Hydrogen Fluoride Dimer Research Articles

A basis set superposition error‐free second‐order perturbation theory from Hermitian chemical Hamiltonian approach self‐consistent field canonic orbitals

AbstractWe present an alternative perturbational approach free of basis set superposition error (BSSE) within the framework of the chemical Hamiltonian approach (CHA). The new formulation (CHA‐S‐MP2) is based on canonic (and orthogonal) CHA orbitals obtained from a hermitized CHA Fock operator. The final expression shows a considerable simplification of the method as compared to the previous CHA‐MP2 formalism. Numerical full geometry optimizations of water and hydrogen fluoride dimers and potential energy surfaces for helium and argon dimers for several basis sets are presented. The present method is compared to both the counterpoise and previous CHA‐MP2 BSSE correction schemes, showing a remarkable agreement between all three methods. However, the wrong behavior using the aug‐cc‐pVDZ basis set indicates that the present method is not as robust as the original non‐hermitian CHA‐MP2 formulation.

New Molecular-Mechanics Model for Simulations of Hydrogen Fluoride in Chemistry and Biology.

Hydrogen fluoride (HF) is the most polar diatomic molecule and one of the simplest molecules capable of hydrogen-bonding. HF deviates from ideality both in the gas phase and in solution and is thus of great interest from a fundamental standpoint. Pure and aqueous HF solutions are broadly used in chemical and industrial processes, despite their high toxicity. HF is a stable species also in some biological conditions, because it does not readily dissociate in water unlike other hydrogen halides; yet, little is known about how HF interacts with biomolecules. Here, we set out to develop a molecular-mechanics model to enable computer simulations of HF in chemical and biological applications. This model is based on a comprehensive high-level ab initio quantum chemical investigation of the structure and energetics of the HF monomer and dimer; (HF)n clusters, for n = 3-7; various clusters of HF and H2O; and complexes of HF with analogs of all 20 amino acids and of several commonly occurring lipids, both neutral and ionized. This systematic analysis explains the unique properties of this molecule: for example, that interacting HF molecules favor nonlinear geometries despite being diatomic and that HF is a strong H-bond donor but a poor acceptor. The ab initio data also enables us to calibrate a three-site molecular-mechanics model, with which we investigate the structure and thermodynamic properties of gaseous, liquid, and supercritical HF in a wide range of temperatures and pressures; the solvation structure of HF in water and of H2O in liquid HF; and the free diffusion of HF across a lipid bilayer, a key process underlying the high cytotoxicity of HF. Despite its inherent simplifications, the model presented significantly improves upon previous efforts to capture the properties of pure and aqueous HF fluids by molecular-mechanics methods and to our knowledge constitutes the first parameter set calibrated for biomolecular simulations.

Quantum Stabilization of the Frustrated Hydrogen Bonding Structure in the Hydrogen Fluoride Trimer.

We performed ab initio path integral molecular dynamics (PIMD) and molecular dynamics (MD) simulations to discuss the thermal and nuclear quantum effects on the stabilities of hydrogen bonding network in a hydrogen fluoride trimer (HF)3 cluster. By the conventional molecular orbital calculation, the (HF)3 cluster has an equilateral triangle shape, which has a frustration in the chemical structure of the hydrogen bonds, whereas the hydrogen bonding structure of a hydrogen fluoride dimer (HF)2 cluster is nearly perpendicular to the acceptor molecule. The ratio of the triangular structures with the three hydrogen bondings in the PIMD simulation is larger than that in the MD one, whereas nonhydrogen bonding conformations such as a dimerlike structure are often found in MD simulation. The nuclear quantum effect stabilizes the frustrated hydrogen bonding network of the triangular (HF)3 cluster.

A new full-dimensional ab initio intermolecular potential energy surface and vibrational states for (HF)2 and (DF)2.

We present a new full-dimensional ab initio potential energy surface (PES) of a hydrogen fluoride dimer [(HF)2] using the supermolecular approach. The calculations were performed at the coupled-cluster single double triple level, with an augmented correlation-consistent polarized valence quadruple-zeta basis set plus bond functions. The basis set superposition error was corrected by a full counterpoise procedure. With the exchange symmetry of the two HF molecules, the permutation invariant polynomial neural network approach was used to fit the hypersurface with a root-mean-square-error of 0.465 cm-1 for about 110 000 points. The ab initio noise of intermolecular potential in the long range was smoothed by the long-range coefficients method. The equilibrium configuration of the complex was found to be a Cs structure located at two equivalent minima with the well depth of 1573.495 cm-1. The eigenstates were calculated by employing a symmetry-adapted Lanczos propagation algorithm in the mixed radical discrete variable representation/angular finite basis representation. The tunneling splitting for the ground state of (HF)2 is 0.665 cm-1, agreeing well with experimental value of 0.65869 cm-1. Vibrational fundamentals are also very close to the observed values. The results of vibrational states calculations demonstrate the high accuracy of our new PES.

Effect of Electron Correlation on Intermolecular Interactions: A Pair Natural Orbitals Coupled Cluster Based Local Energy Decomposition Study.

The development of post-Hartree-Fock (post-HF) energy decomposition schemes that are able to decompose the HF and correlation components of the interaction energy into chemically meaningful contributions is a very active field of research. One of the challenges is to provide a clear-cut quantification to the elusive London dispersion component of the intermolecular interaction. London dispersion is well-known to be a pure correlation effect, and as such it is not properly described by mean field theories. In this context, we have recently developed the local energy decomposition (LED) analysis, which provides a chemically meaningful decomposition of the interaction energy between two or more fragments computed at the domain-based local pair natural orbitals coupled cluster (DLPNO-CCSD(T)) level of theory. In this work, this scheme is used in conjunction with other interpretation tools to study a series of molecular adducts held together by intermolecular interactions of different natures. The HF and correlation components of the interaction energy are thus decomposed into a series of chemically meaningful contributions. Emphasis is placed on discussing the physical effects associated with the inclusion of electron correlation. It is found that four distinct physical effects can contribute to the magnitude of the correlation part of intermolecular binding energies (Δ EintC): (i) London dispersion, (ii) the correlation correction to the reference induction energy, (iii) the correlation correction to the electron sharing process, and (iv) the correlation correction to the permanent electrostatics. As expected, the largest contribution to the correlation binding energy of neutral, apolar molecules is London dispersion, as in the argon dimer case. In contrast, the correction for the HF induction energy dominates Δ EintC in systems in which an apolar molecule interacts with charged or strongly polar species, as in Ar-Li+. This effect has its origin in the systematic underestimation of polarizabilities at the HF level of theory. For similar reasons, electron sharing largely contributes to the correlation binding energy of covalently bound molecules, as in the beryllium dimer case. Finally, the correction for HF permanent electrostatics significantly contributes to Δ EintC in molecules with strong dipoles, such as water and hydrogen fluoride dimers. This effect originates from the characteristic overestimation of dipole moments at the HF level of theory, leading in some cases to positive Δ EintC values. Our results are apparently in contrast to the widely accepted view that Δ EintC is typically dominated by London dispersion, at least, in the strongly interacting region. Clearly, post-HF energy decomposition schemes are very powerful tools to analyze, categorize, and understand the various contributions to the intermolecular interaction energy. Hopefully, this will eventually lead to insights that are helpful in designing systems with tailored properties. All analysis tools presented in this work will be available free of charge in the next release of the ORCA program package.

Local energy decomposition analysis of hydrogen-bonded dimers within a domain-based pair natural orbital coupled cluster study.

The local energy decomposition (LED) analysis allows for a decomposition of the accurate domain-based local pair natural orbital CCSD(T) [DLPNO-CCSD(T)] energy into physically meaningful contributions including geometric and electronic preparation, electrostatic interaction, interfragment exchange, dynamic charge polarization, and London dispersion terms. Herein, this technique is employed in the study of hydrogen-bonding interactions in a series of conformers of water and hydrogen fluoride dimers. Initially, DLPNO-CCSD(T) dissociation energies for the most stable conformers are computed and compared with available experimental data. Afterwards, the decay of the LED terms with the intermolecular distance (r) is discussed and results are compared with the ones obtained from the popular symmetry adapted perturbation theory (SAPT). It is found that, as expected, electrostatic contributions slowly decay for increasing r and dominate the interaction energies in the long range. London dispersion contributions decay as expected, as r−6. They significantly affect the depths of the potential wells. The interfragment exchange provides a further stabilizing contribution that decays exponentially with the intermolecular distance. This information is used to rationalize the trend of stability of various conformers of the water and hydrogen fluoride dimers.

Third-Order Nonlinear Optical Properties of Endohedral Fullerene (H2)2@C70 and (H2O)2@C70 Accompanied by the Prospective of Novel (HF)2@C70

In view of the experimental observation of (H2)2@C70 and (H2O)2@C70, it has been suggested that hydrogen fluoride (HF) dimer can be completely localized within the sub-nanospace inside the fullerene C70 cage. With the aim of quantum chemical prospective of (HF)2@C70, electronic structure calculations of C60 hosting H2, HF, and H2O monomers, as well as C70 hosting H2, HF, and H2O monomers and dimers, were performed by using the density functional theory, together with the quantum theory of atoms in molecules, the natural population, and interaction energy calculation. The F–H···F bonding energy inside C70 was estimated at −13.25 kcal/mol, which is smaller than that of free dimer in the gas phase (−8.37 kcal/mol). Exploration of various featured properties suggests that (HF)2@C70 may be also regarded as a unique system composed of both inter- and intramolecular interactions like (H2)2@C70 and (H2O)2@C70. In addition, absorption spectroscopy and linear and nonlinear optical coefficients of C60 hosting H2, HF...

Ab Initio Quantum Mechanical Description of Noncovalent Interactions at Its Limits: Approaching the Experimental Dissociation Energy of the HF Dimer.

Hydrogen fluoride dimer is a perfect model system for studying hydrogen bonding. Its size makes it possible to apply the most advanced theoretical methods available, yet it is a full-featured complex of molecules with nontrivial electronic structure and dynamic properties. Moreover, the dissociation energy of the HF dimer has been measured experimentally with an unparalleled accuracy of ±1 cm(-1)(Bohac et al. J. Chem. Phys. 1992, 9, 6681). In this work, we attempt to reproduce it by purely ab initio means, using advanced quantum-mechanical computational methods free of any empiricism. The purpose of this study is to demonstrate the capabilities of today's computational chemistry and to point out its limitations by identifying the contributions that introduce the largest uncertainty into the result. The dissociation energy is calculated using a composite scheme including large basis set CCSD(T) calculations, contributions of higher excitations up to CCSDTQ, relativistic and diagonal Born-Oppenheimer corrections and anharmonic vibrational calculations. The error of the calculated dissociation energy is 0.07 kcal/mol (25 cm(-1), 2.5%) when compared to the experiment. The major part of this error can be attributed to the inaccuracy of the calculations of the zero-point vibrational energy.

Quantum theory of atoms in molecules/charge-charge flux-dipole flux models for fundamental vibrational intensity changes on H-bond formation of water and hydrogen fluoride

The Quantum Theory of Atoms In Molecules/Charge-Charge Flux-Dipole Flux (QTAIM/CCFDF) model has been used to investigate the electronic structure variations associated with intensity changes on dimerization for the vibrations of the water and hydrogen fluoride dimers as well as in the water-hydrogen fluoride complex. QCISD/cc-pVTZ wave functions applied in the QTAIM/CCFDF model accurately provide the fundamental band intensities of water and its dimer predicting symmetric and antisymmetric stretching intensity increases for the donor unit of 159 and 47 km mol(-1) on H-bond formation compared with the experimental values of 141 and 53 km mol(-1). The symmetric stretching of the proton donor water in the dimer has intensity contributions parallel and perpendicular to its C2v axis. The largest calculated increase of 107 km mol(-1) is perpendicular to this axis and owes to equilibrium atomic charge displacements on vibration. Charge flux decreases occurring parallel and perpendicular to this axis result in 42 and 40 km mol(-1) total intensity increases for the symmetric and antisymmetric stretches, respectively. These decreases in charge flux result in intensity enhancements because of the interaction contributions to the intensities between charge flux and the other quantities. Even though dipole flux contributions are much smaller than the charge and charge flux ones in both monomer and dimer water they are important for calculating the total intensity values for their stretching vibrations since the charge-charge flux interaction term cancels the charge and charge flux contributions. The QTAIM/CCFDF hydrogen-bonded stretching intensity strengthening of 321 km mol(-1) on HF dimerization and 592 km mol(-1) on HF:H2O complexation can essentially be explained by charge, charge flux and their interaction cross term. Atomic contributions to the intensities are also calculated. The bridge hydrogen atomic contributions alone explain 145, 237, and 574 km mol(-1) of the H-bond stretching intensity enhancements for the water and HF dimers and their heterodimer compared with total increments of 149, 321, and 592 km mol(-1), respectively.

On the Calculations of Interaction Energies and Induced Electric Properties within the Polarizable Continuum Model

In this work we investigate the influence of a polarizable environment on the interaction energies and the interaction-induced (excess) static electric dipole properties for the selected model hydrogen-bonded complexes. The excess properties were estimated for water and hydrogen fluoride dimers using the supermolecular approach and assuming the polarizable continuum model (PCM) as a representation of the polarizable environment. We analyze in this context the performance of the counterpoise correction and the consequences of various possible monomer cavity choices. The polarizable environment reduces the absolute magnitudes of interaction energies and interaction-induced dipole moments, whereas an increase is observed for the absolute magnitudes of induced polarizabilities and first hyperpolarizabilities. Our results indicate that the use of either monomeric (MC) or dimeric (DC) cavities in calculations of monomer properties does not change qualitatively the resultant excess properties. We conclude that the DC scheme is more consistent with the definition of the interaction energy and consequently also the interaction-induced property, whereas the MC scheme corresponds to the definition of stabilization energy. Our results indicate also a good performance of the counterpoise correction scheme for the self-consistent methods in the case of all studied properties.

Assessing group-based cutoffs and the Ewald method for electrostatic interactions in clusters and in saturated, superheated, and supersaturated vapor phases of dipolar molecules

Monte Carlo simulations in the canonical, isobaric-isothermal, grand canonical, and Gibbs ensembles were used to assess whether the computationally expensive Ewald summation method for the computation of the first-order electrostatic energy can be replaced with a simpler truncation approach for accurate simulations of the saturated, superheated, and supersaturated vapor phases of dipolar and hydrogen-bonding molecules. Rotationally averaged hydrogen fluoride dimer and trimer energies, thermophysical properties and aggregation in the superheated vapor phase of hydrogen fluoride, nucleation free energy barriers for water, and the vapor–liquid coexistence properties of hydrogen fluoride and water were investigated over a wide range of state points. We find that for densities not too close to the critical density, results obtained from simulations using a spherical potential truncation based on neutral groups (molecules or fragments) for the Coulomb interactions are statistically identical to those obtained using the Ewald summation method. Use of the simpler spherical truncation results in a significant reduction of the computational effort for simulations employing molecular mechanics force fields and also allows for straightforward implementation of many-body expansion methods to compute the potential energy from electronic structure calculations of subsystems of the entire vapor-phase system.

Coupled Cluster in Condensed Phase. Part I: Static Quantum Chemical Calculations of Hydrogen Fluoride Clusters.

A multiscale approach with roots in electronic structure calculations relies on the good description of intermolecular forces. In this study we lay the foundations for a condensed phase treatment based on the electronic structure of hydrogen fluoride on a very high level of theory. This investigation comprises cluster calculations in a static quantum chemical approach employing density functional theory, second order Møller-Plesset perturbation theory (MP2) and the coupled cluster singles, doubles with perturbative triples method in combination with several basis sets as well as at the complete basis set limit. The clusters we considered are up to 12 monomer units large and consist of ring and chain structures. We find a good agreement of the intramolecular distance obtained from the MP2 approach and the largest basis set. The binding energy of the hydrogen fluoride dimer calculated from coupled cluster at the basis set limit agrees excellently with experiment, whereas the calculated frequencies at all levels agree reasonably well with different experimental values. Large cooperative effects are observed for the ring structures as compared to the chain clusters. The energy per monomer unit indicates the most stable structures to be the ring clusters.

Dissociation Rate Constant of the Hydrogen Fluoride Dimer by the ab Initio Anharmonic RRKM Theory

The dissociation rate constants for hydrogen fluoride dimers are calculated using the YL method, proposed by Yao and Lin (Yao, L.; Mebel, A. M.; Lu, H. F.; Neusser, H. J.; Lin, S. H. J. Phys. Chem. A 2007, 111, 6722). The dividing surface method and RRKM theory are also used to obtain pseudo-transition states and rate constants, respectively. For the dissociation of HF dimers, the anharmonic rate constants are around 3.02 x 10(10) to 3.46 x 10(12) s(-1), while the harmonic rate constants are in the range of 2.93 x 10(10) to 1.66 x 10(13) s(-1), at a temperature range of 243-1000 K, for the canonical case. For the microcanonical case, the anharmonic rate constants are in the range of 1.91 x 10(12) to 1.32 x 10(13) s(-1) and the harmonic rate constants are in the range of 1.89 x 10(12) to 4.93 x 10(13) s(-1), with a total energy range of 1338-4500 cm(-1). Both harmonic and anharmonic rate constants are only comparable to the experimental results 0.5 x 10(10) s(-1) and 1 x 10(11) s(-1) in an inert gas bath at room temperature. In this case the harmonic and anharmonic rate constants show similar results in this calculation. The results of this paper indicate that the YL method is suitable for calculating dissociation rate constants of small flexible HF dimers and the anharmonic effect should be included.

The Important Role of Lone-Pairs in Force Field (MM4) Calculations on Hydrogen Bonding in Alcohols

An expanded treatment of hydrogen bonding has been developed for MM4 force field calculations, which is an extension from the traditional van der Waals-electrostatic model. It adds explicit hydrogen-bond angularity by the inclusion of lone-pair directionality. The vectors that account for this directionality are placed along the hydrogen acceptor and its chemically intuitive electron pairs. No physical lone-pairs are used in the calculations. Instead, an H-bond angularity function, and a lone-pair directionality function, are incorporated into the hydrogen-bond term. The inclusion of the lone-pair directionality results in improved accuracy in hydrogen-bonded geometries and interaction energies. In this work is described hydrogen bonding in alcohols, and also in water and hydrogen fluoride dimer. The extension to other compounds such as aldehydes, ketones, amides, and so on is straightforward and will be discussed in future work. The conformational energies of ethylene glycol are discussed.

High-Resolution Continuous-Wave-Diode Laser Cavity Ring-Down Spectroscopy of the Hydrogen Fluoride Dimer in a Pulsed Slit Jet Expansion: Two Components of the N = 2 Triad near 1.3 μm

The near-infrared overtone spectra of the prototypical hydrogen-bond hydrogen fluoride dimer (HF)2 contain rich information on hydrogen bond dynamics. We report a study of the N=2 triad involving excitations with two quanta of HF stretching in (HF)2 around 1.3 microm (7500-7800 cm(-1)) by means of continuous-wave-diode laser cavity ring-down spectroscopy in a pulsed supersonic slit jet expansion. The analysis of the rotationally resolved overtone spectra allows the study of vibrational mode-selective kinetics, such as hydrogen bond predissociation with lifetimes tauPD and tunneling rearrangement (switching) processes with periods tausw obtained from the tunneling splitting DeltaT in highly excited vibrational states. The Ka=1<--0 transition of the Nj=22 band of (HF)2 has been reinvestigated by us in a supersonic jet expansion; the much improved data obtained here are in excellent agreement with several previous experimental results. Our analysis provides subband-level positions and properties 0(Ka=1(A+))=7711.37956(66) cm(-1), DeltaT=0.0936(10) cm(-1), and tauPD=1.3-1.9x10(-9) s, depending on the level symmetry A+ and B+. We have also analyzed spectra of the Nj=21 band, which we have observed for the first time in a supersonic jet with rotational resolution. For the Ka=0<--0 transition of this band, we find the band center at 0(A+)=7550.3555(26) cm(-1) and a tunneling splitting of DeltaT=0.0150(37) cm(-1). This level involves mostly excitation of the H-bonded HF stretching with two quanta. The mode-selective tunneling switching is in agreement with a simple picture of inhibited tunneling. These experimental values are close to those calculated on the "SO(-3)" potential energy hypersurface of Klopper, Quack, and Suhm. The N=2 triad also exhibits a strongly mode-selective predissociation dynamics, with a predissociation lifetime tauPD=4.99(84)x10(-11) s in the Nj=21 level, which is more than 20 times shorter than that for the Nj=22 level.

A polarizable high‐rank quantum topological electrostatic potential developed using neural networks: Molecular dynamics simulations on the hydrogen fluoride dimer

AbstractThe HF molecule is a simple polar system that serves as a prototype for developing new potentials. Here we build on earlier work [Liem and Popelier, J Chem Phys 2003, 119, 4560] in which a high‐rank multipolar potential was used to simulate liquid HF. That work was the first example of high‐rank multipole moments (up to hexadecapole) being employed in conjunction with multipolar Ewald summation in a molecular dynamics simulation. This potential is now extended with polarization, which is delivered by artificial neural networks. The neural nets predict how atomic multipole moments change as the position of neighboring molecules vary. This novel approach is successfully tested on the HF dimer in vacuum. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Stabilization of an excess electron on molecular surfaces by a pair of HF molecules

AbstractThis work presents the results of solvation of electrons on several hypothetical cyclooctane and cyclohexane molecular surfaces, using the hydrogen fluoride (HF) dimer. These complexes were constructed with extensive OH groups on one side of a hydrocarbon surface (i.e., cyclohexane sheets), which creates hydrogen‐bonded networks that can form, increasing the dipole moment of the system. Concurrently, the hydrogen atoms on the opposite side of the surface form a pocket of positive charge that can attract excess electrons. Two possible orientations for HF dimer solvation on eight molecular surfaces that have been demonstrated to be stable toward electron detachment are examined. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

BSSE‐free hardness profiles of hydrogen bond exchange in the hydrogen fluoride dimer

AbstractHardness profiles are calculated for the hydrogen bond interchange in the hydrogen fluoride dimer; the study is carried out at the Hartree–Fock and second‐order Møller–Plesset (MP2) levels of theory with three different basis sets. The basis set superposition error (BSSE) is corrected by means of the chemical Hamiltonian approach (CHA). Uncorrected and BSSE‐corrected energy and hardness profiles are compared. Their analysis shows that hardness profiles do not lead to spurious minima as energy profiles do in various cases. The CHA methodology is shown to be valuable in the analysis of intermolecular interactions, with BSSE observed to carry smaller modifications on hardness profiles than on energy curves. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Impact of Nuclear Quantum Effects on the Molecular Structure of Bihalides and the Hydrogen Fluoride Dimer

The structural impact of nuclear quantum effects is investigated for a set of bihalides, [XHX](-), X = F, Cl, and Br, and the hydrogen fluoride dimer. Structures are calculated with the vibrational self-consistent-field (VSCF) method, the second-order vibrational perturbation theory method (VPT2), and the nuclear-electronic orbital (NEO) approach. In the VSCF and VPT2 methods, the vibrationally averaged geometries are calculated for the Born-Oppenheimer electronic potential energy surface. In the NEO approach, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wave functions are calculated variationally with molecular orbital methods. Electron-electron and electron-proton dynamical correlation effects are included in the NEO approach using second-order perturbation theory (NEO-MP2). The nuclear quantum effects are found to alter the distances between the heavy atoms by 0.02-0.05 A for the systems studied. These effects are of similar magnitude as the electron correlation effects. For the bihalides, inclusion of the nuclear quantum effects with the NEO-MP2 or the VSCF method increases the X-X distance. The bihalide X-X distances are similar for both methods and are consistent with two-dimensional grid calculations and experimental values, thereby validating the use of the computationally efficient NEO-MP2 method for these types of systems. For the hydrogen fluoride dimer, inclusion of nuclear quantum effects decreases the F-F distance with the NEO-MP2 method and increases the F-F distance with the VSCF and VPT2 methods. The VPT2 F-F distances for the hydrogen fluoride dimer and the deuterated form are consistent with the experimentally determined values. The NEO-MP2 F-F distance is in excellent agreement with the distance obtained experimentally for a model that removes the large amplitude bending motions. The analysis of these calculations provides insight into the significance of electron-electron and electron-proton correlation, anharmonicity of the vibrational modes, and nonadiabatic effects for hydrogen-bonded systems.

Basis set effects on the energy and hardness profiles of the hydrogen fluoride dimer

In earlier work, the present authors have shown that hardness profiles are less dependent on the level of calculation than energy profiles for potential energy surfaces (PESs) having pathological behaviors. At variance with energy profiles, hardness profiles always show the correct number of stationary points. This characteristic has been used to indicate the existence of spurious stationary points on the PESs. In the present work, we apply this methodology to the hydrogen fluoride dimer, a classical difficult case for the density functional theory methods.