Toward a realistic density functional theory potential energy surface for the H5+ cluster

Computational Modeling: Up-to-Date Approaches and Cutting-Edge Applications from Clusters, Nanostructures to Bulk Systems

The contributions in this special theme collection, in honor to Prof. P. Villarreal, cover a broad variety of computational methodologies and experimental techniques, containing studies on gas phase, clusters and condensed phase systems.

Infrared spectroscopy and anharmonic theory of H3 +Ar2,3 complexes: The role of symmetry in solvation

The vibrational spectra of H₃ ⁺Ar_2,3 and D₃ ⁺Ar_2,3 are investigated in the 2000 cm^-1 to 4500 cm^-1 region through a combination of mass-selected infrared laser photodissociation spectroscopy and computational work including the effects of anharmonicity. In the reduced symmetry of the di-argon complex, vibrational activity is detected in the regions of both the symmetric and antisymmetric hydrogen stretching modes of H₃ ⁺. The tri-argon complex restores the D_3h symmetry of the H₃ ⁺ ion, with a concomitant reduction in the vibrational activity that is limited to the region of the antisymmetric stretch. Throughout these spectra, additional bands are detected beyond those predicted with harmonic vibrational theory. Anharmonic theory is able to reproduce some of the additional bands, with varying degrees of success.

The smallest proton-bound dimer H5+: theoretical progress

The protonated hydrogen dimer, H5+, is the smallest system including proton transfer, and has been of long-standing interest since its first laboratory observation in 1962. H5+ and its isotopologues are the intermediate complexes in deuterium fractionation reactions, and are of central importance in molecular astrophysics. The recently recorded infrared spectra of both H5+ and D5+ reveal a rich vibrational dynamics of the cations, which presents a challenge for standard theoretical approaches. Although H5+ is a four-electron ion, which makes highly accurate electronic structure calculations tractable, the construction of ab initio-based potential energy and dipole moment surfaces has proved a hard task. In the same vein, the difficulties in treating the nuclear motion could also become cumbersome due to their high dimensionality, floppiness and/or symmetry. These systems are prototypical examples for studying large-amplitude motions, as they are highly delocalized, interconverting between equivalent minima through internal rotation and proton transfer motions requiring state-of-the-art treatments. Recent advances in the computational vibrational spectroscopy of the H5+ cation and its isotopologues are reported from full quantum spectral simulations, providing important information in a rigorous manner, and open perspectives for further future investigations. This article is part of a discussion meeting issue 'Advances in hydrogen molecular ions: H3+, H5+ and beyond'.

Communication: Trapping a proton in argon: Spectroscopy and theory of the proton-bound argon dimer and its solvation

Ion-molecule complexes of the form H⁺Ar_n are produced in pulsed-discharge supersonic expansions containing hydrogen and argon. These ions are analyzed and mass-selected in a reflectron spectrometer and studied with infrared laser photodissociation spectroscopy. Infrared spectra for the n = 3-7 complexes are characterized by a series of strong bands in the 900-2200 cm^-1 region. Computational studies at the MP2/aug-cc-pVTZ level examine the structures, binding energies, and infrared spectra for these systems. The core ion responsible for the infrared bands is the proton-bound argon dimer, Ar-H⁺-Ar, which is progressively solvated by the excess argon. Anharmonic vibrational theory is able to reproduce the vibrational structure, identifying it as arising from the asymmetric proton stretch in combination with multiple quanta of the symmetric argon stretch. Successive addition of argon shifts the proton vibration to lower frequencies, as the charge is delocalized over more ligands. The Ar-H⁺-Ar core ion has a first solvation sphere of five argons.

Theoretical predictions on the role of the internal H3(+) rotation in the IR spectra of the H5(+) and D5(+) cations

The IR spectra of the H5(+) and D5(+) cations in the mid- and far-IR spectral regions have been recently reported by experimentalists. These spectra show very rich vibrational patterns representing a challenge for state-of-the-art theoretical methods to provide definitive interpretations of them. Using a full-dimensional quantum anharmonic treatment, within the MCTDH approach, together with ab initio potential and dipole moment surfaces, the predominant features in the spectra are assigned, completing an important part in previous theoretical and experimental comparisons. The internal rotation of the H3(+) unit by exciting the H3(+)-H2 stretching mode is found to correspond to the new calculated features at 1182, 1876, and 2139 cm(-1) of the H5(+) spectrum, leading to a consistent assignment with the experimental spectra. In the calculated spectra of both H5(+) and D5(+) clusters, the progressions in the H3(+)-H2 stretch of the shared proton and the in- and out-of- plane H3(+) rotation are demonstrated to be the main features. Such states are expected to play a central role in the low temperature hydrogen/deuterium proton hop/exchange H3(+) + H2 reactions.

First-principles simulations of vibrational states and spectra for H5(+) and D5(+) clusters using multiconfiguration time-dependent Hartree approach

Simulations of the infrared (IR) spectra of the H5(+) and D5(+) clusters are carried out in the whole energy range, using a recent, reliable "on the fly" DFT-based potential energy surface, and its corresponding dipole moment surface. For the present study we adopted a recently proposed four-dimensional quantum model to describe the proton transfer motion between the two vibrating H2 or D2 units. Time-dependent and time-independent approaches within the multiconfiguration time-dependent Hartree method are employed for investigating the vibrational dynamics of the complexes. The obtained spectra are compared with recent experimental data available for energies up to 4500 and 3500 cm(-1) for the H5(+) and D5(+), respectively. Even though the present results are based on a reduced dimensional model, the infrared spectra are shown to be in good qualitative accord with those observed experimentally. Also as the reported data are subject to the potential energy surface, comparisons with previous theoretical calculations based on an analytical ab initio parameterized surface are also presented. The differences on the topology of the potentials are discussed in connection with their effect on the spectral features. We found that the main characteristics of the experimentally observed spectra are reproduced by both surfaces, evaluating in this way the sensitivity of such computations on the quality of the underlying potential. This finding serves to connect aspects of the potential surface of these systems to their spectral complexity, and could be indicative to calibrate intrinsic errors in their calculation for future studies.

Full-dimensional quantum calculations of the dissociation energy, zero-point, and 10 K properties of H7+/D7+ clusters using an ab initio potential energy surface

Diffusion Monte Carlo (DMC) and path-integral Monte Carlo computations of the vibrational ground state and 10 K equilibrium state properties of the H7 (+)/D7 (+) cations are presented, using an ab initio full-dimensional potential energy surface. The DMC zero-point energies of dissociated fragments H5 (+)(D5 (+))+H2(D2) are also calculated and from these results and the electronic dissociation energy, dissociation energies, D0, of 752 ± 15 and 980 ± 14 cm(-1) are reported for H7 (+) and D7 (+), respectively. Due to the known error in the electronic dissociation energy of the potential surface, these quantities are underestimated by roughly 65 cm(-1). These values are rigorously determined for first time, and compared with previous theoretical estimates from electronic structure calculations using standard harmonic analysis, and available experimental measurements. Probability density distributions are also computed for the ground vibrational and 10 K state of H7 (+) and D7 (+). These are qualitatively described as a central H3 (+)/D3 (+) core surrounded by "solvent" H2/D2 molecules that nearly freely rotate.

Full-dimensional quantum calculations of the vibrational states of H5(+)

Full-dimensional quantum calculations of the vibrational states of H5(+) have been performed on the accurate potential energy surface developed by Xie et al. [J. Chem. Phys. 122, 224307 (2005)]. The zero point energies of H5(+), H4D(+), D4H(+), and D5(+) and their ground-state geometries are presented and compared with earlier theoretical results. The first 10 low-lying excited states of H5(+) are assigned to the fundamental, overtone, and combination of the H2-H3(+) stretch, the shared proton hopping and the out-of-plane torsion. The ground-state torsional tunneling splitting, the fundamental of the photon hopping mode and the first overtone of the torsion mode are 87.3 cm(-1), 354.4 cm(-1), and 444.0 cm(-1), respectively. All of these values agree well with the diffusion Monte Carlo and multi-configuration time-dependent Hartree results where available.

Vibrational dynamics of the H5(+) and its isotopologues from multiconfiguration time-dependent Hartree calculations

Full-dimensional multiconfiguration time-dependent Hartree (MCTDH) computations are reported for the vibrational states of the H(5)(+) and its H(4)D(+), H(3)D(2)(+), H(2)D(3)(+), HD(4)(+), D(5)(+) isotopologues employing two recent analytical potential energy surfaces of Xie et al. [J. Chem. Phys. 122, 224307 (2005)] and Aguado et al. [J. Chem. Phys. 133, 024306 (2010)]. The potential energy operators are constructed using the n-mode representation adapted to a four-combined mode cluster expansion, including up to seven-dimensional grids, chosen adequately to take advantage in representing the MCTDH wavefunction. An error analysis is performed to quantify the convergence of the potential expansion to reproduce the reference surfaces at the energies of interest. An extensive analysis of the vibrational ground state properties of these isotopes and comparison with the reference diffusion Monte Carlo results by Acioli et al. [J. Chem. Phys. 128, 104318 (2008)] are presented. It is found that these systems are highly delocalized, interconverting between equivalent minima through rotation and internal proton transfer motions even at their vibrational ground state. Isotopic substitution affects the zero-point energy and structure, showing preference in the arrangements of the H and D within the mixed clusters, and the most stable conformers of each isotopomer are the ones with the H in the central position. Vibrational excited states are also computed and by comparing the energies and structures predicted from the two surfaces, the effect of the potential topology on them is discussed.

IR photodissociation spectroscopy of H7(+), H9(+), and their deuterated analogues

Cluster ions of H7(+)/D7(+) and H9(+)/D9(+) produced in a supersonic molecular beam with a pulsed discharge source are mass selected and studied with infrared laser photodissociation spectroscopy. Photodissociation occurs by the loss of H2 (D2) from each cluster, producing resonances in the 2000-4500 cm(-1) region. Vibrational patterns indicate that these ions consist of an H3(+) (D3(+)) core ion solvated by H2 (D2) molecules. There is no evidence for the shared proton structure seen previously for H5(+). The H3(+) ion core vibrational bands are weakened and broadened significantly, presumably by enhanced rates of intramolecular vibrational relaxation. Computational studies at the DFT/B3LYP or MP2 levels of theory (including scaling) are adequate to reproduce qualitative details of the vibrational spectra, but neither provides quantitative agreement with vibrational frequencies.

Mid- and Far-IR Spectra of H5(+) and D5(+) Compared to the Predictions of Anharmonic Theory

H5(+) is the smallest proton-bound dimer. As such, its potential energy surface and spectroscopy are highly complex, with extreme anharmonicity and vibrational state mixing; this system provides an important benchmark for modern theoretical methods. Unfortunately, previous measurements covered only the higher-frequency region of the infrared spectrum. Here, spectra for H5(+) and D5(+) are extended to the mid- and far-IR, where the fundamental of the proton stretch and its combinations with other low-frequency vibrations are expected. Ions in a supersonic molecular beam are mass-selected and studied with multiple-photon dissociation spectroscopy using the FELIX free electron laser. A transition at 379 cm(-1) is assigned tentatively to the fundamental of the proton stretch of H5(+), and bands throughout the 300-2200 cm(-1) region are assigned to combinations of this mode with bending and torsional vibrations. Coupled vibrational calculations, using ab initio potential and dipole moment surfaces, account for the highly anharmonic nature of these complexes.

Full-dimensional (15-dimensional) ab initio analytical potential energy surface for the H7+ cluster

Full-dimensional ab initio potential energy surface is constructed for the H(7)(+) cluster. The surface is a fit to roughly 160,000 interaction energies obtained with second-order MöllerPlesset perturbation theory and the cc-pVQZ basis set, using the invariant polynomial method [B. J. Braams and J. M. Bowman, Int. Rev. Phys. Chem. 28, 577 (2009)]. We employ permutationally invariant basis functions in Morse-type variables for all the internuclear distances to incorporate permutational symmetry with respect to interchange of H atoms into the representation of the surface. We describe how different configurations are selected in order to create the database of the interaction energies for the linear least squares fitting procedure. The root-mean-square error of the fit is 170 cm(-1) for the entire data set. The surface dissociates correctly to the H(5)(+) + H(2) fragments. A detailed analysis of its topology, as well as comparison with additional ab initio calculations, including harmonic frequencies, verify the quality and accuracy of the parameterized potential. This is the first attempt to present an analytical representation of the 15-dimensional surface of the H(7)(+) cluster for carrying out dynamics studies.

Quantum-dynamics study of the H5+ cluster: full dimensional benchmark results on its vibrational states

A full-dimensional quantum dynamics study is carried out for the highly fluxional H(5)(+) cation on a recent reference potential energy surface by using the multi configuration time-dependent Hartree method. With five equivalent light atoms and shallow barriers between various low-lying stationary points on the surface, the spectroscopic characterization of H(5)(+) represents a huge challenge for accurate quantum dynamics simulations. The present calculation is the first such a study on this cation, which together with its isotope analogies are of primary importance in the interstellar chemistry. The vibrational ground state properties and several vibrationally excited states corresponding to low vibrational frequency motions, not yet directly observable by the experiment, are presented and analyzed.

Internal proton transfer and H2 rotations in the H5(+) cluster: a marked influence on its thermal equilibrium state

Classical and path integral Monte Carlo (CMC, PIMC) "on the fly" calculations are carried out to investigate anharmonic quantum effects on the thermal equilibrium structure of the H5(+) cluster. The idea to follow in our computations is based on using a combination of the above-mentioned nuclear classical and quantum statistical methods, and first-principles density functional (DFT) electronic structure calculations. The interaction energies are computed within the DFT framework using the B3(H) hybrid functional, specially designed for hydrogen-only systems. The global minimum of the potential is predicted to be a nonplanar configuration of C(2v) symmetry, while the next three low-lying stationary points on the surface correspond to extremely low-energy barriers for the internal proton transfer and to the rotation of the H2 molecules, around the C2 axis of H5(+), connecting the symmetric C(2v) minima in the planar and nonplanar orientations. On the basis of full-dimensional converged PIMC calculations, results on the quantum vibrational zero-point energy (ZPE) and state of H5(+) are reported at a low temperature of 10 K, and the influence of the above-mentioned topological features of the surface on its probability distributions is clearly demonstrated.