Molecular-frame differential photoelectron circular dichroism of O 1s-photoelectrons of triﬂuoromethyloxirane

The differential photoelectron circular dichroism (PECD) of O 1 s photoelectrons of R -triﬂuoromethyloxirane enantiomers as a function of the photoelectron emission direction in the molecular frame of reference and the direction from which circularly polarized light hits the molecule, is studied experimentally and theoretically for different photoelectron kinetic energies. A coincident detection of the photoelectrons and two ionic molecular fragments, performed with cold target recoil ion momentum spectroscopy, allows us to determine the orientation of the molecule in the laboratory frame and to obtain in addition the molecular-frame photo-electron diffraction patterns. From these we deduce the differential PECD. For given molecular orientations and photoelectron emission directions, we observe a normalized PECD strength clearly beyond 50%. These observations are in agreement with respective relaxed-core Hartree-Fock calculations, performed by employing the single center method. The present results support our recent observation of a huge differential PECD in O 1 s photoemission of the methyloxirane molecule.

the laboratory-frame differential photoionization cross section over Legendre polynomials [11][12][13].Those expansion coefficients are differential quantities, which result from an incomplete compensation between different partial photoionization amplitudes [2].Thus, for randomly oriented molecules, the effect partially cancels out leading to a much reduced signal strength.On the contrary, a circular dichroism in the angular distribution [14,15] and even an apparent PECD [16] on the order of 100% can be observed when fixing achiral diatomic molecules in space.Thus, as predicted theoretically in our previous work [17], PECD becomes an effect as strong as 100% when certain (fixed) orientations of chiral molecules with respect to the propagation direction of the ionizing light are considered.Such a huge dichroic contrast considerably increases the sensitivity of PECD, e.g., for enantiomeric-excess determination.
In order to confirm this prediction, we first studied in the past uniaxially oriented chiral molecules.As test bench systems, we chose methyloxirane (MOx, C 3 H 6 O) in Ref. [17] and trifluoromethyloxirane (TFMOx, C 3 H 3 F 3 O) in Refs.[18,19].These joint experimental and theoretical studies [17][18][19] confirmed that fixing already one molecular orientation axis in space increases the differential PECD by almost an order of magnitude.In a subsequent study of MOx, we were able to finally perform measurements of a chiral molecule fully fixed in space and reported enantiosensitive polarization-averaged molecular-frame photoelectron diffraction patterns [20].Very recently, we demonstrated a fourfold differential PECD of fully fixed-in-space MOx clearly beyond 50% [21], confirming thereby theoretical predictions of Ref. [17].
This article is part of our series of studies [18,19] of angleresolved photoionization of TFMOx.Here, we investigate its differential PECD in the molecular frame of reference, similar to our previous studies of MOx [20,21].In particular, we report the polarization-averaged molecular-frame and fourfold differential photoemission patterns with the respective PECDs for O 1s photoelectrons of TFMOx at different kinetic energies.

A. Experiment
The experimental data for the O 1s photoionization of TFMOx were recorded at Synchrotron SOLEIL (Saint-Aubin, France) at beamline SEXTANTS using cold target recoil ion momentum spectroscopy (COLTRIMS, [22][23][24]) during the same experimental campaign as the data presented in Ref. [18].The details on the experiment can be found therein.Briefly, the measurements were performed using enantiopure R-TFMOx molecules obtained commercially from SynQuest Laboratories.The molecular gas jet was introduced in the reaction chamber via expansion of the molecular vapor using a closed-loop recycling system [25].The emitted electrons and ionic fragments were accelerated to opposite directions towards time-and position-sensitive microchannel-plate detectors with enhanced efficiency on the ion detector [26].We employed a mesh-free COLTRIMS spectrometer with electrostatic lenses and (quasi) field-free drift regions.From the measured particles' positions of impact and times of flight, we reconstructed their initial vector momenta.The electron energy was calibrated via separate calibration measurements of Ar 2p photoelectrons.The measurements on TFMOx were performed at five selected photon energies, which yielded O 1s photoelectrons centered at kinetic energies of 3.1, 4.1, 6.1, 8.1, and 11.7 eV, respectively.
The measured momenta of the photoelectrons and ionic fragments provide the information on the spatial orientation of the molecule at the instant of the photoionization and allow thereby to access the photoelectron emission distributions in the molecular frame of reference (under the assumption of the axial-recoil approximation [27]).In more detail, we determined the spatial orientation of the indvidual TFMOx molecules in the gas phase by using the following fragmentation channel: For this channel, we recorded between 3 × 10 5 and 6 × 10 5 valid events for each light helicity and photon energy.The neutral fragment was not recorded by our ion detection.Instead, we reconstructed the momentum vector of the neutral OH 0 fragment by using the measured momenta of the charged fragments and exploiting momentum conservation (while neglecting the recoil from the electrons and the linear momentum of the photon).In order to distinguish this fragmentation channel from background and to ensure that three fragment momenta define a plane (i.e., are not collinear), the following momentum gates The momenta of all fragments and the derived fragment coordinate system are schematically depicted in Fig. 1.

B. Theory
The molecular-frame photoelectron angular distribution (MFPAD), induced by the one-photon ionization of a molecule using circularly polarized light (q = ±1), can be described by the following expansion of the differential photoionization cross section over (complex conjugated) spherical harmonics Y LM : Here, k = {θ, φ} are the photoelectron emission angles in the molecular frame of reference, and {α, β} are the Euler angles describing the orientation of that molecular coordinate frame with respect to the light propagation direction.Note that the third angle γ , which describes the rotation around the light propagation direction (laboratory z axis), is irrelevant for circularly polarized light and can be set to γ = 0.The remaining orientation angles can also be considered as polar β and azimuthal α spherical angles, describing the direction from which incident light hits the molecule.
The analytic expression for the expansion coefficients B q LM in Eq. ( 1) reads ) where D are the Wigner rotation matrices.The photoelectron scattering phase e −iδ m is included in the photoionization transition amplitude A ε mk for the emission of a photoelectron partial wave with energy ε, angular momentum , and its projection m, after absorption of a photon with polarization k, altogether given in the frame of molecular reference.
The simplest way to obtain polarization-averaged molecular-frame photoelectron angular distributions (PA-MFPADs) is to average over all molecular orientation Euler angles (i.e., by averaging over all possible directions of the ionizing light), which can be performed analytically using the following orthogonality properties of the rotational matrices: As a consequence, the corresponding expansion coefficients B LM become independent of the light polarization q [20].The partial transition amplitudes A ε mk for the O 1s photoionization of R-TFMOx, entering Eq. ( 2), were computed for the equilibrium internuclear geometry of the neutral electronic ground state with the single center (SC) method and code [28][29][30].Calculations were performed in the relaxed-core Hartree-Fock approximation using the SC expansions restricted by c 79 and ε 29 for the occupied and continuum orbitals, respectively, as described in detail in our previous work on the O 1s photoionization of this molecule [18].

A. Polarization-averaged MFPAD
In the case of a molecular breakup into fragments consisting of several atoms (where usually the fragments' center of mass does not coincide with the center of charge), the fragmentation dynamics is accompanied by relative rotations of the fragments.Thereby, the detected asymptotic momenta (see Fig. 1) and the fragment coordinate system {X, Y, Z} derived from them cannot be related to the orientation of the molecule at the instant of photoemission in a straightforward manner.In order to infer such a molecular coordinate system {x , y , z }, we applied a procedure introduced in our previous work [20].In particular, we search for a molecular orientation, which provides the best agreement between the computed and the measured PA-MFPADs.For this purpose, first, the minimum value of each PA-MFPAD was subtracted from the distribution, and the integral emission signal was normalized to unity.We then quantified the amount of agreement between the renormalized theoretical I T (θ, ϕ) and experimental I E (θ, ϕ) emission patterns, using the following figure of merit, This parameter depends on the relative orientation of the molecular {x , y , z } and fragment {X, Y, Z} coordinate systems, which is given by the roll, pitch, and yaw angles: R = {R X , R Y , R Z }.Based on the axial-recoil approximation [27] and using the assumption that the molecular and fragment coordinate systems coincide for the initial parameter, we computed the PA-MFPAD and, by rotating this computed pattern in small steps, searched for a molecular orientation that yields smallest values of d 2 .
The procedure described above was applied to the data obtained for all five photoelectron kinetic energies and yielded very similar molecular coordinate systems, after applying this optimization scheme.This corresponding coordinate system is illustrated in Fig. 1 by the black {x , y , z } arrows.As one can see, this molecular coordinate system is considerably skewed with respect to the fragment coordinate system (depicted by the green {X, Y, Z} arrows), suggesting strong (yet deterministic) relative rotation of the fragments in the course of this breakup.We observed a very similar result in our previous study on O 1s photoelectrons of uniaxially oriented TFMOx [18].There, owing to intricate rotations of two fragments with respect to each other occurring in the course of a slow dissociation, a fragmentation axis (which connected the trifluoromethyl-group and oxirane-ring fragments) deviated from a theoretically found optimal axis (which was almost parallel to the plane of the oxirane ring).
The PA-MFPADs of the O 1s photoelectrons of R-TFMOx are depicted in Fig. 2 for five photoelectron kinetic energies as functions of the photoelectron emission angles.For the experiment, those angles are defined in the fragment coordinate system, and for theory, in the molecular system.As one can see from this figure, the theory provides a good semiquantitative description of the experiment, in particular, when considering the complexity of the problem.It is worth noting that the fragment coordinate system alone does not provide access to the handedness of the molecule.This is because the measured momenta of the fragments define only a plane.However, PA-MFPADs are enantiosensitive objects [20] and switching between the enantiomers mirrors the emission distributions via reflection at cos θ = 0.

B. Differential MFPADs and PECDs
We divided our data set with respect to the orientation of the molecule.Considering the available statistics recorded in the experiment, we binned that division in constant steps of cos θ = 1 3 and ϕ = 30 • into 72 subsets.We present these distributions as functions of the two photoelectron emission angles {θ, ϕ} and transformed the photon propagation direction into the same coordinates.Accordingly, the photoemission and light-propagation directions are defined in the fragment coordinate system for the experimental data and in the molecular system for the theoretical ones, just as in the preceding sections.However, for clarity, we will focus in the following on measured and computed MFPADs and PECDs for one representative molecular orientation, whereas the full set of results can be found in the Supplemental Material document [31].
The measured and computed MFPADs of the O 1s photoelectrons of R-TFMOx are depicted in Fig. 3 for five photoelectron kinetic energies.The experimental results are shown in the left column, the computed MFPADs on the right.These are the distributions highlighted by the red-colored border in the overview Figs.S1-S5 in the Supplemental Material [31].The selected representative orientation of the molecule with respect to the propagation direction of circularly polarized light (negative helicity) is indicated in the bottom of Fig. 3.The relative skew between the fragment and the molecular coordinate systems, seen from those insets with respect to the light propagation direction, provides an estimate for the relative rotation of the fragments, which occurs during the breakup of the molecule and prior to the detection of the fragments.
As one can see from this figure, the measured and computed differential MFPADs are in a reasonable semiquantitative agreement.In order to obtain such MFPADs for the opposite enantiomer (not measured in this paper) and light polarization, the following two symmetry operations (based on the electric-dipole approximation) can be used.First, the reflection of the photoelectron-emission and light-propagation coordinates at cos θ = 0 mediates simultaneous switching between the two enantiomers and the light helicities.Second, keeping the enantiomer and swapping helicity is equivalent to reversing the light propagation direction, which is mediated by transforming ϕ → ϕ + 180 • and cos θ → − cos θ .
Similar to our previous works [17][18][19]21], the differential PECD is defined as the normalized difference of the photoelectron distributions recorded for circularly polarized light with positive and negative helicities (I + and I − , respectively), which corresponds to the half of the usually used 2b 1 value [11][12][13].The measured and computed differential PECDs of the O 1s photoelectrons of R-TFMOx are depicted in Fig. 4 for five photoelectron kinetic energies.The orientation of the molecule with respect to the light propagation direction is the same as in Fig. 3.The present theoretical predictions reproduce the measured PECD landscapes including their signs.However, the contrast of the computed PECDs (clearly above ±80%) is somewhat larger than that of the measured ones (which is only around ±50%).

IV. SUMMARY
The work presented in this paper is a logical continuation of our previous experimental and theoretical studies of the differential PECD of partly and fully fixed-in-space chiral molecules [17][18][19][20][21]. Here, we explored the differential PECD of O 1s photoelectrons in body-fixed-frame R-TFMOx molecules.In particular, we measured and computed the PA-MFPADs and the MFPADs of spatially oriented TFMOx for five photoelectron kinetic energies.In order to achieve a reasonable agreement between the present theory and the experiment, a substantial rotation of the fragments during the respective breakup was assumed.Thereby, the main emission patterns, observed in the experimental MFPADs of fixedin-space TFMOx, are reproduced in the theory at similar photoelectron emission directions but still with a somewhat higher relative contrast.The found average skewing between the theoretically and experimentally deduced molecular and fragment coordinate frames, respectively, can be used to estimate the dynamics and times of a relative rotation of the fragments during the considered breakup.Possible reasons for a quantitative disagreement between the measured and computed MFPADs are discussed in great detail in our previous work on this molecule [18].With the help of those MFPADs, we observe a differential PECD of fully fixed-inspace TFMOx molecules well beyond 50%, similar to our recent analogous study of the closely related molecule MOx [21].Our studies confirm again that orienting the molecule increases the dichroic contrast of this chiral asymmetry by up to two orders of magnitude (from a few to almost 100%), which considerably improves its sensitivity for chiral recognition in the gas phase.

FIG. 1 .
FIG.1.The fragment coordinate system (green {X, Y, Z} axes) for R-TFMOx, obtained with the help of the fragments' momenta (red arrows with yellow spots at their tips representing measured distributions), together with the constructed molecular coordinate system that corresponds to the instant of photoionization (black {x , y , z } axes, see the text for details).

FIG. 2 .
FIG. 2. Experimental (left panels A, C, E, G, and I) and theoretical (right panels B, D, F, H, and J) polarization-averaged molecular-frame photoelectron angular distributions (PA-MFPADs) as functions of the photoelectron emission angles {θ, φ} for different kinetic energies of the O 1s photoelectrons of R-TFMOx.Panels A and B for 3.1 eV; C and D for 4.1 eV; E and F for 6.1 eV; G and H for 8.1 eV; I and J for 11.7 eV.

FIG. 3 .
FIG. 3. Experimental (left panels A, C, E, G, and I) and theoretical (right panels B, D, F, H, and J) MFPADs as functions of the photoelectron emission angles {θ, φ}, obtained for different kinetic energies of the O 1s photoelectrons of R-TFMOx.The selected orientation of the molecule (with respect to the propagation direction of the circularly polarized light with negative helicity) is visualized at the bottom.This direction differs in the fragment and molecular coordinate systems (see the text for details).The photoelectron energy is 3.1 eV in panels A and B; 4.1 eV in C and D; 6.1 eV in E and F; 8.1 eV in G and H; 11.7 eV in I and J.

FIG. 4 .
FIG. 4. Experimental (left panels A, C, E, G, and I) and theoretical (right panels B, D, F, H, and J) molecular-frame differential PECDs as functions of the photoelectron emission angles {θ, φ}, obtained for different kinetic energies of the O 1s photoelectrons of R-TFMOx.The molecular orientation with respect to the light propagation direction is the same as in Fig. 3.The photoelectron energy is 3.1 eV in panels A and B; 4.1 eV in C and D; 6.1 eV in E and F; 8.1 eV in G and H; 11.7 eV in I and J.