Exploring the Spatial Features of Electronic Transitions in Molecular and Biomolecular Systems by Swift Electrons

We devise a new kind of experiment that extends the technology of electron energy loss spectroscopy to probe (supra-)molecular systems: by using an electron beam in a configuration that avoids molecular damage and a very recently introduced electron optics setup for the analysis of the outcoming electrons, one can obtain information on the spatial features of the investigated excitations. Physical insight into the proposed experiment is provided by means of a simple but rigorous model to obtain the transition rate and selection rule. Numerical simulations of DNA G-quadruplexes and other biomolecular systems, based on time dependent density functional theory calculations, point out that the conceived new technique can probe the multipolar components and even the chirality of molecular transitions, superseding the usual optical spectroscopies for those cases that are problematic, such as dipole-forbidden transitions, at a very high spatial resolution.


Numerical integration procedure of the OAM-EELS rate
The numerical integration of the energy loss rate per unit of angular momentum, eq(10), imply the discretization of the molecular transition potential V 0n (r). Due to the dimension of the large systems treated, we use a linear response TD-DFT approach: 1 indeed, the optimal compromise between accuracy and computational cost makes TD-DFT the most widely used method of calculating excitation energies of chemically relevant systems. The transition potential is therefore expressed as: X ia (ω 0n ) and Y ia (ω 0n ) refer to the excitation and de-excitation coefficients involving all the possible pair of occupied φ i (r ) and virtual φ a (r ) orbitals that describe a transition from the ground (|0 ) to an excited state (|n ) associated to an energy difference of ω 0n . The quantity in eq.(1) have been discretized over a cubic grid (with a length side of L, fig.1) by slightly modifying the input of the G16 software, 2 and then averaged along the direction The eq.(2) of the manuscript have been finally integrated over the grid cube, by a homemade Matlab script.

Computational details
We report in this section the computational details used in the simulations, the geometrical structures, the excitation energies, the oscillator strengths and the final OAM resolved transition probabilities of studied systems.

Coordinates of the structures
The coordinates ( in Angstrom) of the geometrical structures of different molecular systems are here reported.

Optimized Guanine
The geometrical structure have been optimized using the B3LYP excahnge-correlation (xc) functional and the cc-pVTZ basis set. TD-DFT simulations have been done using the CAM-B3LYP xc-functional and the aug-cc-pVTZ basis set. The cubic grid, centered on the molecular structure, have a spacing of 0.1 a.u., a side of 40 a.u., and 68921 grid points. The annular shaped electron beam have the following dimensions: 7 a.u. of radius and 3 a.u.
thick.  the structure of the tetramer, we used the same computational protocol of the total system including the grid cube: therefore this last was centered on the total structure.

Alanine enantiomers
The geometrical structure have been optimized using the B3LYP excahnge-correlation (xc) functional and the cc-pVTZ basis set. TD-DFT simulations have been done using the CAM-B3LYP xc-functional and the cc-pVTZ basis set. The cubic grid, centered on the molecular structure, have a spacing of 0.1 a.u., a side of 80 a.u. and 531441 grid points. The annular shaped electron beam have the following dimensions: 7 a.u. of radius and 3 a.u. thick.

G-quadruplex structures
The geometries of the guanine core of parallel and anti-parallel G-quadruplexes have been extracted from the NMR structures of PDB files: PDB ID 2MB2 4 and 143D 5 , respectively. Therefore, they have been refined by projecting the MP2/cc-pVDZ optimized geometry of  All the TD-DFT calculations have been done employing the CAM-B3LYP xc-functional and the 6-31G(d) basis set. The cubic grid, centered on each supramolecular structure, have a spacing of 0.1 a.u., a side of 85 a.u. and 753571 grid points for both the structures. The annular shaped electron beam have the following dimensions: 24 a.u. of radius and 3 a.u. thick.