Excited Vibronic States Research Articles

(Invited) Computational Models of Proton-Coupled Electron Transfer in Energy Conversion and Storage

Proton-coupled electron transfer (PCET) is a fundamental class of chemical reactions that is central to many electrochemical energy conversion and storage processes. In this talk, I will discuss recent theoretical model development for interfacial and intermolecular PCET reactions. First, I will describe a rate constant model for the reaction between reduced anatase TiO2 surfaces and a 4-MeO-TEMPO nitroxyl radical. The rate constants are modeled using vibronically nonadiabatic PCET theory, where the transferring proton and electron are treated quantum mechanically. Hybrid functional periodic density functional theory (DFT) calculations are used to derive model parameters such as driving forces, reorganization energies, and proton vibrational wavefunctions. This modeling strategy highlights the role of inner-sphere bond reorganization and excited vibronic states on the PCET reactivity of metal oxides. Next, I will show how similar computational approaches can be used to predict whether electrochemical PCET is likely to proceed through concerted electron–proton transfer or sequential electron transfer and proton transfer steps. Using PCET reactions between redox-active quinones and functionalized imidazole bases as an example, mechanistic predictions are informed by DFT-calculated potential energy surfaces. These PCET modeling strategies have broad applicability to kinetic studies of electrochemical energy conversion and storage processes.

Proton-Coupled Electron Transfer upon Oxidation of Tyrosine in a De Novo Protein: Analysis of Proton Acceptor Candidates.

Redox-active residues, such as tyrosine and tryptophan, play important roles in a wide range of biological processes. The α3Y de novo protein, which is composed of three α helices and a tyrosine residue Y32, provides a platform for investigating the redox properties of tyrosine in a well-defined protein environment. Herein, the proton-coupled electron transfer (PCET) reaction that occurs upon oxidation of tyrosine in this model protein by a ruthenium photosensitizer is studied by using a vibronically nonadiabatic PCET theory that includes hydrogen tunneling and excited vibronic states. The input quantities to the analytical nonadiabatic rate constant expression, such as the diabatic proton potential energy curves and associated proton vibrational wave functions, reorganization energy, and proton donor-acceptor distribution functions, are obtained from density functional theory calculations on model systems and molecular dynamics simulations of the solvated α3Y protein. Two possible proton acceptors, namely, water or a glutamate residue in the protein scaffold, are explored. The PCET rate constant is greater when glutamate is the proton acceptor, mainly due to the more favorable driving force and shorter equilibrium proton donor-acceptor distance, although contributions from excited vibronic states mitigate these effects. Nevertheless, water could be the dominant proton acceptor if its equilibrium constant associated with hydrogen bond formation is significantly greater than that for glutamate. Although these calculations do not definitively identify the proton acceptor for this PCET reaction, they elucidate the conditions under which each proton acceptor can be favored. These insights have implications for tyrosine-based PCET in a wide variety of biochemical processes.

Theory for proton-coupled energy transfer.

In the recently discovered proton-coupled energy transfer (PCEnT) mechanism, the transfer of electronic excitation energy between donor and acceptor chromophores is coupled to a proton transfer reaction. Herein, we develop a general theory for PCEnT and derive an analytical expression for the nonadiabatic PCEnT rate constant. This theory treats the transferring hydrogen nucleus quantum mechanically and describes the PCEnT process in terms of nonadiabatic transitions between reactant and product electron-proton vibronic states. The rate constant is expressed as a summation over these vibronic states, and the contribution of each pair of vibronic states depends on the square of the vibronic coupling as well as the spectral convolution integral, which can be viewed as a generalization of the Förster-type spectral overlap integral for vibronic rather than electronic states. The convolution integral also accounts for the common vibrational modes shared by the donor and acceptor chromophores for intramolecular PCEnT. We apply this theory to model systems to investigate the key features of PCEnT processes. The excited vibronic states can contribute significantly to the total PCEnT rate constant, and the common modes can either slow down or speed up the process. Because the pairs of vibronic states that contribute the most to the PCEnT rate constant may correspond to spectroscopically dark states, PCEnT could occur even when there is no apparent overlap between the donor emission and acceptor absorption spectra. This theory will assist in the interpretation of experimental data and will guide the design of additional PCEnT systems.

(Keynote) Electrochemical Proton-Coupled Electron Transfer Theory

Proton-coupled electron transfer (PCET) plays a vital role in a wide range of electrochemical processes. This talk will describe theoretical and computational methods that have been developed to study electrochemical PCET and a variety of applications to molecular and heterogeneous electrocatalysis. My group has formulated a general PCET theory that includes the quantum mechanical effects of the electrons and transferring protons, as well as the motions of the donor-acceptor modes and solvent or protein environment. This PCET theory enables the calculation of rate constants and kinetic isotope effects for comparison to experiment. Our extension of this theory to electrochemical PCET incorporates the electronic structure of the electrode and the interfacial electric fields arising from the electrical double layer. Theoretical formulations for both homogeneous and heterogeneous electrochemical PCET provide analytical expressions for the rate constants and current densities as functions of applied potential. This theory has been applied to proton discharge on metal electrodes, as well as PCET at metal oxides and graphite-conjugated catalysts. These applications highlight the importance of using a theory that quantizes the transferring proton and includes the effects of hydrogen tunneling and excited electron-proton vibronic states. The insights from these theoretical studies are useful for the design of electrocatalytic systems to control the movement and coupling of electrons and protons for energy conversion processes.References Venkataraman, A. V. Soudackov, and S. Hammes-Schiffer, Theoretical formulation of nonadiabatic electrochemical proton-coupled electron transfer at metal-solution interfaces, J. Phys. Chem. C 112, 12386-12397 (2008).K. Goldsmith, Y. C. Lam, A V. Soudackov, and S. Hammes-Schiffer, Proton discharge on a gold electrode from triethylammonium in acetonitrile: Theoretical modeling of potential-dependent kinetic isotope effects, J. Am. Chem. Soc. 141, 1084-1090 (2019).C. Lam, A. V. Soudackov, and S. Hammes-Schiffer, Kinetics of proton discharge on metal electrodes: Effects of vibrational nonadiabaticity and solvent dynamics, J. Phys. Chem. Lett. 10, 5312-5217 (2019).E. Warburton, P. Hutchison, M. N. Jackson, M. L. Pegis, Y. Surendranath, and S. Hammes-Schiffer, “Interfacial field-driven proton-coupled electron transfer at graphite-conjugated organic acids,” J. Am. Chem. Soc. 142, 20855-20864 (2020).E. Warburton, A. V. Soudackov, and S. Hammes-Schiffer, Theoretical modeling of electrochemical proton-coupled electron transfer, Chem. Rev. 122, 10599-10650 (2022).

Illuminating Ligand Field Contributions to Molecular Qubit Spin Relaxation via T1 Anisotropy.

Electron spin relaxation in paramagnetic transition metal complexes constitutes a key limitation on the growth of molecular quantum information science. However, there exist very few experimental observables for probing spin relaxation mechanisms, leading to a proliferation of inconsistent theoretical models. Here we demonstrate that spin relaxation anisotropy in pulsed electron paramagnetic resonance is a powerful spectroscopic probe for molecular spin dynamics across a library of highly coherent Cu(II) and V(IV) complexes. Neither the static spin Hamiltonian anisotropy nor contemporary computational models of spin relaxation are able to account for the experimental T1 anisotropy. Through analysis of the spin-orbit coupled wave functions, we derive an analytical theory for the T1 anisotropy that accurately reproduces the average experimental anisotropy of 2.5. Furthermore, compound-by-compound deviations from the average anisotropy provide a promising approach for observing specific ligand field and vibronic excited state coupling effects on spin relaxation. Finally, we present a simple density functional theory workflow for computationally predicting T1 anisotropy. Analysis of spin relaxation anisotropy leads to deeper fundamental understanding of spin-phonon coupling and relaxation mechanisms, promising to complement temperature-dependent relaxation rates as a key metric for understanding molecular spin qubits.

Reorganization Energies for Interfacial Proton-Coupled Electron Transfer to a Water Oxidation Catalyst.

The reorganization energy (λ) for interfacial electron transfer (ET) and proton-coupled ET (PCET) from a conductive metal oxide (In2O3:Sn, ITO) to a surface-bound water oxidation catalyst was extracted from kinetic data measured as a function of the thermodynamic driving force. Visible light excitation resulted in rapid excited-state injection (kinj > 108 s-1) to the ITO, which photo-initiated the two interfacial reactions of interest. The rate constants for both reactions increased with the driving force, -ΔG°, to a saturating limit, kmax, with rate constants consistently larger for ET than for PCET. Marcus-Gerischer analysis of the kinetic data provided the reorganization energy for interfacial PCET (0.90 ± 0.02 eV) and ET (0.40 ± 0.02 eV), respectively. The magnitude of kmax for PCET was found to decrease with pH, behavior that was absent for ET. Both the decrease in kmax and the larger reorganization energy for an unwanted competing PCET reaction from the ITO to the oxidized catalyst showcases a significant kinetic advantage for driving solar water oxidation at high pH. Computational analysis revealed a larger inner-sphere reorganization energy contribution for PCET than for ET arising from a more significant change in the Ru-O bond length for the PCET reaction. Extending the Marcus-Gerischer theory to PCET by including the excited electron-proton vibronic states and the proton donor-acceptor motion provided an apparent reorganization energy of 1.01 eV. This study demonstrates that the Marcus-Gerischer theory initially developed for ET can be reliably extended to PCET for quantifying and interpreting reorganization energies observed experimentally.

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.

Theoretical Modeling of Electrochemical Proton-Coupled Electron Transfer.

Proton-coupled electron transfer (PCET) plays an essential role in a wide range of electrocatalytic processes. A vast array of theoretical and computational methods have been developed to study electrochemical PCET. These methods can be used to calculate redox potentials and pKa values for molecular electrocatalysts, proton-coupled redox potentials and bond dissociation free energies for PCET at metal and semiconductor interfaces, and reorganization energies associated with electrochemical PCET. Periodic density functional theory can also be used to compute PCET activation energies and perform molecular dynamics simulations of electrochemical interfaces. Various approaches for maintaining a constant electrode potential in electronic structure calculations and modeling complex interactions in the electric double layer (EDL) have been developed. Theoretical formulations for both homogeneous and heterogeneous electrochemical PCET spanning the adiabatic, nonadiabatic, and solvent-controlled regimes have been developed and provide analytical expressions for the rate constants and current densities as functions of applied potential. The quantum mechanical treatment of the proton and inclusion of excited vibronic states have been shown to be critical for describing experimental data, such as Tafel slopes and potential-dependent kinetic isotope effects. The calculated rate constants can be used as input to microkinetic models and voltammogram simulations to elucidate complex electrocatalytic processes.

Nuclear-electronic orbital methods: Foundations and prospects.

The incorporation of nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations and molecular dynamics simulations is a longstanding challenge. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons with wave function and density functional theory methods. This approach inherently includes nuclear delocalization and zero-point energy in molecular energy calculations, geometry optimizations, reaction paths, and dynamics. It can also provide accurate descriptions of excited electronic, vibrational, and vibronic states as well as nuclear tunneling and nonadiabatic dynamics. Nonequilibrium nuclear-electronic dynamics simulations beyond the Born-Oppenheimer approximation can be used to investigate a wide range of excited state processes. This Perspective provides an overview of the foundational NEO methods and enumerates the prospects for using these methods as building blocks for future developments. The conceptual simplicity and computational efficiency of the NEO approach will enhance its accessibility and applicability to diverse chemical and biological systems.

Theoretical Study of Dynamic Stark-Induced π-Electron Rotations in Low-Symmetry Aromatic Ring Molecules beyond the Frozen Nuclear Approximation.

The effects of vibrational motions on dynamic Stark-induced π-electron rotations in a low-symmetry aromatic ring molecule are theoretically studied in the adiabatic approximation. We adopt a simplified three-electronic state model with a few vibronic states. A pair of the lowest vibronic states in two electronic excited states is set degenerate by irradiation of two linearly polarized UV lasers. The resultant degenerate state is named the dynamic Stark-induced degenerate vibronic state (DSIDVS). The laser parameters (intensities and central frequencies) are determined under the conditions of DSIDVS formation. The aromatic ring molecules of interest are supposed to belong to the weak coupling case. The analytical expressions for the DSIDVS and coherent angular momentum LZ(t) are derived in the displaced harmonic oscillator (DHO) model. Two horizontal potential displacements (δα, δβ) between the two electronic excited states (α and β) and the ground state are the parameters in the DHO model. The LZ(t) calculated with δα = δβ is characterized by a regular sequence of the angular momentum pulses with a positive (or negative) constant. For a more general case with δα ≠ δβ, the regular sequence is broken down because of the contribution of the first excited vibronic state in each electronic state to LZ(t).

Influence of Vibronic Coupling on Ultrafast Singlet Fission in a Linear Terrylenediimide Dimer.

Singlet fission (SF) is a photophysical process capable of boosting the efficiency of solar cells. Recent experimental investigations into the mechanism of SF provide evidence for coherent mixing between the singlet, triplet, and charge transfer basis states. Up until now, this interpretation has largely focused on electronic interactions; however, nuclear motions resulting in vibronic coupling have been suggested to support rapid and efficient SF in organic chromophore assemblies. Further information about the complex interactions between vibronic excited states is needed to understand the potential role of this coupling in SF. Here, we report mixed singlet and correlated triplet pair states giving rise to sub-50 fs SF in a terrylene-3,4:11,12-bis(dicarboximide) (TDI) dimer in which the two TDI molecules are covalently linked by a direct N-N connection at one of their imide positions, leading to a linear dimer with perpendicular TDI π systems. We observe the transfer of low-frequency coherent wavepackets between the initial predominantly singlet states to the product triplet-dominated states. This implies a non-negligible dependence of SF on nonadiabatic coupling in this dimer. We interpret our experimental results in the framework of a modified Holstein Hamiltonian, which predicts that vibronic interactions between low-frequency singlet modes and high-frequency correlated triplet pair motions lead to mixing of the pure basis states. These results highlight how nonadiabatic mixing can shape the complex potential energy landscape underlying ultrafast SF.

Nuclear-Electronic Orbital Multistate Density Functional Theory.

Hydrogen tunneling is essential for a wide range of chemical and biological processes. The description of hydrogen tunneling with multicomponent quantum chemistry approaches, where the transferring hydrogen nucleus is treated on the same level as the electrons, is challenging due to the importance of both static and dynamical electron-proton correlation. Herein the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is presented as a strategy to include both types of correlation. In this approach, two localized nuclear-electronic wave functions obtained with the NEO-DFT method are combined with a nonorthogonal configurational interaction approach to produce bilobal, delocalized ground and excited vibronic states. By including a correction function, the NEO-MSDFT approach can produce quantitatively accurate hydrogen tunneling splittings for fixed geometries of systems such as malonaldehyde and acetoacetaldehyde. This approach is computationally efficient and can be combined with methods such as vibronic coupling theory to describe tunneling dynamics and to compute vibronic couplings in many types of systems.

Magnetic Resonance Study of p‐Type 3C SiC Microparticles

The silicon carbide (SiC) 3C polytype microparticles with p‐type conductivity are investigated by X‐band electron paramagnetic resonance (EPR) technique in a wide temperature interval. Morphological and Raman studies show that the mean particle diameter is ≈8 μm, and the SiC polytype is purely 3C. At T = 10 K, the EPR spectrum of the shallow boron acceptor substituting cubic site (BK), with C3V symmetry and anisotropic g‐ and A‐tensors resulting from static Jahn–Teller effect, is observed. At T > 30 K, a continuous transition from static to dynamic Jahn–Teller effect for BK is detected. The temperature‐dependent BK principal g‐values are explained by the transitions between the ground and excited vibronic states of BK. At T = 130–150 K, the BK center is characterized by Td symmetry and isotropic g‐ and A‐tensors. At T = 60–190 K, an additional single EPR Lorentzian line with short spin relaxation times and isotropic g‐value close to BK average g‐value is observed. It was explained by averaging the anisotropic spectrum caused by rapid transitions between the boron ground and excited vibronic states split by Jahn–Teller potential. In addition, the EPR signal from carbon vacancy with characteristic superhyperfine structure is observed at T = 10–298 K.

Frequency and Time Domain Nuclear-Electronic Orbital Equation-of-Motion Coupled Cluster Methods: Combination Bands and Electronic-Protonic Double Excitations.

The accurate description of excited vibronic states is important for modeling a wide range of photoinduced processes. The nuclear-electronic orbital (NEO) approach, which treats specified protons on the same level as the electrons, can describe excited electronic-protonic states. Herein the multicomponent equation-of-motion coupled cluster with singles and doubles (NEO-EOM-CCSD) method and its time-domain counterpart, TD-NEO-EOM-CCSD, are developed and implemented. The application of these methods to the HCN molecule highlights their capabilities. These methods predict qualitatively reasonable energies and intensities for a combination band corresponding to simultaneous excitation of two vibrational modes, as well as an overtone. These methods also describe states with double excitation character, such as excited electronic-protonic states corresponding to the simultaneous excitation of an electron and a proton. The ability of the NEO-EOM-CCSD method and its time-dependent counterpart to describe combination bands, overtones, and double excitations will enable a wide range of photochemical applications.

Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear-Electronic Orbital Method.

In multicomponent quantum chemistry, more than one type of particle is treated quantum mechanically with either density functional theory or wave function based methods. In particular, the nuclear-electronic orbital (NEO) approach treats specified nuclei, typically hydrogen nuclei, on the same level as the electrons. This approach enables the incorporation of nuclear quantum effects, such as nuclear delocalization, anharmonicity, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects directly into quantum chemistry calculations. Such effects impact optimized geometries, molecular vibrational frequencies, reaction paths, isotope effects, and dynamical simulations. Multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) achieve an optimal balance between computational efficiency and accuracy for computing ground and excited state properties, respectively. Multicomponent wave function based methods, such as the coupled cluster singles and doubles (NEO-CCSD) method for ground states and the equation-of-motion counterpart (NEO-EOM-CCSD) for excited states, attain similar accuracy without requiring any parametrization and can be systematically improved but are more computationally expensive. Variants of the orbital-optimized perturbation theory (NEO-OOMP2) method achieve nearly the accuracy of NEO-CCSD for ground states with significantly lower computational cost. Additional approaches for computing excited electronic, vibrational, and vibronic states include the delta self-consistent field (NEO-ΔSCF), complete active space SCF (NEO-CASSCF), and nonorthogonal configuration interaction methods. Multireference methods are particularly important for describing hydrogen tunneling processes. Other types of multicomponent systems, such as those containing electrons and positrons, have also been studied within the NEO framework. The NEO approach allows the incorporation of nuclear quantum effects and non-Born-Oppenheimer effects for specified nuclei into quantum chemistry calculations in an accessible and computationally efficient manner.

Theoretical Treatment of Ultrafast Decay of Excited Vibronic States in the Improved Adiabatic Approximation

Theoretical Treatment of Ultrafast Decay of Excited Vibronic States in the Improved Adiabatic Approximation

Rotational assignment of the 14NO3 high-resolution spectrum in the 15,100 cm−1 region

Rotational assignment of the high-resolution fluorescence excitation spectrum of the 14NO3 radical in the B∼2E′←X∼2A2′ transition 0–0 band region located around 15,100cm−1 was made with the help of the ground state combination differences and the observation of Zeeman effect. We successfully identified several rotational line series based on the newly obtained rotational assignments of 162 lines with |k″|⩽15 and J′⩽18.5 quantum numbers, and we estimated the effective molecular constants of the excited vibronic states.

A proposed new scheme for vibronically resolved time-dependent photoelectron spectroscopy: pump-repump-continuous wave-photoelectron spectroscopy (prp-cw-pes).

We propose a new scheme for time-resolved photoelectron spectroscopy denoted as pump-repump-continuous wave-photoelectron spectroscopy (prp-cw-pes). This scheme is comprised of two femtosecond laser (pump) pulses under cw illumination (probe). By changing the time-delay between pump and repump lasers one can manipulate the populations of vibronic levels in electronic excited states. The cw laser acts on for a long time and establishes resonance between excited states and the continuum photo-ionized states. Sharp spectra can be obtained from the resonance condition [formula: see text]. The intensities in the spectra are sensitive to the time-delay between the pump-repump pulses, but only depend on the populations of excited states, not the phase relations (coherence). As a result, each time-delayed snapshot spectrum is a weighted sum of so-called fingerprints, where a fingerprint is the vibrationally resolved photoelectron spectrum for a single vibronic excited state. The latter information can potentially be simulated reliably using vibronic models and wave packet propagation methods. In the easiest application of the experiment, different time-delays produce different spectra, for a single molecular system. This wealth of experimental data can be fitted to an, ideally small, set of theoretical fingerprints by adjusting the populations as fitting parameters. This technique might be able to distinguish between closely related molecular species. Adopting a different viewpoint, the proposed scheme can also be employed to monitor the time-dependent dynamics by changing the phase relationship between the pump and repump laser that can be viewed as a "control mechanism" employed in wave packet interferometry. Simplifications arise as the change in the spectra is due to the changing populations, not because of the coherence. In this paper, we outline the ideas behind the scheme and illustrate the ideas theoretically using simple model systems.

Revealing the radiative and non-radiative relaxation rates of the fluorescent dye Atto488 in a λ/2 Fabry-Pérot-resonator by spectral and time resolved measurements.

Using a Fabry-Pérot-microresonator with controllable cavity lengths in the λ/2-regime, we show the controlled modification of the vibronic relaxation dynamics of a fluorescent dye molecule in the spectral and time domain. By altering the photonic mode density around the fluorophores we are able to shape the fluorescence spectrum and enhance specifically the probability of the radiative transitions from the electronic excited state to distinct vibronic excited states of the electronic ground state. Analysis and correlation of the spectral and time resolved measurements by a theoretical model and a global fitting procedure allows us to reveal quantitatively the spectrally distributed radiative and non-radiative relaxation dynamics of the respective dye molecule under ambient conditions at the ensemble level.

Vibronic resonances facilitate excited-state coherence in light-harvesting proteins at room temperature.

Until recently it was believed that photosynthesis, a fundamental process for life on earth, could be fully understood with semiclassical models. However, puzzling quantum phenomena have been observed in several photosynthetic pigment-protein complexes, prompting questions regarding the nature and role of these effects. Recent attention has focused on discrete vibrational modes that are resonant or quasi-resonant with excitonic energy splittings and strongly coupled to these excitonic states. Here we unambiguously identify excited state coherent superpositions in photosynthetic light-harvesting complexes using a new experimental approach. Decoherence on the time scale of the excited state lifetime allows low energy (56 cm(-1)) oscillations on the signal intensity to be observed. In conjunction with an appropriate model, these oscillations provide clear and direct experimental evidence that the persistent coherences observed originate from quantum superpositions among vibronic excited states.