Oppenheimer Separation Research Articles

Insights into the Mechanism of Nonadiabatic Photodissociation from Product Vibrational Distributions. The Remarkable Case of Phenol.

The fate of a photoexcited molecule is often strongly influenced by electronic degeneracies, such as conical intersections, which break the Born-Oppenheimer separation of electronic and nuclear motion. Detailed information concerning internal energy redistribution in a nonadiabatic process can be extracted from the product state distribution of a photofragment in photodissociation. Here, we focus on the nonadiabatic photodissociation of phenol and discuss the internal excitation of the phenoxyl fragment using both symmetry analysis and wave packet dynamics. It is shown that unique and general selection rules exist, which can be attributed to the geometric phase in the adiabatic representation. Further, our results provide a reinterpretation of the experimental data, shedding light on the impact of conical intersections on the product state distribution.

On the Validity of the Born−Oppenheimer Separation and the Accuracy of Diagonal Corrections in Anharmonic Molecular Vibrations

The energies and wave functions of several lowest-lying vibrational states of FHF(-), ClHCl(-), and BrHBr(-) have been computed by a finite-difference method with and without the Born-Oppenheimer (BO) separation between the heavy (halogen) and light (hydrogen) particle motion. The so-called diagonal BO correction (DBOC), which includes the effect of the heavy particles' kinetic energy operator acting on the light particles' wave functions, has also been made to the energies. The errors caused by the BO approximation are found to be remarkably small (ca. 10(-5) au) and can be systematically and effectively reduced by the DBOC except for states excited in the heavy particle motion. When the bare mass of the light particle is used instead of the reduced mass in the BO approximation and, therefore, the translational degrees of freedom are not correctly decoupled, the errors in the BO treatment become greater by a factor of 2-7. However, these additional errors are almost completely erased by the DBOC. Analytical and numerical results suggest that the remaining errors in the BO and DBOC treatments be proportional to epsilon(1) and epsilon(2), where epsilon is the mass ratio of the light to heavy particles, when the corrections are made to the potential energy surfaces and the wave functions for these surfaces are determined variationally. When the DBOC is applied in the first-order perturbation approximation, the remaining errors scale as epsilon(3/2).

The chemist’s concept of molecular structure

The concept of molecular structure is fundamental to the practice and understanding of chemistry, but the meaning of this term has evolved and is still evolving. The Born–Oppenheimer separation of electronic and nuclear motions lies at the heart of most modern quantum chemical models of molecular structure. While this separation introduces a great computational and practical simplification, it is neither essential to the conceptual formulation of molecular structure nor universally valid. Going beyond the Born–Oppenheimer approximation introduces new paradigms, bringing fresh insight into the chemistry of fluxional molecules, proteins, superconductors and macroscopic dielectrics, thus opening up new avenues for exploration. But it requires that our ideas of molecular structure need to evolve beyond simple ball-and-stick-type models.

Exact, Born–Oppenheimer, and quantum-chemistry-like calculations in helium clusters doped with light molecules: The He2N2(X) system

Helium clusters doped with diatomic molecules, He(N)-BC, have been recently studied by means of a quantum-chemistry-like approach. The model treats He atoms as "electrons" and dopants as "nuclei" in standard electronic structure calculations. Due to the large mass difference between He atoms and electrons, and to the replacement of Coulomb interactions by intermolecular potentials, it is worth assessing up to what extent are the approximations involved in this model, i.e., decoupling of the BC rotation from the He-atom orbital angular momenta and Born-Oppenheimer separation of the BC stretch versus the He motions, accurate enough. These issues have been previously tackled elsewhere for the (4)He(2)-Br(2)(X) system, which contains a heavy dopant [Roncero et al., Int. J. Quantum Chem. 107, 2756 (2007)]. Here, we consider a similar cluster but with a much lighter dopant such as N(2)(X). Although the model does not provide the correct energy levels for the cluster, positions and intensities of the main detectable lines of the vibrotational Raman spectrum at low temperature are accurately reproduced.

Theory of Diatomic Molecules in an External Electromagnetic Field from First Quantum Mechanical Principles

We study a general problem of the translational/rotational/vibrational/electronic dynamics of a diatomic molecule exposed to an interaction with an arbitrary external electromagnetic field. The theory developed in this paper is relevant to a variety of specific applications, such as alignment or orientation of molecules by lasers, trapping of ultracold molecules in optical traps, molecular optics and interferometry, rovibrational spectroscopy of molecules in the presence of intense laser light, or generation of high order harmonics from molecules. Starting from the first quantum mechanical principles, we derive an appropriate molecular Hamiltonian suitable for description of the center of mass, rotational, vibrational, and electronic molecular motions driven by the field within the electric dipole approximation. Consequently, the concept of the Born-Oppenheimer separation between the electronic and the nuclear degrees of freedom in the presence of an electromagnetic field is introduced. Special cases of the dc/ac-field limits are then discussed separately. Finally, we consider a perturbative regime of a weak dc/ac field, and obtain simple analytic formulas for the associated Born-Oppenheimer translational/rotational/vibrational molecular Hamiltonian.

Comparison Between a Generalized Electronic Diabatic Approach and the Born–Oppenheimer Separation Scheme in Inertial Frames

Comparison between a generalized electronic diabatic approach and the Born-Oppenheimer separation scheme in inertial frames

Conical Intersections: The New Conventional Wisdom

During the past decade the perception of conical intersections has changed. It is now appreciated that what was once viewed largely as a theoretical curiosity is an essential aspect of electronically nonadiabatic processes. Concomitantly, our understanding of this singular consequence of the Born−Oppenheimer separation of nuclear and electronic motion has grown enormously. In this work the theory of conical intersections is reviewed.

Electron−Nuclear Coupling in the Classical Limit for the Electronic Degrees of Freedom

The use of a classical limit for the electronic degrees of freedom avoids the need to keep the nuclei clamped while solving for the dynamics of the electrons. The Hamiltonian for the electrons will then depend on the nuclear coordinates as dynamical variables. The resulting (classical) electron−nuclear coupled equations of motion exhibit dynamical symmetry and are shown to depend only on the ratio, κ4, of the electron to nuclear mass. We explore the coupled electron−nuclear dynamics as a function of κ for the special case of a single electron moving between two centers. In the dynamical regime where the nuclei are heavy and the Born−Oppenheimer separation should work, the full dynamical procedure is in excellent agreement with the nuclear dynamics as computed using the Born−Oppenheimer separation. In the opposite regime where the period of the electronic motion is long, a case that can be physically realized for very high Rydberg states, one reaches an ‘inverse' behavior where the nuclei adiabatically adj...

Beyond the Potential Energy Surface: Ab initio Corrections to the Born−Oppenheimer Approximation for H2O

It is customary when computing ro−vibrational transitions in molecules to invoke the Born−Oppenheimer separation between nuclear and electronic motion. However, it is known from accurate calculations on H2+ and H2 that the first-order (diagonal adiabatic) and second-order (nonadiabatic) corrections are not negligible and are both important. In the present work, we have made an ab initio implementation of the Bunker and Moss formalism for the nonadiabatic correction and applied it to H2 and H2O. From comparison to accurate calculations for H2, we find that we can obtain good results for the nonadiabatic correction using CI singles to treat the electronically excited states if we scale the results, but we must go beyond the SCF approximation to obtain an accurate diagonal adiabatic correction. For H2O, we find that the first-order correction is more important than the second-order correction for bending energy levels, but the second-order correction is more important than the first-order correction for stre...

Rovibrational Energies of Triatomic Molecules by Means of the Rayleigh–Schrödinger Perturbation Theory

A Rayleigh–Schrödinger perturbation theory approach based on the adiabatic (Born–Oppenheimer) separation of vibrational motions was previously developed and used to evaluate for a system of coupled oscillators the adiabatic energy levels and their nonadiabatic corrections. This method is applied here to calculate rotation–vibration energies of the triatomic molecular ions HeH+2 and ArNO+ consisting of a strongly bound diatomic fragment and a relatively loosely bound rare gas atom. In these systems the high-frequency stretching motion of the diatomic fragment can be separated from the other two low-frequency motions without substantial loss of accuracy. Treating the diatomic fragment as a rigid rotor, the low-frequency stretching motion is decoupled from the bending motion in analogy to the concept of the adiabatic (Born–Oppenheimer) separation of motions and the strong nonadiabatic couplings between these two motions are accounted for perturbationally. Although the resulting perturbation series may show poor convergence, they turn out to be accurately summable by applying standard techniques for the summation of divergent series. Comparison with the results obtained from full-dimensional calculations for the two ions shows that the approach is capable of providing accurate energies for quite a few of the bound rotation–vibration states and that in the case of the HeH+2 ion it is even able to predict the positions and widths of some low-lying resonance states with good accuracy. The perturbation approach yields zeroth-order energies and corrections in terms of the relevant quantum numbers. It thus allows a direct assignment of the energy levels without any reference to the corresponding eigenfunctions. The weak couplings between the high- and low-frequency motions can easily be treated by the same perturbative approach and numerically exact energies can finally be obtained.

Multidimensional stochastic models for the study of molecular motions in liquids and liquid crystals

Multidimensional stochastic equations are set up to interpret rotational molecular dynamics in isotropic solutions and liquid crystalline phases. The description of the dynamics is based on standard or augmented Fokker—Planck type equations, containing a classical Liouville term and dissipation terms including fluctuating interactions with the local environment. Numerically exact solutions of the equations are obtained by using orthogonal functions of the phase space coordinates (Euler angles, conjugated momenta, and solvent collective variables) and special algorithms for handling large and sparse non-Hermitian matrices. Under regimes of particular dynamics, Born—Oppenheimer separation is employed. Solutions of the dynamic problem are provided in the time-correlation function language, suitable for direct interpretation of spectroscopic observables. As an example, the interpretation of quasi-elastic neutron scattering experiments for molecules with a single scattering centre in a nematic liquid crystalline phase is presented.

A review on the derivation of the spin-Restricted Hartree—Fock (RHF) Self-Consistent Field (SCF) equations for open-shell systems. Description of different methods to handle the off-diagonal Lagrangian multipliers coupling closed and open shells

A detailed review is presented on the spin-Restricted Hartree—Fock (RHF) Self-Consistent Field (SCF) method (i) for high-spin ( M s = S) open-shell systems and (ii) for singlet systems with two occupied open-shell orbitals, either of two different irreducible representations (Irreps) or of the same Irrep. Basic knowledge, such as the Born—Oppenheimer separation and approximation, the model of independent particles, Slater determinants and important rules of Group Theory, is shortly summarized. The RHF equations are derived by a minimization of the energy expectation value for the lowest state of a given Irrep and spin multiplicity, described by a spin-adapted wavefunction composed of a single Slater determinant in the high-spin case and a linear combination of two determinants in the open-shell singlet case. Molecular symmetry is considered throughout the paper. The Roothaan method as well as two other widely used methods to handle the orthogonality constraints for the spin orbitals of the respective Slater determinant(s) are described. Detailed algorithms for performing an RHF—SCF procedure with each of these methods are presented.

An internal coordinate model of coupling between the torsion and C–H vibrations in methanol

The torsional tunneling splittings of the asymmetric C–H stretches (ν2 and ν9) in methanol are inverted with the E level lower in energy than the A level, whereas the symmetric C–H stretch (ν3) is normal with A below E. An internal coordinate model, which treats the torsion and the three C–H stretches simultaneously, accounts for the observed tunneling splittings. The model parameters are the local stretching frequency ω=2934.0 cm−1, the direct local–local coupling λ=−42.2 cm−1, and a single stretch-torsion coupling parameter μ=12.9 cm−1. The torsion-vibration coupling is nonadiabatic in the sense that it is not consistent with a Born–Oppenheimer separation of the torsion from the other vibrations. The fact that the model is based largely on the G6 molecular symmetry suggests that tunneling inversion may be common in torsional molecules. The torsionally mediated couplings among the C–H stretches do not conserve symmetry in the Cs point group and are strong enough to contribute to rapid intramolecular vibrational redistribution (IVR).

An adiabatic model for rare gas–halogen van der Waals complexes: application to HeBr2(B).

An adiabatic approach to compute quasibound states of rare gas–halogen van der Waals molecules is presented where the interhalogen vibrational coordinate is considered as the `slow' mode (i.e. it takes the place of the nuclear coordinates in the traditional Born–Oppenheimer separation). Although it might not seem intuitive (as far as the halogen vibrational motion is not slower than the intermolecular one), it is shown that a satisfactory (zero order) picture is provided for those states where the halogen subunit is near dissociation. The method is applied to HeBr2 (B,v), for which experimental data at very high v are available. Results are discussed in light of comparisons with `exact' close-coupling calculations.

On the inverse Born–Oppenheimer separation for high Rydberg states of molecules

On the inverse Born–Oppenheimer separation for high Rydberg states of molecules

On the inverse Born-Oppenheimer separation for high Rydberg states of molecules

The separation of radial electronic and nuclear motions is discussed with special reference to high Rydberg states of molecules. An inverse separation is obtained when the rapid nuclear motion instantaneously adjusts itself to the position of the Rydberg electron. The electron moves in the potential averaged over the position of the nuclei (and their valence electrons). This inverse separation is useful when ωn3 > 1, where ω is the spacing of nuclear energy states (in au) and n is the principal quantum number of the Rydberg electron whose orbital period increases as n3. The inverse Born–Oppenheimer separation can break down owing to the finite kinetic energy of the Rydberg electron. Like the Born–Oppenheimer separation, its inverse can also be formulated in an adiabatic or a diabatic basis. The diabatic inverse Born–Oppenheimer is practical both for interpretation of zero electron kinetic energy (ZEKE) spectra and for computations. Explicit results are given for a model system of an electron orbiting a vibrating dipole, identifying the relevant coupling constants. The discussion emphasizes the radial motion and the limits discussed here are not quite equivalent to the four (or, actually, five) Hund's coupling cases relevant to angular momentum coupling schemes. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 85–100, 1998

Calculation of Rotation–Vibration Energy Levels in Ground State C3by a Born–Oppenheimer-Type Separation of the Vibrational Motions

The MORBID (Morse Oscillator Rigid Bender Internal Dynamics) model [P. Jensen,J. Mol. Spectrosc.128,478–501 (1988)] has been used to determine effective bending potentials and reduced masses for C3in the lowest stretching vibrational states by performing a Born–Oppenheimer separation of the bending and stretching motions. These calculations were based on a potential energy function obtained in a least-squares fit (using the MORBID program) to experimental data for C3. For the lowest stretching states of C3, the vibrational energies obtained from the effective Hamiltonians are in a very close agreement with their “numerically exact” counterparts (i.e., the energies determined variationally in the MORBID approach), indicating a high accuracy of the Born–Oppenheimer separation employed.

A three-dimensional study of the ClHCl dynamics using the light—heavy nucleus Born—Oppenheimer separation

The Born—Oppenheimer type separation between the light and heavy nuclear motions is applied to the dynamics of the ClHCl system. Among the hydrogenic states which describe the motion of the H atom for clamped chlorine nuclei, certain specific states are shown to explain the main structures of the photodetachment spectrum of ClHCl −. Three resonances are obtained at scattering energies SE=0.639, 0.899 and 0.932 eV and are analyzed in terms of the normal modes of a linear symmetrical triatomic molecule. The calculated photodetachment spectrum is compared to previous theoretical and experimental data.

Study of the IH+Br dynamics using a Born—Oppenheimer separation between light and heavy nucleus motions

The study of the dynamics of heavy—light + heavy systems using the Born—Oppenheimer-type separation between the light and the heavy nuclear motions is presently extended to the IH+Br system. The hydrogenic wavefunctions that describe the H nucleus motion for clamped I and Br nuclei, and the potential energy curves that govern the I—Br relative motion are reported. Particular attention has been paid to the hydrogenic state for which the potential energy curve has a well. Surprisingly, it is found that this state does not participate in the H-atom exchange process in the reactive collision: IH ( v=0, j) + Br → I+HBr( v′, j′). This reaction mainly proceeds via crossings between diabatic potential energy curves correlated to the IH ( v=0, j) + Br and I + HBr( v′=1, v′=2, j′) limits. In addition, the photodetachment spectrum of the IHBr − anion has been calculated. The present 3D result fairly agrees with the recent calculation of Metz and Neumark (J. Chem. Phys. 97 (1992) 962). Salient features of the photodetachment spectrum stem from the hydrogenic wavefunctions properties.

Adiabatic three-dimensional simulations of the IHI−, BrHI−, and BrHBr− photoelectron spectra

In order to better characterize the transition-state region for the I + HI, Br + HI, and Br + HBr reactions, the photoelectron spectra of IHI−, IDI−, BrHI−, BrHBr−, and BrDBr− have been simulated using a three-dimensional adiabatic approach. This method of simulation uses a Born–Oppenheimer separation in time scales between the fast hydrogen-atom motion and the slow halogen-atom motion to greatly simplify the computation of the photoelectron spectrum. The resulting simulations are compared to the experimental photoelectron and threshold photodetachment spectra of these anions, and to ‘‘exact’’ simulations of the IHI− and IDI− spectra. The comparison with the exact simulations shows that the adiabatic method is reasonably accurate, and is a considerable improvement over previous approximate simulation schemes. Potential-energy surfaces for the I + HI and Br + HI reactions are evaluated based on a comparison between the simulated and experimental spectra. A three-dimensional surface for the Br + HBr reaction that reproduces the experimental photoelectron spectrum is constructed by extending a fitted collinear surface to three dimensions.