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.

The potential energy surfaces are obtained within the adiabatic or Born–Oppenheimer approximation. This corresponds to a representation of the molecular wave function where the variables corresponding to the molecular geometry and the electronic degrees of freedom are separated. The non separable terms are due to the nuclear kinetic energy terms in the molecular Hamiltonian and are said to couple the potential energy surfaces. Nearby an avoided crossing or conical intersection, these terms are substantive. Therefore one unitary transformation is performed from the adiabatic representation to the so-called diabatic representation in which the nuclear kinetic energy operator is diagonal. In this representation, the coupling is due to the electronic energy and is a scalar quantity that is significantly easier to estimate numerically.

In the diabatic representation, the potential energy surfaces are smoother, so that low order Taylor series expansions of the surface capture much of the complexity of the original system. However strictly diabatic states do not exist in the general case. 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. Hence, pseudo-diabatic potentials are synonymous with diabatic potentials.

Applicability

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. For these systems, it is necessary to go beyond the Born–Oppenheimer approximation. This is often the terminology used to refer to the study of nonadiabatic systems.

A well-known approach involves recasting the molecular Schrödinger equation into a set of coupled eigenvalue equations. 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. A diabatic transformation of the adiabatic states replaces these off-diagonal kinetic energy terms by potential energy terms. Sometimes, this is called the "adiabatic-to-diabatic transformation", abbreviated ADT.

Diabatic transformation of two electronic surfaces

. 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). By applying the Leibniz rule are (where coordinates are suppressed for clarity):

may be neglected except for k = 1 and p = 2. Upon making the expansion

the coupled Schrödinger equations for the nuclear part take the form (see the article Born–Oppenheimer approximation)

\begin{pmatrix}
E_1(\mathbf{R})+ \mathrm{T_n}(\mathbf{R})_{11}&\mathrm{T_n}(\mathbf{R})_{12}\\
\mathrm{T_n}(\mathbf{R})_{21}&E_2(\mathbf{R})+\mathrm{T_n}(\mathbf{R})_{22}\\
\end{pmatrix}
\boldsymbol{\Phi}(\mathbf{R})
= E \,\boldsymbol{\Phi}(\mathbf{R})
\quad \mathrm{with}\quad 
\boldsymbol{\Phi}(\mathbf{R})\equiv
\begin{pmatrix}
\Phi_1(\mathbf{R}) \\
\Phi_2(\mathbf{R}) \\
\end{pmatrix} .

In order to remove the problematic off-diagonal kinetic energy terms, define two new orthonormal states by a diabatic transformation of the adiabatic states

is the diabatic angle gives for diagonal matrix elements

is Hermitian and pure-imaginary. The off-diagonal elements of the momentum operator satisfy,

exists, such that to a good approximation

diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition of Smith are diabatic states. (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 is irrotational,

leading to pseudo diabatic states.

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

\begin{pmatrix}
T_\mathrm{n}+
    \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2} & 0 \\
0  & T_\mathrm{n} +
    \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2}
\end{pmatrix}
\tilde{\boldsymbol{\Phi}}(\mathbf{R})
+
\tfrac{E_{2}(\mathbf{R})-E_{1}(\mathbf{R})}{2}
\begin{pmatrix}
\cos2\gamma
 & \sin2\gamma \\
\sin2\gamma &
-\cos2\gamma
\end{pmatrix}
\tilde{\boldsymbol{\Phi}}(\mathbf{R})
= E \tilde{\boldsymbol{\Phi}}(\mathbf{R}).

has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is

Adiabatic-to-diabatic transformation

Here, in contrast to previous treatments, the non-Abelian case is considered.

Felix Smith in his article, but it is restricted to two states. Such a system is defined as Abelian (see Comment below), known also as the ADT 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>:

is the force-matrix operator, introduced in Diabatic, also known as the Non-Adiabatic Coupling Transformation (NACT) matrix:

is the N-dimensional (nuclear) grad-operator:

.

:

; j=1,M are the Born–Oppenheimer has to be analytic, have to satisfy the following equation:

is a tensor field. This equation is known as the non-Abelian form of the Curl can be shown to be of the form:

(see also Geometric phase is an ordering operator, the dot stands for a scalar product.

-matrix can be expressed as a product of Euler matrices. (i<j=2, is of the form:

are:

) becomes an ordinary (line) integral:

.

is presented as a product of six 4 x 4 Euler matrices (for the six quasi-Euler angles) and the relevant six differential equations form one set of three coupled equations, whereas the other three become, as before, ordinary line integrals.

A comment concerning the two-state (Abelian) case

Since the treatment of the two-state case as presented in Diabatic raised numerous doubts we consider it here as a special case of the Non-Abelian case just discussed. For this purpose we assume the 2 to be of the form:

fulfills the corresponding first order differential equation as well as the subsequent line integral:

is the relevant NACT is a chosen contour in configuration space (usually a planar one) along which the integration is performed. The line integral yields meaningful results if and only if the corresponding (previously derived) Curl-equation is zero for every point in the region of interest (ignoring the pathological points).

References