Nuclear-electronic Orbital Research Articles

Nuclear-Electronic Orbital Time-Dependent Configuration Interaction Method.

Combining real-time electronic structure with the nuclear-electronic orbital (NEO) method has enabled the simulation of complex nonadiabatic chemical processes. However, accurate descriptions of hydrogen tunneling and double excitations require multiconfigurational treatments. Herein, we develop and implement the real-time NEO time-dependent configuration interaction (NEO-TDCI) approach. Comparison to NEO-full CI calculations of absorption spectra for a molecular system shows that the NEO-TDCI approach can accurately capture the tunneling splitting associated with the electronic ground state as well as vibronic progressions corresponding to double electron-proton excitations associated with excited electronic states. Both of these features are absent from spectra obtained with single reference real-time NEO methods. Our simulations of hydrogen tunneling dynamics illustrate the oscillation of the proton density from one side to the other via a delocalized, bilobal proton wave function. These results indicate that the NEO-TDCI approach is highly suitable for studying hydrogen tunneling and other inherently multiconfigurational systems.

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 Many-Body Perspective of Nuclear Quantum Effects in Aqueous Clusters.

Nuclear quantum effects play an important role in the structure and thermodynamics of aqueous systems. By performing a many-body expansion with nuclear-electronic orbital (NEO) theory, we show that proton quantization can give rise to significant energetic contributions for many-body interactions spanning several molecules in single-point energy calculations of water clusters. Although zero-point motion produces a large increase in energy at the one-body level, nuclear quantum effects serve to stabilize higher-order molecular interactions. These results are significant because they demonstrate that nuclear quantum effects play a nontrivial role in many-body interactions of aqueous systems. Our approach also provides a pathway for incorporating nuclear quantum effects into water potential energy surfaces. The NEO approach is advantageous for many-body expansion analyses because it includes nuclear quantum effects directly in the energies.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals.

Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.

Assessing Implicit and Explicit Polarizable Solvation Models for Nuclear-Electronic Orbital Systems: Quantum Proton Polarization and Solvation Energetics.

Accurate simulations of many chemical processes require the inclusion of both nuclear quantum effects and a solvent environment. The nuclear-electronic orbital (NEO) approach, which treats electrons and select nuclei quantum mechanically on the same level, combined with a polarizable continuum model (PCM) for the solvent environment, addresses this challenge in a computationally practical manner. In this work, the NEO-PCM approach is extended beyond the IEF-PCM (integral equation formalism PCM) and C-PCM (conductor PCM) approaches to the SS(V)PE (surface and simulation of volume polarization for electrostatics) and ddCOSMO (domain decomposed conductor-like screening model) approaches. IEF-PCM, SS(V)PE, C-PCM, and ddCOSMO all exhibit similar solvation energies as well as comparable nuclear polarization within the NEO framework. The calculations show that the nuclear density does not leak out of the molecular cavity because it is much more localized than the electronic density. Finally, the polarization of quantized protons is analyzed in both continuum solvent and explicit solvent environments described by the polarizable MB-pol model, illustrating the impact of specific hydrogen-bonding interactions captured only by explicit solvation. These calculations highlight the relationship among solvation formalism, nuclear polarization, and energetics.

Multicomponent Cholesky Decomposition: Application to Nuclear-Electronic Orbital Theory.

The Cholesky decomposition technique is commonly used to reduce the memory requirement for storing two-particle repulsion integrals in quantum chemistry calculations that use atomic orbital bases. However, when quantum methods use multicomponent bases, such as nuclear-electronic orbitals, additional challenges are introduced due to asymmetric two-particle integrals. This work proposes several multicomponent Cholesky decomposition methods for calculations using nuclear-electronic orbital density functional theory. To analyze the errors in different Cholesky decomposition components, benchmark calculations using water clusters are carried out. The largest benchmark calculation is a water cluster (H2O)27 where all 54 protons are treated quantum mechanically. This study provides energetic and complexity analyses to demonstrate the accuracy and performance of the proposed multicomponent Cholesky decomposition method.

Nuclear-Electronic Orbital QM/MM Approach: Geometry Optimizations and Molecular Dynamics.

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods allow simulations of chemical reactions in atomistic solvent and heterogeneous environments such as proteins. Herein, the nuclear-electronic orbital (NEO) QM/MM approach is introduced to enable the quantization of specified nuclei, typically protons, in the QM region using a method such as NEO-density functional theory (NEO-DFT). This approach includes proton delocalization, polarization, anharmonicity, and zero-point energy in geometry optimizations and dynamics. Expressions for the energies and analytical gradients associated with the NEO-QM/MM method, as well as the previously developed polarizable continuum model (NEO-PCM), are provided. Geometry optimizations of small organic molecules hydrogen bonded to water in either dielectric continuum solvent or explicit atomistic solvent illustrate that aqueous solvation can strengthen hydrogen-bonding interactions for the systems studied, as indicated by shorter intermolecular distances at the hydrogen-bond interface. We then performed a real-time direct dynamics simulation of a phenol molecule in explicit water using the NEO-QM/MM method. These developments and initial examples provide the foundation for future studies of nuclear-electronic quantum dynamics in complex chemical and biological environments.

A quantum computing implementation of nuclearelectronic orbital (NEO) theory: Toward an exact pre-Born-Oppenheimer formulation of molecular quantum systems.

Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10-6 hartree and entanglement entropy to within 10-4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.

Electronic Born-Oppenheimer approximation in nuclear-electronic orbital dynamics.

Within the nuclear-electronic orbital (NEO) framework, the real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics. Herein, the electronic Born-Oppenheimer (BO) approximation within the NEO framework is presented. In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of-magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting. For the intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.

Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field.

In an ultrastrong magnetic field, with field strength B ≈ B0 = 2.35 × 105 T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.

Simultaneous Optimization of Nuclear-Electronic Orbitals.

Accurate modeling of important nuclear quantum effects, such as nuclear delocalization, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects, requires treatment of both nuclei and electrons quantum mechanically. The nuclear-electronic orbital (NEO) method provides an elegant framework to treat specified nuclei, typically protons, on the same level as the electrons. In conventional electronic structure theory, finding a converged ground state can be a computationally demanding task; converging NEO wavefunctions, due to their coupled electronic and nuclear nature, is even more demanding. Herein, we present an efficient simultaneous optimization method that uses the direct inversion in the iterative subspace method to simultaneously converge wavefunctions for both the electrons and quantum nuclei. Benchmark studies show that the simultaneous optimization method can significantly reduce the computational cost compared to the conventional stepwise method for optimizing NEO wavefunctions for multicomponent systems.

Triple electron-electron-proton excitations and second-order approximations in nuclear-electronic orbital coupled cluster methods.

The accurate description of nuclear quantum effects, such as zero-point energy, is important for modeling a wide range of chemical and biological processes. Within the nuclear-electronic orbital (NEO) approach, such effects are incorporated in a computationally efficient way by treating electrons and select nuclei, typically protons, quantum mechanically with molecular orbital techniques. Herein, we implement and test a NEO coupled cluster method that explicitly includes the triple electron-electron-proton excitations, where two electrons and one proton are excited simultaneously, using automatic differentiation Our calculations show that this NEO-CCSDTeep method provides highly accurate proton densities and proton affinities, outperforming any previously studied NEO method. These examples highlight the importance of the triple electron-electron-proton excitations for an accurate description of nuclear quantum effects. Additionally, we also implement and test the second-order approximate coupled cluster with singles and doubles (NEO-CC2) method as well as its scaled-opposite-spin (SOS) versions. The NEO-SOS'-CC2 method, which scales the electron-proton correlation energy as well as the opposite-spin and same-spin components of the electron-electron correlation energy, achieves nearly the same accuracy as the NEO-CCSDTeep method for the properties studied. Because of its low computational cost, this method will enable a wide range of chemical and photochemical applications for large molecular systems. This work sets the stage for a variety of developments and applications within the NEO framework.

Multicomponent Orbital-Optimized Perturbation Theory with Density Fitting: Anharmonic Zero-Point Energies in Protonated Water Clusters.

Nuclear quantum effects such as zero-point energy are important in a wide range of chemical and biological processes. The nuclear-electronic orbital (NEO) framework intrinsically includes such effects by treating electrons and specified nuclei quantum mechanically on the same level. Herein, we implement the NEO scaled-opposite-spin orbital-optimized second-order Møller-Plesset perturbation theory with electron-proton correlation scaling (NEO-SOS'-OOMP2) using density fitting. This efficient implementation allows applications to larger systems with multiple quantum protons. Both the NEO-SOS'-OOMP2 method and its counterpart without orbital optimization predict proton affinities to within experimental precision and relative energies of protonated water tetramer isomers in agreement with previous NEO coupled cluster calculations. Applications to protonated water hexamers and heptamers illustrate that anharmonicity is critical for computing accurate relative energies. The NEO-SOS'-OOMP2 approach captures anharmonic zero-point energies at any geometry in a computationally efficient manner and hence will be useful for investigating reaction paths and dynamics in chemical systems.

Nuclear-electronic orbital approach to quantization of protons in periodic electronic structure calculations.

The nuclear-electronic orbital (NEO) method is a well-established approach for treating nuclei quantum mechanically in molecular systems beyond the usual Born-Oppenheimer approximation. In this work, we present a strategy to implement the NEO method for periodic electronic structure calculations, particularly focused on multicomponent density functional theory (DFT). The NEO-DFT method is implemented in an all-electron electronic structure code, FHI-aims, using a combination of analytical and numerical integration techniques as well as a resolution of the identity scheme to enhance computational efficiency. After validating this implementation, proof-of-concept applications are presented to illustrate the effects of quantized protons on the physical properties of extended systems, such as two-dimensional materials and liquid-semiconductor interfaces. Specifically, periodic NEO-DFT calculations are performed for a trans-polyacetylene chain, a hydrogen boride sheet, and a titanium oxide-water interface. The zero-point energy effects of the protons as well as electron-proton correlation are shown to noticeably impact the density of states and band structures for these systems. These developments provide a foundation for the application of multicomponent DFT to a wide range of other extended condensed matter systems.

Semiclassical Real-Time Nuclear-Electronic Orbital Dynamics for Molecular Polaritons: Unified Theory of Electronic and Vibrational Strong Couplings.

Molecular polaritons have become an emerging platform for remotely controlling molecular properties through strong light-matter interactions. Herein, a semiclassical approach is developed for describing molecular polaritons by self-consistently propagating the real-time dynamics of classical cavity modes and a quantum molecular subsystem described by the nuclear-electronic orbital (NEO) method, where electrons and specified nuclei are treated quantum mechanically on the same level. This semiclassical real-time NEO approach provides a unified description of electronic and vibrational strong couplings and describes the impact of the cavity on coupled nuclear-electronic dynamics while including nuclear quantum effects. For a single o-hydroxybenzaldehyde molecule under electronic strong coupling, this approach shows that the cavity suppression of excited state intramolecular proton transfer is influenced not only by the polaritonic potential energy surface but also by the time scale of the chemical reaction. This work provides the foundation for exploring collective strong coupling in nuclear-electronic quantum dynamical systems within optical cavities.

Solvated Nuclear-Electronic Orbital Structure and Dynamics.

Nonadiabatic dynamical processes such as proton-coupled electron transfer and excited state intramolecular proton transfer have been the subject of much research. One of the promising theoretical methods to describe these processes is the nuclear-electronic orbital (NEO) approach. This approach inherently accounts for nuclear quantum effects within quantum chemistry calculations, and it has recently been extended to directly simulate nonadiabatic processes with the development of real-time NEO methods. These processes can also be significantly dependent on the surrounding chemical environment, however, and capturing the effects of the environment is often necessary for analyzing experimentally relevant systems. This work couples the NEO density functional theory and real-time time-dependent density functional theory approaches with solvation through the polarizable continuum model. The effects of this coupling are investigated for ground state properties, solvent-dependent vibrational frequencies, and direct excited state intramolecular proton transfer dynamics.

Direct Dynamics with Nuclear-Electronic Orbital Density Functional Theory.

Direct dynamics simulations of chemical reactions typically require the selection of a method for generating the potential energy surfaces and a method for the dynamical propagation of the nuclei on these surfaces. The nuclear-electronic orbital (NEO) framework avoids this Born-Oppenheimer separation by treating specified nuclei on the same level as the electrons with wave function methods or density functional theory (DFT). The NEO approach is particularly applicable to proton, hydride, and proton-coupled electron transfer reactions, where the transferring proton(s) and all electrons are treated quantum mechanically. In this manner, the zero-point energy, density delocalization, and anharmonicity of the transferring protons are inherently and efficiently included in the energies, optimized geometries, and dynamics.This Account describes how various NEO methods can be used for direct dynamics simulations on electron-proton vibronic surfaces. The strengths and limitations of these approaches are discussed, and illustrative examples are presented. The NEO-DFT method can be used to simulate chemical reactions on the ground state vibronic surface, as illustrated by the application to hydride transfer in C4H9+. The NEO multistate DFT (NEO-MSDFT) method is useful for simulating ground state reactions in which the proton density becomes bilobal during the dynamics, a characteristic of hydrogen tunneling, as illustrated by proton transfer in malonaldehyde. The NEO time-dependent DFT (NEO-TDDFT) method produces excited electronic, vibrational, and vibronic surfaces. The application of linear-response NEO-TDDFT to H2 and H3+, as well as the partially and fully deuterated counterparts, shows that this approach produces accurate fundamental vibrational excitation energies when all nuclei and all electrons are treated quantum mechanically. Moreover, when only specified nuclei are treated quantum mechanically, this approach can be used to optimize geometries on excited state vibronic surfaces, as illustrated by photoinduced single and double proton transfer systems, and to conduct adiabatic dynamics on these surfaces. The real-time NEO-TDDFT method provides an alternative approach for simulating nonequilibrium nuclear-electronic dynamics of such systems. These various NEO methods can be combined with nonadiabatic dynamics methods such as Ehrenfest and surface hopping dynamics to include the nonadiabatic effects between the quantum and classical subsystems. The real-time NEO-TDDFT Ehrenfest dynamics simulation of excited state intramolecular proton transfer in o-hydroxybenzaldehyde illustrates the power of this type of combined approach. The field of multicomponent quantum chemistry is in the early stages, and the methods discussed herein provide the foundation for a wide range of promising future directions to be explored. An appealing future direction is the expansion of the real-time NEO-TDDFT method to describe the dynamics of all nuclei and electrons on the same level. Direct dynamics simulations using NEO wave function methods such as equation-of-motion coupled cluster or multiconfigurational approaches are also attractive but computationally expensive options. The further development of NEO direct dynamics methods will enable the simulation of the nuclear-electronic dynamics for a vast array of chemical and biological processes that extend beyond the Born-Oppenheimer approximation.

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.

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.

Multicomponent Unitary Coupled Cluster and Equation-of-Motion for Quantum Computation.

The variational quantum eigensolver (VQE) algorithm combined with the unitary coupled cluster (UCC) ansatz has been developed for the quantum computation of molecular energies and wave functions within the Born-Oppenheimer approximation. Herein, this approach is extended to multicomponent systems to enable the quantum mechanical treatment of more than one type of particle, such as electrons and positrons or electrons and nuclei, without invoking the Born-Oppenheimer approximation. Specifically, we introduce the multicomponent unitary coupled cluster (mcUCC) method combined with the VQE algorithm for the calculation of ground-state energies and wave functions as well as the multicomponent equation-of-motion (mcEOM) method for the calculation of excitation energies. These methods are developed within the nuclear-electronic orbital (NEO) framework and are formulated in the qubit basis to enable implementations on quantum computers. Moreover, these methods are used to calculate the ground-state energy and excitation energies of positronium hydride, where both electrons and the positron are treated quantum mechanically, as well as the H2 molecule, where both electrons and one proton are treated quantum mechanically. These applications validate the implementation and provide benchmark data for future calculations. The errors due to Trotterization of the mcUCC ansatz are also analyzed. This formalism, as well as the accompanying computer code, will serve as the basis for applications to more complex multicomponent systems, such as simulations of photoinduced nonadiabatic molecular processes, on both classical and quantum computers.