a molecule-frame photoelectron CD prediction for the O–O stretch mode PADs.

FIG. 5.

Molecule-frame photoelectron dichroism, ACD (normalized difference LCP-RCP) for light propagating along C2 symmetry axis (z) and detected at θ in the xz plane (Fig. 1). The starred curves omit the vibrational phase (see text).

FIG. 5.

Molecule-frame photoelectron dichroism, ACD (normalized difference LCP-RCP) for light propagating along C2 symmetry axis (z) and detected at θ in the xz plane (Fig. 1). The starred curves omit the vibrational phase (see text).

Close modal

In the foregoing it is the chiral b1 parameters that display the greater sensitivity to assumed geometry, as previously established.16–18 We have previously argued that this is because b1 depends on the sine of the relative phase between interfering partial waves, while b2 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, b1 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 b1(v+ = 1) for small ion equilibrium displacements is surprising, seemingly exceeding the variability in the fixed geometry b1 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 = re and this antisymmetry is approximately maintained, at least for small displacements, in the term

$\mathcal {X}_{v=0}\mathcal {X}^+_{v^+}$
Xv=0Xv++ (see Fig. 2); a consequent negation of complex matrix elements across certain regions of the integration of Eq. (2a) is equivalent to introducing to them an additional phase of π. However, the impact on the phase of the fully averaged matrix element
$T_{i,f,v,v^+}$
Ti,f,v,v+
is both more subtle and potentially more dramatic. Consider splitting the integral Eq. (2a) for the case v+ = 1 into parts over the positive and negative loops of
$\mathcal {X}_{v=0}\mathcal {X}^+_{1^+}$
Xv=0X1++
, with a final step combining the two sub-results. Viewed in the complex plane this latter corresponds to finding the difference of two vectors which, if nearly commensurate in magnitude and phase, will have a resultant of small magnitude (i.e., producing low ionization yield) and angle (phase) that can rapidly swing with relatively small variations of the input amplitudes. The phase of the resultant averaged matrix element can thus be greatly transformed in the averaging operation.

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

$|\mathcal {X}\mathcal {X}^+|$
|XX+|⁠, i.e., neglecting the vibrational phase. As seen in Figs. 5 and 6, b1 does then revert towards the reference norm. For larger geometry displacements the parity effect is already reduced since the pseudo-antisymmetry of
$\mathcal {X}\mathcal {X}^+$
XX+
is lessened, and so the average is less a small difference between two large commensurate complex quantities—hence the resultant phase is better behaved, and b1 variations less extreme.

FIG. 6.

Re-plotted parameters

$b^{\lbrace 0\rbrace }_{2}, b^{\lbrace +1\rbrace }_{1}$
b2{0},b1{+1} as Fig. 4(a) but with vibrational phase factors omitted from the calculation.

FIG. 6.

Re-plotted parameters

$b^{\lbrace 0\rbrace }_{2}, b^{\lbrace +1\rbrace }_{1}$
b2{0},b1{+1} as Fig. 4(a) but with vibrational phase factors omitted from the calculation.

Close modal

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,

$b_1^{+1}$
b1+1 can deviate enormously from expectations given by either a fixed equilibrium geometry or a zero point motion averaged, but non-state-resolved, calculation. This has been attributed to the vibrational phase of
$\mathcal {X}^{+}$
X+
. At low energies, where PECD is anyway more pronounced, b1 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 vv+ 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.

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