Toward Reliable Computation of Vibrational Signatures of Complex Molecular Systems/Trendbericht Physikalische Chemie 2025 (3/3)

Dem Ursprung des Lebens nähern sich Forschende aus zwei Richtungen: von der frühen Erde zu den ersten Biomolekülen und von heute zum Genom alter Organismen. CO₂ lässt sich elektrochemisch reduzieren; damit sich das lohnt, müssen Katalysatormaterialien Moleküle mit mindestens zwei Kohlenstoffatomen bilden. Für die elektrokatalytischen Grenzflächen gibt es nun neue additive Fertigungsverfahren. Um Infrarotspektren von Molekülen vollständig zu simulieren, braucht es viel Rechenkapazität – daher gibt es Kniffe, die sogar IR-Signaturen von Proteinen zuverlässig simulierbar machen könnten.

Toward Reliable Computation of Vibrational Signatures of Complex Molecular Systems

From the very beginning of theoretical chemistry, the aim has been to reproduce, explain, and predict experiments. One example that requires theoretical assistance is the interpretation of infrared spectra. Nowadays, infrared spectra of small molecules obtained experimentally and theoretically are often consistent with one another.¹⁾ However, the underlying vibrational wave function methods are computationally so demanding that they are hardly applicable to any system larger than a couple of atoms; an issue when the molecular systems of interest are often more complex than single molecules and highly dynamic: infrared spectroscopy is invaluable in monitoring even small changes within protein structures, e.g. in the hydrogen-bonding network, providing valuable insight into biomolecular processes.^2,3) For instance, difference infrared spectroscopy can resolve subtle changes in the hydrogen bonding pattern of the central flavin molecule of the blue light sensor using flavin (Bluf) upon illumination.⁴⁾

Tackling large systems

For specific and often large biomolecular systems, spectroscopic maps have been established semi-empirically.⁵⁾ However, these are parameterized for certain scenarios and thereby not universally applicable. A more general approach to tackle such system sizes and their dynamics is to generate vibrational spectra from ab-initio dynamics simulations.^6–8) These approaches are, by design, global, i.e. they consider the full system and frequency range. Explaining subtle changes in structurally more well-defined protein systems requires accuracy beyond the typical errors of ab-initio molecular dynamics. For instance, the peaks in the experimental infrared spectrum of Bluf only have a difference of a few reciprocal centimeters between the dark and light state due to different hydrogen-bonding patterns.⁴⁾ This is challenging for computational interpretation of the experimental spectra: the size of these biomolecular systems is beyond the reach of global accurate vibrational treatments. Moreover, they have highly complex potential energy surfaces (PES) with multiple accessible minima. Finally, hydrogen-bonded systems are known to be anharmonic and subject to quantum-mechanical effects.⁹⁾ The theoretical methods meeting these requirements for all degrees of freedom, however, would require utopian amounts of computing power for a full protein.

Fortunately, experimental interest usually focuses on certain spectroscopic signatures. This can for example be the vibrational band dominated by the C=O stretching vibrations in protein backbones, denoted amide-I band. Such a focus is even more pronounced in difference infrared spectroscopy^2,3) for slightly different samples, like the dark and light states of the Bluf protein.

In these cases, only the spectral regions that show a difference are relevant for interpretation (Figure 1, p. 73, calculated spectrum of a small molecule).

Multi-level and embedding schemes

Similarly, vibrational multi-level approaches¹¹⁾ concentrate the computational effort on the relevant regions. Multi-level or embedding schemes are established for the electronic problem and often lead to conclusive calculations on complex systems.^12–14) Alongside multi-level methods focusing on a particular spatial region, fragmentation approaches similarly reduce the computational effort.^15–19) These approaches are particularly suited for global properties, like the electronic energy, but can also be combined with local multi-level ideas.^20,21)

Most multi-level or fragmentation schemes for the electronic problem divide the real space into regions that are treated separately, on different footings, or both. In contrast to that, multi-level and embedding schemes in vibrational-structure theory allow the partitioning of the 3N–6-dimensional vibrational space into different regions. By this, detailed handles for the optimal computational setup can be provided. The handles to tailor vibrational structure calculations can be associated with the underlying potential energy and the vibrational wave function itself. However, to gain experience and tailor the parameterization for a concrete example, reliable reference data is needed, which is rare, even for small organic molecules.^22–25)

Matching theory and experiment

The more complex the structural target, the more sophisticated the computational model needs to become. The many accessible minima in the PES of biomolecular systems can be accounted for by weight-averaging,²⁶⁾ counting the population of the different minima in molecular dynamics simulations and then averaging the spectra for the respective minima. These local spectra may then be obtained via tailored multi-level vibrational-structure calculations for each minimum. The individual vibrational calculations themselves already require many choices in the computational setup to be feasible. This includes the vibrational space to be considered in the calculation as well as its partition into parts that are treated on a different footing. For instance, describing be the spectral region between 1700 and 1800 cm^–1 of uracil requires the carbonyl stretching vibrations. But, those alone do not match with the experiment (Figure 2, 2-mode spectrum). So, which additional degrees of freedom and which couplings need to be considered on what accuracy of the PES to achieve sufficient agreement with experiments?

Figure 2 shows our best tailored spectrum in vacuum compared to an experimental spectrum in an Ar matrix at 10 K. There, the experiment agrees with the calculation. This includes the resonance band at about 1700 – 1710 cm^–1 as well as the relative intensities of the peaks. We found that recovering the resonance band requires at least third-order couplings between certain modes in the PES, though a semi-empirical r²scan-3c electronic structure description of these couplings was sufficient. Obtaining positions of the theoretical peaks within about 10 cm^–1 of the experimental value, however, called for a more expensive electronic structure description on a selected part of the PES using the hybrid functional B2PLYP/aug-cc-pVTZ. The bottom of Figure 2 depicts the vibrational modes spanning this part of the PES. They have been chosen based on couplings observed in lower-level calculations.

The theoretical spectrum is subject to errors, as effects impacting the experimental spectrum, such as temperature²⁷⁾ or matrix effects^22,28), are not considered in the computational calculations. Hence, an agreement within 10 cm^–1 is considered within the uncertainties.

The individual electronic calculations for the PES of larger systems, like the aforementioned proteins, become too expensive to calculate. Here, multi-level³⁰⁾ approaches, fragmentation approaches^31–34), or a combination of both, are indispensable.

For a water wire in the protein bacteriorhodopsin, Yagi et al. yielded accurate spectroscopic signatures by combining weight-averaging, selected spaces, and multi-level and fragmentation for the PES.³⁵⁾

Vibrational embedding theory

The reduced-space model of the water wire in bacteriorhodopsin contains 16 vibrational degrees of freedom and is, therefore, small enough for standard accurate vibrational wave function approaches. If the vibrational space under consideration gets larger, it is beneficial to also tailor the vibrational wave function to the vibrational signature of interest – for an overview, see review ref. 11) and references therein. While most of the approaches mentioned there focus on small molecules, we recently applied our vibrational embedding theory to the water wire in bacteriorhodopsin.³⁶⁾

In combination with a local description of the vibrational coordinates, these embedding schemes may also work in first-principle vibrational exciton models for the description of two-dimensional infrared spectra,^37–39) providing yet more information on the biological systems.

Outlook

With the above-mentioned method developments and approaches, the calculation of infrared signatures of complex biomolecular systems with rigorous quantum-mechanical footings are soon within reach. In combination with experiments, this will enable unprecedented understanding of subtle, but important structural processes in biomolecular systems. Developing and assessing robust, physically sound models, providing “the right answer for the right reason”, will benefit from accurate experimental references of varying size and complexity. This has similarly been pointed out for electronic structure methods by Mata and Suhm.⁴⁰⁾ For vibrational calculations, this may even be more important as high-accurate theoretical references are out of reach even for medium-sized molecules. Such well-tested and accurate multi-level vibrational methods will then not only enable a deeper understanding of biomolecular processes, they may provide more reliable reference data than currently available for mixed quantum-classical spectroscopic maps applicable for even larger systems.

Drei Fragen an die Autorin: Carolin König

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Carolin König ist seit dem Jahr 2020 Tenure-Track-Professorin an der Universität Hannover. Davor hatte sie ab 2018 eine Juniorprofessur für Theoretische Chemie an der Universität Kiel inne. Zuvor hat sie als Postdoc in Aarhus und Stockholm gearbeitet. Ihr Doktorat in Theoretischer Chemie begann sie in Leiden, und nach einem Zwischenstopp an der TU Braunschweig schloss sie Ihre Doktorarbeit 2013 an der Universität Münster ab. König war unter anderem Alexander-von-Humboldt-Stipendiatin und Marie-Skłodowska-Curie-Fellow. Sie ist zudem Emmy-Noether-Gruppenleiterin und erhielt 2023 den Hellmann-Preis der Arbeitsgemeinschaft Theoretische Chemie.

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