In quantum chemistry, 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. In the neighbourhood of an avoided crossing or conical intersection, these terms cannot be neglected. One therefore usually performs one unitary transformation 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 which is significantly easier to estimate numerically.
Contents
- Applicability
- Diabatic transformation of two electronic surfaces
- Adiabatic to diabatic transformation
- A comment concerning the two state Abelian case
- References
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 does not hold, 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 expansion of 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 order to introduce the diabatic transformation we assume now, for the sake of argument, that only two Potential Energy Surfaces (PES), 1 and 2, approach each other and that all other surfaces are well separated; the argument can be generalized to more surfaces. Let the collection of electronic coordinates be indicated by
The nuclear kinetic energy is a sum over nuclei A with mass MA,
(Atomic units are used here). By applying the Leibniz rule for differentiation, the matrix elements of
The subscript
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, we define two new orthonormal states by a diabatic transformation of the adiabatic states
where
These elements are zero because
Assume that a diabatic angle
i.e.,
By a small change of notation these differential equations for
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")
It can be shown that these conditions are rarely ever satisfied, so that a strictly diabatic transformation rarely ever exists. It is common to use approximate functions
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. The surfaces
Adiabatic-to-diabatic transformation
Here, in contrast to previous treatments, the non-Abelian case is considered.
Felix Smith in his article considers the adiabatic-to-diabatic transformation (ADT) for a multi-state system but a single coordinate,
where
Here
and
To derive the matrix
where
where
(see also Geometric phase). Here
A different type of solutions is based on quasi-Euler angles according to which any
The product
whereas the third equation (for
expressed solely in terms of
Similarly, in case of a four-state system
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 × 2 ADT matrix
Substituting this matrix in the above given first order differential equation (for
where