Hydrogen-Atom Electronic Basis Sets for Multicomponent Quantum Chemistry.

Multicomponent methods are a conceptually simple way to include nuclear quantum effects into quantum chemistry calculations. In multicomponent methods, the electronic molecular orbitals are described using the linear combination of atomic orbitals approximation. This requires the selection of a one-particle electronic basis set which, in practice, is commonly a correlation-consistent basis set. In multicomponent method studies, it has been demonstrated that large electronic basis sets are required for quantum hydrogen nuclei to accurately describe electron-nuclear correlation. However, as we show in this study, much of the need for large electronic basis sets is due to the correlation-consistent electronic basis sets not being optimized to describe nuclear properties and electron-nuclear correlation. Herein, we introduce a series of correlation-consistent electronic basis sets for hydrogen atoms called cc-pVnZ-mc with additional basis functions optimized to reproduce multicomponent density functional theory protonic densities. These new electronic basis sets are shown to yield better protonic densities with fewer electronic basis functions than the standard correlation-consistent basis sets and reproduce other protonic properties such as proton affinities and protonic excitation energies, even though they were not optimized for these purposes. The cc-pVnZ-mc basis sets should enable multicomponent many-body calculations on larger systems due to the improved computational efficiency they provide for a given level of accuracy.

(T) Correction for Multicomponent Coupled-Cluster Theory for a Single Quantum Proton.

(T) and [T] perturbative corrections are derived for multicomponent coupled-cluster theory with single and double excitations (CCSD). Benchmarking for systems with a single quantum proton shows that multicomponent CCSD methods that include perturbative corrections are more accurate than multicomponent CCSD for the calculation of proton affinities and absolute energies. An approximation is introduced that includes only (T) or [T] contributions from mixed electron-nuclear excitations.

Multicomponent heat-bath configuration interaction with the perturbative correction for the calculation of protonic excited states.

In this study, we extend the multicomponent heat-bath configuration interaction (HCI) method to excited states. Previous multicomponent HCI studies have been performed using only the variational stage of the HCI algorithm as they have largely focused on the calculation of protonic densities. Because this study focuses on energetic quantities, a second-order perturbative correction after the variational stage is essential. Therefore, this study implements the second-order Epstein-Nesbet correction to the variational stage of multicomponent HCI for the first time. Additionally, this study introduces a new procedure for calculating reference excitation energies for multicomponent methods using the Fourier-grid Hamiltonian (FGH) method, which should allow the one-particle electronic basis set errors to be better isolated from errors arising from an incomplete description of electron-proton correlation. The excited-state multicomponent HCI method is benchmarked by computing protonic excitations of the HCN and FHF- molecules and is shown to be of similar accuracy to previous excited-state multicomponent methods such as the multicomponent time-dependent density-functional theory and equation-of-motion coupled-cluster theory relative to the new FGH reference values.

Multicomponent MP4 and the inclusion of triple excitations in multicomponent many-body methods.

This study implements the full multicomponent third-order (MP3) and fourth-order (MP4) many-body perturbation theory methods for the first time. Previous multicomponent studies have only implemented a subset of the full contributions, and the present implementation is the first multicomponent many-body method to include any connected triples contribution to the electron-proton correlation energy. The multicomponent MP3 method is shown to be comparable in accuracy to the multicomponent coupled-cluster doubles method for the calculation of proton affinities, while the multicomponent MP4 method is of similar accuracy as the multicomponent coupled-cluster singles and doubles method. From the results in this study, it is hypothesized that the relative accuracy of multicomponent methods is more similar to their single-component counterparts than previously assumed. It is demonstrated that for multicomponent MP4, the fourth-order triple-excitation contributions can be split into electron-electron and electron-proton contributions and the electron-electron contributions ignored with very little loss of accuracy of protonic properties.

Multicomponent CASSCF Revisited: Large Active Spaces Are Needed for Qualitatively Accurate Protonic Densities.

Multicomponent methods seek to treat select nuclei, typically protons, fully quantum mechanically and equivalent to the electrons of a chemical system. In such methods, it is well-known that due to the neglect of electron-proton correlation, a Hartree-Fock (HF) description of the electron-proton interaction catastrophically fails leading to qualitatively incorrect protonic properties. In single-component quantum chemistry, the qualitative failure of HF is normally indicative of the need for multireference methods such as complete active space self-consistent field (CASSCF). While a multicomponent CASSCF method was implemented nearly 20 years ago, it is only able to perform calculations with very small active spaces (∼105 multicomponent configurations). Therefore, in order to extend the realm of applicability of the multicomponent CASSCF method, this study derives and implements a new two-step multicomponent CASSCF method that uses multicomponent heat-bath configuration interaction for the configuration interaction step, enabling calculations with very large active spaces (up to 16 electrons in 48 orbitals). We find that large electronic active spaces are needed to obtain qualitatively accurate protonic densities for the HCN and FHF- molecules. Additionally, the multicomponent CASSCF method implemented here should have further applications for double-well protonic potentials and systems that are inherently electronically multireference.

Mixed Quantum-Classical Dynamics with Machine Learning-Based Potentials via Wigner Sampling.

Machine learning-based approaches for surface hopping (SH) offer the prospect of SH simulations with ab initio accuracy, but with a computational cost more similar to classical molecular dynamics simulations. However, such approaches in the adiabatic basis are difficult due to the need to fit a machine learning model to reproduce the nonadiabatic coupling, which rapidly changes in the vicinity of a conical intersection. Previous approaches have typically dealt with this difficulty by either computing the hopping probabilities using methods that do not require the explicit nonadiabatic coupling or by employing adaptive sampling. In this study, we introduce a new approach using a simple modification of Wigner sampling to generate appropriate training data. Test SH simulations on the two-state spin-boson Hamiltonian system show that Wigner sampling with an appropriately selected data set can reduce the size of the training data set by up to a factor of 7.5 per degree of freedom compared to previously linear sampling-based approaches.

Quantifying Multireference Character in Multicomponent Systems with Heat-Bath Configuration Interaction.

Multicomponent quantum chemical methods seek to include nuclear quantum effects of select nuclei in quantum chemistry calculations by not invoking the Born-Oppenheimer approximation for these nuclei. In multicomponent methods, the inclusion of electron-proton correlation is essential for obtaining even qualitatively accurate protonic densities. However, most of the recently developed multicomponent methods have either used or obtained molecular orbitals from a single-reference mean-field wave function that neglects all electron-proton correlation that is analogous to using Hartree-Fock orbitals in a single-component framework. We examine the consequences of using Hartree-Fock orbitals in multicomponent calculations by developing the multicomponent heat-bath configuration interaction (HCI) method. Multicomponent HCI is a multicomponent selected configuration interaction (CI) technique that enables an accurate approximation of a complete active space or truncated CI wave function for systems with large active spaces. The multicomponent HCI method is shown to reproduce the ground-state protonic density of the HeHHe+, HCN, and FHF- systems when compared to reference grid-based calculations. For all three systems, the coefficient of the leading configuration in the wave function expansion is less than 0.95, indicating that all systems have multireference character. This is highly noteworthy as none of the systems have multireference character in a single-component framework and suggests that multireference character appears inherent to or at least more commonly in a multicomponent framework than a single-component framework. Even when natural orbitals are used rather than Hartree-Fock orbitals for the multicomponent HCI calculations, aspects of the multireference character remain for FHF- and HCN. Consequences and implications of the multireference character of multicomponent quantum chemical systems are discussed.

Separation of electron-electron and electron-proton correlation in multicomponent orbital-optimized perturbation theory.

The multicomponent orbital-optimized second-order Møller-Plesset perturbation theory (OOMP2) method is the first multicomponent MP2 method that is able to calculate qualitatively accurate protonic densities, protonic affinities, and geometrical changes due to nuclear quantum effects in multicomponent systems. In this study, two approximations of the multicomponent OOMP2 method are introduced in an effort to demonstrate that, in orbital-optimized multicomponent methods, performing the orbital-optimization process with only electron-proton correlation is sufficient to obtain accurate protonic properties. Additionally, these approximations should reduce the computational expense of the multicomponent OOMP2 method. In the first approximation, the first-order wave function is written as a linear combination of one-electron one-proton excitations rather than as a linear combination of one-electron one-proton and two-electron excitations as in the original multicomponent OOMP2 method. Electron-electron correlation is included perturbatively after the orbital-optimization procedure has converged. In the second approach, the first approximation is further modified to neglect all terms in the orbital-rotation gradients that depend on the two-electron molecular-orbital integrals, which, assuming a fixed-sized protonic basis set, reduces the computational scaling for the orbital-optimization iterations to Ne 3, where Ne is a measure of the electronic system size, compared to the Ne 5 scaling of the original multicomponent OOMP2 method. The second approximation requires one Ne 5 step after orbital convergence to compute the electron-electron correlation energy. The accuracy of the calculated protonic densities, protonic affinities, and optimized geometries of these approximations is similar or improved relative to the original multicomponent OOMP2 method.

Vibrational adaptive sampling configuration interaction.

Selected configuration interaction plus perturbation theory approaches have long been used to solve both the electronic and vibrational Schrödinger equations. In the last few years, many new selection algorithms have been developed for these approaches and applied to solve the electronic Schrödinger equation, but these algorithms have seen little to no use for solving the vibrational Schrödinger equation. Herein, we adapt one of the recently developed approaches, the adaptive sampling configuration interaction (ASCI) method, to calculate the vibrational excitations of molecules. The vibrational ASCI method has accuracy comparable to other high-accuracy approaches for solving the vibrational Schrödinger equation while requiring only modest computer resources. We study two different approaches for calculating excited states with vibrational ASCI and benchmark the method on acetonitrile and ethylene oxide. Finally, we demonstrate the applicability of the vibrational ASCI method to large systems by calculating the 128 lowest energy vibrational states of naphthalene, which has 48 vibrational degrees of freedom.

Reproducing global potential energy surfaces with continuous-filter convolutional neural networks.

Neural networks fit to reproduce the potential energy surfaces of quantum chemistry methods offer a realization of analytic potential energy surfaces with the accuracy of ab initio methods at a computational cost similar to classical force field methods. One promising class of neural networks for this task is the SchNet architecture, which is based on the use of continuous-filter convolutional neural networks. Previous work has shown the ability of the SchNet architecture to reproduce density functional theory energies and forces for molecular configurations sampled during equilibrated molecular dynamics simulations. Due to the large change in energy when bonds are broken and formed, the fitting of global potential energy surfaces is normally a more difficult task than fitting the potential energy surface in the region of configurational space sampled during equilibrated molecular dynamics simulations. Herein, we demonstrate the ability of the SchNet architecture to reproduce the energies and forces of the potential energy surfaces of the H + H2 and Cl + H2 reactions and the OCHCO+ and H2CO/cis-HCOH/trans-HCOH systems. The SchNet models reproduce the potential energy surface of the reactions well with the best performing SchNet model having a test set root-mean-squared error of 0.52 meV and 2.01 meV for the energies of the H + H2 and Cl + H2 reactions, respectively, and a test set mean absolute error for the force of 0.44 meV/bohr for the H + H2 reaction. For the OCHCO+ and H2CO/cis-HCOH/trans-HCOH systems, the best performing SchNet model has a test set root-mean-squared error of 2.92 meV and 13.55 meV, respectively.

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.

Development of a practical multicomponent density functional for electron-proton correlation to produce accurate proton densities.

Multicomponent density functional theory (DFT) enables the consistent quantum mechanical treatment of both electrons and protons. A major challenge has been the design of electron-proton correlation (epc) functionals that produce even qualitatively accurate proton densities. Herein an electron-proton correlation functional, epc17, is derived analogously to the Colle-Salvetti formalism for electron correlation and is implemented within the nuclear-electronic orbital (NEO) framework. The NEO-DFT/epc17 method produces accurate proton densities efficiently and is promising for diverse applications.

Multicomponent Density Functional Theory: Impact of Nuclear Quantum Effects on Proton Affinities and Geometries.

Nuclear quantum effects such as zero point energy play a critical role in computational chemistry and often are included as energetic corrections following geometry optimizations. The nuclear-electronic orbital (NEO) multicomponent density functional theory (DFT) method treats select nuclei, typically protons, quantum mechanically on the same level as the electrons. Electron-proton correlation is highly significant, and inadequate treatments lead to highly overlocalized nuclear densities. A recently developed electron-proton correlation functional, epc17, has been shown to provide accurate nuclear densities for molecular systems. Herein, the NEO-DFT/epc17 method is used to compute the proton affinities for a set of molecules and to examine the role of nuclear quantum effects on the equilibrium geometry of FHF-. The agreement of the computed results with experimental and benchmark values demonstrates the promise of this approach for including nuclear quantum effects in calculations of proton affinities, pKa's, optimized geometries, and reaction paths.

Communication: Density functional theory embedding with the orthogonality constrained basis set expansion procedure.

Density functional theory (DFT) embedding approaches have generated considerable interest in the field of computational chemistry because they enable calculations on larger systems by treating subsystems at different levels of theory. To circumvent the calculation of the non-additive kinetic potential, various projector methods have been developed to ensure the orthogonality of molecular orbitals between subsystems. Herein the orthogonality constrained basis set expansion (OCBSE) procedure is implemented to enforce this subsystem orbital orthogonality without requiring a level shifting parameter. This scheme is a simple alternative to existing parameter-free projector-based schemes, such as the Huzinaga equation. The main advantage of the OCBSE procedure is that excellent convergence behavior is attained for DFT-in-DFT embedding without freezing any of the subsystem densities. For the three chemical systems studied, the level of accuracy is comparable to or higher than that obtained with the Huzinaga scheme with frozen subsystem densities. Allowing both the high-level and low-level DFT densities to respond to each other during DFT-in-DFT embedding calculations provides more flexibility and renders this approach more generally applicable to chemical systems. It could also be useful for future extensions to embedding approaches combining wavefunction theories and DFT.

Is the Accuracy of Density Functional Theory for Atomization Energies and Densities in Bonding Regions Correlated?

The development of approximate exchange-correlation functionals is critical for modern density functional theory. A recent analysis of atomic systems suggested that some modern functionals are straying from the path toward the exact functional because electron densities are becoming less accurate while energies are becoming more accurate since the year 2000. To investigate this trend for more chemically relevant systems, the electron densities in the bonding regions and the atomization energies are analyzed for a series of diatomic molecules with 90 different functionals. For hybrid generalized gradient approximation functionals developed since the year 2000, the errors in densities and atomization energies are decoupled; the accuracy of the energies remains relatively consistent while the accuracy of the densities varies significantly. Such decoupling is not observed for generalized gradient and meta-generalized gradient approximation functionals. Analysis of electron densities in bonding regions is found to be important for the evaluation of functionals for chemical systems.

Calculation of Positron Binding Energies and Electron-Positron Annihilation Rates for Atomic Systems with the Reduced Explicitly Correlated Hartree-Fock Method in the Nuclear-Electronic Orbital Framework.

Although the binding of a positron to a neutral atom has not been directly observed experimentally, high-level theoretical methods have predicted that a positron will bind to a neutral atom. In the present study, the binding energies of a positron to lithium, sodium, beryllium, and magnesium, as well as the electron-positron annihilation rates for these systems, are calculated using the reduced explicitly correlated Hartree-Fock (RXCHF) method within the nuclear-electronic orbital (NEO) framework. Due to the lack of explicit electron-positron correlation, NEO Hartree-Fock and full configuration interaction calculations with reasonable electronic and positronic basis sets do not predict positron binding to any of these atoms. In contrast, the RXCHF calculations predict positron binding energies and electron-positron annihilation rates in qualitative agreement with previous highly accurate but computationally expensive stochastic variational method calculations. These results illustrate that the RXCHF method can successfully describe the binding of a positron to a neutral species with no dipole moment. Moreover, the RXCHF method will be computationally tractable for calculating positron binding to molecular systems. The RXCHF approach offers a balance of accuracy and computational tractability for studying these types of positronic systems.

Multicomponent density functional theory embedding formulation.

Multicomponent density functional theory (DFT) methods have been developed to treat two types of particles, such as electrons and nuclei, quantum mechanically at the same level. In the nuclear-electronic orbital (NEO) approach, all electrons and select nuclei, typically key protons, are treated quantum mechanically. For multicomponent DFT methods developed within the NEO framework, electron-proton correlation functionals based on explicitly correlated wavefunctions have been designed and used in conjunction with well-established electronic exchange-correlation functionals. Herein a general theory for multicomponent embedded DFT is developed to enable the accurate treatment of larger systems. In the general theory, the total electronic density is separated into two subsystem densities, denoted as regular and special, and different electron-proton correlation functionals are used for these two electronic densities. In the specific implementation, the special electron density is defined in terms of spatially localized Kohn-Sham electronic orbitals, and electron-proton correlation is included only for the special electron density. The electron-proton correlation functional depends on only the special electron density and the proton density, whereas the electronic exchange-correlation functional depends on the total electronic density. This scheme includes the essential electron-proton correlation, which is a relatively local effect, as well as the electronic exchange-correlation for the entire system. This multicomponent DFT-in-DFT embedding theory is applied to the HCN and FHF(-) molecules in conjunction with two different electron-proton correlation functionals and three different electronic exchange-correlation functionals. The results illustrate that this approach provides qualitatively accurate nuclear densities in a computationally tractable manner. The general theory is also easily extended to other types of partitioning schemes for multicomponent systems.

Ab initio investigation of the aqueous solvation of the nitrate ion.

The surface affinity of the nitrate ion in aqueous clusters is investigated with a variety of theoretical methods. A sampling of structures in which the nitrate ion is solvated by 32 water molecules is optimized using second order Møller-Plesset perturbation theory (MP2). Four of these MP2 optimized structures are used as starting points for fully ab initio molecular dynamics simulations at the dispersion corrected restricted Hartree-Fock (RHF-D) level of theory. The nitrate ion solvated by 16, 32, and 64 water molecules is also investigated with umbrella sampling molecular dynamics simulations using QM/MM methodology, where the nitrate ion is modeled with MP2 and the water molecules are described using either the non-empirical effective fragment potential (EFP) or the empirical TIP5P potential. The turning point between surface and interior solvation of the nitrate ion is predicted to lie around a cluster size of 64 water molecules.

Quantum treatment of protons with the reduced explicitly correlated Hartree-Fock approach.

The nuclear-electronic orbital (NEO) approach treats select nuclei quantum mechanically on the same level as the electrons and includes nonadiabatic effects between the electrons and the quantum nuclei. The practical implementation of this approach is challenging due to the significance of electron-nucleus dynamical correlation. Herein, we present a general extension of the previously developed reduced NEO explicitly correlated Hartree-Fock (RXCHF) approach, in which only select electronic orbitals are explicitly correlated to each quantum nuclear orbital via Gaussian-type geminal functions. Approximations of the electronic exchange between the geminal-coupled electronic orbitals and the other electronic orbitals are also explored. This general approach enables computationally tractable yet accurate calculations on molecular systems with quantum protons. The RXCHF method is applied to the hydrogen cyanide (HCN) and FHF(-) systems, where the proton and all electrons are treated quantum mechanically. For the HCN system, only the two electronic orbitals associated with the CH covalent bond are geminal-coupled to the proton orbital. For the FHF(-) system, only the four electronic orbitals associated with the two FH covalent bonds are geminal-coupled to the proton orbital. For both systems, the RXCHF method produces qualitatively accurate nuclear densities, in contrast to mean field-based NEO approaches. The development and implementation of the RXCHF method provide the framework to perform calculations on systems such as proton-coupled electron transfer reactions, where electron-proton nonadiabatic effects are important.

Nuclear-electronic orbital reduced explicitly correlated Hartree-Fock approach: Restricted basis sets and open-shell systems.

The nuclear electronic orbital (NEO) reduced explicitly correlated Hartree-Fock (RXCHF) approach couples select electronic orbitals to the nuclear orbital via Gaussian-type geminal functions. This approach is extended to enable the use of a restricted basis set for the explicitly correlated electronic orbitals and an open-shell treatment for the other electronic orbitals. The working equations are derived and the implementation is discussed for both extensions. The RXCHF method with a restricted basis set is applied to HCN and FHF(-) and is shown to agree quantitatively with results from RXCHF calculations with a full basis set. The number of many-particle integrals that must be calculated for these two molecules is reduced by over an order of magnitude with essentially no loss in accuracy, and the reduction factor will increase substantially for larger systems. Typically, the computational cost of RXCHF calculations with restricted basis sets will scale in terms of the number of basis functions centered on the quantum nucleus and the covalently bonded neighbor(s). In addition, the RXCHF method with an odd number of electrons that are not explicitly correlated to the nuclear orbital is implemented using a restricted open-shell formalism for these electrons. This method is applied to HCN(+), and the nuclear densities are in qualitative agreement with grid-based calculations. Future work will focus on the significance of nonadiabatic effects in molecular systems and the further enhancement of the NEO-RXCHF approach to accurately describe such effects.