Beyond the Born-Oppenheimer Approximation

Apart from the further refinements of the theoretical grounds for the B-0 approach which is closely related to the notion of the Potential Energy Surface (PES), there has been a continuing interest in theoretical consideration of molecular systems where the motions of both nuclei and electrons are treated equivalently. Before we turn our attention to this approach, it should be mentioned that there is a significant body of work, recently reviewed by Langsfield and Yarkony [7], where the departure from the B-0 approximation 

In this review we will first describe two approaches which we have used to represent atomic and molecular systems without resorting to the B-0 approximations. Next, we will describe two numerical applications of the theory, which led to determining interesting non-adiabatic contributions. In the last section we will consider future theoretical work on a general non-adiabatic approach to an N-particle system with any isotropic interaction potential, including coulombic interaction, which is presently being developed in our group. 

Explicit separation of the center-of-mass motion (Method I) 

To model the physical systems, i.e., write the Hamiltonian, particles are considered to be non-relativistic, charged, point masses interacting under an isotropic potential. The 

The particles are numbered from 1 to N with Mi the mass of particle i, R, = [A, Yt Zi) a column vector of Cartesian coordinates for particle i in the external, laboratory fixed, frame, Vr the Laplacian in the coordinates of R, and Ri — Rj the distance between particles i and j. The total Hamiltonian, eqn.(l), is, of course, separable into an operator describing the translational motion of the center of mass and an operator describing the internal energy. This separation is realized by a transformation to center-of-mass and internal (relative) coordinates. Let R be the vector of particle coordinates in the laboratory fixed reference frame. 

---

Molecular Dynamics Beyond the Born-Oppenheimer Approximation. 

H. Koppel, W. Domcke, and L. S. Cederbaum, Multimode molecular dynamics beyond the Born-Oppenheimer approximation, Adv. Chem. Phys. 57, 59-246(1984). 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at... 

Woolley, R. G. 1991. "Quantum Chemistry Beyond the Born-Oppenheimer Approximation." Journal of Molecular Structure (Theochem), 230 17—46. 

L. J. Butler, Chemical Reaction Dynamics Beyond the Born-Oppenheimer Approximation, Annual Review of Physical Chemistry. 49 (1998), 125-171. 

Electro-Nuclear Quantum Mechanics Beyond the Born-Oppenheimer Approximation. Towards a Quantum Electronic Theory of Chemical Reaction Mechanisms... 

Electro-nuclear quantum mechanics beyond the Born-Oppenheimer approximation. Towards a quantum electronic theory of chemical reaction mechanisms... 

Non-Born-Oppenheimer effects. The assumption of a rigorous separation of nuclear and electronic motions is in most cases a quite good approximation and there is a good understanding of when it will fail. Methods for going beyond the Born-Oppenheimer approximation are still somewhat limited in term of generality and applicability. 

Butler LJ Chemical reaction dynamics beyond the Born-Oppenheimer approximation. Annu Rev Phys Chem 1998, 49 125—171. 

Although eqn (2.33) represents a highly simplified model of conjugated molecules, it still remains a considerable challenge to solve, understand and predict its physical beha viour. We discuss various additional approximations to in Section 2.8. However, in the next section we discuss going beyond the Born-Oppenheimer approximation to include explicit electron-phonon coupling. 

Thus this book describes the recent theories of chemical dynamics beyond the Born-Oppenheimer framework from a fundamental perspective of quantum wavepacket dynamics. To formulate these issues on a clear theoretical basis and to develop the novel theories beyond the Born-Oppenheimer approximation, however, we should first learn a basic classical and quantum nuclear dynamics on an adiabatic (the Born-Oppenheimer) potential energy surface. So we learn much from the classic theories of nonadiabatic transition such as the Landau-Zener theory and its variants. 

Koppel H, Domcke W, Cederbaum LS (1984) Multi-mode molecular dynamics beyond the born-oppenheimer approximation. Adv Chem Phys 57 59... 

Nafie, L.A. (1983) Adiabatic behavior beyond the Born-Oppenheimer approximation. Complete adiabatic wavefunctions and vibrationally induced electronic current density. J. Chem. Phys., 79, 4950-4957. 

BEYOND THE BORN-OPPENHEIMER APPROXIMATION A SUM OVER STATES APPROACH... 

One learns directly that the converted Bom-Handy formula leads to a curiosity, viz. the hydrogen molecule does not move and does not rotate. Nevertheless the final Born-Handy formula contains contributions from vibrational as well as from translational and rotational degrees of freedom in contrast to our previous theory, based on the quantization of the electron-vibrational Hamiltonian, which contained solely the contributions from the vibrational degrees. As it will be shown below, this understanding has a profound significance for aU systems and phenomena beyond the Born-Oppenheimer approximation. Moreover, the interpretation of the... 

Woolley, 1991] R. G. Woolley. Quantum chemistry beyond the Born-Oppenheimer approximation, J. Molec. Struct (Theochem)., 230, 17-46, 1991. 

---