Diagonal Born-Oppenheimer Correction Research Articles

Numerical convergence of the Sinc discrete variable representation for solving molecular vibrational states with a conical intersection in adiabatic representation

Within the Born-Oppenheimer (BO) approximation, nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface (PES). However, this single PES picture should be reconsidered if a conical intersection (CI) is present, although the energy is well below the CI. The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation, i.e., the diagonal BO correction (DBOC) and the geometric phase (GP), which are divergent at the CI. At the same time, there are cusps in the adiabatic PESs. Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation. A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation (DVR) method. We examine the numerical accuracy of the Sinc DVR method for solving the Schrödinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation. The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points, without special treatment of the divergence of the DBOC and the GP. The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP, whose accurate form usually is not easy to obtain.

Thermochemistry of Uracil, Thymine, Cytosine, and Adenine.

The goal of this study is to give reliable and accurate thermochemical data for uracil, thymine, cytosine, and adenine. The gas-phase heats of formation of these compounds were determined with the diet-HEAT-F12 protocol, which uses explicitly correlated coupled-cluster calculations along with anharmonic vibrational, scalar relativistic, and diagonal Born-Oppenheimer corrections. The thermochemically relevant tautomers of cytosine were also investigated. To derive heats of formation in the gas and solid phases as well as sublimation enthalpies, the thermochemical network approach was utilized, i.e., the available literature data were collected, reviewed, and combined with our theoretical calculations. Solvation free energies were also determined with various methods and compared to experimental data.

A discontinuous basis enables numerically exact solution of the Schrödinger equation around conical intersections in the adiabatic representation.

Solving the vibrational Schrödinger equation in the neighborhood of conical intersections in the adiabatic representation is a challenge. At the intersection point, first- and second-derivative nonadiabatic coupling matrix elements become singular, with the singularity in the second-derivative coupling (diagonal Born-Oppenheimer correction) being non-integrable. These singularities result from discontinuities in the vibronic functions associated with the individual adiabatic states, and our group has recently argued that these divergent matrix elements cancel when discontinuous adiabatic vibronic functions sum to a continuous total nonadiabatic wave function. Here we describe the realization of this concept: a novel scheme for the numerically exact solution of the Schrödinger equation in the adiabatic representation. Our approach is based on a basis containing functions that are discontinuous at the intersection point. We demonstrate that the individual adiabatic nuclear wave functions are themselves discontinuous at the intersection point. This proves that discontinuous basis functions are essential to any tractable method that solves the Schrödinger equation around conical intersections in the adiabatic representation with high numerical precision. We establish that our method provides numerically exact results by comparison to reference calculations performed in the diabatic representation. In addition, we quantify the energetic error associated with constraining the density to be zero at the intersection point, a natural approximation. Prospects for extending the present treatment of a two-dimensional model to systems of higher dimensionality are discussed.

Non-adiabatic mass-correction functions and rovibrational states of 4 ( ).

The mass-correction functions in the second-order non-adiabatic Hamiltonian are computed for the 4 molecular ion using the variational method, floating explicitly correlated Gaussian functions, and a general coordinate-transformation formalism. When non-adiabatic rovibrational energy levels are computed using these (coordinate-dependent) mass-correction functions and a highly accurate potential energy and diagonal Born-Oppenheimer correction curve, significantly improved theoretical results are obtained for the nine rotational and two rovibrational intervals known from high-resolution spectroscopy experiments.

Calculations of electronic properties and vibrational parameters of alkaline-earth lithides: MgLi+ and CaLi+

ABSTRACTThe electronic ground states of MgLi and CaLi molecular ions are investigated for their spectroscopic constants and properties such as the dipole - and quadrupole moments, and static dipole polarisabilities. The quadrupole moments and the static dipole polarisabilities for these ions have been calculated and reported here, for the first time. The maximum possible error bars, arising due to the finite basis set and the exclusion of higher correlation effects beyond partial triples, are quoted for reliability. Further, the adiabatic effects such as diagonal Born-Oppenheimer corrections are also calculated for these molecules. The vibrational energies, the wavefunctions, and the relevant vibrational parameters are obtained by solving the vibrational Schrödinger equation using the potential energy curve and the permanent dipole moment curve of the molecular electronic ground state. Thereafter, spontaneous and black-body radiation induced transition rates are calculated to obtain the lifetimes of the vibrational states. The lifetime of rovibronic ground state for MgLi, at room temperature, is found to be and for CaLi it is . It has been observed that the lifetime of the highly excited vibrational state is several times larger than (comparable to) that of the vibrational ground state of MgLi (CaLi). In addition, a few low-lying electronic excited states of Σ and Π symmetries have been investigated for their electronic and vibrational properties, using EOM-CCSD method together with the QZ basis sets.

Impact of Post-CCSD(T) Corrections on Reaction Energetics and Rate Constants of the OH• + HCl Reaction.

High level ab initio calculations have been performed to predict the reaction energy and barrier height for the OH• + HCl reaction. After including the effect of full quadratic excitations at the coupled cluster level, in addition to core, relativistic, spin-orbit, and diagonal Born-Oppenheimer corrections, we found the values of reaction energy and barrier height to be -15.29 and +2.38 kcal mol-1, respectively. Employing this reaction energy and barrier height, we used variational transition state theory in conjunction with small curvature tunneling to calculate the rate constants within a temperature range from 138 to 1000 K. The calculated rate constants were found to be in good agreement with available experimental results throughout the whole temperature range.

High Accuracy Quantum Chemical and Thermochemical Network Data for the Heats of Formation of Fluorinated and Chlorinated Methanes and Ethanes.

Reliable heats of formation are reported for numerous fluorinated and chlorinated methane and ethane derivatives by means of an accurate thermochemical protocol, which involves explicitly correlated coupled-cluster calculations augmented with anharmonic, scalar relativistic, and diagonal Born-Oppenheimer corrections. The theoretical results, along with additional experimental data, are further enhanced with the help of the thermochemical network approach. For 28 species, out of 50, this study presents the best estimates, and discrepancies with previous reports are also highlighted. Furthermore, the effects of the less accurate theoretical data on the results yielded by thermochemical networks are discussed.

Diagonal Born–Oppenheimer correction for coupled-cluster wave-functions

ABSTRACTWe examine how geometry-dependent normalisation freedom of electronic wave-functions affects extraction of a meaningful diagonal Born–Oppenheimer correction (DBOC) to the ground-state Born–Oppenheimer potential energy surface (PES). By viewing this freedom as a kind of gauge-freedom, it is shown that DBOC and the resulting associated mass-dependent adiabatic PES are gauge-invariant quantities. A sum-over-states (SOS) formula for DBOC which explicitly exhibits this invariance is derived. A biorthogonal formulation suitable for DBOC computations using standard unnormalised coupled-cluster (CC) wave-functions is presented. This is shown to lead to a biorthogonal version of SOS formula with similar properties. On this basis, different computational schemes for evaluating DBOC using approximate CC wave-functions are derived. One of this agrees with the formula used in the current literature. The connection to adiabatic-to-diabatic transformations in non-adiabatic dynamics is explored and complications arising from biorthogonal nature of CC theory are identified.

Accurate ab initio calculations of spectroscopic constants and properties of BeLi+

We have calculated the spectroscopic constants and the molecular properties, for the ground-state (X1Σ+) of BeLi+, such as dipole moment, quadrupole moment and dipole polarizability at different levels of correlation: many-body perturbation theory (MP2), coupled cluster method with single and double excitations (CCSD) and CCSD with perturbative triples (CCSD(T)). The correlation consistent polarized valence cc-pVXZ (X = D, T, Q) basis sets and also their augmented counterparts are used together with the non-relativistic and relativistic Hamiltonians. The results are extrapolated to the complete basis set limit (CBS) using exponential-Gaussian functions. Thus, accurate and reliable results for BeLi+ with the maximum possible error bars on them are reported. The effects of diagonal Born-Oppenheimer corrections (DBOC) and of electron correlations beyond the CCSD(T) level of approximation are also estimated. In addition, by solving the vibrational Schrödinger equation using the potential energy curve obtained at CCSD(T)/QZ level, we have calculated the vibrational parameters such as energies, rotational constants, permanent dipole moments, spontaneous and black-body radiation (BBR) induced transition rates between different vibrational levels of the electronic ground state. The lifetime of the lowest rovibrational state (v=0 and J=0) at room temperature, is found to be 4.67s. Further, the electronic and the vibrational parameters of a few low-lying electronic excited states: 13Σ+, 13Π, 2–41Σ+, and 1–21Π are calculated using the EOM-CCSD/QZ method. The dipole moments for the electronic transitions from the ground- to the excited singlet states are also reported.

The multichannel n-propyl + O2 reaction surface: Definitive theory on a model hydrocarbon oxidation mechanism

The n-propyl + O2 reaction is an important model of chain branching reactions in larger combustion systems. In this work, focal point analyses (FPAs) extrapolating to the ab initio limit were performed on the n-propyl + O2 system based on explicit quantum chemical computations with electron correlation treatments through coupled cluster single, double, triple, and perturbative quadruple excitations [CCSDT(Q)] and basis sets up to cc-pV5Z. All reaction species and transition states were fully optimized at the rigorous CCSD(T)/cc-pVTZ level of theory, revealing some substantial differences in comparison to the density functional theory geometries existing in the literature. A mixed Hessian methodology was implemented and benchmarked that essentially makes the computations of CCSD(T)/cc-pVTZ vibrational frequencies feasible and thus provides critical improvements to zero-point vibrational energies for the n-propyl + O2 system. Two key stationary points, n-propylperoxy radical (MIN1) and its concerted elimination transition state (TS1), were located 32.7 kcal mol−1 and 2.4 kcal mol−1 below the reactants, respectively. Two competitive β-hydrogen transfer transition states (TS2 and TS2′) were found separated by only 0.16 kcal mol−1, a fact unrecognized in the current combustion literature. Incorporating TS2′ in master equation (ME) kinetic models might reduce the large discrepancy of 2.5 kcal mol−1 between FPA and ME barrier heights for TS2. TS2 exhibits an anomalously large diagonal Born-Oppenheimer correction (ΔDBOC = 1.71 kcal mol−1), which is indicative of a nearby surface crossing and possible nonadiabatic reaction dynamics. The first systematic conformational search of three hydroperoxypropyl (QOOH) intermediates was completed, uncovering a total of 32 rotamers lying within 1.6 kcal mol−1 of their respective lowest-energy minima. Our definitive energetics for stationary points on the n-propyl + O2 potential energy surface provide key benchmarks for future studies of hydrocarbon oxidation.

High Accuracy ab Initio Calculations of Rotational-Vibrational Levels of the HCN/HNC System.

Highly accurate ab initio calculations of vibrational and rotational-vibrational energy levels of the HCN/HNC (hydrogen cyanide/hydrogen isocyanide) isomerising system are presented for several isotopologues. All-electron multireference configuration interaction (MRCI) electronic structure calculations were performed using basis sets up to aug-cc-pCV6Z on a grid of 1541 geometries. The ab initio energies were used to produce an analytical potential energy surface (PES) describing the two minima simultaneously. An adiabatic Born-Oppenheimer diagonal correction (BODC) correction surface as well as a relativistic correction surface were also calculated. These surfaces were used to compute vibrational and rotational-vibrational energy levels up to 25 000 cm-1 which reproduce the extensive set of experimentally known HCN/HNC levels with a root-mean-square deviation σ = 1.5 cm-1. We studied the effect of nonadiabatic effects by introducing opportune radial and angular corrections to the nuclear kinetic energy operator. Empirical determination of two nonadiabatic parameters results in observed energies up to 7000 cm-1 for four HCN isotopologues (HCN, DCN, H13CN, and HC15N) being reproduced with σ = 0.37 cm-1. The height of the isomerization barrier, the isomerization energy and the dissociation energy were computed using a number of models; our best results are 16 809.4, 5312.8, and 43 729 cm-1, respectively.

Moderate-Cost Ab Initio Thermochemistry with Chemical Accuracy.

A moderate-cost ab initio composite model chemistry including the explicitly correlated CCSD(T*)(F12) and conventional coupled-cluster methods up to perturbative quadruple excitations along with correlation consistent basis sets is developed. The model, named diet-HEAT-F12, is also augmented with diagonal Born-Oppenheimer and scalar relativistic corrections. The methods and basis sets used for the calculation of the individual components are selected to reproduce, as close as possible, without using any fitted parameters, the benchmark HEAT contributions. A well-defined recipe for calculating size-dependent 95% confidence intervals was also worked out for the model. The reliability of the protocol was checked using the W4-11 data set as well as a disjoint set of 23 accurate atomization energies collected from the literature and obtained by the procedure of Feller, Peterson, and Dixon. The best error statistics for the test set was yielded by the diet-HEAT-F12 protocol among the models W3X, W3X-L, and W3-F12 considered.

Note: On the inclusion of a diagonal Born-Oppenheimer correction in the reduced dimensional treatment of the H2O-para-H2 complex.

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Constructive and Destructive Interference in Nonadiabatic Tunneling via Conical Intersections.

As a manifestation of the molecular Aharonov-Bohm effect, tunneling-facilitated dissociation under a conical intersection (CI) requires the inclusion of the geometric phase (GP) to ensure a single-valued adiabatic wave function encircling the CI. In this work, we demonstrate using a simple two-dimensional model that the GP induces destructive interference for vibrational states with even quanta in the coupling mode, but it leads to constructive interference for those with odd quanta. The interference patterns are manifested in tunneling wave functions and clearly affect the tunneling lifetime. It is further shown that the inclusion of the diagonal Born-Oppenheimer correction is necessary for agreement with exact results.

Photodissociation of phenol via nonadiabatic tunneling: Comparison of two ab initio based potential energy surfaces

The nonadiabatic tunneling-facilitated photodissociation of phenol is investigated using a reduced-dimensional quantum model on two ab initio-based coupled potential energy surfaces (PESs). Although dynamics occurs largely on the lower adiabat, the proximity to a conical intersection between the S1 and S2 states requires the inclusion of both the geometric phase (GP) and diagonal Born-Oppenheimer correction (DBOC). The lifetime of the lowest-lying vibronic state is computed using the diabatic and various adiabatic models. The GP and DBOC terms are found to be essential on one set of PESs, but have a small impact on the other.

Benchmark-Quality Atomization Energies for BeH and BeH2.

The total atomization energies for BeH and BeH2 have been calculated using the Feller-Peterson-Dixon approach to better than ±1 kcal/mol. The calculations are based on CCSD(T) all-electron calculations extrapolated to the complete limit, and CCSDT and CCSDTQ corrections are included. A scalar relativistic correction and a diagonal Born-Oppenheimer correction are included. Accurate zero-point energies are used. The total atomization energies at 0 K are 47.7 kcal/mol for BeH and 140.0 kcal/mol for BeH2 with error of at most ±0.3 kcal/mol.

Accurate Theoretical Thermochemistry for Fluoroethyl Radicals.

An accurate coupled-cluster (CC) based model chemistry was applied to calculate reliable thermochemical quantities for hydrofluorocarbon derivatives including radicals 1-fluoroethyl (CH3-CHF), 1,1-difluoroethyl (CH3-CF2), 2-fluoroethyl (CH2F-CH2), 1,2-difluoroethyl (CH2F-CHF), 2,2-difluoroethyl (CHF2-CH2), 2,2,2-trifluoroethyl (CF3-CH2), 1,2,2,2-tetrafluoroethyl (CF3-CHF), and pentafluoroethyl (CF3-CF2). The model chemistry used contains iterative triple and perturbative quadruple excitations in CC theory, as well as scalar relativistic and diagonal Born-Oppenheimer corrections. To obtain heat of formation values with better than chemical accuracy perturbative quadruple excitations and scalar relativistic corrections were inevitable. Their contributions to the heats of formation steadily increase with the number of fluorine atoms in the radical reaching 10 kJ/mol for CF3-CF2. When discrepancies were found between the experimental and our values it was always possible to resolve the issue by recalculating the experimental result with currently recommended auxiliary data. For each radical studied here this study delivers the best heat of formation as well as entropy data.

The best of both Reps-Diabatized Gaussians on adiabatic surfaces.

When simulating nonadiabatic molecular dynamics, choosing an electronic representation requires consideration of well-known trade-offs. The uniqueness and spatially local couplings of the adiabatic representation come at the expense of an electronic wave function that changes discontinuously with nuclear motion and associated singularities in the nonadiabatic coupling matrix elements. The quasi-diabatic representation offers a smoothly varying wave function and finite couplings, but identification of a globally well-behaved quasi-diabatic representation is a system-specific challenge. In this work, we introduce the diabatized Gaussians on adiabatic surfaces (DGAS) approximation, a variant of the ab initio multiple spawning (AIMS) method that preserves the advantages of both electronic representations while avoiding their respective pitfalls. The DGAS wave function is expanded in a basis of vibronic functions that are continuous in both electronic and nuclear coordinates, but potentially discontinuous in time. Because the time-dependent Schrödinger equation contains only first-order derivatives with respect to time, singularities in the second-derivative nonadiabatic coupling terms (i.e., diagonal Born-Oppenheimer correction; DBOC) at conical intersections are rigorously absent, though singular time-derivative couplings remain. Interpolation of the electronic wave function allows the accurate prediction of population transfer probabilities even in the presence of the remaining singularities. We compare DGAS calculations of the dynamics of photoexcited ethene to AIMS calculations performed in the adiabatic representation, including the DBOC. The 28 fs excited state lifetime observed in DGAS simulations is considerably shorter than the 50 fs lifetime observed in the adiabatic simulations. The slower decay in the adiabatic representation is attributable to the large, repulsive DBOC in the neighborhood of conical intersections. These repulsive DBOC terms are artifacts of the discontinuities in the individual adiabatic vibronic basis functions and therefore cannot reflect the behavior of the exact molecular wave function, which must be continuous.

Investigating the ground-state rotamers of n-propylperoxy radical.

The n-propylperoxy radical has been described as a molecule of critical importance to studies of low temperature combustion. Ab initio methods were used to study this three-carbon alkylperoxy radical, normal propylperoxy. Reliable CCSD(T) (coupled-cluster theory, incorporating single, double, and perturbative triple)/ANO0 geometries were predicted for the molecule's five rotamers. For each rotamer, energetic predictions were made using basis sets as large as the cc-pV5Z in conjunction with coupled cluster levels of theory up to CCSDT(Q). Along with the extrapolations, corrections for relativistic effects, zero-point vibrational energies, and diagonal Born-Oppenheimer corrections were used to further refine energies. The results indicate that the lowest conformer is the gauche-gauche (GG) rotamer followed by the gauche-trans (0.12 kcal mol-1 above GG), trans-gauche (0.44 kcal mol-1), gauche'-gauche (0.47 kcal mol-1), and trans-trans (0.57 kcal mol-1). Fundamental vibrational frequencies were obtained using second-order vibrational perturbation theory. This is the first time anharmonic frequencies have been computed for this system. The most intense IR features include all but one of the C-H stretches. The O-O fundamental (1063 cm-1 for the GG structure) also has a significant IR intensity, 19.6 km mol-1. The anharmonicity effects on the potential energy surface were also used to compute vibrationally averaged rg,0K bond lengths, accounting for zero-point vibrations present within the molecule.

Theoretical investigations on mechanisms and kinetics of the O+CF3I reaction

Both singlet and bifurcated triplet (3A″ and 3A′) potential energy surfaces for the O+CF3I reaction were explored systematically using density functional M06-2X method with the triple- and quadruple-ζ quality and small-core energy-consistent relativistic pseudopotentials. Single-point energies were calculated using the explicitly correlated CCSD(T)-F12 method. On the triplet surface, the association of O(3P) with CF3I leads to the weakly bound complexes. The subsequent decomposition or isomerization surmounts high barriers, indicative of the little importance of the triplet energetic routes for the title reaction. On the singlet surface, a deep well corresponding to the s-CF3IO intermediate exists. Moreover, the singlet-triplet surface crossing to form s-CF3IO from the O(3P)+CF3I has been characterized by locating the minimum on the seam of intersection using the multireference SA-CASSCF method and the single-point energy computations with RS2 and MRCISD methods. It was identified that the subsequent CI bond decomposition of singlet s-CF3IO is the dominant mechanism for the IO radical production in the O(3P)+CF3I reaction. Neither atomic iodine nor IF molecule production is significant because of the high barrier for the isomerization of s-CF3IO to s-CF3OI. The energies of some key stationary points on the dominant reaction routes were corrected with CCSDT, CCSDT(Q), and an extrapolation to FCI, together with spin-orbit coupling and diagonal Born-Oppenheimer correction. The theoretical rate constants are in good agreement with the available experimental data. The overall rate constants exhibit positive temperature dependence in the range 200–3000Kand could be expressed as k(T)=6.6×10−12(T/298)0.74e−108/Tcm3molecule−1s−1. No pressure dependence of the rate constant has been found.