Chemical bond breaking/forming as a Franck–Condon electronic process

We describe chemical bond changes as Franck–Condon electronic processes within a new theoretical ansatz that we call ‘rigged’ Born–Oppenheimer (R-BO) approach. The notion of the separability of nuclear and electron states implied in the standard Born–Oppenheimer (BO) scheme is retained. However, in the present scheme the electronic wave functions do not depend upon the nuclear coordinate (R-space). The new functions are obtained from an auxiliary Hamiltonian corresponding to the electronic system (r-coordinates) submitted to a Coulomb potential generated by external sources of charges in real space (α-coordinates) instead of massive nuclear objects. A stationary arrangement characterized by the coordinates α_0A, is determined by a particular electronic wave function, ψ(r;α_0A); it is only at this stationary point, where an electronic Schrödinger equation: H_e(r,α_0A)|Ψ(r;α_0A)=E(α_0A)|Ψ(r;α_0A) must hold. This equation permits us to use modern electronic methods based upon analytic first and second derivatives to construct model electronic wave functions and stationary geometry for external sources. If the set of wave functions {Ψ(r;α_0A)} is made orthogonal, the energy functional in α-space, E(α;α_0A)=Ψ(r;α_0A)|H_e(r,α_0A)|Ψ(r;α_0A) is isomorphic to a potential energy function in R-space: E(R;α_0A)=Ψ(r;α_0A)|H_e(r,R)|Ψ(r;α_0A). This functional defines, by hypothesis, a trapping convex potential in R-space and the nuclear quantum states are determined by a particular Schrödinger equation. The total wave function for the chemical species A reads as a product of our electronic wave function with the nuclear wave function (Ξ_ik(R;α_0A)): Φ_ik(r,R)=Ψ_i(r,α_0A)Ξ_ik(R;α_0A). This approach facilitates the introduction of molecular frame without restrictions in the R-space. Two molecules (characterized with different electronic spectra) that are decomposable into the same number of particles (isomers) have the same Coulomb Hamiltonian and they are then characterized by different electronic wave functions for which no R-coordinate ‘deformation’ can possibly change its electronic structure. A bond breaking/forming process must be formally described as a spectroscopic-like electronic process. The theory provides an alternative to the adiabatic as well as the diabatic scheme for understanding molecular processes. As an illustration of the present ideas, the reaction of H₂+CO leading to formaldehyde is examined in some detail.     

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.

Exactly Solvable Model Mimicking the H<Subscript>2</Subscript> Molecule in the Limit of Large Nuclear Masses

Avoiding the limitations of the Born–Oppenheimer approximation remains an issue of continuing importance in molecular physics. If we take the H₂ molecule as a prototype, then one uses, in some sense, the ratio of electron mass m to nuclear mass M as an expansion parameter, sometimes cited to be explicitly (m/M)^1/4. Here, we employ a model of such a two-electron diatomic molecule set up by Makarewicz [Am. J. Phys. 54 (1986) 178] to study the exact ground-state wave function obtained by treating electrons and nuclei on the same footing, Of course, to obtain such a wave function analytically means adopting model force laws, both for confinement and for the interelectronic interaction. From the exact wave function and ground-state energy, contact is finally established with the large nuclear mass limit.     

Formulation of N- and v-representable density functional theory. IV. Non-Born–Oppenheimer approach

Local-scaling transformations are employed in order to formulate a desity functional theory where both electrons and nuclei are treated quantum mechanically. Because of the properties of these transformations, the ensuing version of density functional theory is N- and v- representable. In particular, we study the effect of these transformations on both densities and generating wave functions, and as a result, a functional for the energy expressed in terms of the electronic and nuclear densities is constructed. Plausible physical approximations for simplifying this electronic-nuclear functional are considered, and upon variation of the fuctional with respect to the electronic and nuclear densities, a system of coupled Euler–Lagrange equations is obtained.     

Sensitivity of the NTB phase formation to the molecular structure of imino-linked dimers

ABSTRACT

Here we report on the synthesis and mesomorphic properties of a series of imino-linked dimeric molecules. In order to improve our understanding of the structure–N_TB phase correlations, we have studied the impact of geometric and electronic factors arising from varying mesogenic units, different spacer lengths and from the ratio (n/m) between the lengths of terminal chains (n) and spacer (m). From the perspective of the molecular geometry, the results show that the stability of the N_TB phase results from increasing effective molecular bending and with the broadening of the mesogenic unit, in particular near the spacer, and that the n/m ratio plays a substantial role in conjunction with the specific mesogenic unit. A computational study of the electronic properties shows that a broadening of the mesogenic core in the vicinity of the spacer is associated with an increased anisotropy of the electrostatic potential distribution. Within a given series of materials our study suggests that the incidence of the N_TB phase and its thermal stability are governed by the synergy of specific geometrical factors and the anisotropy of the electrostatic potential distribution of the mesogenic core.     

Benchmark calculations for He2 and LiH molecules using explicitly correlated Gaussian functions

Explicitly correlated Gaussian (ECG) functions with carefully optimized non-linear parameters are used to calculate the electronic energies of He₂⁺ and LiH at their equilibrium internuclear distances. The obtained variational upper bounds (−4.99464392 and −8.070538 hartree, respectively) are the lowest reported to date. By extrapolating results obtained with various expansion lengths, the estimations of the Born–Oppenheimer limits are made.     

A Non-Born–Oppenheimer Self-consistent Field Method

This communication outlines the development of a novel method for the ab initio computation of molecular systems wherein the Born–Oppenheimer approximation is not invoked. In the current method, a common Hamiltonian is employed to operate on the electrons and nuclei simultaneously. In addition, an inseparable wavefunction is generated to describe the non-Born–Oppenheimer behaviour. The physical implications of the new method are discussed.     

Behavior of molecular potential energy curves for large nuclear separations

We prove by elementary geometric methods and within the Born–Oppenheimer approximation that as the nuclei of a molecule are dissociated into spatially separated clusters, the discrete molecular energies approach sums of the energies of isolated subsystems. Our methods also show that the spectral projections associated with the discrete molecular spectrum asymptotically approach direct sums of suitable spectral projections for the isolated subsystems. These results apply to any system of particles interacting by asymptotically vanishing pair potentials. We prove that the 1/R expansion for discrete molecular potential curves is asymptotic as R → ∞, and we discuss the behavior of the coefficients of the 1/R expansion for the ground state of H₂⁺.     

Ab initio dynamics with wave-packets and density matrices

Efficient methodologies to conduct simultaneous dynamics of electrons and nuclei are discussed. Particularly, attention is directed to a recent development that combines quantum dynamics with ab initio molecular dynamics. The two components of the methodology, namely, quantum dynamics and ab initio molecular dynamics, are harnessed together using a time-dependent self-consistent field-like coupling procedure. An approach to conduct quantum dynamics using an accurate banded, sparse and Toeplitz representation for the discrete free propagator is highlighted with suitable review of other related approaches. One notable feature of the method is that all important quantum dynamical effects including zero-point effects, tunneling as well as over-barrier reflections are accurately treated. Computational methodologies for improved efficiency of the quantum dynamics are also discussed. There exists a number of ways to carry out simultaneous ab initio molecular dynamics (such as Born–Oppenheimer dynamics and extended Lagrangian dynamics, Car–Parrinello dynamics being a prime example of the latter); our prime focus remains on atom-centered density-matrix propagation and Born–Oppenheimer dynamics. The electronic degrees of freedom are handled at accurate levels of density functional theory, using hybrid or gradient corrected approximations. Benchmark calculations are provided for a prototypical proton transfer system. Future generalizations and goals are discussed.     

The physics of the born–oppenheimer approximation

A new derivation of the Born–Oppenheimer separation of electronic and nuclear motion is presented. The arguments used differ from those in earlier works in not being specially designed for molecules. Instead they aim at an intuitive understanding of the qualitative behavior of the low energy bound states of any, real or hypothetical, Coulomb interacting system of particles. The virial theorem is the starting point of the discussion. After a brief explanation of how it can be used to understand atomic structure it is applied to molecules. It is found that coordinates of collective and individual motion are natural coordinates for the approximate separation, rather than nuclear and electronic. It is also shown that it is the form of the interaction between the particles that is responsible for the separation; the smallness of m_et/M_Nu is irrelevant.     

Molecular structure and Bader's theory

 It is shown that a supposed catastrophe of Bader's theory of atoms in molecules, suggested by Cassam-chena? and Jayatilaka [Theor Chem Acc (2001) 105: 213] is merely a consequence of the approximate character of the adiabatic Born–Oppenheimer theory of molecular structure, and that nonadiabatic approaches could be in accordance with Bader's ideas. Received: 4 April 2001 / Accepted: 5 September 2001 / Published online: 3 June 2002     

Atoms in molecules: beyond Born–Oppenheimer paradigm

This contribution presents the first atoms in molecules study that goes beyond the Born–Oppenheimer paradigm employing the newly developed two-component quantum theory of atoms in molecules (TC-QTAIM). The LiH, LiD, and LiT systems containing quantum instead of clamped hydrogen nuclei are used as typical examples. The computational analysis that is done on non-adiabatic wavefunctions derived from the fully variational multicomponent molecular orbital approach (FV-MC-MO) results in hydrogen atomic basins without any clamped nucleus. The topological analysis of the Γ-field, the field that replaces the usual one-electron density used in the orthodox topological analysis, reveals delicate differences among the considered systems. The calculation of basin properties also demonstrates that the TC-QTAIM differentiates among atomic basins containing isotopes. Since the nuclear dynamics is contained intrinsically in non-adiabatic wavefunctions, the nuclear contribution to both topological analysis and basin properties naturally emerges from the TC-QTAIM analysis resolving the long-standing obstacle of consistent incorporation of nuclear dynamics within the context of the orthodox QTAIM. Also, a similar analysis is done on non-adiabatic wavefunctions describing excited instead of ground nuclear vibrations of the considered systems demonstrating the fact that TC-QTAIM is capable of being employed for both ground and excited nuclear vibrational states.     

Site-directed electronic tunneling through a vibrating molecular network

The effect of electronic-nuclear coupling on electronic transport through a complex molecular network is studied. Electronic tunneling dynamics in a network of N donor/acceptor sites, connected by molecular bridges, is shown to be controlled by electronic-nuclear coupling at the bridges. Particularly, electronic coupling to an accepting nuclear mode at the contact site between the donor and the rest of the network is shown to affect the tunneling path selection to specific acceptors. In the "deep" tunneling regime, the network is mapped onto an N-level system using a recursive perturbation expansion, enabling analytical treatment of the electronic dynamics. The analytic formulation is applied for two model systems, demonstrating site-directed tunneling by electronic-nuclear coupling. Numerical simulations suggest that this phenomenon is not limited to the deep tunneling regime.