Investigation of isotope effects with the nuclear-electronic orbital approach

Capturing Correlation Effects in Positron Binding to Atoms and Molecules

A major challenge in contemporary electronic structure theory involves the development of methods to describe in a balanced manner the contribution of correlation effects to energy differences. This challenge can be even greater for multicomponent systems containing more than one type of quantum particle. In the present work, we describe a flexible code for carrying out self-consistent field and configuration interaction (CI) calculations on multicomponent systems and use it to generate trial wave functions for use in diffusion Monte Carlo (DMC) calculations of the positron affinity of Be, Be2, Be4, Mg, CS2, and benzene. The resulting positron affinities (PAs) are in good agreement with the best values from the literature.

Optimization of Quantum Nuclei Positions with the Adaptive Nuclear-Electronic Orbital Approach

The use of multicomponent methods has become increasingly popular over the last years. Under this framework, nuclei (commonly protons) are treated quantum mechanically on the same footing as the electronic structure problem. Under the use of atomic-centered orbitals, this can lead to some complications as the ideal location of the nuclear basis centers must be optimized. In this contribution, we propose a straightforward approach to determine the position of such centers within the self-consistent cycle of a multicomponent calculation, making use of individual proton charge centroids. We test the method on model systems including the water dimer, a protonated water tetramer, and a porphine system. Comparing to numerical gradient calculations, the adaptive nuclear-electronic orbital (NEO) procedure is able to converge the basis centers to within a few cents of an Ångström and with less than 0.1 kcal/mol differences in absolute energies. This is achieved in one single calculation and with a small added computational effort of up to 80% compared to a regular NEO- self-consistent field run. An example application for the human transketolase proton wire is also provided.

A nuclear configuration interaction approach to study nuclear spin effects: an application to ortho- and para-3 He2 @C60

We introduce a non-orthogonal configuration interaction approach to investigate nuclear quantum effects on energies and densities of confined fermionic nuclei. The Hamiltonian employed draws parallels between confined systems and many-electron atoms, where effective non-Coulombic potentials represent the interactions of the trapped particles. One advantage of this method is its generality, as it offers the potential to study the nuclear quantum effects of various confined species affected by effective isotropic or anisotropic potentials. As a first application, we analyze the quantum states of two 3 He atoms encapsulated in C60 . At the Hartree-Fock level, we observe the breaking of spin and spatial symmetries. To ensure wavefunctions with the correct symmetries, we mix the broken-symmetry Hartree-Fock states within the non-orthogonal configuration interaction expansion. Our proposed approach predicts singly and triply degenerate ground states for the singlet (para-3 He2 @C60 ) and triplet (ortho-3 He2 @C60 ) nuclear spin configurations, respectively. The ortho-3 He2 @C60 ground state is 5.69 cm-1 higher in energy than the para-3 He2 @C60 ground state. The nuclear densities obtained for these states exhibit the icosahedral symmetry of the C60 embedding potential. Importantly, our calculated energies for the lowest 85 states are in close agreement with perturbation theory results based on a harmonic oscillator plus rigid rotor model of 3 He2 @C60 .

Generalized quantum cumulant dynamics

A means of unifying some semiclassical models of computational chemistry is presented; these include quantized Hamiltonian dynamics, quantal cumulant dynamics, and semiclassical Moyal dynamics (SMD). A general method for creating the infinite hierarchy of operator dynamics in the Heisenberg picture is derived together with a general method for truncation (or closure) of that series, and in addition, we provide a simple link to the phase space methods of SMD. Operator equations of arbitrary order may be created readily, avoiding the tedious algebra identified previously. Truncation is based on a simple recurrence formula which is related to, but avoids the more complex contractions of, Wick's theorem. This generalized method is validated against a number of trial problems considered using the previous methods. We also touch on some of the limitations involved using such methods, noting, in particular, that any truncation will lead to a state which is in some sense unphysical. Finally, we briefly introduce our quantum algebra package QuantAL which provides an automated method for the generation of the required equation set, the initial conditions for all variables from any start, and all the higher order approximations necessary for truncation of the series, at essentially arbitrary order.

Multicomponent density functional theory: Including the density gradient in the electron-proton correlation functional for hydrogen and deuterium

Multicomponent density functional theory (DFT) has many practical advantages for incorporating nuclear quantum effects into quantum chemistry calculations. Within the nuclear-electronic orbital (NEO) framework, specified nuclei, typically protons, are treated quantum mechanically on the same level as the electrons. Previously, electron-proton correlation functionals based on the local density approximation (LDA), denoted epc17 and epc18, were developed and shown to provide more accurate proton densities and energies compared to the neglect of electron-proton correlation, but a quantitatively accurate description of both densities and energies simultaneously has remained elusive. Herein, an electron-proton correlation functional that depends on the electron and proton density gradients, as well as the densities, is derived and implemented. Compared to the LDA functionals, the resulting generalized gradient approximation functional, denoted epc19, is able to simultaneously provide accurate proton densities and energies, as well as reproduce the impact of nuclear quantum effects on optimized geometries. In addition, without further parameterization, the NEO-DFT/epc19 method provides accurate densities and energies for deuterium as well as hydrogen. These results demonstrate that the form of the epc19 functional is able to capture the essential aspects of electron-proton correlation and highlights the importance of including gradient terms. This approach will enable the exploration of nuclear quantum effects and isotope effects in a wide range of systems.

Toward a muon-specific electronic structure theory: effective electronic Hartree-Fock equations for muonic molecules

An effective set of Hartree-Fock (HF) equations are derived for electrons of muonic systems, i.e., molecules containing a positively charged muon, conceiving the muon as a quantum oscillator, which are completely equivalent to the usual two-component HF equations used to derive stationary states of the muonic molecules. In these effective equations, a non-Coulombic potential is added to the orthodox coulomb and exchange potential energy terms, which describes the interaction of the muon and the electrons effectively and is optimized during the self-consistent field cycles. While in the two-component HF equations a muon is treated as a quantum particle, in the effective HF equations it is absorbed into the effective potential and practically transformed into an effective potential field experienced by electrons. The explicit form of the effective potential depends on the nature of muon's vibrations and is derivable from the basis set used to expand the muonic spatial orbital. The resulting effective Hartree-Fock equations are implemented computationally and used successfully, as a proof of concept, in a series of muonic molecules containing all atoms from the second and third rows of the Periodic Table. To solve the algebraic version of the equations muon-specific Gaussian basis sets are designed for both muon and surrounding electrons and it is demonstrated that the optimized exponents are quite distinct from those derived for the hydrogen isotopes. The developed effective HF theory is quite general and in principle can be used for any muonic system while it is the starting point for a general effective electronic structure theory that incorporates various types of quantum correlations into the muonic systems beyond the HF equations.

Prediction of proton affinities of organic molecules using the any-particle molecular-orbital second-order proton propagator approach

We assess the performance of the recently developed any-particle molecular-orbital second-order proton propagator (APMO/PP2) scheme [M. Díaz-Tinoco, J. Romero, J. V. Ortiz, A. Reyes and R. Flores-Moreno, J. Chem. Phys., 2013, 138, 194108] on the calculation of gas phase proton affinities (PAs) of a set of 150 organic molecules comprising several functional groups: amines, alcohols, aldehydes, amides, ketones, esters, ethers, carboxylic acids and carboxylate anions. APMO/PP2 PAs display an overall mean absolute error of 0.68 kcal mol^-1 with respect to experimental data. These results suggest that the APMO/PP2 method is an alternative approach for the quantitative prediction of gas phase proton affinities. One novel feature of the method is that a PA can be obtained from a single calculation of the optimized protonated molecule.

Fast and accurate prediction of proton affinities: revisiting the extended Koopmans' theorem for protons

In this work we propose schemes based on the extended Koopmans' theorem for quantum nuclei (eKT), in the framework of the any particle molecular orbital approach (APMO/KT), for the quantitative prediction of gas phase proton affinities (PAs). The performance of these schemes has been tested on a set of 300 organic molecules containing diverse functional groups. The APMO/KT scheme scaled by functional group (APMO/KT-SC-FG) displays an overall mean absolute error of 1.1 kcal mol^-1 with respect to experimental data. Its performance in PA calculations is similar to that of post-Hartree-Fock composite methods or that of the APMO second order proton propagator (APMO/PP2) approach. The APMO/KT-SC-FG scheme is also employed to predict PAs of polyfunctional molecules such as the Nerve Agent VX and the 20 common α-amino acids, finding excellent agreement with available theoretical and/or experimental data. The accuracy of the predictions demonstrates that the APMO/KT-SC-FG scheme is a low-cost alternative to adiabatic methods for the calculation of accurate PAs. One of the most appealing features of the APMO/KT-SC-FG scheme, is that PAs can be derived from one single-point APMO Hartree-Fock calculation.

On the physical interpretation of the nuclear molecular orbital energy

Recently, several groups have extended and implemented molecular orbital (MO) schemes to simultaneously obtain wave functions for electrons and selected nuclei. Many of these schemes employ an extended Hartree-Fock approach as a first step to find approximate electron-nuclear wave functions and energies. Numerous studies conducted with these extended MO methodologies have explored various effects of quantum nuclei on physical and chemical properties. However, to the best of our knowledge no physical interpretation has been assigned to the nuclear molecular orbital energy (NMOE) resulting after solving extended Hartree-Fock equations. This study confirms that the NMOE is directly related to the molecular electrostatic potential at the position of the nucleus.

2nd coordination sphere controlled electron transfer of iron hangman complexes on electrodes probed by surface enhanced vibrational spectroscopy

Surface enhanced vibrational spectroscopy shows the correlation between electron transfer kinetics and protonation degree of Fe Hangman complexes on electrodes.

Iron hangman complexes exhibit improved catalytic properties regarding O₂ and H₂O₂ reduction, which are attributed to the presence of a proton donating group in defined vicinity of the catalytic metal centre. Surface enhanced resonance Raman (SERR) and IR (SEIRA) spectro-electrochemistry has been applied concomitantly for the first time to analyse such iron hangman porphyrin complexes attached to electrodes in aqueous solution. While the SERR spectra yield information about the redox state of the central iron, the SEIRA spectra show protonation and deprotonation events of the 2^nd coordination sphere. To investigate the influence of a proton active hanging group on the heterogeneous electron transfer between the iron porphyrin and the electrode, two hangman complexes with either an acid or ester functional group were compared. Using time resolved SERR spectroscopy the electron transfer rates of both complexes were determined. Complexes with an acid group showed a slow electron transfer rate at neutral pH that increased significantly at pH 4, while complexes with an ester group exhibited a much faster, but pH independent rate. SEIRA measurements were able to determine directly for the first time a pK_a value of 3.4 of a carboxylic hanging group in the immobilized state that shifted to 5.2 in D₂O buffer solution. The kinetic data showed an increase of the heterogeneous electron transfer rate with the protonation degree of the acid groups. From these results, we propose a PCET which is strongly modulated by the protonation state of the acid hanging group via hydrogen bond interactions.

Hidden aspects of the Structural theory of chemistry: MC-QTAIM analysis reveals "alchemical" transformation from a triatomic to a diatomic structure

The Structural theory of chemistry introduces chemical/molecular structure as a combination of relative arrangement and bonding patterns of atoms in a molecule. Nowadays, the structure of atoms in molecules is derived from the topological analysis of the quantum theory of atoms in molecules (QTAIM). In this context, a molecular structure is varied by large geometrical variations and concomitant reorganization of electronic structure that usually take place in chemical reactions or under extreme hydrostatic pressure. In this report, a new mode of structural variation is introduced within the context of the newly proposed multi-component QTAIM (MC-QTAIM) that originates from the mass variation of nuclei. Accordingly, XCN and CNX series of species are introduced where X stands for a quantum particle with a unit of positive charge and a variable mass that is varied in discrete steps between the mass of a proton and a positron. Ab initio non-Born-Oppenheimer (non-BO) calculations are done on both series of species and the resulting non-BO wavefunctions are used for the MC-QTAIM analysis, revealing a triatomic structure for the proton mass and a diatomic structure for the positron mass. In both series of species, a critical mass between that of proton and positron mass is discovered where the transition from triatomic to diatomic structure takes place. This abrupt structural transformation has a topological nature resembling the usual phase transitions in thermodynamics. The discovered mass-induced structural transformation is a hidden aspect of the Structural theory which is revealed only beyond the BO paradigm, when nuclei are treated as quantum waves instead of clamped point charges.

The role of electrostatic induction in secondary isotope effects on acidity: theory and computational confirmation

How well can secondary isotope effects on acidity in aqueous solution be approximated by gas-phase computations on the hydrated components?

Interpretation of intermolecular geometric isotope effect in hydrogen bonds: nuclear orbital plus molecular orbital study

The intermolecular geometric isotope effect (GIE) in hydrogen bond A-X···B (X = H and D) is investigated theoretically using the nuclear orbital plus molecular orbital (NOMO) theory. To interpret the GIE in terms of physically meaningful energy components such as electrostatic and exchange-repulsion interactions, the reduced variational space self-consistent-field method is extended to the NOMO scheme. The intermolecular GIE is analyzed as a two-stage process: the intramolecular bond shrinkage and the intermolecular bond elongation. According to the isotopic shifts of energy components described by the NOMO/MP2 method, the intermolecular GIE is approximately interpreted as a process reducing the exchange-repulsion interaction after the decrease of electrostatic interaction.

On the physical origin of the cation-anion intermediate bond in ionic liquids Part I. Placing a (weak) hydrogen bond between two charges

The intermediate bond forces in ionic liquids are investigated from static quantum chemical calculations at various methods and two basis sets. The experimentally observed red-shift of the donor-proton bond stretching frequency due to a bond elongation is confirmed by all methods. Comparing Hartree-Fock to second-order Møller-Plesset perturbation theory, the Hartree-Fock method gives in many cases an erroneous description of the geometries. Furthermore, the Hartree-Fock interaction energies can deviate up to 60 kJ mol(-1) from Møller-Plesset perturbation theory indicating the importance of dispersion interaction. While the usual trends of decreasing stability or interaction energies with increasing ion sizes are found, the geometries involving hydrogen atoms do not change this order of total interaction energies. Therefore, the hydrogen bond is not the most important interaction for ion pairs with regard to the total interaction energy. On the other hand, the different established analysis methods give rise to hydrogen bonding in several ion pairs. Charge analysis reveals the hydrogen-bonding character of the ion pair and shows, depending on the type of ions combined and further on the type of conformers considered, that a hydrogen bond can be present. The possibility of hydrogen bonding is also shown by an analysis of the frontier orbitals. Calculating potential energy surfaces and observing from this the change in the donor proton bond indicates that regular hydrogen bonds are possible in ion pairs of ionic liquids. Thereby, the maximum of bond elongation exceeds the one of a usual hydrogen bond by far. The more salt-like hydrogen-bonded ion pair [NH(4)][BF(4)] exhibits a steeper maximum than the more ionic liquid like ion pair [EtNH(3)][BF(4)]. The fact that imidazolium-based ionic liquids as [Emim][Cl] can display two faces, hydrogen bonding and purely ionic bonding, points to a disturbing rather than stabilizing role of hydrogen bonding on the interaction of the counterions in imidazolium-based ionic liquids. While geometry and charge analysis provides attributes of weak (blue-shifted) hydrogen bonds, large bond elongations accompanied by red-shifts are obtained for the ion pairs investigated. This can be understood by the simple fact that these imidazolium-based ionic liquid ion pairs constitute weak hydrogen bonds placed between two delocalized charges.

Secondary Deuterium Isotope Effects on the Acidity of Carboxylic Acids and Phenols

Secondary deuterium isotope effects (IEs) on acidities have been accurately measured by an NMR titration method applicable to a mixture of isotopologues. Deuteration definitely decreases the acidity of carboxylic acids and phenols, by up to 0.031 in the DeltapK per D. For aliphatic acids, the IEs decrease as the site of deuteration becomes more distant from the OH, as expected, but a surprising result is that IEs in both phenol and benzoic acid do not decrease as the site of deuteration moves from ortho to meta to para. The experimental data are supported by ab initio computations, which, however, substantially overestimate the IEs. The discrepancy does not seem to be due to solvation. The IEs originate in isotope-sensitive vibrations whose frequencies and zero-point energies are lowered upon deprotonation. In the simplest case, formate, the key vibration can be recognized as the C-H stretch, which is weakened by delocalization of the oxygen lone pairs. For the aromatic acids, delocalization cannot account for the near constancy of IEs from ortho, meta, and para deuteriums, but the observed IEs are consistent with calculated vibrational frequencies and electron densities. Moreover, the ability of the frequency analysis to account for the IEs is evidence against an inductive origin.

Darwin and mass-velocity relativistic corrections in non-Born-Oppenheimer variational calculations

The Pauli approach to account for the mass-velocity and Darwin relativistic corrections has been applied to the formalism for quantum mechanical molecular calculations that does not assume the Born-Oppenheimer (BO) approximation regarding separability of the electronic and nuclear motions in molecular systems. The corrections are determined using the first order perturbation theory and are derived for the non-BO wave function of a diatomic system expressed in terms of explicitly correlated Gaussian functions with premultipliers in the form of even powers of the internuclear distance. As a numerical example we used calculations of the transition energies for pure vibrational states of the HD(+) ion.

Explicit Dynamical Electron−Proton Correlation in the Nuclear−Electronic Orbital Framework

A method that includes explicit electron-proton correlation directly into the nuclear-electronic orbital self-consistent-field framework is presented. This nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) scheme is formulated using Gaussian basis functions for the electrons and the quantum nuclei in conjunction with Gaussian-type geminal functions. The NEO approach is designed for the quantum treatment of a relatively small number of nuclei, such as the hydrogen nuclei involved in key hydrogen bonding interactions or hydrogen transfer reactions. The conventional nuclear-electronic-orbital-based methods produce nuclear wave functions that are too localized, leading to severe overestimations of hydrogen vibrational frequencies, as well as inaccuracies in geometries, isotope effects, couplings, and tunneling splittings. The application of the NEO-XCHF approach to a model system illustrates that the description of the nuclear wave function is significantly improved by the inclusion of explicit electron-proton correlation. In contrast to the NEO-HF method, the NEO-XCHF method leads to hydrogen vibrational stretch frequencies that are in excellent agreement with those calculated from grid-based methods. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically and only electron-proton correlation is treated explicitly. Electron-electron dynamical correlation can be included with density functional theory or perturbation theory methods.

Semiquantal analysis of hydrogen bond

The semiquantal time-dependent Hartree (SQTDH) theory is applied to the coupled Morse and modified Lippincott-Schroeder (LS) model potentials of hydrogen bond. The structural correlation between the heavy atoms distance and the proton position, the geometric isotope effect, the energy of hydrogen bond formation, and the proton vibrational frequency shift are examined in a broad range of structural parameters. In particular, the geometric isotope effect is found to depend notably on the choice of the potential model, for which the LS potential gives the isotope shift of the heavy atoms distance in the range of 0.02-0.04 A, in quantitative agreement with the experimental findings from assortment of hydrogen bonding crystals. The fourth-order expansion approximation to the semiquantal extended potential was confirmed to be highly accurate in reproducing the full SQTDH results. The approximation is computationally efficient and flexible enough to be applied to general models of hydrogen bond.

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.