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We present the first rigorous 12D quantum calculations of the intermolecular vibration-tunneling (VT) states of (D2O)3, in which the monomers are treated as rigid. They are performed using the recently developed methodology [Simkó, Felker, and Bačić, J. Chem. Phys. 162, 034301 (2025)], which was utilized in the 12D quantum treatment of (H2O)3. A high-quality ab initio 2 + 3-body water potential is employed. The 12D eigenstates of (D2O)3 are used to simulate the low-frequency absorption spectrum of this trimer. Through the comparison of the results obtained here for (D2O)3 with those computed previously for (H2O)3, we elucidate the effects of the substitution of all H atoms of water trimer with D atoms on the VT states and the low-frequency spectrum. They are substantial particularly for the large-amplitude torsional vibrations, whose energy levels in (D2O)3 are, by a factor of two, lower than those of (H2O)3. The entire low-frequency absorption spectrum of (D2O)3, which is dominated by the bending transitions, is strongly redshifted in comparison with the spectrum of (H2O)3. The deuteration of the intermolecular stretching vibrations lowers their energy by only a few wave numbers, since it does not change much the masses of the water monomers. In addition, calculations are performed for (D2O)3, which directly reveal the significant effects of the three-body interactions on the vibrational energy levels and the low-frequency spectrum of the trimer. 

In a recent work, we reported on the computation of intermolecular bending states in water trimer, a 9D problem when the monomers are treated as rigid and the intermonomer distances are fixed. In this paper, we present the results of an effort to ascertain the vibrational character of those states. The large-amplitude, highly coupled nature of these vibrations presents significant challenges with respect to their assignment. We introduce a new methodology, based on the use of symmetry coordinates for all nine of the torsional and librational degrees of freedom that constitute the intermolecular bending modes in the trimer, which addresses those challenges. The approach ultimately yields expressions for the bend eigenstates as expansions over small numbers of physically meaningful terms, from which the contributions of the torsional and librational excitations to a bending state can be identified and quantified. Results for approximately one hundred bending states with excitation energies below ∼560 cm−1 are presented. 

The water trimer, as the smallest water cluster in which the three-body interactions can manifest, is arguably the most important hydrogen-bonded trimer. Accurate, fully coupled quantum treatment of its excited intermolecular vibrations has long been an elusive goal. Here, we present the methodology that for the first time allows rigorous twelve-dimensional (12D) quantum calculation of the intermolecular vibration-tunneling eigenstates of the water trimer, with the monomers treated as rigid. These 12D eigenstates are used to simulate the low-frequency absorption spectrum of the trimer for direct comparison with the measured far-infrared (FIR) spectrum of the water trimer in helium nanodroplets. The 12D calculations reveal weak coupling between the large-amplitude torsional and intermolecular stretching vibrations. The calculated torsional tunneling splittings are in excellent agreement with spectroscopic results. There are visible differences between the spectrum simulated using the 12D eigenstates and that based on our earlier 9D calculations where the stretching vibrations are not included. The peaks in the 12D spectrum are generally shifted to slightly lower energies relative to those in the 9D spectrum, as well as the measured FIR spectrum, and are often split by intermolecular stretch–bend Fermi resonances that the 9D treatment cannot capture. 

We present fully coupled, full-dimensional quantum calculations of the inter- and intra-molecular vibrational states of HCl trimer, a paradigmatic hydrogen-bonded molecular trimer. They are performed utilizing the recently developed methodology for the rigorous 12D quantum treatment of the vibrations of the noncovalently bound trimers of flexible diatomic molecules [Felker and Bačić, J. Chem. Phys. 158, 234109 (2023)], which was previously applied to the HF trimer by us. In this work, the many-body 12D potential energy surface (PES) of (HCl)3 [Mancini and Bowman, J. Phys. Chem. A 118, 7367 (2014)] is employed. The calculations extend to the intramolecular HCl-stretch excited vibrational states of the trimer with one- and two-quanta, together with the low-energy intermolecular vibrational states in the two excited v = 1 intramolecular vibrational manifolds. They reveal significant coupling between the intra- and inter-molecular vibrational modes. The 12D calculations also show that the frequencies of the v = 1 HCl stretching states of the HCl trimer are significantly redshifted relative to those of the isolated HCl monomer. Detailed comparison is made between the results of the 12D calculations on the two-body PES, obtained by removing the three-body term from the original 2 + 3-body PES, and those computed on the 2 + 3-body PES. It demonstrates that the three-body interactions have a strong effect on the trimer binding energy as well as on its intra- and inter-molecular vibrational energy levels. Comparison with the available spectroscopic data shows that good agreement with the experiment is achieved only if the three-body interactions are included. Some low-energy vibrational states localized in a secondary minimum of the PES are characterized as well. 

In this work the H2O–HCN complex is quantitatively characterized in two ways. First, we report a new rigid-monomer 5D intermolecular potential energy surface (PES) for this complex, calculated using the symmetry-adapted perturbation theory based on density functional theory method. The PES is based on 2833 ab initio points computed employing the aug-cc-pVQZ basis set, utilizing the autoPES code, which provides a site-site analytical fit with the long-range region given by perturbation theory. Next, we present the results of the quantum 5D calculations of the fully coupled intermolecular rovibrational states of the H2O–HCN complex for the total angular momentum J values of 0, 1, and 2, performed on the new PES. These calculations rely on the quantum bound-state methodology developed by us recently and applied to a variety of noncovalently bound binary molecular complexes. The vibrationally averaged ground-state geometry of H2O–HCN determined from the quantum 5D calculations agrees very well with that from the microwave spectroscopic measurements. In addition, the computed ground-state rotational transition frequencies, as well as the B and C rotational constants calculated for the ground state of the complex, are in excellent agreement with the experimental values. The assignment of the calculated intermolecular vibrational states of the H2O–HCN complex is surprisingly challenging. It turns out that only the excitations of the intermolecular stretch mode can be assigned with confidence. The coupling among the angular degrees of freedom (DOFs) of the complex is unusually strong, and as a result most of the excited intermolecular states are unassigned. On the other hand, the coupling of the radial, intermolecular stretch mode and the angular DOFs is weak, allowing straightforward assignment of the excitation of the former. 

We present the computational methodology, which for the first time allows rigorous twelve-dimensional (12D) quantum calculations of the coupled intramolecular and intermolecular vibrational states of hydrogen-bonded trimers of flexible diatomic molecules. Its starting point is the approach that we introduced recently for fully coupled 9D quantum calculations of the intermolecular vibrational states of noncovalently bound trimers comprised of diatomics treated as rigid. In this paper, it is extended to include the intramolecular stretching coordinates of the three diatomic monomers. The cornerstone of our 12D methodology is the partitioning of the full vibrational Hamiltonian of the trimer into two reduced-dimension Hamiltonians, one in 9D for the intermolecular degrees of freedom (DOFs) and another in 3D for the intramolecular vibrations of the trimer, and a remainder term. These two Hamiltonians are diagonalized separately, and a fraction of their respective 9D and 3D eigenstates is included in the 12D product contracted basis for both the intra- and intermolecular DOFs, in which the matrix of the full 12D vibrational Hamiltonian of the trimer is diagonalized. This methodology is implemented in the 12D quantum calculations of the coupled intra- and intermolecular vibrational states of the hydrogen-bonded HF trimer on an ab initio calculated potential energy surface (PES). The calculations encompass the one- and two-quanta intramolecular HF-stretch excited vibrational states of the trimer and low-energy intermolecular vibrational states in the intramolecular vibrational manifolds of interest. They reveal several interesting manifestations of significant coupling between the intra- and intermolecular vibrational modes of (HF)3. The 12D calculations also show that the frequencies of the v = 1, 2 HF stretching states of the HF trimer are strongly redshifted in comparison to those of the isolated HF monomer. Moreover, the magnitudes of these trimer redshifts are much larger than that of the redshift for the stretching fundamental of the donor-HF moiety in (HF)2, most likely due to the cooperative hydrogen bonding in (HF)3. The agreement between the 12D results and the limited spectroscopic data for the HF trimer, while satisfactory, leaves room for improvement and points to the need for a more accurate PES. 

We present the computational methodology that allows rigorous and efficient nine-dimensional (9D) quantum calculations of the intermolecular vibrational states of noncovalently bound trimers of diatomic molecules, with the monomers treated as rigid. The full 9D vibrational Hamiltonian of the trimer is partitioned into a 3D “frame” (or stretching) Hamiltonian and a 6D “bend” Hamiltonian. These two Hamiltonians are diagonalized separately, and a certain number of their lowest-energy eigenstates is included in the final 9D product contracted basis in which the full 9D intermolecular vibrational Hamiltonian is diagonalized. This methodology is applied to the 9D calculations of the intermolecular vibrational levels of (HF) 3 , a prototypical hydrogen-bonded trimer, on the rigid-monomer version of an ab initio calculated potential energy surface (PES). They are the first to include fully the stretch-bend coupling present in the trimer. The frequencies of all bending fundamentals considered from the present 9D calculations are about 10% lower than those from the earlier quantum 6D calculations that considered only the bending modes of the HF trimer. This means that the stretch-bend coupling is strong, and it is imperative to include it in any accurate treatment of the (HF) 3 vibrations aiming to assess the accuracy of the PES employed. Moreover, the 9D results are in better agreement with the limited available spectroscopic data that those from the 6D calculations. In addition, the 9D results show sensitivity to the value of the HF bond length, equilibrium or vibrationally averaged, used in the calculations. The implication is that full-dimensional 12D quantum calculations will be required to obtain definitive vibrational excitation energies for a given PES. Our study also demonstrates that the nonadditive three-body interactions are very significant in (HF) 3 and have to be included in order to obtain accurate intermolecular vibrational energy levels of the trimer.