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Extended Lagrangian Born-Oppenheimer molecular dynamics (XL-BOMD) [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] is formulated for orbital-free Hohenberg-Kohn density-functional theory and for charge equilibration and polarizable force-field models that can be derived from the same orbital-free framework. The purpose is to introduce the most recent features of orbital-based XL-BOMD to molecular dynamics simulations based on charge equilibration and polarizable force-field models. These features include a metric tensor generalization of the extended harmonic potential, preconditioners, and the ability to use only a single Coulomb summation to determine the fully equilibrated charges and the interatomic forces in each time step for the shadow Born-Oppenheimer potential energy surface. The orbital-free formulation has a charge-dependent, short-range energy term that is separate from long-range Coulomb interactions. This enables local parameterizations of the short-range energy term, while the long-range electrostatic interactions can be treated separately. The theory is illustrated for molecular dynamics simulations of an atomistic system described by a charge equilibration model with periodic boundary conditions. The system of linear equations that determines the equilibrated charges and the forces is diagonal, and only a single Ewald summation is needed in each time step. The simulations exhibit the same features in accuracy, convergence, and stability as are expected from orbital-based XL-BOMD.

In this work, we describe a computer program called ATOM-MOL-nonBO for performing bound state calculations of small atoms and molecules without assuming the Born–Oppenheimer approximation. All particles forming the systems, electrons and nuclei, are treated on equal footing. The wave functions of the bound states are expanded in terms of all-particle one-center complex explicitly correlated Gaussian functions multiplied by Cartesian angular factors. As these Gaussian functions are eigenfunctions of the operator representing the square of the total angular momentum of the system, the problem separates and calculations of states corresponding to different values of the total rotational quantum number can be solved independently from each other. Due to thorough variational optimization of the Gaussian exponential parameters, the method allows us to generate very accurate wave functions. The optimization is aided by analytically calculated energy gradient determined with respect to the parameters. Three examples of calculations performed for diatomic and triatomic molecules are shown as an illustration of calculations that can be performed with this program. Finally, we discuss the limitations, applicability range, and bottlenecks of the program.

We employ theoretically “exact” and numerically “accurate” Beyond Born–Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C6H6+) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn–Teller interactions and accidental conical intersections/seams to pseudo Jahn–Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states (X2E1g, B2E2g, and C2A2u) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X2E1g and B2E2g−C2A2u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches.

It is shown how the electronic equations of motion in extended Lagrangian Born-Oppenheimer molecular dynamics simulations [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008); J. Chem. Phys. 147, 054103 (2017)] can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born-Oppenheimer molecular dynamics. The low-rank approximation of the Jacobian is directly related to Broyden's class of quasi-Newton algorithms and Jacobian-free Newton-Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.

It is commonly assumed that the time-dependent electron flux calculated within the Born-Oppenheimer (BO) approximation vanishes. This is not necessarily true if the flux is directly determined from the continuity equation obeyed by the electron density. This finding is illustrated for a one-dimensional model of coupled electronic-nuclear dynamics. There, the BO flux is in perfect agreement with the one calculated from a solution of the time-dependent Schrodinger equation for the coupled motion. A reflection principle is derived where the nuclear BO flux is mapped onto the electronic flux.

In the case of an adiabatic motion in molecules, electrons adjust to the smoothly changing geometry of the nuclei. Although then the Born-Oppenheimer (BO) approximation is valid, it fails in predicting the time-dependence of electron momenta because, within its product ansatz for the wave function, the respective expectation values are zero. It is shown that this failure can be circumvented using the Ehrenfest theorem. Here we extend our former work [T. Schaupp et al., Eur. Phys. J. B 91, 97 (2018)] and regard models in higher dimensions and for more particles. We solve the time-dependent Schrödinger equation for the combined nuclear-electronic motion and compare the results to those derived from BO wave functions. For all situations, it is found that the time-dependent BO electronic momenta are in excellent agreement with the numerically exact results.

The high-order harmonic generation of the HD molecule is studied in non-Born-Oppenheimer approximation. It is found that there are only the odd harmonics in the harmonic spectrum of the HD molecule though the generation of even harmonics is possible in principle. Theoretical analysis [T. Kreibich et al., Phys. Rev. Lett. 87, 103901 (2001)] reveals that the nuclear dipole moment can contribute to the generation of the even harmonics, but the acceleration of the nucleus is about three orders of magnitude less than that of the electron. Hence, the even harmonics cannot be observed in the harmonic spectrum of the HD molecule.

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrodinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C6H3F3+ (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X2E′′, A2A2′′, B2E′, and C2A2′), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C–C asymmetric stretching, C–C symmetric stretching, and C–C–C scissoring motion (Q9x, Q9y, Q10x, Q10y, Q12x, and Q12y) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs.In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrodinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C6H3F3+ (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electro...

All available "conventional" absorption/emission spectroscopic data have been combined with photodissociation data and translational spectroscopy data in a global analysis that yields analytic potential energy and Born-Oppenheimer breakdown functions for the X(1)Σ(+) and A(1)Π states of CH(+) and its isotopologues that reproduce all of the data (on average) within their assigned uncertainties. For the ground X(1)Σ(+) state, this fully quantum mechanical "Direct-Potential-Fit" analysis yielded an improved empirical well depth of 𝔇e = 34 362.8(3) cm(-1) and equilibrium bond length of re = 1.128 462 5 (58) Å. For the A(1)Π state, the resulting well depth and equilibrium bond length are 𝔇e = 10 303.7(3) cm(-1) and re = 1.235 896 (14) Å, while the electronic isotope shift from the hydride to the deuteride is ΔTe = - 5.99(±0.08) cm(-1).

A first-principle theoretical approach to study the process of radiative electron attachment is developed and applied to the negative molecular ions CN(-), C4H(-), and C2H(-). Among these anions, the first two have already been observed in the interstellar space. Cross sections and rate coefficients for formation of these ions by direct radiative electron attachment to the corresponding neutral radicals are calculated. For the CN molecule, we also considered the indirect pathway, in which the electron is initially captured through non-Born-Oppenheimer coupling into a vibrationally resonant excited state of the anion, which then stabilizes by radiative decay. We have shown that the contribution of the indirect pathway to the formation of CN(-) is negligible in comparison to the direct mechanism. The obtained rate coefficients for the direct mechanism at 30 K are 7 × 10(-16) cm(3)/s for CN(-), 7 × 10(-17) cm(3)/s for C2H(-), and 2 × 10(-16) cm(3)/s for C4H(-). These rates weakly depend on temperature between 10 K and 100 K. The validity of our calculations is verified by comparing the present theoretical results with data from recent photodetachment experiments.

Semi-quartic force fields (QFF) rely on a Taylor-expansion of the multi-dimensional Born-Oppenheimer potential energy surface (PES) and are frequently used within the calculation of anharmonic vibrational frequencies based on 2nd order vibrational perturbation theory (VPT2). As such they are usually determined by differentiation of the electronic energy with respect to the nuclear coordinates. Alternatively, potential energy surfaces can be expanded in terms of multi-mode expansions, which typically do not require any derivative techniques. The computational effort to retrieve QFF from size-reduced multi-mode expansions has been studied and has been compared with standard Taylor-expansions. As multi-mode expansions allow for the convenient introduction of subtle approximations, these will be discussed in some detail. In addition, a preliminary study about the applicability of a generalized Duschinsky transformation to QFFs is provided. This transformation allows for the efficient evaluation of VPT2 frequencies of isotopologues from the PES of the parent compound and thus avoids the recalculation of PESs in different axes systems.

We present extensive density functional theory (DFT) calculations dedicated to analyze the adsorption behavior of CO molecules on small AlxOy (±) clusters. Following the experimental results of Johnson et al. [J. Phys. Chem. A 112, 4732 (2008)], we consider structures having the bulk composition Al2O3, as well as smaller Al2O2 and Al2O units. Our electron affinity and total energy calculations are consistent with aluminum oxide clusters having two-dimensional rhombus-like structures. In addition, interconversion energy barriers between two- and one-dimensional atomic arrays are of the order of 1 eV, thus clearly defining the preferred isomers. Single CO adsorption on our charged AlxOy (±) clusters exhibits, in general, spontaneous oxygen transfer events leading to the production of CO2 in line with the experimental data. However, CO can also bind to both Al and O atoms of the clusters forming aluminum oxide complexes with a CO2 subunit. The vibrational spectra of AlxOy + CO2 provides well defined finger prints that may allow the identification of specific isomers. The AlxOy (+) clusters are more reactive than the anionic species and the final Al2O(+) + CO reaction can result in the production of atomic Al and carbon dioxide as observed from experiments. We underline the crucial role played by the local atomic environment, charge density distribution, and spin-multiplicity on the oxidation behavior of CO molecules. Finally, we analyze the importance of coadsorption and finite temperature effects by performing DFT Born-Oppenheimer molecular dynamics. Our calculations show that CO oxidation on AlxOy (+) clusters can be also promoted by the binding of additional CO species at 300 K, revealing the existence of fragmentation processes in line with the ones experimentally inferred.

The influence of thermally induced nuclear dynamics (molecular vibrations) in the initial electronic ground state on the valence orbital momentum profiles of furan has been theoretically investigated using two different approaches. The first of these approaches employs the principles of Born-Oppenheimer molecular dynamics, whereas the so-called harmonic analytical quantum mechanical approach resorts to an analytical decomposition of contributions arising from quantized harmonic vibrational eigenstates. In spite of their intrinsic differences, the two approaches enable consistent insights into the electron momentum distributions inferred from new measurements employing electron momentum spectroscopy and an electron impact energy of 1.2 keV. Both approaches point out in particular an appreciable influence of a few specific molecular vibrations of A1 symmetry on the 9a1 momentum profile, which can be unravelled from considerations on the symmetry characteristics of orbitals and their energy spacing.

A dynamical extension of the "curvy-steps" approach to linear-scaling self-consistent field calculations is presented, which yields an extended-Lagrangian formulation of ab initio molecular dynamics. An exponential parametrization of the one-electron density matrix, expressed in terms of atom-centered Gaussian basis functions, facilitates propagation along the manifold of density matrices in a geometrically correct fashion that automatically enforces idempotency constraints. The extended Lagrangian itself is constraint free, thus neither density matrix purification nor expensive, iterative solution for Lagrange multipliers is required. Propagation is highly efficient, and time steps compare favorably to those used in Car-Parrinello molecular dynamics simulations. The behavior of the method, especially with regard to the maintenance of adiabatic decoupling of nuclei and electrons, is examined for a sequence of diatomic molecules, and comparison is made to trajectories propagated on the converged Born-Oppenheimer surface. Certain claims to the contrary notwithstanding, our results demonstrate that vibrational frequencies may depend on the value of the fictitious mass parameter, even in an atom-centered basis. Light-atom stretching frequencies can be significantly redshifted, even when the nuclear and electronic energy scales are well separated. With a sufficiently small fictitious mass and a short time step, accurate frequencies can be obtained; we characterize appropriate values of these parameters for a wide range of vibrational frequencies.

Observation of infrared electronic transitions involving the 1 (1)Deltag state of 7Li2 has instigated an investigation of Born-Oppenheimer breakdown in four singlet electronic states correlating with (2s+2s), (2s+2p), and (2p+2p) lithium atoms. The 1 (1)Deltag state, which correlates at long range with (2p+2p) atoms, has been observed in emission from the (5p) (1)Piu Rydberg state and in 1 (1)Deltag-B (1)Piu bands, in both instances following optical-optical double-resonance excitation. The latter transition was observed previously for the lighter isotopomer, 6Li2 [C. Linton, F. Martin, P. Crozet, A. J. Ross, and R. Bacis, J. Mol. Spectrosc. 158, 445 (1993)]. By analyzing multiple-isotopomer data for several electronic systems simultaneously, we have determined the electronic isotope shifts and the leading vibrational and/or rotational Born-Oppenheimer breakdown terms for the X (1)Sigmag+, A (1)Sigmau+, B (1)Piu, and 1 (1)Deltag states of the lithium dimer. This paper also reports Fourier transform measurements of the B-X absorption spectra of 6Li2 and 7Li2, which were required to better define the bottom portion of the B (1)Piu state potential.

Very accurate non-Born-Oppenheimer variational calculations of the ground state of e(+)LiH have been performed using explicitly correlated Gaussian functions with preexponential factors dependent on powers of the internuclear distance. In order to determine the positron detachment energy of e(+)LiH and the dissociation energy corresponding to the e(+)LiH fragmentation into HPs and Li(+) we also calculated non-BO energies of HPs, LiH, and Li(+). For all the systems the calculations provided the lowest ever-reported variational upper-bounds to the ground state energies. Annihilation rates of HPs and e(+)LiH were also computed. The dissociation energy of e(+)LiH into HPs and Li(+) was determined to be 0.036 548 hartree.

Fourier transform microwave spectroscopy has been used to investigate vibrational excitation and relaxation of diatomic molecules produced by an electric discharge in the throat of a supersonic nozzle. Rotational transitions of SO, SiO, and SiS, in vibrational states up to v=33 for 32S16O, v=45 for 28Si16O, and v=51 for 28Si32S in their ground electronic states have been detected. The isotopic species 33S16O, 34S16O, 29Si16O, 28Si18O, 29Si32S, and 28Si34S have also been observed in highly excited vibrational states. Microwave transitions include up to v=22 for the second lowest excited electronic state b 1Σ+ of SO (∼10 510 cm−1 above ground) have also been detected. Effective vibrational temperatures have been derived for each species, and a general model is proposed to qualitatively explain the observations. Vibrational excitation is caused by inelastic collisions with the hot electrons produced in the discharge. The subsequent vibrational populations are largely determined by vibration–vibration energy transfer via molecule–molecule binary collisions. Two regions can be inferred from the data: one characterized by a temperature of around 1000 K and a second region with a temperature of several thousand degrees Kelvin. Improved Dunham coefficients and correction terms for the breakdown of the Born–Oppenheimer approximation have been determined for b 1Σ+ SO, X 1Σ+ SiO, and X 1Σ+ SiS. Nuclear spin-rotation hyperfine structure for the 29Si isotopic species of SiO and SiS has been observed in all highly excited vibrational states.

Threshold ion-pair production spectroscopy (TIPPS) has been applied to two isotopomers, HCl and DCl. From the high-resolution TIPP spectra the ion-pair thresholds of the two molecules have been precisely measured. Combined with the known ionization energy of H(D) and the electron affinity of Cl, the difference between their bond dissociation energies is calculated, and therefore an experimental determination of the effect of Born–Oppenheimer breakdown on the dissociation limit of the ground state potential curve has been obtained. The difference in De for the two isotopomers was found to be: De(H–Cl)−De(D–Cl)=3.2±1.0 cm−1. The bond energy for HCl was in agreement with our previous determination, D0(H–Cl)=35748.2±0.8 cm−1. These results are compared to a recent study of Born–Oppenheimer breakdown in HCl by Coxon and Hajigeorgiou, where high resolution spectroscopic data was used to fit Born–Oppenheimer breakdown correction terms for the intramolecular potential function. The present study also measured the high resolution spectra for photoion-pair production for HCl and DCl in the threshold region for ion-pair production (∼86 nm). Although there is qualitative agreement between the current results and previous experimental and theoretical work, there are some important differences. The possible mechanism for ion-pair formation in HCl and DCl is discussed in light of these high resolution results.

The potential energy curve for the B 1Πu state of Li2 has a rotationless barrier which protrudes above its energy asymptote. A direct fit to spectroscopic data for all three isotopomers of this species, including Λ-doubling splittings and tunneling predissociation line widths, is used to determine an accurate analytic potential energy function plus Born–Oppenheimer breakdown and Λ-doubling perturbation radial strength functions for this system. This analysis introduces an analytic model for representing a potential function with a rotationless barrier, and shows that a radial perturbation function treatment can determine the symmetry of the perturbing state giving rise to Λ-doubling splittings.

We reconsider the Born–Oppenheimer–Huang treatment of systems of electrons and nuclei for the case of their interaction with time-dependent fields. Initially, we present a framework in which all expressions derived are formally exact since no truncations are introduced. The objective is to explore the general structure of the equations under the most unrestricted conditions, including the possibility that the electronic basis is dependent both on the nuclear coordinates and on time. We then derive an application of the theory applicable to cases of interaction with strong time-dependent fields. The method truncates the electronic basis only after the time-dependent interaction is taken into account in the electronic wave functions. This leads to theory which is similar to a Born–Oppenheimer-type truncation within the interaction picture.

The full Born–Oppenheimer potential energy hypersurfaces of the Ag3 system have been explored by the local spin-density scheme using an analytic potential. Our calculated physical properties, such as dissociation energies and barriers to isomerization, compare well with the available high-level configuration interaction calculations and experiments at the equilibrium geometry. Despite its simplicity, the analytic potential provides an excellent description of the neutral trimer system for silver and the other transition metals.

The fourfold way is a general algorithm for generating diabatic electronic wave functions that span the same space as a small set of variationally optimized adiabatic electronic wave functions and for using the resulting diabatic wave functions to generate diabatic potential energy surfaces and their couplings. In this paper we extend the fourfold way so it is applicable to more complex polyatomic systems and in particular to the calculation of global potential energy surfaces for such systems. The extension involves partitioning the active space into three blocks, introducing restricted orbital rotation within two of the blocks, introducing a specific resolution of the subspace containing molecular orbitals that are doubly occupied in all dominant configuration state functions, and introducing specific orientations of the coordinate systems for reference molecular orbitals and resolution molecular orbitals. The major strength of the improved method presented in this paper is that it allows the diabatic molecular orbitals to exhibit a gradual change of chemical character with smooth deformation along the reaction coordinate for a change of chemical arrangement while preserving the orbital character required for a physical ordering of the orbitals. This feature is required for the convenient construction of global potential energy surfaces for non-Born–Oppenheimer rearrangements. The resulting extended algorithm is illustrated by calculating diabatic potential energy surfaces and couplings for the two lowest singlet potential energy surfaces of HNCO.

We report on evaluation of the diagonal Born–Oppenheimer correction (DBOC) to the electronic energy with Hartree–Fock (HF) and conventional correlated wave functions for general molecular systems. Convergence of both HF and configuration interaction (CI) DBOC with the one-particle basis seems to be rather fast, with triple-ζ quality correlation consistent sets of Dunning et al. sufficiently complete to approach the respective basis set limits for the DBOC of the ground state of H2 within 0.1 cm−1. Introduction of electron correlation via the CI singles and doubles method has a substantial effect on the absolute value of the DBOC for H2, H2O, and BH in their ground states (ca. +13 cm−1 out of 115 cm−1, +22 cm−1 out of 622 cm−1, and +11 cm−1 out of 370 cm−1, respectively). The effect of the correlation correction to the DBOC on relative energies is small, e.g., the barrier to linearity of water changes by ca. 1 cm−1; however, the value is difficult to converge to the ab initio limit. Based on recent results...

Very accurate, rigorous and fully variational, all-particle, non-Born–Oppenheimer calculations of the vibrational spectrum of the H2 molecule have been performed. Very high accuracy has been achieved by expanding the wave functions in terms of explicitly correlated Gaussian functions with preexponential powers of the internuclear distance. An indicator of the high accuracy of the calculations is the new upper bound for the H2 nonrelativistic nonadiabatic ground state energy equal to −1.164 025 030 0 hartree.

A potential energy surface for the major isotopomer of water is constructed by fitting to observed vibration–rotation energy levels of the system using the exact kinetic energy operator nuclear motion program DVR3D. The starting point for the fit is the ab initio Born–Oppenheimer surface of Partridge and Schwenke [J. Chem. Phys. 106, 4618 (1997)] and corrections to it: both one- and two-electron relativistic effects, a correction to the height of the barrier to linearity, allowance for the Lamb shift and the inclusion of both adiabatic and nonadiabatic non-Born–Oppenheimer corrections. Fits are made by scaling the starting potential by a morphing function, the parameters of which are optimized. Two fitted potentials are presented which only differ significantly in their treatment of rotational nonadiabatic effects. Energy levels up to 25 468 cm−1 with J=0, 2, and 5 are fitted with only 20 parameters. The resulting potentials predict experimentally known levels with J⩽10 with a standard deviation of 0.1 cm−1, and are only slightly worse for J=20, for which rotational nonadiabatic effects are significant. The fits showed that around 100 known energy levels are probably the result of misassignments. Analysis of misassigned levels above 20 000 cm−1 leads to the reassignment of 23 transitions.

A direct-potential-fit analysis of all accessible data for the A (1)Σ(+) - X (1)Σ(+) system of NaH and NaD is used to determine analytic potential energy functions incorporating the correct theoretically predicted long-range behaviour. These potentials represent all of the data (on average) within the experimental uncertainties and yield an improved estimate for the ground-state NaH well depth of 𝔇e = 15797.4 (±4.3) cm(-1), which is ∼20 cm(-1) smaller than the best previous estimate. The present analysis also yields the first empirical determination of centrifugal (non-adiabatic) and potential-energy (adiabatic) Born-Oppenheimer breakdown correction functions for this system, with the latter showing that the A-state electronic isotope shift is -1.1(±0.6) cm(-1) going from NaH to NaD.

New high resolution polarization data have been obtained for the A–X band system of 6,7Li2 , and new Fourier transform data for the homonuclear lithium dimers. They are combined with earlier data for 6,6Li2 and 7,7Li2 in the first systematic combined-isotopomer analysis of data for Li2 . This analysis of 8445 rovibrational transitions yields an improved and internally consistent set of molecular constant for the three Li2 isotopomers, and determines the electronic isotope shift and leading vibrational and rotational Born–Oppenheimer breakdown correction terms for both electronic states.

We introduce an analytical model designed to capture the most important features of the electronic matrix elements arising in non-Born–Oppenheimer couplings between a bound anion state and a neutral-molecule-plus-ejected-electron state. In this particle-in-a-radial-box model, vibrations are assumed to cause modulations in the depth (U0) and length (L) parameters of the box. The most important elements of this model are that L is chosen to reproduce the proper dependence of the radial size of the anion’s orbital on electron binding energy, and U0 is chosen to produce the correct electron affinity. Within this model, which is shown to be consistent with trends seen in ab initio calculations of associated electron ejection rates, the coupling matrix elements can be evaluated analytically to provide closed-form expressions for how the rates depend upon (1) the kinetic energy of the ejected electron, (2) the energy spacing between the anion and neutral energy surfaces as a function of geometry, (3) the differe...

In a recently developed approach to ab initio molecular dynamics (ADMP), we used an extended Lagrangian to propagate the density matrix in a basis of atom centered Gaussian functions. Results of trajectory calculations obtained by this method are compared with the Born–Oppenheimer approach (BO), in which the density is converged at each step rather than propagated. For NaCl, the vibrational frequency with ADMP is found to be independent of the fictitious electronic mass and to be equal to the BO trajectory result. For the photodissociation of formaldehyde, H2CO→H2+CO, and the three body dissociation of glyoxal, C2H2O2→H2+2CO, very good agreement is found between the Born–Oppenheimer trajectories and the extended Lagrangian approach in terms of the rotational and vibrational energy distributions of the products. A 1.2 ps simulation of the dynamics of chloride ion in a cluster of 25 water molecules was used as a third test case. The Fourier transform of the velocity–velocity autocorrelation function showed ...

Whenever a quantum chemist extracts the eigenstate of an electronic Hamiltonian, he makes, consciously or not, a decision concerning the phase of the wave function. This is done for each calculated state at each nuclear position. Thus he defines a Born–Oppenheimer (BO) frame of reference. There is no absolute phase just as there is no absolute position or time in mechanics. This leads naturally to the question: What are the quantities which do not depend on the arbitrary phases, i.e., what are the BO invariants? In this article we identify BO invariants with respect to an arbitrary path in nuclear configuration space. We identify invariant electronic states along these paths and their Aharonov–Anandan geometric phases. For closed loops not passing through electronic energy degeneracies these invariant states are the BO adiabatic wave functions and the phases are the Berry phases. The results establish rigorous relations between the full nonadiabatic couplings matrix and the geometric phases.

Multiscale wavelets are used to solve the quantum eigenvalue equations for the hydrogen molecular ion H2+ in the Born–Oppenheimer approximation. Normally restricted to Cartesian systems, “wavelets on the interval” (a normal wavelet family augmented by special edge functions) have recently been applied to such boundary value problems as the hydrogen atom in spherical polar coordinates [J. Mackey, J. L. Kinsey, and B. R. Johnson, J. Comp. Phys. 168, 356 (2001)]. These methods are extended here to ground and excited electronic states of the simplest molecule, for which the electronic Hamiltonian is separable in confocal elliptic coordinates. The set of curvilinear coordinate quantum systems for which wavelet bases have been applied is thus enlarged.

The nuclear motion, Born–Oppenheimer and Breit–Pauli corrections to the diamagnetic susceptibility χd of an isolated ground-state helium atom are evaluated and found to be less than 0.1% of the static nonrelativistic result χd(0).

A method of calculation of rotation–vibration states for a general triatomic that places the body-fixed z axis perpendicular to the plane of the molecule is implemented within a discrete variable representation (DVR) for the vibrational motion. Calculations are presented for water and H3+. For H3+ the new method improves on previous high accuracy ab initio treatments of the rotation–vibration energies of the molecule both in accuracy and the range of rotational states that can be treated. Reliable treatment of quasilinear geometries means that the method is also promising for treating very highly excited states.

Transitions to overtone 2ν2 and 2ν3, and combination ν2+ν3 vibrations in jet-cooled H2D+ and D2H+ molecular ions have been measured for the first time by high-resolution IR spectroscopy. The source of these ions is a pulsed slit jet supersonic discharge, which allows for efficient generation, rotational cooling, and high frequency (100 KHz) concentration modulation for detection via sensitive lock-in detection methods. Isotopic substitution and high-resolution overtone spectroscopy in this fundamental molecular ion permit a systematic, first principles investigation of Born–Oppenheimer “breakdown” effects due to large amplitude vibrational motion as well as provide rigorous tests of approximate theoretical methods beyond the Born–Oppenheimer level. The observed overtone transitions are in remarkably good agreement (

We present non-Born–Oppenheimer calculations of the electrical static polarizability of the LiH molecule. This is the first calculation of the non-Born–Oppenheimer polarizability for a system with more than two electrons. The polarizability is calculated by numerical differentiation of the energy obtained at different electric field strengths. The values for the energy are obtained by variational optimization with analytical gradients of the wave function expanded in a basis of explicitly correlated Gaussian functions. We also present a derivation of the integrals and energy gradients needed to perform these calculations. The result for the polarizability is 29.57 a.u.

A hierarchical family of analytical Born–Oppenheimer potential energy surfaces has been developed for the H+H2 system. Ab initio calculations of near full configuration interaction (FCI) quality (converged to within ≈1 μEh) were performed for a set of 4067 configurations with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets. The complete basis set (CBS) limit energies were obtained using a highly accurate many-body basis set extrapolation scheme. Surfaces were fitted for the estimated CBS limit, as well as for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets using a robust new functional form. The mean unsigned fitting error for the CBS surface is a mere 0.0023 kcal/mol, and deviations for data not included in the fitting process are of similarly small magnitudes. Highly accurate calculations of the saddle point and van der Waals minimum configurations were performed using basis sets as large as aug-mcc-pV7Z, and these data show excellent agreement with the results of the extrapolated pote...

Using Born-Oppenheimer molecular dynamics (BOMD), we explore the nature of interactions between H2 and the activated carbonyl carbon, C(carbonyl), of the acetone-B(C6F5)3 adduct surrounded by an explicit solvent (1,4-dioxane). BOMD simulations at finite (non-zero) temperature with an explicit solvent produced long-lasting instances of significant vibrational perturbation of the H-H bond and H2-polarization at C(carbonyl). As far as the characteristics of H2 are concerned, the dynamical transient state approximates the transition-state of the heterolytic H2-cleavage. The culprit is the concerted interactions of H2 with C(carbonyl) and a number of Lewis basic solvent molecules-i.e., the concerted C(carbonyl)⋯H2⋯solvent interactions. On one hand, the results presented herein complement the mechanistic insight gained from our recent transition-state calculations, reported separately from this article. But on the other hand, we now indicate that an idea of the sufficiency of just one simple reaction coordinate in solution-phase reactions can be too simplistic and misleading. This article goes in the footsteps of the rapidly strengthening approach of investigating molecular interactions in large molecular systems via "computational experimentation" employing, primarily, ab initio molecular dynamics describing reactants-interaction without constraints of the preordained reaction coordinate and/or foreknowledge of the sampling order parameters.

The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calculations to extract accurate integral cross sections (ICSs) and rate constants for comparison with most recent experimental quantities. We calculate non-adiabatic coupling terms among the lowest three singlet states of H3+ system (11A', 21A', and 31A') using MRCI level of calculation and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D+ + H2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H3+. We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum numbers. Finally, total and state-to-state ICSs are calculated as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H2 molecule, and consequently, those quantities are compared with previous theoretical and experimental results.

Using Born-Oppenheimer molecular dynamics (BOMD) with density functional theory, transition-state (TS) calculations, and the quantitative energy decomposition analysis (EDA), we examined the mechanism of H2-liberation from LB—H(+) + (−)H—LA ion-pair, 1, in which the Lewis base (LB) is (o-C6H4Me)3P and the Lewis acid (LA) is B(p-C6F4H)3. BOMD simulations indicate that the path of H2 liberation from the ion-pair 1 goes via the short-lived transient species, LB⋯H2⋯LA, which are structurally reminiscent of the TS-structure in the minimum-energy-path describing the reversible reaction between H2 and (o-C6H4Me)3P/B(p-C6F4H)3 frustrated Lewis pair (FLP). With electronic structure calculations performed on graphics processing units, our BOMD data-set covers more than 1 ns of evolution of the ion-pair 1 at temperature T ≈ 400 K. BOMD simulations produced H2-recombination events with various durations of H2 remaining fully recombined as a molecule within a LB/LA attractive “pocket”—from very short vibrational-time ...

The 33Πg state of the dicarbon molecule, C2, has been identified for the first time by a combination of resonant ionization spectroscopy, mass spectrometry, and high-level ab initio quantum chemical calculations. This marks the discovery of the final valence triplet state of C2 spectroscopically accessible from the lowest triplet state. It is found to be vibronically coupled to the recently discovered 43Πg state, necessitating vibronic calculations beyond the Born-Oppenheimer approximation to reconcile calculated rotational constants with observations. The 33Πg state of C2 is observed to have a much shorter fluorescence lifetime than expected, possibly pointing to predissociation by coupling to the unbound d3Πg state.

We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non-Adiabatic Coupling Terms (NACTs) for the three lowest singlet states of H3 (+) in hyperspherical coordinates as functions of hyperangles (θ and ϕ) for a grid of fixed values of hyperradius (1.5 ⩽ ρ ⩽ 20 bohrs) using the MRCI level of methodology employing ab initio quantum chemistry package (MOLPRO). The NACT between the ground and the first excited state translates along the seams on the θ - ϕ space, i.e., there are six Conical Intersections (CIs) at each θ (60° ⩽ θ ⩽ 90°) within the domain, 0 ⩽ ϕ ⩽ 2π. While transforming the adiabatic PESs to the diabatic ones, such surfaces show up six crossings along those seams. Our beyond Born-Oppenheimer approach could incorporate the effect of NACTs accurately and construct single-valued, continuous, smooth, and symmetric diabatic PESs. Since the location of CIs and the spatial amplitudes of NACTs are most prominent around ρ = 10 bohrs, generally only those results are depicted.

Extended Lagrangian Born-Oppenheimer molecular dynamics based on Kohn-Sham density functional theory is generalized in the limit of vanishing self-consistent field optimization prior to the force evaluations. The equations of motion are derived directly from the extended Lagrangian under the condition of an adiabatic separation between the nuclear and the electronic degrees of freedom. We show how this separation is automatically fulfilled and system independent. The generalized equations of motion require only one diagonalization per time step and are applicable to a broader range of materials with improved accuracy and stability compared to previous formulations., 4 figures 13 pages

The BO neutral, cationic, and anionic molecular species have been painstakingly studied through multireference configuration interaction and single reference coupled cluster methods employing basis sets of quintuple cardinality. Potential energy curves have been constructed for 38 (BO), 37 (BO(+)), and 12 (BO(-)) states and the usual molecular parameters have been extracted most of which are in very good agreement with the scarce experimental data. Numerous avoided crossings appear on more or less all of the studied states of the neutral and cationic species challenging the validity of the Born Oppenheimer approximation. Finally, all excited states of the anionic system lie above the ground state of the neutral BO system and are therefore resonances.

The role of nuclear quantum effects on the adsorption of molecular hydrogen in metal-organic frameworks (MOFs) has been investigated on grounds of Grand-Canonical Quantized Liquid Density-Functional Theory (GC-QLDFT) calculations. For this purpose, we have carefully validated classical H2 -host interaction potentials that are obtained by fitting Born-Oppenheimer ab initio reference data. The hydrogen adsorption has first been assessed classically using Liquid Density-Functional Theory (LDFT) and the Grand-Canonical Monte Carlo (GCMC) methods. The results have been compared against the semi-classical treatment of quantum effects by applying the Feynman-Hibbs correction to the Born-Oppenheimer-derived potentials, and by explicit treatment within the Grand-Canonical Quantized Liquid Density-Functional Theory (GC-QLDFT). The results are compared with experimental data and indicate pronounced quantum and possibly many-particle effects. After validation calculations have been carried out for IRMOF-1 (MOF-5), GC-QLDFT is applied to study the adsorption of H2 in a series of MOFs, including IRMOF-4, -6, -8, -9, -10, -12, -14, -16, -18 and MOF-177. Finally, we discuss the evolution of the H2 quantum fluid with increasing pressure and lowering temperature.

The formation of collision complexes, as a first step towards reaction, in collisions between two open-electronic shell radicals is treated within an adiabatic channel approach. Adiabatic channel potentials are constructed on the basis of asymptotic electrostatic, induction, dispersion, and exchange interactions, accounting for spin-orbit coupling within the multitude of electronic states arising from the separated reactants. Suitable coupling schemes (such as rotational + electronic) are designed to secure maximum adiabaticity of the channels. The reaction between C((3)P) and OH((2)Π) is treated as a representative example. The results show that the low temperature association rate coefficients in general cannot be represented by results obtained with a single (generally the lowest) potential energy surface of the adduct, asymptotically reaching the lowest fine-structure states of the reactants, and a factor accounting for the thermal population of the latter states. Instead, the influence of non-Born-Oppenheimer couplings within the multitude of electronic states arising during the encounter markedly increases the capture rates. This effect extends up to temperatures of several hundred K.

Multiple time-scale algorithms exploit the natural separation of time-scales in chemical systems to greatly accelerate the efficiency of molecular dynamics simulations. Although the utility of these methods in systems where the interactions are described by empirical potentials is now well established, their application to ab initio molecular dynamics calculations has been limited by difficulties associated with splitting the ab initio potential into fast and slowly varying components. Here we show that such a timescale separation is possible using two different schemes: one based on fragment decomposition and the other on range separation of the Coulomb operator in the electronic Hamiltonian. We demonstrate for both water clusters and a solvated hydroxide ion that multiple time-scale molecular dynamics allows for outer time steps of 2.5 fs, which are as large as those obtained when such schemes are applied to empirical potentials, while still allowing for bonds to be broken and reformed throughout the dynamics. This permits computational speedups of up to 4.4x, compared to standard Born-Oppenheimer ab initio molecular dynamics with a 0.5 fs time step, while maintaining the same energy conservation and accuracy., Comment: 28 pages, 7 Figures

A classical molecular dynamics simulation including three-body corrections was compared with combined ab initio quantum mechanics/molecular mechanics molecular dynamics simulations (QM/MM–MD), which were carried out at Hartree–Fock (HF) and density functional theory (DFT) level for Ca2+ in water. In the QM approach the region of primary interest—the first hydration sphere of the calcium ion—was treated by Born–Oppenheimer quantum mechanics, while the rest of the system was described by classical pair potentials. Coordination numbers of 7.1, 7.6, and 8.1 were found in the classical, the HF, and the DFT simulation, respectively, using the same double-ζ basis set in both QM methods. The CPU time for one DFT step was about 50% above the time for a HF step, but due to a smaller number of steps needed for equilibration in the DFT case, there was no significant difference in the overall simulation time.

Ab initio semiglobal potential energy and dipole moment hypersurfaces for the isomerising HCN–HNC system are computed, using a grid of 242 points, principally at the all-electron cc-pCVQZ CCSD(T) level. Several potential energy hypersurfaces (PES) are presented including one which simultaneously fits 1527 points from earlier ab initio, smaller basis CCSD(T) calculations of Bowman et al. [J. Chem. Phys. 99, 308 (1993)]. The resulting potential is then morphed with 17 aug-cc-pCVQZ CCSD(T) points calculated at HNC geometries to improve the representation of the HNC part of the surface. The PES is further adjusted to coincide with three ab initio points calculated, at the cc-pCV5Z CCSD(T) level, at the critical points of the system. The final PES includes relativistic and adiabatic corrections. Vibrational band origins for HCN and HNC with energy up to 12 400 cm−1 above the HCN zero-point energy are calculated variationally with the new surfaces. Band transition dipoles for the fundamentals of HCN and HNC, and a few overtone and hot band transitions for HCN have been calculated with the new dipole surface, giving generally very good agreement with experiment. The rotational levels of ground and vibrationally excited states are reproduced to high accuracy.

We present a new method called the natural decay of mixing (NDM) method for introducing decoherence effects into the semiclassical Ehrenfest self-consistent potential method. The NDM method is similar in spirit to two recently developed methods, the continuous surface switching (CSS) and continuous surface switching II (CSS2) methods, but, like the pure semiclassical Ehrenfest method, it involves only a single variable that serves as both the weight of an electronic state and its electronic population. We demonstrate how this allows the NDM method to be applied to systems where the CSS and CSS2 methods cannot be applied, and also to cases where the CSS and CSS2 methods would be prohibitively expensive. The method is tested for electronically nonadiabatic processes, both reactive and nonreactive, and in a wider context it contributes to the rapidly blossoming fields of quantum measurement and hybrid quantum/classical algorithms for the dynamics of complex systems.

We report the first ever non-Born–Oppenheimer variational calculations on the ground state of a four electron molecular system. The basis set used in the calculations consists of explicitly correlated Gaussians multiplied by powers of the internuclear distance. To accelerate the optimization of the many nonlinear variational parameters involved in the variational wave function, we performed the calculations on a cluster of Linux workstations using MPI and a parallel implementation of the formulas. Results for the nonadiabatic ground state energy of LiH, as well as expectation values for the kinetic and potential energies, the internuclear and square of the internuclear distance, the virial coefficient, and the square of the energy gradient norm are reported.