# Local Approaches in Vibrational Structure Theory

Vibrational spectra of complex systems, such as proteins, are very sensitive to structural changes of the investigated molecules. The interpretation of these spectra, however, requires theoretical assistanc and the most standard harmonic approximation does often leave ambiguities. In vibrational structure theory we go beyond the harmonic approximation and use vibrational wave functions in anharmonic potential energy surfaces to calculate optical spectra. These calculations are, however, considerable more demanding and exhibit steeper computational scaling than the standard harmonic choice. This restricts the applicability of the standard models to a few tens of vibrational modes. Especially for larger systems, we expect that the locality of the interaction within a system can be exploited to devise highly efficient vibrational structure approaches. With these developments we expect to push the size limitations of these methods to much larger system sizes.

Our work in this field comprises all three components, i.e., the vibrational coordinates, the generation of the potential energy surface as well as the parameterization of the vibrational wave function itself. The software platform of these projects is the MidasCpp code by the group of Ove Christiansen at Aarhus University, which we are collaborating with.

**Local vibrational coordinates**

Our work in the generation of alternative vibrational coordinates includes the combination of localization and optimization schemes for rectilinear vibrational coordinates as well as the so-called flexible adaptation of local coordinates of nuclei (FALCON). The latter FALCON-type coordinates are due to their well-defined spatial locality key ingredients for the efficient generation of PESs in the double incremental approach described below. The FALCON scheme can furthermore been used to generate spatially motivated vibrational subspaces that can be expanded systematically. We could, for example, show that for alpha-conotoxin a subspace of 146 strictly local modes (out of the total space of 519 vibrations) is capable of approximating the full harmonic solutions with an error that is clearly below the errors expected from other error sources.

E.L. Klinting, C. König, and O. Christiansen, *Hybrid Optimized and Localized Vibrational Coordinates*, J. Phys. Chem. A, **119**, 11007-11021 (2015).

C. König, M.B. Hansen, I.H. Godtliebsen, and O. Christiansen, *FALCON: A Method for Flexible Adaptation of Local Coordinates of Nuclei*, J. Chem. Phys., **144**, 074108 (2016).

**Fragmented description of the potential energy surface**

We have shown that the so-called double incremental expansion in combination with the above-mentioned FALCON coordinates (DIF) can be used to devise a computationally highly efficient scheme for PES generations. Adding an approximate transformation step for the some of the vibrational coordinates in the PES (DIFACT), even linear scaling of the accumulated cost for all single-point calculations in the PES generation can be achieved.

This work pushes the previous size limitations for PES generations from quantum-chemical electronic energy points to significantly larger systems. This means that vibrational structure calculations for covalently bound systems with several hundreds of degrees of freedom are now within reach: In the hexa-phenyl case, we treated all 180 vibrational degrees of freedom explicitly in the PES as well as in the anharmonic vibrational calculations including infrared spectra.

C. König and O. Christiansen,* Linear-Scaling Generation of Potential Energy Surfaces Using a Double Incremental Expansion*, J. Chem. Phys., **145**, 064105 (2016).

D. Madsen, O. Christiansen, and C. König, *Anharmonic Vibrational Spectra from Double Incremental Potential Energy and Dip**ole Surfaces*, Phys. Chem. Chem. Phys.,** 20**,3445-3456 (2018).