Nuclear Quantum Effects in Water and Aqueous Systems: Experiment, Theory, and Current Challenges.

Nuclear quantum effects influence the structure and dynamics of hydrogen-bonded systems, such as water, which impacts their observed properties with widely varying magnitudes. This review highlights the recent significant developments in the experiment, theory, and simulation of nuclear quantum effects in water. Novel experimental techniques, such as deep inelastic neutron scattering, now provide a detailed view of the role of nuclear quantum effects in water's properties. These have been combined with theoretical developments such as the introduction of the principle of competing quantum effects that allows the subtle interplay of water's quantum effects and their manifestation in experimental observables to be explained. We discuss how this principle has recently been used to explain the apparent dichotomy in water's isotope effects, which can range from very large to almost nonexistent depending on the property and conditions. We then review the latest major developments in simulation algorithms and theory that have enabled the efficient inclusion of nuclear quantum effects in molecular simulations, permitting their combination with on-the-fly evaluation of the potential energy surface using electronic structure theory. Finally, we identify current challenges and future opportunities in this area of research.

Isotopic fractionation in proteins as a measure of hydrogen bond length.

If a deuterated molecule containing strong intramolecular hydrogen bonds is placed in a hydrogenated solvent, it may preferentially exchange deuterium for hydrogen. This preference is due to the difference between the vibrational zero-point energy for hydrogen and deuterium. It is found that the associated fractionation factor Φ is correlated with the strength of the intramolecular hydrogen bonds. This correlation has been used to determine the length of the H-bonds (donor-acceptor separation) in a diverse range of enzymes and has been argued to support the existence of short low-barrier H-bonds. Starting with a potential energy surface based on a simple diabatic state model for H-bonds, we calculate Φ as a function of the proton donor-acceptor distance R. For numerical results, we use a parameterization of the model for symmetric O-H⋯O bonds [R. H. McKenzie, Chem. Phys. Lett. 535, 196 (2012)]. We consider the relative contributions of the O-H stretch vibration, O-H bend vibrations (both in plane and out of plane), tunneling splitting effects at finite temperature, and the secondary geometric isotope effect. We compare our total Φ as a function of R with NMR experimental results for enzymes, and in particular with an earlier model parametrization Φ(R), used previously to determine bond lengths.

Electron-vibration entanglement in the Born-Oppenheimer description of chemical reactions and spectroscopy.

Entanglement is sometimes regarded as the quintessential measure of the quantum nature of a system and its significance for the understanding of coupled electronic and vibrational motions in molecules has been conjectured. Previously, we considered the entanglement developed in a spatially localized diabatic basis representation of the electronic states, considering design rules for qubits in a low-temperature chemical quantum computer. We extend this to consider the entanglement developed during high-energy processes. We also consider the entanglement developed using adiabatic electronic basis, providing a novel way for interpreting effects of the breakdown of the Born-Oppenheimer (BO) approximation. We consider: (i) BO entanglement in the ground-state wavefunction relevant to equilibrium thermodynamics, (ii) BO entanglement associated with low-energy wavefunctions relevant to infrared and tunneling spectroscopies, (iii) BO entanglement in high-energy eigenfunctions relevant to chemical reaction processes, and (iv) BO entanglement developed during reactive wavepacket dynamics. A two-state single-mode diabatic model descriptive of a wide range of chemical phenomena is used for this purpose. The entanglement developed by BO breakdown correlates simply with the diameter of the cusp introduced by the BO approximation, and a hierarchy appears between the various BO-breakdown correction terms, with the first-derivative correction being more important than the second-derivative correction which is more important than the diagonal correction. This simplicity is in contrast to the complexity of BO-breakdown effects on thermodynamic, spectroscopic, and kinetic properties. Further, processes poorly treated at the BO level that appear adequately treated using the Born-Huang adiabatic approximation are found to have properties that can only be described using a non-adiabatic description. For the entanglement developed between diabatic electronic states and the nuclear motion, qualitatively differently behavior is found compared to traditional properties of the density matrix and hence entanglement provides new information about system properties. For chemical reactions, this type of entanglement simply builds up as the transition-state region is crossed. It is robust to small changes in parameter values and is therefore more attractive for making quantum qubits than is the related fragile ground-state entanglement, provided that coherent motion at the transition state can be sustained.

A unified diabatic description for electron transfer reactions, isomerization reactions, proton transfer reactions, and aromaticity.

While diabatic approaches are ubiquitous for the understanding of electron-transfer reactions and have been mooted as being of general relevance, alternate applications have not been able to unify the same wide range of observed spectroscopic and kinetic properties. The cause of this is identified as the fundamentally different orbital configurations involved: charge-transfer phenomena involve typically either 1 or 3 electrons in two orbitals whereas most reactions are typically closed shell. As a result, two vibrationally coupled electronic states depict charge-transfer scenarios whereas three coupled states arise for closed-shell reactions of non-degenerate molecules and seven states for the reactions implicated in the aromaticity of benzene. Previous diabatic treatments of closed-shell processes have considered only two arbitrarily chosen states as being critical, mapping these states to those for electron transfer. We show that such effective two-state diabatic models are feasible but involve renormalized electronic coupling and vibrational coupling parameters, with this renormalization being property dependent. With this caveat, diabatic models are shown to provide excellent descriptions of the spectroscopy and kinetics of the ammonia inversion reaction, proton transfer in N2H7(+), and aromaticity in benzene. This allows for the development of a single simple theory that can semi-quantitatively describe all of these chemical phenomena, as well as of course electron-transfer reactions. It forms a basis for understanding many technologically relevant aspects of chemical reactions, condensed-matter physics, chemical quantum entanglement, nanotechnology, and natural or artificial solar energy capture and conversion.

Non-adiabatic effects in thermochemistry, spectroscopy and kinetics: the general importance of all three Born-Oppenheimer breakdown corrections.

Using a simple model Hamiltonian, the three correction terms for Born-Oppenheimer (BO) breakdown, the adiabatic diagonal correction (DC), the first-derivative momentum non-adiabatic correction (FD), and the second-derivative kinetic-energy non-adiabatic correction (SD), are shown to all contribute to thermodynamic and spectroscopic properties as well as to thermal non-diabatic chemical reaction rates. While DC often accounts for >80% of thermodynamic and spectroscopic property changes, the commonly used practice of including only the FD correction in kinetics calculations is rarely found to be adequate. For electron-transfer reactions not in the inverted region, the common physical picture that diabatic processes occur because of surface hopping at the transition state is proven inadequate as the DC acts first to block access, increasing the transition state energy by (ℏω)(2)λ/16J(2) (where λ is the reorganization energy, J the electronic coupling and ω the vibration frequency). However, the rate constant in the weakly-coupled Golden-Rule limit is identified as being only inversely proportional to this change rather than exponentially damped, owing to the effects of tunneling and surface hopping. Such weakly-coupled long-range electron-transfer processes should therefore not be described as "non-adiabatic" processes as they are easily described by Born-Huang ground-state adiabatic surfaces made by adding the DC to the BO surfaces; instead, they should be called just "non-Born-Oppenheimer" processes. The model system studied consists of two diabatic harmonic potential-energy surfaces coupled linearly through a single vibration, the "two-site Holstein model". Analytical expressions are derived for the BO breakdown terms, and the model is solved over a large parameter space focusing on both the lowest-energy spectroscopic transitions and the quantum dynamics of coherent-state wavepackets. BO breakdown is investigated pertinent to: ammonia inversion, aromaticity in benzene, the Creutz-Taube ion, the bacterial photosynthetic reaction centre, BNB, the molecular conductor Alq3, and inverted-region charge recombination in a ferrocene-porphyrin-fullerene triad photosynthetic model compound. Throughout, the fundamental nature of BO breakdown is linked to the properties of the cusp catastrophe: the cusp diameter is shown to determine the magnitudes of all couplings, numerical basis-set and trajectory-integration requirements, and to determine the transmission coefficient κ used to understand deviations from transition-state theory.

Bond angle variations in XH3 [X = N, P, As, Sb, Bi]: the critical role of Rydberg orbitals exposed using a diabatic state model.

Ammonia adopts sp(3) hybridization (HNH bond angle 108°) whereas the other members of the XH3 series PH3, AsH3, SbH3, and BiH3 instead prefer octahedral bond angles of 90-93°. We use a recently developed general diabatic description for closed-shell chemical reactions, expanded to include Rydberg states, to understand the geometry, spectroscopy and inversion reaction profile of these molecules, fitting its parameters to results from Equation of Motion Coupled-Cluster Singles and Doubles (EOM-CCSD) calculations using large basis sets. Bands observed in the one-photon absorption spectrum of NH3 at 18.3 eV, 30 eV, and 33 eV are reassigned from Rydberg (formally forbidden) double excitations to valence single-excitation resonances. Critical to the analysis is the inclusion of all three electronic states in which two electrons are placed in the lone-pair orbital n and/or the symmetric valence σ* antibonding orbital. An illustrative effective two-state diabatic model is also developed containing just three parameters: the resonance energy driving the high-symmetry planar structure, the reorganization energy opposing it, and HXH bond angle in the absence of resonance. The diabatic orbitals are identified as sp hybrids on X; for the radical cations XH3(+) for which only 2 electronic states and one conical intersection are involved, the principle of orbital following dictates that the bond angle in the absence of resonance is acos(-1/5) = 101.5°. The multiple states and associated multiple conical intersection seams controlling the ground-state structure of XH3 renormalize this to acos[3 sin(2)(2(1/2)atan(1/2))/2 - 1/2] = 86.7°. Depending on the ratio of the resonance energy to the reorganization energy, equilibrium angles can vary from these limiting values up to 120°, and the anomalously large bond angle in NH3 arises because the resonance energy is unexpectedly large. This occurs as the ordering of the lowest Rydberg orbital and the σ* orbital swap, allowing Rydbergization to compresses σ* to significantly increase the resonance energy. Failure of both the traditional and revised versions of the valence-shell electron-pair repulsion (VSEPR) theory to explain the ground-state structures in simple terms is attributed to exclusion of this key physical interaction.

Valence-bond non-equilibrium solvation model for a twisting monomethine cyanine.

We propose and analyze a two-state valence-bond model of non-equilibrium solvation effects on the excited-state twisting reaction of monomethine cyanines. Suppression of this reaction is thought responsible for environment-dependent fluorescence yield enhancement in these dyes. Fluorescence is quenched because twisting is accompanied via the formation of dark twisted intramolecular charge-transfer (TICT) states. For monomethine cyanines, where the ground state is a superposition of structures with different bond and charge localizations, there are two possible twisting pathways with different charge localizations in the excited state. For parameters corresponding to symmetric monomethines, the model predicts two low-energy twisting channels on the excited-state surface, which leads to a manifold of TICT states. For typical monomethines, twisting on the excited state surface will occur with a small barrier or no barrier. Changes in the solvation configuration can differentially stabilize TICT states in channels corresponding to different bonds, and that the position of a conical intersection between adiabatic states moves in response to solvation to stabilize either one channel or the other. There is a conical intersection seam that grows along the bottom of the excited-state potential with increasing solvent polarity. For monomethine cyanines with modest-sized terminal groups in moderately polar solution, the bottom of the excited-state potential surface is completely spanned by a conical intersection seam.

A diabatic state model for double proton transfer in hydrogen bonded complexes.

Four diabatic states are used to construct a simple model for double proton transfer in hydrogen bonded complexes. Key parameters in the model are the proton donor-acceptor separation R and the ratio, D1/D2, between the proton affinity of a donor with one and two protons. Depending on the values of these two parameters the model describes four qualitatively different ground state potential energy surfaces, having zero, one, two, or four saddle points. Only for the latter are there four stable tautomers. In the limit D2 = D1 the model reduces to two decoupled hydrogen bonds. As R decreases a transition can occur from a synchronous concerted to an asynchronous concerted to a sequential mechanism for double proton transfer.

Effect of quantum nuclear motion on hydrogen bonding.

This work considers how the properties of hydrogen bonded complexes, X-H⋯Y, are modified by the quantum motion of the shared proton. Using a simple two-diabatic state model Hamiltonian, the analysis of the symmetric case, where the donor (X) and acceptor (Y) have the same proton affinity, is carried out. For quantitative comparisons, a parametrization specific to the O-H⋯O complexes is used. The vibrational energy levels of the one-dimensional ground state adiabatic potential of the model are used to make quantitative comparisons with a vast body of condensed phase data, spanning a donor-acceptor separation (R) range of about 2.4-3.0 Å, i.e., from strong to weak hydrogen bonds. The position of the proton (which determines the X-H bond length) and its longitudinal vibrational frequency, along with the isotope effects in both are described quantitatively. An analysis of the secondary geometric isotope effect, using a simple extension of the two-state model, yields an improved agreement of the predicted variation with R of frequency isotope effects. The role of bending modes is also considered: their quantum effects compete with those of the stretching mode for weak to moderate H-bond strengths. In spite of the economy in the parametrization of the model used, it offers key insights into the defining features of H-bonds, and semi-quantitatively captures several trends.

Signatures of the Berry curvature in the frequency dependent interlayer magnetoresistance in tilted magnetic fields.

We show that in a layered metal, the angle dependent, finite frequency, interlayer magnetoresistance is altered due to the presence of a non-zero Berry curvature at the Fermi surface. At zero frequency, we find a conservation law which demands that the 'magic angle' condition for interlayer magnetoresistance extrema as a function of magnetic field tilt angle is essentially both field and Berry curvature independent. In the finite frequency case, however, we find that surprisingly large signatures of a finite Berry curvature occur in the periodic orbit resonances. We outline a method whereby the presence and magnitude of the Berry curvature at the Fermi surface can be extracted.

Thermodynamics of a bad metal-Mott insulator transition in the presence of frustration.

Thermodynamic properties of the Hubbard model on the anisotropic triangular lattice at half filling are calculated by the finite-temperature Lanczos method. The charge susceptibility exhibits clear signatures of a Mott metal-insulator transition. The metallic phase is characterized by a small charge susceptibility, large entropy, large renormalized quasiparticle mass, and large spin susceptibility. The fluctuating local magnetic moment in the metallic phase is large and comparable to that in the insulating phase. These bad metallic characteristics occur above a relatively low coherence temperature, as seen in organic charge transfer salts.

A two-state model of twisted intramolecular charge-transfer in monomethine dyes.

A two-state model Hamiltonian is proposed, which can describe the coupling of twisting displacements to charge-transfer behavior in the ground and excited states of a general monomethine dye molecule. This coupling may be relevant to the molecular mechanism of environment-dependent fluorescence yield enhancement. The model is parameterized against quantum chemical calculations on different protonation states of the green fluorescent protein chromophore, which are chosen to sample different regimes of detuning from the cyanine (resonant) limit. The model provides a simple yet realistic description of the charge transfer character along two possible excited state twisting channels associated with the methine bridge. It describes qualitatively different behavior in three regions that can be classified by their relationship to the resonant (cyanine) limit. The regimes differ by the presence or absence of twist-dependent polarization reversal and the occurrence of conical intersections. We find that selective biasing of one twisting channel over another by an applied diabatic biasing potential can only be achieved in a finite range of parameters near the cyanine limit.

A three-state effective Hamiltonian for symmetric cationic diarylmethanes.

We analyze the low-energy electronic structure of a series of symmetric cationic diarylmethanes, which are bridge-substituted derivatives of Michler's Hydrol Blue. We use a four-electron, three-orbital complete active space self-consistent field and multi-state multi-reference perturbation theory model to calculate a three-state diabatic effective Hamiltonian for each dye in the series. We exploit an isolobal analogy between the active spaces of the self-consistent field solutions for each dye to represent the electronic structure in a set of analogous diabatic states. The diabatic states can be identified with the bonding structures in classical resonance-theoretic models of cyanine dyes. We identify diabatic states with opposing charge and bond-order localization, analogous to the classical resonance structures, and a third state with charge on the bridge. While the left- and right-charged structures are similar for all dyes, the structure of the bridge-charged diabatic state, and the Hamiltonian matrix elements connected to it, change significantly across the series. The change is correlated with an inversion of the sign of the charge carrier on the bridge, which changes from an electron pair to a hole as the series is traversed.

Consistent description of the metallic phase of overdoped cuprate superconductors as an anisotropic marginal Fermi liquid.

We consider a model self-energy consisting of an isotropic Fermi liquid term and a marginal Fermi liquid term which is anisotropic over the Fermi surface, vanishing in the same directions as the superconducting gap and the pseudogap. This model self-energy gives a consistent description of experimental results from angle-dependent magnetoresistance, specific heat, de Haas-van Alphen, and measurements of the quasiparticle dispersion near the Fermi surface from photoemission. In particular, we reconcile the strongly doping-dependent anomalous scattering rate observed in angle-dependent magnetoresistance with the almost doping-independent specific heat.

Quantum entanglement between a nonlinear nanomechanical resonator and a microwave field.

We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semiclassical approximation of this system has an interesting fixed-point bifurcation structure. In the semiclassical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positive to negative detuning. We show that signatures of this bifurcation structure are also present in the full dissipative quantum system and further show that the bifurcation structure leads to mixed-state entanglement between the nanomechanical resonator and the microwave cavity in the dissipative quantum system that is a maximum close to the semiclassical bifurcation. Quantum signatures of the semiclassical limit cycles are presented.

Bond alternation, polarizability, and resonance detuning in methine dyes.

We derive structure-property relationships for methine ("Brooker") dyes relating the color of the dye and its symmetric parents to its bond alternation in the ground state and also to the dipole properties associated with its low-lying charge-resonance (or charge-transfer) transition. We calibrate and test these relationships on an array of different protonation states of the green fluorescent protein chromophore motif (an asymmetric halochromic methine dye) and its symmetric parent dyes. The relationships rely on the assumption that the diabatic states that define the Platt model for methine dye color [J. R. Platt, J. Chem. Phys. 25, 80 (1956)] can also be distinguished by their single-double bond alternation and by their charge localization character. These assumptions are independent of the primary constraint that defines the diabatic states in the Platt model--specifically, the Brooker deviation rule for methine dyes [L. G. S. Brooker, Rev. Mod. Phys. 14, 275 (1942)]. Taking these assumptions, we show that the Platt model offers an alternate route to known structure-property relationships between the bond length alternation and the quadratic nonlinear polarizability ß. We show also that the Platt model can be parameterized without the need for synthesis of the symmetric parents of a given dye, using the dipole data obtained through spectroscopic measurements. This suggests that the Platt model parameters may be used as independent variables in free-energy relationships for chromophores whose symmetric parents cannot be synthesized or chromophores strongly bound to biomolecular environments. The latter category includes several recently characterized biomolecular probe constructs. We illustrate these concepts by an analysis of previously reported electroabsorption and second-harmonic generation experiments on green fluorescent proteins.

Quantum entanglement between electronic and vibrational degrees of freedom in molecules.

We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.

Sensitivity of the photophysical properties of organometallic complexes to small chemical changes.

We investigate an effective model Hamiltonian for organometallic complexes that are widely used in optoelectronic devices. The two most important parameters in the model are J, the effective exchange interaction between the π and π* orbitals of the ligands, and ε*, the renormalized energy gap between the highest occupied orbitals on the metal and on the ligand. We find that the degree of metal-to-ligand charge transfer character of the lowest triplet state is strongly dependent on the ratio ε*/J. ε* is purely a property of the complex and can be changed significantly by even small variations in the complex's chemistry, such as replacing substituents on the ligands. We find that small changes in ε*/J can cause large changes in the properties of the complex, including the lifetime of the triplet state and the probability of injected charges (electrons and holes) forming triplet excitations. These results give some insight into the observed large changes in the photophysical properties of organometallic complexes caused by small changes in the ligands.

Charge distribution near bulk oxygen vacancies in cerium oxides.

Understanding the electronic charge distribution around oxygen vacancies in transition metal and rare earth oxides is a scientific challenge of considerable technological importance. We show how significant information about the charge distribution around vacancies in cerium oxide can be gained from a study of high resolution crystal structures of higher order oxides which exhibit ordering of oxygen vacancies. Specifically, we consider the implications of a bond valence sum analysis of Ce7O12 and Ce11O20. To illuminate our analysis we show alternative representations of the crystal structures in terms of orderly arrays of coordination defects and in terms of fluorite-type modules. We found that in Ce7O12, the excess charge resulting from removal of an oxygen atom delocalizes among all three triclinic Ce sites closest to the O vacancy. In Ce11O20, the charge localizes on the next nearest neighbour Ce atoms. Our main result is that the charge prefers to distribute itself so that it is farthest away from the O vacancies. This contradicts the standard picture of charge localization which assumes that each of the two excess electrons localizes on one of the cerium ions nearest to the vacancy. This standard picture is assumed in most calculations based on density functional theory (DFT). Based on the known crystal structure of Pr6O11, we also predict that the charge in Ce6O11 will be found in the second coordination shell of the O vacancy. We also extend the analysis to the Magnéli phases of titanium and vanadium oxides (M(n)O(2n₋1), where M = Ti, V) and consider the problem of metal-insulator transitions (MIT) in these oxides. We found that the bond valence analysis may provide a useful predictive tool in structures where the MIT is accompanied by significant changes in the metal-oxygen bond lengths. Although this review focuses mainly on bulk cerium oxides with some extension to the Magnéli phases of titanium and vanadium, our approach to characterizing electronic properties of oxygen vacancies and the physical insights gained should also be relevant to surface defects and to other rare earth and transition metal oxides.