The mathematical representation of large data sets of electronic energies has seen substantial progress in the past 10 years. The so-called Permutationally Invariant Polynomial (PIP) representation is one established approach. This approach dates from 2003, when a global potential energy surface (PES) for CH was reported using a basis of polynomials that are invariant with respect to the 120 permutations of the five equivalent H atoms. More recently, several approaches from "machine learning" have been applied to fit these large data sets. Gaussian Process (GP) regression is such an approach. Here, we consider the implementation of the (full) GP due to Krems and co-workers, with a modification that renders it permutationally invariant, which we denote by PIP-GP. This modification uses the approach of Guo and co-workers and later extended by Zhang and co-workers, to achieve permutational invariance for neural-network fits. The PIP, GP, and PIP-GP approaches are applied to four case studies for fitting data sets of electronic energies: HO, OCHCO, and HCO/ cis-HCOH/ trans-HCOH with the goal of assessing precision, accuracy in normal-mode analysis and barrier heights, and timings. We also report an application to (HCOOH), where the full PIP approach is possible but where the PIP-GP one is not feasible. However, by replicating data, which is feasible in this case, the GP approach is able to represent the data with precision comparable to that of the PIP approach. We examine these assessments for varying sizes of data sets in each case to determine the dependence of properties of the fits on the training data size. We conclude with some comments on the different aspects of computational effort of the PIP, GP, and PIP-GP approaches and also challenges these methods face for more "rugged" PESs, exemplified here by HCO/ cis-HCOH/ trans-HCOH.

Medium and strong hydrogen bonds are common in biological systems. Here, they provide structural support and can act as proton transfer relays to drive electron and/or energy transfer. Infrared spectroscopy is a sensitive probe of molecular structure and hydrogen bond strength but strongly hydrogen-bonded structures often exhibit very broad and complex vibrational bands. As an example, strong hydrogen bonds between carboxylic acids and nitrogen-containing aromatic bases commonly display a 900 cm(-1) broad feature with a remarkable double-hump structure. Although previous studies have assigned this feature to the OH, the exact origin of the shape and width of this unusual feature is not well understood. In this study, we present ab initio calculations of the contributions of the OH stretch and bend vibrational modes to the vibrational spectrum of strongly hydrogen-bonded heterodimers of carboxylic acids and nitrogen-containing aromatic bases, taking the 7-azaindole--acetic acid and pyridine--acetic acid dimers as examples. Our calculations take into account coupling between the OH stretch and bend modes as well as how both of these modes are affected by lower frequency dimer stretch modes, which modulate the distance between the monomers. Our calculations reproduce the broadness and the double-hump structure of the OH vibrational feature. Where the spectral broadness is primarily caused by the dimer stretch modes strongly modulating the frequency of the OH stretch mode, the double-hump structure results from a Fermi resonance between the out of the plane OH bend and the OH stretch modes.

Medium and strong hydrogen bonds give rise to broad vibrational features frequently spanning several hundred wavenumbers and oftentimes exhibiting unusual substructures. These broad vibrational features can be modeled from first principles, in a reduced dimensional calculation, that adiabatically separates low-frequency modes, which modulate the hydrogen bond length, from high-frequency OH stretch and bend modes that contribute to the vibrational structure. Previously this method was used to investigate the origin of an unusual vibrational feature frequently found in the spectra of dimers between carboxylic acids and nitrogen-containing aromatic bases that spans over 900 cm and contains two broad peaks. It was found that the width of this feature largely originates from low-frequency modes modulating the hydrogen bond length and that the structure results from Fermi resonance interactions. In this report, we examine how these features change with the relative acid and base strength of the components as reflected by their aqueous pK values. Dimers with large pK differences are found to have features that can extend to frequencies below 1000 cm. The relationships between mean OH/NH frequency, aqueous pK, and O-N distance are examined in order to obtain a more rigorous understanding of the origin and shape of the vibrational features. The mean OH/NH frequencies are found to correlate well with O-N distances. The lowest OH stretch frequencies are found in dimer geometries with O-N distances between 2.5 and 2.6 Å. At larger O-N distances, the hydrogen bonding interaction is not as strong, resulting in higher OH stretch frequencies. When the O-N distance is smaller than 2.5 Å, the limited space between the O and N determines the OH stretch frequency, which gives rise to frequencies that decrease with O-N distances. These two effects place a lower limit on the OH stretch frequency which is calculated to be near 700 cm. Understanding how the vibrational features of strongly hydrogen-bonded structures depend on the relative pK and other structural parameters will guide studies of biological structures and analysis of proton transfer studies using photoacids.