Theoretical Methods for Vibrational Spectroscopy and Collision Induced Dissociation in the Gas Phase

Abstract

In this chapter we review recent advances in theoretical methods to understand and rationalize anharmonic vibrational spectroscopy (IR-MPD and IR-PD) and collision induced dissociations (CID) in the gas phase. We focused our attention on the application of molecular dynamics-based methods. DFT-based molecular dynamics was shown to be able to reproduce InfraRed Multi-Photon Dissociation (IR-MPD) and InfraRed Pre-Dissociation (IR-PD) action spectroscopy experiments, and help assign the vibrational bands, taking into account finite temperature, conformational dynamics, and various anharmonicities. Crucial examples of dynamical vibrational spectroscopy are given on the protonated Ala _n H⁺ series (related to IR-MPD in the 800–4,000 cm⁻¹ domain), ionic clusters (related to IR-PD in the 3,000–4,000 cm⁻¹ region), and neutral peptides (related to IR-MPD in the far-IR). We give examples from simple (e.g., cationized urea) to more complex (e.g., peptides and carbohydrates) molecular systems where molecular dynamics was particularly suited to understanding CID experiments.

1 Introduction

Vibrational spectroscopy is one of the most important and crucial experimental tools broadly applied in chemistry, physics, and biology. Vibrational spectroscopy is unique in the sense that it provides a relationship between the three-dimensional organization of molecular structures and their vibrational fingerprints. This book provides a snapshot of the tremendous developments that have been made in the field of gas phase spectroscopy for the structural characterization of (bio)-molecules and assemblies of molecules within the past decade. A preceding snapshot was given in the book by Schermann [1]. Vibrational action spectroscopy has in particular experienced an amazingly successful era, with many new experimental developments and applications in many different areas in chemistry, analytical chemistry, physics, and biology. Previous reviews include [2–6], and updated reviews can be found in the present book.

Vibrational action spectroscopy can be roughly divided into experiments performed at low (and extremely low) temperatures and experiments performed at finite temperature. A further division concerns neutral and charged molecules (and clusters) being investigated, in relation to the initial stage of production of the molecules. One more division amongst the experimental set-ups is the number of photons used for laser interrogation and subsequent molecular fragmentation. IR-MPD (InfraRed Multi-Photon Dissociation) experiments use several photons before the fragmentation of the investigated molecule is achieved, whereas IR-PD (InfraRed Photo-Dissociation or InfraRed Pre-Dissociation) experiments use one photon for the departure of the tagged entity from the investigated molecular assembly. IR-MPD experiments have been more traditionally applied in the domain of assemblies of biomolecules, possibly interacting with metal ions and/or a few solvent molecules, whereas IR-PD experiments have been more traditionally applied in the domain of clusters and small biomolecules. See again [1–6] and the present chapters for more details and illustrations.

As in many other experimental areas, vibrational action spectroscopy strongly relies on theoretical calculations to provide a clear and definitive picture of the structures of the molecular assemblies. The synergy between experiments and theoretical calculations is visible in the publications: almost 100% of the papers in the gas phase IR-PD/IR-MPD communities are now published with associated theoretical calculations to help analyze the experimental fingerprints.

In this chapter, our goal is to present theoretical methods applied to gas phase vibrational spectroscopy. This is reviewed in Sect. 3 where we present harmonic and anharmonic spectra calculations, with special emphasis on dynamical approaches to anharmonic spectroscopy. In particular, we present the many reasons and advantages of dynamical anharmonic theoretical spectroscopy over harmonic/anharmonic non-dynamical methods in Sect. 3.1. Illustrations taken from our work on dynamical theoretical spectroscopy are then presented in Sects. 4–6 in relation to action spectroscopy experiments. All examples presented here are conducted either in relation to finite temperature IR-MPD and IR-PD experiments, or to cold IR-MPD experiments. Beyond the conformational dynamics provided by the finite temperature trajectories, the chosen examples illustrate how the dynamical spectra manage to capture vibrational anharmonicities of different origins, of different strengths, in various domains of the vibrations from 100 to 4,000 cm⁻¹, and on various molecular systems. Other comprehensive reviews on theoretical anharmonic spectroscopy can be found in [7–9].

The examples we have taken from our work have been chosen to illustrate a few issues, where anharmonicities are the main ingredients and where dynamical theoretical spectroscopy plays a crucial role in elucidating the fingerprints. The first examples presented in Sect. 4 on protonated alanine peptides Ala _n H⁺ of increasing size and structural complexity illustrate the importance of temperature and entropic effects in structural equilibria, and how conformational dynamics at finite temperature is the key ingredient for unraveling the structures-vibrational fingerprints relationships for floppy peptide molecules. Also illustrated in these examples is the importance of vibrational anharmonicities in $\mathrm{N}\;\hbox{--} {\mathrm{H}}^{+}\cdots \mathrm{O}=\mathrm{C}$ hydrogen bonds, and how their precise characterization in the dynamical spectra is a definitive asset in unraveling the peptidic structures. These theoretical investigations have been carried in relation with finite temperature IR-MPD action spectroscopy experiments, where temperatures varied from ~300 K to about 500 K, depending on the actual set-ups.

Our second example in Sect. 5 illustrates how theoretical dynamical spectroscopy is successful at capturing strong anharmonic hydrogen bonds of the type M⁻⋯solvent and the associated subtly enhanced solvent–solvent H-bonds of varying strengths in ionic clusters (where M⁻ is an ion and the solvent can be water or methanol). Strong anharmonicities of ~700–900 cm⁻¹ have been correctly captured by the dynamical spectra, in relation to IR-PD experiments conducted at ~100 K temperature.

The third example presented in Sect. 6 illustrates our preliminary theoretical dynamical spectra in the far-IR domain below 1,000 cm⁻¹, where vibrational anharmonicities arising from mode-couplings and delocalized motions have been captured with great accuracy on a model phenylalanine neutral di-peptide, in relation to cold IR-MPD experiments.

Gas phase vibrational spectroscopy methods are often coupled to activation methods which allow molecular dissociation. After, e.g., an activated reaction, vibrational spectroscopy can be used to characterize fragments obtained and to help in defining reaction mechanisms and products. A popular way to activate ions in the gas phase is collision induced dissociation (CID). The experimental side is reviewed in one of the chapters (see Patrick and Polfer [10]), whereas we review here theoretical methods to be applied for CID modeling.

From a theoretical point of view, we can distinguish two large categories of activation processes and subsequent dissociation: (1) dissociation obtained in a single collision limit; (2) multiple collision activation. In the first activation mechanism, the energy is put into the ion in a short time; because a unique shock can dissociate the ion. This is also what happens in surface induced dissociation (SID) experiments, where the ion collides with an inert surface and can easily dissociate after the shock. When using collision with an inert gas, as in CID, the single collision regime is a limit regime not totally reached in practice, but which can be used as a good approximation when the gas pressure is very low. In this regime the molecule gets the energy and then two mechanisms can be responsible for subsequent reactivity: (1) a shattering mechanism, where dissociation occurs over a shorter time and the bond where the energy is initially stored is broken before the internal vibrational relaxation (IVR) takes place; (2) an IVR (or almost IVR) mechanism, where the energy has time to flow through the modes being redistributed and thus the dissociation channel is activated and reaction occurs. This second regime, or, better, the full IVR regime, is, on the other hand, characteristic of multiple collisions which “softly” heat the system towards the reaction channel. These two activation modes are schematically shown in Fig. 1.

Fig. 1

Different activation schemes leading to dissociation: (a) single reactive collision; (b) multiple collisions activation

Collision-induced dissociation is a particular case of unimolecular reaction, such that in the statistical limit the unimolecular RRKM theory can be applied [11]. This theory underlines that the micro canonical distribution is preserved along the reaction path, and thus the IVR regime is supposedly achieved within the characteristic reaction time. As pointed out by Schag and Levin some years ago, the unimolecular dissociation of large molecules reflects bottlenecks to intramolecular vibrational energy distribution and the available phase space is too large to allow dissociation from a uniform distribution of the excitation [12]. As we have discussed in the literature, and discuss more extensively in Sect. 7, reactivity induced by dissociation is a dynamical process which can or cannot be described by statistical methods.

Thus, we have divided our discussion in Sect. 7 into three parts: (1) on the identification of the potential energy surface (PES) on which the dynamics occur (Sect. 7.1); (2) on the statistical methods employed and on the framework of their applicability (Sect. 7.2); (3) on direct dynamics applied to understand CID (Sect. 7.3).

In particular, we show how direct dynamics can be useful for a more complete understanding of the fragmentation processes. This is essentially related to four aspects: (1) exploring the PES “on-the-fly” is particularly useful for complex and large molecules whose full description (all possible conformers and reaction channels) cannot be achieved manually; (2) revealing the transferred energy after collision and how it is partitioned between rotational and vibrational energy – this information can then be plugged into RRKM equations (once the PES is known) and thus provide the lifetime distributions of reactant, products and intermediates; (3) obtaining information on energy flow through modes after collision and before dissociation; (4) achieving direct access to dissociation, thus identifying the reaction mechanisms and whether or not reactivity is statistical.

2 Brief Survey of Some Relevant Theoretical Methods for Vibrational Spectroscopy and Collision-Induced Dissociation in the Gas Phase

As we see in the next sections, the Potential Energy Surface (PES) of a gas phase molecule or cluster is the crucial entity for unraveling molecular structures. A PES is a complex multidimensional surface composed of 3N−6 dimensions, where N is the number of atoms in the system; see a schematic illustration of a PES in Fig. 2. Of special interest in a PES are the local minima (valleys) which represent energetically stable structures, and the saddle points or transition states. The transition states connect two local minima. Minima and transition states are topologically characterized, respectively, by all harmonic vibrational frequencies being real and by one imaginary frequency. Harmonic frequencies are calculated by diagonalization of the Hessian matrix of the system, i.e., a matrix composed of second derivatives of the potential energy with respect to the 3N coordinates of the system. Six of these frequencies are zero, corresponding to global translational and rotational motions of the system in space.

Fig. 2

Schematical potential energy surface landscape showing a typical minimum and transition state connecting two local minima

Exploration of the PES to localize the minima and the associated structures is now a routine job performed by all well-known quantum chemistry packages (Gaussian, ADF, Gamess, NWChem, ORCA, …); see for instance [13] for detailed discussions on the methods and algorithms for conformational search on the PES. Conformational search is also called geometry optimization. This is a rather “easy” task for molecular systems with a reasonable number of internal degrees of freedom, but such a conformational search is complex and time-consuming as soon as the size (and number of internal degrees of freedom) of the system increases. The number of conformations to be searched to sample the PES properly indeed increases dramatically as soon as the molecular system becomes larger. Only the lowest energy structures are of interest anyway, but it might be time-consuming and difficult to find these specific structures on the complex PES.

Nowadays, a preliminary conformational search is generally performed with a low-level representation of the interactions, typically employing classical force fields or semi-empirical-based quantum methods, followed by clustering of structures in terms of structural families, and providing a first ranking of molecular structures in terms of energy order, from lowest to highest energies. It should be noted that this conformational search can be performed with standard algorithms locating minima on a PES as described in the book by Cramer [13], or by Monte-Carlo (MC) exploration of the PES, thus allowing relatively large motions in the conformational sampling, or by Molecular Dynamics (MD) simulations usually coupled with elaborate Parallel Tempering methods using replicas of the same system at different temperatures and allowing MC jumps between replicas, thus enhancing the PES sampling. MC and MD methods, together with elaborate enhancement of PES sampling, are described in the book by Smit and Frenkel [14]. Any other theoretical method for enhancing the PES sampling can be applied, for instance metadynamics [15].

For structural families within a certain energy range, obtained with this first low-level screening, subsequent geometry optimizations are performed at a higher level of theory, i.e., using electronic ab initio methods. The DFT (Density Functional Theory) electronic representation is nowadays probably the most widely used in these calculations, as it provides an electronic framework capable of dealing with molecular systems composed of (several) hundreds of atoms. The energy ordering obtained with electronic representations is usually very different from the ordering obtained with lower levels of representations, and very often minima obtained at the low level PES representations no longer exist on the quantum PES, whereas new local conformations appear with the quantum representation. The low level PES exploration is therefore considered more as a first screening of the PES, providing guidelines for performing higher level quantum calculations on selected and supposedly relevant portions of the PES. For the lowest energy structures obtained with the quantum representation, harmonic frequencies are calculated through the diagonalization of a Hessian matrix, and are used to discriminate minima, transition states, or any conformation with more than one imaginary frequency (which represents no specific topological point on the PES).

To go beyond the knowledge of the structures at the minima and transition states, and learn more about the evolution with time of the structures and/or the exploration of the PES “on the fly,” one has to employ Molecular Dynamics (MD) methods. See again the book by Smit and Frenkel [14] for further details, the book by Marx and Hutter [16] on MD based on the electronic DFT framework, and the book by Cramer [13] for semi-empirical and QM-MM MD methods.

The applications of quantum-based MD which we present throughout this chapter in the context of infrared vibrational spectroscopy and collision-induced dissociation rely on Born–Oppenheimer MD simulations, briefly outlined below. Within Born–Oppenheimer MD, the static (time-independent) Schrödinger equation is solved at each step of the dynamics, i.e., for each configuration of the nuclei:

$$ {\widehat{H}}_e\left(\mathbf{r};\mathbf{q}\right)\Psi \left(\mathbf{r};\mathbf{q}\right)={E}_e\left(\mathbf{q}\right)\Psi \left(\mathbf{r};\mathbf{q}\right) $$

(1)

where r are the electron coordinates, Ĥ _e (r; q) the electronic Hamiltonian operator which depends parametrically on the nuclei positions, and E _e (q) the ground state energy at a given atomic/nuclei configuration. E _e can be calculated at the semi-empirical, DFT, or higher ab initio levels (typically MP2). Once this is known, forces acting on the nuclei can be calculated and the nuclear degrees of freedom are propagated with classical molecular dynamics according to

$$ -\frac{\partial {E}_e}{\partial {q}_i}={m}_i\frac{d^2{q}_i}{d{t}^2} $$

(2)

where positions q _i (and momenta p _i ) of each of the nuclei of mass m _i evolve on the potential energy surface E _e obtained from (1).

We also encourage readers to go to [1] where one chapter surveys in slightly more details the methods mentioned above.

3 Theoretical Methods for Vibrational IR Spectroscopy

3.1 Many Good Reasons to Prefer Dynamical Anharmonic Spectroscopy

The Fermi Golden Rule provides the definition for the calculation of an infrared (IR) spectrum I(ω) as a function of the reciprocal wave-number ω [17]:

$$ I\left(\omega \right)=3{\displaystyle \sum_i{\displaystyle \sum_f{\rho}_i{\left|\left\langle f\left|\mathbf{E}.\mathbf{M}\right|i\right\rangle \right|}^2\delta \left({\omega}_{fi}-\omega \right)}} $$

(3)

where E is the applied external field vector, M is the dipole vector of the absorbing molecular system, |i〉 and |f〉 are, respectively, the initial and final vibrational states of the system (eigenstates of the system excluding the radiation), ρ _i is the density of molecules in the initial vibrational state |i〉, and ω _fi is the reciprocal wave-number of the transition between the initial and final vibrational states.

In the double harmonic approximation (mechanical and electrical), the Fermi Golden Rule reduces to (4) [18]:

$$ \kappa ={\displaystyle \sum_{\alpha =x,y,z}{\displaystyle \sum_{k=1}^{3N-6}{\left|{\left(\frac{\partial {\mu}_{\alpha }}{\partial {q}_k}\right)}_{eq}\right|}^2{\delta}_{v_k,{v}_k\pm 1}}.} $$

(4)

where κ is the absolute intensity of the active infrared transitions. In the above expression, the sums run over the three directions of space and over the 3N−6 normal modes q _k of the system and ${\left(\frac{\partial {\mu}_{\alpha }}{\partial {q}_k}\right)}_{eq}$ are the transition dipole derivatives in all three directions of space. Normal modes and transition dipole moments are defined at the equilibrium geometry (eq.) of the system. See for instance [18–20] for full derivations. The normal modes q _k give the frequency values at which the 0 → 1 (v _k  → v _k+1) vibrational transitions take place and the associated atomic movements, whereas dipole derivatives give the infrared intensities associated with each mode. These two quantities are routinely obtained with geometry optimization and harmonic frequency calculations (diagonalization of the Hessian matrix), typically performed by quantum chemistry packages (in case of ab initio electronic representations) such as the well-known Gaussian, ADF, Gamess, NWChem, ORCA (…) packages. DFT (Density Functional Theory) electronic representation is nowadays probably the most used in these calculations, as it provides an electronic framework capable of dealing with molecular systems composed of (several) hundreds of atoms.

The conventional way to proceed for the evaluation of (4) is to perform a conformational search on the potential energy surface (PES) of the molecular system to assess stable structures (see Sect. 2 in this chapter for more details). For the lowest energy structures, harmonic frequencies are calculated, and are used to eliminate transition states or any conformation with more than one imaginary frequency, and thus keep only the structures associated with minima on the PES for further analyses. For these minima, harmonic frequencies of the normal modes are scaled according to the electronic representation and basis sets used in the calculations [21–23], possibly also depending on the frequency range [24], and a final vibrational spectrum is obtained, either IR or Raman depending on rules applied ((4) is for IR). The main purpose of scaling the calculated frequencies is to compensate for the double harmonic approximations (potential energy surface and dipole moment; see (4)) and compensate for the approximations in the level of the ab initio representation. The calculated absorption spectra are finally convoluted with a Gaussian or Lorentzian band profile, generally adjusted for experimental conditions. Once these steps have been achieved, comparisons between the calculated IR vibrational spectra of the identified lower energy conformers and the experimental spectrum can be made. A match is searched, i.e., a one-to-one match that one structure gives the experimental signatures. To that end, the calculated band-positions and band-intensities are compared with the experiment.

The paradigm here is that the lowest energy conformation should be the one providing the (best) match with the experimental signatures. Experimentally, the formation of the lowest energy conformation is indeed believed to be driven by kinetic and entropic effects. Although this might prove right in many experiments performed in the gas phase, especially when they are based on supersonic expansions for the production of the molecules and clusters, this is not a systematic statement. Exceptions in supersonic expansions have been reported (see for instance [25, 26]), and clear deviations have been observed by the group of J. Lisy in IR-PD argon-tagging experiments where high energy conformers can be kinetically trapped [27, 28].

The main limitations of the theoretical harmonic approach (referred to in the remainder of the text as “static harmonic”) to vibrational spectroscopy are:

• The search for the minima of lower energy on the potential energy surface, which can typically be tricky for floppy molecules. This is also a difficult search to perform for large and complex molecular assemblies, and for molecules surrounded by a few solvent molecules. See Sect. 2. Last but not least, in order to fulfill the paradigm mentioned above, one has to make sure that the lowest energy conformer has indeed been found, which is by no means trivial.

• The double harmonic approximations entering (4).

• The lack of temperature. By construction, the search for the minima on the PES provides frozen structures. This is especially a very crude approximation for floppy molecules which can undergo conformational dynamics at finite temperature, as is likely to occur in the IR-MPD and IR-PD experiments of interest to us.

We return to these points in more detail later on.

In Statistical Mechanics, the Fermi Golden Rule (3) can be reformulated using Linear Response Theory [17, 29], and can thereby be rewritten as the Fourier Transform of the time correlation function of the fluctuating dipole moment vector of the absorbing molecular system:

$$ I\left(\omega \right)=\frac{2\uppi \beta {\omega}^2}{3cV}{\displaystyle {\int}_{-\infty}^{\infty}\mathrm{d}t\left\langle \delta \mathbf{M}(t)\cdot \delta \mathbf{M}(0)\right\rangle}\kern0.5em \exp \left(i\omega t\right) $$

(5)

where β = 1/kT, T is the temperature, c is the speed of light in vacuum, and V is the volume. The angular brackets represent a statistical average of the correlation function, where $\delta \mathbf{M}(t)=\mathbf{M}(t)-<\mathbf{M}>$ with < M > the time average of M(t). The calculation in (5) is done in the absence of an applied external field. For the prefactor in (5), we have taken into account an empirical quantum correction factor (multiplying the classical line shape) of the form $\beta \hslash \omega /\left(1- \exp \left(-\beta \hslash \omega \right)\right)$, which was shown by us and others to give accurate results on calculated IR intensities [30–32]. For more detailed discussions on quantum corrections, see for instance [33–36].

The main advantages of the molecular dynamics (MD) approach in (5) for the calculation of infrared spectra (also called “dynamical spectra” in the remainder of the text) can be listed as follows:

• There are no approximations made in (5) apart from the hypothesis of linear response theory, i.e., a small perturbation from the applied external electric field on the absorbing molecular system. Such a condition is always fulfilled in vibrational spectroscopy of interest here. There are no harmonic approximations made, be they on the potential energy surface or on the dipole moment, contrary to the static calculations entering into (4). These approximations are not needed in (5). As a consequence, vibrational anharmonicities are naturally taken into account in (5). One thus needs to know the time evolution of the dipole moment of the system to calculate an anharmonic IR spectrum. This is naturally done with molecular dynamics simulations. In fact, the finite temperature dynamics takes place on all accessible parts of the potential energy surface, be they harmonic or anharmonic. The quality of the potential energy surface is entirely contained in the “ab-initio” force field used in the dynamics, calculated at the DFT/BLYP (+dispersion when needed) level in works presented here (see details in Sect. 3.2). The good to excellent agreements of the absolute (and relative) positions of the different active bands obtained in our theoretical works (see for instance dynamical spectra in the gas phase [37–41], in the liquid phase [31, 42–45], and at solid–liquid and liquid–air interfaces [46–48]) is a demonstration (though a posteriori) that this level of theory is correct.

• Even more crucial in this discussion, the calculation of IR spectra with MD is related only to the time-dependent dipole moment of the molecular system, requiring neither any harmonic expansion of the transition dipole moment nor the knowledge of normal modes, in contrast to harmonic calculations. Therefore, if the dipole moments and their fluctuations are accurately calculated along the trajectory, the resulting IR spectrum should be reliable. The vibrations therefore do not rely directly on the curvature of the potential energy surface at the minima on the PES (i.e., normal modes and derivatives using these normal modes) but rather on the time evolution of the electric dipole moment of the molecular system, which is governed by the conformational dynamics at the finite temperature of the simulation. As a consequence, dynamical anharmonic spectra from (5) and harmonic spectra from (4) rely on strictly different properties, and presumably require different levels of accuracy for the evaluation of these properties. See a more detailed discussion on these points later.

• Equation (5) gives the whole infrared spectrum of a molecular system in a single calculation, i.e., the band-positions, the band-intensities, and the band-shapes, through the Fourier transform of a time correlation function. There are no approximations applied, particularly the shape and broadening of the vibrational bands result from the underlying dynamics and mode-couplings of the system at a given temperature.

• Dynamics simulations are performed at finite temperature. At a given temperature, and depending on the energy barriers on the potential energy surface (PES), conformational dynamics between different isomeric forms of the absorbing gas phase molecular entity can be sampled by MD. All conformations populated over time are thus taken into account in the final calculation of the infrared spectrum from (5). Beyond the conformational dynamics between conformers and isomers of the molecule of interest, any population dynamics (around the minima structures), typically H-bond dynamics, local geometry rearrangements, also give rise to a natural broadening of the calculated IR active bands, which is essential for the comparison to the experimental spectra. These points have been demonstrated in some of our work, typically in our seminal case-study of the gas phase floppy protonated peptide Ala₂H⁺ [40, 49], as reviewed in Sect. 4. Very good agreement of the ~300 K dynamical infrared spectra of this peptide with the IR-MPD experimental spectra, could be achieved thanks to the simulated room-temperature dynamics of the gas phase peptide, thus enabling us to capture and to take into account the continual conformational dynamics between the two major conformers of the molecule over time. This could not be achieved with standard static calculations [49, 50]. One pivotal issue we have emphasized in this work is that we found the most populated conformations of Ala₂H⁺ not to be the geometries at the bottom wells on the PES, but rather all conformations explored over time in going from one isomer to the other. A Boltzmann-weighted IR spectrum based on the harmonic spectra of the two minima structures on the PES was therefore of no use for matching the experimental signatures.

Here we pause for a more general discussion on vibrational anharmonicities. As already pointed out in the introduction, static quantum chemistry calculations can be performed beyond the vibrational harmonic approximation, though they are more rarely applied in the communities of IR-MPD and IR-PD action spectroscopy. There are exceptions however; see works by the group of A.B. McCoy in collaboration with IR-PD experiments from the group of M.A. Johnson at Yale University [51, 52], or works from R.B. Gerber’s group in collaboration with the IR-MPD experiments of van Helden and Oomens on the IR-MPD FELIX set-up [53].

Static vibrational anharmonic calculations have been reviewed very recently by Gerber [8], a pioneer in the field. See also [54, 55] for slightly older reviews in the domain, showing how fast the field is still evolving. The vibrational self-consistent field (VSCF) method is probably the most extensively used method, providing accuracy and moderate computational cost. Developed in the late 1970s [56–59], its renewed interest lies in recent algorithmic developments of VSCF and its variant form VSCF-PT2 (based on Perturbation Theory), also called CC-VSCF in the literature (Correlation Corrected), thus making the method amenable to larger biomolecular systems at affordable computational cost. See for instance [60, 61] for the latest developments concerning linear scaling of the VSCF-PT2 method with an increase in the number of degrees of freedom and pairwise normal interactions simplifications.

Advantages and disadvantages of these theoretical methods over the dynamical method in accounting for vibrational anharmonicities can be roughly summarized as follows.

The anharmonic methods cited above are still based on geometry optimizations (0 K structures), therefore not including temperature, whereas MD does include temperature in the final spectrum. This is obviously an important attractive argument in favor of MD when the objective of the computations is the interpretation of finite temperature spectroscopy as in IR-MPD and IR-PD experiments. As already pointed out above in the case of the Ala₂H⁺ peptide, a Boltzmann-weighted IR spectrum based on the anharmonic spectra of the two minima structures on the PES did not help match the experimental signatures. Finite temperature dynamics was mandatory to achieve a match and provide understanding of the experimental signatures [40]. See Sect. 4 for more details.

Size of the molecules is another issue. Resolution of the full-dimensional vibrational Schrödinger equation for systems larger than a few atoms is too complex to be solved directly and some approximations are needed, where VSCF-PT2 is based on the replacement of the explicit correlation in the N-body system by a series of single-particle problems coupled through an effective potential which is dependent on all the other degrees of freedom. Even in this case, the size of the molecular systems which can be investigated remains limited, on the order of typically 20–30 atoms [55]. A further simplified VSCF-PT2 algorithm has been developed by Gerber [61], based on a criterion related to geometric properties of the normal modes so as to keep a priori only the relevant mode–mode couplings. Tri-peptides have hence been investigated [61] and bigger peptides should also be accessible with this method. Size of the peptides is less critical in DFT-based MD, especially (see description of DFT-based MD in Sect. 3.2) using the CP2K package [62–64] where a dual electronic representation in Gaussian and Plane Wave (GPW) basis sets is employed, and clever and fast algorithms have been implemented to solve the electronic Schrödinger equation at each time-step of the dynamics. Hence, vibrational spectroscopy of peptides containing up to 10–20 residues is currently investigated in our group and others with DFT-MD.

One crucial point in a comparative study of VSCF-PT2 and DFT-MD for anharmonic spectroscopy, is spectroscopic accuracy, or in other words the quality of the underlying PES. This is crucial for VSCF calculations, as they require a description of the PES curvature(s) at the minima. On the one hand, VSCF-PT2 anharmonic frequencies can be performed with high quality PES, i.e., at least B3LYP and even at the MP2 level of theory (though much more costly), and very accurate spectroscopic data have been obtained [55]. When increasing the size of the systems, i.e., towards peptides of reasonable size (for the IR-MPD/IR-PD experimental communities), these authors nonetheless often resort to much simpler and less expensive PES, such as the semi-empirical PM3 method [61]. On the other hand, DFT-MD is based on the DFT electronic representation and the final quality of the PES depends on the functional used. All investigations performed in our group used the BLYP functional, augmented by dispersion whenever it is needed for hydrogen bonded systems, and one might argue that this level is not sufficient (see [55] for comments on this issue). As reviewed in the present chapter, DFT-MD/BLYP (+dispersion) has been successful and accurate at capturing the main vibrational anharmonicities relevant to the spectroscopy of gas phase protonated peptide chains [39–41], highly anharmonic ionic clusters [37, 65], and vibrational couplings arising from solute-solvent interactions [31, 44] but also solid–liquid and liquid–air interfaces [46, 47]. In other words, this electronic representation has been found to be accurate for various molecular systems, in various environments, and in various domains of vibrational frequencies from 100 to 4,000 cm⁻¹. Note that B3LYP-MD trajectories are computationally more costly than non-hybrid functionals-based MD, typically a factor of ~40 per time-step (using the CP2K package).

We are thus tempted to argue that we have been able to achieve spectroscopic accuracy with the DFT/BLYP representation in most applications to date, although some failures have also been noted [66].

Here again we emphasize that the DFT-MD spectra are not calculated from normal modes, and thus are not calculated from the anharmonic curvatures at the minima on the PES, but they are calculated from the time evolution of dipole moments. The quality of the PES is obviously of importance for DFT-MD trajectories, in terms of energy differences between conformers and transition states, in terms of forces acting on the atoms, properties thus governing the conformational dynamics, and consequently the relevance of the conformations dipole moments over time. These are the driving forces of DFT-MD-based vibrational spectroscopy rather than the intrinsic quality of the curvatures at the minima on the PES. In other words, the requirements on accuracy to be reached in the time evolution of dipole moments entering (5) for dynamical spectroscopy are different from what is required in (4) from the PES curvatures at the minima (static harmonic and anharmonic spectra calculations alike). As a consequence, any debate on the BLYP functional used in our DFT-MD trajectories for dynamical spectroscopy in comparison to the B3LYP functional widely used in static harmonic/anharmonic spectra calculations might not be so relevant. Furthermore, as illustrated in Sect. 6, where we employ DFT-MD for spectra in the anharmonic 100–1,000 cm⁻¹ domain, mode couplings are amazingly accurately obtained by DFT-MD at the BLYP level, once again proving that the time evolution of the dipole moment of the peptide, even on a “low level BLYP PES representation,” is extremely well provided by this electronic representation.

Gas phase dynamical anharmonic vibrational spectroscopy using DFT-based MD in relation to action spectroscopy has been initiated in our group, with work combining IR-MPD experiment and DFT-MD spectroscopy on the protonated Ala₂H⁺ peptide [40]. This is reviewed in Sect. 4. We believe this investigation and the subsequent associated investigations on Ala₃H⁺ [39] and Ala₇H⁺ [41] have triggered interest not only in the IR-MPD/IR-PD experimental communities but also in the theoretical community. Hence, the group of V. Blum in Berlin later followed our strategies on Ala₁₅H⁺ [67]. In relation to IR-MPD experiments, the group of R.B. Gerber has also applied DFT-MD to the anharmonic spectroscopy of protonated sugars [68, 69]. In a different area of experiments, Spiegelman et al. applied these strategies to gas phase polycyclic aromatic hydrocarbons (PAH) and their hydrates [70]. In our group, IR-MPD and anharmonic DFT-MD spectroscopy have been combined on the flexible peptides ${\mathrm{Ala}}_{\left(n=2,3,7\right)}{\mathrm{H}}^{+}$ [39–41], on the protonated and deprotonated phosphorylated amino acid serine [38, 66], and more recently we have been interested in highly anharmonic ionic clusters [37, 65] in collaboration with IR-PD argon-tagging experiments. Some remarkable results on these clusters are reviewed in Sect. 5. Also of pivotal interest to us is the combination of IR-MPD and DFT-MD spectroscopy in the far-IR region, where anharmonicities arising from delocalized modes and from mode couplings are crucial, thus pushing forward the quest of the DFT-MD theoretical methodology for anharmonic spectroscopy. Such investigations are highlighted in Sect. 6. Quantum effects of the (hydrogen) nuclei might be of importance for molecules in which O–H and N–H vibrational signatures are the main features used for assigning structures. The group of D. Marx is pioneering the application of Path Integral DFT-based MD to that end; see for instance the publication on the vibrational spectroscopy of ${\mathrm{CH}}_5^{+}$ in that context [71]. Rather than performing these accurate but rather expensive simulations, we have chosen to incorporate quantum effects in the initial conditions of classical nuclear dynamics, especially through the insertion of zero point vibrational energy (ZPE) for initial positions and velocities of the dynamics [37, 72].

As stated, (5) gives the whole infrared spectrum of a molecular system with a single calculation, i.e., band-positions, band-intensities, and band-shapes. Some more comments are required.

The positions of the vibrational bands directly reflect the quality of the representation of the intramolecular interactions. As already pointed out, our work has shown that the DFT/BLYP (+dispersion) representation works extremely well, not only for band-spacings but also for the absolute values of vibrational bands. Intermolecular hydrogen bonding leads to vibrational shifts, the extent of which reflects the strengths of the interactions. Our work has shown how dynamical spectra can be accurate in reproducing bandshifts produced by hydrogen bonding of various strengths, as illustrated in the next sections. One very recent illustration concerns the very strong anharmonic bandshift of the N–H stretch motion in Cl⁻⋯NMA(H₂O) _n clusters, where the dynamical spectra were able to account for the large 900 cm⁻¹ red-shift upon hydrogen bonding.

Band-intensities and their matching to the experiments are commonly seen as a probe of the quality of the electrostatic interactions of the theoretical model. On one hand, (5) is strictly valid for one-photon linear IR absorption spectroscopy: this signal is thus identical to the signal measured in linear IR absorption spectroscopy in the liquid phase. Theoretical and experimental signals are thus directly comparable in the liquid condensed phase, in terms of band-positions, band-intensities, and band-shapes, and work on liquid water and solutes immersed in liquid water has indeed shown how excellent this agreement can be [31, 43, 44, 73]. On the other hand, temperature and energy distribution (equipartition) among the vibrational degrees of freedom play a non-trivial role in the accuracy of band-intensities derived from MD simulations. This can become crucial in gas phase spectroscopy calculations. Equipartition of the energy among the 3N−6 vibrational degrees of freedom is indeed not an easy task to achieve. We have discussed this point for low-temperature 20 K (quasi-)harmonic dynamics of gas phase amino acids and peptides [38, 74]. If modes are difficult to couple, equipartition is almost impossible to achieve, especially during the rather short timescales affordable in DFT-MD. The thermalization process is improved by surrounding solvent, as this increases the possibilities of energy transfers, not only through vibrational couplings but also through intermolecular interactions. Some discrepancies in band-intensities between dynamical spectra and experiments can therefore possibly be traced back to the equipartition of energy. Tavan et al. [75] have for instance devised a thermal correction to dynamical intensities based on the ratio of the average temperature of the dynamics and the actual temperature of the individual vibrational modes which might be able to correct deficiencies arising in the equipartition of energy in short timescale dynamics.

The situation in the gas phase is even more complicated by the actual signals recorded in the action spectroscopy experiments. IR-MPD and IR-PD experiments are multi-photon IR absorption processes leading to fragmentation of the molecule or cluster. The recorded signal is the fragmentation yield as a function of the IR excitation wavelength. It is thus an indirect measurement of IR absorption, in contrast to the linear IR absorption. So, although they reflect the same underlying vibrational properties, stationary IR absorption ((4) and (5)) and IR-MPD/IR-PD experiments are by no means equivalent. Calculations and experiments are therefore not directly comparable for band-intensities, giving rise to another source of possible discrepancies. The direct simulation of IR-MPD fragmentation spectra, with a clear theoretical expression of signal intensity in terms of dynamical quantities, remains an open question. See [76, 77] for kinetic models attempting to provide such a theoretical framework. It should be noted that it is commonly believed that the relative intensities of the active bands in IR-MPD are governed by the absorption of the few first photons, which would then be closer to calculations. The agreement between our dynamical spectra and IR-MPD/IR-PD experiments nonetheless appears very satisfactory from the point of view of spectral intensities.

DFT electronic representation routinely allows the treatment of molecular systems containing hundreds of heavy atoms, and the related DFT-MD trajectories can be reasonably accumulated over a few tens of picoseconds for systems composed of typically 100–500 atoms. Going beyond these scales of time and length would require simpler electronic representations, such as semi-empirical, or classical force fields.

In order for classical force fields to be reliable for vibrational spectroscopy, they have to be based on anharmonic analytical expressions for the intramolecular stretch, bend, and torsional motions, as well as stretch-bend (and all possible combinations between the modes) cross-terms for the intramolecular interactions. Anharmonic expressions and mode-couplings are indeed mandatory to be included in the force field expression to be able to achieve reliable vibrational band-positions. Such refinements are not included in the well-established CHARMM, AMBER, GROMOS, or OPLS force fields, routinely employed in MD simulations of biomolecules and organic molecules. Furthermore, as shown in [78, 79], any force field for spectroscopy should include at least fluctuating atomic charges in the electrostatic model, and also possibly fluctuating dipole moments (and maybe beyond), in order to recover reliable IR band-intensities. Again, such refinements are not included in the classical force fields routinely used in the literature. Any advance in the area of classical force fields for vibrational spectroscopy thus requires new force field developments, which is a huge amount of work. Such developments can be found in [80, 81] for instance, related to specific molecules and/or specific vibrational modes.

Semi-empirical dynamics seem more appealing for increasing timescales and lengths, because they are still based on an electronic representation, although simplified. The group of R.B. Gerber has in particular shown that PM3 representations can be accurate enough for vibrational spectroscopy of simple molecules [82] (static calculations). DFTB (DFT Tight Binding) also shows promising results for vibrational spectroscopy; see for instance [83–87].

DFT-MD simulations can also be applied to IR, Raman, and nonlinear Sum Frequency Generation vibrational spectroscopies in the liquid phase and at interfaces. This is beyond the scope of the present review, but we refer the reader to [7, 9, 31, 47, 73] for more information on the time-dependent signals to be calculated, and results.

3.2 DFT-Based MD

Our methodology consists of DFT-based molecular dynamics simulations, performed within the Born–Oppenheimer (BOMD) framework [62–64] (mainly using the CP2K package) or the Car-Parrinello [62, 88] framework (using the CPMD package). All applications presented in Sects. 4–6 employ BOMD, so that we only present here simulation details related to BOMD performed with the CP2K package. The methods and algorithms employed in the CP2K package are described in detail in [64]. A detailed presentation of BOMD and CPMD methods can be found in [89, 90].

In our dynamics, the nuclei are treated classically and the electrons quantum mechanically within the DFT formalism. Dynamics consist of solving Newton’s equations of motion at finite temperature, with the forces acting on the nuclei deriving from the Kohn–Sham energy. In BOMD the Schrödinger equation for the electronic configuration of the system is solved at each time step of the dynamics (i.e., at each new configuration of the nuclei); see Sect. 2. Mixed plane waves and Gaussian basis sets are used in CP2K. Only the valence electrons are taken into account and pseudo-potentials of the Goedecker–Tetter–Hutter (GTH) form are used [91–93]. We use the Becke, Lee, Yang, and Parr (BLYP) gradient-corrected functional [94, 95] for the exchange and correlation terms. Dispersion interactions have been included with the Grimme D2 and D3 corrections [96] in certain applications. Calculations are restricted to the Γ point of the Brillouin zone. We employ plane-wave basis sets with a kinetic energy cut-off usually around 340 Ry and Gaussian basis sets of double-ζ (DZVP) to triple-ζ (TZVP) qualities from the CP2K library. The kinetic energy cut-off and basis set sizes are systematically optimized on each investigated system by careful energy convergence tests.

Our dynamics are strictly microcanonical (NVE ensemble), once thermalization has been achieved (through NVE and velocity rescaling periods of time). Our gas phase simulations use the decoupling technique of Martyna and Tuckerman [97] to eliminate the effect of the periodic images of the charge density, relevant to charged gas phase systems.

The knowledge of the evolution with time of the molecular dipole moments is mandatory for the calculation of IR spectra with MD simulations. In the modern theory of polarization, the dipole moment of the (periodic) box cell is calculated with the Berry phase representation, as implemented in the CPMD and CP2K packages [98]. Briefly, in the limit where the Γ point approximation applies, the electronic contribution to the cell dipole moment M ^el _α (where α = x, y, z) is given by [99]

$$ {\mathbf{M}}_{\alpha}^{el}=\frac{e}{\left|{\mathbf{G}}_{\alpha}\right|}\Im \ln {z}_N $$

(6)

where ℑ ln z _N is the imaginary part of the logarithm of the dimensionless complex number ${z}_N=\left\langle \varPsi \left|{e}^{-i{\mathbf{G}}_{\alpha}\cdot \widehat{\mathbf{R}}}\right|\varPsi \right\rangle$, G _α is a reciprocal lattice vector of the simple cubic supercell of length L (G ₁ = 2π/L(1,0,0), G ₂ = 2π/L(0,1,0), G ₃ = 2π/L(0,0,1)), and $\widehat{\mathbf{R}}={\displaystyle {\sum}_{i=1}^N{\widehat{\mathbf{r}}}_i}$ denotes the collective position operator of the N electrons (or in other words the center of the electronic charge distribution). Ψ is the ground-state wave function. The quantity ℑ ln z _N is the Berry phase, which in terms of a set of occupied Kohn–Sham orbitals ψ _k (r) is computed as ℑ ln ${z}_N=2\Im$ ln det S with elements of the matrix S given by ${S}_{kl}=\left\langle {\psi}_k\left|{e}^{-i{\mathbf{G}}_{\alpha}\cdot \widehat{\mathbf{r}}}\right|{\psi}_l\right\rangle$ [99].

IR spectra are given as products α(ω)n(ω) expressed in cm⁻¹ (decadic linear absorption coefficient) as a function of wavenumber, ω, in cm⁻¹. The spectra are smoothed with a window filtering applied in the time domain, i.e., each term of the correlation function C(t) is multiplied by a Gaussian function exp(−0.5σ(t/t _max )²), where t _max is the length of the simulation, and σ is usually chosen around a value of 10 for gas phase simulations. This convolution only has the purpose to remove the numerical noise arising from the finite length of the Fourier transform of (5). These calculations are performed with our home-made code.

The average times easily affordable with DFT-based MD are of the order of a few tens of picoseconds for systems composed of a few hundred atoms, running on massively parallel machines. This is the timescale on which stretching, bending, and torsional motions can be properly sampled. For rather rigid molecules in the gas phase or molecules, the geometry of which is somewhat ‘constrained’ by the surrounding aqueous phase environment, one trajectory accumulated over this timescale is enough to get a reliable theoretical infrared spectrum [31, 38, 44]. When floppy peptides undergo conformational dynamics over the tens of picoseconds period of time, which is especially the case at room temperature in the gas phase, several trajectories have to be accumulated in order to take into account the conformational diversity. The final IR spectra presented in our work are averaged over all trajectories, so the conformational diversity and heterogeneity of the dynamics at finite temperature can be accounted for, in a natural way. Each of the dynamics performed gives slightly different IR features, reflecting the actual conformational sampling and properties of the isomeric/conformational family, and the statistical average over all trajectories is the only meaningful and relevant quantity, which can thus be compared to the spectrum recorded in the experiments. Such statistics is also relevant for H-bond dynamics.

The length of trajectories is also related to the vibrational domain to be sampled. One has to keep in mind that the length of time has to be commensurate with the investigated vibrational motions. Hence, trajectories around 5 ps are just enough to sample stretching motions in the high frequency domain of 3,000–4,000 cm⁻¹, provided that several trajectories starting from different initial conformations (structure and/or velocities) are accumulated and averaged for the final IR dynamical spectrum. In the mid-IR domain, trajectories of at least 10 ps each are needed to sample the slower stretching and bending motions of the 1,000–2,000 cm⁻¹ domain. In the far-IR below 1,000 cm⁻¹, much longer trajectories are needed to sample properly the much slower motions typical of that domain, i.e., torsional motions and possibly opening/closure of structures (typically for peptide chains). These delocalized and highly coupled motions require trajectories of 20–50 ps on average to be safely sampled, again with a final average over a few trajectories. We illustrate dynamical spectroscopy for all these three domains in the next sections of that review.

All investigations presented in Sects. 4–6 have employed Born–Oppenheimer dynamics (BOMD). We apply no scaling factor of any kind to the vibrations extracted from the dynamics. The sampling of vibrational anharmonicities, i.e., potential energy surface, dipole anharmonicities, mode couplings, anharmonic modes, being included in our simulations, by construction, application of a scaling factor to the band-positions is therefore not required. As reviewed afterwards, excellent agreements between dynamical spectra and IR-MPD and IR-PD experiments have been achieved. Any remaining discrepancies between dynamical and experiment spectra should mainly be because of the choice of the DFT/BLYP (+dispersion) functional, as DFT-based dynamics are only as good as the functional itself allows. Conformational sampling might be another source of discrepancy: sampling is limited by the length of the trajectories we can afford and the number of initial isomers we are including in the investigation. It is also clear that vibrational anharmonic effects probed in molecular dynamics are intrinsically linked to the temperature of the simulation, as pointed out in [37, 74, 100]. Closely related is the influence of quantum effects such as Zero Point Vibrational Energy (ZPE) in the initial conditions of the dynamics, playing a role in the dynamics temperature and in the final dynamical vibrational frequency values. This has been discussed in [72] in the context of semi-classical dynamics for anharmonic spectroscopy and in [37] in the context of highly anharmonic red-shifts in ionic clusters from DFT-MD simulations.

3.3 Assignment of Modes with Vibrational Density of States VDOS

An accurate calculation of anharmonic infrared spectra is one goal to achieve, the assignment of the active bands into individual atomic displacements or vibrational modes is another. This issue is essential to the understanding of the underlying molecular structural and dynamical properties. In molecular dynamics simulations, interpretation of the infrared active bands into individual atomic displacements is traditionally and easily done using the vibrational density of states (VDOS) formalism. The VDOS is obtained through the Fourier transform of the atomic velocity auto-correlation function:

$$ V\;DOS\left(\omega \right)={\displaystyle \sum_{i=1,N}{\displaystyle {\int}_{-\infty}^{\infty}\left\langle {\mathbf{v}}_{\mathbf{i}}(t)\cdot {\mathbf{v}}_{\mathbf{i}}(0)\right\rangle}\kern0.5em \exp \left(i\omega t\right)\kern0.5em dt} $$

(7)

where i runs over all atoms of the investigated system and v _i (t) is the velocity vector of atom i at time t. As in (5), the angular brackets in (7) represent a statistical average of the correlation function. The VDOS spectrum provides all vibrational modes of the molecular system. However, only some of these modes are either infrared or Raman active, so VDOS spectra can by no means directly substitute IR or Raman spectra.

The VDOS can be further decomposed according to atom types, or to groups of atoms, or to chemical groups of interest, to obtain a detailed assignment of the vibrational bands in terms of individual atomic motions. This is done by restricting the sum over i in (7) to the atoms of interest only. Such individual signatures are easy to interpret in terms of movements for localized vibrational modes involving only a few atomic groups, typically in the 3,000–4,000 cm⁻¹ domain, but the interpretation of the VDOS becomes more complicated for delocalized modes or for highly coupled modes. In the far-IR domain below 1,000 cm⁻¹, where such motions are typical, we resort to the Fourier transforms of intramolecular coordinates (IC) time correlation functions instead of the VDOS: ${\displaystyle {\int}_{-\infty}^{\infty}\left\langle IC(t)\cdot IC(0)\right\rangle}\kern0.5em \exp \left(i\omega t\right)\kern0.5em dt$. Of course, such an approach requires some a priori knowledge of the relevant IC(s), as one cannot easily analyze all possible combinations.

We and others have developed methods to extract vibrational modes from the dynamics, especially bypassing the limitations from VDOS assignments [19, 101–103]. Such methods usually provide “effective normal modes,” similar to the well-known normal modes obtained by a Hessian diagonalization in static harmonic calculations, but maintaining a certain degree of mode couplings and temperature from the dynamics in the final modes. Describing these methods is beyond the scope of this chapter; all applications discussed in Sects. 4–6 use the simplest VDOS and ICs for assigning the vibrational modes.

4 IR-MPD Spectroscopy and Conformational Dynamics of Floppy Ala _n H⁺ Protonated Peptides

The vibrational spectroscopy of gas phase protonated peptides Ala _n H⁺ is the first example that illustrates the crucial need to take into account the conformational dynamics of molecules into the final vibrational spectrum calculation. See [39–41, 49] for a complete description of our theoretical investigations on the spectroscopy of Ala₂H⁺, Ala₃H⁺, and Ala₇H⁺.

Our room temperature trajectories on Ala₂H⁺ showed a highly floppy molecule with continuous isomerization dynamics between transA1 and transA2 conformers (see Fig. 3, right). This is followed by the time evolution of the dihedral angle Φ = C ₂ − N − C ₃ − C ₄ (top of Fig. 3): this angle distinguishes the two lower energy conformers of Ala₂H⁺, transA1 with Φ = 198.1° and transA2 with Φ = 284.2° (from geometry optimizations at the BLYP/6-31G* level [40]). The continuous evolution of Φ seen on the plot roughly between 180° and 300° during the dynamics reveals the continuous exploration of the two wells of transA1 and transA2 at room temperature.

Fig. 3

Ala₂H⁺ gas phase dynamics from [40]. Top: time evolution of the dihedral angle Φ = C ₂ − N − C ₃ − C ₄ for a typical trajectory of the trans isomer at 300 K. Bottom: free energy profile along the Φ dihedral angle averaged over all trans trajectories. Right: atom labeling for Ala₂H⁺ and schematic representations of transA1/transA2 conformers

The free-energy curve along the dihedral coordinate Φ extracted from the dynamics (calculated as G(Φ) = −k _B T lnP(Φ), where P(Φ) is the probability of occurrence of a certain angle Φ in the course of the dynamics, bottom of Fig. 3) shows a rather flat well for angles between 190° and 300°. There is thus no evident barrier between transA1 and transA2, only an overall free energy difference of ~0.3 kcal/mol between the two structures. This value is quite a distance away from the ~2 kcal/mol energy difference between the two optimized conformers, and from the ~2.5 kcal/mol [50] energy difference between transA1 and the transA1-transA2 transition state, computed at 0 K. This emphasizes the importance of entropic contributions in structural equilibria, naturally taken into account in the dynamics.

Overall, we found that the conformations where the N-terminal of the peptide is protonated $\left({\mathrm{NH}}_3^{+}\right)$ are predominantly populated, with a strong hydrogen bond between ${\mathrm{NH}}_3^{+}$ and the neighboring carbonyl group. Moreover, there is enough energy for the ${\mathrm{NH}}_3^{+}$ group to rotate and exchange the hydrogen atom that can be hydrogen bonded to the neighboring carbonyl, at room temperature. This is done a few times over the tens of picoseconds trajectories.

The dynamical anharmonic infrared spectra of gas phase Ala₂H⁺ are presented in Figs. 4 and 5 (using (5)) and compared to the IR-MPD experiments from [104] in the 2,800–4,000 cm⁻¹ domain (Fig. 4) and from [50] in the 1,000–2,000 cm⁻¹ domain (Fig. 5). See [39, 40] for a more detailed description of results and assignments.

Fig. 4

Infrared spectrum of gas phase Ala₂H⁺ in the trans form in the 2,800–4,000 cm⁻¹ vibrational domain. Taken from [39]. IR-MPD Experiment (top), theoretical calculation from DFT-MD (bottom)

Fig. 5

Infrared spectrum of gas phase Ala₂H⁺ in the trans form in the 800–2,000 cm⁻¹ vibrational domain. Taken from [40]. Squares: IR-MPD experiment taken from [50]. Solid curve: dynamical spectrum from DFT-based molecular dynamics at ~300 K. The band assignment deduced from the vibrational density of state (VDOS) analysis is illustrated with schemes on top of the bands

In the 3,000–4,000 cm⁻¹ domain we can see that our dynamical spectrum reproduces very well the IR-MPD spectrum. The band located at 3,560 cm⁻¹ corresponds to the O–H stretch of the C-terminal COOH group of the peptide, the calculation matching the experimental band in terms of position and band shape. The shoulder located at ~3,590 cm⁻¹ nicely reflects the feature also present in the experiment. The OH group is free of hydrogen bonding throughout the dynamics. The broad band located between 3,300 and 3,500 cm⁻¹ (including the small feature at ~3,490 cm⁻¹) is produced by the N–H stretches from the amide N–H group and from the free N–H groups of the ${\mathrm{NH}}_3^{+}$ N-terminal. The symmetric and antisymmetric stretches of the free N–H of the ${\mathrm{NH}}_3^{+}$ can be seen at 3,415 and 3,370 cm⁻¹, respectively. The backbone amide N–H stretch also appears on the higher frequency part of the band, around 3,415 cm⁻¹. Although the calculated band nicely reflects the two parts which can be observed in the experimental band, it is nonetheless not broad enough in comparison to the experiment. The width of IR bands from molecular dynamics arises from a combination of temperature, conformational dynamics of the molecule, and anharmonicities (mode couplings, dipole anharmonicity, PES anharmonicity). The difference observed here is likely to be because the temperature of the simulations is not high enough.

Interestingly, the IR signature of the ${\mathrm{NH}}_3^{+}$ hydrogen atom involved in the N–H⁺⋯O=C H-bond with the neighboring C=O carbonyl throughout the trajectories is spread over the broad 2,500–2,850 cm⁻¹ region of the spectrum, strongly down-shifted from the symmetric and antisymmetric modes of ${\mathrm{NH}}_3^{+}$. Strong anharmonicity of the N–H⁺⋯O=C hydrogen bond are responsible for the displacement of this band to low frequencies, whereas mode-couplings anharmonicities and the dynamics of the hydrogen bond at finite temperature are responsible for its ~350 cm⁻¹ spread. Indeed, the N–H⁺⋯O=C H-bond distance fluctuates by 0.3–0.4 Å around its mean value and there is enough energy for the ${\mathrm{NH}}_3^{+}$ group to rotate and exchange the hydrogen atom, which can be hydrogen bonded to the neighboring carbonyl. These complex H-bond dynamics therefore lead to vibrational signatures which are spread over a large frequency domain. At the time of our publications [39, 40] there was no experimental signal recorded in this domain of 2,500–2,850 cm⁻¹ to validate our theoretical spectral data. This has been recorded by the group of M.A. Johnson in 2013 [52] with tagging IR-PD experiments, confirming our dynamical spectral assignments.

The double band around 3,000–3,100 cm⁻¹ is produced by the combined C_α–H and C–H stretch modes of the methyl groups of the peptide. These bands are up-shifted by ~40 cm⁻¹ from experiment, and the band spacing (~50 cm⁻¹) is slightly bigger than the experimental one (~30 cm⁻¹). It is nonetheless remarkable that our calculation predicts an intensity of the band so close to experiment. The harmonic calculation [105] does not give such intensity to this mode.

In the 1,000–2,000 cm⁻¹ mid-IR (see Fig. 5) the dynamical spectrum again matches the IR-MPD experiment extremely well, again without any scaling factor applied. Both experiment and calculation display three separate absorption domains, which are reproduced with very good accuracy by the dynamical spectrum in terms of both relative positions of the bands and bandwidths. The relative positions of the different bands and fine details such as the two sub-bands of the 1,100–1,200 cm⁻¹ domain or the two distinct IR active bands of the 1,300–1,600 cm⁻¹ domain are also very well reproduced. The two peaks located at 1,130 cm⁻¹ and 1,150 cm⁻¹ are produced by stretching and bending of the N- and C-terminals of the peptide. The band between 1,340 and 1,500 cm⁻¹ arises from vibrations on the N-terminal side of the peptide, namely δ(N − H) bending motions of the ${\mathrm{NH}}_3^{+}$ group coupled with skeletal ν(C − C) stretchings. The narrow band extending between 1,500 and 1,600 cm⁻¹ is composed of the δ(N − H) amide II motion coupled with ν(N − C ₂) stretching of the backbone. The 1,620–1,800 cm⁻¹ domain is composed of three separate peaks in the dynamical spectrum: the 1,670 cm⁻¹ peak comes from the amide I of the C₂=O₂ carbonyl group, the 1,720 cm⁻¹ is related to a superposition of the two amide I motions, and the peak at 1,770 cm⁻¹ involves C₄=O₄. The amide I domain is clearly composed of three bands of high intensity in our calculation, whereas the experiment offers only two such bands, although a third sub-peak of far lower intensity may be distinguished in the 1,660 cm⁻¹ tail of this domain. Discrepancies in intensities between experiment and simulated dynamical spectrum can be traced back to the difference in the signals experimentally recorded (fragmentation yield) and calculated (linear IR absorption). See discussion in Sect. 3.1.

An important aspect from Fig. 3 that we have not discussed yet but now becomes pivotal in the discussion of the dynamical IR spectra concerns the sampling of the two lower energy conformers of Ala₂H⁺ over time. This sampling can be seen in Fig. 3 by the number of times dihedral angle values typical of transA1 and transA2 conformers (respectively Φ ~ 198° and ~280°) have been obtained over the length of the trajectory. One can see that the majority of conformations sampled over time have angle values different from these two targets: in other words, the dynamics sample conformations outside the minima on the PES, which is also clear from the free energy profile in Fig. 3 with the rather flat transA1-transA2 well. The direct conclusion is that a Boltzmann weighted spectrum of the harmonic spectra of transA1 and transA2 is not useful for the interpretation of the IR-MPD experiment. These two conformers are indeed clearly not the ones of interest once temperature is included, but rather all conformers explored during the continuous transA1-transA2 conformational dynamics participate in the final vibrational features. The weight of each conformation explored during the dynamics is naturally taken into account in the vibrational spectrum.

We continue the exploration of the vibrational signatures of protonated alanine peptides by investigating Ala₃H⁺. Three main structural families have been identified by Vaden et al. [104] and McMahon et al. [106] which can be summarized in the “NH₂ family” where the proton is located on the N-terminal carbonyl group of the tripeptide, “elongated ${\mathrm{NH}}_3^{+}$” (denoted as the ‘${\mathrm{NH}}_3^{+}$’ family) where the proton is located on the N-terminal ${\mathrm{NH}}_3^{+}$ and the peptide chain is extended (similar to transA1/transA2 structures for Ala₂H⁺ above), and “folded ${\mathrm{NH}}_3^{+}$” (denoted as ‘folded’ family) where the peptide chain is folded through an ${\mathrm{NH}}_3^{+}\dots \mathrm{O}=\mathrm{C}$ H-bond. See Fig. 6 for illustrations. Energy barriers have not been characterized, but are presumably high enough with respect to 380–530 K temperatures of relevance for the IR-MPD experiments performed by the group of J.P. Simons in Oxford [107], so that conformational dynamics are not likely to occur during the timescale of our trajectories. Overall, we indeed do not observe spontaneous conformational interconversion/isomerization during the dynamics around room temperatures, but this becomes permitted around 500 K.

Fig. 6

Infrared spectrum of gas phase Ala₃H⁺ for the 2,500–4,000 cm⁻¹. Taken from [39]. Left: conformational families of Ala₃H⁺ taken into account for the dynamics. Right: (top) harmonic calculations from [104] and (bottom) comparison of IR-MPD experiment (exp) and dynamical spectrum from DFT-MD simulations (calc) [39]

Considering the initial photochemical scheme used for the production of the ions in the experiment [107], trajectories have been accumulated at temperatures in the range 380–530 K, and have all been used in the statistical average for the dynamical IR spectrum (each trajectory has the same weight in the final average). The main dynamical behaviors seen in the trajectories correspond to H-bond dynamics and local geometrical reorganizations, including rotations of ${\mathrm{NH}}_3^{+}$, CH₃, and COOH groups, which influence the vibrational features of the peptide as detailed below. The final spectrum of the protonated alanine tripeptide obtained from our DFT-MD trajectories is presented in Fig. 6 (bottom right). It agrees remarkably well with the IR-MPD experimental spectrum (see discussion on band-intensities in Sect. 3.1).

The band located at 3,560 cm⁻¹ is produced by the O–H stretch of the C-terminus, identical to that in Ala₂H⁺ and Ala₃H⁺, corresponding to conformations with a similar free C-Terminal O–H. It should be noted that the slight asymmetry of this band is correctly reproduced by the calculations. The 3,100–3,500 cm⁻¹ region can be separated into two parts. The 3,300–3,500 cm⁻¹ frequency region is produced by the stretches of the N–H groups of the tripeptide that are not involved in hydrogen bonds along the dynamics, i.e., the symmetric and antisymmetric stretch of the N-terminal amine NH₂, the stretches of the free N–H of the ${\mathrm{NH}}_3^{+}$, and the stretches of the C-terminal amide N–H, depending on the peptide family. The 3,100–3,300 cm⁻¹ domain is uniquely produced by the stretching motion of the N-terminal amide N–H group. Our trajectories show that this amide group can be weakly hydrogen bonded to the amine N-terminal in the NH₂ tripeptide family (through a distorted H-bond), and also weakly hydrogen bonded to the C-terminal carbonyl of the peptide in the ${\mathrm{NH}}_3^{+}$ family (again distorted H-bond). These H-bonds are, however, strong enough to induce the shift towards lower energy of the N-terminal N–H stretch in comparison to the positions of the free N–Hs.

The conformational diversity of Ala₃H⁺ and the conformational dynamics of the N–H groups within these structures thus nicely show up in the broad N–H vibrational band. The overall shape of the N–H band is well reproduced by our dynamical spectrum, and the complex vibrational patterns of this band are therefore a result of the local dynamics of the N–H groups in the different Ala₃H⁺ families.

Identically to Ala₂H⁺, the 2,800–3,100 cm⁻¹ domain of Ala₃H⁺ is assigned to the C–H stretches arising from the methyls and the C _α –H groups of the tripeptide. Also, similar to the Ala₂H⁺ peptide, the vibrational signatures of N–H⁺⋯O=C in the ${\mathrm{NH}}_3^{+}$ families are strongly red-shifted to a lower frequency compared to the other N–H stretches, and appear over the extended 2,000–2,800 cm⁻¹ domain. Such a large 800 cm⁻¹ spread is again entirely caused by the vibrational anharmonicities, mode-couplings, and dynamics of the N–H⁺⋯O=C H-bond at finite temperature. Remarkably, the stretching of the protonated skeleton C–O–H⁺ in the NH₂ family is superimposed on the N–H⁺ stretches of the ${\mathrm{NH}}_3^{+}$ families. It is consequently impossible to distinguish both families using the stretching patterns of the H-bonded N–H⁺ or C–O–H⁺ groups in the high frequency domain alone.

It is interesting to compare the scaled harmonic spectra of the three Ala₃H⁺ peptide families with the dynamical anharmonic spectrum discussed above. This is done in Fig. 6, where the scaled harmonic spectra are taken from [104] for the four lowest optimized geometries (top right of Fig. 6). As can be immediately observed, the N–H broad band of the experimental spectrum is systematically associated with only two main intense harmonic bands, greatly separated by 200 and 150 cm⁻¹ for the NH₂ and ${\mathrm{NH}}_3^{+}$ families, respectively; hints of a third band located close to the ~3,490 cm⁻¹ experimental band can also be seen, with a very low intensity. The interpretation given here from MD simulations that the broad and complex N–H vibrational band comes from the intrinsic local dynamics of the N–H groups in the different conformers/isomers of Ala₃H⁺, including the breaking and forming of these H-bonds, can only be achieved when performing molecular dynamics simulations. Scaled harmonic spectra, possibly including a Boltzmann weighted average, are unable to give such a broad dynamical band. Last but not least, C–H harmonic modes predicted around 3,000 cm⁻¹ have no intensity in the harmonic spectra, whereas the anharmonic spectrum extracted from MD correctly predicts the intensity in this region.

Comparing the spectra of Ala₂H⁺ and Ala₃H⁺ peptides, the first instructive result is that both peptides display a C-terminal O–H group free of any hydrogen bonding. This band is maintained with the size of the Ala _n H⁺ peptide [104]. The N–H band is broadened in going from Ala₂H⁺ to Ala₃H⁺, which reflects more diverse conformations and conformational dynamics of the hydrogen bonds of the N–H groups of these peptides. For both peptides, the amides not involved in any H-bond (the single N–H of Ala₂H⁺, the C-terminal N–H of Ala₃H⁺, and the NH₂ group of Ala₃H⁺) and the N–H groups of ${\mathrm{NH}}_3^{+}$ also free of H-bonds, are collectively responsible for the 3,500–3,300 cm⁻¹ vibrational band. The longer chain Ala₃H⁺ has one amide at the N-terminal which can now be involved in several kinds of hydrogen bond patterns depending on the conformations. It is remarkable that this amide group is responsible for the extension of the N–H band towards the lower 3,100 cm⁻¹ range. Such an extended band is experimentally observed [104] for increasing size of Ala _n H⁺, even gaining more intensity for Ala₇H⁺. The C–H vibrational band is conserved from Ala₂H⁺ to Ala₃H⁺ and throughout larger sizes, greatly extending between 3,100 and 2,900 cm⁻¹.

Going to the larger Ala₇H⁺ peptide, the initial structures for the trajectories were taken from [104], and the three lower energy conformers were used as starting conformations for the dynamics (they lie within less than 20 kJ/mol of energy [104]). Ala₇H⁺ are globular folded structures in which the N-terminal ${\mathrm{NH}}_3^{+}$ is the central element for the folding. Considering the photochemical scheme used for the production of Ala₇H⁺ [104], it is very likely that these conformations would equally participate in the IR-MPD spectrum. The three lower energy optimized structures contain charge-solvating NH⁺ → O=C hydrogen bonds involving one (A₇1), two (A₇2), and three (A₇3) NH groups. Seven trajectories have been accumulated, two initiated from A₇3 optimized structure, two from A₇2, and three from A₇1, each for 3–5 ps.

The average conformation at room temperature differs from the frozen 0 K well-defined structures. Although the 0 K temperature drives the formation of simultaneous multiple hydrogen bonds between the ${\mathrm{NH}}_3^{+}$ charged group and its immediate surroundings, dynamics at finite temperature does not favor such multiple H-bonds. Out of the seven BOMD trajectories, two provide a single NH⁺⋯O=C H-bond formed on average as the most relevant statistical event, five provide two such H-bonds on average, and only one trajectory provides a three NH⋯O=C H-bonds situation over transient periods of time.

The two NH⋯O=C H-bonded situation is thus statistically the most probable at the temperature of 350 K. These H-bonds can form and break easily. It is not surprising that maintaining three NH⋯O=C H-bonds at finite temperature is rather difficult, as such a configuration provides a huge constraint over the whole geometry of the peptide: the peptide does not comply easily with such a constraint, once temperature and entropic effects are taken into account. The (CO)OH C-terminal hydroxyl of Ala₇H⁺ is also on average not hydrogen bonded to its surroundings. As expected, over the rather limited periods of time of the dynamics (5 ps), only the dynamical behavior of forming and breaking H-bonds has been observed in the dynamics and no other remarkable structural reorganization of the chain could be seen (this would require much longer simulations and possibly higher temperatures, as already seen for Ala₃H⁺).

Figure 7 presents the IR-MPD experimental data extracted from [104] and the IR dynamical spectrum extracted from the seven trajectories of Ala₇H⁺ peptide. Bands have been colored according to the nature of the stretching assignments. It is clear that all experimental spectral features are obtained in the dynamical spectrum. When comparing IR-MPD and calculated spectra, one has to be cautious discussing IR intensities, as they are not comparable between experiments and simulations; see Sect. 3.1. The band shapes and positions agree well. In particular, the new experimental feature appearing at 3,100–3,230 cm⁻¹ for Ala₇H⁺, not seen for smaller peptide chains [39], is present in the dynamical spectrum.

Fig. 7

Experimental IR-MPD spectrum from [104] (bottom, black line) and dynamical spectrum (top, red line) obtained as an average over the seven trajectories of Ala₇H⁺ protonated peptide. Taken from [41]. The left figure has color-codings and labels explaining the assignments of the three vibrational domains in terms of average number of NH⁺⋯O = C H-bonds formed in Ala₇H⁺ peptide. The right figure displays the assignments in terms of stretchings of the O–H, N–H, N–H⁺, and C–H (C_α–H and CH₃) motions

We do not discuss the vibrational assignments in detail; they can be found in [41] and they follow the trends found for the shorter peptides. One band discussed here, though, is the band observed in the 3,030–3,300 cm⁻¹ which was absent for the lower peptide chain lengths. It is remarkably reproduced by the present dynamical spectrum, though slightly less broad. This band is entirely produced by N–H⁺ stretches arising from conformations where the ${\mathrm{NH}}_3^{+}$ N-terminal is involved in two hydrogen bonds on average. The region between 2,500 and 3,000 cm⁻¹ is more complex for Ala₇H⁺, as the highly anharmonic vibrations of the singly H-bonded ${\mathrm{NH}}_3^{+}$ conformations and H-bonded hydroxyl groups provide vibrational signatures in that domain on top of the C–H stretchings.

One can go one step further in the discussion of the N–H⁺⋯O=C vibrational signatures in the Ala _n H⁺ series. Ala₂H⁺, Ala₃H⁺, and some Ala₇H⁺ conformers have one such H-bond on average at room temperature: its vibrational signature appears in the 2,000–3,000 cm⁻¹ domain. This N–H⁺⋯O=C H-bond is highly non-harmonic, and the vibrational domain reflects the strength and degree of anharmonicity of such H-bond. The strongest H-bonds are formed in the Ala₃H⁺ peptide, where vibrational signatures down to 2,000 cm⁻¹ have been characterized, whereas the two other peptides have signatures above 2,500 cm⁻¹. For Ala₇H⁺, we have seen that the N–H⁺⋯O=C strength decreases with increase in the number of H-bonds formed between ${\mathrm{NH}}_3^{+}$ and the C=O groups. Hence, 1-H-bond conformers provide signatures in the 2,500–3,000 cm⁻¹, whereas 2-H-bonds conformers provide signatures in the 3,000–3,300 cm⁻¹, and 3-H-bonds conformers in the 3,300–3,400 cm⁻¹. In this latter case, the N–H⁺⋯O=C vibrational signatures overlap with those arising from H-bonded N–H (neutral) amide groups.

It is also clear from the above analyses of the dynamics of Ala _n H⁺ peptides that the knowledge of finite temperature properties is mandatory to characterize vibrational signatures of floppy molecules. Entropic effects are naturally taken into account in the dynamics, with consequences on conformational equilibria. Anharmonic vibrational effects, i.e., mode couplings, anharmonicities from H-bonds, and anharmonic motions on the PES, are naturally taken into account in the dynamics, without a priori knowledge. Furthermore, at finite temperatures, conformers with simultaneous N–H⋯O=C H-bonds might not be statistically relevant, as shown here for Ala₇H⁺. Although the 0 K frozen picture provided conformers in which the ${\mathrm{NH}}_3^{+}$ group was involved in three simultaneous H-bonds as the most stable structure, the finite temperature shows that only 2- and 1-H-bonds structures are statistically relevant. Once this conformational property is taken into account, the IR dynamical vibrational spectrum is in good agreement with the experiment.

5 IR-PD and Highly Anharmonic Ionic Clusters

The above examples of Ala _n H⁺ peptides demonstrate that dynamical spectra are capable of capturing the dynamical behavior of the N–H⁺⋯O=C H-bonds and their vibrational anharmonicities with remarkable accuracy, thus providing definitive assignments of the protonated alanine peptide structures produced in the gas phase. We now turn to the vibrational spectroscopy of even more challenging molecular systems, ionic clusters, displaying large vibrational anharmonicities. Details of our combined IR-PD experimental and theoretical dynamical investigations can be found in [37, 65], where the structures of Cl⁻-(Methanol)_1,2 and Cl⁻NMA(H₂O) _{n = 0–2} clusters (NMA = N-methyl-acetamide) have been unraveled combining InfraRed Predissociation (IR-PD) experiments from the group of J. Lisy (University of Illinois at Urbana-Champaign) and DFT-based molecular dynamics simulations (DFT-MD). We highlight below some results on Cl⁻-(Methanol) _{n = 1–2} systems.

The experimental IR-PD and the dynamical spectrum of Cl⁻-CH₃OH complex are reported in Fig. 8 (left). The dynamical spectrum shows a prominent peak at 3,085 cm⁻¹ closely matching the experimental 3,087–3,109 cm⁻¹ band. The experiment displays a doublet, believed to result either from multiple argon binding sites or from the simultaneous presence of the staggered and eclipsed conformations of Cl⁻-CH₃OH(Ar) in the experiment [108]. At the 100 K temperature of the dynamics, we have shown that there is a continuous conformational dynamics (rotation) of the methyl group without geometrical distinction between staggered and non-staggered orientations of this group over time. There are thus no distinct vibrational signatures of these specific orientations to be expected in the vibrational signatures of the cluster at finite temperature. The multiple binding sites of argon in the cluster (not taken into account in the dynamics) therefore remain the most probable source of the doublet.

Fig. 8

Dynamical IR spectrum (100 K) of Cl⁻–CH₃OH (top left) and Cl⁻–(CH₃OH)₂ (top right) compared to the associated IR-PD experimental spectra of Cl⁻–(CH₃OH)Ar and Cl⁻–(CH₃OH)₂Ar (bottom of each figure). The inserts are illustrations of the structures of Cl⁻–CH₃OH and Cl⁻–(CH₃OH)₂. Taken from [65]

The O–H stretch calculated from the dynamical IR spectrum is red-shifted from 3,682 cm⁻¹ in uncomplexed methanol (our calculation with the same BOMD representation to be compared to 3,681 cm⁻¹ in gas phase experiment [109]) to 3,085 cm⁻¹ when the chloride anion binds to the methanol OH. This is in agreement with experiment: the dynamical IR spectrum thus provides a proper account of the large 600 cm⁻¹ down shift of the O–H stretching once O–H is bound to Cl⁻. No scaling factor has been applied in the dynamical spectrum. The methyl stretching motions provide the 2,840–2,920 cm⁻¹ broad band in the dynamical spectrum, also in agreement with the IRPD features. Remarkably, one can distinguish a feature of low intensity in the dynamical spectrum located at 2,980 cm⁻¹, in agreement with the IRPD feature at 3,005 cm⁻¹. This was interpreted as a combination hot band in [110].

The IR-PD spectrum and dynamical spectra of the two isomers of Cl⁻-(Methanol)₂ are presented in Fig. 8 (right). The BOMD IR spectrum of isomer 2A shows two main bands in the 2,500–3,500 cm⁻¹ domain: a very intense band located at 3,195 cm⁻¹ produced by the O–H stretching motions of the two equivalent O–H groups, and two bands between 2,800 and 3,000 cm⁻¹ produced by the methyl C–H motions. The peak of very small amplitude between 3,000 and 3,100 cm⁻¹ is also produced by O–H. For isomer 2B, the double peak (3,256 and 3,283 cm⁻¹) arises from the O–H stretching motion of the hydrogen-bonded O–H group of the second methanol molecule. The broad 2,750–3,000 cm⁻¹ band has overlapping contributions from the CH₃ stretching motions (2,850–3,000 cm⁻¹) and from the O–H stretching of the methanol hydrogen bonded to the Cl⁻ anion. This last contribution is at 2,844 cm⁻¹ and is broadly active in the 2,750–2,900 cm⁻¹ domain.

Clearly, the addition of a second methanol to Cl⁻-(Methanol)₁ in the second shell of Cl⁻ in isomer 2B, strengthens the strong OH⋯Cl⁻ hydrogen bond and thus induces a strong red-shift of the O–H stretch from 3,085 cm⁻¹ (Cl⁻-(Methanol)₁) to 2,750–2,900 cm⁻¹ (isomer 2B of Cl⁻-(Methanol)₂). For isomer 2A, the addition of a second methanol to Cl⁻-(Methanol)₁, with the two methanol molecules located in the first hydration shell of the anion, forming two ionic hydrogen bonds with Cl⁻, then results in a weakening of the H-bond and therefore provides a shift of the O–H stretch to higher frequency, from 3,085 cm⁻¹ (Cl⁻-(Methanol)₁) to 3,195 cm⁻¹ (isomer 2A of Cl⁻-(Methanol)₂).

The O–H bands above 3,000 cm⁻¹ in isomers 2A (a single intense peak) and 2B (double peak) match the positions and band-shapes of the bands observed experimentally between 3,150 and 3,300 cm⁻¹. The calculated O–H stretch for isomer 2A might be too intense in comparison to the experiment, but the presence of one band (3,193 cm⁻¹ in the IR-PD) and a doublet (3,225 and 3,242 cm⁻¹ in the IR-PD) are well reproduced by the dynamical spectra arising from the superposition of the spectra of both isomers. Intensity depends on the ratio of population of the two isomers; this ratio has been taken as 1:1 here. Similarly, the O–H stretch band coming from the O–H hydrogen bonded to Cl⁻ in isomer 2B, localized at 2,844 cm⁻¹, is very close to the 2,819 cm⁻¹ band in the IR-PD.

Our dynamical spectra thus clearly suggest that the two isomers 2A and 2B of Cl⁻-(Methanol)₂ are simultaneously present in the experiment. Our dynamics and the RRKM calculations presented in [65] show that they do not spontaneously inter-convert over the short 10 ps timescale of the ab initio dynamics, but that they can possibly interconvert over the hundred picosecond and nanosecond timescales at the 100 K temperature relevant to the experiment.

6 Cold IR-MPD and Vibrational Anharmonicities Below 1,000 cm⁻¹

The previous examples illustrated the strength of DFT-based molecular dynamics for vibrational spectroscopy, in particular for getting vibrational anharmonicities with good accuracy in the 1,000–4,000 cm⁻¹ domain. In these examples, anharmonicities resulted from the formation of medium to very strong hydrogen bonds, from their associated local dynamics at finite temperature, and from mode-couplings in this high frequency domain. We now illustrate how dynamical spectroscopy can provide agreement with experiments in the domain below 1,000 cm⁻¹. That domain is expected to be strongly anharmonic, so dynamics-based theoretical spectroscopy should be a relevant method to apply.

This is demonstrated here with DFT-MD simulations to extract the vibrational spectrum of the Ac-Phe-Ala-NH₂ peptide (capped peptide composed of phenyl and alanine residues, labeled FA in the remainder of the text; see Fig. 9) in the far-IR domain, in relation to IR-UV ion-dip experiments [111]. With dynamical spectroscopy, one has to keep in mind that the length of the trajectories has to be commensurate with the domain investigated. Hence, to sample the vibrational modes and their couplings in the 100–1,000 cm⁻¹ region, we have accumulated trajectories of 30 ps and make an average for the final theoretical spectrum over two trajectories (i.e., 60 ps sampling in total).

Fig. 9

IR-MPD experiment (black), static harmonic IR spectrum (red), and dynamical IR spectrum (blue) of Ac-Phe-Ala-NH₂ peptide in the far-IR domain. For clarity, amplitudes are multiplied by 5 below 400 cm⁻¹ in all spectra. Currently submitted to Angew Chemie Int [111]

Figure 9 presents the dynamical spectrum calculated for structure FA1 along with the experimental spectrum in the 100–800 cm⁻¹ region. The far-IR vibrations are extremely well reproduced by the DFT-MD simulation. Especially the vibrations below 500 cm⁻¹ are in excellent agreement, with maximum deviations below 10 cm⁻¹, almost within the spectral experimental resolution, and the agreement in the 500–1,000 cm⁻¹ domain is within 30 cm⁻¹. The dynamical anharmonic spectrum undoubtedly proves that the FA1 structure is responsible for the experimental vibrational features. See for comparison the harmonic spectrum of this structure, calculated with the same BLYP representation, where it can be seen that neither the positions nor the number of active bands match the experiment. In contrast, the dynamical anharmonic far-IR spectrum is in excellent agreement with the experiment, for the number of active bands, band-positions and shapes. Absolute band-intensities seem underestimated, but relative intensities are well reproduced by the simulations.

DFT-MD-based IR spectroscopy can be obtained with high accuracy in the far-IR vibrational domain, providing a definitive assignment of the experimental vibrational features. No scaling factors entered our final theoretical spectrum. Such an achievement is out of reach of harmonic spectra in that vibrational domain. DFT-MD simulations combined with experimental far-IR spectra are thus a promising accurate tool for the structural characterization of peptides structures in the far-IR vibrational region. Particularly for larger peptides, where the diagnostic amide vibrations in the mid-IR spectrum (amide I, II and III) may suffer from spectral congestion, the set of delocalized vibrations in the far-IR spectrum may form an interesting alternative for unraveling peptides structures from vibrational spectroscopy. For the small peptide shown here, DFT-MD has been able to discriminate between similar secondary structures using far-IR anharmonic signatures only (not shown; see [111]), which cannot be differentiated in the mid-IR spectrum.

7 Collision Induced Dissociation

In collision induced dissociation (CID) the ions selected by choosing a given m/z are sent with a given translational energy to the collision room where they collide with an inert gas (Ar, Ne, N₂, Xe …). Thus, translational energy is converted in internal rovibrational energy, and then the ion can subsequently fragment. There are different techniques for activating ions which differ for the energy range employed, for the instrument, and for the activation mechanism, as reviewed by Sleno and Volmer [112], Laskin and Futrell [113], or Mayer and Poon [114]. In this domain again, theoretical calculations can help one to understand the CID processes and in particular provide an atomistic comprehension of the mechanisms and pathways leading to the final fragments. We thus review in Sect. 7 the most popular theoretical tools applied in the domain, i.e., RRKM statistical theory and direct molecular dynamics, and simultaneously review how and when these methods can be used and what knowledge can be gained. We illustrate with some examples taken from the literature and from our works.

As we have discussed in the introduction of this chapter, dissociation of ions induced by collision (with an inert gas, CID, or with a surface, SID) can be theoretically described by following the PES (Potential Energy Surface) from reactants to products. On this PES the reactivity can be statistical, thus being described by RRKM theory, or not-statistical. Here we thus review how the knowledge of PES is used to rationalize CID (Sect. 7.1), then the RRKM theory and its application to CID reactivity (Sect. 7.2), and finally how chemical dynamics can provide a framework for an interpretation of experiments taking into account reactivity on both short and long timescales (Sect. 7.3).

7.1 Potential Energy Surface

Following Born–Oppenheimer approximation, the reactivity of an activated system can be described as dynamically evolving on the potential energy surface, i.e., the multidimensional surface (a 3N−6 surface in internal coordinates, where N is the number of atoms of the dissociating molecule) obtained by solving the time-independent Schroedinger equation for the molecular system at fixed nuclei positions. The well-known methods of quantum chemistry are used [13] and they give access to energies and vibrational frequencies of reactants, products, intermediates, and transition states connecting them. A schematical PES is shown in Fig. 2, representing a model system. The reactants and (eventually) intermediates are geometries characterized by 3N−6 real frequencies (i.e., minima in the PES) and are obtained by means of well-know geometry optimization procedures. The transition states are points on the PES that connect two minima and, from a topological point of view, are obtained as saddle points on the PES characterized by one imaginary frequency (these are called “tight” transition states; we see later on how another type of TS can be defined).

As previously mentioned, the energy is obtained by solving the electronic structure problem. This gives the internal electronic energy to which the zero-point energy should be added to describe the actual potential felt by the molecule during the reaction. The zero point energy correction is obtained from vibrational normal modes, adding the energy of each harmonic oscillator associated to normal modes:

$$ {E}^{ZPE}={\displaystyle \sum_i^{3N-6}\frac{1}{2}}\hslash {\omega}_i $$

(8)

where ω _i is the frequency of the i normal mode and $\hslash =h/2\pi$ with h the Plank constant. Solving the PES problem thus needs an efficient algorithm to identify minima and saddle points (a problem common to the conformational sampling) and also the reaction channel(s). Furthermore, to have a full picture of the multidimensional surface, several energy points should be calculated, and thus the electronic structure problem should be solved in a relatively fast way. Thus the PES problem can be divided into two (connected) problems: (1) an efficient sampling of different conformers and reaction channels; (2) efficient way of calculating electronic energy of the molecular system. For small molecules both problems are solved by considering all possible isomerizations and reaction channels. For large systems, enhanced sampling techniques should be considered, whereas generally the conformational sampling is done by molecular dynamics at high temperature (with a simple Hamiltonian) and fragmentation pathways are obtained by moving the bonds and angles manually.

The reaction pathways identification (quite a tricky problem for large systems) is often guided by experiments: only products compatible with m/z observed in CID experiments are studied. This was the case for Ca²⁺/formamide [115], Ca²⁺/urea [116], or uracil interacting with Ca²⁺, Pb²⁺, Cu²⁺, or Na⁺ [117] dissociation studies of Yanez, Salpin, and co-workers.

Traditionally, computational studies of reaction paths are performed by constructing a PES as extended as possible, i.e., locating minima and transition states. With this approach it was possible to understand the fragmentation mechanisms of metal-biomolecular systems, as reported in a series of studies by Ohanessian and co-workers on glycine-Cu⁺ [118], Zn²⁺ binding amino acids [119] or for a series of singly and doubly charged cations binding glycine [120].

Peptide fragmentation mechanism is non-trivial [113, 121, 122] and some studies have used PES to understand the different complex reaction channels. Several authors have thus studied PES of protonated glycine [123], glycylglycine [124], diglycine [125], small peptide models to understand the competition between ${\mathrm{b}}_2^{+}$ and ${\mathrm{a}}_2^{+}$ formation [126–128], glycylglycylglycine [129], or to investigate the model of mobile proton suggested to be at the basis of peptide gas phase reactivity [130, 131]. These calculations are often coupled with vibrational spectroscopy in the gas phase, which is used to validate the structure of the reaction products [132, 133].

When calculating the electronic energy, the level of theory chosen depends mainly on the system size. Although for small systems high correlated methods (such as CCSD(T) or CASPT2) can be used, when the system is larger DFT becomes compulsory. MP2 is also an option, but it needs an extended basis set to provide converged results. Furthermore, because PES deals with dissociation, basis set superposition error should be considered, and this is generally done by employing the Boys and Bernardi method [134]. In this field, in particular for systems containing metals, often the DFT method is first compared with high level calculations on small systems of the same kind (e.g., organic molecules in interaction with a metal) to identify the best functional (and basis set) which can be then applied to larger systems. This was done, for example, by the Yanez group when studying Sr²⁺–formamide, pointing out that the G96LYP functional reproduces CCSD(T) calculations [135] and then uses this functional when studying the PES [136]. An important aspect is that to calculate the ZPE it is necessary to obtain harmonic frequencies. To this end the Hessian matrix has to be constructed and then diagonalized, and this is often possible only at a low level of theory. Thus, for example, in the case of La³⁺, Gd³⁺, and Sm³⁺ interacting with NH₃, the DFT and MP2 electronic energies (without inclusion of ZPE) were first validated against CCSD(T) and CASPT2 calculations and then used to describe reactivity [137]. Furthermore, when dealing with large systems, it is possible to obtain the electronic energy at a level of theory and then add the ZPE obtained at a lower one.

Before discussing RRKM statistical theory and how it is applied to study CID reactivity, it should be noted that transition states located by topological analysis of the PES (i.e., saddle points) are a particular kind of transition state. In fact, a more general definition of transition state comes from the variational transition state theory (V-TST) which is related to reaction kinetics [138, 139]. Saddle points are always transition states but they are special cases, called tight transition states. In the case of dissociation pathways, when the two parts of the molecule are broken apart there is no saddle point in the reaction coordinate but a loose transition state exists. This is defined better in the next section. Finally, it should be mentioned that a relatively different formulation of transition state is given by Vanden-Eijnden, Ciccotti, and co-workers to study rare event sampling [140]. In this framework, a transition state is defined as an iso-committor surface (in both position and momenta) where the probability to go to the products is equal to those to come back to the reactants. Although it has never been applied to CID reactivity, it would be particularly intriguing, especially in the most general formulation which considers both position and momenta of the system.

7.2 RRKM Theory

RRKM theory is the well-known and consolidated statistical theory for unimolecular dissociation. It was developed in the late 1920s by Rice and Ramsperger [141, 142] and Kassel [143], who treated a system as an assembly of s identical harmonic oscillators. One oscillator is truncated at the activation energy E ₀. The theory disregards any quantum effect and the approximation of having all identical is too crude, such that the derived equation for micro canonical rate constant, k(E),

$$ k(E)=\nu {\left(\frac{E-{E}_0}{E}\right)}^{s-1} $$

(9)

is incapable of providing the correct rate. The theory is based on a concept still present in the actual formulation: energy flows statistically among all the oscillators and the chance of finding the system with a particular arrangement of its internal energy is equivalent to any other. Later, Marcus and Rice [144] and Rosenstock, Wallenstein, Wahrhaftig, and Eyring [145] developed the actual version of the theory which takes into account the vibrational (and rotational) degrees of freedom in detail, leading to the well-known RRKM/QET (for quasi-equilibrium theory) expression. In its simple formulation (i.e., neglecting rotational or tunneling effects), reads simply

$$ k(E)=\frac{\sigma {N}^{\ddagger}\left(E-{E}_0\right)}{h\rho (E)} $$

(10)

where ρ(E) is the reactant vibrational density of states, N ^‡(E − E ₀) is the sum of states in the transition state, and σ is the reaction symmetry factor. This expression is very simple and powerful: from geometry and vibrational frequencies of reactant and transition state it is possible to obtain reaction rate constants as a function of the energy injected into the system. The calculation of sum and density of states can be done using the direct count method with the implementation proposed by Beyer and Swineheart [146]. Earlier, Whitten and Rabinovich proposed a method based on classical sum of states formulation with empirical parameterization to save computing time [147, 148]. Nowadays it is rarely used because the computational power available makes both calculation of frequencies and direct counting accessible. Thus, the question, from theoretical point of view, is to identify minima and transition states; this comes back to correct exploration of the PES (see Sect. 7.1), considering minima and transition states for all possible isomerizations and reaction channels, and having access to the energy given to the system in the collision process. An important aspect to highlight is that the resulting rate constant depends only on the total energy E and the total angular momentum in the more general formulation that takes into account rotational energy. In fact, it is assumed that the rate constant does not depend upon where the energy is initially located and that a microcanonical ensemble is maintained as the molecule dissociates. This is equivalent to the assumption that IVR is rapid compared to the lifetime with respect to dissociation. That is, vibrations are assumed to be strongly coupled by higher order terms in the expansion of the potential energy function. This means that after a collision the energy is likely to be in proportion to its equilibrium probability and that the states at that energy are all equally probable. To what extent this “ideal” is attained and what are the possible other situations was described in detail by Bunker and Hase [149]. Despite the intrinsic limitation of an “ideal” theory, RRKM is able finally to take into account many examples of unimolecular dissociation produced by collisional activation, as reviewed, for example by Baer and Mayer [150]. The simple RRKM formulation of (10) can be extended, taking into account rotational energy [151], tunneling effects [152, 153], and non-adiabatic transitions between different spin states [154].

RRKM theory is also at the basis of localization of “loose” transition states in the PES. Another assumption of the theory is that a “critical configuration” exists (commonly called transition state or activated complex) which separates internal states of the reactant from those of the products. In classical dynamics this is what is represented by a dividing surface separating reactant and product phase spaces. Furthermore, RRKM theory makes use of the transition state theory assumption: once the system has passed this barrier it never comes back. Here we do not want to discuss the limits of this assumption (this was done extensively for the liquid phase [155] but less in the gas phase; for large molecules we can have a situation similar to systems in a dynamical solvent, where the non-reacting sub-system plays the role of the solvent in liquid phase), but how RRKM theory is useful for TS searching. We have to remember that a reaction in statistical theory is represented by a flux in phase space and the TS corresponds to a dividing surface at which this flux is a minimum. In the previous section we described the TS localization on the PES, a point with a particular topological feature: one imaginary and 3N−7 real frequencies. This is called “tight” transition state and the localization of this dividing surface is dominated by the energy contribution to determining the TS sum of states. However, in dissociation phenomena, we often have a PES from reactant to product without any saddle point. In this case, the transition state is defined and thus located by the “variational transition state theory” (V-TST) [138, 139]. The TS is a minimum in the sum of states along the reaction coordinate:

$$ \frac{d{N}^{\ddagger}\left(E,J,R\right)}{dR}=0 $$

(11)

Thus all points of the PES can be obtained (in principle) and unimolecular dissociation rate constants expressed as a function of internal energy. Nowadays, the vibrational frequencies can be calculated at a high level of theory, and the sum and density of states obtained by direct count without any particular problem for small or medium sized systems, and thus RRKM kinetics were employed to examine several unimolecular dissociations [150]. Even complex kinetic schemes were solved to obtain the rate constant for product formation through different reaction channels [156, 157].

7.3 Chemical Dynamics

Static (PES) and kinetic (RRKM) information can be complemented by chemical dynamics simulations, which are able to fill some aspects of gas phase reactivity not considered by the previous approaches. In particular, chemical dynamics can be used to model explicitly the collision between the ion and the target atom and thus it is possible to obtain the energy transferred in the collision and (eventually) the reactions. The molecular system, represented as an ensemble of atoms each bearing a mass m _i , evolves on the Born–Oppenheimer potential energy surface through Newton’s equation of motions:

$$ -\frac{\partial V}{\partial {q}_i}={m}_i\frac{d^2{q}_i}{d{t}^2} $$

(12)

This means that the molecule’s nuclei positions, q _i , and momenta, p _i , evolve on the potential energy obtained by solving the time-independent Schroedinger equation at each configuration.

This general chemical dynamics (or direct dynamics) approach can be applied to study different molecular problems (see, e.g., the recent review by Hase, Song, and Gordon [158]), and the application to problems related to CID was pioneered mainly by Hase, Bunker, and co-workers. In particular, the random collision of an atom with a molecule was discussed (and an operative algorithm is given) for atom–molecule reaction on pre-determined potential energy surfaces [159] and the initial conditions of the molecule were discussed in the framework of state-selected unimolecular dissociation [160, 161]. The same procedure, also valid for non-reactive collisions, and the general chemical dynamics code VENUS, developed by Hase and co-workers [162, 163], can be used for different reactivity situations. In particular for CID, one has to consider initial conditions of the ion and the rare gas atom (or molecule). Initial internal energy of the ion can be considered in different ways: (1) sampling normal mode vibrational energies from a Boltzmann distribution at a given temperature (T _vib ); (2) sampling a microcanonical ensemble by orthant or normal mode sampling [164, 165]; (3) sampling fixed vibrational normal modes; and (4) adding energy to a particular mode (local mode sampling [166]) on a previously determined total energy. These initial conditions are generally classically sampled (or semi-classically because one considers normal mode zero point energies) and recently the effect of quantum initial conditions on unimolecular reactions was also considered [167].

Rotational energy and angular momentum for the polyatomic molecule is selected by assuming separability of vibrational and rotational motion. Thus initial rotational conditions are obtained either by assuming a thermal partitioning of RT/2 about each internal rotational axis or by assuming that the molecule is a symmetric top (I _x  < I _y  = I _z ) and then the total angular momentum and its x component are sampled from the probability distributions:

$$ \begin{array}{ll}P\left({J}_x\right)= \exp \left[-{J}_x^2/2{I}_x{k}_bT\right]\hfill & 0\le {J}_x\le \infty \hfill \end{array} $$

(13)

$$ \begin{array}{ll}P(J)=J \exp \left[-{J}^2/2{I}_z{k}_bT\right]\hfill & {J}_x\le J\le \infty \hfill \end{array} $$

(14)

as described by Bunker and Goring-Simpson [168].

Then the ion-projectile relative energy is set and random orientations in Euler angles between the (rigid body) ion and the projectile (often an atom, in the case of N₂ the center of mass is considered and vibrational and rotational sampling is done also to set its internal initial conditions) are sampled. Then the possible impact parameters are considered. They can either be set to a unique value or an ensemble of possible values can be sampled in a defined range generally corresponding to the molecular size. Finally, the collision is carried out at a given energy defined in the center-of-mass of the system composed by the ion and the projectile, E _CM , which is in relation to the laboratory framework energy, E _LAB (directly set in the instrument):

$$ {E}_{CM}=\frac{m_2}{m_1+{m}_2}{E}_{LAB} $$

(15)

where m ₁ is the mass of ion and m ₂ of the projectile. A detailed review of collision dynamics in CID is reported by Douglas [169].

This procedure implemented in the chemical dynamics code VENUS [162, 163] is suited to mimic the collision (in the single collision limit hypothesis). Simulation parameters (temperature, energy, etc.…) can be tuned as needed. A schematic picture chemical dynamics set-up for simulating collisions is shown in Fig. 10.

Fig. 10

Schematic representation of how the CID molecular dynamics is generally set-up (see Hase et al. [159] for details)

Initially, direct dynamics simulations were performed using analytical potential energy functions for both intra- and intermolecular interactions. Reactivity (generally of a cluster) was initially taken into account by analytical Morse functions. Energy transfer was studied by using non-reactive molecular mechanics force fields, examining its dependence on the collision energy, the nature of the projectile, or the molecular shape of the ion. Energy transfer was studied, for example, in the case of trans-stilbene by Bolton and Nordholm [170]. Lim and Garret studied the collision between different gases and vibrationally excited azulene, finding that using simple ion-projectile analytical function results are in good agreement with experiments in particular for heavy gases (Ar, Kr, Xe) and that the energy transfer is dominated by the repulsive part of the potential [171]. In this way, energy transfer in Al clusters was studied as a function of cluster size and shape, pointing out that, when in the spherical structure, the translational energy is mainly transferred into internal vibrational energy; on the other hand, in planar clusters the rotational excitation is relevant [172]. A clear characterization of rotational vs vibrational activation is important when the RRKM reactivity is considered and in particular if the reaction channels have different rotational barriers (rotational energy is important, particularly when reactant and TS do not have the same moment of inertia). Furthermore, the energy transfer in octahedral Al₆ clusters was studied with different projectiles [173] (Ne, Ar, Xe) finding similar results, and the energy transfer could be qualitatively interpreted by means of the simple refined impulsive model developed by Mahan [174]. More recently, Hase and co-workers applied chemical dynamics to peptide CID, using AMBER force field [175] for intramolecular potential energy and an analytical function for the ion–projectile (Ar) interaction obtained from high level quantum chemistry calculations on building blocks [176], being purely repulsive:

$$ {V}_{Ar-i}={A}_{Ar-i} \exp \left(-{b}_{Ar-i}{r}_{Ar-i}\right)+\frac{C_{Ar-i}}{r_{Ar-i}^9}. $$

(16)

In this way, they were able to study energy transfer in peptide CID as a function of peptide shape [177].

By employing an analytical potential describing Cr⁺–CO interaction, Martinez-Nunez et al. [178] have studied the CID of a $\mathrm{C}\mathrm{r}{\left(\mathrm{C}\mathrm{O}\right)}_6^{+}$ cluster colliding with Xe, where they found that reaction dynamics is highly affected by “impulsive character.” This corresponds to the so-called shattering reaction mechanism, where the bond is broken largely before energy transfer between vibrational modes. It should be noted that the shattering mechanism can be important in driving the reactivity, as was, for example, noticed experimentally and theoretically by studying the dissociation of CH₃SH⁺ and ${\mathrm{CH}}_3{\mathrm{SCH}}_3^{+}$ [179, 180]. Shattering is a reaction mechanism that shirks the basic assumptions of statistical theory and thus any prediction based on this theoretical framework. Dynamics can thus be fundamental to point out whether this kind of direct dissociation mechanism plays a role in the appearance of reaction products which cannot be explained from merely statistical considerations.

Later, Hase and co-workers developed couplings between VENUS and quantum chemistry codes such as Gaussian, GAMESS-US, MOPAC, and NW-Chem, such that the ion reactivity could be considered directly (i.e., without any prior knowledge of the PES). A QM description can be used to treat either the whole system (ion and projectile) or only the intramolecular ion potential, whereas the ion-projectile intermolecular interaction is treated via an analytical potential, similar to that of (16). A full QM description, at B3LYP level, was employed in studying fragmentation of H₂CO⁺ colliding with Ne [181], finding good agreement with experiments for the collisional cross section, σ _CID . This quantity is obtained from simulations, simply by evaluating the reactive probability as a function of impact parameter, P(b), and then integrating over the whole b range:

$$ {\sigma}_{CID}=2\pi {\displaystyle {\int}_0^{b_{max}}P}(b)bdb $$

(17)

This study was able to show the main processes that can be activated by collision: (1) conversion of collisional energy into internal energy of scattered H₂CO⁺; (2) sequential activation of a bond and reactivity observed in the simulation time length (200 fs); (3) direct reaction after the collision (here H atom knock out). It is also interesting to note how the energy transfer is affected by collision orientation and, to some extent, the initial vibrational preparation of the ion. Full QM studies can be applied to relatively small systems, with the advantage of being able to describe in detail the effects of projectile nature in reactivity without any parameterization and taking into account the possibility of charge transfer to the ion, as pointed out by Anderson and co-workers studying the reactivity of ${\mathrm{NO}}_2^{+}$ with rare gases [182].

By using a QM + MM approach it was possible to treat bigger systems, with a reasonable statistical sampling. With this approach our group studied the reactivity of protonated urea (the QM part was done at the MP2 level of theory) showing how the shattering mechanism is responsible for opening the reaction channel, leading to the high energy product observed experimentally [183]. We have further studied the role of the projectile in energy transfer: a diatomic projectile (N₂) transfers less energy with respect to a monoatomic gas (Ar) [184]. Interestingly, it was found that the rotational activation decreases as the initial rotational quantum number of the projectile increases. This was qualitatively justified by the constraint of conserving total angular momentum, but more studies on rotational activation are needed. The same approach was extended to study reactivity of doubly-charged cations, CaUrea²⁺ [185]. Here, dynamics showed how the neutral loss pathway is obtained in direct activation whereas the Coulomb explosion, occurring after molecular reorganization, is obtained at longer timescales (schematically shown in Fig. 11). In the same study, energy transfer obtained from dynamics was used in statistical RRKM approaches.

Fig. 11

Schematic representation of reactivity obtained in CID of [CaUrea]²⁺ by Spezia et al. [185], where neutral loss (Ca²⁺ + urea) pathway is obtained by a “direct” mechanism whereas the Coulomb explosion $\left({\mathrm{caNH}}_2^{+}+{\mathrm{NH}}_2{\mathrm{CO}}^{+}\right)$ pathway is obtained by a structural reorganization

To improve statistics, in both length of time and system size, the QM part can be treated by employing semi-empirical Hamiltonians. In this way, it was possible to study peptide reactivity, as done by Hase and co-workers for reactivity of N-protonated glycine [186] in both CID and SID, suggesting that “nonstatistical fragmentation dynamics may be important in the collisional dissociation of protonated amino acids and peptides.” In this study the CID impact parameter was set to zero, and the shattering was observed mainly in SID, showing that, also in this case, it is initiated by particular collisional orientations. AM1 results were in agreement with both Amber- (for energy transfer) and (for reactivity) MP2-based simulations [187], thus paving the way for using semi-empirical Hamiltonians in the puzzling study of gas phase reactivity of peptides. Protonated octaglycine SID was studied by Park et al. [188], showing a dependence on the relevance of shattering on the collisional orientation between the ion and the surface. More recently, in our group we have studied the CID reactivity of N-formylalanylamide (a model dipeptide), pointing out that the semi-empirical Hamiltonian is able to provide a picture of reactivity in agreement with experiments, such that it becomes possible to provide a semi-quantitative theoretical MS-MS spectra [189]. Furthermore, these results strengthen the “mobile” proton model [190, 191] which triggers the peptides gas phase reactivity. Currently, we are studying different related systems to understand better the molecular details of such a rich reactivity.

The use of a semi-empirical Hamiltonian also makes possible the study of negative polyanions. By coupling QM + MM chemical dynamics simulations with ESI-MS/MS experiments, we have recently investigated the fragmentation of a sulfated saccharide (the galactose-6-sulfate), determining for the first time reaction mechanisms of a sugar by a combined computational and experimental approach [192]. Furthermore, Bednarski et al. studied dissociation of poly[(R,S)-3-hydroxybutanoic acid] anions, in which molecular dynamics was able to rationalize the discrepancy between experiments and RRKM analysis [193]. The reactivity of Li⁺–Uracil was recently studied with a specifically adjusted semi-empirical AM1 Hamiltonian, as reported by Martínez-Nunez and co-workers [194]. It should be noted that in this case, as well as for the protonated urea reactivity [184], once the transition state was identified and the energy transfer obtained, it was possible to use dynamics to study how a system evolves from the transition states towards different reaction channels. In particular, post-transition state dynamics show that, starting from a transition state, other products can also be obtained and not just that identified by the transition state.

A QM (or QM + MM) approach in studying CID reactivity makes possible, in principle, the study of the whole process: collision, energy transfer, energy flow through modes (if fast enough), and eventually fragmentation. Thus, we require a tool to provide us with a complete answer to the question of what is the reaction of an ion in the CID process: unfortunately there are several issues making this goal difficult to achieve. The main limitation is in the time available in simulations and in a lack of sufficient statistical sampling. Even when employing faster methods (such as a semi-empirical Hamiltonian), we can observe processes in the picoseconds timescale, whereas reactivity (especially for lager systems) may occur on much longer timescales. In other words, direct dynamics can be used to obtain, on the one hand, fast direct reactivity (from hundreds of femtoseconds to tens of picoseconds) and on the other the energy transfer which can be used in statistical kinetics theory (such as RRKM) to evaluate reactivity. The combination of fast and slow reaction timescales is what physically provides the final product distribution obtained in tandem mass spectrometry experiments, and chemical dynamics can be used to rationalize the appearance of different fragments. For example, as stated in the previously mentioned studies from the groups of Hase, Anderson, and ourselves, the “shattering” reaction mechanisms are often strongly related to the orientation of the collision. Thus the shape of the ion can be determinant for the appearance (and the relative population) of some reaction pathways.

8 Conclusions and Some Future Directions

This chapter was intended to give some flavors of molecular dynamics-based methods for the calculation of gas phase vibrational spectra and collision induced dissociations of gas phase molecules and complexes. The examples were taken from our own work. We hope we have convincingly shown the essential ingredients entering these simulations, their usefulness in relation to IR-MPD and IR-PD action spectroscopic experiments, their usefulness in relation to CID experiments, and how essential these methods are to produce definitive assignments and microscopic interpretation of experimental features.

We now conclude this chapter by a discussion of what we think are the challenges to be tackled by theoreticians in the years to come in the two domains of gas phase spectroscopy and collision-induced dissociation modeling, but also some challenges that, we theoreticians, would like to suggest to the experimentalists to strengthen our knowledge of structural and dynamical information of gas phase molecular assemblies.

One systematic issue in molecular dynamics (MD) simulations is the timescales and lengths. Our anharmonic spectra were calculated with DFT-based MD, where typical timescales and lengths are a few 10′ of picoseconds and molecular systems composed of ~500 atoms. Investigations of anharmonic spectra of peptides in the far-IR domain require trajectories of at least 30–50 ps for relevant sampling of the mode-couplings. As shown in Sect. 6, dynamical spectra are of excellent accuracy and thus can help unravel peptidic conformations using a vibrational domain where the vibrational signatures are expected to be less congested than in the 1,000–2,000 cm⁻¹ and 3,000–4,000 cm⁻¹ domains. Going beyond these timescales, which might evidently be necessary for more complex peptides than those presented in Sect. 6, and beyond the current sizes in order to investigate peptides composed of several tens of residues, semi-empirical-based MD should be the method to use. Its advantage is that it is still based on electronic representations, though simplified, and does not rely as heavily as classical force fields on large scale parameterization of the analytical force field expressions. Semi-empirical MD has been applied to our trajectories for collision induced dissociation, with great success. This thus seems a promising avenue for dynamical spectroscopy which we have already engaged.

Quantum effects of the nuclei have not been taken into account in the classical nuclear dynamics presented here. We currently follow strategies based on the introduction of Zero Point Energy (ZPE) within the modes at the initial time of the dynamics to include quantum effects in the classical nuclear trajectories. We have implemented such strategies [37, 72] and applied them to the dynamical spectroscopy of ionic clusters [37] in relation to IR-PD experiments. Theoretical issues such as Zero Point Leak [195, 196] can be encountered, however, and we are currently working in that domain. Closely related are Fermi resonances, also investigated by our group.

The experimental methods employed for the production of the molecular assemblies in the gas phase can be an issue for the subsequent spectroscopic characterization. In some of the IR-MPD experiments, ions are produced with the typical techniques of mass spectrometry, typically ESI (Electro Spray Ionization) and MALDI (Matrix-Assisted Laser Desorption/Ionization). In others, supersonic expansions are used to form peptides at rather low temperature. In some of the IR-PD experiments, the ionic clusters are formed by a combination of supersonic expansion and subsequent collisional processes. In these experiments in particular, it has been shown that higher energy conformers can be kinetically trapped. Dynamics simulations and associated RRKM calculations can be used to clarify the processes and mechanisms leading to the formation of higher energy conformers, and prevent isomerization towards lower energy conformers. Such simulations are currently underway in our group.

Certainly of importance to theoreticians (and surely to experimentalists) is the direct modeling of the action spectroscopy intensity signals. It should be remembered that static and dynamical spectra calculations are related to the linear one-photon absorption signal, whereas the experiments record the consequent action of this absorption in terms of molecular fragmentation. Theory and experiments are by no means equivalent, and, as discussed in the introduction of this chapter, deviation between experiment and theory is expected on band-intensities. Theoretical work has to be done for a direct modeling of the signal. Kinetic-based models and kinetic modeling are certainly the approaches worth following.

What challenges can we suggest to IR-MPD/IR-PD experimentalists? We strongly encourage developments of set-ups in the area of temperature-dependent spectroscopy. This would open up the path to the direct experimental sampling of the different conformers of molecules and clusters with an increase of internal energy (of course this is highly dependent on energy barriers) through the direct temperature-evolution of the IR signatures. This would also be one way to probe vibrational anharmonicities gradually, and the appearance of different kinds of anharmonicities. Temperature is the natural parameter of molecular dynamics simulations, so the interplay with MD simulations is undoubtedly pivotal for the interpretation of temperature-dependent IR features. This is already under way in our group in synergy with experiments of ions trapping from the Lisy group, where temperature effects can be investigated by switching between different fragmentation channels. Another crucial challenge is multi-dimensional gas phase IR spectroscopy, following the spirit of the now rather well-established 2D-IR spectroscopy conducted in the liquid phase and at solid-liquid and liquid-air interfaces. These two-dimensional spectroscopies provide direct information on mode-couplings, and they simultaneously reduce the possible congestion of spectral information seen in linear (1D) spectra of complex and large peptides. There again, MD-based spectroscopy is an essential theoretical tool for the interpretation of the experiments, as already seen in several publications of 2D-IR spectroscopy in the liquid phase. Closely related are vibrational excitations and subsequent time-dependent IR spectroscopy, the development of which we also strongly encourage, following the spirit of the latest experimental developments of time-dependent linear IR in liquids, or time-dependent 2D-IR in liquids and at interfaces. There again, MD time-dependent simulations are essential as theoretical support. Two-dimensional and time-dependent spectroscopic experiments not only reveal time-dependent properties but also reveal couplings, both properties being the essential ingredients of MD-based theoretical spectroscopy.

In CID modeling we believe chemical dynamics simulations can be the basis for defining a complete theoretical procedure to obtain the tandem mass spectrometry (MS/MS) spectrum of a given molecule from nothing. In fact, although for other experiments (e.g., InfraRed, NMR, Raman, UV-Vis) theoretical and computational methods can provide the final spectrum of a given molecular species, this is not currently the case for MS/MS. Thus, if we write down a new or experimentally unknown species, it is almost impossible to infer a theoretical MS/MS spectrum prior to experiments. We believe this is the challenge for theoretical modeling in collision-induced dissociation for years to come. Combining different dynamics approaches for both short and long timescales (as already presented here), it should be possible to devise a theoretical procedure to obtain the MS/MS spectrum of any given molecular species from its chemical and reactive properties. To accomplish this, chemical dynamics and RRKM statistical theory of unimolecular dissociation can be combined, providing the means for the theoretical framework of the different reaction pathways leading to gas phase fragmentation. To this end, developments in the quantum description, in particular for complex biomolecules, are needed, together with a theoretical framework for ro-vibrational partitioning of energy transfer. Connected with the energy transfer problem, particular care should be devoted to model multiple collisions. This further bridges the gap between different timescales involved in the fragmentation processes. These are general directions in which we are currently working.