Analysis of nuclear quantum effects on hydrogen bonding

The impact of nuclear quantum effects on hydrogen bonding is investigated for a series of hydrogen fluoride (HF)n clusters and a partially solvated fluoride anion, F-(H2O). The nuclear quantum effects are included using the path integral formalism in conjunction with the Car-Parrinello molecular dynamics (PICPMD) method and using the second-order vibrational perturbation theory (VPT2) approach. For the HF clusters, a directional change in the impact of nuclear quantum effects on the hydrogen-bonding strength is observed as the clusters evolve toward the condensed phase. Specifically, the inclusion of nuclear quantum effects increases the F-F distances for the (HF)n=2-4 clusters and decreases the F-F distances for the (HF)n>4 clusters. This directional change occurs because the enhanced electrostatic interactions between the HF monomers become more dominant than the zero point energy effects of librational modes as the size of the HF clusters increases. For the F-(H2O) system, the inclusion of nuclear quantum effects decreases the F-O distance and strengthens the hydrogen bonding interaction between the fluoride anion and the water molecule because of enhanced electrostatic interactions. The vibrationally averaged 19F shielding constant for F-(H2O) is significantly lower than the value for the equilibrium geometry, indicating that the electronic density on the fluorine decreases as a result of the quantum delocalization of the shared hydrogen. Deuteration of this system leads to an increase in the vibrationally averaged F-O distance and nuclear magnetic shielding constant because of the smaller degree of quantum delocalization for deuterium.     

Density functional theory treatment of electron correlation in the nuclear-electronic orbital approach

This paper presents the nuclear-electronic orbital density functional theory [NEO-DFT(ee)] method for including electron-electron correlation and nuclear quantum effects self-consistently in quantum chemical calculations. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEO-DFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEO-DFT(ee) B3LYP method. The excellent agreement of the NEO-DFT(ee)-optimized bihalide structures with the vibrationally averaged geometries from grid-based quantum dynamical methods provides validation for the NEO-DFT(ee) approach. Electron-proton correlation could be included by the development of an electron-nucleus correlation functional. Alternatively, explicit electron-proton correlation could be included directly into the NEO self-consistent-field framework with Gaussian-type geminal functions.     

Modeling positrons in molecular electronic structure calculations with the nuclear-electronic orbital method

The nuclear-electronic orbital (NEO) method was modified and extended to positron systems for studying mixed positronic-electronic wavefunctions, replacing the mass of the proton with the mass of the positron. Within the modified NEO framework, the NEO-HF (Hartree-Fock) method provides the energy corresponding to the single-configuration mixed positronic-electronic wavefunction, minimized with respect to the molecular orbitals expressed as linear combinations of Gaussian basis functions. The electron-electron and electron-positron correlation can be treated in the NEO framework with second-order perturbation theory (NEO-MP2) or multiconfigurational methods such as the full configuration interaction (NEO-FCI) and complete active space self-consistent-field (NEO-CASSCF) methods. In addition to implementing these methods for positronic systems, strategies for calculating electron-positron annihilation rates using NEO-HF, NEO-MP2, and NEO-FCI wavefunctions were also developed. To apply the NEO method to the positronium hydride (PsH) system, positronic and electronic basis sets were optimized at the NEO-FCI level and used to compute NEO-MP2 and NEO-FCI energies and annihilation rates. The effects of basis set size on NEO-MP2 and NEO-FCI correlation energies and annihilation rates were compared. Even-tempered electronic and positronic basis sets were also optimized for the e+LiH molecule at the NEO-MP2 level and used to compute the equilibrium bond length and vibrational energy.     

Analysis of the nuclear-electronic orbital method for model hydrogen transfer systems

Fundamental issues associated with the application of the nuclear-electronic orbital (NEO) approach to hydrogen transfer systems are addressed. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated with molecular orbital methods. The positions of the nuclear basis function centers are optimized variationally. In the application of the NEO approach to hydrogen transfer systems, the hydrogen nuclei and all electrons are treated quantum mechanically. Within the NEO framework, the transferring hydrogen atom can be represented by two basis function centers to allow delocalization of the proton vibrational wavefunction. In this paper, the NEO approach is applied to the [He-H-He]+ and [He-H-He]++ model systems. Analyses of technical issues pertaining to flexibility of the basis set to describe both single and double well proton potential energy surfaces, linear dependency of the hydrogen basis functions, multiple minima in the basis function center optimization, convergence of the number of hydrogen basis function centers, and basis set superposition error are presented. The accuracy of the NEO approach is tested by comparison to grid calculations for these model systems.     

Multicomponent density functional theory study of the interplay between electron-electron and electron-proton correlation

The interplay between electron-electron and electron-proton correlation is investigated within the framework of the nuclear-electronic orbital density functional theory (NEO-DFT) approach, which treats electrons and select protons quantum mechanically on the same level. Recently two electron-proton correlation functionals were developed from the electron-proton pair densities obtained from explicitly correlated wavefunctions. In these previous derivations, the kinetic energy contribution arising from electron-proton correlation was neglected. In this paper, an electron-proton correlation functional that includes this kinetic energy contribution is derived using the adiabatic connection formula in multicomponent DFT. The performance of the NEO-DFT approach using all three electron-proton correlation functionals in conjunction with three well-established electronic exchange-correlation functionals is assessed. NEO-DFT calculations with these electron-proton correlation functionals capture the increase in the hydrogen vibrational stretching frequencies arising from the inclusion of electron-electron correlation in model systems. Electron-proton and electron-electron correlation are found to be uncoupled and predominantly additive effects to the total energy for the model systems studied. Thus, electron-proton correlation functionals and electronic exchange-correlation functionals can be developed independently and subsequently combined together without re-parameterization.     

19F-19F spin-spin coupling constant surfaces for (HF)2 clusters: the orientation and distance dependence of the sign and magnitude of J(F-F)

Ab initio calculations using the equation-of-motion coupled cluster method have been carried out to investigate 19F-19F spin-spin coupling constants for a pair of HF molecules. The overall features of the J(F-F) coupling surface with respect to the F-F distance and the orientation of the pair of HF molecules reflect those of the Fermi-contact (FC) surface, although the FC term may not be a good quantitative estimate of J(F-F). The hydrogen-bonded HF dimer exhibits unusual behavior compared to other hydrogen-bonded complexes, since both the FC term and 2hJ(F-F) exhibit variations in sign and magnitude as the F-F distance changes and the linearity of the hydrogen bond is destroyed. The FC term for F-F coupling is relative small and negative for the equilibrium dimer. At the dimer F-F distance, the maximum negative value for the FC term is found for the linear arrangement F-H...H-F, while the maximum positive value is found for the linear H-F...F-H arrangement, despite the fact that neither of these structures is bound. Changes in the sign and magnitude of the FC term are analyzed using the nuclear magnetic resonance triplet wave function model, which relates the orientation of magnetic nuclei to the phases of the wave functions for excited triplet states that couple to the ground state. The FC term for a particular orientation is a result of competing positive and negative contributions from different triplet states, the sign of each contribution being determined by the alignment of the nuclear magnetic moments in that state. Factors are identified which must play a role in determining which types of wave functions dominate.     

Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+G?), where G? is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+G?) rather than (1+G?). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.     

Nuclear-electronic orbital nonorthogonal configuration interaction approach

The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) approach is presented. In this framework, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear-electronic time-independent Schrodinger equation is solved with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wave function. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between the two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEO-full CI and Fourier grid calculations. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wave functions and the corresponding hydrogen tunneling splittings.     

On the extent of intramolecular hydrogen bonding in gas-phase and hydrated 1,2-ethanediol

We investigate the quantum dynamical nature of hydrogen bonding in 1,2-ethanediol and monohydrated 1,2-ethanediol using different levels of ab initio theory. Global full-dimensional potential energy surfaces were constructed from PW91/cc-pVDZ, B3LYP/cc-pVDZ, and MP2/cc-pVDZ ab initio data for gas-phase and monohydrated 1,2-ethanediol, using a modified Shepard interpolation scheme. Zero-point energies and nuclear vibrational wave functions were calculated on these surfaces using the quantum diffusion Monte Carlo algorithm. The nature of intra- and intermolecular hydrogen bonding in these molecules was investigated by considering a ground-state nuclear vibrational wavefunction with reduced complete nuclear permutation and inversion (CNPI) symmetry. Separate wavefunction histograms were determined from the ground-state nuclear vibrational wavefunction by projection into bondlength coordinates. The O-H and O-O wavefunction histograms and vibrationally averaged distances were then used to probe the extent of intra- and intermolecular hydrogen bonding. From these data, we conclude that gas-phase ethanediol may possess a weak hydrogen bond, with a relatively short O-O distance but no detectable proton delocalization. Monohydrated ethanediol was found to exhibit no intramolecular hydrogen bonding but instead possessed two intermolecular hydrogen bonds, indicated by both shortening of the O-O distance and significant proton delocalization. The degree of proton delocalization and shortening of the vibrationally averaged O-O distance was found to be dependent on the ab initio method used to generate the potential energy surface (PES) data set.     

Calculating molecular vibrational spectra beyond the harmonic approximation

We present a new approach for calculating anharmonic corrections to vibrational frequency calculations. The vibrational wavefunction is modelled using translated Hermite functions thus allowing anharmonic effects to be incorporated directly into the wavefunction whilst still retaining the simplicity of the Hermite basis. We combine this new method with an optimised finite-difference grid for computing the necessary third and fourth nuclear derivatives of the energy. We compare our combined approach to existing anharmonic models—vibrational self-consistent field theory (VSCF), vibrational perturbation theory (VPT), and vibrational configuration interaction theory (VCI)—and find that it is more cost effective than these alternatives. This makes our method well-suited to computing anharmonic corrections for frequencies in medium-sized molecules. Contribution of the Mark S. Gordon 65th Birthday Festschrift Issue.     

Nuclear quantum effects and hydrogen bonding in liquids

We have employed ab initio path integral molecular dynamics simulations to investigate the role of nuclear quantum effects on the strength of hydrogen bonds in liquid hydrogen fluoride. Nuclear quantum effects are shown to be responsible for a stronger hydrogen bond and an enhanced dipole-dipole interaction, which lead, in turn, to a shortening of the H...F intrachain distance. The simulation results are analyzed in terms of the electronic density shifts with respect to a purely classical treatment of the nuclei. The observed enhanced hydrogen-bond interaction, which arises from a coupling of intra- and intermolecular effects, should be a general phenomenon occurring in all hydrogen-bonded systems.     

A second quantization formulation of multimode dynamics

A new formalism for calculating and analyzing many-mode quantum dynamics is presented. The formalism is similar in spirit to the second quantization formulation of electronic structure theory. The similarity means that similar techniques can be employed for calculating the many-mode nuclear wave function. As a consequence a new formulation of the vibrational self-consistent-field (VSCF) method can be developed. Another result is that the formalism opens up for the construction of new methods that go beyond the VSCF level. A vibrational coupled cluster (VCC) theory is constructed using the new formalism. The size-extensivity concept is introduced in the context of multimode wave functions and the size extensivity of approximate VCC methods is illustrated in comparison with the non-size-extensive vibrational configuration interaction method.     

Nuclear momentum distribution in solid and liquid HF from ab initio calculation

A calculation of nuclear momentum distribution of liquid and solid hydrogen fluoride was performed. In both systems, density functional theory generalized gradient approximation functional of Perdew, Burke, and Ernzerhof was used for the calculation: for liquid hydrogen fluoride, using an atom centered basis set for an isolated molecule with optimized geometry, and for solid hydrogen fluoride using plane-wave basis sets on optimized orthorhombic crystal cell. For liquid hydrogen fluoride, a semiclassical approach was adopted with the vibrational contribution to momentum distribution obtained from the density functional theory calculation and translational and rotational contributions calculated classically. Nuclear momentum distribution in the solid hydrogen fluoride was calculated entirely quantum mechanically using phonon dispersion and vibrational density of states calculated in the framework of plane-wave density functional theory. Theoretical results were contrasted with recently obtained results of Compton (deep inelastic) neutron scattering on liquid and solid hydrogen fluoride. In case of liquid hydrogen fluoride, almost a perfect agreement between theory and experiment was achieved within the harmonic Born-Oppenheimer approximation. For the solid system under investigation, the harmonic approximation leads to small (4%) overestimation of the square root of the second moment indicating that neutron Compton scattering technique is sensitive to proton delocalization due to hydrogen bonding in solid hydrogen fluoride.     

The sign and magnitude of 2hJ(F,F) and 1hJ(F,H) in FH...FH. A CLOPPA analysis of their distance dependence

The sign change of the intermolecular (2h)J(F,F) coupling in the (HF)2 dimer as a function of the F-F distance is discussed by means of the CLOPPA method. It is found that it is due to the competition of positive and negative contributions involving the interaction of the sigma lone pair of the acceptor nucleus with vacant molecular orbitals localized in the F-H...F moiety and with other molecular orbitals localized in the donor molecule. The origin of the sign of each contribution is fully determined by analyzing the response of the electronic system to the magnetic perturbation at the acceptor F nucleus. (2h)J(F,F) coupling in the FH...F-, which is positive for all F-F distances, is also analyzed in order to look for the differences with the former case.     

Opposing Electronic and Nuclear Quantum Effects on Hydrogen Bonds in H2O and D2O

The effect of extending the O−H bond length(s) in water on the hydrogen-bonding strength has been investigated using static ab initio molecular orbital calculations. The “polar flattening” effect that causes a slight σ-hole to form on hydrogen atoms is strengthened when the bond is stretched, so that the σ-hole becomes more positive and hydrogen bonding stronger. In opposition to this electronic effect, path-integral ab initio molecular-dynamics simulations show that the nuclear quantum effect weakens the hydrogen bond in the water dimer. Thus, static electronic effects strengthen the hydrogen bond in H₂O relative to D₂O, whereas nuclear quantum effects weaken it. These quantum fluctuations are stronger for the water dimer than in bulk water.     

Vibrational structure theory: new vibrational wave function methods for calculation of anharmonic vibrational energies and vibrational contributions to molecular properties

A number of recently developed theoretical methods for the calculation of vibrational energies and wave functions are reviewed. Methods for constructing the appropriate quantum mechanical Hamilton operator are briefly described before reviewing a particular branch of theoretical methods for solving the nuclear Schr?dinger equation. The main focus is on wave function methods using the vibrational self-consistent field (VSCF) as starting point, and includes vibrational configuration interaction (VCI), vibrational M?ller-Plesset (VMP) theory, and vibrational coupled cluster (VCC) theory. The convergence of the different methods towards the full vibrational configuration interaction (FVCI) result is discussed. Finally, newly developed vibrational response methods for calculation of vibrational contributions to properties, energies, and transition probabilities are discussed.