Supplemental Material for Site-dependent nuclear dynamics in core-excited butadiene

In the present study we investigate nuclear dynamics upon terminal and central core-electron excitation to the π∗ orbital in butadiene with special focus on differences in the local geometry at the site where the core-electron excitation takes place and how these connect to nuclear dynamics. We utilize the equivalent-core model1 to calculate the geometries of the molecule in the core-excited states for terminal and central carbon excitation, and to understand how the localized core hole affects nuclear dynamics. The model is relatively straightforward, and intuitive. Upon creation of a core hole the valence electrons see an effective nuclear charge which increases by one, and this affects bond distances and angles

as well as potential surfaces. Practically speaking the terminal carbon excited state is simulated by the NH 2 CHCHCH 2 molecule, while the central carbon excitation is represented by the CH 2 NHCHCH 2 . molecule The optimized geometries are calculated using the electron correlation at the CASPT2 level and a restricted active space (RAS)SCF as implemented in the MOLCAS software program. 2 The orbitals were obtained using the cc-pVDZ basis set and self-consistent field (SCF) calculation. 3 The resulting geometries for the groundstate molecule, the terminal carbon 1s-π * , and the central carbon 1s-π * are shown in Fig.   S1. Once the optimized geometries are obtained, the force constant matrix and analysis of the harmonic frequency are produced by computing the analytical Hessian at the optimized geometry. This calculation is done using the MCKINLEY program as implemented in the MOLCAS software program. 2 The infrared (IR) absorption intensity of the harmonic frequencies for the optimized wave-functions are shown in Fig. S1 (a). The frequencies are expressed in reciprocal centimeters. In Table S1 the assignment of the harmonic frequencies with the normal modes is presented.
As shown in Fig.S1 (a), the optimized geometries of the terminal and the central coreexcited states are different. For excitation of the terminal carbon 1s electron the conjugated C=C bond distance is reduced from 1.3496 to 1.4012 Å due to the anti-bonding character of the π * orbital on the terminal C=C bonds. This weakening of the C=C bond at the excitation site results in C-H bond contraction at both central and terminal carbon sites.
Apart from the bond dynamics upon π * orbital occupation, a site-selective behaviour is also visible. This is reflected on the increased vibrational frequency of the C * T H 2 (NH 2 ) and C * C (NH) bonds, clearly seen for instance in the asymmetric stretch which exceeds 3600 cm −1 , reflecting changes in the NH 2 and NH bond lengths. In addition, a different out-of-plane motion appears in both core-excited states; we find that the hydrogen atoms connected to the terminal core-excited carbon (nitrogen in the NH 2 CHCHCH 2 ) are 35.7 o out of the molecular plane, while the hydrogen atoms connected to the terminal carbon next to the central core-excited carbon (CH 2 -N in the CH 2 NHCHCH 2 ) are only 8.5 o out of the molecular plane. This difference is also reflected in the out-of-plane bending vibrational frequency upon core-excitation; on the NH 2 out-of-plane vibrational frequency of -452 cm−1 for the NH 2 CHCHCH 2 , and on the CH 2 -N out-of-plane vibrational frequency value of -220 cm−1 for the CH 2 NHCHCH 2 . Negative frequencies should be used for zero-point energy correction of the equilibrium geometry of the terminal and the central core-excited states.
i.e., the equilibrium geometry of these states is bent on the site close to the excited carbon.
This result confirms dynamics upon core-hole localization conclusively and the π * orbital occupation; while the excitation to the π * leaves the molecule with a freedom for an outof-plane motion around the terminal bonds, the localized core-hole induces an out-of-plane bending around the site of excitation.
Independently of the geometry optimization, we have performed a calculation of the poten-tial energy surface using the "Z+1" model along the N-H out-of-plane bending and stretching coordinates, with all other coordinates kept fixed. The 2D localized representation of the potential energy surfaces (PESs) for the terminal C1s(a g )-π * (a u ) and central C1s(b u )π * (b g )states are shown in Fig.S1 (b). The calculation indicates a chemical shift of 0.45 eV, which is 0.2 eV less than the experimental value of 0.65 eV.
The left panel shows the PESs along the N-H out-of-plane-bending coordinate. From the PESs we can see that the potential energy of the terminal core-excited state slightly decreases along the bending coordinate up to an angle of 35 degrees. This is in good agreement with the calculated angle using the single-shot geometry optimization. This is why the potential energy of the central core-excited state increases along the op-bending coordinate, suggesting that the planar trans geometry survives upon the central core-excitation. In the right panel the PESs along the N-H stretching coordinate shows identical behaviour for both states; where the N-H bond lengths decrease by about 0.05 Å upon core-electron excitation.
Nuclear dynamics studying hydrogen evaporation One of the advantages of multi-coincidence measurements is that the fragment KER correlation reflects the mechanism and sequence of three-body dissociation channels. The KER correlation between three fragments of channel 4 H is depicted in the regression distribution of the Dalitz plot 5 shown in Fig. S2 (a). An identical regression distribution is measured for channel 4 D . Each axis presents a normalized squared momentum, In a sample of randomly-oriented molecules excited by radiation with linear polarization the ensemble of resonantly-excited species will be anisotropic with respect to the direction of the polarization vector. The angular distribution of the fragments emitted from such an ensemble provides a fingerprint of nuclear dynamics. [6][7][8][9] The molecular anisotropy parameter, β, can be determined for any set of data, also after reconstructing the residual momenta of the corresponding ion-pairs dissociated in channels 4 D and 4 H (P 2 +P 3 =-P 1 ). This assumes that the C-C bond break is fast enough with respect to the rotational period of the molecule. The angular distribution for the deuterium and hydrogen atoms is given by the well-established expression for the differential cross section: where σ is the total cross section (integrated over all angles), θ is the angle between the molecular bond and the polarization direction and P 2 is the second-order Legendre polynomial, P 2 (x) = (3x 2 − 1)/2. Note that for two-body dissociation the anisotropy parameter is related to the angle between the broken bond and the polarization vector of the light, θ, by this expression β=3cos 2 θ -1. 10 The highest probability for butadiene C1s-π * resonant excitation by radiation with linear polarization is when the molecular plane is perpendicular to the polarization (A g ⊗ A u ⊗ A u ).
The alignment of the excited molecule with respect to the polarization vector is shown in Fig.   S2 (b). This alignment gives for reference anisotropic angular distribution with anisotropy parameter 11 β −→ −1 in the case of rapid dissociation; taking place much faster than molecular rotation. In the three-body channels 4 D and 4 H the rapid deuterium and hydrogen evaporation, that are very early after core-excitation, thus deviation from the anisotropic angular distribution (β −→ 0) is an indication for the out-of-plane nuclear motions on the terminal and central sites, respectively. For the C T (0−→0 ) transition, features 0 and A, however, the anisotropy parameters of deuterium evaporation shows deviation toward isotropic distribution with anisotropy parameters of β -0.39. As the initial evaporation can be seen as a 2-body breakup, this would correspond to an average C-D bond angle of 15 • with respect to the molecular plane. Such a deviation indicates out-of-plane nuclear motion taking place during the core-hole lifetime, in the the low energy region of the terminal core-excited state. predicted by the "Z+1" model for the terminal core-excited states, where the hydrogen atoms (NH 2 ) undergo a localized out-of-plane bending motion.by a localized core hole on the terminal carbon. C T 1s→ π * C C 1s→ π * Figure S1: Nuclear dynamics following terminal and central core-excitation to the π * orbital is studied using the equivalent-cores model. (a) The geometry optimization was carried out for 1,3 trans butadiene, for the NH 2 CHCHCH 2 , and the CH 2 NHCHCH 2 to simulate the terminal and central core-excited butadiene. (b) The potential energy curves for the terminal and central core-excited states have been calculated along the C * -H 2 bending (left) and C * -H stretching (right) coordinates. This shows an identical sequential three-body break-up; where H/D atoms evaporates early after the core-excitation and it continues with C-C dissociation. (b) The C1s total ion yield spectrum of the partially deuteriated butadiene with vibrational features indicated on the spectrum are shown together with the angular anisotropies for the hydrogen and deuterium evaporation. The angular anisotropy, β value, and the corresponding angle between the hydrogen/deuterium momentum and the polarization vectors are plotted for photon energies within the resonance region. The accuracy of the anisotropy values is determined by detector efficiency, errors in the definition of the spatial and temporal center of the data, and failure to properly define the direction of the polarization vector of the incident light relative to the detector orientation. For strong channels with good statistics, this leads to an estimate of about ±0.05 error on β value.