Symmetry Breaking and Non-Born-Oppenheimer Effects in Radical Cations

The various electronic states in which molecules can exist are often considered to be independent. In this article we turn our attention to the limitations of this assumption, namely the interaction between different electronic states through the nuclear motion. This interaction can have several important consequences, two of which are discussed in some detail. One is a distortion of the molecular framework, leading to a lowering of the symmetry of a molecule in excited or ionic states compared to the neutral ground state. General aspects of this symmetry lowering are outlined and interpreted with the aid of typical examples. The other consequence considered is the ability of the nuclei to “jump” between different molecular potential energy surfaces (non-Born-Oppenheimer effects). The nature of this behavior is analyzed and it is argued that the “jumping” can be very fast and efficient, dominating completely the nuclear motion. To exemplify our general ideas we refer to the photoelectron spectra of ethylene and related compounds and demonstrate that they are governed by strong non-Born-Oppenheimer effects. It emerges that the Franck-Condon principle fails in the analysis of their vibronic structure.     

Non-Born-Oppenheimer electronic and nuclear densities for a Hooke-Calogero three-particle model: non-uniqueness of density-derived molecular structure

We consider the calculation of non-Born-Oppenheimer, nBO, one-particle densities for both electrons and nuclei. We show that the nBO one-particle densities evaluated in terms of translationally invariant coordinates are independent of the wavefunction describing the motion of center of mass of the whole system. We show that they depend, however, on an arbitrary reference point from which the positions of the vectors labeling the particles are determined. We examine the effect that this arbitrary choice has on the topology of the one-particle density by selecting the Hooke-Calogero model of a three-body system for which expressions for the one-particle densities can be readily obtained in analytic form. We extend this analysis to the one-particle densities obtained from full Coulomb interaction wavefunctions for three-body systems. We conclude, in view of the fact that there is a close link between the choice of the reference point and the topology of one-particle densities that the molecular structure inferred from the topology of these densities is not unique. We analyze the behavior of one-particle densities for the Hooke-Calogero Born-Oppenheimer, BO, wavefunction and show that topological transitions are also present in this case for a particular mass value of the light particles even though in the BO regime the nuclear masses are infinite. In this vein, we argue that the change in topology caused by variation of the mass ratio between light and heavy particles does not constitute a true indication in the nBO regime of the emergence of molecular structure.     

Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions

An ab initio electronic wave-packet dynamics coupled with the simultaneous classical dynamics of nuclear motions in a molecule is studied. We first survey the dynamical equations of motion for the individual components. Reflecting the nonadiabatic dynamics that electrons can respond to nuclear motions only with a finite speed, the equations of motion for nuclei include a force arising from the kinematic (nuclear momentum) coupling from electron cloud. To materialize these quantum effects in the actual ab initio calculations, we study practical implementation of relevant electronic matrix elements that are related to the derivatives with respect to the nuclear coordinates. Applications of the present scheme are performed in terms of the configuration state functions (CSF) using the canonical molecular orbitals as basis functions without transformation to particular diabatic basis. In the CSF representation, the nonadiabatic interaction due to the kinematic coupling is anticipated to be rather small, and instead it should be well taken into account through the off-diagonal elements of the electronic Hamiltonian matrix. Therefore it is expected that the nonadiabatic dynamics based on this CSF basis neglecting the kinematic coupling may work. To verify this anticipation and to quantify the actual effects of the kinematic coupling, we compare the dynamics with and without the kinematic-coupling terms using the same CSF set. Applications up to the fifth electronically excited states in a nonadiabatic collision between H(2) and B(+) shows that the overall behaviors of these two calculations are surprisingly similar to each other in an average sense except for a fast fluctuation reflecting the electronic time scale. However, at the same time, qualitative differences in the collision events are sometimes observed. Therefore it turns out after all that the kinematic-coupling terms cannot be neglected in the CSF-basis representation. The present applications also demonstrate that the nonadiabatic electronic wave-packet dynamics within ab initio quantum chemical calculation is feasible.     

Vibration-induced energy relaxation in two-level system

Energy decay in a general two-level electronic system coupled to a vibrational harmonic mode interacting with a thermal bath is studied theoretically. The model assumes a general form of the off-diagonal elements (in the electronic basis) of the vibrational-electronic interaction. The cases for constant, linear, and quadratic dependence with respect to the vibrational displacement are investigated. For short-time regime, fast oscillations corresponding to a coherent energy exchange between electrons and vibrations appear. Their frequency Ω corresponds to the energy difference (ħΩ) between electronic levels. Additionally, for the case of linear or quadratic coupling, the amplitude modulation of the oscillations with the frequency ω equal to that of vibrational motion is found.     

Non-Born-Oppenheimer molecular structure and one-particle densities for H2D+

We show that the nonadiabatic (non-Born-Oppenheimer) ground state of a three-nuclei system can be effectively calculated with the use of an explicitly correlated Gaussian basis set with floating centers. Sample calculations performed for the H2D+ system with various basis set sizes show good convergence with respect to both the total energy and the expectation values of the internuclear distances (molecular geometry), the distances between the nuclei and the electrons, and between the electrons. We also provide a derivation of the formulas for one-particle density calculations and some density plots showing the spatial distribution of the H2D+ nuclear and electronic densities.     

Electron correlation: the many-body problem at the heart of chemistry

The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schr?dinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation.     

Dimeres composes de molecules dont l'excitation electronique entraine un deplacement de la configuration nucleaire d'equilibre accompagne d'un changement de constante de force

We have studied the validity of the traditional model of a dimer that has been treated exactly by Merrifield and Fulton and Gouterman, solving the vibronic coupled equations by a numerical method. This model takes into account the modification of the nuclear equilibrium configuration, but it neglects the variation of the force constant when the monomer is electronically excited from the fundamental to a given excited state (the corresponding electronic potentials are both considered as harmonic). We have shown by inspection of the absorption and fluorescence spectra calculated by solving the vibronic equation exactly that the variation force constant cannot be neglected, even if it is weak, particularly in the weak coupling region. The weak, intermediate and strong coupling criteria have been deduced, for the model studied, by examination of the dimeric electronic potential surfaces for different cases of intermolecular interactions.     

Exploring dynamical electron theory beyond the Born-Oppenheimer framework: from chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field

Chemical theory and its application to dynamical electrons in molecules under intense electromagnetic fields is explored, in which we take an explicit account of nuclear nonadiabatic (kinematic) interactions along with simultaneous coupling with intense optical interactions. All the electronic wavefunctions studied here are necessarily time-dependent, and thereby beyond stationary state quantum chemistry based on the Born-Oppenheimer framework. As a general and tractable alternative framework with which to track the electronic and nuclear simultaneous dynamics, we propose an on-the-fly method to calculate the electron and nuclear wavepackets coupled along the branching non-Born-Oppenheimer paths, through which their bifurcations, strong quantum entanglement between nuclear electronic motions, and coherence and decoherence among the phases associated with them are properly represented. Some illustrative numerical examples are also reported, which are aimed at our final goals; real time tracking of nonadiabatic electronic states, chemical dynamics in densely degenerate electronic states coupled with nuclear motions and manipulation and/or creation of new electronic states in terms of intense lasers, and so on. Other examples are also presented as to how the electron wavepacket dynamics can be used to analyze chemical reactions, shedding a new light on some typical and conventional chemical reactions such as proton transfer followed by tautomerization.     

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: an ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.     

Surface reaction dynamics and the harmonic oscillator

One of the fundamental steps in chemical reaction dynamics involves breaking reactant bonds. This is facilitated by placement of energy into the vibrational degrees of freedom associated with the bond. Here we present a model for vibrational excitation in molecule-surface collisions in which the equilibrium geometry of the (diatomic) molecule varies with distance from the surface. The special feature of this model is that the potential energy surfaces for bound nuclear motion are constructed from quadratic potentials, thus enabling analytic solutions. Comparisons are made between exact results obtained from a purely classical trajectory model and various hybrid models in which the internal vibrational modes are treated quantum mechanically in the harmonic limit.     

Ab initio electron propagators in molecules with strong electron-phonon interaction. I. Phonon averages

Ab initio electron propagators in molecular systems with strong electron-electron and electron-phonon interactions are considered to study molecular electronic properties. This research is important in electron transfer reactions where the electron transition is not considered any longer as a single electron transfer process or in temperature dependences of current-voltage characteristics in molecular wires or aggregates. To calculate electron Green's functions, the authors apply a small polaron canonical transformation that intrinsically contains strong electron-phonon effects. According to this transformation, the excitation energies of the noninteracting Hamiltonian are shifted down by the relaxation (solvation) energy for each state. The electron-electron interaction is also renormalized by the electron-phonon coupling. For some values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative resulting in the attraction between two electrons. Within this transformation, they develop a diagrammatic expansion for electron Green's function in which the electron-phonon interaction is included into the multiple phonon correlation functions. The multiple phonon correlation functions are exactly found. It is pointed out that Wick's theorem for such correlation functions is invalid. Consequently, there is no Dyson equation for electron Green's functions. The proposed approach can be considered for future method developments for quantum chemical calculations that include strong nonadiabatic (non-Born-Oppenheimer) effects.     

Vibrational motion in isotopomers of the HeH+ molecular ion: An application of the electron nuclear dynamics method

The importance of isotopic substitution as a tool for elucidation of chemical reaction events originates in the fact that the Coulombic Hamiltonian is isotopically invariant except for the nuclear kinetic energy term. Thus, in theories of isotope effects based on the Born-Oppenheimer scheme, the basic presumption is the invariance of the potential energy surface (PES). We use, however, a fully dynamic approach, called Electron Nuclear Dynamics (END), which does not require a preconstructed PES. Since the END formalism is rather different from commonly used procedures, we study the anharmonic nuclear vibration in isotopic species of the HeH⁺ molecular ion as a model problem. A single time-dependent complex parametrized determinantal wave function is used for the electrons and the nuclei are treated classically. The time evolution of the nuclear and electronic dynamical variables obtained by integration of equations of motion are reported as bond length, nuclear kinetic energy, and Mulliken populations. The molecule vibrates as a classical object. The product of the reduced mass and the square of the vibrational frequency is isotopomer invariant for any common total energy. The difference between the total energy and the nuclear kinetic energy as a function of the internuclear distance is interpreted as the average dynamic potential. © 1997 John Wiley & Sons, Inc.     

The E x e dynamic Jahn-Teller problem: a new insight from the strong coupling limit

Correct boundary conditions for the E x e dynamic Jahn-Teller problem are considered explicitly for the first time to obtain approximate analytical solutions in the strong coupling limit. Numerical solutions for the decoupled equations using the finite difference method are also presented. The numerical solutions for the decoupled equations exhibit avoided crossings in the weak coupling region, which explains the oscillating behavior of the solutions obtained by Longuet-Higgins et al. for the coupled equations. The obtained analytical energy expressions show improved agreement with the numerical calculations as compared with the previous treatment in which the potentials were assumed to be harmonic. We demonstrate that the pseudorotational energy j(2)/(2g(2)), where g is the dimensionless vibronic coupling constant, and j total angular momentum: j=+/-1/2,+/-3/2,..., in the conventional strong coupling expression for the vibronic levels of the lower sheet is exact. Non-Hermitian first-order perturbation theory gives the energy which is correct up to 1/g(4). The asymptotic behavior of the wave function at the origin does not influence the corrected energy up to order of 1/g(4). At the same time the treatment of the upper sheet with correct boundary conditions gives solutions which are entirely different from the corresponding Slonczewski's solutions. Besides, the correct boundary conditions enable us to evaluate the nonadiabatic coupling between the lower and upper potential sheets. The energy correction due to the nonadiabatic coupling is estimated to be of order 1/g(6).     

Trajectory-based solution of the nonadiabatic quantum dynamics equations: an on-the-fly approach for molecular dynamics simulations

The non-relativistic quantum dynamics of nuclei and electrons is solved within the framework of quantum hydrodynamics using the adiabatic representation of the electronic states. An on-the-fly trajectory-based nonadiabatic molecular dynamics algorithm is derived, which is also able to capture nuclear quantum effects that are missing in the traditional trajectory surface hopping approach based on the independent trajectory approximation. The use of correlated trajectories produces quantum dynamics, which is in principle exact and computationally very efficient. The method is first tested on a series of model potentials and then applied to study the molecular collision of H with H(2) using on-the-fly TDDFT potential energy surfaces and nonadiabatic coupling vectors.     

Quantum investigation of linear triatomic exchange reactions. A computational method

The model hamiltonian for a linear triatomic exchange reaction is derived in natural reaction coordinates, and a method is developed to solved the system of wave equations in close coupling approximation. The interaction zone is divided into subintervals, in each of which the coefficients of the system are assumed to be constant; this assumption provides an exact analystical solution over the interval. Special boundary conditions which account for the derivatives of the coefficients are used to match the solutions for adjacent intervals. The method converges to the exact solution with an increasing number of intervals of decreasing length. Results obtained for certain model system illustrate the merits of the method and its convergence.     

Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.     

Solution of quantum Langevin equation: approximations, theoretical and numerical aspects

Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.