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Modern electronic structure theory is built around the Born–Oppenheimer approximation and the construction of an electronic Hamiltonian H ̂ e l (X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ′) and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, H ̂ P S (X , P). While classical Born–Oppenheimer dynamics that run along the eigensurfaces of the operator H ̂ e l (X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of H ̂ P S (X , P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general. [ABSTRACT FROM AUTHOR]

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, "Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions," J. Chem. Phys. 122, 084113 (2005) and V. Krishna, "Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit," J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree–Fock with spin–orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born–Oppenheimer approximation when using ab initio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results. [ABSTRACT FROM AUTHOR]

We propose an efficient version of ensemble Hartree–Fock/density functional theory to calculate a set of two charge-transfer states for systems with odd-numbers of electrons. The approach does require definitions of donor/acceptor fragments; however, the approach is not very sensitive to such definitions—even in the limit of very strong electronic coupling. The key ansatz is that, by mandating that the vector space spanned by the active orbitals projects equally onto the donor and acceptor fragments, such a constraint eliminates all intra-molecular local excitations and makes it far easier to generate potential energy surfaces that are smooth over a wide region of configuration space. The method is fast, working with only two electron configurations, and should be useful for ab initio non-adiabatic dynamics in the near future. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K . In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operator O ̂ whose matrix element J O ̂ K was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations. [ABSTRACT FROM AUTHOR]

Multicomponent density functional theory (DFT) has many practical advantages for incorporating nuclear quantum effects into quantum chemistry calculations. Within the nuclear-electronic orbital (NEO) framework, specified nuclei, typically protons, are treated quantum mechanically on the same level as the electrons. Previously, electron-proton correlation functionals based on the local density approximation (LDA), denoted epc17 and epc18, were developed and shown to provide more accurate proton densities and energies compared to the neglect of electron-proton correlation, but a quantitatively accurate description of both densities and energies simultaneously has remained elusive. Herein, an electron-proton correlation functional that depends on the electron and proton density gradients, as well as the densities, is derived and implemented. Compared to the LDA functionals, the resulting generalized gradient approximation functional, denoted epc19, is able to simultaneously provide accurate proton densities and energies, as well as reproduce the impact of nuclear quantum effects on optimized geometries. In addition, without further parameterization, the NEO-DFT/epc19 method provides accurate densities and energies for deuterium as well as hydrogen. These results demonstrate that the form of the epc19 functional is able to capture the essential aspects of electron-proton correlation and highlights the importance of including gradient terms. This approach will enable the exploration of nuclear quantum effects and isotope effects in a wide range of systems. [ABSTRACT FROM AUTHOR]

For a system without spin–orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e., the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors) and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass [which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors below, cf. Eq. (96)]. The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing the new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The recently developed real-time nuclear–electronic orbital (RT-NEO) approach provides an elegant framework for treating electrons and selected nuclei, typically protons, quantum mechanically in nonequilibrium dynamical processes. However, the RT-NEO approach neglects the motion of the other nuclei, preventing a complete description of the coupled nuclear–electronic dynamics and spectroscopy. In this work, the dynamical interactions between the other nuclei and the electron–proton subsystem are described with the mixed quantum–classical Ehrenfest dynamics method. The NEO-Ehrenfest approach propagates the electrons and quantum protons in a time-dependent variational framework, while the remaining nuclei move classically on the corresponding average electron–proton vibronic surface. This approach includes the non-Born–Oppenheimer effects between the electrons and the quantum protons with RT-NEO and between the classical nuclei and the electron–proton subsystem with Ehrenfest dynamics. Spectral features for vibrational modes involving both quantum and classical nuclei are resolved from the time-dependent dipole moments. This work shows that the NEO-Ehrenfest method is a powerful tool to study dynamical processes with coupled electronic and nuclear degrees of freedom. [ABSTRACT FROM AUTHOR]

Multicomponent quantum chemistry allows the quantum mechanical treatment of electrons and specified protons on the same level. Typically the goal is to identify a self-consistent-field (SCF) solution that is the global minimum associated with the molecular orbital coefficients of the underlying Hartree-Fock (HF) or density functional theory (DFT) calculation. To determine whether the solution is a minimum or a saddle point, herein we derive the stability conditions for multicomponent HF and DFT in the nuclear-electronic orbital (NEO) framework. The gradient is always zero for an SCF solution, whereas the Hessian must be positive semi-definite for the solution to be a minimum rather than a saddle point. The stability matrices for NEO-HF and NEO-DFT have the same matrix structures, which are identical to the working matrices of their corresponding linear response time-dependent theories (NEO-TDHF and NEO-TDDFT) but with a different metric. A negative eigenvalue of the stability matrix is a necessary but not sufficient condition for the corresponding NEO-TDHF or NEO-TDDFT working equation to have an imaginary eigenvalue solution. Electron-proton systems could potentially exhibit three types of instabilities: electronic, protonic, and electron-proton vibronic instabilities. The internal and external stabilities for theories with different constraints on the spin and spatial orbitals can be analyzed. This stability analysis is a useful tool for characterizing SCF solutions and is helpful when searching for lower-energy solutions. Initial applications to HCN, HNC, and 2-cyanomalonaldehyde, in conjunction with NEO ∆SCF calculations, highlight possible connections between stationary points in nuclear coordinate space for conventional electronic structure calculations and stationary points in orbital space for NEO calculations. [ABSTRACT FROM AUTHOR]

The multicomponent extension of time-dependent density functional theory (TDDFT) within the nuclear-electronic orbital (NEO) framework enables the calculation of both electronic and vibrational excitations simultaneously. In this NEO-TDDFT approach, all electrons and select nuclei, typically protons, are treated quantum mechanically on the same level. Herein, the dependence of the proton vibrational excitation energies on the nuclear and electronic basis sets is examined. Protonic basis sets that include f basis functions in conjunction with substantial electronic basis sets for the quantum hydrogen are found to produce accurate proton vibrational excitation energies that are mostly within ∼30 cm−1 of reference values for the molecules studied. The NEO-TDDFT approach is shown to be effective for open-shell as well as closed-shell systems. Additionally, an approach for computing and visualizing the nuclear transition densities associated with the proton vibrational excitations is implemented. These nuclear transition densities are important for characterizing the proton vibrational excitations and determining the spatial orientations of the corresponding vibrational modes. These capabilities are essential for a variety of applications, including the incorporation of anharmonic effects into molecular vibrational frequency calculations. [ABSTRACT FROM AUTHOR]