Analytical gradients for nuclear-electronic orbital multistate density functional theory: Geometry optimizations and reaction paths.

Hydrogen tunneling plays a critical role in many biologically and chemically important processes. The nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed to describe hydrogen transfer systems. In this approach, the transferring proton is treated quantum mechanically on the same level as the electrons within multicomponent DFT, and a nonorthogonal configuration interaction scheme is used to produce delocalized vibronic states from localized vibronic states. The NEO-MSDFT method has been shown to provide accurate hydrogen tunneling splittings for fixed molecular systems. Herein, the NEO-MSDFT analytical gradients for both ground and excited vibronic states are derived and implemented. The analytical gradients and semi-numerical Hessians are used to optimize and characterize equilibrium and transition state geometries and to generate minimum energy paths (MEPs), for proton transfer in the deprotonated acetylene dimer and malonaldehyde. The barriers along the resulting MEPs are lower when the transferring proton is quantized because the NEO-MSDFT method inherently includes the zero-point energy of the transferring proton. Analysis of the proton densities along the MEPs illustrates that the proton density can exhibit symmetric or asymmetric bilobal character associated with symmetric or slightly asymmetric double-well potential energy surfaces and hydrogen tunneling. Analysis of the contributions to the intrinsic reaction coordinate reveals that changes in the C-O bond lengths drive proton transfer in malonaldehyde. This work provides the foundation for future reaction path studies and direct nonadiabatic dynamics simulations of a wide range of hydrogen transfer reactions.

Analytical Gradients for Nuclear-Electronic Orbital Time-Dependent Density Functional Theory: Excited-State Geometry Optimizations and Adiabatic Excitation Energies.

The computational investigation of photochemical processes often entails the calculation of excited-state geometries, energies, and energy gradients. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons, thereby including the associated nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations. The multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) methods allow efficient calculations of ground and excited states, respectively. Herein, the analytical gradients are derived and implemented for the NEO-TDDFT method and the associated Tamm-Dancoff approximation (NEO-TDA). The programmable equations for these analytical gradients as well as the NEO-DFT analytical Hessian are provided. The NEO approach includes the anharmonic zero-point energy (ZPE) and density delocalization associated with the quantum protons as well as vibronic mixing in geometry optimizations and energy calculations of ground and excited states. The harmonic ZPE associated with the other nuclei can be computed via the NEO Hessian. This approach is used to compute the 0-0 adiabatic excitation energies for a set of nine small molecules with all protons quantized, exhibiting slight improvement over the conventional electronic approach. Geometry optimizations of two excited-state intramolecular proton-transfer systems, [2,2'-bipyridyl]-3-ol and [2,2'-bipyridyl]-3,3'-diol, are performed with one and two quantized protons, respectively. The NEO calculations for these systems produce electronically excited-state geometries with stronger intramolecular hydrogen bonds and similar relative stabilities compared to conventional electronic methods. This work provides the foundation for nonadiabatic dynamics simulations of fundamental processes such as photoinduced proton transfer and proton-coupled electron transfer.

Transition states, reaction paths, and thermochemistry using the nuclear-electronic orbital analytic Hessian.

The nuclear-electronic orbital (NEO) method is a multicomponent quantum chemistry theory that describes electronic and nuclear quantum effects simultaneously while avoiding the Born-Oppenheimer approximation for certain nuclei. Typically specified hydrogen nuclei are treated quantum mechanically at the same level as the electrons, and the NEO potential energy surface depends on the classical nuclear coordinates. This approach includes nuclear quantum effects such as zero-point energy and nuclear delocalization directly into the potential energy surface. An extended NEO potential energy surface depending on the expectation values of the quantum nuclei incorporates coupling between the quantum and classical nuclei. Herein, theoretical methodology is developed to optimize and characterize stationary points on the standard or extended NEO potential energy surface, to generate the NEO minimum energy path from a transition state down to the corresponding reactant and product, and to compute thermochemical properties. For this purpose, the analytic coordinate Hessian is developed and implemented at the NEO Hartree-Fock level of theory. These NEO Hessians are used to study the SN2 reaction of ClCH3Cl- and the hydride transfer of C4H9 +. For each system, analysis of the single imaginary mode at the transition state and the intrinsic reaction coordinate along the minimum energy path identifies the dominant nuclear motions driving the chemical reaction. Visualization of the electronic and protonic orbitals along the minimum energy path illustrates the coupled electronic and protonic motions beyond the Born-Oppenheimer approximation. This work provides the foundation for applying the NEO approach at various correlated levels of theory to a wide range of chemical reactions.

Molecular Vibrational Frequencies with Multiple Quantum Protons within the Nuclear-Electronic Orbital Framework.

The nuclear-electronic orbital (NEO) approach treats all electrons and specified nuclei, typically protons, on the same quantum mechanical level. Proton vibrational excitations can be calculated using multicomponent time-dependent density functional theory (NEO-TDDFT) for fixed classical nuclei. Recently the NEO-DFT(V) approach was developed to enable the calculation of molecular vibrational frequencies for modes composed of both classical and quantum nuclei. This approach uses input from NEO-TDDFT to construct an extended NEO Hessian that depends on the expectation values of the quantum protons as well as the classical nuclear coordinates. Herein strategies are devised for extending these approaches to molecules with multiple quantum protons in a self-contained, effective, and computationally practical manner. The NEO-TDDFT method is shown to describe vibrational excitations corresponding to collective nuclear motions, such as linear combinations of proton vibrational excitations. The NEO-DFT(V) approach is shown to incorporate the most significant anharmonic effects in the molecular vibrations, particularly for the hydrogen stretching modes. These theoretical strategies pave the way for a wide range of multicomponent quantum chemistry applications.

Diagonal Born-Oppenheimer Corrections within the Nuclear-Electronic Orbital Framework.

The nuclear-electronic orbital (NEO) method treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons. This approach invokes the Born-Oppenheimer separation between the quantum and classical nuclei, as well as the conventional separation between the electrons and classical nuclei. To test the validity of this additional adiabatic approximation, herein the diagonal Born-Oppenheimer correction (DBOC) within the NEO framework is derived, analyzed, and calculated numerically for a set of eight molecules. Inclusion of the NEO DBOC is found to change the equilibrium bond lengths by only ∼10-4 Šand the heavy atom vibrational stretching frequencies by ∼1-2 cm-1 per quantum proton bonded to an atom participating in the vibrational mode. These results imply that the DBOC does not significantly impact molecular properties computed with the NEO approach, although it can be included when necessary. Understanding the physical characteristics and quantitative contributions of the DBOC has broad implications for applications of multicomponent density functional theory and wave function methods.

Brønsted Acid Scaling Relationships Enable Control Over Product Selectivity from O2 Reduction with a Mononuclear Cobalt Porphyrin Catalyst.

The selective reduction of O2, typically with the goal of forming H2O, represents a long-standing challenge in the field of catalysis. Macrocyclic transition-metal complexes, and cobalt porphyrins in particular, have been the focus of extensive study as catalysts for this reaction. Here, we show that the mononuclear Co-tetraarylporphyrin complex, Co(porOMe) (porOMe = meso-tetra(4-methoxyphenyl)porphyrin), catalyzes either 2e-/2H+ or 4e-/4H+ reduction of O2 with high selectivity simply by changing the identity of the Brønsted acid in dimethylformamide (DMF). The thermodynamic potentials for O2 reduction to H2O2 or H2O in DMF are determined and exhibit a Nernstian dependence on the acid pK a, while the CoIII/II redox potential is independent of the acid pK a. The reaction product, H2O or H2O2, is defined by the relationship between the thermodynamic potential for O2 reduction to H2O2 and the CoIII/II redox potential: selective H2O2 formation is observed when the CoIII/II potential is below the O2/H2O2 potential, while H2O formation is observed when the CoIII/II potential is above the O2/H2O2 potential. Mechanistic studies reveal that the reactions generating H2O2 and H2O exhibit different rate laws and catalyst resting states, and these differences are manifested as different slopes in linear free energy correlations between the log(rate) versus pK a and log(rate) versus effective overpotential for the reactions. This work shows how scaling relationships may be used to control product selectivity, and it provides a mechanistic basis for the pursuit of molecular catalysts that achieve low overpotential reduction of O2 to H2O.

Molecular Vibrational Frequencies within the Nuclear-Electronic Orbital Framework.

A significant challenge for multicomponent quantum chemistry methods is the calculation of vibrational frequencies for comparison to experiment. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically key protons, quantum mechanically. The Born-Oppenheimer separation between the quantum and classical nuclei prevents the direct calculation of vibrational frequencies corresponding to modes composed of both types of nuclei. Herein an effective strategy for calculating the vibrational frequencies of the entire molecule within the NEO framework is devised and implemented. This strategy requires diagonalization of an extended NEO Hessian that depends on the expectation values of the quantum nuclei as well as the coordinates of the classical nuclei and is constructed with input from multicomponent time-dependent density functional theory (NEO-TDDFT). Application of this NEO-DFT(V) approach to molecular systems illustrates that it accurately incorporates the most significant anharmonic effects. This general theoretical formulation opens up a broad spectrum of new directions for multicomponent quantum chemistry.

Alternative forms and transferability of electron-proton correlation functionals in nuclear-electronic orbital density functional theory.

Multicomponent density functional theory (DFT) allows the consistent quantum mechanical treatment of both electrons and nuclei. Recently the epc17 electron-proton correlation functional was derived using a multicomponent extension of the Colle-Salvetti formalism and was implemented within the nuclear-electronic orbital (NEO) framework for treating electrons and specified protons quantum mechanically. Herein another electron-proton correlation functional, denoted epc18, is derived using a different form for the functional parameter interpreted as representing the correlation length for electron-proton interactions. The epc18 functional is shown to perform similarly to the epc17 functional for predicting three-dimensional proton densities and proton affinities. Both functionals are shown to be transferable for use with a series of diverse electronic exchange-correlation functionals, indicating that any reasonable electronic exchange-correlation functional may be used in tandem with the epc17 and epc18 electron-proton correlation functionals. Understanding the impact of different forms of the electron-proton correlation functional, as well as the interplay between electron-proton and electron-electron correlation, is critical for the general applicability of NEO-DFT.

Kinetic and Mechanistic Characterization of Low-Overpotential, H2O2-Selective Reduction of O2 Catalyzed by N2O2-Ligated Cobalt Complexes.

A soluble, bis-ketiminate-ligated Co complex [Co(N2O2)] was recently shown to catalyze selective reduction of O2 to H2O2 with an overpotential as low as 90 mV. Here we report experimental and computational mechanistic studies of the Co(N2O2)-catalyzed O2 reduction reaction (ORR) with decamethylferrocene (Fc*) as the reductant in the presence of AcOH in MeOH. Analysis of the Co/O2 binding stoichiometry and kinetic studies support an O2 reduction pathway involving a mononuclear cobalt species. The catalytic rate exhibits a first-order kinetic dependence on [Co(N2O2)] and [AcOH], but no dependence on [Fc*] or [O2]. Differential pulse voltammetry and computational studies support CoIII-hydroperoxide as the catalyst resting state and protonation of this species as the rate-limiting step of the catalytic reaction. These results contrast previous mechanisms proposed for other Co-catalyzed ORR systems, which commonly feature rate-limiting protonation of a CoIII-superoxide adduct earlier in the catalytic cycle. Computational studies show that protonation is strongly favored at the proximal oxygen of the CoIII(OOH) species, accounting for the high selectivity for formation of hydrogen peroxide. Further analysis shows that a weak dependence of the ORR rate on the p Ka values of the protonated CoIII(OOH) species across a series of Co(N2O2) catalysts provides a rationale for the unusually low overpotential observed for O2 reduction to H2O2.