The diabatic representation as a mathematical tool for theoretical calculations of atomic collisions and of molecular interactions.
One of the guiding principles in modern chemical dynamics and spectroscopy is that the motion of the nuclei in a molecule is slow compared to that of its electrons.
This is justified by the large disparity between the mass of an electron, and the typical mass of a nucleus and leads to the Born–Oppenheimer approximation and the idea that the structure and dynamics of a chemical species are largely determined by nuclear motion on potential energy surfaces.
Nearby an avoided crossing or conical intersection, these terms are substantive.
In this representation, the coupling is due to the electronic energy and is a scalar quantity that is significantly easier to estimate numerically.
Hence, diabatic potentials generated from transforming multiple electronic energy surfaces together are generally not exact.
These can be called pseudo-diabatic potentials, but generally the term is not used unless it is necessary to highlight this subtlety.
The motivation to calculate diabatic potentials often occurs when the Born–Oppenheimer approximation breaks down, or is not justified for the molecular system under study.
This is achieved by expanding the exact wave function in terms of products of electronic and nuclear wave functions (adiabatic states) followed by integration over the electronic coordinates.
The coupled operator equations thus obtained depend on nuclear coordinates only.
Off-diagonal elements in these equations are nuclear kinetic energy terms.
In the absence of magnetic interactions these electronic states, which depend parametrically on the nuclear coordinates, may be taken to be real-valued functions.
The nuclear kinetic energy is a sum over nuclei A with mass MA, (Atomic units are used here).
Upon making the expansion the coupled Schrödinger equations for the nuclear part take the form (see the article Born–Oppenheimer approximation) In order to remove the problematic off-diagonal kinetic energy terms, define two new orthonormal states by a diabatic transformation of the adiabatic states
The off-diagonal elements of the momentum operator satisfy, Assume that a diabatic angle
(Smith was the first to define this concept; earlier the term diabatic was used somewhat loosely by Lichten).
can be rewritten in the following more familiar form: It is well known that the differential equations have a solution (i.e., the "potential" V exists) if and only if the vector field ("force")
Under the assumption that the momentum operators are represented exactly by 2 x 2 matrices, which is consistent with neglect of off-diagonal elements other than the (1,2) element and the assumption of "strict" diabaticity, it can be shown that On the basis of the diabatic states the nuclear motion problem takes the following generalized Born–Oppenheimer form It is important to note that the off-diagonal elements depend on the diabatic angle and electronic energies only.
are adiabatic PESs obtained from clamped nuclei electronic structure calculations and
has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is Here, in contrast to previous treatments, the non-Abelian case is considered.
Felix Smith in his article[1] considers the adiabatic-to-diabatic transformation (ADT) for a multi-state system but a single coordinate,
Such a system is defined as Abelian and the ADT matrix is expressed in terms of an angle,
In the present treatment a system is assumed that is made up of M (> 2) states defined for an N-dimensional configuration space, where N = 2 or N > 2.
To discuss the non-Abelian case the equation for the just mentioned ADT angle,
is the force-matrix operator, introduced in Diabatic, also known as the Non-Adiabatic Coupling Transformation (NACT) matrix:[4] Here
to be analytic (excluding the pathological points), the components of the vector matrix,
[10][11] For instance in case of a tri-state system this matrix can be presented as a product of three such matrices,
fulfills the corresponding first order differential equation as well as the subsequent line integral:[3][15][16][17][18] where
is the relevant NACT matrix element, the dot stands for a scalar product and
is a chosen contour in configuration space (usually a planar one) along which the integration is performed.